research article approximations for equilibrium problems...
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Research ArticleApproximations for Equilibrium Problems andNonexpansive Semigroups
Huan-chun Wu and Cao-zong Cheng
Department of Mathematics Beijing University of Technology Beijing 100124 China
Correspondence should be addressed to Huan-chun Wu wuhuanchunemailsbjuteducn
Received 29 November 2013 Accepted 4 February 2014 Published 16 March 2014
Academic Editor Hassen Aydi
Copyright copy 2014 H-c Wu and C-z Cheng This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the setof all common fixed points of a nonexpansive semigroup and prove the strong convergence theorem in Hilbert spaces Our resultextends the recent result of Zegeye and Shahzad (2013) In the last part of the paper by the way we point out that there is a slightflaw in the proof of the main result in Shehursquos paper (2012) and perfect the proof
1 Introduction
Let119867 be a real Hilbert space and let 119862 be a nonempty closedconvex subset of 119867 A mapping 119879 119862 rarr 119862 is callednonexpansive if 119879119909 minus 119879119910 le 119909 minus 119910 for all 119909 119910 isin 119862 Wedenote the set of fixed points of 119879 by 119865(119879) It is known that119865(119879) is closed and convex A family S = 119879(119905) 0 le 119905 lt
infin of mappings from 119862 into itself is called a nonexpansivesemigroup on 119862 if it satisfies the following conditions
(i) 119879(0)119909 = 119909 for all 119909 isin 119862(ii) 119879(119904 + 119905) = 119879(119904)119879(119905) for all 119904 119905 ge 0(iii) 119879(119904)119909 minus 119879(119904)119910 le 119909 minus 119910 for all 119909 119910 isin 119862 and 119904 ge 0(iv) for all 119909 isin 119862 119904 997891rarr 119879(119904)119909 is continuous
We denote by 119865(S) the set of all common fixed points of Sthat is119865(S) = ⋂
0le119905ltinfin119865(119879(119905)) It is clear that119865(S) is a closed
convex subsetThe equilibrium problem for 119891 119862 times 119862 rarr R is to
find 119909 isin 119862 such that 119891(119909 119910) ge 0 for all 119910 isin 119862 The set ofsuch solutions is denoted by EP(119891) Numerous problems inphysics optimization and economics can be reduced to finda solution of the equilibrium problem (for instance see [1])
For solving equilibrium problem we assume that thebifunction 119891 satisfies the following conditions
(A1) 119891(119909 119909) = 0 for all 119909 isin 119862
(A2) 119891 is monotone that is 119891(119909 119910) + 119891(119910 119909) le 0 for any119909 119910 isin 119862
(A3) for each 119909 119910 119911 isin 119862 lim sup119905darr0119891(119905119911 + (1 minus 119905)119909 119910) le
119891(119909 119910)(A4) 119891(119909 sdot) is convex and lower semicontinuous for each
119909 isin 119862
Several methods have been proposed to solve the equilib-rium problem see [1ndash7] For finding common fixed points ofa nonexpansive semigroup Nakajo and Takahashi [8] intro-duced a convergent sequence for nonexpansive semigroupS = 119879(119905) 0 le 119905 lt infin as follows
1199090= 119909 isin 119862
119910119899= 120572119899119909119899+ (1 minus 120572
119899)1
119905119899
int
119905119899
0
119879 (119904) 119909119899119889119904
119862119899= 119911 isin 119862
1003817100381710038171003817119910119899 minus 1199111003817100381710038171003817 le
1003817100381710038171003817119909119899 minus 1199111003817100381710038171003817
119876119899= 119911 isin 119862 ⟨119909
119899minus 119911 119909
0minus 119909119899⟩ ge 0
119909119899+1
= 119875119862119899⋂119876119899
(1199090)
(1)
Some authors have paidmore attention to find an element119901 isin 119865(S)capEP(119891) Buong and Duong [9] constructed the fol-lowing iterative sequence and proved the weak convergence
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 351372 7 pageshttpdxdoiorg1011552014351372
2 Abstract and Applied Analysis
theorem for an equilibrium problem and a nonexpansivesemigroup in Hilbert spaces
1199090isin 119867
119906119896isin 119862 119891 (119906
119896 119910) +
1
119903119896
⟨119910 minus 119906119896 119906119896minus 119909119896⟩ ge 0 forall119910 isin 119862
119909119896+1
= 120583119896119909119896+ (1 minus 120583
119896)1
119905119896
int
119905119896
0
119879 (119904) 119906119896119889119904
(2)
In 2012 Shehu [10] studied iterative methods for fixedpoint problem variational inequality and generalized mixedequilibrium problem and introduced a new algorithm whichdoes not involve the CQ algorithm and viscosity approxima-tion method However we discover that there is a slight flawin the proof of Theorem 31 in [10]
Motivated by Nakajo and Takahashi [8] Buong andDuong [9] and especially Shehu [10] andZegeye and Shahzad[11] we present a new iterative method for finding a commonelement of the set of solutions of an equilibrium problemand the set of all common fixed points of a nonexpansivesemigroup and prove the strong convergence theorem inHilbert spaces Our result extends the recent result of [11] Inthe last part of the paper we perfect and simplify the proof ofTheorem 31 in [10]
2 Preliminaries
Throughout this paper let119867be a realHilbert spacewith innerproduct ⟨sdot sdot⟩ and norm sdot and let 119862 be a nonempty closedconvex subset of 119867 We write 119909
119899rarr 119909 to indicate that the
sequence 119909119899 converges strongly to 119909 Similarly 119909
119899 119909 will
symbolize weak convergence It is well known that119867 satisfiesOpialrsquos condition that is for any sequence 119909
119899 sub 119867 with
119909119899 119909 we have
lim inf119899rarrinfin
1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817 lt lim inf119899rarrinfin
1003817100381710038171003817119909119899 minus 1199101003817100381710038171003817 forall119910 = 119909 (3)
For any 119909 isin 119867 there exists a unique point 119875119862119909 isin 119862 such that
1003817100381710038171003817119909 minus 1198751198621199091003817100381710038171003817 le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119910 isin 119862 (4)
119875119862is called the metric projection of119867 onto 119862 We know that
119875119862is a nonexpansive mapping of119867 onto 119862 and 119875
119862satisfies
⟨119909 minus 119910 119875119862119909 minus 119875119862119910⟩ ge
1003817100381710038171003817119875119862119909 minus 1198751198621199101003817100381710038171003817
2 forall119909 119910 isin 119867 (5)
For 119909 isin 119867 and 119911 isin 119862 we have
119911 = 119875119862119909 lArrrArr ⟨119909 minus 119911 119910 minus 119911⟩ le 0 for every119910 isin 119862 (6)
The following lemmas will be used in the proof of ourresults
Lemma 1 (see [1]) Let 119862 be a nonempty closed convex subsetof119867 and let119891 be a bifunction from119862times119862 toR satisfying (A1)ndash(A4) If 119903 gt 0 and 119909 isin 119867 then there exists 119911 isin 119862 such that
119891 (119911 119910) +1
119903⟨119910 minus 119911 119911 minus 119909⟩ ge 0 forall119910 isin 119862 (7)
Lemma2 (see [2]) For 119903 gt 0 define amapping 119879119903 119867 rarr 2
119862
as follows
119879119903 (119909) = 119911 isin 119862 119891 (119911 119910) +
1
119903⟨119910 minus 119911 119911 minus 119909⟩ ge 0 forall119910 isin 119862
(8)
Then the following hold
(i) 119879119903is single valued
(ii) 119879119903is firmly nonexpansive that is for any 119909 119910 isin 119867
⟨119909 minus 119910 119879119903119909 minus 119879119903119910⟩ ge 119879
119903119909 minus 1198791199031199102
(iii) 119865(119879119903) = 119864119875(119891)
(iv) 119864119875(119891) is closed and convex
Lemma 3 (see [12]) Suppose that (A1)ndash(A4) hold If 119909 119910 isin 119867and 1199031 1199032gt 0 then
100381710038171003817100381710038171198791199032
119910 minus 1198791199031
11990910038171003817100381710038171003817le1003817100381710038171003817119910 minus 119909
1003817100381710038171003817 +
10038161003816100381610038161199032 minus 11990311003816100381610038161003816
1199032
100381710038171003817100381710038171198791199032
119910 minus 11991010038171003817100381710038171003817 (9)
Lemma 4 (see [13]) Let 119862 be a nonempty bounded closedsubset of 119867 and let 119879(119904) 0 le 119904 lt infin be a nonexpansivesemigroup on 119862 Then for every ℎ ge 0
lim119905rarr+infin
sup119909isin119862
10038171003817100381710038171003817100381710038171003817
1
119905int
119905
0
119879 (119904) 119909 119889119904 minus 119879 (ℎ)1
119905int
119905
0
119879 (119904) 119909 119889119904
10038171003817100381710038171003817100381710038171003817
= 0 (10)
Lemma 5 (see [14]) Let 119909119899 and 119910
119899 be bounded sequences
in a Banach space 119883 and let 120573119899 be a sequence in [0 1]
with 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1 Suppose
that 119909119899+1
= (1 minus 120573119899)119909119899+ 120573119899119910119899for all integers 119899 ge 1
and lim sup119899rarrinfin
(119910119899+1
minus 119910119899 minus 119909
119899+1minus 119909119899) le 0 Then
lim119899rarrinfin
119910119899minus 119909119899 = 0
Lemma 6 (see [15]) Let 119886119899 be a sequence of nonnegative real
numbers satisfying 119886119899+1
le (1 minus 120572119899)119886119899+ 120572119899120573119899 where
(i) 120572119899 sub (0 1) suminfin
119899=1120572119899= infin
(ii) lim sup119899rarrinfin
120573119899le 0
Then lim119899rarrinfin
119886119899= 0
3 Strong Convergence Theorems
In this section we introduce a new iterative method forfinding a common element of the set of solutions of anequilibriumproblemand the set of all commonfixedpoints ofa nonexpansive semigroup and prove the strong convergencetheorem in Hilbert spaces
Theorem 7 Let 119862 be a nonempty closed convex subset of areal Hilbert space 119867 and let 119891 be a bifunction from 119862 times 119862 toR satisfying (A1)ndash(A4) Suppose that S = 119879(119905) 0 le 119905 lt infin
Abstract and Applied Analysis 3
is a nonexpansive semigroup on 119862 such that 119865(S) cap119864119875(119891) = 0For 119906 isin 119867 let 119909
119899 119910119899 and 119911
119899 be generated by
1199091isin 119862 chosen arbitrarily
119910119899= 120572119899119906 + (1 minus 120572
119899) 119909119899
119911119899isin 119862 such that119891 (119911
119899 119910) +
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 0
forall119910 isin 119862
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
(11)
where the real sequences 120572119899 120573119899 in (0 1) and 119903
119899 sub (0infin)
satisfy the following conditions
(1) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(2) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(3) 0 lt 119888 le 119903119899lt infin lim
119899rarrinfin|119903119899+1
minus 119903119899| = 0
(4) 119905119899 sub (0infin) lim
119899rarrinfin119905119899= infin and lim
119899rarrinfin(|119905119899+1
minus
119905119899|119905119899+1) = 0
Then the sequence 119909119899 converges strongly to 119875
119865(S)cap119864119875(119891)119906
Proof Note that the set 119865(S) cap EP(119891) is closed and convexsince 119865(S) and EP(119891) are closed and convex For simplicitywe writeΩ = 119865(S) cap EP(119891)
From Lemmas 1 and 2 we have 119911119899= 119879119903119899
119910119899 and for any
119901 isin Ω
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817 =
10038171003817100381710038171003817119879119903119899
119910119899minus 119879119903119899
11990110038171003817100381710038171003817le1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817 (12)
Observe that1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817 le 1205721198991003817100381710038171003817119906 minus 119901
1003817100381710038171003817 + (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 (13)
It follows that
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817 =
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119901
10038171003817100381710038171003817100381710038171003817
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
+ 120573119899
1
119905119899
int
119905119899
0
1003817100381710038171003817119879 (119904) 119911119899 minus 119879 (119904) 1199011003817100381710038171003817 119889119904
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 + 1205731198991003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
+ 120573119899[120572119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817 + (1 minus 120572119899)
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817]
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 + 1205721198991205731198991003817100381710038171003817119906 minus 119901
1003817100381710038171003817
le max 1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 1003817100381710038171003817119906 minus 119901
1003817100381710038171003817
(14)
From a simple inductive process one has1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817 le max 10038171003817100381710038171199091 minus 1199011003817100381710038171003817 1003817100381710038171003817119906 minus 119901
1003817100381710038171003817 (15)
which yields that 119909119899 is bounded So are 119910
119899 and 119911
119899
Set 120590119899= (1119905
119899) int119905119899
0119879(119904)119911119899119889119904 For any 119901 isin Ω we have
1003817100381710038171003817120590119899+1 minus 1205901198991003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899+1119889119904 minus
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899+1119889119904 minus
1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899119889119904
+1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899119889119904 minus
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
(119879 (119904) 119911119899+1
minus 119879 (119904) 119911119899) 119889119904
+(1
119905119899+1
minus1
119905119899
)int
119905119899
0
119879 (119904) 119911119899119889119904 +1
119905119899+1
int
119905119899+1
119905119899
119879 (119904) 119911119899119889119904
100381710038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
(119879 (119904) 119911119899+1
minus 119879 (119904) 119911119899) 119889119904
+ (1
119905119899+1
minus1
119905119899
)int
119905119899
0
(119879 (119904) 119911119899minus 119879 (119904) 119901) 119889119904
+(1
119905119899+1
minus1
119905119899
)int
119905119899
0
119879 (119904) 119901 119889119904 +1
119905119899+1
int
119905119899+1
119905119899
119879 (119904) 119911119899119889119904
100381710038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
(119879 (119904) 119911119899+1
minus 119879 (119904) 119911119899) 119889119904
+ (1
119905119899+1
minus1
119905119899
)int
119905119899
0
(119879 (119904) 119911119899minus 119879 (119904) 119901) 119889119904
+1
119905119899+1
int
119905119899+1
119905119899
(119879 (119904) 119911119899 minus 119879 (119904) 119901) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le1003817100381710038171003817119911119899+1 minus 119911119899
1003817100381710038171003817 +21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
(16)
It follows from Lemma 3 that
1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817 le
1003817100381710038171003817119910119899+1 minus 1199101198991003817100381710038171003817 +
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119903119899+1
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817
(17)
Hence
1003817100381710038171003817120590119899+1 minus 1205901198991003817100381710038171003817
le1003817100381710038171003817119910119899+1 minus 119910119899
1003817100381710038171003817 +
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119903119899+1
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817
+21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
le1003817100381710038171003817120572119899+1119906 + (1 minus 120572119899+1) 119909119899+1 minus 120572119899119906 minus (1 minus 120572119899) 119909119899
1003817100381710038171003817
4 Abstract and Applied Analysis
+
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119903119899+1
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817 +
21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
le1003817100381710038171003817119909119899+1 minus 119909119899
1003817100381710038171003817 + 120572119899+1 (119906 +1003817100381710038171003817119909119899+1
1003817100381710038171003817) + 120572119899 (119906 +1003817100381710038171003817119909119899
1003817100381710038171003817)
+
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119888
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817 +
21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
(18)
This implies that
lim sup119899rarrinfin
(1003817100381710038171003817120590119899+1 minus 120590119899
1003817100381710038171003817 minus1003817100381710038171003817119909119899+1 minus 119909119899
1003817100381710038171003817) le 0 (19)
It follows from Lemma 5 that lim119899rarrinfin
120590119899minus 119909119899 = 0 Thus
lim119899rarrinfin
1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 = lim119899rarrinfin
120573119899
1003817100381710038171003817120590119899 minus 1199091198991003817100381710038171003817 = 0 (20)
For any 119901 isin Ω we have1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2=10038171003817100381710038171003817119879119903119899
119910119899minus 119879119903119899
11990110038171003817100381710038171003817
2
le ⟨119910119899minus 119901 119911
119899minus 119901⟩
=1
2[1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2+1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
(21)
Thus1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2le1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2 (22)
