Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 307913 9 pageshttpdxdoiorg1011552013307913

Research ArticleInfinitely Many Quasi-Coincidence Point Solutions ofMultivariate Polynomial Problems

Yi-Chou Chen

Department of General Education National Army Academy Taoyuan 320 Taiwan

Correspondence should be addressed to Yi-Chou Chen cycuchougmailcom

Received 9 November 2013 Accepted 5 December 2013

Academic Editor Wei-Shih Du

Copyright copy 2013 Yi-Chou ChenThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Let 119865 R119899 times R rarr R be a real-valued polynomial function of the form 119865(119909 119910) = 119886119904(119909)119910119904

+ 119886119904minus1

(119909)119910119904minus1

+ sdot sdot sdot + 1198860(119909) where the

degree 119904 of 119910 in 119865(119909 119910) is greater than 1 For arbitrary polynomial function 119891(119909) isin R[119909] 119909 isin R119899 we will find a polynomialsolution 119910(119909) isin R[119909] to satisfy the following equation (⋆) 119865(119909 119910(119909)) = 119886119891(119909) where 119886 isin R is a constant depending on thesolution 119910(119909) namely a quasi-coincidence (point) solution of (⋆) and 119886 is called a quasi-coincidence value of (⋆) In this paperwe prove that (i) the number of all solutions in (⋆) does not exceed deg

119910119865(119909 119910)((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1) provided those

solutions are of finitely many exist (ii) if all solutions are of infinitely many exist then any solution is represented as the form119910(119909) = minus119886

119904minus1(119909)119904119886

119904(119909) + 120582119901(119909) where 120582 is arbitrary and 119901(119909) = (119891(119909)119886

119904(119909))1119904 is also a factor of 119891(119909) provided the equation (⋆)

has infinitely many quasi-coincidence (point) solutions

1 Introduction

In 1987 Lenstra [1] researched a polynomial function119865(119909 119910) isin Q(120572) [119909 119910] (120572 is an algebraic number) andattempted to search the factorization of 119865(119909 119910) Continuinghis job many scientists tried to find the roots of the polyno-mial equations (cf [2ndash6]) Later many authors also studiedfixed point theory and fixed coincidence theory (cf [7ndash11])Recently Lai and Chen ([12ndash15]) research the quasi-fixed(point) polynomial problem they assumed 119865 R119899 timesR rarr R

a polynomial function and solved 119910(119909) isin R[119909] to satisfy thepolynomial equation as the form

119865 (119909 119910 (119909)) = 119886119901119898

(119909) 119909 isin R (1)

where 119886 isin R 119901(119909) is an irreducible polynomial in R[119909] andthe polynomial function 119865(119909 119910) is written by

119865 (119909 119910) =

119904

sum

119894=0

119886119894(119909) 119910119894 with 119904 ge 1 (2)

where 119904 = deg119910119865 denotes the degree of 119910 in 119865(119909 119910)

Definition 1 (Lai and Chen [12]) A polynomial function119910 = 119910(119909) satisfying (1) is called a quasi-fixed solution

corresponding to some real number 119886 This number 119886 iscalled a quasi-fixed value corresponding to the polynomialsolutions 119910 = 119910(119909)

Moreover Chen and Lai [16] extended (1) to a more gen-eral coincidence (point) problem in which the119891(119909) isin R[119909] isreplaced by the irreducible polynomial power 119901119898(119909) isin R[119909]where 119891(119909) is an arbitrary polynomial Then we restate (1) asthe following equation

119865 (119909 119910) = 119886119891 (119909) (3)

It is a new development coincidence point-like problem Wecall the polynomial solution 119910 = 119910(119909) for (3) a quasi-coincidence (point) solution Precisely we give the followingdefinition like Definition 1

Definition 2 (Chen and Lai [16]) A polynomial function119910 = 119910(119909) satisfying (3) is called a quasi-coincidence (point)solution corresponding to some real number 119886 This number119886 is called a quasi-coincidence value corresponding to thepolynomial solutions 119910 = 119910(119909)

2 Abstract and Applied Analysis

Furthermore we consider a multivariate polynomialfunction 119865 R119899 times R rarr R and extend (3) as a more generalcoincidence (point) problem inwhich the119909 = (119909

1 1199092 119909

119899)

is replaced by 119909 throughout this paper where 119891(119909) is anonzero arbitrary polynomial inR[119909] Then we restate (3) asthe following equation

119865 (119909 119910) = 119886119891 (119909) (4)

Thus we can give some definitions like Definition 2 asfollows

Definition 3 A polynomial function 119910 = 119910(119909) satisfying (4)is called a quasi-coincidence (point) solution correspondingto some real number 119886 This number 119886 is called a quasi-coincidence value corresponding to the polynomial solutions119910 = 119910(119909)

The number of all solutions in (4) may be infinitely manyfinitely many or not solvable In this paper we solve allsolutions of (4) if the number is infinitely many Moreoverwe provide an upper bound for the number of all solutions ifthe number is finitely many

In Section 2 we derive some properties of quasi-coincidence solutions If (4) has infinitely many quasi-coincidence solutions the form of 119865(119909 119910) will be describedin Section 3 In the last section we solve all solutions if (4)has infinitely many solutions

2 Preliminaries

For convenience we denote the polynomial function by

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198861(119909) 119910 + 119886

0(119909)

=

119904

sum

119894=0

119886119894(119909) 119910119894

(5)

throughout this paper and since there may exist manysolutions corresponding to the same number 119886 we use thesimilar notations like (Definition 2 [11]) to represent them

Notation 1 (1) Qcs119865 the set of all solutions satisfying

equation (4) the solution in Qcs119865is also called a quasi-

coincidence solution in (4) (like Definition 2)(2) Qcv

119865 the set of all solutions 119886 satisfying equation (4)

the solution in Qcv119865is also called a quasi-coincidence value

in (4) (like Definition 2)(3) Qcs

119865(119886) the set of all quasi-coincidence solutions

119910(119909) corresponding to a quasi-coincidence value 119886(4) For each 119886 isin R we denote |Qcs

119865(119886)| as the cardinal

number of Qcs119865(119886)

Evidently by Notation 1 we have the following lemma

Lemma 4 (i) 119876119888119904119865= ⋃119886isin119876119888V119865

119876119888119904119865(119886)

(ii) 119876119888119904119865(119886)⋂119876119888119904

119865(119887) = 0 for any 119886 = 119887 in 119876119888V

119865

(iii) |119876119888119904119865(119886)| le deg

119910119865(119909 119910) for any 119886 isin R

(iv) |119876119888119904119865| le |119876119888V

119865||119876119888119904119865(119886)| for any 119886 isin 119876119888V

119865

Proof (i)⋃119886isinQcv119865 Qcs

119865(119886) sube Qcs

119865is obvious

Conversely for any 119910(119909) isin Qcs119865 by Notation 1(1) we

have

119865 (119909 119910 (119909)) = 119886119891 (119909) (6)

for some 119886 isin Qcv119865Thismeans119910(119909) isin Qcs

119865(119886) andwe obtain

Qcs119865sube ⋃

119886isinQcv119865

Qcs119865(119886) (7)

(ii) Let 119886 = 119887 in Qcv119865 if there exists 119910(119909) isin R[119909] such that

119910 (119909) isin Qcs119865(119886)⋂Qcs

119865(119887) (8)

by Notation 1(3) we have

119886119891 (119909) = 119865 (119909 119910 (119909)) = 119887119891 (119909) (9)

This leads a contradiction to 119886 = 119887 and we haveQcs119865(119886)⋂Qcs

119865(119887) = 0

(iii) For each 119886 isin R the number of all solutions 119910 =

119910(119909) to the polynomial equation 119865(119909 119910) minus 119886119891(119909) = 0 is atmost deg

119910(119865(119909 119910) minus 119886119891(119909)) = deg

119910119865(119909 119910) then the result is

obtained(iv) By (i) we have

Qcs119865= ⋃

119886isinQcv119865

Qcs119865(119886) (10)

It follows that

1003816100381610038161003816Qcs119865

1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

⋃

119886isinQcv119865

Qcs119865(119886)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816Qcv119865

1003816100381610038161003816

1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816 for any 119886 isin Qcv119865

(11)

In the following lemma we explain some interestingproperties of the relations of quasi-coincidence point solu-tions Throughout this paper we consider (4) for polynomialfunction (5) and nonzero arbitrary polynomial 119891(119909) inR[119909]

Lemma 5 Let the cardinal number |119876119888V119865| ge 2 and 119886 = 119887 in

119876119888V119865 Then for any 119910

1(119909) isin 119876119888119904

119865(119886) and 119910

2(119909) isin 119876119888119904

119865(119887) one

has

1199101(119909) minus 119910

2(119909) = 119889119901 (119909) 119891119900119903 119904119900119898119890 119889 isin R (12)

and this 119901(119909) is a factor of 119891(119909) that is 119901(119909) | 119891(119909)

Proof Since 1199101(119909) 119910

2(119909) correspond to 119886 119887 respectively we

have

119865 (119909 1199101(119909)) = 119886119891 (119909) (13)

119865 (119909 1199102(119909)) = 119887119891 (119909) (14)

Abstract and Applied Analysis 3

Subtracting (14) from (13) and using binomial formula ityields that

(119886 minus 119887) 119891 (119909) = 119865 (119909 1199101(119909)) minus 119865 (119909 119910

2(119909))

= 119886119904(119909) [119910

119904

1(119909) minus 119910

119904

2(119909)]

+ 119886119904minus1

(119909) [119910119904minus1

1(119909) minus 119910

119904minus1

2(119909)]

+ sdot sdot sdot + 1198861(119909) [119910

1(119909) minus 119910

2(119909)]

= [1199101(119909) minus 119910

2(119909)]

times [119886119904(119909) 119866119904(1199101(119909) 119910

2(119909))

+ 119886119904minus1

(119909) 119866119904minus1

(1199101(119909) 119910

2(119909))

+ sdot sdot sdot + 1198861(119909)]

= [1199101(119909) minus 119910

2(119909)] 119876 (119909 119910

1(119909) 119910

2(119909))

(15)

where

119866119895(1199101(119909) 119910

2(119909)) = 119910

119895minus1

1(119909)

+ 119910119895minus2

1(119909) 1199102(119909) + sdot sdot sdot + 119910

119895minus1

2(119909)

(16)

for 119895 = 119904 119904 minus 1 2 1 Evidently the factor 1199101(119909) minus 119910

2(119909) is

divisible to the term (119886 minus 119887)119891(119909) and since 119886 = 119887 we obtain

1199101(119909) minus 119910

2(119909) = 119889119901 (119909) (17)

for some real number 119889 isin R and factor 119901(119909) of 119891(119909)

In Lemma 5 the difference of any two distinct quasi-coincidence solutions corresponding to distinct values is afactor of 119891(119909) Thus we may define a class of those factors inthe following

Notation 2 (i) DenoteΦ(119901(119909)) = 120572119901(119909) 120572 isin R(ii) Let 119910(119909) be an arbitrary polynomial in R[119909] and we

denote 119910(119909) + Φ(119901(119909)) = 119910(119909) + 120572119901(119909) 120572 isin R(iii) deg119891(119909) = sum

119899

119894=1deg119909119894

119891(119909)

If 1199101(119909) 119910

2(119909) isin Qcs

119865correspond to distinct quasi-

coincidence values by Lemma 5 and Notation 2 we have

1199101(119909) minus 119910

2(119909) isin ⋃

119901(119909)|119891(119909)

Φ(119901 (119909)) (18)

Since the number of all factors 119901(119909) to 119891(119909) is at most2deg119891(119909) by the definitions of ldquothe pigeonhole principlerdquo in[17] we have the following results

Lemma 6 Suppose that1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge 119896 sdot 2deg119891(119909)

+ 2 (19)

Then there exists 119910(119909) isin 119876119888119904119865and 119901(119909) is a factor of119891(119909) such

that1003816100381610038161003816(119910 (119909) + Φ (119901 (119909))) cap 119876119888119904

119865

1003816100381610038161003816 ge 119896 (20)

Proof Since |Qcv119865| ge 1198962

deg119891(119909)+2 by (18) there exists 119910

1(119909)

1199102(119909) 119910

1198962deg119891(119909)+2

(119909) isin Qcs119865such that

119910119895(119909) minus 119910

1(119909) isin Φ (119901

119895(119909)) (21)

for some factor 119901119895(119909) of 119891(119909) for 119895 = 2 3 1198962

deg119891(119909)+ 2

Moreover we have that the number of all factors to 119891(119909) isat most 2deg119891(119909) By ldquothe pigeonhole principlerdquo there exists asubset 119910

119895119894(119909)119896

119894=1sube 119910119895(119909)1198962

deg119891(119909)+2

119895=2such that

119910119895119894(119909) minus 119910

1(119909) isin Φ (119901 (119909)) (22)

for 119894 = 1 2 119896 and some factor 119901(119909) of 119865(119909) (this 119901(119909) isindependent of the choice of 119894) Thus

119910119895119894(119909)119896

119894=1

sube (1199101(119909) + Φ (119901 (119909))) capQcs

119865

1003816100381610038161003816(1199101 (119909) + Φ (119901 (119909))) capQcs119865

1003816100381610038161003816 ge 119896

(23)

For convenience we explain the relations of Qcs119865and

Φ(119901(119909)) in the following lemma

Lemma 7 Let 119910(119909) isin 119876119888119904119865(119886) for some 119886 isin R Then

119876119888119904119865= 119876119888119904

119865(119886)⋃( ⋃

119901(119909)|119891(119909)

(119910 (119909) + Φ (119901 (119909))) cap 119876119888119904119865)

(24)

for some factor 119901(119909) of 119891(119909)

Proof For any 1199101(119909) isin Qcs

119865 Qcs119865(119886) then 119910

1(119909) isin Qcs(119887)

for some 119887 isin Qcv119865 By Lemma 5 we have

1199101(119909) minus 119910 (119909) isin Φ (119901 (119909)) (25)

for some factor 119901(119909) of 119891(119909) Then

1199101(119909) isin ⋃

119901(119909)|119891(119909)

119910 (119909) + Φ (119901 (119909)) (26)

and it follows that

Qcs119865sube Qcs

119865(119886)⋃( ⋃

119901(119909)|119891(119909)

119910 (119909) + Φ (119901 (119909))) (27)

Moreover by Lemma 4(i) Qcs119865(119886) sube Qcs

119865 then we obtain

Qcs119865

= Qcs119865(119886)⋃( ⋃

119901(119909)|119891(119909)

(119910 (119909) + Φ (119901 (119909))) capQcs119865)

(28)

In order to let the number of all elements in the intersec-tion of sets 119910

1(119909) + Φ(119901

1(119909)) and 119910

2(119909) + Φ(119901

2(119909)) be large

enough we find a lower bound for |Qcv119865| in the following

theorem

4 Abstract and Applied Analysis

Theorem 8 Suppose that the cardinal number1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (29)

Then for any 1199101(119909) = 119910

2(119909) isin 119876119888119904

119865 there exist two factors

1199011(119909) and 119901

2(119909) of 119891(119909) such that

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) cap 119876119888119904

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(30)

Proof Let 1199101(119909) isin Qcs

119865and by assumption

1003816100381610038161003816Qcv119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (31)

and by Lemma 6 there exists a factor 1199011(119909) of 119891(119909) such that

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) capQcs

119865

1003816100381610038161003816 ge 2deg119891(119909)

+ 119904 + 3 (32)

This implies that1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865minus 1199101(119909)

1003816100381610038161003816 ge 2deg119891(119909)

+ 119904 + 2

(33)

Moreover for any 1199102(119909) isin Qcs

119865 we have

(1199101(119909) + Φ (119901

1(119909)))

capQcs119865minus 1199101(119909) sube (119910

1(119909) + Φ (119901

1(119909))) capQcs

119865

sube Qcs119865

(by Lemma 7)

= Qcs119865(119887)⋃( ⋃

119901(119909)|119891(119909)

(1199102(119909) + Φ (119901 (119909))) capQcs

119865)

(34)

for some constant 119887 isin R and it follows that

2deg119891(119909)

+ 119904 + 2

=1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865minus 1199101(119909)

1003816100381610038161003816

le1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865

1003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

Qcs119865(119887)⋃( ⋃

119901(119909)|119891(119909)

(1199102(119909) + Φ (119901 (119909))) capQcs

119865)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816Qcs119865(119887)

1003816100381610038161003816 + sum

119901(119909)|119891(119909)

1003816100381610038161003816(1199102 (119909) + Φ (119901 (119909))) capQcs119865

1003816100381610038161003816

(by Lemma 4 (iii))

le 119904 + sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865

1003816100381610038161003816

(35)

Canceling both sides of the above inequality by ldquo119904rdquo it followsthat

2deg119891(119909)

+ 2 le sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865

1003816100381610038161003816 (36)

this implies that

2deg119891(119909)

+ 1 le sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865minus 1199102(119909)

1003816100381610038161003816

(37)

By the pigeonholersquo principle and since the number of allfactors 119901(119909) to 119891(119909) is at most 2deg119891(119909) we have

1003816100381610038161003816(1199102 (119909) + Φ (1199012(119909))) capQcs

119865minus 1199102(119909)

1003816100381610038161003816 ge 2

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909)) minus 119910

1(119909))

cap (1199102(119909) + Φ (119901

2(119909)) minus 119910

2(119909)) capQcs

119865

1003816100381610038161003816 ge 2

(38)

for some factor 1199012(119909) of 119891(119909) Thus we obtain

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) capQcs

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(39)

Up to now we have not shown that the factor 119901(119909)

uniquely existed eventually In the following theorem wewould show the uniqueness property for the factor 119901(119909) of119891(119909) if the number of all quasi-coincidence values is largeenough

Theorem 9 Assume that the cardinal number1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (40)

Then for any 1199101(119909) 1199102(119909) isin 119876119888119904

119865 one has

1199101(119909) minus 119910

2(119909) = 120582119901 (119909) (41)

where 120582 isin R and 119901(119909) is a factor of 119891(119909) (this 119901(119909) isindependent of the choice of 119910

1(119909) and 119910

2(119909))

Proof Let 1199101(119909) = 119910

2(119909) isin Qcs

119865 by Theorem 8 we have

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) capQcs

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(42)

for some factors 1199011(119909) and 119901

2(119909) of 119891(119909) There exists

1198921(119909) = 119892

2(119909) isin Qcs

119865 such that

1198921(119909) 119892

2(119909) isin (119910

1(119909) + Φ (119901

1(119909)))

cap (1199102(119909) + Φ (119901

2(119909)))

capQcs119865minus 1199101(119909) 119910

2(119909)

(43)

This implies that

1198921(119909) isin 119910

1(119909) + Φ (119901

1(119909))

1198922(119909) isin 119910

1(119909) + Φ (119901

1(119909))

1198921(119909) isin 119910

2(119909) + Φ (119901

2(119909))

1198922(119909) isin 119910

2(119909) + Φ (119901

2(119909))

(44)

Abstract and Applied Analysis 5

By Notation 2(ii) it yields that

1198921(119909) minus 119910

1(119909) = 120582

11199011(119909)

1198922(119909) minus 119910

1(119909) = 120582

21199011(119909)

1198921(119909) minus 119910

2(119909) = 120582

31199012(119909)

1198922(119909) minus 119910

2(119909) = 120582

41199012(119909)

(45)

for some constants 1205821 1205822 1205823 and 120582

4isin R and consequently

1198922(119909) minus 119892

1(119909) = (119892

2(119909) minus 119910

1(119909)) minus (119892

1(119909) minus 119910

1(119909))

= (1205822minus 1205821) 1199011(119909)

1198922(119909) minus 119892

1(119909) = (119892

2(119909) minus 119910

2(119909)) minus (119892

1(119909) minus 119910

2(119909))

= (1205824minus 1205823) 1199012(119909)

(46)

This implies that

(1205822minus 1205821) 1199011(119909) = (120582

4minus 1205823) 1199012(119909) (47)

and 1199011(119909) = 119901

2(119909) Therefore

1199101(119909) minus 119910

2(119909) = (119892

1(119909) minus 119910

2(119909)) minus (119892

1(119909) minus 119910

1(119909))

= (1205823minus 1205821) 1199011(119909)

(48)

and this means that the factor 119901(119909) of 119891(119909) is uniquely deter-mined independent of the choice of 119910

1(119909) and 119910

2(119909)

Corollary 10 Assume that the cardinal number

1003816100381610038161003816119876119888119904119865

1003816100381610038161003816 ge deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) (2

deg119891(119909)) + 2)

(49)

Then for any 119910(119909) isin 119876119888119904119865 there exists ℎ(119909) 119901(119909) isin R[119909] such

that

119910 (119909) = ℎ (119909) + 120582119901 (119909) (50)

for some 120582 isin R (ℎ(119909) 119901(119909) are independent of the choice of119910(119909))

Proof By Lemma 4(iv) we have

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcv119865

1003816100381610038161003816

1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816 for any 119886 isin Qcv119865

(by Lemma 4 (iii))

le deg119910119865 (119909 119910)

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(51)

By assumption it follows that

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) (2

deg119891(119909)) + 2)

le deg119910119865 (119909 119910)

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(52)

Dividing both sides of the above equation by deg119910119865(119909 119910) we

get

1003816100381610038161003816Qcv119865

1003816100381610038161003816 ge ((2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2) (53)

If ℎ(119909) isin Qcs119865 for any 119910(119909) isin Qcs

119865 by Theorem 9 we have

119910 (119909) = ℎ (119909) + 119889119901 (119909) (54)

for some factor 119901(119909) of 119891(119909)

3 The Type of 119865(119909119910) If the Numberof All Quasi-Coincidence SolutionsIs Infinitely Many

In this section we consider (4) for polynomial function119865(119909 119910) in (5) that is let

119865 (119909 119910) =

119904

sum

119894=0

119886119894(119909) 119910119894 with 119904 ge 2 (55)

119891(119909) isin R[119909] and we assume that 119865(119909 119910) has at least119904 + 1 distinct quasi-coincidence solutions satisfying someconditions that is 119910

1(119909) 1199102(119909) 1199103(119909) 119910

119904+1(119909) in the

following theorem According to the above assumptions wecould derive the following result

Theorem 11 Suppose that the cardinal number |119876119888119904119865| ge 119904 + 1

and for each 119910(119909) isin 119876119888119904119865can be represented as the form

119910 (119909) = 1199101(119909) + 120582

119894119901 (119909) 120582

119894isin R (56)

for some 1199101(119909) 119901(119909) isin R[119909] and 120582

119894isin R for 119894 = 1 2 119904 +

1 Then 119901119904

(119909) | 119891(119909) and the polynomial 119865(119909 119910) can berepresented as

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894 (57)

for constants 119888119894isin R 119894 = 0 1 119904

Proof Let 119910119894(119909) be distinct quasi-coincidence solutions of

119865(119909 119910) corresponding to quasi-coincidence values 119886119894 1 le 119894 le

119904 + 1 such that

119865 (119909 119910119894(119909)) = 119886

119894119891 (119909) (58)

Choose 119894 = 1 119865(119909 1199101(119909)) = 119886

1119891(119909) When 119910 minus 119910

1(119909) divides

the function 119865(119909 119910) we get

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909) (59)

where 1198651(119909 119910) is the quotient and 119886

1119891(119909) is the remainder

From the above identity take 119910 = 1199102(119909) and it becomes

119865 (119909 1199102(119909)) = (119910

2(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) + 119886

1119891 (119909)

= 1198862119891 (119909)

(60)

6 Abstract and Applied Analysis

Then

(1199102(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) = (119886

2minus 1198861) 119891 (119909) (61)

By (56) 1199102(119909) minus 119910

1(119909) = 120582

2119901(119909) it yields that

1198651(119909 1199102(119909)) = (

(1198862minus 1198861)

1205822

)119891 (119909)

119901 (119909)

= 1198892

119891 (119909)

119901 (119909)isin R [119909] for 119889

2=

(1198862minus 1198861)

1205822

(62)

Hence

1198651(119909 119910) = (119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909) (63)

