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Research ArticleBifurcation Problems for Generalized Beam Equations
Fosheng Wang
Department of Mathematics Sichuan University Chengdu 610064 China
Correspondence should be addressed to Fosheng Wang fosheng321163com
Received 4 October 2014 Accepted 4 December 2014 Published 22 December 2014
Academic Editor Ricardo Weder
Copyright copy 2014 Fosheng Wang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We investigate a class of bifurcation problems for generalized beam equations and prove that the one-parameter family of problemshave exactly two bifurcation points via a unified elementary approach The proof of the main results relies heavily on calculusfacts rather than such complicated arguments as Lyapunov-Schmidt reduction technique or Morse index theory from nonlinearfunctional analysis
1 Introduction and Main Results
In physics the vibration of an elastic beam with length 1 andone endpoint hinged at 119909 = 0 which is compressed at the freeedge (119909 = 1) by a force of intensity proportional to 120582 gt 0 isgoverned by the so-called beam equation
12059310158401015840+ 120582 sin120593 = 0 in (0 1) (1)
see [1] The beam maintains its shape when the ldquoforcerdquo 120582 issufficiently small but it will buckle once 120582 exceeds a certainvalue In mathematics the set of such values can be studiedby exploiting the homogeneous Neumann boundary valueproblem
12059310158401015840+ 120582 sin120593 = 0 in (0 1)
1205931015840
(0) = 1205931015840
(1) = 0
(2)
Before stating precisely the properties which we will explorein BVP (2) we embed this problem into a family of suchboundary value problems that is we introduce the family ofproblems
12059310158401015840+ 120582ℎ ∘ 120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(3)
where 119886 119887 isin R with 119886 lt 119887 120582 belongs to a certain nonemptysubset of R and 120593 = 120593(119909) is the unknown the function ℎ isin
119862infin(RR) satisfies the following there exists an 119897 gt 0 such
that
(H1) ℎ(119909 + 119897) = minusℎ(119909) for all 119909 isin R(H2) 119909
0isin R for which
ℎ1015840(1199090) gt 0 (4)
ℎ (1199090+ 119909) = minusℎ (119909
0minus 119909) gt 0 forall119909 isin [0 119897] (5)
119870 [0 119897] ni 119909 997891997888rarr radicint
1199090+119909
1199090
ℎ (119905) d119905
isin [0infin) is a concave function
(6)
Remark 1 We call the equation occurring in BVP (3) ldquogen-eralizedrdquo beam equation such equations are widely used todescribe various physical phenomena
Remark 2 It follows immediately from the hypothesis (5) thatℎ(1199090) = ℎ(119909
0+ 119897) = 0 and from (H1) that ℎ is 2119897-periodic
Remark 3 It is easy to see that the function ℎ(119909) = sin119909119909 isin R satisfies the hypotheses (H1)-(H2) with 119897 = 120587 119909
0= 0
Trivially BVP (3) admits the trivial solution 120593 = 0 for any120582 isin R Here we are focused on the bifurcation theory for BVP(3) The bifurcation points are determined by eigenvaluesassociated with the differential operator 12059310158401015840 + 120582ℎ ∘ 120593 At such
Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2014 Article ID 635731 6 pageshttpdxdoiorg1011552014635731
2 Advances in Mathematical Physics
points the number of solutions to (3) may change Howeververy little further work has been done to determine whetherthe number of solutions changes at these points In this paperwe give such a criterion for a class of nonlinear problems
Theorem 4 Let 119886 119887 120582 isin R with 119886 lt 119887 Assume (H1)-(H2)Then
plusmn1205872
((119887 minus 119886)2ℎ1015840(1199090))
(7)
are two bifurcation points for BVP (3) Besides (3) has non-constant solutions if and only if
1205872lt |120582| (119887 minus 119886)
2ℎ1015840(1199090) (8)
The proof of a bifurcation assertion of a nonlinear equa-tion often has as ingredients such topological arguments asKrasnoselskiirsquos and Rabinowitzrsquos theorems on bifurcationThese arguments usually have the assumption that the alge-braic multiplicity of the associated linear eigenvalue problemis odd see [1ndash3] and the references therein Since then severalauthors have also attempted to remove such oddness assump-tion see [1 2 4] In particular Ma and Wang [2] developedan elaborate algorithm to prove steady state bifurcationassertions concerning nonlinear equations this algorithmdoes not assume the oddness of the algebraic multiplicitySee [5ndash13] for more studies on bifurcation problems Ourapproach to prove Theorem 4 does not assume such paritycondition on the algebraic multiplicity