From the convexity of sdot 2 it follows that1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2
=
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) (119909119899minus 119901) + 120573
119899(1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119901)
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2
+ 120573119899
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
(119879 (119904) 119911119899 minus 119879 (119904) 119901) 119889119904
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[1003817100381710038171003817120572119899119906 + (1 minus 120572119899) 119909119899 minus 119901
1003817100381710038171003817
2
minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2
+120572119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
(23)
Hence
120573119899
1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2+ 120572119899120573119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2
le1003817100381710038171003817119909119899 minus 119909119899+1
1003817100381710038171003817 (1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817)
+ 120572119899120573119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2
(24)
Since lim119899rarrinfin
119909119899+1
minus 119909119899 = 0 and lim
119899rarrinfin120572119899= 0 one has
lim119899rarrinfin
1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (25)
Observe thatlim119899rarrinfin
1003817100381710038171003817119910119899 minus 1199091198991003817100381710038171003817 = lim119899rarrinfin
120572119899
1003817100381710038171003817119906 minus 1199091198991003817100381710038171003817 = 0 (26)
As 119911119899minus 119909119899 le 119911
119899minus 119910119899 + 119910
119899minus 119909119899 the following equality
holdslim119899rarrinfin
1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (27)
Now we show thatlim119899rarrinfin
1003817100381710038171003817119879 (119904) 119911119899 minus 1199111198991003817100381710038171003817 = 0 forall0 le 119904 lt infin (28)
In fact we have1003817100381710038171003817119879 (119904) 119911119899 minus 119911119899
1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
119879 (119904) 119911119899 minus 119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
+ 119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
+1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le
10038171003817100381710038171003817100381710038171003817
119879 (119904) 119911119899minus 119879 (119904)
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le 2
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
(29)
Notice that10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119909
119899
10038171003817100381710038171003817100381710038171003817
+1003817100381710038171003817119909119899 minus 119911119899
1003817100381710038171003817
=1
120573119899
1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 +
1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(30)
For any 119901 isin Ω let 119866 = 119909 isin 119862 119909 minus 119901 le max1199091minus 119901 119906 minus
119901 It is easy to see that119866 is a bounded closed convex subsetand 119879(119904)119866 is a subset of 119866 Since
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817 =
10038171003817100381710038171003817119879119903119899
119910119899minus 119901
10038171003817100381710038171003817
le (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 + 1205721198991003817100381710038171003817119906 minus 119901
1003817100381710038171003817
le max 10038171003817100381710038171199091 minus 1199011003817100381710038171003817 1003817100381710038171003817119906 minus 119901
1003817100381710038171003817
(31)
Abstract and Applied Analysis 5
the sequence 119911119899 is contained in 119866 It follows from Lemma 4
that
lim119899rarrinfin
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119879 (119904)
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
= 0 (32)
From (29) (30) and (32) the expression (28) is obtainedNext we prove that
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ le 0 (33)
where 119909 = 119875Ω119906 In order to show this inequality we can
choose a subsequence 119909119899119895
of 119909119899 such that
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ = lim
119895rarrinfin
⟨119906 minus 119909 119909119899119895
minus 119909⟩ (34)
Due to the boundedness of 119909119899119895
there exists a subsequence119909119899119895119894
of 119909119899119895
such that 119909119899119895119894
120596 Without loss of generalitywe assume that 119909
119899119895
120596 From (27) we see that 119911119899119895
120596Since 119911
119899119895
sub 119862 and 119862 is closed and convex we get 120596 isin 119862We first show that 120596 isin EP(119891) By 119911
119899= 119879119903119899
119910119899 we have
119891 (119911119899 119910) +
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 0 forall119910 isin 119862 (35)
It follows from the monotonicity of 119891 that
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 119891 (119910 119911
119899) forall119910 isin 119862 (36)
Replacing 119899 by 119899119895 we obtain
⟨119910 minus 119911119899119895
119911119899119895
minus 119910119899119895
119903119899119895
⟩ ge 119891(119910 119911119899119895
) forall119910 isin 119862 (37)
From (25) (27) and (A4) we have
119891 (119910 120596) le 0 forall119910 isin 119862 (38)
For 0 lt 119905 le 1 119910 isin 119862 set 119910119905= 119905119910 + (1 minus 119905)120596 We have 119910
119905isin 119862
and 119891(119910119905 120596) le 0 Hence
0 = 119891 (119910119905 119910119905) le 119905119891 (119910
119905 119910) + (1 minus 119905) 119891 (119910
119905 120596) le 119905119891 (119910
119905 119910)
(39)
Dividing by 119905 we see that
119891 (119910119905 119910) ge 0 (40)
Letting 119905 darr 0 and from (A3) we get
119891 (120596 119910) ge 0 forall119910 isin 119862 (41)
That is 120596 isin EP(119891)Second we prove that 120596 isin 119865(S) Note that the equality
(27) implies that 119911119899119895
120596 Suppose for contradiction that 120596 notin119865(S) that is
there exists 1199040gt 0 such that 119879 (119904
0) 120596 = 120596 (42)
Then from Opialrsquos condition and (28) we obtain
lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
= lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 119879 (1199040) 119911119899119895
+ 119879 (1199040) 119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
= lim inf119895rarrinfin
100381710038171003817100381710038171003817119879 (1199040) 119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
le lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 120596100381710038171003817100381710038171003817
(43)
which is a contradictionTherefore 120596 isin 119865(S) Consequentlyone gets 120596 isin Ω
From (34) and the property of metric projection we have
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ = lim
119895rarrinfin
⟨119906 minus 119909 119909119899119895
minus 119909⟩
= ⟨119906 minus 119909 120596 minus 119909⟩ le 0
(44)
The inequality (33) arrivesFinally we show that 119909
119899rarr 119909 From (11) we have
1003817100381710038171003817119909119899+1 minus 1199091003817100381710038171003817
2
=
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119909
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119911119899 minus 1199091003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1199091003817100381710038171003817
2
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨119906 minus 119909 119909
119899minus 119909⟩ + 120572
119899119906 minus 1199092]
(45)
It follows from (33) and Lemma 6 that 119909119899 converges strongly
to 119909
Remark 8 Let119867 = R and 119862 = [0 1] Setting 119891(119909 119910) = 1199102 minus1199092 we see that 119891(119909 119910) satisfies (A1)ndash(A4) For 0 le 119905 lt +infin
let
119879 (119905) 119909 =119909
1 + 119905119909 forall119909 isin [0 1] (46)
Thus it follows thatS = 119879(119905) 0 le 119905 lt infin is a nonexpansivesemigroup such that 119865(S) cap EP(119891) = 0 If we take
120572119899=
1
119899 + 1 1205731= 1205732=1
2 120573119899=1
2minus1
119899 forall119899 ge 3
119903119899equiv 119888 gt 0 119905
119899= 119899
(47)
then all assumptions and conditions in Theorem 7 aresatisfied
Remark 9 Taking 119906 = 0 inTheorem 7 we obtain the iterativemethod for minimum-norm solution of an equilibriumproblem and a nonexpansive semigroup
6 Abstract and Applied Analysis
As a direct consequence of Theorem 7 we obtain thefollowing corollary
Corollary 10 Let 119862 be a nonempty closed convex subset of areal Hilbert space119867 and assume that S = 119879(119905) 0 le 119905 lt infin
is a nonexpansive semigroup on 119862 such that 119865(S) = 0 Let 120572119899
and 120573119899 be real sequences in (0 1) and let 119909
119899 and 119911
119899 be
generated by
1199091isin 119862 chosen arbitrarily
119911119899= 119875119862[(1 minus 120572
119899) 119909119899]
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
(48)
Suppose that the following conditions are satisfied
(1) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(2) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(3) 119905119899 sub (0infin) lim
119899rarrinfin119905119899= infin and lim
119899rarrinfin(|119905119899+1
minus
119905119899|119905119899+1) = 0
Then the sequence 119909119899 converges strongly to 119875
119865(S)0
Proof Letting 119891(119909 119910) = 0 for all 119909 119910 isin 119862 119903119899= 1 and 119906 = 0
in Theorem 7 we get the result
Remark 11 Corollary 10 extends the recent results of Zegeyeand Shahzad [11 Corollaries 32 and 33] from finite family ofnonexpansive mappings to a nonexpansive semigroup
4 A Note on Shehursquos Paper lsquolsquoIterative Methodfor Fixed Point Problem VariationalInequality and Generalized MixedEquilibrium Problems with Applicationsrsquorsquo
In 2012 Shehu [10] studied iterative methods for fixedpoint problem variational inequality and generalized mixedequilibrium problem and gave an interesting convergencetheorem However there is a slight flaw in the proof of themain result (Theorem 31 in [10])
Shehu obtained the following result (for more details see[10])
Theorem 12 (see [10]) Let 119870 be a closed convex subset ofa real Hilbert space 119867 let 119865 be a bifunction from 119870 times 119870
satisfying (A1)ndash(A4) let 120593 119870 rarr R cup +infin be a properlower semicontinuous and convex function with assumption(B1) or (B2) let A be a 120583-Lipschitzian relaxed (120582 120574)-cocoercivemapping of 119870 into 119867 and let 120595 be an 120572-inverse stronglymonotone mapping of 119870 into 119867 Suppose that 119879 119870 rarr 119870
is a nonexpansive mapping of 119870 into itself such that Ω =
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875 = 0 Let 120572119899 and 120573
119899 be two real
sequences in (0 1) and 119903119899 119904119899 sub (0infin) Let 119909
119899 119910119899 and
119906119899 be generated by 119909
1isin 119870
119910119899= 119875119870[(1 minus 120572
119899) 119909119899]
119906119899= 119879(119865120593)
119903119899
(119910119899minus 119903119899120595119910119899)
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899119879119875119870(119906119899minus 119904119899119860119906119899)
(49)
Suppose that the following conditions are satisfied
(a) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(b) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(c) 0 lt 119886 le 119904119899le 119887 lt 2(120574minus120582120583
2)1205832 lim119899rarrinfin
|119904119899+1minus119904119899| = 0
(d) 0 lt 119888 le 119903119899le 119889 lt 2120572 lim
119899rarrinfin|119903119899+1
minus 119903119899| = 0
Then the sequence 119909119899 converges strongly to an element of
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875
This theorem is proved in [10] by the following steps
Step 1 The sequence 119909119899 is bounded
Step 2 The following equalities hold
lim119899rarrinfin
1003817100381710038171003817119860119906119899 minus 119860119909lowast1003817100381710038171003817 = 0 lim
119899rarrinfin
1003817100381710038171003817120595119910119899 minus 120595119909lowast1003817100381710038171003817 = 0
forall119909lowastisin Ω
(50)
Step 3 If 120596 is a weak limit of 119909119899119895
which is a subsequence of119909119899 then 120596 isin Ω
Step 4 The sequence 119909119899 converges strongly to 120596
In Step 4 in order to show that the sequence 119909119899 con-
verges strongly to 120596 the author shows the inequalitylim sup
119899rarrinfin⟨minus120596 119909
119899minus 120596⟩ le 0 by defining a mapping 120601
119867 rarr 119877 as follows 120601(119909) = 120583119899119909119899minus 1199092 where 120583 is a Banach
limit It is proved that the set 119870lowast = 119909 isin 119867 120601(119909) =
min119906isin119867
120601(119906) = 0 and119870lowastcap119865(119879) = 0 An element of119870lowastcap119865(119879)is taken arbitrarily and is denoted by120596 Of course the element120596 is not necessarily the weak sequential cluster point of 119909
119899
However in Step 3 the symbol 120596 stands for the weak limit of119909119899119895
which is a subsequence of 119909119899 In the sequel the author
obtains
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2le (1 minus 120573
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2 (51)
The meaning of the element 120596 in (51) is ambiguous It isdifficult to ensure consistency
Now we perfect and simplify the proof of Step 4 Accord-ing to the equality in Step 2 lim
119899rarrinfin119860119906119899minus119860119909lowast = 0 for all
119909lowastisin Ω we see that the set 119860119909lowast 119909lowast isin Ω contains only one
element Since 119860 is a relaxed (120582 120574)-cocoercive mapping of 119870into119867 that is there exist 120582 120574 gt 0 such that
⟨119860119909 minus 119860119910 119909 minus 119910⟩ ge minus1205821003817100381710038171003817119860119909 minus 119860119910
1003817100381710038171003817
2+ 120574
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
2
forall119909 119910 isin 119870
(52)
Abstract and Applied Analysis 7
it follows that themapping119860 is one-to-oneTherefore the setΩ is a singleton By Step 3 the sequence 119909
119899 possesses only
oneweak sequential cluster point It follows fromLemma238in [16] that 119909
119899 converges weakly to 120596 and so
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817(1 minus 120572119899) 119909119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+2120572119899(1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
2
1198991205962]
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
1198991205962]
(53)
Since 119909119899 converges weakly to 120596 it follows from Lemma 22
in [10] or Lemma 6 in this paper that 119909119899 converges strongly
to 120596 isin Ω
Conflict of Interests
The authors declare that there is no conflict of interests
Acknowledgment
The authors would like to thank referees and editors for theirvaluable comments and suggestions
References
[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1-4 pp 123ndash145 1994
[2] P L Combettes and S A Hirstoaga ldquoEquilibrium program-ming in Hilbert spacesrdquo Journal of Nonlinear and ConvexAnalysis vol 6 no 1 pp 117ndash136 2005
[3] Y Liu ldquoA general iterative method for equilibrium problemsand strict pseudo-contractions in Hilbert spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 10 pp4852ndash4861 2009
[4] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[5] D-N Qu and C-Z Cheng ldquoA strong convergence theoremon solving common solutions for generalized equilibriumproblems and fixed-point problems in Banach spacerdquo FixedPoint Theory and Applications vol 2011 article 17 2011
[6] R D Chen X L Shen and S J Cui ldquoWeak and strongconvergence theorems for equilibrium problems and countablestrict pseudocontractions mappings in Hilbert spacerdquo Journalof Inequalities and Applications vol 2010 Article ID 474813 11pages 2010
[7] C Jaiboon and P Kumam ldquoStrong convergence for general-ized equilibrium problems fixed point problems and relaxedcocoercive variational inequalitiesrdquo Journal of Inequalities andApplications vol 2010 Article ID 728028 43 pages 2010
[8] K Nakajo and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and nonexpansive semigroupsrdquoJournal of Mathematical Analysis and Applications vol 279 no2 pp 372ndash379 2003
[9] N Buong and N D Duong ldquoWeak convergence theorem for anequilibrium problem and a nonexpansive semigroup in Hilbertspacesrdquo InternationalMathematical Forum vol 5 no 53ndash56 pp2775ndash2786 2010
[10] Y Shehu ldquoIterative method for fixed point problem variationalinequality and generalized mixed equilibrium problems withapplicationsrdquo Journal of Global Optimization vol 52 no 1 pp57ndash77 2012
[11] H Zegeye and N Shahzad ldquoApproximation of the commonminimum-norm fixed point of a finite family of asymptoticallynonexpansive mappingsrdquo Fixed Point Theory and Applicationsvol 2013 article 1 2013
[12] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[13] T Shimizu andW Takahashi ldquoStrong convergence to commonfixed points of families of nonexpansive mappingsrdquo Journal ofMathematical Analysis and Applications vol 211 no 1 pp 71ndash83 1997
[14] T Suzuki ldquoStrong convergence theorems for infinite families ofnonexpansive mappings in general Banach spacesrdquo Fixed PointTheory and Applications vol 2005 Article ID 685918 2005
[15] H-K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