Continuing this process from 119894 = 2 to 119904 minus 1 we obtain

119865119894(119909 119910) = (119910 minus 119910

119894+1(119909)) 119865

119894+1(119909 119910) + 119889

119894+1

119891 (119909)

119901119894 (119909) (64)

for some 119889119894+1

isin R 119894 = 1 2 119904 minus 1 Finally we could get

119865119904minus1

(119909 119910) = (119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909) (65)

119865119904(119909) does not contain the variable 119910 since deg

119910119865 = 119904 By the

assumption (58) 119865(119909 119910119904+1

(119909)) = 119886119904+1

119891(119909) It follows that

119865119904(119909) = 120582

119891 (119909)

119901119904 (119909)isin R [119909] for some constant 120582 isin R

(66)

Consequently

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909)

= (119910 minus 1199101(119909)) ((119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) = sdot sdot sdot = (119910 minus 119910

1(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ((119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot )

+ 1198892

119891 (119909)

119901 (119909)) + 1198861119891 (119909)

= (119910 minus 1199101(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ( (119910 minus 119910119904(119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909)

(67)

By (56) we have119910119894(119909) = 119910

1(119909)+120582

119894119901(119909) 119894 = 2 3 119904+1Then

119865(119909 119910) can be expanded to a power series in the expression

119865 (119909 119910) = (119910 minus 1199101(119909))

times ((119910 minus 1199101(119909) minus 120582

2119901 (119909))

times (sdot sdot sdot ( (119910 minus 1199101(119909) minus 120582

119904119901 (119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(68)

for some real numbers 119888119895 119895 = 0 1 119904 Moreover the

leading coefficient of 119865(119909 119910) 119888119904(119891(119909)119901

119904

(119909)) is contained toR[119909] and it follows 119901119904(119909) | 119891(119909)

In the above theorem if there exist at least 119904 + 1 quasi-coincidence solutions with some relations then 119865(119909 119910) hasa fixed type In the following theorem if 119865(119909 119910) has a fixedtype expressed as in Theorem 11 then the cardinal number|Qcs119865| = infin

Theorem 12 The following three conditions are equivalent

(i) 119865(119909 119910) = sum119904

119894=0119888119894(119891(119909)119901

119894

(119909))(119910 minus 1199101(119909))119894 for some

1199101(119909) isin R[119909] some factor 119901(119909) of 119891(119909) and 119888

119894isin R

119894 = 0 1 119904

(ii) |119876119888119904119865| = infin

(iii) |119876119888V119865| ge (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 2

(In fact if |119876119888119904119865| = infin then |119876119888119904

119865| = the cardinal number of

R)

Proof (i) rArr (ii) Suppose that (i) holds Let 119910 = 1199101(119909)+120582119901(119909)

for any constant 120582 isin R we have

119865 (119909 1199101(119909) + 120582119901 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(120582119901 (119909))

119894

= (

119904

sum

119894=0

119888119894120582119894

)119891 (119909)

= 119888119901119898

(119909) for 119888 =119904

sum

119894=0

119888119894120582119894

isin R

(69)

This means that 1199101(119909) + 120582119901(119909) isin Qcs

119865for all 120582 isin R and we

obtain

infin =100381610038161003816100381610038161199101(119909) + 120582119901 (119909)

120582isinR

10038161003816100381610038161003816le1003816100381610038161003816Qcs119865

1003816100381610038161003816 (70)

It follows that the cardinal number |Qcs119865| = infin

Abstract and Applied Analysis 7

(ii) rArr (iii) can be obtained obvious from Lemma 4(iv)(iii) rArr (i) For any 119910(119909) 119910

1(119909) isin Qcs

119865 by Theorem 9 we

have

119910 (119909) minus 1199101(119909) = 119889119901 (119909) (71)

for some fixed factor 119901(119909) of 119891(119909) and by Theorem 11 weobtain

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(72)

for some 1199101(119909) and 119888

119894isin R 119894 = 0 1 119904

Corollary 13 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed values does notexceed an integer ℓ Actually

ℓ = (2deg119891(119909)

+ 119904 + 3) sdot 2deg119891(119909)

+ 1 (73)

Proof By the contrapositive of Theorem 12(ii) rArr (iii) wehave ldquoif |Qcs

119865| lt infin then |Qcv

119865| lt (2

deg119891(119909)+119904+3)sdot2

deg119891(119909)+

2rdquoHence the number of all quasi-fixed values is atmost ℓ thatis ℓ le (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1

Corollary 14 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed solutions does notexceed

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1) (74)

Proof By Lemma 4(iv) we have for any 119886 isin Qcv119865

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(by Lemma 4 (iii) and Corollary 13)

le deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1)

(75)

4 Main Theorems and Some Corollaries

If the 119865(119909 119910) can be represented as the form (57) then anyquasi-coincidence solution can be formed in this section

Lemma 15 Let 119865(119909 119910) be represented as in (57) Then ℎ(119909) isin

R[119909] is a quasi-coincidence solution of 119865(119909 119910) if and only if

ℎ (119909) = 119910 (119909) + 119889119901 (119909) (76)

for some 119889 isin R and some factor 119901(119909) of 119891(119909)

Proof Since

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

(77)

we let 119910 = 119910(119909) and then

119865 (119909 119910 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910(119909) minus 119910(119909))

119894

= 1198880119891 (119909)

(78)

this means 119910(119909) isin Qcs119865

By Theorem 12

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

then 1003816100381610038161003816Qcs119865

1003816100381610038161003816 = infin

(79)

Assume that ℎ(119909) is a quasi-coincidence solution of 119865(119909 119910)and by Corollary 10 we obtain that for any quasi-coincidencesolution ℎ(119909) we have

ℎ (119909) = 119910 (119909) + 119889119901 (119909) for some 119889 isin R (80)

Conversely suppose ℎ(119909) = 119910(119909) + 119889119901(119909) for some factor119901(119909) of 119891(119909) and 119889 isin R Substituting this ℎ(119909) as 119910 in (57)we have

119865 (119909 ℎ (119909)) = 119865 (119909 119910 (119909) + 119889119901 (119909))

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119889119901 (119909))

119894

= (

119904

sum

119894=0

119888119894119889119894

)119891 (119909)

(81)

Therefore ℎ(119909) isin Qcs119865

Note that not any polynomial function 119865(119909 119910) can bewritten as (57) Actually almost all 119865(119909 119910) are expressed asthe form of the next theorem In this situation any solutioncan be written as the next form (lowast) in this theorem undersome condition

Theorem 16 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (82)

and 119891(119909) a polynomial If the cardinal number |Qcs119865| is

infinitely many then each quasi-coincidence solution of (4)must be of the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (lowast)

for arbitrary 120582 isin R where 119901(119909) = (119891(119909)119886119904(119909))1119904 is a factor

of 119891(119909)

Proof Assume |Qcs119865| = infin By Theorem 12 we have

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909)

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910(119909))

119894

(83)

8 Abstract and Applied Analysis

for some 119888119894isin R 119894 = 0 119904 and119910(119909) isin Qcs

119865 Comparing the

coefficients of 119910119904 and 119910119904minus1 in both sides of the above equation

we get

119886119904(119909) = 119888

119904

119891 (119909)

119901119904 (119909)

119886119904minus1

(119909) = minus119904119886119904(119909) 119910 (119909) + 119888

119904minus1

119891 (119909)

119901119904minus1 (119909)

(84)

Consequently by (84) we get

119901119904

(119909) = 119888119904

119891 (119909)

119886119904(119909)

119910 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

isin R [119909]

(85)

By Lemma 15 for any 119889 isin R we have that any quasi-coincidence solution is represented by

119910 (119909) + 119889119901 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

+ 119889119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ (119889 minus119888119904minus1

119904119888119904

)119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909)

(86)

where 119901(119909) = (119888119904)1119904

(119891(119909)119886119904(119909))1119904 (note that since 119889 is

arbitrary then 120582 is arbitrary)This completes the proof

Corollary 17 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (87)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886 (88)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

120582 minus119886119904minus1

(119909)

119904119886119904(119909)

(995333)

for arbitrary 120582 isin R

Corollary 18 Let 119865 R times R rarr R be a polynomial functionwith

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (89)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886119909 (90)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

1205821+ 1205822

119886119904minus1

(119909)

119909(995333)

for arbitrary 1205821and some 120582

2isin R

Proof Assume that there exist infinitely many solutions byTheorem 16 any solution of (4) has the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (91)

for arbitrary 120582 isin R and 119901(119909) = 119888(119909119886119904(119909))1119904

isin R[119909]and then 119886

119904(119909) is a factor of 119909 This means that 119886

119904(119909) isin R

or 119886119904(119909) = 119896119909 for some constant 119896 isin R if 119886

119904(119909) isin R

this implies 119901(119909) = 119888(119909119886119904(119909))1119904

notin R[119909] and this leads acontradiction So we have 119886

119904(119909) = 119896119909 for some 119896 isin R then

119901(119909) = 1198881198961119904

isin R and any solution (91) is represented as

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) = 1205821+ 1205822

119886119904minus1

(119909)

119909(92)

for arbitrary 1205821= 120582119888119896

1119904 and some 1205822= minus1119904119896 isin R

Finally we provide one example to explain Theorem 16

Example 19 Let 119909 = (1199091 1199092) isin R2 119891(119909) = (119909

1+ 1199092)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091minus 1199092) 119910

+ (1199092

11199092

2minus 1199092

11199092minus 11990911199092

2+ 1199092

1+ 211990911199092+ 1199092

2)

(93)

Can we solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909)This polynomial function has exactly 5(ge 119904 + 3 since 119904 = 2)

quasi-fixed solutions as follows

119865 (1199091 1199092 11990911199092minus 1199091minus 1199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 1199091+ 1199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 21199091+ 21199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+1199091

2+1199092

2) =

3

4(1199091+ 1199092)2

(94)

In fact by Theorem 16 we can find any quasi-coincidencesolution written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091minus 1199092

2+ 120582119901 (119909)

= 11990911199092+ (120582 minus

1

2) (1199091minus 1199092)

= 11990911199092+ 120583 (119909

1minus 1199092)

(95)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091+ 1199092 This shows the quasi-coincidence

(point) solutions have cardinal |Qs119865| = infin

In practice we have no idea to check the number of thisequation is infinitely many or finitely many But we providean easy method to solve all solutions if the number of all

Abstract and Applied Analysis 9

solutions is infinitely many in this paper Thus we can solvethose solutions directly and checkwhether those solutions arethe solutions of 119865(119909 119910) = 119886119891(119909) and give an example in thefollowing

Example 20 Let 119909 = (1199091 1199092) isin R2 119891(119909) = 119909

2

1(1199092+ 1199093)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091(1199092+ 1199093)) 119910

+ (1199092

11199092

2+ 1199092

111990921199093+ 1199092

11199092

3)

(96)

We will solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909) ifthe number of all solutions is infinitely many

By Theorem 16 we can find that any quasi-coincidencesolution can be written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091(1199092+ 1199093)

2+ 120582119901 (119909)

= 11990911199092+ 1205831199091(1199092+ 1199093)

(97)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091(1199092+ 1199093) We let 119910(119909) = 119909

11199092+ 1205831199091(1199092+

1199093) and calculate 119865(119909 119910(119909)) and obtain

119865 (119909 119910 (119909)) = (1205832

+ 120583 + 1) 1199092

1(1199092+ 1199093)2

(98)

This means that the quasi-coincidence (point) solutions havecardinal |Qs

119865| = infin

We would like to provide one open problem as follows

Further Development Let Q be a quotient field Consider aquotient-valued polynomial function

119865 Q119899

timesQ 997888rarr Q (99)

Can we find all quasi-coincidence solutions 119910 = 119910(119909) isin Q[119909]

to satisfy

119865 (119909 119910) = 119886119891 (119909) (100)

for some polynomials 119891(119909) isin Q[119909] by a co-NP hardnessalgorithm

References

[1] A K Lenstra ldquoFactoring multivariate polynomials over alge-braic number fieldsrdquo SIAM Journal on Computing vol 16 no 3pp 591ndash598 1987

[2] E P Tsigaridas and I Z Emiris ldquoUnivariate polynomial realroot isolation continued fractions revisitedrdquo in AlgorithmsmdashESA 2006 vol 4168 of Lecture Notes in Computer Science pp817ndash828 Springer Berlin Germany 2006

[3] J von zur Gathen and J Gerhard Modern Computer AlgebraCambridge University Press Cambridge UK 2nd edition2003

[4] M G Marinari H M Moller and T Mora ldquoOn multiplicitiesin polynomial system solvingrdquo Transactions of the AmericanMathematical Society vol 348 no 8 pp 3283ndash3321 1996

[5] P Gopalan V Guruswami and R J Lipton ldquoAlgorithmsfor modular counting of roots of multivariate polynomialsrdquoAlgorithmica vol 50 no 4 pp 479ndash496 2008

[6] V Y Pan ldquoUnivariate polynomials nearly optimal algorithmsfor numerical factorization and root-findingrdquo Journal of Sym-bolic Computation vol 33 no 5 pp 701ndash733 2002

[7] C-M Chen T-H Chang and Y-P Liao ldquoCoincidence the-orems generalized variational inequality theorems and min-imax inequality theorems for the 120601-mapping on G-convexspacesrdquo Fixed Point Theory and Applications vol 2007 ArticleID 078696 13 pages 2007

[8] C-M Chen T-H Chang and C-W Chung ldquoCoincidence the-orems on nonconvex sets and its applicationsrdquo The TaiwaneseJournal of Mathematics vol 13 no 2 pp 501ndash513 2009

[9] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinearmultivaluedmapsrdquoTopology and Its Applications vol159 no 1 pp 49ndash56 2012

[10] W-S Du ldquoOn approximate coincidence point properties andtheir applications to fixed point theoryrdquo Journal of AppliedMathematics vol 2012 Article ID 302830 17 pages 2012

[11] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[12] H-C Lai and Y-C Chen ldquoA quasi-fixed polynomial problemfor a polynomial functionrdquo Journal of Nonlinear and ConvexAnalysis vol 11 no 1 pp 101ndash114 2010

[13] H-C Lai and Y-C Chen ldquoQuasi-fixed polynomial for vector-valued polynomial functions on R119899 times Rrdquo Fixed Point Theoryvol 12 no 2 pp 391ndash400 2011 (Romanian)

[14] Y C Chen and H C Lai ldquoQuasi-fixed point solutions of real-valued polynomial equationrdquo in Proceedings of the 9th Interna-tional Conference on Fixed PointTheory and Its Applications pp16ndash22 Changhua Taiwan 2009

[15] Y-C Chen and H-C Lai ldquoA non-NP-complete algorithm for aquasi-fixed polynomial problemrdquoAbstract andAppliedAnalysisvol 2013 Article ID 893045 10 pages 2013

[16] Y C Chen and H C Lai ldquoNew quasi-coincidence pointpolynomial problemsrdquo Journal of Applied Mathematics vol2013 Article ID 959464 8 pages 2013

[17] R P Grimaldi Discrete and Combinatorial Mathematics Pear-son Secaucus NJ USA 5th edition 2003

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

Furthermore we consider a multivariate polynomialfunction 119865 R119899 times R rarr R and extend (3) as a more generalcoincidence (point) problem inwhich the119909 = (119909

1 1199092 119909

119899)

is replaced by 119909 throughout this paper where 119891(119909) is anonzero arbitrary polynomial inR[119909] Then we restate (3) asthe following equation

119865 (119909 119910) = 119886119891 (119909) (4)

Thus we can give some definitions like Definition 2 asfollows

Definition 3 A polynomial function 119910 = 119910(119909) satisfying (4)is called a quasi-coincidence (point) solution correspondingto some real number 119886 This number 119886 is called a quasi-coincidence value corresponding to the polynomial solutions119910 = 119910(119909)

The number of all solutions in (4) may be infinitely manyfinitely many or not solvable In this paper we solve allsolutions of (4) if the number is infinitely many Moreoverwe provide an upper bound for the number of all solutions ifthe number is finitely many

In Section 2 we derive some properties of quasi-coincidence solutions If (4) has infinitely many quasi-coincidence solutions the form of 119865(119909 119910) will be describedin Section 3 In the last section we solve all solutions if (4)has infinitely many solutions

2 Preliminaries

For convenience we denote the polynomial function by

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198861(119909) 119910 + 119886

0(119909)

=

119904

sum

119894=0

119886119894(119909) 119910119894

(5)

throughout this paper and since there may exist manysolutions corresponding to the same number 119886 we use thesimilar notations like (Definition 2 [11]) to represent them

Notation 1 (1) Qcs119865 the set of all solutions satisfying

equation (4) the solution in Qcs119865is also called a quasi-

coincidence solution in (4) (like Definition 2)(2) Qcv

119865 the set of all solutions 119886 satisfying equation (4)

the solution in Qcv119865is also called a quasi-coincidence value

in (4) (like Definition 2)(3) Qcs

119865(119886) the set of all quasi-coincidence solutions

119910(119909) corresponding to a quasi-coincidence value 119886(4) For each 119886 isin R we denote |Qcs

119865(119886)| as the cardinal

number of Qcs119865(119886)

Evidently by Notation 1 we have the following lemma

Lemma 4 (i) 119876119888119904119865= ⋃119886isin119876119888V119865

119876119888119904119865(119886)

(ii) 119876119888119904119865(119886)⋂119876119888119904

119865(119887) = 0 for any 119886 = 119887 in 119876119888V

119865

(iii) |119876119888119904119865(119886)| le deg

119910119865(119909 119910) for any 119886 isin R

(iv) |119876119888119904119865| le |119876119888V

119865||119876119888119904119865(119886)| for any 119886 isin 119876119888V

119865

Proof (i)⋃119886isinQcv119865 Qcs

119865(119886) sube Qcs

119865is obvious

Conversely for any 119910(119909) isin Qcs119865 by Notation 1(1) we

have

119865 (119909 119910 (119909)) = 119886119891 (119909) (6)

for some 119886 isin Qcv119865Thismeans119910(119909) isin Qcs

119865(119886) andwe obtain

Qcs119865sube ⋃

119886isinQcv119865

Qcs119865(119886) (7)

(ii) Let 119886 = 119887 in Qcv119865 if there exists 119910(119909) isin R[119909] such that

119910 (119909) isin Qcs119865(119886)⋂Qcs

119865(119887) (8)

by Notation 1(3) we have

119886119891 (119909) = 119865 (119909 119910 (119909)) = 119887119891 (119909) (9)

This leads a contradiction to 119886 = 119887 and we haveQcs119865(119886)⋂Qcs

119865(119887) = 0

(iii) For each 119886 isin R the number of all solutions 119910 =

119910(119909) to the polynomial equation 119865(119909 119910) minus 119886119891(119909) = 0 is atmost deg

119910(119865(119909 119910) minus 119886119891(119909)) = deg

119910119865(119909 119910) then the result is

obtained(iv) By (i) we have

Qcs119865= ⋃

119886isinQcv119865

Qcs119865(119886) (10)

It follows that

1003816100381610038161003816Qcs119865

1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

⋃

119886isinQcv119865

Qcs119865(119886)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816Qcv119865

1003816100381610038161003816

1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816 for any 119886 isin Qcv119865

(11)

In the following lemma we explain some interestingproperties of the relations of quasi-coincidence point solu-tions Throughout this paper we consider (4) for polynomialfunction (5) and nonzero arbitrary polynomial 119891(119909) inR[119909]

Lemma 5 Let the cardinal number |119876119888V119865| ge 2 and 119886 = 119887 in

119876119888V119865 Then for any 119910

1(119909) isin 119876119888119904

119865(119886) and 119910

2(119909) isin 119876119888119904

119865(119887) one

has

1199101(119909) minus 119910

2(119909) = 119889119901 (119909) 119891119900119903 119904119900119898119890 119889 isin R (12)

and this 119901(119909) is a factor of 119891(119909) that is 119901(119909) | 119891(119909)

Proof Since 1199101(119909) 119910

2(119909) correspond to 119886 119887 respectively we

have

119865 (119909 1199101(119909)) = 119886119891 (119909) (13)

119865 (119909 1199102(119909)) = 119887119891 (119909) (14)

Abstract and Applied Analysis 3

Subtracting (14) from (13) and using binomial formula ityields that

(119886 minus 119887) 119891 (119909) = 119865 (119909 1199101(119909)) minus 119865 (119909 119910

2(119909))

= 119886119904(119909) [119910

119904

1(119909) minus 119910

119904

2(119909)]

+ 119886119904minus1

(119909) [119910119904minus1

1(119909) minus 119910

119904minus1

2(119909)]

+ sdot sdot sdot + 1198861(119909) [119910

1(119909) minus 119910

2(119909)]

= [1199101(119909) minus 119910

2(119909)]

times [119886119904(119909) 119866119904(1199101(119909) 119910

2(119909))

+ 119886119904minus1

(119909) 119866119904minus1

(1199101(119909) 119910

2(119909))

+ sdot sdot sdot + 1198861(119909)]

= [1199101(119909) minus 119910

2(119909)] 119876 (119909 119910

1(119909) 119910

2(119909))

(15)

where

119866119895(1199101(119909) 119910

2(119909)) = 119910

119895minus1

1(119909)

+ 119910119895minus2

1(119909) 1199102(119909) + sdot sdot sdot + 119910

119895minus1

2(119909)

(16)

for 119895 = 119904 119904 minus 1 2 1 Evidently the factor 1199101(119909) minus 119910

2(119909) is

divisible to the term (119886 minus 119887)119891(119909) and since 119886 = 119887 we obtain

1199101(119909) minus 119910

2(119909) = 119889119901 (119909) (17)

for some real number 119889 isin R and factor 119901(119909) of 119891(119909)

In Lemma 5 the difference of any two distinct quasi-coincidence solutions corresponding to distinct values is afactor of 119891(119909) Thus we may define a class of those factors inthe following

Notation 2 (i) DenoteΦ(119901(119909)) = 120572119901(119909) 120572 isin R(ii) Let 119910(119909) be an arbitrary polynomial in R[119909] and we

denote 119910(119909) + Φ(119901(119909)) = 119910(119909) + 120572119901(119909) 120572 isin R(iii) deg119891(119909) = sum

119899

119894=1deg119909119894

119891(119909)

If 1199101(119909) 119910

2(119909) isin Qcs

119865correspond to distinct quasi-

coincidence values by Lemma 5 and Notation 2 we have

1199101(119909) minus 119910

2(119909) isin ⋃

119901(119909)|119891(119909)

Φ(119901 (119909)) (18)

Since the number of all factors 119901(119909) to 119891(119909) is at most2deg119891(119909) by the definitions of ldquothe pigeonhole principlerdquo in[17] we have the following results

Lemma 6 Suppose that1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge 119896 sdot 2deg119891(119909)

+ 2 (19)

Then there exists 119910(119909) isin 119876119888119904119865and 119901(119909) is a factor of119891(119909) such

that1003816100381610038161003816(119910 (119909) + Φ (119901 (119909))) cap 119876119888119904

119865

1003816100381610038161003816 ge 119896 (20)

Proof Since |Qcv119865| ge 1198962

deg119891(119909)+2 by (18) there exists 119910

1(119909)

1199102(119909) 119910

1198962deg119891(119909)+2

(119909) isin Qcs119865such that

119910119895(119909) minus 119910

1(119909) isin Φ (119901

119895(119909)) (21)

for some factor 119901119895(119909) of 119891(119909) for 119895 = 2 3 1198962

deg119891(119909)+ 2

Moreover we have that the number of all factors to 119891(119909) isat most 2deg119891(119909) By ldquothe pigeonhole principlerdquo there exists asubset 119910

119895119894(119909)119896

119894=1sube 119910119895(119909)1198962

deg119891(119909)+2

119895=2such that

119910119895119894(119909) minus 119910

1(119909) isin Φ (119901 (119909)) (22)

for 119894 = 1 2 119896 and some factor 119901(119909) of 119865(119909) (this 119901(119909) isindependent of the choice of 119894) Thus

119910119895119894(119909)119896

119894=1

sube (1199101(119909) + Φ (119901 (119909))) capQcs

119865

1003816100381610038161003816(1199101 (119909) + Φ (119901 (119909))) capQcs119865

1003816100381610038161003816 ge 119896

(23)

For convenience we explain the relations of Qcs119865and

Φ(119901(119909)) in the following lemma

Lemma 7 Let 119910(119909) isin 119876119888119904119865(119886) for some 119886 isin R Then

119876119888119904119865= 119876119888119904

119865(119886)⋃( ⋃

119901(119909)|119891(119909)

(119910 (119909) + Φ (119901 (119909))) cap 119876119888119904119865)

(24)

for some factor 119901(119909) of 119891(119909)

Proof For any 1199101(119909) isin Qcs

119865 Qcs119865(119886) then 119910

1(119909) isin Qcs(119887)

for some 119887 isin Qcv119865 By Lemma 5 we have

1199101(119909) minus 119910 (119909) isin Φ (119901 (119909)) (25)

for some factor 119901(119909) of 119891(119909) Then

1199101(119909) isin ⋃

119901(119909)|119891(119909)

119910 (119909) + Φ (119901 (119909)) (26)

and it follows that

Qcs119865sube Qcs

119865(119886)⋃( ⋃

119901(119909)|119891(119909)

119910 (119909) + Φ (119901 (119909))) (27)

Moreover by Lemma 4(i) Qcs119865(119886) sube Qcs

119865 then we obtain

Qcs119865

= Qcs119865(119886)⋃( ⋃

119901(119909)|119891(119909)

(119910 (119909) + Φ (119901 (119909))) capQcs119865)

(28)

In order to let the number of all elements in the intersec-tion of sets 119910

1(119909) + Φ(119901

1(119909)) and 119910

2(119909) + Φ(119901

2(119909)) be large

enough we find a lower bound for |Qcv119865| in the following

theorem

4 Abstract and Applied Analysis

Theorem 8 Suppose that the cardinal number1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (29)

Then for any 1199101(119909) = 119910

2(119909) isin 119876119888119904

119865 there exist two factors

1199011(119909) and 119901

2(119909) of 119891(119909) such that

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) cap 119876119888119904

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(30)

Proof Let 1199101(119909) isin Qcs

119865and by assumption

1003816100381610038161003816Qcv119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (31)

and by Lemma 6 there exists a factor 1199011(119909) of 119891(119909) such that

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) capQcs

119865

1003816100381610038161003816 ge 2deg119891(119909)

+ 119904 + 3 (32)

This implies that1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865minus 1199101(119909)

1003816100381610038161003816 ge 2deg119891(119909)

+ 119904 + 2

(33)

Moreover for any 1199102(119909) isin Qcs

119865 we have

(1199101(119909) + Φ (119901

1(119909)))

capQcs119865minus 1199101(119909) sube (119910

1(119909) + Φ (119901

1(119909))) capQcs

119865

sube Qcs119865

(by Lemma 7)

= Qcs119865(119887)⋃( ⋃

119901(119909)|119891(119909)

(1199102(119909) + Φ (119901 (119909))) capQcs

119865)

(34)

for some constant 119887 isin R and it follows that

2deg119891(119909)

+ 119904 + 2

=1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865minus 1199101(119909)

1003816100381610038161003816

le1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865

1003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

Qcs119865(119887)⋃( ⋃

119901(119909)|119891(119909)

(1199102(119909) + Φ (119901 (119909))) capQcs

119865)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816Qcs119865(119887)

1003816100381610038161003816 + sum

119901(119909)|119891(119909)

1003816100381610038161003816(1199102 (119909) + Φ (119901 (119909))) capQcs119865

1003816100381610038161003816

(by Lemma 4 (iii))

le 119904 + sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865

1003816100381610038161003816

(35)

Canceling both sides of the above inequality by ldquo119904rdquo it followsthat

2deg119891(119909)

+ 2 le sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865

1003816100381610038161003816 (36)

this implies that

2deg119891(119909)

+ 1 le sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865minus 1199102(119909)

1003816100381610038161003816

(37)

By the pigeonholersquo principle and since the number of allfactors 119901(119909) to 119891(119909) is at most 2deg119891(119909) we have

1003816100381610038161003816(1199102 (119909) + Φ (1199012(119909))) capQcs

119865minus 1199102(119909)

1003816100381610038161003816 ge 2

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909)) minus 119910

1(119909))

cap (1199102(119909) + Φ (119901

2(119909)) minus 119910

2(119909)) capQcs

119865

1003816100381610038161003816 ge 2

(38)

for some factor 1199012(119909) of 119891(119909) Thus we obtain

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) capQcs

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(39)

Up to now we have not shown that the factor 119901(119909)

uniquely existed eventually In the following theorem wewould show the uniqueness property for the factor 119901(119909) of119891(119909) if the number of all quasi-coincidence values is largeenough

Theorem 9 Assume that the cardinal number1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (40)

Then for any 1199101(119909) 1199102(119909) isin 119876119888119904

119865 one has

1199101(119909) minus 119910

2(119909) = 120582119901 (119909) (41)

where 120582 isin R and 119901(119909) is a factor of 119891(119909) (this 119901(119909) isindependent of the choice of 119910

1(119909) and 119910

2(119909))

Proof Let 1199101(119909) = 119910

2(119909) isin Qcs

119865 by Theorem 8 we have

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) capQcs

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(42)

for some factors 1199011(119909) and 119901

2(119909) of 119891(119909) There exists

1198921(119909) = 119892

2(119909) isin Qcs

119865 such that

1198921(119909) 119892

2(119909) isin (119910

1(119909) + Φ (119901

1(119909)))

cap (1199102(119909) + Φ (119901

2(119909)))

capQcs119865minus 1199101(119909) 119910

2(119909)

(43)

This implies that

1198921(119909) isin 119910

1(119909) + Φ (119901

1(119909))

1198922(119909) isin 119910

1(119909) + Φ (119901

1(119909))

1198921(119909) isin 119910

2(119909) + Φ (119901

2(119909))

1198922(119909) isin 119910

2(119909) + Φ (119901

2(119909))

(44)

Abstract and Applied Analysis 5

By Notation 2(ii) it yields that

1198921(119909) minus 119910

1(119909) = 120582

11199011(119909)

1198922(119909) minus 119910

1(119909) = 120582

21199011(119909)

1198921(119909) minus 119910

2(119909) = 120582

31199012(119909)

1198922(119909) minus 119910

2(119909) = 120582

41199012(119909)

(45)

for some constants 1205821 1205822 1205823 and 120582

4isin R and consequently

1198922(119909) minus 119892

1(119909) = (119892

2(119909) minus 119910

1(119909)) minus (119892

1(119909) minus 119910

1(119909))

= (1205822minus 1205821) 1199011(119909)

1198922(119909) minus 119892

1(119909) = (119892

2(119909) minus 119910

2(119909)) minus (119892

1(119909) minus 119910

2(119909))

= (1205824minus 1205823) 1199012(119909)

(46)

This implies that

(1205822minus 1205821) 1199011(119909) = (120582

4minus 1205823) 1199012(119909) (47)

and 1199011(119909) = 119901

2(119909) Therefore

1199101(119909) minus 119910

2(119909) = (119892

1(119909) minus 119910

2(119909)) minus (119892

1(119909) minus 119910

1(119909))

= (1205823minus 1205821) 1199011(119909)

(48)

and this means that the factor 119901(119909) of 119891(119909) is uniquely deter-mined independent of the choice of 119910

1(119909) and 119910

2(119909)

Corollary 10 Assume that the cardinal number

1003816100381610038161003816119876119888119904119865

1003816100381610038161003816 ge deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) (2

deg119891(119909)) + 2)

(49)

Then for any 119910(119909) isin 119876119888119904119865 there exists ℎ(119909) 119901(119909) isin R[119909] such

that

119910 (119909) = ℎ (119909) + 120582119901 (119909) (50)

for some 120582 isin R (ℎ(119909) 119901(119909) are independent of the choice of119910(119909))

Proof By Lemma 4(iv) we have

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcv119865

1003816100381610038161003816

1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816 for any 119886 isin Qcv119865

(by Lemma 4 (iii))

le deg119910119865 (119909 119910)

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(51)

By assumption it follows that

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) (2

deg119891(119909)) + 2)

le deg119910119865 (119909 119910)

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(52)

Dividing both sides of the above equation by deg119910119865(119909 119910) we

get

1003816100381610038161003816Qcv119865

1003816100381610038161003816 ge ((2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2) (53)

If ℎ(119909) isin Qcs119865 for any 119910(119909) isin Qcs

119865 by Theorem 9 we have

119910 (119909) = ℎ (119909) + 119889119901 (119909) (54)

for some factor 119901(119909) of 119891(119909)

3 The Type of 119865(119909119910) If the Numberof All Quasi-Coincidence SolutionsIs Infinitely Many

In this section we consider (4) for polynomial function119865(119909 119910) in (5) that is let

119865 (119909 119910) =

119904

sum

119894=0

119886119894(119909) 119910119894 with 119904 ge 2 (55)

119891(119909) isin R[119909] and we assume that 119865(119909 119910) has at least119904 + 1 distinct quasi-coincidence solutions satisfying someconditions that is 119910

1(119909) 1199102(119909) 1199103(119909) 119910

119904+1(119909) in the

following theorem According to the above assumptions wecould derive the following result

Theorem 11 Suppose that the cardinal number |119876119888119904119865| ge 119904 + 1

and for each 119910(119909) isin 119876119888119904119865can be represented as the form

119910 (119909) = 1199101(119909) + 120582

119894119901 (119909) 120582

119894isin R (56)

for some 1199101(119909) 119901(119909) isin R[119909] and 120582

119894isin R for 119894 = 1 2 119904 +

1 Then 119901119904

(119909) | 119891(119909) and the polynomial 119865(119909 119910) can berepresented as

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894 (57)

for constants 119888119894isin R 119894 = 0 1 119904

Proof Let 119910119894(119909) be distinct quasi-coincidence solutions of

119865(119909 119910) corresponding to quasi-coincidence values 119886119894 1 le 119894 le

119904 + 1 such that

119865 (119909 119910119894(119909)) = 119886

119894119891 (119909) (58)

Choose 119894 = 1 119865(119909 1199101(119909)) = 119886

1119891(119909) When 119910 minus 119910

1(119909) divides

the function 119865(119909 119910) we get

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909) (59)

where 1198651(119909 119910) is the quotient and 119886

1119891(119909) is the remainder

From the above identity take 119910 = 1199102(119909) and it becomes

119865 (119909 1199102(119909)) = (119910

2(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) + 119886

1119891 (119909)

= 1198862119891 (119909)

(60)

6 Abstract and Applied Analysis

Then

(1199102(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) = (119886

2minus 1198861) 119891 (119909) (61)

By (56) 1199102(119909) minus 119910

1(119909) = 120582

2119901(119909) it yields that

1198651(119909 1199102(119909)) = (

(1198862minus 1198861)

1205822

)119891 (119909)

119901 (119909)

= 1198892

119891 (119909)

119901 (119909)isin R [119909] for 119889

2=

(1198862minus 1198861)

1205822

(62)

Hence

1198651(119909 119910) = (119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909) (63)

Continuing this process from 119894 = 2 to 119904 minus 1 we obtain

119865119894(119909 119910) = (119910 minus 119910

119894+1(119909)) 119865

119894+1(119909 119910) + 119889

119894+1

119891 (119909)

119901119894 (119909) (64)

for some 119889119894+1

isin R 119894 = 1 2 119904 minus 1 Finally we could get

119865119904minus1

(119909 119910) = (119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909) (65)

119865119904(119909) does not contain the variable 119910 since deg

119910119865 = 119904 By the

assumption (58) 119865(119909 119910119904+1

(119909)) = 119886119904+1

119891(119909) It follows that

119865119904(119909) = 120582

119891 (119909)

119901119904 (119909)isin R [119909] for some constant 120582 isin R

(66)

Consequently

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909)

= (119910 minus 1199101(119909)) ((119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) = sdot sdot sdot = (119910 minus 119910

1(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ((119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot )

+ 1198892

119891 (119909)

119901 (119909)) + 1198861119891 (119909)

= (119910 minus 1199101(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ( (119910 minus 119910119904(119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909)

(67)

By (56) we have119910119894(119909) = 119910

1(119909)+120582

119894119901(119909) 119894 = 2 3 119904+1Then

119865(119909 119910) can be expanded to a power series in the expression

119865 (119909 119910) = (119910 minus 1199101(119909))

times ((119910 minus 1199101(119909) minus 120582

2119901 (119909))

times (sdot sdot sdot ( (119910 minus 1199101(119909) minus 120582

119904119901 (119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(68)

for some real numbers 119888119895 119895 = 0 1 119904 Moreover the

leading coefficient of 119865(119909 119910) 119888119904(119891(119909)119901

119904

(119909)) is contained toR[119909] and it follows 119901119904(119909) | 119891(119909)

In the above theorem if there exist at least 119904 + 1 quasi-coincidence solutions with some relations then 119865(119909 119910) hasa fixed type In the following theorem if 119865(119909 119910) has a fixedtype expressed as in Theorem 11 then the cardinal number|Qcs119865| = infin

Theorem 12 The following three conditions are equivalent

(i) 119865(119909 119910) = sum119904

119894=0119888119894(119891(119909)119901

119894

(119909))(119910 minus 1199101(119909))119894 for some

1199101(119909) isin R[119909] some factor 119901(119909) of 119891(119909) and 119888

119894isin R

119894 = 0 1 119904

(ii) |119876119888119904119865| = infin

(iii) |119876119888V119865| ge (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 2

(In fact if |119876119888119904119865| = infin then |119876119888119904

119865| = the cardinal number of

R)

Proof (i) rArr (ii) Suppose that (i) holds Let 119910 = 1199101(119909)+120582119901(119909)

for any constant 120582 isin R we have

119865 (119909 1199101(119909) + 120582119901 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(120582119901 (119909))

119894

= (

119904

sum

119894=0

119888119894120582119894

)119891 (119909)

= 119888119901119898

(119909) for 119888 =119904

sum

119894=0

119888119894120582119894

isin R

(69)

This means that 1199101(119909) + 120582119901(119909) isin Qcs

119865for all 120582 isin R and we

obtain

infin =100381610038161003816100381610038161199101(119909) + 120582119901 (119909)

120582isinR

10038161003816100381610038161003816le1003816100381610038161003816Qcs119865

1003816100381610038161003816 (70)

It follows that the cardinal number |Qcs119865| = infin

Abstract and Applied Analysis 7

(ii) rArr (iii) can be obtained obvious from Lemma 4(iv)(iii) rArr (i) For any 119910(119909) 119910

1(119909) isin Qcs

119865 by Theorem 9 we

have

119910 (119909) minus 1199101(119909) = 119889119901 (119909) (71)

for some fixed factor 119901(119909) of 119891(119909) and by Theorem 11 weobtain

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(72)

for some 1199101(119909) and 119888

119894isin R 119894 = 0 1 119904

Corollary 13 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed values does notexceed an integer ℓ Actually

ℓ = (2deg119891(119909)

+ 119904 + 3) sdot 2deg119891(119909)

+ 1 (73)

Proof By the contrapositive of Theorem 12(ii) rArr (iii) wehave ldquoif |Qcs

119865| lt infin then |Qcv

119865| lt (2

deg119891(119909)+119904+3)sdot2

deg119891(119909)+

2rdquoHence the number of all quasi-fixed values is atmost ℓ thatis ℓ le (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1

Corollary 14 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed solutions does notexceed

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1) (74)

Proof By Lemma 4(iv) we have for any 119886 isin Qcv119865

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(by Lemma 4 (iii) and Corollary 13)

le deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1)

(75)

4 Main Theorems and Some Corollaries

If the 119865(119909 119910) can be represented as the form (57) then anyquasi-coincidence solution can be formed in this section

Lemma 15 Let 119865(119909 119910) be represented as in (57) Then ℎ(119909) isin

R[119909] is a quasi-coincidence solution of 119865(119909 119910) if and only if

ℎ (119909) = 119910 (119909) + 119889119901 (119909) (76)

for some 119889 isin R and some factor 119901(119909) of 119891(119909)

Proof Since

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

(77)

we let 119910 = 119910(119909) and then

119865 (119909 119910 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910(119909) minus 119910(119909))

119894

= 1198880119891 (119909)

(78)

this means 119910(119909) isin Qcs119865

By Theorem 12

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

then 1003816100381610038161003816Qcs119865

1003816100381610038161003816 = infin

(79)

Assume that ℎ(119909) is a quasi-coincidence solution of 119865(119909 119910)and by Corollary 10 we obtain that for any quasi-coincidencesolution ℎ(119909) we have

ℎ (119909) = 119910 (119909) + 119889119901 (119909) for some 119889 isin R (80)

Conversely suppose ℎ(119909) = 119910(119909) + 119889119901(119909) for some factor119901(119909) of 119891(119909) and 119889 isin R Substituting this ℎ(119909) as 119910 in (57)we have

119865 (119909 ℎ (119909)) = 119865 (119909 119910 (119909) + 119889119901 (119909))

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119889119901 (119909))

119894

= (

119904

sum

119894=0

119888119894119889119894

)119891 (119909)

(81)

Therefore ℎ(119909) isin Qcs119865

Note that not any polynomial function 119865(119909 119910) can bewritten as (57) Actually almost all 119865(119909 119910) are expressed asthe form of the next theorem In this situation any solutioncan be written as the next form (lowast) in this theorem undersome condition

Theorem 16 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (82)

and 119891(119909) a polynomial If the cardinal number |Qcs119865| is

infinitely many then each quasi-coincidence solution of (4)must be of the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (lowast)

for arbitrary 120582 isin R where 119901(119909) = (119891(119909)119886119904(119909))1119904 is a factor

of 119891(119909)

Proof Assume |Qcs119865| = infin By Theorem 12 we have

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909)

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910(119909))

119894

(83)

8 Abstract and Applied Analysis

for some 119888119894isin R 119894 = 0 119904 and119910(119909) isin Qcs

119865 Comparing the

coefficients of 119910119904 and 119910119904minus1 in both sides of the above equation

we get

119886119904(119909) = 119888

119904

119891 (119909)

119901119904 (119909)

119886119904minus1

(119909) = minus119904119886119904(119909) 119910 (119909) + 119888

119904minus1

119891 (119909)

119901119904minus1 (119909)

(84)

Consequently by (84) we get

119901119904

(119909) = 119888119904

119891 (119909)

119886119904(119909)

119910 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

isin R [119909]

(85)

By Lemma 15 for any 119889 isin R we have that any quasi-coincidence solution is represented by

119910 (119909) + 119889119901 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

+ 119889119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ (119889 minus119888119904minus1

119904119888119904

)119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909)

(86)

where 119901(119909) = (119888119904)1119904

(119891(119909)119886119904(119909))1119904 (note that since 119889 is

arbitrary then 120582 is arbitrary)This completes the proof

Corollary 17 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (87)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886 (88)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

120582 minus119886119904minus1

(119909)

119904119886119904(119909)

(995333)

for arbitrary 120582 isin R

Corollary 18 Let 119865 R times R rarr R be a polynomial functionwith

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (89)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886119909 (90)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

1205821+ 1205822

119886119904minus1

(119909)

119909(995333)

for arbitrary 1205821and some 120582

2isin R

Proof Assume that there exist infinitely many solutions byTheorem 16 any solution of (4) has the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (91)

for arbitrary 120582 isin R and 119901(119909) = 119888(119909119886119904(119909))1119904

isin R[119909]and then 119886

119904(119909) is a factor of 119909 This means that 119886

119904(119909) isin R

or 119886119904(119909) = 119896119909 for some constant 119896 isin R if 119886

119904(119909) isin R

this implies 119901(119909) = 119888(119909119886119904(119909))1119904

notin R[119909] and this leads acontradiction So we have 119886

119904(119909) = 119896119909 for some 119896 isin R then

119901(119909) = 1198881198961119904

isin R and any solution (91) is represented as

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) = 1205821+ 1205822

119886119904minus1

(119909)

119909(92)

for arbitrary 1205821= 120582119888119896

1119904 and some 1205822= minus1119904119896 isin R

Finally we provide one example to explain Theorem 16

Example 19 Let 119909 = (1199091 1199092) isin R2 119891(119909) = (119909

1+ 1199092)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091minus 1199092) 119910

+ (1199092

11199092

2minus 1199092

11199092minus 11990911199092

2+ 1199092

1+ 211990911199092+ 1199092

2)

(93)

Can we solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909)This polynomial function has exactly 5(ge 119904 + 3 since 119904 = 2)

quasi-fixed solutions as follows

119865 (1199091 1199092 11990911199092minus 1199091minus 1199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 1199091+ 1199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 21199091+ 21199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+1199091

2+1199092

2) =

3

4(1199091+ 1199092)2

(94)

In fact by Theorem 16 we can find any quasi-coincidencesolution written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091minus 1199092

2+ 120582119901 (119909)

= 11990911199092+ (120582 minus

1

2) (1199091minus 1199092)

= 11990911199092+ 120583 (119909

1minus 1199092)

(95)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091+ 1199092 This shows the quasi-coincidence

(point) solutions have cardinal |Qs119865| = infin

In practice we have no idea to check the number of thisequation is infinitely many or finitely many But we providean easy method to solve all solutions if the number of all

Abstract and Applied Analysis 9

solutions is infinitely many in this paper Thus we can solvethose solutions directly and checkwhether those solutions arethe solutions of 119865(119909 119910) = 119886119891(119909) and give an example in thefollowing

Example 20 Let 119909 = (1199091 1199092) isin R2 119891(119909) = 119909

2

1(1199092+ 1199093)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091(1199092+ 1199093)) 119910

+ (1199092

11199092

2+ 1199092

111990921199093+ 1199092

11199092

3)

(96)

We will solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909) ifthe number of all solutions is infinitely many

By Theorem 16 we can find that any quasi-coincidencesolution can be written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091(1199092+ 1199093)

2+ 120582119901 (119909)

= 11990911199092+ 1205831199091(1199092+ 1199093)

(97)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091(1199092+ 1199093) We let 119910(119909) = 119909

11199092+ 1205831199091(1199092+

1199093) and calculate 119865(119909 119910(119909)) and obtain

119865 (119909 119910 (119909)) = (1205832

+ 120583 + 1) 1199092

1(1199092+ 1199093)2

(98)

This means that the quasi-coincidence (point) solutions havecardinal |Qs

119865| = infin

We would like to provide one open problem as follows

Further Development Let Q be a quotient field Consider aquotient-valued polynomial function

119865 Q119899

timesQ 997888rarr Q (99)

Can we find all quasi-coincidence solutions 119910 = 119910(119909) isin Q[119909]

to satisfy

119865 (119909 119910) = 119886119891 (119909) (100)

for some polynomials 119891(119909) isin Q[119909] by a co-NP hardnessalgorithm

References

[1] A K Lenstra ldquoFactoring multivariate polynomials over alge-braic number fieldsrdquo SIAM Journal on Computing vol 16 no 3pp 591ndash598 1987

[2] E P Tsigaridas and I Z Emiris ldquoUnivariate polynomial realroot isolation continued fractions revisitedrdquo in AlgorithmsmdashESA 2006 vol 4168 of Lecture Notes in Computer Science pp817ndash828 Springer Berlin Germany 2006

[3] J von zur Gathen and J Gerhard Modern Computer AlgebraCambridge University Press Cambridge UK 2nd edition2003

[4] M G Marinari H M Moller and T Mora ldquoOn multiplicitiesin polynomial system solvingrdquo Transactions of the AmericanMathematical Society vol 348 no 8 pp 3283ndash3321 1996

[5] P Gopalan V Guruswami and R J Lipton ldquoAlgorithmsfor modular counting of roots of multivariate polynomialsrdquoAlgorithmica vol 50 no 4 pp 479ndash496 2008

[6] V Y Pan ldquoUnivariate polynomials nearly optimal algorithmsfor numerical factorization and root-findingrdquo Journal of Sym-bolic Computation vol 33 no 5 pp 701ndash733 2002