[16] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator Theory in Hilbert Spaces Springer NewYork NY USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
theorem for an equilibrium problem and a nonexpansivesemigroup in Hilbert spaces
1199090isin 119867
119906119896isin 119862 119891 (119906
119896 119910) +
1
119903119896
⟨119910 minus 119906119896 119906119896minus 119909119896⟩ ge 0 forall119910 isin 119862
119909119896+1
= 120583119896119909119896+ (1 minus 120583
119896)1
119905119896
int
119905119896
0
119879 (119904) 119906119896119889119904
(2)
In 2012 Shehu [10] studied iterative methods for fixedpoint problem variational inequality and generalized mixedequilibrium problem and introduced a new algorithm whichdoes not involve the CQ algorithm and viscosity approxima-tion method However we discover that there is a slight flawin the proof of Theorem 31 in [10]
Motivated by Nakajo and Takahashi [8] Buong andDuong [9] and especially Shehu [10] andZegeye and Shahzad[11] we present a new iterative method for finding a commonelement of the set of solutions of an equilibrium problemand the set of all common fixed points of a nonexpansivesemigroup and prove the strong convergence theorem inHilbert spaces Our result extends the recent result of [11] Inthe last part of the paper we perfect and simplify the proof ofTheorem 31 in [10]
2 Preliminaries
Throughout this paper let119867be a realHilbert spacewith innerproduct ⟨sdot sdot⟩ and norm sdot and let 119862 be a nonempty closedconvex subset of 119867 We write 119909
119899rarr 119909 to indicate that the
sequence 119909119899 converges strongly to 119909 Similarly 119909
119899 119909 will
symbolize weak convergence It is well known that119867 satisfiesOpialrsquos condition that is for any sequence 119909
119899 sub 119867 with
119909119899 119909 we have
lim inf119899rarrinfin
1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817 lt lim inf119899rarrinfin
1003817100381710038171003817119909119899 minus 1199101003817100381710038171003817 forall119910 = 119909 (3)
For any 119909 isin 119867 there exists a unique point 119875119862119909 isin 119862 such that
1003817100381710038171003817119909 minus 1198751198621199091003817100381710038171003817 le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119910 isin 119862 (4)
119875119862is called the metric projection of119867 onto 119862 We know that
119875119862is a nonexpansive mapping of119867 onto 119862 and 119875
119862satisfies
⟨119909 minus 119910 119875119862119909 minus 119875119862119910⟩ ge
1003817100381710038171003817119875119862119909 minus 1198751198621199101003817100381710038171003817
2 forall119909 119910 isin 119867 (5)
For 119909 isin 119867 and 119911 isin 119862 we have
119911 = 119875119862119909 lArrrArr ⟨119909 minus 119911 119910 minus 119911⟩ le 0 for every119910 isin 119862 (6)
The following lemmas will be used in the proof of ourresults
Lemma 1 (see [1]) Let 119862 be a nonempty closed convex subsetof119867 and let119891 be a bifunction from119862times119862 toR satisfying (A1)ndash(A4) If 119903 gt 0 and 119909 isin 119867 then there exists 119911 isin 119862 such that
119891 (119911 119910) +1
119903⟨119910 minus 119911 119911 minus 119909⟩ ge 0 forall119910 isin 119862 (7)
Lemma2 (see [2]) For 119903 gt 0 define amapping 119879119903 119867 rarr 2
119862
as follows
119879119903 (119909) = 119911 isin 119862 119891 (119911 119910) +
1
119903⟨119910 minus 119911 119911 minus 119909⟩ ge 0 forall119910 isin 119862
(8)
Then the following hold
(i) 119879119903is single valued
(ii) 119879119903is firmly nonexpansive that is for any 119909 119910 isin 119867
⟨119909 minus 119910 119879119903119909 minus 119879119903119910⟩ ge 119879
119903119909 minus 1198791199031199102
(iii) 119865(119879119903) = 119864119875(119891)
(iv) 119864119875(119891) is closed and convex
Lemma 3 (see [12]) Suppose that (A1)ndash(A4) hold If 119909 119910 isin 119867and 1199031 1199032gt 0 then
100381710038171003817100381710038171198791199032
119910 minus 1198791199031
11990910038171003817100381710038171003817le1003817100381710038171003817119910 minus 119909
1003817100381710038171003817 +
10038161003816100381610038161199032 minus 11990311003816100381610038161003816
1199032
100381710038171003817100381710038171198791199032
119910 minus 11991010038171003817100381710038171003817 (9)
Lemma 4 (see [13]) Let 119862 be a nonempty bounded closedsubset of 119867 and let 119879(119904) 0 le 119904 lt infin be a nonexpansivesemigroup on 119862 Then for every ℎ ge 0
lim119905rarr+infin
sup119909isin119862
10038171003817100381710038171003817100381710038171003817
1
119905int
119905
0
119879 (119904) 119909 119889119904 minus 119879 (ℎ)1
119905int
119905
0
119879 (119904) 119909 119889119904
10038171003817100381710038171003817100381710038171003817
= 0 (10)
Lemma 5 (see [14]) Let 119909119899 and 119910
119899 be bounded sequences
in a Banach space 119883 and let 120573119899 be a sequence in [0 1]
with 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1 Suppose
that 119909119899+1
= (1 minus 120573119899)119909119899+ 120573119899119910119899for all integers 119899 ge 1
and lim sup119899rarrinfin
(119910119899+1
minus 119910119899 minus 119909
119899+1minus 119909119899) le 0 Then
lim119899rarrinfin
119910119899minus 119909119899 = 0
Lemma 6 (see [15]) Let 119886119899 be a sequence of nonnegative real
numbers satisfying 119886119899+1
le (1 minus 120572119899)119886119899+ 120572119899120573119899 where
(i) 120572119899 sub (0 1) suminfin
119899=1120572119899= infin
(ii) lim sup119899rarrinfin
120573119899le 0
Then lim119899rarrinfin
119886119899= 0
3 Strong Convergence Theorems
In this section we introduce a new iterative method forfinding a common element of the set of solutions of anequilibriumproblemand the set of all commonfixedpoints ofa nonexpansive semigroup and prove the strong convergencetheorem in Hilbert spaces
Theorem 7 Let 119862 be a nonempty closed convex subset of areal Hilbert space 119867 and let 119891 be a bifunction from 119862 times 119862 toR satisfying (A1)ndash(A4) Suppose that S = 119879(119905) 0 le 119905 lt infin
Abstract and Applied Analysis 3
is a nonexpansive semigroup on 119862 such that 119865(S) cap119864119875(119891) = 0For 119906 isin 119867 let 119909
119899 119910119899 and 119911
119899 be generated by
1199091isin 119862 chosen arbitrarily
119910119899= 120572119899119906 + (1 minus 120572
119899) 119909119899
119911119899isin 119862 such that119891 (119911
119899 119910) +
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 0
forall119910 isin 119862
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
(11)
where the real sequences 120572119899 120573119899 in (0 1) and 119903
119899 sub (0infin)
satisfy the following conditions
(1) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(2) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(3) 0 lt 119888 le 119903119899lt infin lim
119899rarrinfin|119903119899+1
minus 119903119899| = 0
(4) 119905119899 sub (0infin) lim
119899rarrinfin119905119899= infin and lim
119899rarrinfin(|119905119899+1
minus
119905119899|119905119899+1) = 0
Then the sequence 119909119899 converges strongly to 119875
119865(S)cap119864119875(119891)119906
Proof Note that the set 119865(S) cap EP(119891) is closed and convexsince 119865(S) and EP(119891) are closed and convex For simplicitywe writeΩ = 119865(S) cap EP(119891)
From Lemmas 1 and 2 we have 119911119899= 119879119903119899
119910119899 and for any
119901 isin Ω
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817 =
10038171003817100381710038171003817119879119903119899
119910119899minus 119879119903119899
11990110038171003817100381710038171003817le1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817 (12)
Observe that1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817 le 1205721198991003817100381710038171003817119906 minus 119901
1003817100381710038171003817 + (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 (13)
It follows that
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817 =
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119901
10038171003817100381710038171003817100381710038171003817
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
+ 120573119899
1
119905119899
int
119905119899
0
1003817100381710038171003817119879 (119904) 119911119899 minus 119879 (119904) 1199011003817100381710038171003817 119889119904
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 + 1205731198991003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
+ 120573119899[120572119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817 + (1 minus 120572119899)
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817]
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 + 1205721198991205731198991003817100381710038171003817119906 minus 119901
1003817100381710038171003817
le max 1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 1003817100381710038171003817119906 minus 119901
1003817100381710038171003817
(14)
From a simple inductive process one has1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817 le max 10038171003817100381710038171199091 minus 1199011003817100381710038171003817 1003817100381710038171003817119906 minus 119901
1003817100381710038171003817 (15)
which yields that 119909119899 is bounded So are 119910
119899 and 119911
119899
Set 120590119899= (1119905
119899) int119905119899
0119879(119904)119911119899119889119904 For any 119901 isin Ω we have
1003817100381710038171003817120590119899+1 minus 1205901198991003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899+1119889119904 minus
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899+1119889119904 minus
1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899119889119904
+1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899119889119904 minus
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
(119879 (119904) 119911119899+1
minus 119879 (119904) 119911119899) 119889119904
+(1
119905119899+1
minus1
119905119899
)int
119905119899
0
119879 (119904) 119911119899119889119904 +1
119905119899+1
int
119905119899+1
119905119899
119879 (119904) 119911119899119889119904
100381710038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
(119879 (119904) 119911119899+1
minus 119879 (119904) 119911119899) 119889119904
+ (1
119905119899+1
minus1
119905119899
)int
119905119899
0
(119879 (119904) 119911119899minus 119879 (119904) 119901) 119889119904
+(1
119905119899+1
minus1
119905119899
)int
119905119899
0
119879 (119904) 119901 119889119904 +1
119905119899+1
int
119905119899+1
119905119899
119879 (119904) 119911119899119889119904
100381710038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
(119879 (119904) 119911119899+1
minus 119879 (119904) 119911119899) 119889119904
+ (1
119905119899+1
minus1
119905119899
)int
119905119899
0
(119879 (119904) 119911119899minus 119879 (119904) 119901) 119889119904
+1
119905119899+1
int
119905119899+1
119905119899
(119879 (119904) 119911119899 minus 119879 (119904) 119901) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le1003817100381710038171003817119911119899+1 minus 119911119899
1003817100381710038171003817 +21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
(16)
It follows from Lemma 3 that
1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817 le
1003817100381710038171003817119910119899+1 minus 1199101198991003817100381710038171003817 +
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119903119899+1
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817
(17)
Hence
1003817100381710038171003817120590119899+1 minus 1205901198991003817100381710038171003817
le1003817100381710038171003817119910119899+1 minus 119910119899
1003817100381710038171003817 +
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119903119899+1
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817
+21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
le1003817100381710038171003817120572119899+1119906 + (1 minus 120572119899+1) 119909119899+1 minus 120572119899119906 minus (1 minus 120572119899) 119909119899
1003817100381710038171003817
4 Abstract and Applied Analysis
+
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119903119899+1
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817 +
21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
le1003817100381710038171003817119909119899+1 minus 119909119899
1003817100381710038171003817 + 120572119899+1 (119906 +1003817100381710038171003817119909119899+1
1003817100381710038171003817) + 120572119899 (119906 +1003817100381710038171003817119909119899
1003817100381710038171003817)
+
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119888
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817 +
21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
(18)
This implies that
lim sup119899rarrinfin
(1003817100381710038171003817120590119899+1 minus 120590119899
1003817100381710038171003817 minus1003817100381710038171003817119909119899+1 minus 119909119899
1003817100381710038171003817) le 0 (19)
It follows from Lemma 5 that lim119899rarrinfin
120590119899minus 119909119899 = 0 Thus
lim119899rarrinfin
1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 = lim119899rarrinfin
120573119899
1003817100381710038171003817120590119899 minus 1199091198991003817100381710038171003817 = 0 (20)
For any 119901 isin Ω we have1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2=10038171003817100381710038171003817119879119903119899
119910119899minus 119879119903119899
11990110038171003817100381710038171003817
2
le ⟨119910119899minus 119901 119911
119899minus 119901⟩
=1
2[1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2+1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
(21)
Thus1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2le1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2 (22)
From the convexity of sdot 2 it follows that1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2
=
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) (119909119899minus 119901) + 120573
119899(1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119901)
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2
+ 120573119899
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
(119879 (119904) 119911119899 minus 119879 (119904) 119901) 119889119904
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[1003817100381710038171003817120572119899119906 + (1 minus 120572119899) 119909119899 minus 119901
1003817100381710038171003817
2
minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2
+120572119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
(23)
Hence
120573119899
1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2+ 120572119899120573119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2
le1003817100381710038171003817119909119899 minus 119909119899+1
1003817100381710038171003817 (1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817)
+ 120572119899120573119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2
(24)
Since lim119899rarrinfin
119909119899+1
minus 119909119899 = 0 and lim
119899rarrinfin120572119899= 0 one has
lim119899rarrinfin
1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (25)
Observe thatlim119899rarrinfin
1003817100381710038171003817119910119899 minus 1199091198991003817100381710038171003817 = lim119899rarrinfin
120572119899
1003817100381710038171003817119906 minus 1199091198991003817100381710038171003817 = 0 (26)
As 119911119899minus 119909119899 le 119911
119899minus 119910119899 + 119910
119899minus 119909119899 the following equality
holdslim119899rarrinfin
1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (27)
Now we show thatlim119899rarrinfin
1003817100381710038171003817119879 (119904) 119911119899 minus 1199111198991003817100381710038171003817 = 0 forall0 le 119904 lt infin (28)
In fact we have1003817100381710038171003817119879 (119904) 119911119899 minus 119911119899
1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
119879 (119904) 119911119899 minus 119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
+ 119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
+1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le
10038171003817100381710038171003817100381710038171003817
119879 (119904) 119911119899minus 119879 (119904)
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le 2
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
(29)
Notice that10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119909
119899
10038171003817100381710038171003817100381710038171003817
+1003817100381710038171003817119909119899 minus 119911119899
1003817100381710038171003817
=1
120573119899
1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 +
1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(30)
For any 119901 isin Ω let 119866 = 119909 isin 119862 119909 minus 119901 le max1199091minus 119901 119906 minus
119901 It is easy to see that119866 is a bounded closed convex subsetand 119879(119904)119866 is a subset of 119866 Since
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817 =
10038171003817100381710038171003817119879119903119899
119910119899minus 119901
10038171003817100381710038171003817
le (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 + 1205721198991003817100381710038171003817119906 minus 119901
1003817100381710038171003817
le max 10038171003817100381710038171199091 minus 1199011003817100381710038171003817 1003817100381710038171003817119906 minus 119901
1003817100381710038171003817
(31)
Abstract and Applied Analysis 5
the sequence 119911119899 is contained in 119866 It follows from Lemma 4
that
lim119899rarrinfin
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119879 (119904)
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