[7] C-M Chen T-H Chang and Y-P Liao ldquoCoincidence the-orems generalized variational inequality theorems and min-imax inequality theorems for the 120601-mapping on G-convexspacesrdquo Fixed Point Theory and Applications vol 2007 ArticleID 078696 13 pages 2007

[8] C-M Chen T-H Chang and C-W Chung ldquoCoincidence the-orems on nonconvex sets and its applicationsrdquo The TaiwaneseJournal of Mathematics vol 13 no 2 pp 501ndash513 2009

[9] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinearmultivaluedmapsrdquoTopology and Its Applications vol159 no 1 pp 49ndash56 2012

[10] W-S Du ldquoOn approximate coincidence point properties andtheir applications to fixed point theoryrdquo Journal of AppliedMathematics vol 2012 Article ID 302830 17 pages 2012

[11] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[12] H-C Lai and Y-C Chen ldquoA quasi-fixed polynomial problemfor a polynomial functionrdquo Journal of Nonlinear and ConvexAnalysis vol 11 no 1 pp 101ndash114 2010

[13] H-C Lai and Y-C Chen ldquoQuasi-fixed polynomial for vector-valued polynomial functions on R119899 times Rrdquo Fixed Point Theoryvol 12 no 2 pp 391ndash400 2011 (Romanian)

[14] Y C Chen and H C Lai ldquoQuasi-fixed point solutions of real-valued polynomial equationrdquo in Proceedings of the 9th Interna-tional Conference on Fixed PointTheory and Its Applications pp16ndash22 Changhua Taiwan 2009

[15] Y-C Chen and H-C Lai ldquoA non-NP-complete algorithm for aquasi-fixed polynomial problemrdquoAbstract andAppliedAnalysisvol 2013 Article ID 893045 10 pages 2013

[16] Y C Chen and H C Lai ldquoNew quasi-coincidence pointpolynomial problemsrdquo Journal of Applied Mathematics vol2013 Article ID 959464 8 pages 2013

[17] R P Grimaldi Discrete and Combinatorial Mathematics Pear-son Secaucus NJ USA 5th edition 2003

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Differential EquationsInternational Journal of

Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Subtracting (14) from (13) and using binomial formula ityields that

(119886 minus 119887) 119891 (119909) = 119865 (119909 1199101(119909)) minus 119865 (119909 119910

2(119909))

= 119886119904(119909) [119910

119904

1(119909) minus 119910

119904

2(119909)]

+ 119886119904minus1

(119909) [119910119904minus1

1(119909) minus 119910

119904minus1

2(119909)]

+ sdot sdot sdot + 1198861(119909) [119910

1(119909) minus 119910

2(119909)]

= [1199101(119909) minus 119910

2(119909)]

times [119886119904(119909) 119866119904(1199101(119909) 119910

2(119909))

+ 119886119904minus1

(119909) 119866119904minus1

(1199101(119909) 119910

2(119909))

+ sdot sdot sdot + 1198861(119909)]

= [1199101(119909) minus 119910

2(119909)] 119876 (119909 119910

1(119909) 119910

2(119909))

(15)

where

119866119895(1199101(119909) 119910

2(119909)) = 119910

119895minus1

1(119909)

+ 119910119895minus2

1(119909) 1199102(119909) + sdot sdot sdot + 119910

119895minus1

2(119909)

(16)

for 119895 = 119904 119904 minus 1 2 1 Evidently the factor 1199101(119909) minus 119910

2(119909) is

divisible to the term (119886 minus 119887)119891(119909) and since 119886 = 119887 we obtain

1199101(119909) minus 119910

2(119909) = 119889119901 (119909) (17)

for some real number 119889 isin R and factor 119901(119909) of 119891(119909)

In Lemma 5 the difference of any two distinct quasi-coincidence solutions corresponding to distinct values is afactor of 119891(119909) Thus we may define a class of those factors inthe following

Notation 2 (i) DenoteΦ(119901(119909)) = 120572119901(119909) 120572 isin R(ii) Let 119910(119909) be an arbitrary polynomial in R[119909] and we

denote 119910(119909) + Φ(119901(119909)) = 119910(119909) + 120572119901(119909) 120572 isin R(iii) deg119891(119909) = sum

119899

119894=1deg119909119894

119891(119909)

If 1199101(119909) 119910

2(119909) isin Qcs

119865correspond to distinct quasi-

coincidence values by Lemma 5 and Notation 2 we have

1199101(119909) minus 119910

2(119909) isin ⋃

119901(119909)|119891(119909)

Φ(119901 (119909)) (18)

Since the number of all factors 119901(119909) to 119891(119909) is at most2deg119891(119909) by the definitions of ldquothe pigeonhole principlerdquo in[17] we have the following results

Lemma 6 Suppose that1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge 119896 sdot 2deg119891(119909)

+ 2 (19)

Then there exists 119910(119909) isin 119876119888119904119865and 119901(119909) is a factor of119891(119909) such

that1003816100381610038161003816(119910 (119909) + Φ (119901 (119909))) cap 119876119888119904

119865

1003816100381610038161003816 ge 119896 (20)

Proof Since |Qcv119865| ge 1198962

deg119891(119909)+2 by (18) there exists 119910

1(119909)

1199102(119909) 119910

1198962deg119891(119909)+2

(119909) isin Qcs119865such that

119910119895(119909) minus 119910

1(119909) isin Φ (119901

119895(119909)) (21)

for some factor 119901119895(119909) of 119891(119909) for 119895 = 2 3 1198962

deg119891(119909)+ 2

Moreover we have that the number of all factors to 119891(119909) isat most 2deg119891(119909) By ldquothe pigeonhole principlerdquo there exists asubset 119910

119895119894(119909)119896

119894=1sube 119910119895(119909)1198962

deg119891(119909)+2

119895=2such that

119910119895119894(119909) minus 119910

1(119909) isin Φ (119901 (119909)) (22)

for 119894 = 1 2 119896 and some factor 119901(119909) of 119865(119909) (this 119901(119909) isindependent of the choice of 119894) Thus

119910119895119894(119909)119896

119894=1

sube (1199101(119909) + Φ (119901 (119909))) capQcs

119865

1003816100381610038161003816(1199101 (119909) + Φ (119901 (119909))) capQcs119865

1003816100381610038161003816 ge 119896

(23)

For convenience we explain the relations of Qcs119865and

Φ(119901(119909)) in the following lemma

Lemma 7 Let 119910(119909) isin 119876119888119904119865(119886) for some 119886 isin R Then

119876119888119904119865= 119876119888119904

119865(119886)⋃( ⋃

119901(119909)|119891(119909)

(119910 (119909) + Φ (119901 (119909))) cap 119876119888119904119865)

(24)

for some factor 119901(119909) of 119891(119909)

Proof For any 1199101(119909) isin Qcs

119865 Qcs119865(119886) then 119910

1(119909) isin Qcs(119887)

for some 119887 isin Qcv119865 By Lemma 5 we have

1199101(119909) minus 119910 (119909) isin Φ (119901 (119909)) (25)

for some factor 119901(119909) of 119891(119909) Then

1199101(119909) isin ⋃

119901(119909)|119891(119909)

119910 (119909) + Φ (119901 (119909)) (26)

and it follows that

Qcs119865sube Qcs

119865(119886)⋃( ⋃

119901(119909)|119891(119909)

119910 (119909) + Φ (119901 (119909))) (27)

Moreover by Lemma 4(i) Qcs119865(119886) sube Qcs

119865 then we obtain

Qcs119865

= Qcs119865(119886)⋃( ⋃

119901(119909)|119891(119909)

(119910 (119909) + Φ (119901 (119909))) capQcs119865)

(28)

In order to let the number of all elements in the intersec-tion of sets 119910

1(119909) + Φ(119901

1(119909)) and 119910

2(119909) + Φ(119901

2(119909)) be large

enough we find a lower bound for |Qcv119865| in the following

theorem

4 Abstract and Applied Analysis

Theorem 8 Suppose that the cardinal number1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (29)

Then for any 1199101(119909) = 119910

2(119909) isin 119876119888119904

119865 there exist two factors

1199011(119909) and 119901

2(119909) of 119891(119909) such that

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) cap 119876119888119904

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(30)

Proof Let 1199101(119909) isin Qcs

119865and by assumption

1003816100381610038161003816Qcv119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (31)

and by Lemma 6 there exists a factor 1199011(119909) of 119891(119909) such that

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) capQcs

119865

1003816100381610038161003816 ge 2deg119891(119909)

+ 119904 + 3 (32)

This implies that1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865minus 1199101(119909)

1003816100381610038161003816 ge 2deg119891(119909)

+ 119904 + 2

(33)

Moreover for any 1199102(119909) isin Qcs

119865 we have

(1199101(119909) + Φ (119901

1(119909)))

capQcs119865minus 1199101(119909) sube (119910

1(119909) + Φ (119901

1(119909))) capQcs

119865

sube Qcs119865

(by Lemma 7)

= Qcs119865(119887)⋃( ⋃

119901(119909)|119891(119909)

(1199102(119909) + Φ (119901 (119909))) capQcs

119865)

(34)

for some constant 119887 isin R and it follows that

2deg119891(119909)

+ 119904 + 2

=1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865minus 1199101(119909)

1003816100381610038161003816

le1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865

1003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

Qcs119865(119887)⋃( ⋃

119901(119909)|119891(119909)

(1199102(119909) + Φ (119901 (119909))) capQcs

119865)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816Qcs119865(119887)

1003816100381610038161003816 + sum

119901(119909)|119891(119909)

1003816100381610038161003816(1199102 (119909) + Φ (119901 (119909))) capQcs119865

1003816100381610038161003816

(by Lemma 4 (iii))

le 119904 + sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865

1003816100381610038161003816

(35)

Canceling both sides of the above inequality by ldquo119904rdquo it followsthat

2deg119891(119909)

+ 2 le sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865

1003816100381610038161003816 (36)

this implies that

2deg119891(119909)

+ 1 le sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865minus 1199102(119909)

1003816100381610038161003816

(37)

By the pigeonholersquo principle and since the number of allfactors 119901(119909) to 119891(119909) is at most 2deg119891(119909) we have

1003816100381610038161003816(1199102 (119909) + Φ (1199012(119909))) capQcs

119865minus 1199102(119909)

1003816100381610038161003816 ge 2

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909)) minus 119910

1(119909))

cap (1199102(119909) + Φ (119901

2(119909)) minus 119910

2(119909)) capQcs

119865

1003816100381610038161003816 ge 2

(38)

for some factor 1199012(119909) of 119891(119909) Thus we obtain

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) capQcs

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(39)

Up to now we have not shown that the factor 119901(119909)

uniquely existed eventually In the following theorem wewould show the uniqueness property for the factor 119901(119909) of119891(119909) if the number of all quasi-coincidence values is largeenough

Theorem 9 Assume that the cardinal number1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (40)

Then for any 1199101(119909) 1199102(119909) isin 119876119888119904

119865 one has

1199101(119909) minus 119910

2(119909) = 120582119901 (119909) (41)

where 120582 isin R and 119901(119909) is a factor of 119891(119909) (this 119901(119909) isindependent of the choice of 119910

1(119909) and 119910

2(119909))

Proof Let 1199101(119909) = 119910

2(119909) isin Qcs

119865 by Theorem 8 we have

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) capQcs

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(42)

for some factors 1199011(119909) and 119901

2(119909) of 119891(119909) There exists

1198921(119909) = 119892

2(119909) isin Qcs

119865 such that

1198921(119909) 119892

2(119909) isin (119910

1(119909) + Φ (119901

1(119909)))

cap (1199102(119909) + Φ (119901

2(119909)))

capQcs119865minus 1199101(119909) 119910

2(119909)

(43)

This implies that

1198921(119909) isin 119910

1(119909) + Φ (119901

1(119909))

1198922(119909) isin 119910

1(119909) + Φ (119901

1(119909))

1198921(119909) isin 119910

2(119909) + Φ (119901

2(119909))

1198922(119909) isin 119910

2(119909) + Φ (119901

2(119909))

(44)

Abstract and Applied Analysis 5

By Notation 2(ii) it yields that

1198921(119909) minus 119910

1(119909) = 120582

11199011(119909)

1198922(119909) minus 119910

1(119909) = 120582

21199011(119909)

1198921(119909) minus 119910

2(119909) = 120582

31199012(119909)

1198922(119909) minus 119910

2(119909) = 120582

41199012(119909)

(45)

for some constants 1205821 1205822 1205823 and 120582

4isin R and consequently

1198922(119909) minus 119892

1(119909) = (119892

2(119909) minus 119910

1(119909)) minus (119892

1(119909) minus 119910

1(119909))

= (1205822minus 1205821) 1199011(119909)

1198922(119909) minus 119892

1(119909) = (119892

2(119909) minus 119910

2(119909)) minus (119892

1(119909) minus 119910

2(119909))

= (1205824minus 1205823) 1199012(119909)

(46)

This implies that

(1205822minus 1205821) 1199011(119909) = (120582

4minus 1205823) 1199012(119909) (47)

and 1199011(119909) = 119901

2(119909) Therefore

1199101(119909) minus 119910

2(119909) = (119892

1(119909) minus 119910

2(119909)) minus (119892

1(119909) minus 119910

1(119909))

= (1205823minus 1205821) 1199011(119909)

(48)

and this means that the factor 119901(119909) of 119891(119909) is uniquely deter-mined independent of the choice of 119910

1(119909) and 119910

2(119909)

Corollary 10 Assume that the cardinal number

1003816100381610038161003816119876119888119904119865

1003816100381610038161003816 ge deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) (2

deg119891(119909)) + 2)

(49)

Then for any 119910(119909) isin 119876119888119904119865 there exists ℎ(119909) 119901(119909) isin R[119909] such

that

119910 (119909) = ℎ (119909) + 120582119901 (119909) (50)

for some 120582 isin R (ℎ(119909) 119901(119909) are independent of the choice of119910(119909))

Proof By Lemma 4(iv) we have

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcv119865

1003816100381610038161003816

1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816 for any 119886 isin Qcv119865

(by Lemma 4 (iii))

le deg119910119865 (119909 119910)

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(51)

By assumption it follows that

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) (2

deg119891(119909)) + 2)

le deg119910119865 (119909 119910)

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(52)

Dividing both sides of the above equation by deg119910119865(119909 119910) we

get

1003816100381610038161003816Qcv119865

1003816100381610038161003816 ge ((2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2) (53)

If ℎ(119909) isin Qcs119865 for any 119910(119909) isin Qcs

119865 by Theorem 9 we have

119910 (119909) = ℎ (119909) + 119889119901 (119909) (54)

for some factor 119901(119909) of 119891(119909)

3 The Type of 119865(119909119910) If the Numberof All Quasi-Coincidence SolutionsIs Infinitely Many

In this section we consider (4) for polynomial function119865(119909 119910) in (5) that is let

119865 (119909 119910) =

119904

sum

119894=0

119886119894(119909) 119910119894 with 119904 ge 2 (55)

119891(119909) isin R[119909] and we assume that 119865(119909 119910) has at least119904 + 1 distinct quasi-coincidence solutions satisfying someconditions that is 119910

1(119909) 1199102(119909) 1199103(119909) 119910

119904+1(119909) in the

following theorem According to the above assumptions wecould derive the following result

Theorem 11 Suppose that the cardinal number |119876119888119904119865| ge 119904 + 1

and for each 119910(119909) isin 119876119888119904119865can be represented as the form

119910 (119909) = 1199101(119909) + 120582

119894119901 (119909) 120582

119894isin R (56)

for some 1199101(119909) 119901(119909) isin R[119909] and 120582

119894isin R for 119894 = 1 2 119904 +

1 Then 119901119904

(119909) | 119891(119909) and the polynomial 119865(119909 119910) can berepresented as

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894 (57)

for constants 119888119894isin R 119894 = 0 1 119904

Proof Let 119910119894(119909) be distinct quasi-coincidence solutions of

119865(119909 119910) corresponding to quasi-coincidence values 119886119894 1 le 119894 le

119904 + 1 such that

119865 (119909 119910119894(119909)) = 119886

119894119891 (119909) (58)

Choose 119894 = 1 119865(119909 1199101(119909)) = 119886

1119891(119909) When 119910 minus 119910

1(119909) divides

the function 119865(119909 119910) we get

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909) (59)

where 1198651(119909 119910) is the quotient and 119886

1119891(119909) is the remainder

From the above identity take 119910 = 1199102(119909) and it becomes

119865 (119909 1199102(119909)) = (119910

2(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) + 119886

1119891 (119909)

= 1198862119891 (119909)

(60)

6 Abstract and Applied Analysis

Then

(1199102(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) = (119886

2minus 1198861) 119891 (119909) (61)

By (56) 1199102(119909) minus 119910

1(119909) = 120582

2119901(119909) it yields that

1198651(119909 1199102(119909)) = (

(1198862minus 1198861)

1205822

)119891 (119909)

119901 (119909)

= 1198892

119891 (119909)

119901 (119909)isin R [119909] for 119889

2=

(1198862minus 1198861)

1205822

(62)

Hence

1198651(119909 119910) = (119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909) (63)

Continuing this process from 119894 = 2 to 119904 minus 1 we obtain

119865119894(119909 119910) = (119910 minus 119910

119894+1(119909)) 119865

119894+1(119909 119910) + 119889

119894+1

119891 (119909)

119901119894 (119909) (64)

for some 119889119894+1

isin R 119894 = 1 2 119904 minus 1 Finally we could get

119865119904minus1

(119909 119910) = (119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909) (65)

119865119904(119909) does not contain the variable 119910 since deg

119910119865 = 119904 By the

assumption (58) 119865(119909 119910119904+1

(119909)) = 119886119904+1

119891(119909) It follows that

119865119904(119909) = 120582

119891 (119909)

119901119904 (119909)isin R [119909] for some constant 120582 isin R

(66)

Consequently

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909)

= (119910 minus 1199101(119909)) ((119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) = sdot sdot sdot = (119910 minus 119910

1(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ((119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot )

+ 1198892

119891 (119909)

119901 (119909)) + 1198861119891 (119909)

= (119910 minus 1199101(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ( (119910 minus 119910119904(119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909)

(67)

By (56) we have119910119894(119909) = 119910

1(119909)+120582

119894119901(119909) 119894 = 2 3 119904+1Then

119865(119909 119910) can be expanded to a power series in the expression

119865 (119909 119910) = (119910 minus 1199101(119909))

times ((119910 minus 1199101(119909) minus 120582

2119901 (119909))

times (sdot sdot sdot ( (119910 minus 1199101(119909) minus 120582

119904119901 (119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(68)

for some real numbers 119888119895 119895 = 0 1 119904 Moreover the

leading coefficient of 119865(119909 119910) 119888119904(119891(119909)119901

119904

(119909)) is contained toR[119909] and it follows 119901119904(119909) | 119891(119909)

In the above theorem if there exist at least 119904 + 1 quasi-coincidence solutions with some relations then 119865(119909 119910) hasa fixed type In the following theorem if 119865(119909 119910) has a fixedtype expressed as in Theorem 11 then the cardinal number|Qcs119865| = infin

Theorem 12 The following three conditions are equivalent

(i) 119865(119909 119910) = sum119904

119894=0119888119894(119891(119909)119901

119894

(119909))(119910 minus 1199101(119909))119894 for some

1199101(119909) isin R[119909] some factor 119901(119909) of 119891(119909) and 119888

119894isin R

119894 = 0 1 119904

(ii) |119876119888119904119865| = infin

(iii) |119876119888V119865| ge (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 2

(In fact if |119876119888119904119865| = infin then |119876119888119904

119865| = the cardinal number of

R)

Proof (i) rArr (ii) Suppose that (i) holds Let 119910 = 1199101(119909)+120582119901(119909)

for any constant 120582 isin R we have

119865 (119909 1199101(119909) + 120582119901 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(120582119901 (119909))

119894

= (

119904

sum

119894=0

119888119894120582119894

)119891 (119909)

= 119888119901119898

(119909) for 119888 =119904

sum

119894=0

119888119894120582119894

isin R

(69)

This means that 1199101(119909) + 120582119901(119909) isin Qcs

119865for all 120582 isin R and we

obtain

infin =100381610038161003816100381610038161199101(119909) + 120582119901 (119909)

120582isinR

10038161003816100381610038161003816le1003816100381610038161003816Qcs119865

1003816100381610038161003816 (70)

It follows that the cardinal number |Qcs119865| = infin

Abstract and Applied Analysis 7

(ii) rArr (iii) can be obtained obvious from Lemma 4(iv)(iii) rArr (i) For any 119910(119909) 119910

1(119909) isin Qcs

119865 by Theorem 9 we

have

119910 (119909) minus 1199101(119909) = 119889119901 (119909) (71)

for some fixed factor 119901(119909) of 119891(119909) and by Theorem 11 weobtain

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(72)

for some 1199101(119909) and 119888

119894isin R 119894 = 0 1 119904

Corollary 13 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed values does notexceed an integer ℓ Actually

ℓ = (2deg119891(119909)

+ 119904 + 3) sdot 2deg119891(119909)

+ 1 (73)

Proof By the contrapositive of Theorem 12(ii) rArr (iii) wehave ldquoif |Qcs

119865| lt infin then |Qcv

119865| lt (2

deg119891(119909)+119904+3)sdot2

deg119891(119909)+

2rdquoHence the number of all quasi-fixed values is atmost ℓ thatis ℓ le (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1

Corollary 14 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed solutions does notexceed

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1) (74)

Proof By Lemma 4(iv) we have for any 119886 isin Qcv119865

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(by Lemma 4 (iii) and Corollary 13)

le deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1)

(75)

4 Main Theorems and Some Corollaries

If the 119865(119909 119910) can be represented as the form (57) then anyquasi-coincidence solution can be formed in this section

Lemma 15 Let 119865(119909 119910) be represented as in (57) Then ℎ(119909) isin

R[119909] is a quasi-coincidence solution of 119865(119909 119910) if and only if

ℎ (119909) = 119910 (119909) + 119889119901 (119909) (76)

for some 119889 isin R and some factor 119901(119909) of 119891(119909)

Proof Since

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

(77)

we let 119910 = 119910(119909) and then

119865 (119909 119910 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910(119909) minus 119910(119909))

119894

= 1198880119891 (119909)

(78)

this means 119910(119909) isin Qcs119865

By Theorem 12

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

then 1003816100381610038161003816Qcs119865

1003816100381610038161003816 = infin

(79)

Assume that ℎ(119909) is a quasi-coincidence solution of 119865(119909 119910)and by Corollary 10 we obtain that for any quasi-coincidencesolution ℎ(119909) we have

ℎ (119909) = 119910 (119909) + 119889119901 (119909) for some 119889 isin R (80)

Conversely suppose ℎ(119909) = 119910(119909) + 119889119901(119909) for some factor119901(119909) of 119891(119909) and 119889 isin R Substituting this ℎ(119909) as 119910 in (57)we have

119865 (119909 ℎ (119909)) = 119865 (119909 119910 (119909) + 119889119901 (119909))

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119889119901 (119909))

119894

= (

119904

sum

119894=0

119888119894119889119894

)119891 (119909)

(81)

Therefore ℎ(119909) isin Qcs119865

Note that not any polynomial function 119865(119909 119910) can bewritten as (57) Actually almost all 119865(119909 119910) are expressed asthe form of the next theorem In this situation any solutioncan be written as the next form (lowast) in this theorem undersome condition

Theorem 16 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (82)

and 119891(119909) a polynomial If the cardinal number |Qcs119865| is

infinitely many then each quasi-coincidence solution of (4)must be of the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (lowast)

for arbitrary 120582 isin R where 119901(119909) = (119891(119909)119886119904(119909))1119904 is a factor

of 119891(119909)

Proof Assume |Qcs119865| = infin By Theorem 12 we have

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909)

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910(119909))

119894

(83)