= 0 (32)
From (29) (30) and (32) the expression (28) is obtainedNext we prove that
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ le 0 (33)
where 119909 = 119875Ω119906 In order to show this inequality we can
choose a subsequence 119909119899119895
of 119909119899 such that
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ = lim
119895rarrinfin
⟨119906 minus 119909 119909119899119895
minus 119909⟩ (34)
Due to the boundedness of 119909119899119895
there exists a subsequence119909119899119895119894
of 119909119899119895
such that 119909119899119895119894
120596 Without loss of generalitywe assume that 119909
119899119895
120596 From (27) we see that 119911119899119895
120596Since 119911
119899119895
sub 119862 and 119862 is closed and convex we get 120596 isin 119862We first show that 120596 isin EP(119891) By 119911
119899= 119879119903119899
119910119899 we have
119891 (119911119899 119910) +
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 0 forall119910 isin 119862 (35)
It follows from the monotonicity of 119891 that
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 119891 (119910 119911
119899) forall119910 isin 119862 (36)
Replacing 119899 by 119899119895 we obtain
⟨119910 minus 119911119899119895
119911119899119895
minus 119910119899119895
119903119899119895
⟩ ge 119891(119910 119911119899119895
) forall119910 isin 119862 (37)
From (25) (27) and (A4) we have
119891 (119910 120596) le 0 forall119910 isin 119862 (38)
For 0 lt 119905 le 1 119910 isin 119862 set 119910119905= 119905119910 + (1 minus 119905)120596 We have 119910
119905isin 119862
and 119891(119910119905 120596) le 0 Hence
0 = 119891 (119910119905 119910119905) le 119905119891 (119910
119905 119910) + (1 minus 119905) 119891 (119910
119905 120596) le 119905119891 (119910
119905 119910)
(39)
Dividing by 119905 we see that
119891 (119910119905 119910) ge 0 (40)
Letting 119905 darr 0 and from (A3) we get
119891 (120596 119910) ge 0 forall119910 isin 119862 (41)
That is 120596 isin EP(119891)Second we prove that 120596 isin 119865(S) Note that the equality
(27) implies that 119911119899119895
120596 Suppose for contradiction that 120596 notin119865(S) that is
there exists 1199040gt 0 such that 119879 (119904
0) 120596 = 120596 (42)
Then from Opialrsquos condition and (28) we obtain
lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
= lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 119879 (1199040) 119911119899119895
+ 119879 (1199040) 119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
= lim inf119895rarrinfin
100381710038171003817100381710038171003817119879 (1199040) 119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
le lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 120596100381710038171003817100381710038171003817
(43)
which is a contradictionTherefore 120596 isin 119865(S) Consequentlyone gets 120596 isin Ω
From (34) and the property of metric projection we have
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ = lim
119895rarrinfin
⟨119906 minus 119909 119909119899119895
minus 119909⟩
= ⟨119906 minus 119909 120596 minus 119909⟩ le 0
(44)
The inequality (33) arrivesFinally we show that 119909
119899rarr 119909 From (11) we have
1003817100381710038171003817119909119899+1 minus 1199091003817100381710038171003817
2
=
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119909
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119911119899 minus 1199091003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1199091003817100381710038171003817
2
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨119906 minus 119909 119909
119899minus 119909⟩ + 120572
119899119906 minus 1199092]
(45)
It follows from (33) and Lemma 6 that 119909119899 converges strongly
to 119909
Remark 8 Let119867 = R and 119862 = [0 1] Setting 119891(119909 119910) = 1199102 minus1199092 we see that 119891(119909 119910) satisfies (A1)ndash(A4) For 0 le 119905 lt +infin
let
119879 (119905) 119909 =119909
1 + 119905119909 forall119909 isin [0 1] (46)
Thus it follows thatS = 119879(119905) 0 le 119905 lt infin is a nonexpansivesemigroup such that 119865(S) cap EP(119891) = 0 If we take
120572119899=
1
119899 + 1 1205731= 1205732=1
2 120573119899=1
2minus1
119899 forall119899 ge 3
119903119899equiv 119888 gt 0 119905
119899= 119899
(47)
then all assumptions and conditions in Theorem 7 aresatisfied
Remark 9 Taking 119906 = 0 inTheorem 7 we obtain the iterativemethod for minimum-norm solution of an equilibriumproblem and a nonexpansive semigroup
6 Abstract and Applied Analysis
As a direct consequence of Theorem 7 we obtain thefollowing corollary
Corollary 10 Let 119862 be a nonempty closed convex subset of areal Hilbert space119867 and assume that S = 119879(119905) 0 le 119905 lt infin
is a nonexpansive semigroup on 119862 such that 119865(S) = 0 Let 120572119899
and 120573119899 be real sequences in (0 1) and let 119909
119899 and 119911
119899 be
generated by
1199091isin 119862 chosen arbitrarily
119911119899= 119875119862[(1 minus 120572
119899) 119909119899]
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
(48)
Suppose that the following conditions are satisfied
(1) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(2) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(3) 119905119899 sub (0infin) lim
119899rarrinfin119905119899= infin and lim
119899rarrinfin(|119905119899+1
minus
119905119899|119905119899+1) = 0
Then the sequence 119909119899 converges strongly to 119875
119865(S)0
Proof Letting 119891(119909 119910) = 0 for all 119909 119910 isin 119862 119903119899= 1 and 119906 = 0
in Theorem 7 we get the result
Remark 11 Corollary 10 extends the recent results of Zegeyeand Shahzad [11 Corollaries 32 and 33] from finite family ofnonexpansive mappings to a nonexpansive semigroup
4 A Note on Shehursquos Paper lsquolsquoIterative Methodfor Fixed Point Problem VariationalInequality and Generalized MixedEquilibrium Problems with Applicationsrsquorsquo
In 2012 Shehu [10] studied iterative methods for fixedpoint problem variational inequality and generalized mixedequilibrium problem and gave an interesting convergencetheorem However there is a slight flaw in the proof of themain result (Theorem 31 in [10])
Shehu obtained the following result (for more details see[10])
Theorem 12 (see [10]) Let 119870 be a closed convex subset ofa real Hilbert space 119867 let 119865 be a bifunction from 119870 times 119870
satisfying (A1)ndash(A4) let 120593 119870 rarr R cup +infin be a properlower semicontinuous and convex function with assumption(B1) or (B2) let A be a 120583-Lipschitzian relaxed (120582 120574)-cocoercivemapping of 119870 into 119867 and let 120595 be an 120572-inverse stronglymonotone mapping of 119870 into 119867 Suppose that 119879 119870 rarr 119870
is a nonexpansive mapping of 119870 into itself such that Ω =
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875 = 0 Let 120572119899 and 120573
119899 be two real
sequences in (0 1) and 119903119899 119904119899 sub (0infin) Let 119909
119899 119910119899 and
119906119899 be generated by 119909
1isin 119870
119910119899= 119875119870[(1 minus 120572
119899) 119909119899]
119906119899= 119879(119865120593)
119903119899
(119910119899minus 119903119899120595119910119899)
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899119879119875119870(119906119899minus 119904119899119860119906119899)
(49)
Suppose that the following conditions are satisfied
(a) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(b) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(c) 0 lt 119886 le 119904119899le 119887 lt 2(120574minus120582120583
2)1205832 lim119899rarrinfin
|119904119899+1minus119904119899| = 0
(d) 0 lt 119888 le 119903119899le 119889 lt 2120572 lim
119899rarrinfin|119903119899+1
minus 119903119899| = 0
Then the sequence 119909119899 converges strongly to an element of
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875
This theorem is proved in [10] by the following steps
Step 1 The sequence 119909119899 is bounded
Step 2 The following equalities hold
lim119899rarrinfin
1003817100381710038171003817119860119906119899 minus 119860119909lowast1003817100381710038171003817 = 0 lim
119899rarrinfin
1003817100381710038171003817120595119910119899 minus 120595119909lowast1003817100381710038171003817 = 0
forall119909lowastisin Ω
(50)
Step 3 If 120596 is a weak limit of 119909119899119895
which is a subsequence of119909119899 then 120596 isin Ω
Step 4 The sequence 119909119899 converges strongly to 120596
In Step 4 in order to show that the sequence 119909119899 con-
verges strongly to 120596 the author shows the inequalitylim sup
119899rarrinfin⟨minus120596 119909
119899minus 120596⟩ le 0 by defining a mapping 120601
119867 rarr 119877 as follows 120601(119909) = 120583119899119909119899minus 1199092 where 120583 is a Banach
limit It is proved that the set 119870lowast = 119909 isin 119867 120601(119909) =
min119906isin119867
120601(119906) = 0 and119870lowastcap119865(119879) = 0 An element of119870lowastcap119865(119879)is taken arbitrarily and is denoted by120596 Of course the element120596 is not necessarily the weak sequential cluster point of 119909
119899
However in Step 3 the symbol 120596 stands for the weak limit of119909119899119895
which is a subsequence of 119909119899 In the sequel the author
obtains
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2le (1 minus 120573
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2 (51)
The meaning of the element 120596 in (51) is ambiguous It isdifficult to ensure consistency
Now we perfect and simplify the proof of Step 4 Accord-ing to the equality in Step 2 lim
119899rarrinfin119860119906119899minus119860119909lowast = 0 for all
119909lowastisin Ω we see that the set 119860119909lowast 119909lowast isin Ω contains only one
element Since 119860 is a relaxed (120582 120574)-cocoercive mapping of 119870into119867 that is there exist 120582 120574 gt 0 such that
⟨119860119909 minus 119860119910 119909 minus 119910⟩ ge minus1205821003817100381710038171003817119860119909 minus 119860119910
1003817100381710038171003817
2+ 120574
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
2
forall119909 119910 isin 119870
(52)
Abstract and Applied Analysis 7
it follows that themapping119860 is one-to-oneTherefore the setΩ is a singleton By Step 3 the sequence 119909
119899 possesses only
oneweak sequential cluster point It follows fromLemma238in [16] that 119909
119899 converges weakly to 120596 and so
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817(1 minus 120572119899) 119909119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+2120572119899(1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
2
1198991205962]
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
1198991205962]
(53)
Since 119909119899 converges weakly to 120596 it follows from Lemma 22
in [10] or Lemma 6 in this paper that 119909119899 converges strongly
to 120596 isin Ω
Conflict of Interests
The authors declare that there is no conflict of interests
Acknowledgment
The authors would like to thank referees and editors for theirvaluable comments and suggestions
References
[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1-4 pp 123ndash145 1994
[2] P L Combettes and S A Hirstoaga ldquoEquilibrium program-ming in Hilbert spacesrdquo Journal of Nonlinear and ConvexAnalysis vol 6 no 1 pp 117ndash136 2005
[3] Y Liu ldquoA general iterative method for equilibrium problemsand strict pseudo-contractions in Hilbert spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 10 pp4852ndash4861 2009
[4] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[5] D-N Qu and C-Z Cheng ldquoA strong convergence theoremon solving common solutions for generalized equilibriumproblems and fixed-point problems in Banach spacerdquo FixedPoint Theory and Applications vol 2011 article 17 2011
[6] R D Chen X L Shen and S J Cui ldquoWeak and strongconvergence theorems for equilibrium problems and countablestrict pseudocontractions mappings in Hilbert spacerdquo Journalof Inequalities and Applications vol 2010 Article ID 474813 11pages 2010
[7] C Jaiboon and P Kumam ldquoStrong convergence for general-ized equilibrium problems fixed point problems and relaxedcocoercive variational inequalitiesrdquo Journal of Inequalities andApplications vol 2010 Article ID 728028 43 pages 2010
[8] K Nakajo and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and nonexpansive semigroupsrdquoJournal of Mathematical Analysis and Applications vol 279 no2 pp 372ndash379 2003
[9] N Buong and N D Duong ldquoWeak convergence theorem for anequilibrium problem and a nonexpansive semigroup in Hilbertspacesrdquo InternationalMathematical Forum vol 5 no 53ndash56 pp2775ndash2786 2010
[10] Y Shehu ldquoIterative method for fixed point problem variationalinequality and generalized mixed equilibrium problems withapplicationsrdquo Journal of Global Optimization vol 52 no 1 pp57ndash77 2012
[11] H Zegeye and N Shahzad ldquoApproximation of the commonminimum-norm fixed point of a finite family of asymptoticallynonexpansive mappingsrdquo Fixed Point Theory and Applicationsvol 2013 article 1 2013
[12] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[13] T Shimizu andW Takahashi ldquoStrong convergence to commonfixed points of families of nonexpansive mappingsrdquo Journal ofMathematical Analysis and Applications vol 211 no 1 pp 71ndash83 1997
[14] T Suzuki ldquoStrong convergence theorems for infinite families ofnonexpansive mappings in general Banach spacesrdquo Fixed PointTheory and Applications vol 2005 Article ID 685918 2005
[15] H-K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
[16] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator Theory in Hilbert Spaces Springer NewYork NY USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
is a nonexpansive semigroup on 119862 such that 119865(S) cap119864119875(119891) = 0For 119906 isin 119867 let 119909
119899 119910119899 and 119911
119899 be generated by
1199091isin 119862 chosen arbitrarily
119910119899= 120572119899119906 + (1 minus 120572
119899) 119909119899
119911119899isin 119862 such that119891 (119911
119899 119910) +
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 0
forall119910 isin 119862
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
(11)
where the real sequences 120572119899 120573119899 in (0 1) and 119903
119899 sub (0infin)
satisfy the following conditions
(1) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(2) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(3) 0 lt 119888 le 119903119899lt infin lim
119899rarrinfin|119903119899+1
minus 119903119899| = 0
(4) 119905119899 sub (0infin) lim
119899rarrinfin119905119899= infin and lim
119899rarrinfin(|119905119899+1
minus
119905119899|119905119899+1) = 0
Then the sequence 119909119899 converges strongly to 119875
119865(S)cap119864119875(119891)119906
Proof Note that the set 119865(S) cap EP(119891) is closed and convexsince 119865(S) and EP(119891) are closed and convex For simplicitywe writeΩ = 119865(S) cap EP(119891)
From Lemmas 1 and 2 we have 119911119899= 119879119903119899
119910119899 and for any
119901 isin Ω
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817 =
10038171003817100381710038171003817119879119903119899
119910119899minus 119879119903119899
11990110038171003817100381710038171003817le1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817 (12)
Observe that1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817 le 1205721198991003817100381710038171003817119906 minus 119901
1003817100381710038171003817 + (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 (13)
It follows that
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817 =
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119901
10038171003817100381710038171003817100381710038171003817
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
+ 120573119899
1
119905119899
int
119905119899
0
1003817100381710038171003817119879 (119904) 119911119899 minus 119879 (119904) 1199011003817100381710038171003817 119889119904
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 + 1205731198991003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
+ 120573119899[120572119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817 + (1 minus 120572119899)
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817]
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 + 1205721198991205731198991003817100381710038171003817119906 minus 119901
1003817100381710038171003817
le max 1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 1003817100381710038171003817119906 minus 119901
1003817100381710038171003817
(14)
From a simple inductive process one has1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817 le max 10038171003817100381710038171199091 minus 1199011003817100381710038171003817 1003817100381710038171003817119906 minus 119901
1003817100381710038171003817 (15)
which yields that 119909119899 is bounded So are 119910
119899 and 119911
119899
Set 120590119899= (1119905
119899) int119905119899
0119879(119904)119911119899119889119904 For any 119901 isin Ω we have
1003817100381710038171003817120590119899+1 minus 1205901198991003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899+1119889119904 minus
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899+1119889119904 minus
1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899119889119904
+1
119905119899+1
int
119905119899+1
0
119879 (119904) 119911119899119889119904 minus
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
(119879 (119904) 119911119899+1
minus 119879 (119904) 119911119899) 119889119904