8 Abstract and Applied Analysis

for some 119888119894isin R 119894 = 0 119904 and119910(119909) isin Qcs

119865 Comparing the

coefficients of 119910119904 and 119910119904minus1 in both sides of the above equation

we get

119886119904(119909) = 119888

119904

119891 (119909)

119901119904 (119909)

119886119904minus1

(119909) = minus119904119886119904(119909) 119910 (119909) + 119888

119904minus1

119891 (119909)

119901119904minus1 (119909)

(84)

Consequently by (84) we get

119901119904

(119909) = 119888119904

119891 (119909)

119886119904(119909)

119910 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

isin R [119909]

(85)

By Lemma 15 for any 119889 isin R we have that any quasi-coincidence solution is represented by

119910 (119909) + 119889119901 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

+ 119889119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ (119889 minus119888119904minus1

119904119888119904

)119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909)

(86)

where 119901(119909) = (119888119904)1119904

(119891(119909)119886119904(119909))1119904 (note that since 119889 is

arbitrary then 120582 is arbitrary)This completes the proof

Corollary 17 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (87)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886 (88)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

120582 minus119886119904minus1

(119909)

119904119886119904(119909)

(995333)

for arbitrary 120582 isin R

Corollary 18 Let 119865 R times R rarr R be a polynomial functionwith

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (89)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886119909 (90)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

1205821+ 1205822

119886119904minus1

(119909)

119909(995333)

for arbitrary 1205821and some 120582

2isin R

Proof Assume that there exist infinitely many solutions byTheorem 16 any solution of (4) has the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (91)

for arbitrary 120582 isin R and 119901(119909) = 119888(119909119886119904(119909))1119904

isin R[119909]and then 119886

119904(119909) is a factor of 119909 This means that 119886

119904(119909) isin R

or 119886119904(119909) = 119896119909 for some constant 119896 isin R if 119886

119904(119909) isin R

this implies 119901(119909) = 119888(119909119886119904(119909))1119904

notin R[119909] and this leads acontradiction So we have 119886

119904(119909) = 119896119909 for some 119896 isin R then

119901(119909) = 1198881198961119904

isin R and any solution (91) is represented as

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) = 1205821+ 1205822

119886119904minus1

(119909)

119909(92)

for arbitrary 1205821= 120582119888119896

1119904 and some 1205822= minus1119904119896 isin R

Finally we provide one example to explain Theorem 16

Example 19 Let 119909 = (1199091 1199092) isin R2 119891(119909) = (119909

1+ 1199092)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091minus 1199092) 119910

+ (1199092

11199092

2minus 1199092

11199092minus 11990911199092

2+ 1199092

1+ 211990911199092+ 1199092

2)

(93)

Can we solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909)This polynomial function has exactly 5(ge 119904 + 3 since 119904 = 2)

quasi-fixed solutions as follows

119865 (1199091 1199092 11990911199092minus 1199091minus 1199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 1199091+ 1199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 21199091+ 21199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+1199091

2+1199092

2) =

3

4(1199091+ 1199092)2

(94)

In fact by Theorem 16 we can find any quasi-coincidencesolution written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091minus 1199092

2+ 120582119901 (119909)

= 11990911199092+ (120582 minus

1

2) (1199091minus 1199092)

= 11990911199092+ 120583 (119909

1minus 1199092)

(95)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091+ 1199092 This shows the quasi-coincidence

(point) solutions have cardinal |Qs119865| = infin

In practice we have no idea to check the number of thisequation is infinitely many or finitely many But we providean easy method to solve all solutions if the number of all

Abstract and Applied Analysis 9

solutions is infinitely many in this paper Thus we can solvethose solutions directly and checkwhether those solutions arethe solutions of 119865(119909 119910) = 119886119891(119909) and give an example in thefollowing

Example 20 Let 119909 = (1199091 1199092) isin R2 119891(119909) = 119909

2

1(1199092+ 1199093)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091(1199092+ 1199093)) 119910

+ (1199092

11199092

2+ 1199092

111990921199093+ 1199092

11199092

3)

(96)

We will solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909) ifthe number of all solutions is infinitely many

By Theorem 16 we can find that any quasi-coincidencesolution can be written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091(1199092+ 1199093)

2+ 120582119901 (119909)

= 11990911199092+ 1205831199091(1199092+ 1199093)

(97)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091(1199092+ 1199093) We let 119910(119909) = 119909

11199092+ 1205831199091(1199092+

1199093) and calculate 119865(119909 119910(119909)) and obtain

119865 (119909 119910 (119909)) = (1205832

+ 120583 + 1) 1199092

1(1199092+ 1199093)2

(98)

This means that the quasi-coincidence (point) solutions havecardinal |Qs

119865| = infin

We would like to provide one open problem as follows

Further Development Let Q be a quotient field Consider aquotient-valued polynomial function

119865 Q119899

timesQ 997888rarr Q (99)

Can we find all quasi-coincidence solutions 119910 = 119910(119909) isin Q[119909]

to satisfy

119865 (119909 119910) = 119886119891 (119909) (100)

for some polynomials 119891(119909) isin Q[119909] by a co-NP hardnessalgorithm

References

[1] A K Lenstra ldquoFactoring multivariate polynomials over alge-braic number fieldsrdquo SIAM Journal on Computing vol 16 no 3pp 591ndash598 1987

[2] E P Tsigaridas and I Z Emiris ldquoUnivariate polynomial realroot isolation continued fractions revisitedrdquo in AlgorithmsmdashESA 2006 vol 4168 of Lecture Notes in Computer Science pp817ndash828 Springer Berlin Germany 2006

[3] J von zur Gathen and J Gerhard Modern Computer AlgebraCambridge University Press Cambridge UK 2nd edition2003

[4] M G Marinari H M Moller and T Mora ldquoOn multiplicitiesin polynomial system solvingrdquo Transactions of the AmericanMathematical Society vol 348 no 8 pp 3283ndash3321 1996

[5] P Gopalan V Guruswami and R J Lipton ldquoAlgorithmsfor modular counting of roots of multivariate polynomialsrdquoAlgorithmica vol 50 no 4 pp 479ndash496 2008

[6] V Y Pan ldquoUnivariate polynomials nearly optimal algorithmsfor numerical factorization and root-findingrdquo Journal of Sym-bolic Computation vol 33 no 5 pp 701ndash733 2002

[7] C-M Chen T-H Chang and Y-P Liao ldquoCoincidence the-orems generalized variational inequality theorems and min-imax inequality theorems for the 120601-mapping on G-convexspacesrdquo Fixed Point Theory and Applications vol 2007 ArticleID 078696 13 pages 2007

[8] C-M Chen T-H Chang and C-W Chung ldquoCoincidence the-orems on nonconvex sets and its applicationsrdquo The TaiwaneseJournal of Mathematics vol 13 no 2 pp 501ndash513 2009

[9] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinearmultivaluedmapsrdquoTopology and Its Applications vol159 no 1 pp 49ndash56 2012

[10] W-S Du ldquoOn approximate coincidence point properties andtheir applications to fixed point theoryrdquo Journal of AppliedMathematics vol 2012 Article ID 302830 17 pages 2012

[11] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[12] H-C Lai and Y-C Chen ldquoA quasi-fixed polynomial problemfor a polynomial functionrdquo Journal of Nonlinear and ConvexAnalysis vol 11 no 1 pp 101ndash114 2010

[13] H-C Lai and Y-C Chen ldquoQuasi-fixed polynomial for vector-valued polynomial functions on R119899 times Rrdquo Fixed Point Theoryvol 12 no 2 pp 391ndash400 2011 (Romanian)

[14] Y C Chen and H C Lai ldquoQuasi-fixed point solutions of real-valued polynomial equationrdquo in Proceedings of the 9th Interna-tional Conference on Fixed PointTheory and Its Applications pp16ndash22 Changhua Taiwan 2009

[15] Y-C Chen and H-C Lai ldquoA non-NP-complete algorithm for aquasi-fixed polynomial problemrdquoAbstract andAppliedAnalysisvol 2013 Article ID 893045 10 pages 2013

[16] Y C Chen and H C Lai ldquoNew quasi-coincidence pointpolynomial problemsrdquo Journal of Applied Mathematics vol2013 Article ID 959464 8 pages 2013

[17] R P Grimaldi Discrete and Combinatorial Mathematics Pear-son Secaucus NJ USA 5th edition 2003

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Theorem 8 Suppose that the cardinal number1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (29)

Then for any 1199101(119909) = 119910

2(119909) isin 119876119888119904

119865 there exist two factors

1199011(119909) and 119901

2(119909) of 119891(119909) such that

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) cap 119876119888119904

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(30)

Proof Let 1199101(119909) isin Qcs

119865and by assumption

1003816100381610038161003816Qcv119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (31)

and by Lemma 6 there exists a factor 1199011(119909) of 119891(119909) such that

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) capQcs

119865

1003816100381610038161003816 ge 2deg119891(119909)

+ 119904 + 3 (32)

This implies that1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865minus 1199101(119909)

1003816100381610038161003816 ge 2deg119891(119909)

+ 119904 + 2

(33)

Moreover for any 1199102(119909) isin Qcs

119865 we have

(1199101(119909) + Φ (119901

1(119909)))

capQcs119865minus 1199101(119909) sube (119910

1(119909) + Φ (119901

1(119909))) capQcs

119865

sube Qcs119865

(by Lemma 7)

= Qcs119865(119887)⋃( ⋃

119901(119909)|119891(119909)

(1199102(119909) + Φ (119901 (119909))) capQcs

119865)

(34)

for some constant 119887 isin R and it follows that

2deg119891(119909)

+ 119904 + 2

=1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865minus 1199101(119909)

1003816100381610038161003816

le1003816100381610038161003816(1199101 (119909) + Φ (119901

1(119909))) capQcs

119865

1003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

Qcs119865(119887)⋃( ⋃

119901(119909)|119891(119909)

(1199102(119909) + Φ (119901 (119909))) capQcs

119865)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003816100381610038161003816Qcs119865(119887)

1003816100381610038161003816 + sum

119901(119909)|119891(119909)

1003816100381610038161003816(1199102 (119909) + Φ (119901 (119909))) capQcs119865

1003816100381610038161003816

(by Lemma 4 (iii))

le 119904 + sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865

1003816100381610038161003816

(35)

Canceling both sides of the above inequality by ldquo119904rdquo it followsthat

2deg119891(119909)

+ 2 le sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865

1003816100381610038161003816 (36)

this implies that

2deg119891(119909)

+ 1 le sum

119901(119909)|119891(119909)

10038161003816100381610038161199102 (119909) + Φ (119901 (119909)) capQcs119865minus 1199102(119909)

1003816100381610038161003816

(37)

By the pigeonholersquo principle and since the number of allfactors 119901(119909) to 119891(119909) is at most 2deg119891(119909) we have

1003816100381610038161003816(1199102 (119909) + Φ (1199012(119909))) capQcs

119865minus 1199102(119909)

1003816100381610038161003816 ge 2

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909)) minus 119910

1(119909))

cap (1199102(119909) + Φ (119901

2(119909)) minus 119910

2(119909)) capQcs

119865

1003816100381610038161003816 ge 2

(38)

for some factor 1199012(119909) of 119891(119909) Thus we obtain

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) capQcs

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(39)

Up to now we have not shown that the factor 119901(119909)

uniquely existed eventually In the following theorem wewould show the uniqueness property for the factor 119901(119909) of119891(119909) if the number of all quasi-coincidence values is largeenough

Theorem 9 Assume that the cardinal number1003816100381610038161003816119876119888V119865

1003816100381610038161003816 ge (2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2 (40)

Then for any 1199101(119909) 1199102(119909) isin 119876119888119904

119865 one has

1199101(119909) minus 119910

2(119909) = 120582119901 (119909) (41)

where 120582 isin R and 119901(119909) is a factor of 119891(119909) (this 119901(119909) isindependent of the choice of 119910

1(119909) and 119910

2(119909))

Proof Let 1199101(119909) = 119910

2(119909) isin Qcs

119865 by Theorem 8 we have

1003816100381610038161003816(1199101 (119909) + Φ (1199011(119909))) cap (119910

2(119909) + Φ (119901

2(119909))) capQcs

119865

minus 1199101(119909) 119910

2(119909)

1003816100381610038161003816 ge 2

(42)

for some factors 1199011(119909) and 119901

2(119909) of 119891(119909) There exists

1198921(119909) = 119892

2(119909) isin Qcs

119865 such that

1198921(119909) 119892

2(119909) isin (119910

1(119909) + Φ (119901

1(119909)))

cap (1199102(119909) + Φ (119901

2(119909)))

capQcs119865minus 1199101(119909) 119910

2(119909)

(43)

This implies that

1198921(119909) isin 119910

1(119909) + Φ (119901

1(119909))

1198922(119909) isin 119910

1(119909) + Φ (119901

1(119909))

1198921(119909) isin 119910

2(119909) + Φ (119901

2(119909))

1198922(119909) isin 119910

2(119909) + Φ (119901

2(119909))

(44)

Abstract and Applied Analysis 5

By Notation 2(ii) it yields that

1198921(119909) minus 119910

1(119909) = 120582

11199011(119909)

1198922(119909) minus 119910

1(119909) = 120582

21199011(119909)

1198921(119909) minus 119910

2(119909) = 120582

31199012(119909)

1198922(119909) minus 119910

2(119909) = 120582

41199012(119909)

(45)

for some constants 1205821 1205822 1205823 and 120582

4isin R and consequently

1198922(119909) minus 119892

1(119909) = (119892

2(119909) minus 119910

1(119909)) minus (119892

1(119909) minus 119910

1(119909))

= (1205822minus 1205821) 1199011(119909)

1198922(119909) minus 119892

1(119909) = (119892

2(119909) minus 119910

2(119909)) minus (119892

1(119909) minus 119910

2(119909))

= (1205824minus 1205823) 1199012(119909)

(46)

This implies that

(1205822minus 1205821) 1199011(119909) = (120582

4minus 1205823) 1199012(119909) (47)

and 1199011(119909) = 119901

2(119909) Therefore

1199101(119909) minus 119910

2(119909) = (119892

1(119909) minus 119910

2(119909)) minus (119892

1(119909) minus 119910

1(119909))

= (1205823minus 1205821) 1199011(119909)

(48)

and this means that the factor 119901(119909) of 119891(119909) is uniquely deter-mined independent of the choice of 119910

1(119909) and 119910

2(119909)

Corollary 10 Assume that the cardinal number

1003816100381610038161003816119876119888119904119865

1003816100381610038161003816 ge deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) (2

deg119891(119909)) + 2)

(49)

Then for any 119910(119909) isin 119876119888119904119865 there exists ℎ(119909) 119901(119909) isin R[119909] such

that

119910 (119909) = ℎ (119909) + 120582119901 (119909) (50)

for some 120582 isin R (ℎ(119909) 119901(119909) are independent of the choice of119910(119909))

Proof By Lemma 4(iv) we have

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcv119865

1003816100381610038161003816

1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816 for any 119886 isin Qcv119865

(by Lemma 4 (iii))

le deg119910119865 (119909 119910)

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(51)

By assumption it follows that

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) (2

deg119891(119909)) + 2)

le deg119910119865 (119909 119910)

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(52)

Dividing both sides of the above equation by deg119910119865(119909 119910) we

get

1003816100381610038161003816Qcv119865

1003816100381610038161003816 ge ((2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2) (53)

If ℎ(119909) isin Qcs119865 for any 119910(119909) isin Qcs

119865 by Theorem 9 we have

119910 (119909) = ℎ (119909) + 119889119901 (119909) (54)

for some factor 119901(119909) of 119891(119909)

3 The Type of 119865(119909119910) If the Numberof All Quasi-Coincidence SolutionsIs Infinitely Many

In this section we consider (4) for polynomial function119865(119909 119910) in (5) that is let

119865 (119909 119910) =

119904

sum

119894=0

119886119894(119909) 119910119894 with 119904 ge 2 (55)

119891(119909) isin R[119909] and we assume that 119865(119909 119910) has at least119904 + 1 distinct quasi-coincidence solutions satisfying someconditions that is 119910

1(119909) 1199102(119909) 1199103(119909) 119910

119904+1(119909) in the

following theorem According to the above assumptions wecould derive the following result

Theorem 11 Suppose that the cardinal number |119876119888119904119865| ge 119904 + 1

and for each 119910(119909) isin 119876119888119904119865can be represented as the form

119910 (119909) = 1199101(119909) + 120582

119894119901 (119909) 120582

119894isin R (56)

for some 1199101(119909) 119901(119909) isin R[119909] and 120582

119894isin R for 119894 = 1 2 119904 +

1 Then 119901119904

(119909) | 119891(119909) and the polynomial 119865(119909 119910) can berepresented as

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894 (57)

for constants 119888119894isin R 119894 = 0 1 119904

Proof Let 119910119894(119909) be distinct quasi-coincidence solutions of

119865(119909 119910) corresponding to quasi-coincidence values 119886119894 1 le 119894 le

119904 + 1 such that

119865 (119909 119910119894(119909)) = 119886

119894119891 (119909) (58)

Choose 119894 = 1 119865(119909 1199101(119909)) = 119886

1119891(119909) When 119910 minus 119910

1(119909) divides

the function 119865(119909 119910) we get

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909) (59)

where 1198651(119909 119910) is the quotient and 119886

1119891(119909) is the remainder

From the above identity take 119910 = 1199102(119909) and it becomes

119865 (119909 1199102(119909)) = (119910

2(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) + 119886

1119891 (119909)

= 1198862119891 (119909)

(60)

6 Abstract and Applied Analysis

Then

(1199102(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) = (119886

2minus 1198861) 119891 (119909) (61)

By (56) 1199102(119909) minus 119910

1(119909) = 120582

2119901(119909) it yields that

1198651(119909 1199102(119909)) = (

(1198862minus 1198861)

1205822

)119891 (119909)

119901 (119909)

= 1198892

119891 (119909)

119901 (119909)isin R [119909] for 119889

2=

(1198862minus 1198861)

1205822

(62)

Hence

1198651(119909 119910) = (119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909) (63)

Continuing this process from 119894 = 2 to 119904 minus 1 we obtain

119865119894(119909 119910) = (119910 minus 119910

119894+1(119909)) 119865

119894+1(119909 119910) + 119889

119894+1

119891 (119909)

119901119894 (119909) (64)

for some 119889119894+1

isin R 119894 = 1 2 119904 minus 1 Finally we could get

119865119904minus1

(119909 119910) = (119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909) (65)

119865119904(119909) does not contain the variable 119910 since deg

119910119865 = 119904 By the

assumption (58) 119865(119909 119910119904+1

(119909)) = 119886119904+1

119891(119909) It follows that

119865119904(119909) = 120582

119891 (119909)

119901119904 (119909)isin R [119909] for some constant 120582 isin R

(66)

Consequently

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909)

= (119910 minus 1199101(119909)) ((119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) = sdot sdot sdot = (119910 minus 119910

1(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ((119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot )

+ 1198892

119891 (119909)

119901 (119909)) + 1198861119891 (119909)

= (119910 minus 1199101(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ( (119910 minus 119910119904(119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909)

(67)

By (56) we have119910119894(119909) = 119910

1(119909)+120582

119894119901(119909) 119894 = 2 3 119904+1Then

119865(119909 119910) can be expanded to a power series in the expression

119865 (119909 119910) = (119910 minus 1199101(119909))

times ((119910 minus 1199101(119909) minus 120582

2119901 (119909))

times (sdot sdot sdot ( (119910 minus 1199101(119909) minus 120582

119904119901 (119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(68)

for some real numbers 119888119895 119895 = 0 1 119904 Moreover the

leading coefficient of 119865(119909 119910) 119888119904(119891(119909)119901

119904

(119909)) is contained toR[119909] and it follows 119901119904(119909) | 119891(119909)

In the above theorem if there exist at least 119904 + 1 quasi-coincidence solutions with some relations then 119865(119909 119910) hasa fixed type In the following theorem if 119865(119909 119910) has a fixedtype expressed as in Theorem 11 then the cardinal number|Qcs119865| = infin

Theorem 12 The following three conditions are equivalent

(i) 119865(119909 119910) = sum119904

119894=0119888119894(119891(119909)119901

119894

(119909))(119910 minus 1199101(119909))119894 for some

1199101(119909) isin R[119909] some factor 119901(119909) of 119891(119909) and 119888

119894isin R

119894 = 0 1 119904

(ii) |119876119888119904119865| = infin

(iii) |119876119888V119865| ge (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 2

(In fact if |119876119888119904119865| = infin then |119876119888119904

119865| = the cardinal number of

R)

Proof (i) rArr (ii) Suppose that (i) holds Let 119910 = 1199101(119909)+120582119901(119909)

for any constant 120582 isin R we have

119865 (119909 1199101(119909) + 120582119901 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(120582119901 (119909))

119894

= (

119904

sum

119894=0

119888119894120582119894

)119891 (119909)

= 119888119901119898

(119909) for 119888 =119904

sum

119894=0

119888119894120582119894

isin R

(69)

This means that 1199101(119909) + 120582119901(119909) isin Qcs

119865for all 120582 isin R and we

obtain

infin =100381610038161003816100381610038161199101(119909) + 120582119901 (119909)

120582isinR

10038161003816100381610038161003816le1003816100381610038161003816Qcs119865

1003816100381610038161003816 (70)

It follows that the cardinal number |Qcs119865| = infin

Abstract and Applied Analysis 7

(ii) rArr (iii) can be obtained obvious from Lemma 4(iv)(iii) rArr (i) For any 119910(119909) 119910

1(119909) isin Qcs

119865 by Theorem 9 we

have

119910 (119909) minus 1199101(119909) = 119889119901 (119909) (71)

for some fixed factor 119901(119909) of 119891(119909) and by Theorem 11 weobtain

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(72)

for some 1199101(119909) and 119888

119894isin R 119894 = 0 1 119904

Corollary 13 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed values does notexceed an integer ℓ Actually

ℓ = (2deg119891(119909)

+ 119904 + 3) sdot 2deg119891(119909)

+ 1 (73)

Proof By the contrapositive of Theorem 12(ii) rArr (iii) wehave ldquoif |Qcs

119865| lt infin then |Qcv

119865| lt (2

deg119891(119909)+119904+3)sdot2

deg119891(119909)+

2rdquoHence the number of all quasi-fixed values is atmost ℓ thatis ℓ le (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1

Corollary 14 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed solutions does notexceed

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1) (74)

Proof By Lemma 4(iv) we have for any 119886 isin Qcv119865

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(by Lemma 4 (iii) and Corollary 13)

le deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1)

(75)

4 Main Theorems and Some Corollaries

If the 119865(119909 119910) can be represented as the form (57) then anyquasi-coincidence solution can be formed in this section

Lemma 15 Let 119865(119909 119910) be represented as in (57) Then ℎ(119909) isin

R[119909] is a quasi-coincidence solution of 119865(119909 119910) if and only if

ℎ (119909) = 119910 (119909) + 119889119901 (119909) (76)

for some 119889 isin R and some factor 119901(119909) of 119891(119909)

Proof Since

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

(77)

we let 119910 = 119910(119909) and then

119865 (119909 119910 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910(119909) minus 119910(119909))

119894

= 1198880119891 (119909)

(78)

this means 119910(119909) isin Qcs119865

By Theorem 12

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

then 1003816100381610038161003816Qcs119865

1003816100381610038161003816 = infin

(79)

Assume that ℎ(119909) is a quasi-coincidence solution of 119865(119909 119910)and by Corollary 10 we obtain that for any quasi-coincidencesolution ℎ(119909) we have

ℎ (119909) = 119910 (119909) + 119889119901 (119909) for some 119889 isin R (80)

Conversely suppose ℎ(119909) = 119910(119909) + 119889119901(119909) for some factor119901(119909) of 119891(119909) and 119889 isin R Substituting this ℎ(119909) as 119910 in (57)we have

119865 (119909 ℎ (119909)) = 119865 (119909 119910 (119909) + 119889119901 (119909))

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119889119901 (119909))

119894

= (

119904

sum

119894=0

119888119894119889119894

)119891 (119909)

(81)

Therefore ℎ(119909) isin Qcs119865

Note that not any polynomial function 119865(119909 119910) can bewritten as (57) Actually almost all 119865(119909 119910) are expressed asthe form of the next theorem In this situation any solutioncan be written as the next form (lowast) in this theorem undersome condition

Theorem 16 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (82)

and 119891(119909) a polynomial If the cardinal number |Qcs119865| is

infinitely many then each quasi-coincidence solution of (4)must be of the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (lowast)

for arbitrary 120582 isin R where 119901(119909) = (119891(119909)119886119904(119909))1119904 is a factor

of 119891(119909)

Proof Assume |Qcs119865| = infin By Theorem 12 we have

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909)

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910(119909))

119894

(83)

8 Abstract and Applied Analysis

for some 119888119894isin R 119894 = 0 119904 and119910(119909) isin Qcs

119865 Comparing the

coefficients of 119910119904 and 119910119904minus1 in both sides of the above equation

we get

119886119904(119909) = 119888

119904

119891 (119909)

119901119904 (119909)

119886119904minus1

(119909) = minus119904119886119904(119909) 119910 (119909) + 119888

119904minus1

119891 (119909)

119901119904minus1 (119909)

(84)

Consequently by (84) we get

119901119904

(119909) = 119888119904

119891 (119909)

119886119904(119909)

119910 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

isin R [119909]

(85)

By Lemma 15 for any 119889 isin R we have that any quasi-coincidence solution is represented by

119910 (119909) + 119889119901 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

+ 119889119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ (119889 minus119888119904minus1

119904119888119904

)119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909)

(86)

where 119901(119909) = (119888119904)1119904

(119891(119909)119886119904(119909))1119904 (note that since 119889 is

arbitrary then 120582 is arbitrary)This completes the proof

Corollary 17 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (87)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886 (88)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

120582 minus119886119904minus1

(119909)

119904119886119904(119909)

(995333)

for arbitrary 120582 isin R

Corollary 18 Let 119865 R times R rarr R be a polynomial functionwith

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (89)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886119909 (90)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

1205821+ 1205822

119886119904minus1

(119909)

119909(995333)

for arbitrary 1205821and some 120582

2isin R

Proof Assume that there exist infinitely many solutions byTheorem 16 any solution of (4) has the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (91)

for arbitrary 120582 isin R and 119901(119909) = 119888(119909119886119904(119909))1119904

isin R[119909]and then 119886

119904(119909) is a factor of 119909 This means that 119886

119904(119909) isin R

or 119886119904(119909) = 119896119909 for some constant 119896 isin R if 119886

119904(119909) isin R

this implies 119901(119909) = 119888(119909119886119904(119909))1119904

notin R[119909] and this leads acontradiction So we have 119886

119904(119909) = 119896119909 for some 119896 isin R then

119901(119909) = 1198881198961119904

isin R and any solution (91) is represented as

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) = 1205821+ 1205822

119886119904minus1

(119909)

119909(92)

for arbitrary 1205821= 120582119888119896

1119904 and some 1205822= minus1119904119896 isin R

Finally we provide one example to explain Theorem 16

Example 19 Let 119909 = (1199091 1199092) isin R2 119891(119909) = (119909

1+ 1199092)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091minus 1199092) 119910

+ (1199092

11199092

2minus 1199092

11199092minus 11990911199092

2+ 1199092

1+ 211990911199092+ 1199092

2)

(93)

Can we solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909)This polynomial function has exactly 5(ge 119904 + 3 since 119904 = 2)

quasi-fixed solutions as follows

119865 (1199091 1199092 11990911199092minus 1199091minus 1199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 1199091+ 1199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 21199091+ 21199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+1199091

2+1199092

2) =

3

4(1199091+ 1199092)2

(94)

In fact by Theorem 16 we can find any quasi-coincidencesolution written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091minus 1199092

2+ 120582119901 (119909)

= 11990911199092+ (120582 minus

1

2) (1199091minus 1199092)

= 11990911199092+ 120583 (119909

1minus 1199092)

(95)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091+ 1199092 This shows the quasi-coincidence

(point) solutions have cardinal |Qs119865| = infin

In practice we have no idea to check the number of thisequation is infinitely many or finitely many But we providean easy method to solve all solutions if the number of all

Abstract and Applied Analysis 9

solutions is infinitely many in this paper Thus we can solvethose solutions directly and checkwhether those solutions arethe solutions of 119865(119909 119910) = 119886119891(119909) and give an example in thefollowing

Example 20 Let 119909 = (1199091 1199092) isin R2 119891(119909) = 119909

2

1(1199092+ 1199093)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091(1199092+ 1199093)) 119910

+ (1199092

11199092

2+ 1199092

111990921199093+ 1199092

11199092

3)

(96)

We will solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909) ifthe number of all solutions is infinitely many

By Theorem 16 we can find that any quasi-coincidencesolution can be written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091(1199092+ 1199093)

2+ 120582119901 (119909)

= 11990911199092+ 1205831199091(1199092+ 1199093)

(97)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091(1199092+ 1199093) We let 119910(119909) = 119909

11199092+ 1205831199091(1199092+

1199093) and calculate 119865(119909 119910(119909)) and obtain

119865 (119909 119910 (119909)) = (1205832

+ 120583 + 1) 1199092

1(1199092+ 1199093)2

(98)

This means that the quasi-coincidence (point) solutions havecardinal |Qs

119865| = infin

We would like to provide one open problem as follows

Further Development Let Q be a quotient field Consider aquotient-valued polynomial function

119865 Q119899

timesQ 997888rarr Q (99)

Can we find all quasi-coincidence solutions 119910 = 119910(119909) isin Q[119909]

to satisfy

119865 (119909 119910) = 119886119891 (119909) (100)

for some polynomials 119891(119909) isin Q[119909] by a co-NP hardnessalgorithm

References

[1] A K Lenstra ldquoFactoring multivariate polynomials over alge-braic number fieldsrdquo SIAM Journal on Computing vol 16 no 3pp 591ndash598 1987

[2] E P Tsigaridas and I Z Emiris ldquoUnivariate polynomial realroot isolation continued fractions revisitedrdquo in AlgorithmsmdashESA 2006 vol 4168 of Lecture Notes in Computer Science pp817ndash828 Springer Berlin Germany 2006

[3] J von zur Gathen and J Gerhard Modern Computer AlgebraCambridge University Press Cambridge UK 2nd edition2003

[4] M G Marinari H M Moller and T Mora ldquoOn multiplicitiesin polynomial system solvingrdquo Transactions of the AmericanMathematical Society vol 348 no 8 pp 3283ndash3321 1996

[5] P Gopalan V Guruswami and R J Lipton ldquoAlgorithmsfor modular counting of roots of multivariate polynomialsrdquoAlgorithmica vol 50 no 4 pp 479ndash496 2008

[6] V Y Pan ldquoUnivariate polynomials nearly optimal algorithmsfor numerical factorization and root-findingrdquo Journal of Sym-bolic Computation vol 33 no 5 pp 701ndash733 2002

[7] C-M Chen T-H Chang and Y-P Liao ldquoCoincidence the-orems generalized variational inequality theorems and min-imax inequality theorems for the 120601-mapping on G-convexspacesrdquo Fixed Point Theory and Applications vol 2007 ArticleID 078696 13 pages 2007

[8] C-M Chen T-H Chang and C-W Chung ldquoCoincidence the-orems on nonconvex sets and its applicationsrdquo The TaiwaneseJournal of Mathematics vol 13 no 2 pp 501ndash513 2009

[9] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinearmultivaluedmapsrdquoTopology and Its Applications vol159 no 1 pp 49ndash56 2012

[10] W-S Du ldquoOn approximate coincidence point properties andtheir applications to fixed point theoryrdquo Journal of AppliedMathematics vol 2012 Article ID 302830 17 pages 2012

[11] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[12] H-C Lai and Y-C Chen ldquoA quasi-fixed polynomial problemfor a polynomial functionrdquo Journal of Nonlinear and ConvexAnalysis vol 11 no 1 pp 101ndash114 2010

[13] H-C Lai and Y-C Chen ldquoQuasi-fixed polynomial for vector-valued polynomial functions on R119899 times Rrdquo Fixed Point Theoryvol 12 no 2 pp 391ndash400 2011 (Romanian)

[14] Y C Chen and H C Lai ldquoQuasi-fixed point solutions of real-valued polynomial equationrdquo in Proceedings of the 9th Interna-tional Conference on Fixed PointTheory and Its Applications pp16ndash22 Changhua Taiwan 2009

[15] Y-C Chen and H-C Lai ldquoA non-NP-complete algorithm for aquasi-fixed polynomial problemrdquoAbstract andAppliedAnalysisvol 2013 Article ID 893045 10 pages 2013

[16] Y C Chen and H C Lai ldquoNew quasi-coincidence pointpolynomial problemsrdquo Journal of Applied Mathematics vol2013 Article ID 959464 8 pages 2013

[17] R P Grimaldi Discrete and Combinatorial Mathematics Pear-son Secaucus NJ USA 5th edition 2003

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

By Notation 2(ii) it yields that

1198921(119909) minus 119910

1(119909) = 120582

11199011(119909)

1198922(119909) minus 119910

1(119909) = 120582

21199011(119909)

1198921(119909) minus 119910

2(119909) = 120582

31199012(119909)

1198922(119909) minus 119910

2(119909) = 120582

41199012(119909)

(45)

for some constants 1205821 1205822 1205823 and 120582

4isin R and consequently

1198922(119909) minus 119892

1(119909) = (119892

2(119909) minus 119910

1(119909)) minus (119892

1(119909) minus 119910

1(119909))

= (1205822minus 1205821) 1199011(119909)

1198922(119909) minus 119892

1(119909) = (119892

2(119909) minus 119910

2(119909)) minus (119892

1(119909) minus 119910

2(119909))

= (1205824minus 1205823) 1199012(119909)

(46)

This implies that

(1205822minus 1205821) 1199011(119909) = (120582

4minus 1205823) 1199012(119909) (47)

and 1199011(119909) = 119901

2(119909) Therefore

1199101(119909) minus 119910

2(119909) = (119892

1(119909) minus 119910

2(119909)) minus (119892

1(119909) minus 119910

1(119909))

= (1205823minus 1205821) 1199011(119909)

(48)

and this means that the factor 119901(119909) of 119891(119909) is uniquely deter-mined independent of the choice of 119910

1(119909) and 119910

2(119909)

Corollary 10 Assume that the cardinal number

1003816100381610038161003816119876119888119904119865

1003816100381610038161003816 ge deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) (2

deg119891(119909)) + 2)

(49)

Then for any 119910(119909) isin 119876119888119904119865 there exists ℎ(119909) 119901(119909) isin R[119909] such

that

119910 (119909) = ℎ (119909) + 120582119901 (119909) (50)

for some 120582 isin R (ℎ(119909) 119901(119909) are independent of the choice of119910(119909))

Proof By Lemma 4(iv) we have

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcv119865

1003816100381610038161003816

1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816 for any 119886 isin Qcv119865

(by Lemma 4 (iii))

le deg119910119865 (119909 119910)

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(51)

By assumption it follows that

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) (2

deg119891(119909)) + 2)

le deg119910119865 (119909 119910)

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(52)

Dividing both sides of the above equation by deg119910119865(119909 119910) we

get

1003816100381610038161003816Qcv119865

1003816100381610038161003816 ge ((2deg119891(119909)

+ 119904 + 3) (2deg119891(119909)

) + 2) (53)

If ℎ(119909) isin Qcs119865 for any 119910(119909) isin Qcs

119865 by Theorem 9 we have

119910 (119909) = ℎ (119909) + 119889119901 (119909) (54)

for some factor 119901(119909) of 119891(119909)

3 The Type of 119865(119909119910) If the Numberof All Quasi-Coincidence SolutionsIs Infinitely Many

In this section we consider (4) for polynomial function119865(119909 119910) in (5) that is let

119865 (119909 119910) =

119904

sum

119894=0

119886119894(119909) 119910119894 with 119904 ge 2 (55)

119891(119909) isin R[119909] and we assume that 119865(119909 119910) has at least119904 + 1 distinct quasi-coincidence solutions satisfying someconditions that is 119910

1(119909) 1199102(119909) 1199103(119909) 119910

119904+1(119909) in the

following theorem According to the above assumptions wecould derive the following result

Theorem 11 Suppose that the cardinal number |119876119888119904119865| ge 119904 + 1

and for each 119910(119909) isin 119876119888119904119865can be represented as the form

119910 (119909) = 1199101(119909) + 120582

119894119901 (119909) 120582

119894isin R (56)

for some 1199101(119909) 119901(119909) isin R[119909] and 120582

119894isin R for 119894 = 1 2 119904 +

1 Then 119901119904

(119909) | 119891(119909) and the polynomial 119865(119909 119910) can berepresented as

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894 (57)

for constants 119888119894isin R 119894 = 0 1 119904

Proof Let 119910119894(119909) be distinct quasi-coincidence solutions of

119865(119909 119910) corresponding to quasi-coincidence values 119886119894 1 le 119894 le

119904 + 1 such that

119865 (119909 119910119894(119909)) = 119886

119894119891 (119909) (58)

Choose 119894 = 1 119865(119909 1199101(119909)) = 119886

1119891(119909) When 119910 minus 119910

1(119909) divides

the function 119865(119909 119910) we get

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909) (59)

where 1198651(119909 119910) is the quotient and 119886

1119891(119909) is the remainder

From the above identity take 119910 = 1199102(119909) and it becomes

119865 (119909 1199102(119909)) = (119910

2(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) + 119886

1119891 (119909)

= 1198862119891 (119909)

(60)

6 Abstract and Applied Analysis

Then

(1199102(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) = (119886

2minus 1198861) 119891 (119909) (61)

By (56) 1199102(119909) minus 119910

1(119909) = 120582

2119901(119909) it yields that

1198651(119909 1199102(119909)) = (

(1198862minus 1198861)

1205822

)119891 (119909)

119901 (119909)

= 1198892

119891 (119909)

119901 (119909)isin R [119909] for 119889

2=

(1198862minus 1198861)

1205822

(62)

Hence

1198651(119909 119910) = (119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909) (63)

Continuing this process from 119894 = 2 to 119904 minus 1 we obtain

119865119894(119909 119910) = (119910 minus 119910

119894+1(119909)) 119865

119894+1(119909 119910) + 119889

119894+1

119891 (119909)

119901119894 (119909) (64)

for some 119889119894+1

isin R 119894 = 1 2 119904 minus 1 Finally we could get

119865119904minus1

(119909 119910) = (119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909) (65)

119865119904(119909) does not contain the variable 119910 since deg

119910119865 = 119904 By the

assumption (58) 119865(119909 119910119904+1

(119909)) = 119886119904+1

119891(119909) It follows that

119865119904(119909) = 120582

119891 (119909)

119901119904 (119909)isin R [119909] for some constant 120582 isin R

(66)

Consequently

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909)

= (119910 minus 1199101(119909)) ((119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) = sdot sdot sdot = (119910 minus 119910

1(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ((119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot )

+ 1198892

119891 (119909)

119901 (119909)) + 1198861119891 (119909)

= (119910 minus 1199101(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ( (119910 minus 119910119904(119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909)

(67)

By (56) we have119910119894(119909) = 119910

1(119909)+120582

119894119901(119909) 119894 = 2 3 119904+1Then

119865(119909 119910) can be expanded to a power series in the expression

119865 (119909 119910) = (119910 minus 1199101(119909))

times ((119910 minus 1199101(119909) minus 120582

2119901 (119909))

times (sdot sdot sdot ( (119910 minus 1199101(119909) minus 120582

119904119901 (119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(68)

for some real numbers 119888119895 119895 = 0 1 119904 Moreover the

leading coefficient of 119865(119909 119910) 119888119904(119891(119909)119901

119904

(119909)) is contained toR[119909] and it follows 119901119904(119909) | 119891(119909)

In the above theorem if there exist at least 119904 + 1 quasi-coincidence solutions with some relations then 119865(119909 119910) hasa fixed type In the following theorem if 119865(119909 119910) has a fixedtype expressed as in Theorem 11 then the cardinal number|Qcs119865| = infin

Theorem 12 The following three conditions are equivalent

(i) 119865(119909 119910) = sum119904

119894=0119888119894(119891(119909)119901

119894

(119909))(119910 minus 1199101(119909))119894 for some

1199101(119909) isin R[119909] some factor 119901(119909) of 119891(119909) and 119888

119894isin R

119894 = 0 1 119904

(ii) |119876119888119904119865| = infin

(iii) |119876119888V119865| ge (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 2

(In fact if |119876119888119904119865| = infin then |119876119888119904

119865| = the cardinal number of

R)

Proof (i) rArr (ii) Suppose that (i) holds Let 119910 = 1199101(119909)+120582119901(119909)

for any constant 120582 isin R we have

119865 (119909 1199101(119909) + 120582119901 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(120582119901 (119909))

119894

= (

119904

sum

119894=0

119888119894120582119894

)119891 (119909)

= 119888119901119898

(119909) for 119888 =119904

sum

119894=0

119888119894120582119894

isin R

(69)

This means that 1199101(119909) + 120582119901(119909) isin Qcs

119865for all 120582 isin R and we

obtain

infin =100381610038161003816100381610038161199101(119909) + 120582119901 (119909)

120582isinR

10038161003816100381610038161003816le1003816100381610038161003816Qcs119865

1003816100381610038161003816 (70)

It follows that the cardinal number |Qcs119865| = infin

Abstract and Applied Analysis 7

(ii) rArr (iii) can be obtained obvious from Lemma 4(iv)(iii) rArr (i) For any 119910(119909) 119910

1(119909) isin Qcs

119865 by Theorem 9 we

have

119910 (119909) minus 1199101(119909) = 119889119901 (119909) (71)

for some fixed factor 119901(119909) of 119891(119909) and by Theorem 11 weobtain

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(72)

for some 1199101(119909) and 119888

119894isin R 119894 = 0 1 119904

Corollary 13 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed values does notexceed an integer ℓ Actually

ℓ = (2deg119891(119909)

+ 119904 + 3) sdot 2deg119891(119909)

+ 1 (73)

Proof By the contrapositive of Theorem 12(ii) rArr (iii) wehave ldquoif |Qcs

119865| lt infin then |Qcv

119865| lt (2

deg119891(119909)+119904+3)sdot2

deg119891(119909)+

2rdquoHence the number of all quasi-fixed values is atmost ℓ thatis ℓ le (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1

Corollary 14 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed solutions does notexceed

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1) (74)

Proof By Lemma 4(iv) we have for any 119886 isin Qcv119865

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(by Lemma 4 (iii) and Corollary 13)

le deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1)

(75)

4 Main Theorems and Some Corollaries

If the 119865(119909 119910) can be represented as the form (57) then anyquasi-coincidence solution can be formed in this section

Lemma 15 Let 119865(119909 119910) be represented as in (57) Then ℎ(119909) isin

R[119909] is a quasi-coincidence solution of 119865(119909 119910) if and only if

ℎ (119909) = 119910 (119909) + 119889119901 (119909) (76)

for some 119889 isin R and some factor 119901(119909) of 119891(119909)

Proof Since

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

(77)

we let 119910 = 119910(119909) and then

119865 (119909 119910 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910(119909) minus 119910(119909))

119894

= 1198880119891 (119909)

(78)

this means 119910(119909) isin Qcs119865

By Theorem 12

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

then 1003816100381610038161003816Qcs119865

1003816100381610038161003816 = infin

(79)

Assume that ℎ(119909) is a quasi-coincidence solution of 119865(119909 119910)and by Corollary 10 we obtain that for any quasi-coincidencesolution ℎ(119909) we have

ℎ (119909) = 119910 (119909) + 119889119901 (119909) for some 119889 isin R (80)

Conversely suppose ℎ(119909) = 119910(119909) + 119889119901(119909) for some factor119901(119909) of 119891(119909) and 119889 isin R Substituting this ℎ(119909) as 119910 in (57)we have

119865 (119909 ℎ (119909)) = 119865 (119909 119910 (119909) + 119889119901 (119909))

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119889119901 (119909))

119894

= (

119904

sum

119894=0

119888119894119889119894

)119891 (119909)

(81)

Therefore ℎ(119909) isin Qcs119865

Note that not any polynomial function 119865(119909 119910) can bewritten as (57) Actually almost all 119865(119909 119910) are expressed asthe form of the next theorem In this situation any solutioncan be written as the next form (lowast) in this theorem undersome condition

Theorem 16 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (82)

and 119891(119909) a polynomial If the cardinal number |Qcs119865| is

infinitely many then each quasi-coincidence solution of (4)must be of the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (lowast)

for arbitrary 120582 isin R where 119901(119909) = (119891(119909)119886119904(119909))1119904 is a factor

of 119891(119909)

Proof Assume |Qcs119865| = infin By Theorem 12 we have

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909)

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910(119909))

119894

(83)

8 Abstract and Applied Analysis

for some 119888119894isin R 119894 = 0 119904 and119910(119909) isin Qcs

119865 Comparing the

coefficients of 119910119904 and 119910119904minus1 in both sides of the above equation

we get

119886119904(119909) = 119888

119904

119891 (119909)

119901119904 (119909)

119886119904minus1

(119909) = minus119904119886119904(119909) 119910 (119909) + 119888

119904minus1

119891 (119909)

119901119904minus1 (119909)

(84)

Consequently by (84) we get

119901119904

(119909) = 119888119904

119891 (119909)

119886119904(119909)

119910 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

isin R [119909]

(85)

By Lemma 15 for any 119889 isin R we have that any quasi-coincidence solution is represented by

119910 (119909) + 119889119901 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

+ 119889119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ (119889 minus119888119904minus1

119904119888119904

)119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909)

(86)

where 119901(119909) = (119888119904)1119904

(119891(119909)119886119904(119909))1119904 (note that since 119889 is

arbitrary then 120582 is arbitrary)This completes the proof

Corollary 17 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (87)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886 (88)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

120582 minus119886119904minus1

(119909)

119904119886119904(119909)

(995333)

for arbitrary 120582 isin R

Corollary 18 Let 119865 R times R rarr R be a polynomial functionwith

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (89)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886119909 (90)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

1205821+ 1205822

119886119904minus1

(119909)

119909(995333)

for arbitrary 1205821and some 120582

2isin R

Proof Assume that there exist infinitely many solutions byTheorem 16 any solution of (4) has the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (91)

for arbitrary 120582 isin R and 119901(119909) = 119888(119909119886119904(119909))1119904

isin R[119909]and then 119886

119904(119909) is a factor of 119909 This means that 119886

119904(119909) isin R

or 119886119904(119909) = 119896119909 for some constant 119896 isin R if 119886

119904(119909) isin R

this implies 119901(119909) = 119888(119909119886119904(119909))1119904

notin R[119909] and this leads acontradiction So we have 119886

119904(119909) = 119896119909 for some 119896 isin R then

119901(119909) = 1198881198961119904

isin R and any solution (91) is represented as

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) = 1205821+ 1205822

119886119904minus1

(119909)

119909(92)

for arbitrary 1205821= 120582119888119896

1119904 and some 1205822= minus1119904119896 isin R

Finally we provide one example to explain Theorem 16

Example 19 Let 119909 = (1199091 1199092) isin R2 119891(119909) = (119909

1+ 1199092)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091minus 1199092) 119910

+ (1199092

11199092

2minus 1199092

11199092minus 11990911199092

2+ 1199092

1+ 211990911199092+ 1199092

2)

(93)

Can we solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909)This polynomial function has exactly 5(ge 119904 + 3 since 119904 = 2)

quasi-fixed solutions as follows

119865 (1199091 1199092 11990911199092minus 1199091minus 1199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 1199091+ 1199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 21199091+ 21199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+1199091

2+1199092

2) =

3

4(1199091+ 1199092)2

(94)

In fact by Theorem 16 we can find any quasi-coincidencesolution written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091minus 1199092

2+ 120582119901 (119909)

= 11990911199092+ (120582 minus

1

2) (1199091minus 1199092)