+(1
119905119899+1
minus1
119905119899
)int
119905119899
0
119879 (119904) 119911119899119889119904 +1
119905119899+1
int
119905119899+1
119905119899
119879 (119904) 119911119899119889119904
100381710038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
(119879 (119904) 119911119899+1
minus 119879 (119904) 119911119899) 119889119904
+ (1
119905119899+1
minus1
119905119899
)int
119905119899
0
(119879 (119904) 119911119899minus 119879 (119904) 119901) 119889119904
+(1
119905119899+1
minus1
119905119899
)int
119905119899
0
119879 (119904) 119901 119889119904 +1
119905119899+1
int
119905119899+1
119905119899
119879 (119904) 119911119899119889119904
100381710038171003817100381710038171003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
1
119905119899+1
int
119905119899+1
0
(119879 (119904) 119911119899+1
minus 119879 (119904) 119911119899) 119889119904
+ (1
119905119899+1
minus1
119905119899
)int
119905119899
0
(119879 (119904) 119911119899minus 119879 (119904) 119901) 119889119904
+1
119905119899+1
int
119905119899+1
119905119899
(119879 (119904) 119911119899 minus 119879 (119904) 119901) 119889119904
100381710038171003817100381710038171003817100381710038171003817
le1003817100381710038171003817119911119899+1 minus 119911119899
1003817100381710038171003817 +21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
(16)
It follows from Lemma 3 that
1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817 le
1003817100381710038171003817119910119899+1 minus 1199101198991003817100381710038171003817 +
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119903119899+1
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817
(17)
Hence
1003817100381710038171003817120590119899+1 minus 1205901198991003817100381710038171003817
le1003817100381710038171003817119910119899+1 minus 119910119899
1003817100381710038171003817 +
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119903119899+1
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817
+21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
le1003817100381710038171003817120572119899+1119906 + (1 minus 120572119899+1) 119909119899+1 minus 120572119899119906 minus (1 minus 120572119899) 119909119899
1003817100381710038171003817
4 Abstract and Applied Analysis
+
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119903119899+1
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817 +
21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
le1003817100381710038171003817119909119899+1 minus 119909119899
1003817100381710038171003817 + 120572119899+1 (119906 +1003817100381710038171003817119909119899+1
1003817100381710038171003817) + 120572119899 (119906 +1003817100381710038171003817119909119899
1003817100381710038171003817)
+
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119888
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817 +
21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
(18)
This implies that
lim sup119899rarrinfin
(1003817100381710038171003817120590119899+1 minus 120590119899
1003817100381710038171003817 minus1003817100381710038171003817119909119899+1 minus 119909119899
1003817100381710038171003817) le 0 (19)
It follows from Lemma 5 that lim119899rarrinfin
120590119899minus 119909119899 = 0 Thus
lim119899rarrinfin
1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 = lim119899rarrinfin
120573119899
1003817100381710038171003817120590119899 minus 1199091198991003817100381710038171003817 = 0 (20)
For any 119901 isin Ω we have1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2=10038171003817100381710038171003817119879119903119899
119910119899minus 119879119903119899
11990110038171003817100381710038171003817
2
le ⟨119910119899minus 119901 119911
119899minus 119901⟩
=1
2[1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2+1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
(21)
Thus1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2le1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2 (22)
From the convexity of sdot 2 it follows that1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2
=
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) (119909119899minus 119901) + 120573
119899(1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119901)
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2
+ 120573119899
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
(119879 (119904) 119911119899 minus 119879 (119904) 119901) 119889119904
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[1003817100381710038171003817120572119899119906 + (1 minus 120572119899) 119909119899 minus 119901
1003817100381710038171003817
2
minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2
+120572119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
(23)
Hence
120573119899
1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2+ 120572119899120573119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2
le1003817100381710038171003817119909119899 minus 119909119899+1
1003817100381710038171003817 (1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817)
+ 120572119899120573119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2
(24)
Since lim119899rarrinfin
119909119899+1
minus 119909119899 = 0 and lim
119899rarrinfin120572119899= 0 one has
lim119899rarrinfin
1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (25)
Observe thatlim119899rarrinfin
1003817100381710038171003817119910119899 minus 1199091198991003817100381710038171003817 = lim119899rarrinfin
120572119899
1003817100381710038171003817119906 minus 1199091198991003817100381710038171003817 = 0 (26)
As 119911119899minus 119909119899 le 119911
119899minus 119910119899 + 119910
119899minus 119909119899 the following equality
holdslim119899rarrinfin
1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (27)
Now we show thatlim119899rarrinfin
1003817100381710038171003817119879 (119904) 119911119899 minus 1199111198991003817100381710038171003817 = 0 forall0 le 119904 lt infin (28)
In fact we have1003817100381710038171003817119879 (119904) 119911119899 minus 119911119899
1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
119879 (119904) 119911119899 minus 119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
+ 119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
+1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le
10038171003817100381710038171003817100381710038171003817
119879 (119904) 119911119899minus 119879 (119904)
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le 2
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
(29)
Notice that10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119909
119899
10038171003817100381710038171003817100381710038171003817
+1003817100381710038171003817119909119899 minus 119911119899
1003817100381710038171003817
=1
120573119899
1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 +
1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(30)
For any 119901 isin Ω let 119866 = 119909 isin 119862 119909 minus 119901 le max1199091minus 119901 119906 minus
119901 It is easy to see that119866 is a bounded closed convex subsetand 119879(119904)119866 is a subset of 119866 Since
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817 =
10038171003817100381710038171003817119879119903119899
119910119899minus 119901
10038171003817100381710038171003817
le (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 + 1205721198991003817100381710038171003817119906 minus 119901
1003817100381710038171003817
le max 10038171003817100381710038171199091 minus 1199011003817100381710038171003817 1003817100381710038171003817119906 minus 119901
1003817100381710038171003817
(31)
Abstract and Applied Analysis 5
the sequence 119911119899 is contained in 119866 It follows from Lemma 4
that
lim119899rarrinfin
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119879 (119904)
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
= 0 (32)
From (29) (30) and (32) the expression (28) is obtainedNext we prove that
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ le 0 (33)
where 119909 = 119875Ω119906 In order to show this inequality we can
choose a subsequence 119909119899119895
of 119909119899 such that
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ = lim
119895rarrinfin
⟨119906 minus 119909 119909119899119895
minus 119909⟩ (34)
Due to the boundedness of 119909119899119895
there exists a subsequence119909119899119895119894
of 119909119899119895
such that 119909119899119895119894
120596 Without loss of generalitywe assume that 119909
119899119895
120596 From (27) we see that 119911119899119895
120596Since 119911
119899119895
sub 119862 and 119862 is closed and convex we get 120596 isin 119862We first show that 120596 isin EP(119891) By 119911
119899= 119879119903119899
119910119899 we have
119891 (119911119899 119910) +
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 0 forall119910 isin 119862 (35)
It follows from the monotonicity of 119891 that
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 119891 (119910 119911
119899) forall119910 isin 119862 (36)
Replacing 119899 by 119899119895 we obtain
⟨119910 minus 119911119899119895
119911119899119895
minus 119910119899119895
119903119899119895
⟩ ge 119891(119910 119911119899119895
) forall119910 isin 119862 (37)
From (25) (27) and (A4) we have
119891 (119910 120596) le 0 forall119910 isin 119862 (38)
For 0 lt 119905 le 1 119910 isin 119862 set 119910119905= 119905119910 + (1 minus 119905)120596 We have 119910
119905isin 119862
and 119891(119910119905 120596) le 0 Hence
0 = 119891 (119910119905 119910119905) le 119905119891 (119910
119905 119910) + (1 minus 119905) 119891 (119910
119905 120596) le 119905119891 (119910
119905 119910)
(39)
Dividing by 119905 we see that
119891 (119910119905 119910) ge 0 (40)
Letting 119905 darr 0 and from (A3) we get
119891 (120596 119910) ge 0 forall119910 isin 119862 (41)
That is 120596 isin EP(119891)Second we prove that 120596 isin 119865(S) Note that the equality
(27) implies that 119911119899119895
120596 Suppose for contradiction that 120596 notin119865(S) that is
there exists 1199040gt 0 such that 119879 (119904
0) 120596 = 120596 (42)
Then from Opialrsquos condition and (28) we obtain
lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
= lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 119879 (1199040) 119911119899119895
+ 119879 (1199040) 119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
= lim inf119895rarrinfin
100381710038171003817100381710038171003817119879 (1199040) 119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
le lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 120596100381710038171003817100381710038171003817
(43)
which is a contradictionTherefore 120596 isin 119865(S) Consequentlyone gets 120596 isin Ω
From (34) and the property of metric projection we have
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ = lim
119895rarrinfin
⟨119906 minus 119909 119909119899119895
minus 119909⟩
= ⟨119906 minus 119909 120596 minus 119909⟩ le 0
(44)
The inequality (33) arrivesFinally we show that 119909
119899rarr 119909 From (11) we have
1003817100381710038171003817119909119899+1 minus 1199091003817100381710038171003817
2
=
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119909
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119911119899 minus 1199091003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1199091003817100381710038171003817
2
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨119906 minus 119909 119909
119899minus 119909⟩ + 120572
119899119906 minus 1199092]
(45)
It follows from (33) and Lemma 6 that 119909119899 converges strongly
to 119909
Remark 8 Let119867 = R and 119862 = [0 1] Setting 119891(119909 119910) = 1199102 minus1199092 we see that 119891(119909 119910) satisfies (A1)ndash(A4) For 0 le 119905 lt +infin
let
119879 (119905) 119909 =119909
1 + 119905119909 forall119909 isin [0 1] (46)
Thus it follows thatS = 119879(119905) 0 le 119905 lt infin is a nonexpansivesemigroup such that 119865(S) cap EP(119891) = 0 If we take
120572119899=
1
119899 + 1 1205731= 1205732=1
2 120573119899=1
2minus1
119899 forall119899 ge 3
119903119899equiv 119888 gt 0 119905
119899= 119899
(47)
then all assumptions and conditions in Theorem 7 aresatisfied
Remark 9 Taking 119906 = 0 inTheorem 7 we obtain the iterativemethod for minimum-norm solution of an equilibriumproblem and a nonexpansive semigroup
6 Abstract and Applied Analysis
As a direct consequence of Theorem 7 we obtain thefollowing corollary
Corollary 10 Let 119862 be a nonempty closed convex subset of areal Hilbert space119867 and assume that S = 119879(119905) 0 le 119905 lt infin
is a nonexpansive semigroup on 119862 such that 119865(S) = 0 Let 120572119899
and 120573119899 be real sequences in (0 1) and let 119909
119899 and 119911
119899 be
generated by
1199091isin 119862 chosen arbitrarily
119911119899= 119875119862[(1 minus 120572
119899) 119909119899]
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
(48)
Suppose that the following conditions are satisfied
(1) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(2) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(3) 119905119899 sub (0infin) lim
119899rarrinfin119905119899= infin and lim
119899rarrinfin(|119905119899+1
minus
119905119899|119905119899+1) = 0
Then the sequence 119909119899 converges strongly to 119875
119865(S)0
Proof Letting 119891(119909 119910) = 0 for all 119909 119910 isin 119862 119903119899= 1 and 119906 = 0
in Theorem 7 we get the result
Remark 11 Corollary 10 extends the recent results of Zegeyeand Shahzad [11 Corollaries 32 and 33] from finite family ofnonexpansive mappings to a nonexpansive semigroup
4 A Note on Shehursquos Paper lsquolsquoIterative Methodfor Fixed Point Problem VariationalInequality and Generalized MixedEquilibrium Problems with Applicationsrsquorsquo
In 2012 Shehu [10] studied iterative methods for fixedpoint problem variational inequality and generalized mixedequilibrium problem and gave an interesting convergencetheorem However there is a slight flaw in the proof of themain result (Theorem 31 in [10])
Shehu obtained the following result (for more details see[10])
Theorem 12 (see [10]) Let 119870 be a closed convex subset ofa real Hilbert space 119867 let 119865 be a bifunction from 119870 times 119870
satisfying (A1)ndash(A4) let 120593 119870 rarr R cup +infin be a properlower semicontinuous and convex function with assumption(B1) or (B2) let A be a 120583-Lipschitzian relaxed (120582 120574)-cocoercivemapping of 119870 into 119867 and let 120595 be an 120572-inverse stronglymonotone mapping of 119870 into 119867 Suppose that 119879 119870 rarr 119870
is a nonexpansive mapping of 119870 into itself such that Ω =
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875 = 0 Let 120572119899 and 120573
119899 be two real
sequences in (0 1) and 119903119899 119904119899 sub (0infin) Let 119909
119899 119910119899 and
119906119899 be generated by 119909
1isin 119870
119910119899= 119875119870[(1 minus 120572
119899) 119909119899]
119906119899= 119879(119865120593)
119903119899
(119910119899minus 119903119899120595119910119899)
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899119879119875119870(119906119899minus 119904119899119860119906119899)
(49)
Suppose that the following conditions are satisfied
(a) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(b) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(c) 0 lt 119886 le 119904119899le 119887 lt 2(120574minus120582120583
2)1205832 lim119899rarrinfin
|119904119899+1minus119904119899| = 0
(d) 0 lt 119888 le 119903119899le 119889 lt 2120572 lim
119899rarrinfin|119903119899+1
minus 119903119899| = 0
Then the sequence 119909119899 converges strongly to an element of
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875
This theorem is proved in [10] by the following steps
Step 1 The sequence 119909119899 is bounded
Step 2 The following equalities hold
lim119899rarrinfin
1003817100381710038171003817119860119906119899 minus 119860119909lowast1003817100381710038171003817 = 0 lim
119899rarrinfin
1003817100381710038171003817120595119910119899 minus 120595119909lowast1003817100381710038171003817 = 0
forall119909lowastisin Ω
(50)
Step 3 If 120596 is a weak limit of 119909119899119895
which is a subsequence of119909119899 then 120596 isin Ω
Step 4 The sequence 119909119899 converges strongly to 120596
In Step 4 in order to show that the sequence 119909119899 con-
verges strongly to 120596 the author shows the inequalitylim sup
119899rarrinfin⟨minus120596 119909
119899minus 120596⟩ le 0 by defining a mapping 120601
119867 rarr 119877 as follows 120601(119909) = 120583119899119909119899minus 1199092 where 120583 is a Banach
limit It is proved that the set 119870lowast = 119909 isin 119867 120601(119909) =
min119906isin119867
120601(119906) = 0 and119870lowastcap119865(119879) = 0 An element of119870lowastcap119865(119879)is taken arbitrarily and is denoted by120596 Of course the element120596 is not necessarily the weak sequential cluster point of 119909
119899
However in Step 3 the symbol 120596 stands for the weak limit of119909119899119895
which is a subsequence of 119909119899 In the sequel the author
obtains
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2le (1 minus 120573
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2 (51)
The meaning of the element 120596 in (51) is ambiguous It isdifficult to ensure consistency
Now we perfect and simplify the proof of Step 4 Accord-ing to the equality in Step 2 lim
119899rarrinfin119860119906119899minus119860119909lowast = 0 for all
119909lowastisin Ω we see that the set 119860119909lowast 119909lowast isin Ω contains only one
element Since 119860 is a relaxed (120582 120574)-cocoercive mapping of 119870into119867 that is there exist 120582 120574 gt 0 such that
⟨119860119909 minus 119860119910 119909 minus 119910⟩ ge minus1205821003817100381710038171003817119860119909 minus 119860119910
1003817100381710038171003817
2+ 120574
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
2
forall119909 119910 isin 119870
(52)
Abstract and Applied Analysis 7
it follows that themapping119860 is one-to-oneTherefore the setΩ is a singleton By Step 3 the sequence 119909
119899 possesses only
oneweak sequential cluster point It follows fromLemma238in [16] that 119909
119899 converges weakly to 120596 and so
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817(1 minus 120572119899) 119909119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+2120572119899(1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
2
1198991205962]
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
1198991205962]
(53)
Since 119909119899 converges weakly to 120596 it follows from Lemma 22
in [10] or Lemma 6 in this paper that 119909119899 converges strongly
to 120596 isin Ω
Conflict of Interests
The authors declare that there is no conflict of interests
Acknowledgment
The authors would like to thank referees and editors for theirvaluable comments and suggestions
References
[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1-4 pp 123ndash145 1994
[2] P L Combettes and S A Hirstoaga ldquoEquilibrium program-ming in Hilbert spacesrdquo Journal of Nonlinear and ConvexAnalysis vol 6 no 1 pp 117ndash136 2005
[3] Y Liu ldquoA general iterative method for equilibrium problemsand strict pseudo-contractions in Hilbert spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 10 pp4852ndash4861 2009
[4] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[5] D-N Qu and C-Z Cheng ldquoA strong convergence theoremon solving common solutions for generalized equilibriumproblems and fixed-point problems in Banach spacerdquo FixedPoint Theory and Applications vol 2011 article 17 2011
[6] R D Chen X L Shen and S J Cui ldquoWeak and strongconvergence theorems for equilibrium problems and countablestrict pseudocontractions mappings in Hilbert spacerdquo Journalof Inequalities and Applications vol 2010 Article ID 474813 11pages 2010
[7] C Jaiboon and P Kumam ldquoStrong convergence for general-ized equilibrium problems fixed point problems and relaxedcocoercive variational inequalitiesrdquo Journal of Inequalities andApplications vol 2010 Article ID 728028 43 pages 2010
[8] K Nakajo and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and nonexpansive semigroupsrdquoJournal of Mathematical Analysis and Applications vol 279 no2 pp 372ndash379 2003
[9] N Buong and N D Duong ldquoWeak convergence theorem for anequilibrium problem and a nonexpansive semigroup in Hilbertspacesrdquo InternationalMathematical Forum vol 5 no 53ndash56 pp2775ndash2786 2010
[10] Y Shehu ldquoIterative method for fixed point problem variationalinequality and generalized mixed equilibrium problems withapplicationsrdquo Journal of Global Optimization vol 52 no 1 pp57ndash77 2012
[11] H Zegeye and N Shahzad ldquoApproximation of the commonminimum-norm fixed point of a finite family of asymptoticallynonexpansive mappingsrdquo Fixed Point Theory and Applicationsvol 2013 article 1 2013
[12] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[13] T Shimizu andW Takahashi ldquoStrong convergence to commonfixed points of families of nonexpansive mappingsrdquo Journal ofMathematical Analysis and Applications vol 211 no 1 pp 71ndash83 1997
[14] T Suzuki ldquoStrong convergence theorems for infinite families ofnonexpansive mappings in general Banach spacesrdquo Fixed PointTheory and Applications vol 2005 Article ID 685918 2005
[15] H-K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
[16] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator Theory in Hilbert Spaces Springer NewYork NY USA 2011
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Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
+
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119903119899+1
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817 +
21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
le1003817100381710038171003817119909119899+1 minus 119909119899
1003817100381710038171003817 + 120572119899+1 (119906 +1003817100381710038171003817119909119899+1
1003817100381710038171003817) + 120572119899 (119906 +1003817100381710038171003817119909119899
1003817100381710038171003817)
+
1003816100381610038161003816119903119899+1 minus 1199031198991003816100381610038161003816
119888
1003817100381710038171003817119911119899+1 minus 119910119899+11003817100381710038171003817 +
21003816100381610038161003816119905119899+1 minus 119905119899
1003816100381610038161003816
119905119899+1
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
(18)
This implies that
lim sup119899rarrinfin
(1003817100381710038171003817120590119899+1 minus 120590119899
1003817100381710038171003817 minus1003817100381710038171003817119909119899+1 minus 119909119899
1003817100381710038171003817) le 0 (19)
It follows from Lemma 5 that lim119899rarrinfin
120590119899minus 119909119899 = 0 Thus
lim119899rarrinfin
1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 = lim119899rarrinfin
120573119899
1003817100381710038171003817120590119899 minus 1199091198991003817100381710038171003817 = 0 (20)
For any 119901 isin Ω we have1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2=10038171003817100381710038171003817119879119903119899
119910119899minus 119879119903119899
11990110038171003817100381710038171003817
2
le ⟨119910119899minus 119901 119911
119899minus 119901⟩
=1
2[1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2+1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
(21)
Thus1003817100381710038171003817119911119899 minus 119901
1003817100381710038171003817
2le1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2 (22)
From the convexity of sdot 2 it follows that1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2
=
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) (119909119899minus 119901) + 120573
119899(1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119901)
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2
+ 120573119899
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
(119879 (119904) 119911119899 minus 119879 (119904) 119901) 119889119904
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[1003817100381710038171003817120572119899119906 + (1 minus 120572119899) 119909119899 minus 119901
1003817100381710038171003817
2
minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2
+120572119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2minus1003817100381710038171003817119911119899 minus 119910119899
1003817100381710038171003817
2]
(23)
Hence
120573119899
1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817
2le1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2minus1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2+ 120572119899120573119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2
le1003817100381710038171003817119909119899 minus 119909119899+1
1003817100381710038171003817 (1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817)
+ 120572119899120573119899
1003817100381710038171003817119906 minus 1199011003817100381710038171003817
2
(24)
Since lim119899rarrinfin
119909119899+1
minus 119909119899 = 0 and lim
119899rarrinfin120572119899= 0 one has
lim119899rarrinfin
1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (25)
Observe thatlim119899rarrinfin
1003817100381710038171003817119910119899 minus 1199091198991003817100381710038171003817 = lim119899rarrinfin
120572119899
1003817100381710038171003817119906 minus 1199091198991003817100381710038171003817 = 0 (26)
As 119911119899minus 119909119899 le 119911
119899minus 119910119899 + 119910
119899minus 119909119899 the following equality
holdslim119899rarrinfin
1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (27)
Now we show thatlim119899rarrinfin
1003817100381710038171003817119879 (119904) 119911119899 minus 1199111198991003817100381710038171003817 = 0 forall0 le 119904 lt infin (28)
In fact we have1003817100381710038171003817119879 (119904) 119911119899 minus 119911119899
1003817100381710038171003817
=
10038171003817100381710038171003817100381710038171003817
119879 (119904) 119911119899 minus 119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
+ 119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
+1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le
10038171003817100381710038171003817100381710038171003817
119879 (119904) 119911119899minus 119879 (119904)
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le 2
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
+
10038171003817100381710038171003817100381710038171003817
119879 (119904)1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
(29)
Notice that10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119911
119899
10038171003817100381710038171003817100381710038171003817
le
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119909
119899
10038171003817100381710038171003817100381710038171003817
+1003817100381710038171003817119909119899 minus 119911119899
1003817100381710038171003817
=1
120573119899
1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 +
1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(30)
For any 119901 isin Ω let 119866 = 119909 isin 119862 119909 minus 119901 le max1199091minus 119901 119906 minus
119901 It is easy to see that119866 is a bounded closed convex subsetand 119879(119904)119866 is a subset of 119866 Since
1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817 =
10038171003817100381710038171003817119879119903119899
119910119899minus 119901
10038171003817100381710038171003817
le (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817 + 1205721198991003817100381710038171003817119906 minus 119901
1003817100381710038171003817
le max 10038171003817100381710038171199091 minus 1199011003817100381710038171003817 1003817100381710038171003817119906 minus 119901
1003817100381710038171003817
(31)
Abstract and Applied Analysis 5
the sequence 119911119899 is contained in 119866 It follows from Lemma 4
that
lim119899rarrinfin
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119879 (119904)
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
= 0 (32)
From (29) (30) and (32) the expression (28) is obtainedNext we prove that
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ le 0 (33)
where 119909 = 119875Ω119906 In order to show this inequality we can
choose a subsequence 119909119899119895
of 119909119899 such that
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ = lim
119895rarrinfin
⟨119906 minus 119909 119909119899119895
minus 119909⟩ (34)
Due to the boundedness of 119909119899119895
there exists a subsequence119909119899119895119894
of 119909119899119895
such that 119909119899119895119894
120596 Without loss of generalitywe assume that 119909
119899119895
120596 From (27) we see that 119911119899119895
120596Since 119911
119899119895
sub 119862 and 119862 is closed and convex we get 120596 isin 119862We first show that 120596 isin EP(119891) By 119911
119899= 119879119903119899
119910119899 we have
119891 (119911119899 119910) +
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 0 forall119910 isin 119862 (35)
It follows from the monotonicity of 119891 that
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 119891 (119910 119911
119899) forall119910 isin 119862 (36)
Replacing 119899 by 119899119895 we obtain
⟨119910 minus 119911119899119895
119911119899119895
minus 119910119899119895
119903119899119895
⟩ ge 119891(119910 119911119899119895
) forall119910 isin 119862 (37)
From (25) (27) and (A4) we have
119891 (119910 120596) le 0 forall119910 isin 119862 (38)
For 0 lt 119905 le 1 119910 isin 119862 set 119910119905= 119905119910 + (1 minus 119905)120596 We have 119910
119905isin 119862
and 119891(119910119905 120596) le 0 Hence
0 = 119891 (119910119905 119910119905) le 119905119891 (119910
119905 119910) + (1 minus 119905) 119891 (119910
119905 120596) le 119905119891 (119910
119905 119910)
(39)
Dividing by 119905 we see that
119891 (119910119905 119910) ge 0 (40)
Letting 119905 darr 0 and from (A3) we get
119891 (120596 119910) ge 0 forall119910 isin 119862 (41)
That is 120596 isin EP(119891)Second we prove that 120596 isin 119865(S) Note that the equality
(27) implies that 119911119899119895
120596 Suppose for contradiction that 120596 notin119865(S) that is
there exists 1199040gt 0 such that 119879 (119904
0) 120596 = 120596 (42)
Then from Opialrsquos condition and (28) we obtain
lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
= lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 119879 (1199040) 119911119899119895
+ 119879 (1199040) 119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
= lim inf119895rarrinfin
100381710038171003817100381710038171003817119879 (1199040) 119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
le lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 120596100381710038171003817100381710038171003817
(43)
which is a contradictionTherefore 120596 isin 119865(S) Consequentlyone gets 120596 isin Ω
From (34) and the property of metric projection we have
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ = lim
119895rarrinfin
⟨119906 minus 119909 119909119899119895
minus 119909⟩
= ⟨119906 minus 119909 120596 minus 119909⟩ le 0
(44)
The inequality (33) arrivesFinally we show that 119909
119899rarr 119909 From (11) we have
1003817100381710038171003817119909119899+1 minus 1199091003817100381710038171003817
2
=
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119909
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119911119899 minus 1199091003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1199091003817100381710038171003817
2
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨119906 minus 119909 119909
119899minus 119909⟩ + 120572
119899119906 minus 1199092]
(45)
It follows from (33) and Lemma 6 that 119909119899 converges strongly
to 119909
Remark 8 Let119867 = R and 119862 = [0 1] Setting 119891(119909 119910) = 1199102 minus1199092 we see that 119891(119909 119910) satisfies (A1)ndash(A4) For 0 le 119905 lt +infin
let
119879 (119905) 119909 =119909
1 + 119905119909 forall119909 isin [0 1] (46)
Thus it follows thatS = 119879(119905) 0 le 119905 lt infin is a nonexpansivesemigroup such that 119865(S) cap EP(119891) = 0 If we take
120572119899=
1
119899 + 1 1205731= 1205732=1
2 120573119899=1
2minus1
119899 forall119899 ge 3
119903119899equiv 119888 gt 0 119905
119899= 119899
(47)
then all assumptions and conditions in Theorem 7 aresatisfied
Remark 9 Taking 119906 = 0 inTheorem 7 we obtain the iterativemethod for minimum-norm solution of an equilibriumproblem and a nonexpansive semigroup
6 Abstract and Applied Analysis
As a direct consequence of Theorem 7 we obtain thefollowing corollary
Corollary 10 Let 119862 be a nonempty closed convex subset of areal Hilbert space119867 and assume that S = 119879(119905) 0 le 119905 lt infin
is a nonexpansive semigroup on 119862 such that 119865(S) = 0 Let 120572119899
and 120573119899 be real sequences in (0 1) and let 119909
119899 and 119911
119899 be
generated by
1199091isin 119862 chosen arbitrarily
119911119899= 119875119862[(1 minus 120572
119899) 119909119899]
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
(48)
Suppose that the following conditions are satisfied
(1) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(2) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(3) 119905119899 sub (0infin) lim
119899rarrinfin119905119899= infin and lim
119899rarrinfin(|119905119899+1
minus
119905119899|119905119899+1) = 0
Then the sequence 119909119899 converges strongly to 119875
119865(S)0
Proof Letting 119891(119909 119910) = 0 for all 119909 119910 isin 119862 119903119899= 1 and 119906 = 0
in Theorem 7 we get the result
Remark 11 Corollary 10 extends the recent results of Zegeyeand Shahzad [11 Corollaries 32 and 33] from finite family ofnonexpansive mappings to a nonexpansive semigroup
4 A Note on Shehursquos Paper lsquolsquoIterative Methodfor Fixed Point Problem VariationalInequality and Generalized MixedEquilibrium Problems with Applicationsrsquorsquo
In 2012 Shehu [10] studied iterative methods for fixedpoint problem variational inequality and generalized mixedequilibrium problem and gave an interesting convergencetheorem However there is a slight flaw in the proof of themain result (Theorem 31 in [10])
Shehu obtained the following result (for more details see[10])
Theorem 12 (see [10]) Let 119870 be a closed convex subset ofa real Hilbert space 119867 let 119865 be a bifunction from 119870 times 119870
satisfying (A1)ndash(A4) let 120593 119870 rarr R cup +infin be a properlower semicontinuous and convex function with assumption(B1) or (B2) let A be a 120583-Lipschitzian relaxed (120582 120574)-cocoercivemapping of 119870 into 119867 and let 120595 be an 120572-inverse stronglymonotone mapping of 119870 into 119867 Suppose that 119879 119870 rarr 119870
is a nonexpansive mapping of 119870 into itself such that Ω =
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875 = 0 Let 120572119899 and 120573
119899 be two real
sequences in (0 1) and 119903119899 119904119899 sub (0infin) Let 119909
119899 119910119899 and
119906119899 be generated by 119909
1isin 119870
119910119899= 119875119870[(1 minus 120572
119899) 119909119899]
119906119899= 119879(119865120593)
119903119899