= 11990911199092+ 120583 (119909

1minus 1199092)

(95)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091+ 1199092 This shows the quasi-coincidence

(point) solutions have cardinal |Qs119865| = infin

In practice we have no idea to check the number of thisequation is infinitely many or finitely many But we providean easy method to solve all solutions if the number of all

Abstract and Applied Analysis 9

solutions is infinitely many in this paper Thus we can solvethose solutions directly and checkwhether those solutions arethe solutions of 119865(119909 119910) = 119886119891(119909) and give an example in thefollowing

Example 20 Let 119909 = (1199091 1199092) isin R2 119891(119909) = 119909

2

1(1199092+ 1199093)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091(1199092+ 1199093)) 119910

+ (1199092

11199092

2+ 1199092

111990921199093+ 1199092

11199092

3)

(96)

We will solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909) ifthe number of all solutions is infinitely many

By Theorem 16 we can find that any quasi-coincidencesolution can be written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091(1199092+ 1199093)

2+ 120582119901 (119909)

= 11990911199092+ 1205831199091(1199092+ 1199093)

(97)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091(1199092+ 1199093) We let 119910(119909) = 119909

11199092+ 1205831199091(1199092+

1199093) and calculate 119865(119909 119910(119909)) and obtain

119865 (119909 119910 (119909)) = (1205832

+ 120583 + 1) 1199092

1(1199092+ 1199093)2

(98)

This means that the quasi-coincidence (point) solutions havecardinal |Qs

119865| = infin

We would like to provide one open problem as follows

Further Development Let Q be a quotient field Consider aquotient-valued polynomial function

119865 Q119899

timesQ 997888rarr Q (99)

Can we find all quasi-coincidence solutions 119910 = 119910(119909) isin Q[119909]

to satisfy

119865 (119909 119910) = 119886119891 (119909) (100)

for some polynomials 119891(119909) isin Q[119909] by a co-NP hardnessalgorithm

References

[1] A K Lenstra ldquoFactoring multivariate polynomials over alge-braic number fieldsrdquo SIAM Journal on Computing vol 16 no 3pp 591ndash598 1987

[2] E P Tsigaridas and I Z Emiris ldquoUnivariate polynomial realroot isolation continued fractions revisitedrdquo in AlgorithmsmdashESA 2006 vol 4168 of Lecture Notes in Computer Science pp817ndash828 Springer Berlin Germany 2006

[3] J von zur Gathen and J Gerhard Modern Computer AlgebraCambridge University Press Cambridge UK 2nd edition2003

[4] M G Marinari H M Moller and T Mora ldquoOn multiplicitiesin polynomial system solvingrdquo Transactions of the AmericanMathematical Society vol 348 no 8 pp 3283ndash3321 1996

[5] P Gopalan V Guruswami and R J Lipton ldquoAlgorithmsfor modular counting of roots of multivariate polynomialsrdquoAlgorithmica vol 50 no 4 pp 479ndash496 2008

[6] V Y Pan ldquoUnivariate polynomials nearly optimal algorithmsfor numerical factorization and root-findingrdquo Journal of Sym-bolic Computation vol 33 no 5 pp 701ndash733 2002

[7] C-M Chen T-H Chang and Y-P Liao ldquoCoincidence the-orems generalized variational inequality theorems and min-imax inequality theorems for the 120601-mapping on G-convexspacesrdquo Fixed Point Theory and Applications vol 2007 ArticleID 078696 13 pages 2007

[8] C-M Chen T-H Chang and C-W Chung ldquoCoincidence the-orems on nonconvex sets and its applicationsrdquo The TaiwaneseJournal of Mathematics vol 13 no 2 pp 501ndash513 2009

[9] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinearmultivaluedmapsrdquoTopology and Its Applications vol159 no 1 pp 49ndash56 2012

[10] W-S Du ldquoOn approximate coincidence point properties andtheir applications to fixed point theoryrdquo Journal of AppliedMathematics vol 2012 Article ID 302830 17 pages 2012

[11] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[12] H-C Lai and Y-C Chen ldquoA quasi-fixed polynomial problemfor a polynomial functionrdquo Journal of Nonlinear and ConvexAnalysis vol 11 no 1 pp 101ndash114 2010

[13] H-C Lai and Y-C Chen ldquoQuasi-fixed polynomial for vector-valued polynomial functions on R119899 times Rrdquo Fixed Point Theoryvol 12 no 2 pp 391ndash400 2011 (Romanian)

[14] Y C Chen and H C Lai ldquoQuasi-fixed point solutions of real-valued polynomial equationrdquo in Proceedings of the 9th Interna-tional Conference on Fixed PointTheory and Its Applications pp16ndash22 Changhua Taiwan 2009

[15] Y-C Chen and H-C Lai ldquoA non-NP-complete algorithm for aquasi-fixed polynomial problemrdquoAbstract andAppliedAnalysisvol 2013 Article ID 893045 10 pages 2013

[16] Y C Chen and H C Lai ldquoNew quasi-coincidence pointpolynomial problemsrdquo Journal of Applied Mathematics vol2013 Article ID 959464 8 pages 2013

[17] R P Grimaldi Discrete and Combinatorial Mathematics Pear-son Secaucus NJ USA 5th edition 2003

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Then

(1199102(119909) minus 119910

1(119909)) 119865

1(119909 1199102(119909)) = (119886

2minus 1198861) 119891 (119909) (61)

By (56) 1199102(119909) minus 119910

1(119909) = 120582

2119901(119909) it yields that

1198651(119909 1199102(119909)) = (

(1198862minus 1198861)

1205822

)119891 (119909)

119901 (119909)

= 1198892

119891 (119909)

119901 (119909)isin R [119909] for 119889

2=

(1198862minus 1198861)

1205822

(62)

Hence

1198651(119909 119910) = (119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909) (63)

Continuing this process from 119894 = 2 to 119904 minus 1 we obtain

119865119894(119909 119910) = (119910 minus 119910

119894+1(119909)) 119865

119894+1(119909 119910) + 119889

119894+1

119891 (119909)

119901119894 (119909) (64)

for some 119889119894+1

isin R 119894 = 1 2 119904 minus 1 Finally we could get

119865119904minus1

(119909 119910) = (119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909) (65)

119865119904(119909) does not contain the variable 119910 since deg

119910119865 = 119904 By the

assumption (58) 119865(119909 119910119904+1

(119909)) = 119886119904+1

119891(119909) It follows that

119865119904(119909) = 120582

119891 (119909)

119901119904 (119909)isin R [119909] for some constant 120582 isin R

(66)

Consequently

119865 (119909 119910) = (119910 minus 1199101(119909)) 119865

1(119909 119910) + 119886

1119891 (119909)

= (119910 minus 1199101(119909)) ((119910 minus 119910

2(119909)) 119865

2(119909 119910) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) = sdot sdot sdot = (119910 minus 119910

1(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ((119910 minus 119910119904(119909)) 119865

119904(119909) + 119889

119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot )

+ 1198892

119891 (119909)

119901 (119909)) + 1198861119891 (119909)

= (119910 minus 1199101(119909))

times ((119910 minus 1199102(119909))

times (sdot sdot sdot ( (119910 minus 119910119904(119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909)

(67)

By (56) we have119910119894(119909) = 119910

1(119909)+120582

119894119901(119909) 119894 = 2 3 119904+1Then

119865(119909 119910) can be expanded to a power series in the expression

119865 (119909 119910) = (119910 minus 1199101(119909))

times ((119910 minus 1199101(119909) minus 120582

2119901 (119909))

times (sdot sdot sdot ( (119910 minus 1199101(119909) minus 120582

119904119901 (119909)) 120582

119891 (119909)

119901119904 (119909)

+ 119889119904

119891 (119909)

119901119904minus1 (119909)) sdot sdot sdot ) + 119889

2

119891 (119909)

119901 (119909))

+ 1198861119891 (119909) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(68)

for some real numbers 119888119895 119895 = 0 1 119904 Moreover the

leading coefficient of 119865(119909 119910) 119888119904(119891(119909)119901

119904

(119909)) is contained toR[119909] and it follows 119901119904(119909) | 119891(119909)

In the above theorem if there exist at least 119904 + 1 quasi-coincidence solutions with some relations then 119865(119909 119910) hasa fixed type In the following theorem if 119865(119909 119910) has a fixedtype expressed as in Theorem 11 then the cardinal number|Qcs119865| = infin

Theorem 12 The following three conditions are equivalent

(i) 119865(119909 119910) = sum119904

119894=0119888119894(119891(119909)119901

119894

(119909))(119910 minus 1199101(119909))119894 for some

1199101(119909) isin R[119909] some factor 119901(119909) of 119891(119909) and 119888

119894isin R

119894 = 0 1 119904

(ii) |119876119888119904119865| = infin

(iii) |119876119888V119865| ge (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 2

(In fact if |119876119888119904119865| = infin then |119876119888119904

119865| = the cardinal number of

R)

Proof (i) rArr (ii) Suppose that (i) holds Let 119910 = 1199101(119909)+120582119901(119909)

for any constant 120582 isin R we have

119865 (119909 1199101(119909) + 120582119901 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(120582119901 (119909))

119894

= (

119904

sum

119894=0

119888119894120582119894

)119891 (119909)

= 119888119901119898

(119909) for 119888 =119904

sum

119894=0

119888119894120582119894

isin R

(69)

This means that 1199101(119909) + 120582119901(119909) isin Qcs

119865for all 120582 isin R and we

obtain

infin =100381610038161003816100381610038161199101(119909) + 120582119901 (119909)

120582isinR

10038161003816100381610038161003816le1003816100381610038161003816Qcs119865

1003816100381610038161003816 (70)

It follows that the cardinal number |Qcs119865| = infin

Abstract and Applied Analysis 7

(ii) rArr (iii) can be obtained obvious from Lemma 4(iv)(iii) rArr (i) For any 119910(119909) 119910

1(119909) isin Qcs

119865 by Theorem 9 we

have

119910 (119909) minus 1199101(119909) = 119889119901 (119909) (71)

for some fixed factor 119901(119909) of 119891(119909) and by Theorem 11 weobtain

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(72)

for some 1199101(119909) and 119888

119894isin R 119894 = 0 1 119904

Corollary 13 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed values does notexceed an integer ℓ Actually

ℓ = (2deg119891(119909)

+ 119904 + 3) sdot 2deg119891(119909)

+ 1 (73)

Proof By the contrapositive of Theorem 12(ii) rArr (iii) wehave ldquoif |Qcs

119865| lt infin then |Qcv

119865| lt (2

deg119891(119909)+119904+3)sdot2

deg119891(119909)+

2rdquoHence the number of all quasi-fixed values is atmost ℓ thatis ℓ le (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1

Corollary 14 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed solutions does notexceed

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1) (74)

Proof By Lemma 4(iv) we have for any 119886 isin Qcv119865

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(by Lemma 4 (iii) and Corollary 13)

le deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1)

(75)

4 Main Theorems and Some Corollaries

If the 119865(119909 119910) can be represented as the form (57) then anyquasi-coincidence solution can be formed in this section

Lemma 15 Let 119865(119909 119910) be represented as in (57) Then ℎ(119909) isin

R[119909] is a quasi-coincidence solution of 119865(119909 119910) if and only if

ℎ (119909) = 119910 (119909) + 119889119901 (119909) (76)

for some 119889 isin R and some factor 119901(119909) of 119891(119909)

Proof Since

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

(77)

we let 119910 = 119910(119909) and then

119865 (119909 119910 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910(119909) minus 119910(119909))

119894

= 1198880119891 (119909)

(78)

this means 119910(119909) isin Qcs119865

By Theorem 12

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

then 1003816100381610038161003816Qcs119865

1003816100381610038161003816 = infin

(79)

Assume that ℎ(119909) is a quasi-coincidence solution of 119865(119909 119910)and by Corollary 10 we obtain that for any quasi-coincidencesolution ℎ(119909) we have

ℎ (119909) = 119910 (119909) + 119889119901 (119909) for some 119889 isin R (80)

Conversely suppose ℎ(119909) = 119910(119909) + 119889119901(119909) for some factor119901(119909) of 119891(119909) and 119889 isin R Substituting this ℎ(119909) as 119910 in (57)we have

119865 (119909 ℎ (119909)) = 119865 (119909 119910 (119909) + 119889119901 (119909))

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119889119901 (119909))

119894

= (

119904

sum

119894=0

119888119894119889119894

)119891 (119909)

(81)

Therefore ℎ(119909) isin Qcs119865

Note that not any polynomial function 119865(119909 119910) can bewritten as (57) Actually almost all 119865(119909 119910) are expressed asthe form of the next theorem In this situation any solutioncan be written as the next form (lowast) in this theorem undersome condition

Theorem 16 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (82)

and 119891(119909) a polynomial If the cardinal number |Qcs119865| is

infinitely many then each quasi-coincidence solution of (4)must be of the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (lowast)

for arbitrary 120582 isin R where 119901(119909) = (119891(119909)119886119904(119909))1119904 is a factor

of 119891(119909)

Proof Assume |Qcs119865| = infin By Theorem 12 we have

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909)

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910(119909))

119894

(83)

8 Abstract and Applied Analysis

for some 119888119894isin R 119894 = 0 119904 and119910(119909) isin Qcs

119865 Comparing the

coefficients of 119910119904 and 119910119904minus1 in both sides of the above equation

we get

119886119904(119909) = 119888

119904

119891 (119909)

119901119904 (119909)

119886119904minus1

(119909) = minus119904119886119904(119909) 119910 (119909) + 119888

119904minus1

119891 (119909)

119901119904minus1 (119909)

(84)

Consequently by (84) we get

119901119904

(119909) = 119888119904

119891 (119909)

119886119904(119909)

119910 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

isin R [119909]

(85)

By Lemma 15 for any 119889 isin R we have that any quasi-coincidence solution is represented by

119910 (119909) + 119889119901 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

+ 119889119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ (119889 minus119888119904minus1

119904119888119904

)119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909)

(86)

where 119901(119909) = (119888119904)1119904

(119891(119909)119886119904(119909))1119904 (note that since 119889 is

arbitrary then 120582 is arbitrary)This completes the proof

Corollary 17 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (87)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886 (88)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

120582 minus119886119904minus1

(119909)

119904119886119904(119909)

(995333)

for arbitrary 120582 isin R

Corollary 18 Let 119865 R times R rarr R be a polynomial functionwith

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (89)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886119909 (90)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

1205821+ 1205822

119886119904minus1

(119909)

119909(995333)

for arbitrary 1205821and some 120582

2isin R

Proof Assume that there exist infinitely many solutions byTheorem 16 any solution of (4) has the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (91)

for arbitrary 120582 isin R and 119901(119909) = 119888(119909119886119904(119909))1119904

isin R[119909]and then 119886

119904(119909) is a factor of 119909 This means that 119886

119904(119909) isin R

or 119886119904(119909) = 119896119909 for some constant 119896 isin R if 119886

119904(119909) isin R

this implies 119901(119909) = 119888(119909119886119904(119909))1119904

notin R[119909] and this leads acontradiction So we have 119886

119904(119909) = 119896119909 for some 119896 isin R then

119901(119909) = 1198881198961119904

isin R and any solution (91) is represented as

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) = 1205821+ 1205822

119886119904minus1

(119909)

119909(92)

for arbitrary 1205821= 120582119888119896

1119904 and some 1205822= minus1119904119896 isin R

Finally we provide one example to explain Theorem 16

Example 19 Let 119909 = (1199091 1199092) isin R2 119891(119909) = (119909

1+ 1199092)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091minus 1199092) 119910

+ (1199092

11199092

2minus 1199092

11199092minus 11990911199092

2+ 1199092

1+ 211990911199092+ 1199092

2)

(93)

Can we solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909)This polynomial function has exactly 5(ge 119904 + 3 since 119904 = 2)

quasi-fixed solutions as follows

119865 (1199091 1199092 11990911199092minus 1199091minus 1199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 1199091+ 1199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 21199091+ 21199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+1199091

2+1199092

2) =

3

4(1199091+ 1199092)2

(94)

In fact by Theorem 16 we can find any quasi-coincidencesolution written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091minus 1199092

2+ 120582119901 (119909)

= 11990911199092+ (120582 minus

1

2) (1199091minus 1199092)

= 11990911199092+ 120583 (119909

1minus 1199092)

(95)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091+ 1199092 This shows the quasi-coincidence

(point) solutions have cardinal |Qs119865| = infin

In practice we have no idea to check the number of thisequation is infinitely many or finitely many But we providean easy method to solve all solutions if the number of all

Abstract and Applied Analysis 9

solutions is infinitely many in this paper Thus we can solvethose solutions directly and checkwhether those solutions arethe solutions of 119865(119909 119910) = 119886119891(119909) and give an example in thefollowing

Example 20 Let 119909 = (1199091 1199092) isin R2 119891(119909) = 119909

2

1(1199092+ 1199093)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091(1199092+ 1199093)) 119910

+ (1199092

11199092

2+ 1199092

111990921199093+ 1199092

11199092

3)

(96)

We will solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909) ifthe number of all solutions is infinitely many

By Theorem 16 we can find that any quasi-coincidencesolution can be written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091(1199092+ 1199093)

2+ 120582119901 (119909)

= 11990911199092+ 1205831199091(1199092+ 1199093)

(97)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091(1199092+ 1199093) We let 119910(119909) = 119909

11199092+ 1205831199091(1199092+

1199093) and calculate 119865(119909 119910(119909)) and obtain

119865 (119909 119910 (119909)) = (1205832

+ 120583 + 1) 1199092

1(1199092+ 1199093)2

(98)

This means that the quasi-coincidence (point) solutions havecardinal |Qs

119865| = infin

We would like to provide one open problem as follows

Further Development Let Q be a quotient field Consider aquotient-valued polynomial function

119865 Q119899

timesQ 997888rarr Q (99)

Can we find all quasi-coincidence solutions 119910 = 119910(119909) isin Q[119909]

to satisfy

119865 (119909 119910) = 119886119891 (119909) (100)

for some polynomials 119891(119909) isin Q[119909] by a co-NP hardnessalgorithm

References

[1] A K Lenstra ldquoFactoring multivariate polynomials over alge-braic number fieldsrdquo SIAM Journal on Computing vol 16 no 3pp 591ndash598 1987

[2] E P Tsigaridas and I Z Emiris ldquoUnivariate polynomial realroot isolation continued fractions revisitedrdquo in AlgorithmsmdashESA 2006 vol 4168 of Lecture Notes in Computer Science pp817ndash828 Springer Berlin Germany 2006

[3] J von zur Gathen and J Gerhard Modern Computer AlgebraCambridge University Press Cambridge UK 2nd edition2003

[4] M G Marinari H M Moller and T Mora ldquoOn multiplicitiesin polynomial system solvingrdquo Transactions of the AmericanMathematical Society vol 348 no 8 pp 3283ndash3321 1996

[5] P Gopalan V Guruswami and R J Lipton ldquoAlgorithmsfor modular counting of roots of multivariate polynomialsrdquoAlgorithmica vol 50 no 4 pp 479ndash496 2008

[6] V Y Pan ldquoUnivariate polynomials nearly optimal algorithmsfor numerical factorization and root-findingrdquo Journal of Sym-bolic Computation vol 33 no 5 pp 701ndash733 2002

[7] C-M Chen T-H Chang and Y-P Liao ldquoCoincidence the-orems generalized variational inequality theorems and min-imax inequality theorems for the 120601-mapping on G-convexspacesrdquo Fixed Point Theory and Applications vol 2007 ArticleID 078696 13 pages 2007

[8] C-M Chen T-H Chang and C-W Chung ldquoCoincidence the-orems on nonconvex sets and its applicationsrdquo The TaiwaneseJournal of Mathematics vol 13 no 2 pp 501ndash513 2009

[9] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinearmultivaluedmapsrdquoTopology and Its Applications vol159 no 1 pp 49ndash56 2012

[10] W-S Du ldquoOn approximate coincidence point properties andtheir applications to fixed point theoryrdquo Journal of AppliedMathematics vol 2012 Article ID 302830 17 pages 2012

[11] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[12] H-C Lai and Y-C Chen ldquoA quasi-fixed polynomial problemfor a polynomial functionrdquo Journal of Nonlinear and ConvexAnalysis vol 11 no 1 pp 101ndash114 2010

[13] H-C Lai and Y-C Chen ldquoQuasi-fixed polynomial for vector-valued polynomial functions on R119899 times Rrdquo Fixed Point Theoryvol 12 no 2 pp 391ndash400 2011 (Romanian)

[14] Y C Chen and H C Lai ldquoQuasi-fixed point solutions of real-valued polynomial equationrdquo in Proceedings of the 9th Interna-tional Conference on Fixed PointTheory and Its Applications pp16ndash22 Changhua Taiwan 2009

[15] Y-C Chen and H-C Lai ldquoA non-NP-complete algorithm for aquasi-fixed polynomial problemrdquoAbstract andAppliedAnalysisvol 2013 Article ID 893045 10 pages 2013

[16] Y C Chen and H C Lai ldquoNew quasi-coincidence pointpolynomial problemsrdquo Journal of Applied Mathematics vol2013 Article ID 959464 8 pages 2013

[17] R P Grimaldi Discrete and Combinatorial Mathematics Pear-son Secaucus NJ USA 5th edition 2003

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

(ii) rArr (iii) can be obtained obvious from Lemma 4(iv)(iii) rArr (i) For any 119910(119909) 119910

1(119909) isin Qcs

119865 by Theorem 9 we

have

119910 (119909) minus 1199101(119909) = 119889119901 (119909) (71)

for some fixed factor 119901(119909) of 119891(119909) and by Theorem 11 weobtain

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910

1(119909))119894

(72)

for some 1199101(119909) and 119888

119894isin R 119894 = 0 1 119904

Corollary 13 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed values does notexceed an integer ℓ Actually

ℓ = (2deg119891(119909)

+ 119904 + 3) sdot 2deg119891(119909)

+ 1 (73)

Proof By the contrapositive of Theorem 12(ii) rArr (iii) wehave ldquoif |Qcs

119865| lt infin then |Qcv

119865| lt (2

deg119891(119909)+119904+3)sdot2

deg119891(119909)+

2rdquoHence the number of all quasi-fixed values is atmost ℓ thatis ℓ le (2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1

Corollary 14 If the number of all quasi-fixed solutions isfinitely many the number of all quasi-fixed solutions does notexceed

deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1) (74)

Proof By Lemma 4(iv) we have for any 119886 isin Qcv119865

1003816100381610038161003816Qcs119865

1003816100381610038161003816 le1003816100381610038161003816Qcs119865(119886)

1003816100381610038161003816

1003816100381610038161003816Qcv119865

1003816100381610038161003816

(by Lemma 4 (iii) and Corollary 13)

le deg119910119865 (119909 119910) ((2

deg119891(119909)+ 119904 + 3) sdot 2

deg119891(119909)+ 1)

(75)

4 Main Theorems and Some Corollaries

If the 119865(119909 119910) can be represented as the form (57) then anyquasi-coincidence solution can be formed in this section

Lemma 15 Let 119865(119909 119910) be represented as in (57) Then ℎ(119909) isin

R[119909] is a quasi-coincidence solution of 119865(119909 119910) if and only if

ℎ (119909) = 119910 (119909) + 119889119901 (119909) (76)

for some 119889 isin R and some factor 119901(119909) of 119891(119909)

Proof Since

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

(77)

we let 119910 = 119910(119909) and then

119865 (119909 119910 (119909)) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910(119909) minus 119910(119909))

119894

= 1198880119891 (119909)

(78)

this means 119910(119909) isin Qcs119865

By Theorem 12

119865 (119909 119910) =

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910 (119909))

119894

then 1003816100381610038161003816Qcs119865

1003816100381610038161003816 = infin

(79)

Assume that ℎ(119909) is a quasi-coincidence solution of 119865(119909 119910)and by Corollary 10 we obtain that for any quasi-coincidencesolution ℎ(119909) we have

ℎ (119909) = 119910 (119909) + 119889119901 (119909) for some 119889 isin R (80)