(119910119899minus 119903119899120595119910119899)
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899119879119875119870(119906119899minus 119904119899119860119906119899)
(49)
Suppose that the following conditions are satisfied
(a) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(b) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(c) 0 lt 119886 le 119904119899le 119887 lt 2(120574minus120582120583
2)1205832 lim119899rarrinfin
|119904119899+1minus119904119899| = 0
(d) 0 lt 119888 le 119903119899le 119889 lt 2120572 lim
119899rarrinfin|119903119899+1
minus 119903119899| = 0
Then the sequence 119909119899 converges strongly to an element of
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875
This theorem is proved in [10] by the following steps
Step 1 The sequence 119909119899 is bounded
Step 2 The following equalities hold
lim119899rarrinfin
1003817100381710038171003817119860119906119899 minus 119860119909lowast1003817100381710038171003817 = 0 lim
119899rarrinfin
1003817100381710038171003817120595119910119899 minus 120595119909lowast1003817100381710038171003817 = 0
forall119909lowastisin Ω
(50)
Step 3 If 120596 is a weak limit of 119909119899119895
which is a subsequence of119909119899 then 120596 isin Ω
Step 4 The sequence 119909119899 converges strongly to 120596
In Step 4 in order to show that the sequence 119909119899 con-
verges strongly to 120596 the author shows the inequalitylim sup
119899rarrinfin⟨minus120596 119909
119899minus 120596⟩ le 0 by defining a mapping 120601
119867 rarr 119877 as follows 120601(119909) = 120583119899119909119899minus 1199092 where 120583 is a Banach
limit It is proved that the set 119870lowast = 119909 isin 119867 120601(119909) =
min119906isin119867
120601(119906) = 0 and119870lowastcap119865(119879) = 0 An element of119870lowastcap119865(119879)is taken arbitrarily and is denoted by120596 Of course the element120596 is not necessarily the weak sequential cluster point of 119909
119899
However in Step 3 the symbol 120596 stands for the weak limit of119909119899119895
which is a subsequence of 119909119899 In the sequel the author
obtains
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2le (1 minus 120573
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2 (51)
The meaning of the element 120596 in (51) is ambiguous It isdifficult to ensure consistency
Now we perfect and simplify the proof of Step 4 Accord-ing to the equality in Step 2 lim
119899rarrinfin119860119906119899minus119860119909lowast = 0 for all
119909lowastisin Ω we see that the set 119860119909lowast 119909lowast isin Ω contains only one
element Since 119860 is a relaxed (120582 120574)-cocoercive mapping of 119870into119867 that is there exist 120582 120574 gt 0 such that
⟨119860119909 minus 119860119910 119909 minus 119910⟩ ge minus1205821003817100381710038171003817119860119909 minus 119860119910
1003817100381710038171003817
2+ 120574
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
2
forall119909 119910 isin 119870
(52)
Abstract and Applied Analysis 7
it follows that themapping119860 is one-to-oneTherefore the setΩ is a singleton By Step 3 the sequence 119909
119899 possesses only
oneweak sequential cluster point It follows fromLemma238in [16] that 119909
119899 converges weakly to 120596 and so
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817(1 minus 120572119899) 119909119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+2120572119899(1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
2
1198991205962]
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
1198991205962]
(53)
Since 119909119899 converges weakly to 120596 it follows from Lemma 22
in [10] or Lemma 6 in this paper that 119909119899 converges strongly
to 120596 isin Ω
Conflict of Interests
The authors declare that there is no conflict of interests
Acknowledgment
The authors would like to thank referees and editors for theirvaluable comments and suggestions
References
[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1-4 pp 123ndash145 1994
[2] P L Combettes and S A Hirstoaga ldquoEquilibrium program-ming in Hilbert spacesrdquo Journal of Nonlinear and ConvexAnalysis vol 6 no 1 pp 117ndash136 2005
[3] Y Liu ldquoA general iterative method for equilibrium problemsand strict pseudo-contractions in Hilbert spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 10 pp4852ndash4861 2009
[4] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[5] D-N Qu and C-Z Cheng ldquoA strong convergence theoremon solving common solutions for generalized equilibriumproblems and fixed-point problems in Banach spacerdquo FixedPoint Theory and Applications vol 2011 article 17 2011
[6] R D Chen X L Shen and S J Cui ldquoWeak and strongconvergence theorems for equilibrium problems and countablestrict pseudocontractions mappings in Hilbert spacerdquo Journalof Inequalities and Applications vol 2010 Article ID 474813 11pages 2010
[7] C Jaiboon and P Kumam ldquoStrong convergence for general-ized equilibrium problems fixed point problems and relaxedcocoercive variational inequalitiesrdquo Journal of Inequalities andApplications vol 2010 Article ID 728028 43 pages 2010
[8] K Nakajo and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and nonexpansive semigroupsrdquoJournal of Mathematical Analysis and Applications vol 279 no2 pp 372ndash379 2003
[9] N Buong and N D Duong ldquoWeak convergence theorem for anequilibrium problem and a nonexpansive semigroup in Hilbertspacesrdquo InternationalMathematical Forum vol 5 no 53ndash56 pp2775ndash2786 2010
[10] Y Shehu ldquoIterative method for fixed point problem variationalinequality and generalized mixed equilibrium problems withapplicationsrdquo Journal of Global Optimization vol 52 no 1 pp57ndash77 2012
[11] H Zegeye and N Shahzad ldquoApproximation of the commonminimum-norm fixed point of a finite family of asymptoticallynonexpansive mappingsrdquo Fixed Point Theory and Applicationsvol 2013 article 1 2013
[12] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[13] T Shimizu andW Takahashi ldquoStrong convergence to commonfixed points of families of nonexpansive mappingsrdquo Journal ofMathematical Analysis and Applications vol 211 no 1 pp 71ndash83 1997
[14] T Suzuki ldquoStrong convergence theorems for infinite families ofnonexpansive mappings in general Banach spacesrdquo Fixed PointTheory and Applications vol 2005 Article ID 685918 2005
[15] H-K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
[16] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator Theory in Hilbert Spaces Springer NewYork NY USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
Abstract and Applied Analysis 5
the sequence 119911119899 is contained in 119866 It follows from Lemma 4
that
lim119899rarrinfin
10038171003817100381710038171003817100381710038171003817
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119879 (119904)
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
10038171003817100381710038171003817100381710038171003817
= 0 (32)
From (29) (30) and (32) the expression (28) is obtainedNext we prove that
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ le 0 (33)
where 119909 = 119875Ω119906 In order to show this inequality we can
choose a subsequence 119909119899119895
of 119909119899 such that
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ = lim
119895rarrinfin
⟨119906 minus 119909 119909119899119895
minus 119909⟩ (34)
Due to the boundedness of 119909119899119895
there exists a subsequence119909119899119895119894
of 119909119899119895
such that 119909119899119895119894
120596 Without loss of generalitywe assume that 119909
119899119895
120596 From (27) we see that 119911119899119895
120596Since 119911
119899119895
sub 119862 and 119862 is closed and convex we get 120596 isin 119862We first show that 120596 isin EP(119891) By 119911
119899= 119879119903119899
119910119899 we have
119891 (119911119899 119910) +
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 0 forall119910 isin 119862 (35)
It follows from the monotonicity of 119891 that
1
119903119899
⟨119910 minus 119911119899 119911119899minus 119910119899⟩ ge 119891 (119910 119911
119899) forall119910 isin 119862 (36)
Replacing 119899 by 119899119895 we obtain
⟨119910 minus 119911119899119895
119911119899119895
minus 119910119899119895
119903119899119895
⟩ ge 119891(119910 119911119899119895
) forall119910 isin 119862 (37)
From (25) (27) and (A4) we have
119891 (119910 120596) le 0 forall119910 isin 119862 (38)
For 0 lt 119905 le 1 119910 isin 119862 set 119910119905= 119905119910 + (1 minus 119905)120596 We have 119910
119905isin 119862
and 119891(119910119905 120596) le 0 Hence
0 = 119891 (119910119905 119910119905) le 119905119891 (119910
119905 119910) + (1 minus 119905) 119891 (119910
119905 120596) le 119905119891 (119910
119905 119910)
(39)
Dividing by 119905 we see that
119891 (119910119905 119910) ge 0 (40)
Letting 119905 darr 0 and from (A3) we get
119891 (120596 119910) ge 0 forall119910 isin 119862 (41)
That is 120596 isin EP(119891)Second we prove that 120596 isin 119865(S) Note that the equality
(27) implies that 119911119899119895
120596 Suppose for contradiction that 120596 notin119865(S) that is
there exists 1199040gt 0 such that 119879 (119904
0) 120596 = 120596 (42)
Then from Opialrsquos condition and (28) we obtain
lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
= lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 119879 (1199040) 119911119899119895
+ 119879 (1199040) 119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
= lim inf119895rarrinfin
100381710038171003817100381710038171003817119879 (1199040) 119911119899119895
minus 119879 (1199040) 120596100381710038171003817100381710038171003817
le lim inf119895rarrinfin
100381710038171003817100381710038171003817119911119899119895
minus 120596100381710038171003817100381710038171003817
(43)
which is a contradictionTherefore 120596 isin 119865(S) Consequentlyone gets 120596 isin Ω
From (34) and the property of metric projection we have
lim sup119899rarrinfin
⟨119906 minus 119909 119909119899minus 119909⟩ = lim
119895rarrinfin
⟨119906 minus 119909 119909119899119895
minus 119909⟩
= ⟨119906 minus 119909 120596 minus 119909⟩ le 0
(44)
The inequality (33) arrivesFinally we show that 119909
119899rarr 119909 From (11) we have
1003817100381710038171003817119909119899+1 minus 1199091003817100381710038171003817
2
=
10038171003817100381710038171003817100381710038171003817
(1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904 minus 119909
10038171003817100381710038171003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119911119899 minus 1199091003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1199091003817100381710038171003817
2
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 119909
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨119906 minus 119909 119909
119899minus 119909⟩ + 120572
119899119906 minus 1199092]
(45)
It follows from (33) and Lemma 6 that 119909119899 converges strongly
to 119909
Remark 8 Let119867 = R and 119862 = [0 1] Setting 119891(119909 119910) = 1199102 minus1199092 we see that 119891(119909 119910) satisfies (A1)ndash(A4) For 0 le 119905 lt +infin
let
119879 (119905) 119909 =119909
1 + 119905119909 forall119909 isin [0 1] (46)
Thus it follows thatS = 119879(119905) 0 le 119905 lt infin is a nonexpansivesemigroup such that 119865(S) cap EP(119891) = 0 If we take
120572119899=
1
119899 + 1 1205731= 1205732=1
2 120573119899=1
2minus1
119899 forall119899 ge 3
119903119899equiv 119888 gt 0 119905
119899= 119899
(47)
then all assumptions and conditions in Theorem 7 aresatisfied
Remark 9 Taking 119906 = 0 inTheorem 7 we obtain the iterativemethod for minimum-norm solution of an equilibriumproblem and a nonexpansive semigroup
6 Abstract and Applied Analysis
As a direct consequence of Theorem 7 we obtain thefollowing corollary
Corollary 10 Let 119862 be a nonempty closed convex subset of areal Hilbert space119867 and assume that S = 119879(119905) 0 le 119905 lt infin
is a nonexpansive semigroup on 119862 such that 119865(S) = 0 Let 120572119899
and 120573119899 be real sequences in (0 1) and let 119909
119899 and 119911
119899 be
generated by
1199091isin 119862 chosen arbitrarily
119911119899= 119875119862[(1 minus 120572
119899) 119909119899]
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
(48)
Suppose that the following conditions are satisfied
(1) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(2) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(3) 119905119899 sub (0infin) lim
119899rarrinfin119905119899= infin and lim
119899rarrinfin(|119905119899+1
minus
119905119899|119905119899+1) = 0
Then the sequence 119909119899 converges strongly to 119875
119865(S)0
Proof Letting 119891(119909 119910) = 0 for all 119909 119910 isin 119862 119903119899= 1 and 119906 = 0
in Theorem 7 we get the result
Remark 11 Corollary 10 extends the recent results of Zegeyeand Shahzad [11 Corollaries 32 and 33] from finite family ofnonexpansive mappings to a nonexpansive semigroup
4 A Note on Shehursquos Paper lsquolsquoIterative Methodfor Fixed Point Problem VariationalInequality and Generalized MixedEquilibrium Problems with Applicationsrsquorsquo
In 2012 Shehu [10] studied iterative methods for fixedpoint problem variational inequality and generalized mixedequilibrium problem and gave an interesting convergencetheorem However there is a slight flaw in the proof of themain result (Theorem 31 in [10])
Shehu obtained the following result (for more details see[10])
Theorem 12 (see [10]) Let 119870 be a closed convex subset ofa real Hilbert space 119867 let 119865 be a bifunction from 119870 times 119870
satisfying (A1)ndash(A4) let 120593 119870 rarr R cup +infin be a properlower semicontinuous and convex function with assumption(B1) or (B2) let A be a 120583-Lipschitzian relaxed (120582 120574)-cocoercivemapping of 119870 into 119867 and let 120595 be an 120572-inverse stronglymonotone mapping of 119870 into 119867 Suppose that 119879 119870 rarr 119870
is a nonexpansive mapping of 119870 into itself such that Ω =
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875 = 0 Let 120572119899 and 120573
119899 be two real
sequences in (0 1) and 119903119899 119904119899 sub (0infin) Let 119909
119899 119910119899 and
119906119899 be generated by 119909
1isin 119870
119910119899= 119875119870[(1 minus 120572
119899) 119909119899]
119906119899= 119879(119865120593)
119903119899
(119910119899minus 119903119899120595119910119899)
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899119879119875119870(119906119899minus 119904119899119860119906119899)
(49)
Suppose that the following conditions are satisfied
(a) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(b) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(c) 0 lt 119886 le 119904119899le 119887 lt 2(120574minus120582120583
2)1205832 lim119899rarrinfin
|119904119899+1minus119904119899| = 0
(d) 0 lt 119888 le 119903119899le 119889 lt 2120572 lim
119899rarrinfin|119903119899+1
minus 119903119899| = 0
Then the sequence 119909119899 converges strongly to an element of
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875
This theorem is proved in [10] by the following steps
Step 1 The sequence 119909119899 is bounded
Step 2 The following equalities hold
lim119899rarrinfin
1003817100381710038171003817119860119906119899 minus 119860119909lowast1003817100381710038171003817 = 0 lim
119899rarrinfin
1003817100381710038171003817120595119910119899 minus 120595119909lowast1003817100381710038171003817 = 0
forall119909lowastisin Ω
(50)
Step 3 If 120596 is a weak limit of 119909119899119895
which is a subsequence of119909119899 then 120596 isin Ω
Step 4 The sequence 119909119899 converges strongly to 120596
In Step 4 in order to show that the sequence 119909119899 con-
verges strongly to 120596 the author shows the inequalitylim sup
119899rarrinfin⟨minus120596 119909
119899minus 120596⟩ le 0 by defining a mapping 120601
119867 rarr 119877 as follows 120601(119909) = 120583119899119909119899minus 1199092 where 120583 is a Banach
limit It is proved that the set 119870lowast = 119909 isin 119867 120601(119909) =
min119906isin119867
120601(119906) = 0 and119870lowastcap119865(119879) = 0 An element of119870lowastcap119865(119879)is taken arbitrarily and is denoted by120596 Of course the element120596 is not necessarily the weak sequential cluster point of 119909
119899
However in Step 3 the symbol 120596 stands for the weak limit of119909119899119895
which is a subsequence of 119909119899 In the sequel the author
obtains
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2le (1 minus 120573
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2 (51)
The meaning of the element 120596 in (51) is ambiguous It isdifficult to ensure consistency
Now we perfect and simplify the proof of Step 4 Accord-ing to the equality in Step 2 lim
119899rarrinfin119860119906119899minus119860119909lowast = 0 for all
119909lowastisin Ω we see that the set 119860119909lowast 119909lowast isin Ω contains only one
element Since 119860 is a relaxed (120582 120574)-cocoercive mapping of 119870into119867 that is there exist 120582 120574 gt 0 such that
⟨119860119909 minus 119860119910 119909 minus 119910⟩ ge minus1205821003817100381710038171003817119860119909 minus 119860119910
1003817100381710038171003817
2+ 120574
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
2
forall119909 119910 isin 119870
(52)
Abstract and Applied Analysis 7
it follows that themapping119860 is one-to-oneTherefore the setΩ is a singleton By Step 3 the sequence 119909
119899 possesses only