Conversely suppose ℎ(119909) = 119910(119909) + 119889119901(119909) for some factor119901(119909) of 119891(119909) and 119889 isin R Substituting this ℎ(119909) as 119910 in (57)we have

119865 (119909 ℎ (119909)) = 119865 (119909 119910 (119909) + 119889119901 (119909))

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119889119901 (119909))

119894

= (

119904

sum

119894=0

119888119894119889119894

)119891 (119909)

(81)

Therefore ℎ(119909) isin Qcs119865

Note that not any polynomial function 119865(119909 119910) can bewritten as (57) Actually almost all 119865(119909 119910) are expressed asthe form of the next theorem In this situation any solutioncan be written as the next form (lowast) in this theorem undersome condition

Theorem 16 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (82)

and 119891(119909) a polynomial If the cardinal number |Qcs119865| is

infinitely many then each quasi-coincidence solution of (4)must be of the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (lowast)

for arbitrary 120582 isin R where 119901(119909) = (119891(119909)119886119904(119909))1119904 is a factor

of 119891(119909)

Proof Assume |Qcs119865| = infin By Theorem 12 we have

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909)

=

119904

sum

119894=0

119888119894

119891 (119909)

119901119894 (119909)(119910 minus 119910(119909))

119894

(83)

8 Abstract and Applied Analysis

for some 119888119894isin R 119894 = 0 119904 and119910(119909) isin Qcs

119865 Comparing the

coefficients of 119910119904 and 119910119904minus1 in both sides of the above equation

we get

119886119904(119909) = 119888

119904

119891 (119909)

119901119904 (119909)

119886119904minus1

(119909) = minus119904119886119904(119909) 119910 (119909) + 119888

119904minus1

119891 (119909)

119901119904minus1 (119909)

(84)

Consequently by (84) we get

119901119904

(119909) = 119888119904

119891 (119909)

119886119904(119909)

119910 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

isin R [119909]

(85)

By Lemma 15 for any 119889 isin R we have that any quasi-coincidence solution is represented by

119910 (119909) + 119889119901 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

+ 119889119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ (119889 minus119888119904minus1

119904119888119904

)119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909)

(86)

where 119901(119909) = (119888119904)1119904

(119891(119909)119886119904(119909))1119904 (note that since 119889 is

arbitrary then 120582 is arbitrary)This completes the proof

Corollary 17 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (87)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886 (88)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

120582 minus119886119904minus1

(119909)

119904119886119904(119909)

(995333)

for arbitrary 120582 isin R

Corollary 18 Let 119865 R times R rarr R be a polynomial functionwith

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (89)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886119909 (90)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

1205821+ 1205822

119886119904minus1

(119909)

119909(995333)

for arbitrary 1205821and some 120582

2isin R

Proof Assume that there exist infinitely many solutions byTheorem 16 any solution of (4) has the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (91)

for arbitrary 120582 isin R and 119901(119909) = 119888(119909119886119904(119909))1119904

isin R[119909]and then 119886

119904(119909) is a factor of 119909 This means that 119886

119904(119909) isin R

or 119886119904(119909) = 119896119909 for some constant 119896 isin R if 119886

119904(119909) isin R

this implies 119901(119909) = 119888(119909119886119904(119909))1119904

notin R[119909] and this leads acontradiction So we have 119886

119904(119909) = 119896119909 for some 119896 isin R then

119901(119909) = 1198881198961119904

isin R and any solution (91) is represented as

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) = 1205821+ 1205822

119886119904minus1

(119909)

119909(92)

for arbitrary 1205821= 120582119888119896

1119904 and some 1205822= minus1119904119896 isin R

Finally we provide one example to explain Theorem 16

Example 19 Let 119909 = (1199091 1199092) isin R2 119891(119909) = (119909

1+ 1199092)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091minus 1199092) 119910

+ (1199092

11199092

2minus 1199092

11199092minus 11990911199092

2+ 1199092

1+ 211990911199092+ 1199092

2)

(93)

Can we solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909)This polynomial function has exactly 5(ge 119904 + 3 since 119904 = 2)

quasi-fixed solutions as follows

119865 (1199091 1199092 11990911199092minus 1199091minus 1199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 1199091+ 1199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 21199091+ 21199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+1199091

2+1199092

2) =

3

4(1199091+ 1199092)2

(94)

In fact by Theorem 16 we can find any quasi-coincidencesolution written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091minus 1199092

2+ 120582119901 (119909)

= 11990911199092+ (120582 minus

1

2) (1199091minus 1199092)

= 11990911199092+ 120583 (119909

1minus 1199092)

(95)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091+ 1199092 This shows the quasi-coincidence

(point) solutions have cardinal |Qs119865| = infin

In practice we have no idea to check the number of thisequation is infinitely many or finitely many But we providean easy method to solve all solutions if the number of all

Abstract and Applied Analysis 9

solutions is infinitely many in this paper Thus we can solvethose solutions directly and checkwhether those solutions arethe solutions of 119865(119909 119910) = 119886119891(119909) and give an example in thefollowing

Example 20 Let 119909 = (1199091 1199092) isin R2 119891(119909) = 119909

2

1(1199092+ 1199093)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091(1199092+ 1199093)) 119910

+ (1199092

11199092

2+ 1199092

111990921199093+ 1199092

11199092

3)

(96)

We will solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909) ifthe number of all solutions is infinitely many

By Theorem 16 we can find that any quasi-coincidencesolution can be written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091(1199092+ 1199093)

2+ 120582119901 (119909)

= 11990911199092+ 1205831199091(1199092+ 1199093)

(97)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091(1199092+ 1199093) We let 119910(119909) = 119909

11199092+ 1205831199091(1199092+

1199093) and calculate 119865(119909 119910(119909)) and obtain

119865 (119909 119910 (119909)) = (1205832

+ 120583 + 1) 1199092

1(1199092+ 1199093)2

(98)

This means that the quasi-coincidence (point) solutions havecardinal |Qs

119865| = infin

We would like to provide one open problem as follows

Further Development Let Q be a quotient field Consider aquotient-valued polynomial function

119865 Q119899

timesQ 997888rarr Q (99)

Can we find all quasi-coincidence solutions 119910 = 119910(119909) isin Q[119909]

to satisfy

119865 (119909 119910) = 119886119891 (119909) (100)

for some polynomials 119891(119909) isin Q[119909] by a co-NP hardnessalgorithm

References

[1] A K Lenstra ldquoFactoring multivariate polynomials over alge-braic number fieldsrdquo SIAM Journal on Computing vol 16 no 3pp 591ndash598 1987

[2] E P Tsigaridas and I Z Emiris ldquoUnivariate polynomial realroot isolation continued fractions revisitedrdquo in AlgorithmsmdashESA 2006 vol 4168 of Lecture Notes in Computer Science pp817ndash828 Springer Berlin Germany 2006

[3] J von zur Gathen and J Gerhard Modern Computer AlgebraCambridge University Press Cambridge UK 2nd edition2003

[4] M G Marinari H M Moller and T Mora ldquoOn multiplicitiesin polynomial system solvingrdquo Transactions of the AmericanMathematical Society vol 348 no 8 pp 3283ndash3321 1996

[5] P Gopalan V Guruswami and R J Lipton ldquoAlgorithmsfor modular counting of roots of multivariate polynomialsrdquoAlgorithmica vol 50 no 4 pp 479ndash496 2008

[6] V Y Pan ldquoUnivariate polynomials nearly optimal algorithmsfor numerical factorization and root-findingrdquo Journal of Sym-bolic Computation vol 33 no 5 pp 701ndash733 2002

[7] C-M Chen T-H Chang and Y-P Liao ldquoCoincidence the-orems generalized variational inequality theorems and min-imax inequality theorems for the 120601-mapping on G-convexspacesrdquo Fixed Point Theory and Applications vol 2007 ArticleID 078696 13 pages 2007

[8] C-M Chen T-H Chang and C-W Chung ldquoCoincidence the-orems on nonconvex sets and its applicationsrdquo The TaiwaneseJournal of Mathematics vol 13 no 2 pp 501ndash513 2009

[9] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinearmultivaluedmapsrdquoTopology and Its Applications vol159 no 1 pp 49ndash56 2012

[10] W-S Du ldquoOn approximate coincidence point properties andtheir applications to fixed point theoryrdquo Journal of AppliedMathematics vol 2012 Article ID 302830 17 pages 2012

[11] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[12] H-C Lai and Y-C Chen ldquoA quasi-fixed polynomial problemfor a polynomial functionrdquo Journal of Nonlinear and ConvexAnalysis vol 11 no 1 pp 101ndash114 2010

[13] H-C Lai and Y-C Chen ldquoQuasi-fixed polynomial for vector-valued polynomial functions on R119899 times Rrdquo Fixed Point Theoryvol 12 no 2 pp 391ndash400 2011 (Romanian)

[14] Y C Chen and H C Lai ldquoQuasi-fixed point solutions of real-valued polynomial equationrdquo in Proceedings of the 9th Interna-tional Conference on Fixed PointTheory and Its Applications pp16ndash22 Changhua Taiwan 2009

[15] Y-C Chen and H-C Lai ldquoA non-NP-complete algorithm for aquasi-fixed polynomial problemrdquoAbstract andAppliedAnalysisvol 2013 Article ID 893045 10 pages 2013

[16] Y C Chen and H C Lai ldquoNew quasi-coincidence pointpolynomial problemsrdquo Journal of Applied Mathematics vol2013 Article ID 959464 8 pages 2013

[17] R P Grimaldi Discrete and Combinatorial Mathematics Pear-son Secaucus NJ USA 5th edition 2003

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

for some 119888119894isin R 119894 = 0 119904 and119910(119909) isin Qcs

119865 Comparing the

coefficients of 119910119904 and 119910119904minus1 in both sides of the above equation

we get

119886119904(119909) = 119888

119904

119891 (119909)

119901119904 (119909)

119886119904minus1

(119909) = minus119904119886119904(119909) 119910 (119909) + 119888

119904minus1

119891 (119909)

119901119904minus1 (119909)

(84)

Consequently by (84) we get

119901119904

(119909) = 119888119904

119891 (119909)

119886119904(119909)

119910 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

isin R [119909]

(85)

By Lemma 15 for any 119889 isin R we have that any quasi-coincidence solution is represented by

119910 (119909) + 119889119901 (119909) =119888119904minus1

119904119888119904

119901 (119909) minus119886119904minus1

(119909)

119904119886119904(119909)

+ 119889119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ (119889 minus119888119904minus1

119904119888119904

)119901 (119909)

= minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909)

(86)

where 119901(119909) = (119888119904)1119904

(119891(119909)119886119904(119909))1119904 (note that since 119889 is

arbitrary then 120582 is arbitrary)This completes the proof

Corollary 17 Let 119865(119909 119910) be a polynomial function with

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (87)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886 (88)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

120582 minus119886119904minus1

(119909)

119904119886119904(119909)

(995333)

for arbitrary 120582 isin R

Corollary 18 Let 119865 R times R rarr R be a polynomial functionwith

119865 (119909 119910) = 119886119904(119909) 119910119904

+ 119886119904minus1

(119909) 119910119904minus1

+ sdot sdot sdot + 1198860(119909) (89)

If the cardinal number |119876119888119904119865| of equation

119865 (119909 119910) = 119886119909 (90)

is infinitely many then the leading coefficient 119886119904(119909) must be a

real number and each quasi-coincidence point solutionmust beof the form

1205821+ 1205822

119886119904minus1

(119909)

119909(995333)

for arbitrary 1205821and some 120582

2isin R

Proof Assume that there exist infinitely many solutions byTheorem 16 any solution of (4) has the form

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) (91)

for arbitrary 120582 isin R and 119901(119909) = 119888(119909119886119904(119909))1119904

isin R[119909]and then 119886

119904(119909) is a factor of 119909 This means that 119886

119904(119909) isin R

or 119886119904(119909) = 119896119909 for some constant 119896 isin R if 119886

119904(119909) isin R

this implies 119901(119909) = 119888(119909119886119904(119909))1119904

notin R[119909] and this leads acontradiction So we have 119886

119904(119909) = 119896119909 for some 119896 isin R then

119901(119909) = 1198881198961119904

isin R and any solution (91) is represented as

minus119886119904minus1

(119909)

119904119886119904(119909)

+ 120582119901 (119909) = 1205821+ 1205822

119886119904minus1

(119909)

119909(92)

for arbitrary 1205821= 120582119888119896

1119904 and some 1205822= minus1119904119896 isin R

Finally we provide one example to explain Theorem 16

Example 19 Let 119909 = (1199091 1199092) isin R2 119891(119909) = (119909

1+ 1199092)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091minus 1199092) 119910

+ (1199092

11199092

2minus 1199092

11199092minus 11990911199092

2+ 1199092

1+ 211990911199092+ 1199092

2)

(93)

Can we solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909)This polynomial function has exactly 5(ge 119904 + 3 since 119904 = 2)

quasi-fixed solutions as follows

119865 (1199091 1199092 11990911199092minus 1199091minus 1199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092) = 1(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 1199091+ 1199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+ 21199091+ 21199092) = 3(119909

1+ 1199092)2

119865 (1199091 1199092 11990911199092+1199091

2+1199092

2) =

3

4(1199091+ 1199092)2

(94)

In fact by Theorem 16 we can find any quasi-coincidencesolution written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091minus 1199092

2+ 120582119901 (119909)

= 11990911199092+ (120582 minus

1

2) (1199091minus 1199092)

= 11990911199092+ 120583 (119909

1minus 1199092)

(95)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091+ 1199092 This shows the quasi-coincidence

(point) solutions have cardinal |Qs119865| = infin

In practice we have no idea to check the number of thisequation is infinitely many or finitely many But we providean easy method to solve all solutions if the number of all

Abstract and Applied Analysis 9

solutions is infinitely many in this paper Thus we can solvethose solutions directly and checkwhether those solutions arethe solutions of 119865(119909 119910) = 119886119891(119909) and give an example in thefollowing

Example 20 Let 119909 = (1199091 1199092) isin R2 119891(119909) = 119909

2

1(1199092+ 1199093)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091(1199092+ 1199093)) 119910

+ (1199092

11199092

2+ 1199092

111990921199093+ 1199092

11199092

3)

(96)

We will solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909) ifthe number of all solutions is infinitely many

By Theorem 16 we can find that any quasi-coincidencesolution can be written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091(1199092+ 1199093)

2+ 120582119901 (119909)

= 11990911199092+ 1205831199091(1199092+ 1199093)

(97)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091(1199092+ 1199093) We let 119910(119909) = 119909

11199092+ 1205831199091(1199092+

1199093) and calculate 119865(119909 119910(119909)) and obtain

119865 (119909 119910 (119909)) = (1205832

+ 120583 + 1) 1199092

1(1199092+ 1199093)2

(98)

This means that the quasi-coincidence (point) solutions havecardinal |Qs

119865| = infin

We would like to provide one open problem as follows

Further Development Let Q be a quotient field Consider aquotient-valued polynomial function

119865 Q119899

timesQ 997888rarr Q (99)

Can we find all quasi-coincidence solutions 119910 = 119910(119909) isin Q[119909]

to satisfy

119865 (119909 119910) = 119886119891 (119909) (100)

for some polynomials 119891(119909) isin Q[119909] by a co-NP hardnessalgorithm

References

[1] A K Lenstra ldquoFactoring multivariate polynomials over alge-braic number fieldsrdquo SIAM Journal on Computing vol 16 no 3pp 591ndash598 1987

[2] E P Tsigaridas and I Z Emiris ldquoUnivariate polynomial realroot isolation continued fractions revisitedrdquo in AlgorithmsmdashESA 2006 vol 4168 of Lecture Notes in Computer Science pp817ndash828 Springer Berlin Germany 2006

[3] J von zur Gathen and J Gerhard Modern Computer AlgebraCambridge University Press Cambridge UK 2nd edition2003

[4] M G Marinari H M Moller and T Mora ldquoOn multiplicitiesin polynomial system solvingrdquo Transactions of the AmericanMathematical Society vol 348 no 8 pp 3283ndash3321 1996

[5] P Gopalan V Guruswami and R J Lipton ldquoAlgorithmsfor modular counting of roots of multivariate polynomialsrdquoAlgorithmica vol 50 no 4 pp 479ndash496 2008

[6] V Y Pan ldquoUnivariate polynomials nearly optimal algorithmsfor numerical factorization and root-findingrdquo Journal of Sym-bolic Computation vol 33 no 5 pp 701ndash733 2002

[7] C-M Chen T-H Chang and Y-P Liao ldquoCoincidence the-orems generalized variational inequality theorems and min-imax inequality theorems for the 120601-mapping on G-convexspacesrdquo Fixed Point Theory and Applications vol 2007 ArticleID 078696 13 pages 2007

[8] C-M Chen T-H Chang and C-W Chung ldquoCoincidence the-orems on nonconvex sets and its applicationsrdquo The TaiwaneseJournal of Mathematics vol 13 no 2 pp 501ndash513 2009

[9] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinearmultivaluedmapsrdquoTopology and Its Applications vol159 no 1 pp 49ndash56 2012

[10] W-S Du ldquoOn approximate coincidence point properties andtheir applications to fixed point theoryrdquo Journal of AppliedMathematics vol 2012 Article ID 302830 17 pages 2012

[11] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[12] H-C Lai and Y-C Chen ldquoA quasi-fixed polynomial problemfor a polynomial functionrdquo Journal of Nonlinear and ConvexAnalysis vol 11 no 1 pp 101ndash114 2010

[13] H-C Lai and Y-C Chen ldquoQuasi-fixed polynomial for vector-valued polynomial functions on R119899 times Rrdquo Fixed Point Theoryvol 12 no 2 pp 391ndash400 2011 (Romanian)

[14] Y C Chen and H C Lai ldquoQuasi-fixed point solutions of real-valued polynomial equationrdquo in Proceedings of the 9th Interna-tional Conference on Fixed PointTheory and Its Applications pp16ndash22 Changhua Taiwan 2009

[15] Y-C Chen and H-C Lai ldquoA non-NP-complete algorithm for aquasi-fixed polynomial problemrdquoAbstract andAppliedAnalysisvol 2013 Article ID 893045 10 pages 2013

[16] Y C Chen and H C Lai ldquoNew quasi-coincidence pointpolynomial problemsrdquo Journal of Applied Mathematics vol2013 Article ID 959464 8 pages 2013

[17] R P Grimaldi Discrete and Combinatorial Mathematics Pear-son Secaucus NJ USA 5th edition 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

solutions is infinitely many in this paper Thus we can solvethose solutions directly and checkwhether those solutions arethe solutions of 119865(119909 119910) = 119886119891(119909) and give an example in thefollowing

Example 20 Let 119909 = (1199091 1199092) isin R2 119891(119909) = 119909

2

1(1199092+ 1199093)2 and

119865 (119909 119910) = 1198862(119909) 1199102

+ 1198861(119909) 119910 + 119886

0(119909)

= 1199102

minus (211990911199092minus 1199091(1199092+ 1199093)) 119910

+ (1199092

11199092

2+ 1199092

111990921199093+ 1199092

11199092

3)

(96)

We will solve all quasi-fixed solutions of 119865(119909 119910) = 119886119891(119909) ifthe number of all solutions is infinitely many

By Theorem 16 we can find that any quasi-coincidencesolution can be written as

minus1198861(119909)

1199041198862(119909)

+ 120582119901 (119909) =211990911199092minus 1199091(1199092+ 1199093)

2+ 120582119901 (119909)

= 11990911199092+ 1205831199091(1199092+ 1199093)

(97)

where 120583 = 120582 minus 12 isin R is arbitrary and 119901(119909) =

(119891(119909)119886119904(119909))1119904

= 1199091(1199092+ 1199093) We let 119910(119909) = 119909

11199092+ 1205831199091(1199092+

1199093) and calculate 119865(119909 119910(119909)) and obtain

119865 (119909 119910 (119909)) = (1205832

+ 120583 + 1) 1199092

1(1199092+ 1199093)2

(98)

This means that the quasi-coincidence (point) solutions havecardinal |Qs

119865| = infin

We would like to provide one open problem as follows

Further Development Let Q be a quotient field Consider aquotient-valued polynomial function

119865 Q119899

timesQ 997888rarr Q (99)

Can we find all quasi-coincidence solutions 119910 = 119910(119909) isin Q[119909]

to satisfy

119865 (119909 119910) = 119886119891 (119909) (100)

for some polynomials 119891(119909) isin Q[119909] by a co-NP hardnessalgorithm

References

[1] A K Lenstra ldquoFactoring multivariate polynomials over alge-braic number fieldsrdquo SIAM Journal on Computing vol 16 no 3pp 591ndash598 1987

[2] E P Tsigaridas and I Z Emiris ldquoUnivariate polynomial realroot isolation continued fractions revisitedrdquo in AlgorithmsmdashESA 2006 vol 4168 of Lecture Notes in Computer Science pp817ndash828 Springer Berlin Germany 2006

[3] J von zur Gathen and J Gerhard Modern Computer AlgebraCambridge University Press Cambridge UK 2nd edition2003

[4] M G Marinari H M Moller and T Mora ldquoOn multiplicitiesin polynomial system solvingrdquo Transactions of the AmericanMathematical Society vol 348 no 8 pp 3283ndash3321 1996

[5] P Gopalan V Guruswami and R J Lipton ldquoAlgorithmsfor modular counting of roots of multivariate polynomialsrdquoAlgorithmica vol 50 no 4 pp 479ndash496 2008

[6] V Y Pan ldquoUnivariate polynomials nearly optimal algorithmsfor numerical factorization and root-findingrdquo Journal of Sym-bolic Computation vol 33 no 5 pp 701ndash733 2002

[7] C-M Chen T-H Chang and Y-P Liao ldquoCoincidence the-orems generalized variational inequality theorems and min-imax inequality theorems for the 120601-mapping on G-convexspacesrdquo Fixed Point Theory and Applications vol 2007 ArticleID 078696 13 pages 2007

[8] C-M Chen T-H Chang and C-W Chung ldquoCoincidence the-orems on nonconvex sets and its applicationsrdquo The TaiwaneseJournal of Mathematics vol 13 no 2 pp 501ndash513 2009

[9] W-S Du ldquoOn coincidence point and fixed point theorems fornonlinearmultivaluedmapsrdquoTopology and Its Applications vol159 no 1 pp 49ndash56 2012

[10] W-S Du ldquoOn approximate coincidence point properties andtheir applications to fixed point theoryrdquo Journal of AppliedMathematics vol 2012 Article ID 302830 17 pages 2012

[11] W-S Du and S-X Zheng ldquoNew nonlinear conditions andinequalities for the existence of coincidence points and fixedpointsrdquo Journal of Applied Mathematics vol 2012 Article ID196759 12 pages 2012

[12] H-C Lai and Y-C Chen ldquoA quasi-fixed polynomial problemfor a polynomial functionrdquo Journal of Nonlinear and ConvexAnalysis vol 11 no 1 pp 101ndash114 2010

[13] H-C Lai and Y-C Chen ldquoQuasi-fixed polynomial for vector-valued polynomial functions on R119899 times Rrdquo Fixed Point Theoryvol 12 no 2 pp 391ndash400 2011 (Romanian)

[14] Y C Chen and H C Lai ldquoQuasi-fixed point solutions of real-valued polynomial equationrdquo in Proceedings of the 9th Interna-tional Conference on Fixed PointTheory and Its Applications pp16ndash22 Changhua Taiwan 2009

[15] Y-C Chen and H-C Lai ldquoA non-NP-complete algorithm for aquasi-fixed polynomial problemrdquoAbstract andAppliedAnalysisvol 2013 Article ID 893045 10 pages 2013

[16] Y C Chen and H C Lai ldquoNew quasi-coincidence pointpolynomial problemsrdquo Journal of Applied Mathematics vol2013 Article ID 959464 8 pages 2013

[17] R P Grimaldi Discrete and Combinatorial Mathematics Pear-son Secaucus NJ USA 5th edition 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of