oneweak sequential cluster point It follows fromLemma238in [16] that 119909
119899 converges weakly to 120596 and so
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817(1 minus 120572119899) 119909119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+2120572119899(1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
2
1198991205962]
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
1198991205962]
(53)
Since 119909119899 converges weakly to 120596 it follows from Lemma 22
in [10] or Lemma 6 in this paper that 119909119899 converges strongly
to 120596 isin Ω
Conflict of Interests
The authors declare that there is no conflict of interests
Acknowledgment
The authors would like to thank referees and editors for theirvaluable comments and suggestions
References
[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1-4 pp 123ndash145 1994
[2] P L Combettes and S A Hirstoaga ldquoEquilibrium program-ming in Hilbert spacesrdquo Journal of Nonlinear and ConvexAnalysis vol 6 no 1 pp 117ndash136 2005
[3] Y Liu ldquoA general iterative method for equilibrium problemsand strict pseudo-contractions in Hilbert spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 10 pp4852ndash4861 2009
[4] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[5] D-N Qu and C-Z Cheng ldquoA strong convergence theoremon solving common solutions for generalized equilibriumproblems and fixed-point problems in Banach spacerdquo FixedPoint Theory and Applications vol 2011 article 17 2011
[6] R D Chen X L Shen and S J Cui ldquoWeak and strongconvergence theorems for equilibrium problems and countablestrict pseudocontractions mappings in Hilbert spacerdquo Journalof Inequalities and Applications vol 2010 Article ID 474813 11pages 2010
[7] C Jaiboon and P Kumam ldquoStrong convergence for general-ized equilibrium problems fixed point problems and relaxedcocoercive variational inequalitiesrdquo Journal of Inequalities andApplications vol 2010 Article ID 728028 43 pages 2010
[8] K Nakajo and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and nonexpansive semigroupsrdquoJournal of Mathematical Analysis and Applications vol 279 no2 pp 372ndash379 2003
[9] N Buong and N D Duong ldquoWeak convergence theorem for anequilibrium problem and a nonexpansive semigroup in Hilbertspacesrdquo InternationalMathematical Forum vol 5 no 53ndash56 pp2775ndash2786 2010
[10] Y Shehu ldquoIterative method for fixed point problem variationalinequality and generalized mixed equilibrium problems withapplicationsrdquo Journal of Global Optimization vol 52 no 1 pp57ndash77 2012
[11] H Zegeye and N Shahzad ldquoApproximation of the commonminimum-norm fixed point of a finite family of asymptoticallynonexpansive mappingsrdquo Fixed Point Theory and Applicationsvol 2013 article 1 2013
[12] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[13] T Shimizu andW Takahashi ldquoStrong convergence to commonfixed points of families of nonexpansive mappingsrdquo Journal ofMathematical Analysis and Applications vol 211 no 1 pp 71ndash83 1997
[14] T Suzuki ldquoStrong convergence theorems for infinite families ofnonexpansive mappings in general Banach spacesrdquo Fixed PointTheory and Applications vol 2005 Article ID 685918 2005
[15] H-K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
[16] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator Theory in Hilbert Spaces Springer NewYork NY USA 2011
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
6 Abstract and Applied Analysis
As a direct consequence of Theorem 7 we obtain thefollowing corollary
Corollary 10 Let 119862 be a nonempty closed convex subset of areal Hilbert space119867 and assume that S = 119879(119905) 0 le 119905 lt infin
is a nonexpansive semigroup on 119862 such that 119865(S) = 0 Let 120572119899
and 120573119899 be real sequences in (0 1) and let 119909
119899 and 119911
119899 be
generated by
1199091isin 119862 chosen arbitrarily
119911119899= 119875119862[(1 minus 120572
119899) 119909119899]
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899
1
119905119899
int
119905119899
0
119879 (119904) 119911119899119889119904
(48)
Suppose that the following conditions are satisfied
(1) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(2) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(3) 119905119899 sub (0infin) lim
119899rarrinfin119905119899= infin and lim
119899rarrinfin(|119905119899+1
minus
119905119899|119905119899+1) = 0
Then the sequence 119909119899 converges strongly to 119875
119865(S)0
Proof Letting 119891(119909 119910) = 0 for all 119909 119910 isin 119862 119903119899= 1 and 119906 = 0
in Theorem 7 we get the result
Remark 11 Corollary 10 extends the recent results of Zegeyeand Shahzad [11 Corollaries 32 and 33] from finite family ofnonexpansive mappings to a nonexpansive semigroup
4 A Note on Shehursquos Paper lsquolsquoIterative Methodfor Fixed Point Problem VariationalInequality and Generalized MixedEquilibrium Problems with Applicationsrsquorsquo
In 2012 Shehu [10] studied iterative methods for fixedpoint problem variational inequality and generalized mixedequilibrium problem and gave an interesting convergencetheorem However there is a slight flaw in the proof of themain result (Theorem 31 in [10])
Shehu obtained the following result (for more details see[10])
Theorem 12 (see [10]) Let 119870 be a closed convex subset ofa real Hilbert space 119867 let 119865 be a bifunction from 119870 times 119870
satisfying (A1)ndash(A4) let 120593 119870 rarr R cup +infin be a properlower semicontinuous and convex function with assumption(B1) or (B2) let A be a 120583-Lipschitzian relaxed (120582 120574)-cocoercivemapping of 119870 into 119867 and let 120595 be an 120572-inverse stronglymonotone mapping of 119870 into 119867 Suppose that 119879 119870 rarr 119870
is a nonexpansive mapping of 119870 into itself such that Ω =
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875 = 0 Let 120572119899 and 120573
119899 be two real
sequences in (0 1) and 119903119899 119904119899 sub (0infin) Let 119909
119899 119910119899 and
119906119899 be generated by 119909
1isin 119870
119910119899= 119875119870[(1 minus 120572
119899) 119909119899]
119906119899= 119879(119865120593)
119903119899
(119910119899minus 119903119899120595119910119899)
119909119899+1
= (1 minus 120573119899) 119909119899+ 120573119899119879119875119870(119906119899minus 119904119899119860119906119899)
(49)
Suppose that the following conditions are satisfied
(a) lim119899rarrinfin
120572119899= 0 and suminfin
119899=1120572119899= infin
(b) 0 lt lim inf119899rarrinfin
120573119899le lim sup
119899rarrinfin120573119899lt 1
(c) 0 lt 119886 le 119904119899le 119887 lt 2(120574minus120582120583
2)1205832 lim119899rarrinfin
|119904119899+1minus119904119899| = 0
(d) 0 lt 119888 le 119903119899le 119889 lt 2120572 lim
119899rarrinfin|119903119899+1
minus 119903119899| = 0
Then the sequence 119909119899 converges strongly to an element of
119865(119879) cap 119881119868(119870119860) cap 119866119872119864119875
This theorem is proved in [10] by the following steps
Step 1 The sequence 119909119899 is bounded
Step 2 The following equalities hold
lim119899rarrinfin
1003817100381710038171003817119860119906119899 minus 119860119909lowast1003817100381710038171003817 = 0 lim
119899rarrinfin
1003817100381710038171003817120595119910119899 minus 120595119909lowast1003817100381710038171003817 = 0
forall119909lowastisin Ω
(50)
Step 3 If 120596 is a weak limit of 119909119899119895
which is a subsequence of119909119899 then 120596 isin Ω
Step 4 The sequence 119909119899 converges strongly to 120596
In Step 4 in order to show that the sequence 119909119899 con-
verges strongly to 120596 the author shows the inequalitylim sup
119899rarrinfin⟨minus120596 119909
119899minus 120596⟩ le 0 by defining a mapping 120601
119867 rarr 119877 as follows 120601(119909) = 120583119899119909119899minus 1199092 where 120583 is a Banach
limit It is proved that the set 119870lowast = 119909 isin 119867 120601(119909) =
min119906isin119867
120601(119906) = 0 and119870lowastcap119865(119879) = 0 An element of119870lowastcap119865(119879)is taken arbitrarily and is denoted by120596 Of course the element120596 is not necessarily the weak sequential cluster point of 119909
119899
However in Step 3 the symbol 120596 stands for the weak limit of119909119899119895
which is a subsequence of 119909119899 In the sequel the author
obtains
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2le (1 minus 120573
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2 (51)
The meaning of the element 120596 in (51) is ambiguous It isdifficult to ensure consistency
Now we perfect and simplify the proof of Step 4 Accord-ing to the equality in Step 2 lim
119899rarrinfin119860119906119899minus119860119909lowast = 0 for all
119909lowastisin Ω we see that the set 119860119909lowast 119909lowast isin Ω contains only one
element Since 119860 is a relaxed (120582 120574)-cocoercive mapping of 119870into119867 that is there exist 120582 120574 gt 0 such that
⟨119860119909 minus 119860119910 119909 minus 119910⟩ ge minus1205821003817100381710038171003817119860119909 minus 119860119910
1003817100381710038171003817
2+ 120574
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
2
forall119909 119910 isin 119870
(52)
Abstract and Applied Analysis 7
it follows that themapping119860 is one-to-oneTherefore the setΩ is a singleton By Step 3 the sequence 119909
119899 possesses only
oneweak sequential cluster point It follows fromLemma238in [16] that 119909
119899 converges weakly to 120596 and so
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817(1 minus 120572119899) 119909119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+2120572119899(1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
2
1198991205962]
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
1198991205962]
(53)
Since 119909119899 converges weakly to 120596 it follows from Lemma 22
in [10] or Lemma 6 in this paper that 119909119899 converges strongly
to 120596 isin Ω
Conflict of Interests
The authors declare that there is no conflict of interests
Acknowledgment
The authors would like to thank referees and editors for theirvaluable comments and suggestions
References
[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1-4 pp 123ndash145 1994
[2] P L Combettes and S A Hirstoaga ldquoEquilibrium program-ming in Hilbert spacesrdquo Journal of Nonlinear and ConvexAnalysis vol 6 no 1 pp 117ndash136 2005
[3] Y Liu ldquoA general iterative method for equilibrium problemsand strict pseudo-contractions in Hilbert spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 10 pp4852ndash4861 2009
[4] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[5] D-N Qu and C-Z Cheng ldquoA strong convergence theoremon solving common solutions for generalized equilibriumproblems and fixed-point problems in Banach spacerdquo FixedPoint Theory and Applications vol 2011 article 17 2011
[6] R D Chen X L Shen and S J Cui ldquoWeak and strongconvergence theorems for equilibrium problems and countablestrict pseudocontractions mappings in Hilbert spacerdquo Journalof Inequalities and Applications vol 2010 Article ID 474813 11pages 2010
[7] C Jaiboon and P Kumam ldquoStrong convergence for general-ized equilibrium problems fixed point problems and relaxedcocoercive variational inequalitiesrdquo Journal of Inequalities andApplications vol 2010 Article ID 728028 43 pages 2010
[8] K Nakajo and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and nonexpansive semigroupsrdquoJournal of Mathematical Analysis and Applications vol 279 no2 pp 372ndash379 2003
[9] N Buong and N D Duong ldquoWeak convergence theorem for anequilibrium problem and a nonexpansive semigroup in Hilbertspacesrdquo InternationalMathematical Forum vol 5 no 53ndash56 pp2775ndash2786 2010
[10] Y Shehu ldquoIterative method for fixed point problem variationalinequality and generalized mixed equilibrium problems withapplicationsrdquo Journal of Global Optimization vol 52 no 1 pp57ndash77 2012
[11] H Zegeye and N Shahzad ldquoApproximation of the commonminimum-norm fixed point of a finite family of asymptoticallynonexpansive mappingsrdquo Fixed Point Theory and Applicationsvol 2013 article 1 2013
[12] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[13] T Shimizu andW Takahashi ldquoStrong convergence to commonfixed points of families of nonexpansive mappingsrdquo Journal ofMathematical Analysis and Applications vol 211 no 1 pp 71ndash83 1997
[14] T Suzuki ldquoStrong convergence theorems for infinite families ofnonexpansive mappings in general Banach spacesrdquo Fixed PointTheory and Applications vol 2005 Article ID 685918 2005
[15] H-K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
[16] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator Theory in Hilbert Spaces Springer NewYork NY USA 2011
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 7
it follows that themapping119860 is one-to-oneTherefore the setΩ is a singleton By Step 3 the sequence 119909
119899 possesses only
oneweak sequential cluster point It follows fromLemma238in [16] that 119909
119899 converges weakly to 120596 and so
1003817100381710038171003817119909119899+1 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817119910119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2+ 120573119899
1003817100381710038171003817(1 minus 120572119899) 119909119899 minus 1205961003817100381710038171003817
2
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120573119899[(1 minus 120572
119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+2120572119899(1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
2
1198991205962]
le (1 minus 120572119899120573119899)1003817100381710038171003817119909119899 minus 120596
1003817100381710038171003817
2
+ 120572119899120573119899[2 (1 minus 120572
119899) ⟨minus120596 119909
119899minus 120596⟩ + 120572
1198991205962]
(53)
Since 119909119899 converges weakly to 120596 it follows from Lemma 22
in [10] or Lemma 6 in this paper that 119909119899 converges strongly
to 120596 isin Ω
Conflict of Interests
The authors declare that there is no conflict of interests
Acknowledgment
The authors would like to thank referees and editors for theirvaluable comments and suggestions
References
[1] E Blum and W Oettli ldquoFrom optimization and variationalinequalities to equilibriumproblemsrdquoTheMathematics Studentvol 63 no 1-4 pp 123ndash145 1994
[2] P L Combettes and S A Hirstoaga ldquoEquilibrium program-ming in Hilbert spacesrdquo Journal of Nonlinear and ConvexAnalysis vol 6 no 1 pp 117ndash136 2005
[3] Y Liu ldquoA general iterative method for equilibrium problemsand strict pseudo-contractions in Hilbert spacesrdquo NonlinearAnalysis Theory Methods amp Applications vol 71 no 10 pp4852ndash4861 2009
[4] S Takahashi andWTakahashi ldquoViscosity approximationmeth-ods for equilibrium problems and fixed point problems inHilbert spacesrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 506ndash515 2007
[5] D-N Qu and C-Z Cheng ldquoA strong convergence theoremon solving common solutions for generalized equilibriumproblems and fixed-point problems in Banach spacerdquo FixedPoint Theory and Applications vol 2011 article 17 2011
[6] R D Chen X L Shen and S J Cui ldquoWeak and strongconvergence theorems for equilibrium problems and countablestrict pseudocontractions mappings in Hilbert spacerdquo Journalof Inequalities and Applications vol 2010 Article ID 474813 11pages 2010
[7] C Jaiboon and P Kumam ldquoStrong convergence for general-ized equilibrium problems fixed point problems and relaxedcocoercive variational inequalitiesrdquo Journal of Inequalities andApplications vol 2010 Article ID 728028 43 pages 2010
[8] K Nakajo and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and nonexpansive semigroupsrdquoJournal of Mathematical Analysis and Applications vol 279 no2 pp 372ndash379 2003
[9] N Buong and N D Duong ldquoWeak convergence theorem for anequilibrium problem and a nonexpansive semigroup in Hilbertspacesrdquo InternationalMathematical Forum vol 5 no 53ndash56 pp2775ndash2786 2010
[10] Y Shehu ldquoIterative method for fixed point problem variationalinequality and generalized mixed equilibrium problems withapplicationsrdquo Journal of Global Optimization vol 52 no 1 pp57ndash77 2012
[11] H Zegeye and N Shahzad ldquoApproximation of the commonminimum-norm fixed point of a finite family of asymptoticallynonexpansive mappingsrdquo Fixed Point Theory and Applicationsvol 2013 article 1 2013
[12] F Cianciaruso G Marino and L Muglia ldquoIterative methodsfor equilibrium and fixed point problems for nonexpansivesemigroups in Hilbert spacesrdquo Journal of Optimization Theoryand Applications vol 146 no 2 pp 491ndash509 2010
[13] T Shimizu andW Takahashi ldquoStrong convergence to commonfixed points of families of nonexpansive mappingsrdquo Journal ofMathematical Analysis and Applications vol 211 no 1 pp 71ndash83 1997
[14] T Suzuki ldquoStrong convergence theorems for infinite families ofnonexpansive mappings in general Banach spacesrdquo Fixed PointTheory and Applications vol 2005 Article ID 685918 2005
[15] H-K Xu ldquoAnother control condition in an iterative method fornonexpansive mappingsrdquo Bulletin of the Australian Mathemati-cal Society vol 65 no 1 pp 109ndash113 2002
[16] H H Bauschke and P L Combettes Convex Analysis andMonotone Operator Theory in Hilbert Spaces Springer NewYork NY USA 2011
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
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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