As a matter of fact BVP (2) is a special case of Sturm-Liouville problem or boundary value problems for ellipticpartial differential equations Therefore BVP (2) possibly indisguise has been studied extensively in the literature for theexistence of solutions satisfying certain prescribed propertiesfor qualitative properties of solutions and so on see [14ndash17]and the profound references cited therein
The remainder of this paper is organized as follows InSection 2 we introduce some nonlinear functional analysisand formulate the problem in a formal way and in Section 3we give the proof of Theorem 4
2 The Existing Bifurcation Results for BVP (2)
In this section we mainly give a brief review of the existingresults in the literature concerning bifurcation problems forBVP (2) which can be viewed as archetypes of bifurcationproblems for BVP (3) Indeed bifurcation problems for BVP(2) have been often provided as illustration examples to testthe proposed abstract bifurcation-problems-solving methodin the literature see [1 12 13 18]
In particular Ma and Wang [18] proposed an abstractmethod which generalizes slightly the previous one obtainedby Nirenberg [1] In presenting their method the authorsfixed two Banach spaces 119883 and 119884 for which 119883 embedscontinuously and densely into119884The abstract problemwhichthey were concerned with reads
119871120582119906 + 119866
120582(119906) = 0 (9)
where 119871120582 119883 rarr 119884 120582 isin R is a family of bounded linear
operators and 119866120582 119883 rarr 119884 is a family of continuous map-
pings They assumed the following
(H3) 119871120582is in the form 119871
120582= 119860 + 119861
120582with 119860 as a linear
topological isomorphism of 119883 onto 119884 and 119861120582as
compact linear operators hence the spectrum of 119871120582
consists of the exactly countably many eigenvalues120573119896(120582) (listed by algebraic multiplicities) of 119871
120582 there
exists 1205820for which
1205731(1205820) = 120573119895(1205820) 120573
119895(1205820) = 0 forall119895 gt 1
120573119895(1205820) lt 0 if 120582 lt 120582
0
1205731(1205820) = 0 if 120582 = 120582
0
120573119895(1205820) gt 0 if 120582 gt 120582
0
(10)
(H4) For any 120576 gt 0 there exists a 120575 gt 0 such that
1003817100381710038171003817119866120582 (119906)1003817100381710038171003817119884lt 120576 forall119906 isin 119883 with 119906
119883lt 120575 forall120582 isin R (11)
119866120582is analytic in the sense that
119866120582= 1198660
120582+
infin
sum
119899=1
119866119899
120582(119906 119906 119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119899 copies) (12)
119866119899
120582is a continuous symmetric 119899-form on119883The precise problem with which they are concerned is
whether there is a 1205820isin R given in such a way that if 119906
120582= 0
with 120582 in a neighborhood of 1205820is a collection of solutions to
BVP (9) then
119906120582997888rarr 0 in 119883 as 120582 997888rarr 120582
0 (13)
If there exists a 1205820which satisfies the above requirements
then 1205820is called a bifurcation point for nonlinear problem
(9) also problem (9) is said to bifurcate from 1205820
Concerning (9) they proved the followingAssume (H3)-(H4) Then 120582
0is a candidate bifurcation
point of the nonlinear problem (9)The proof of the above theorem provided in [18] utilizes
such complicated methods as Lyapunov-Schmidt reductionmethod Morse index theory and so forth
Ma and Wang [18] used the above theorem to obtainthe bifurcation results for BVP (2) Indeed they wrote firstly119883 = 119906 isin 119867
2(0 1) 119906
1015840(0) = 119906
1015840(1) 119884 = 119871
2(0 1) 119871
120582119906 =
(11988921198891199092)119906 minus 120582119906 and 119866
120582(119906) = 120582 sin 119906 minus 120582119906 thereby recasting
BVP (2) into one of the forms (9) and secondly they solvedthis new bifurcation problem for BVP (2) by utilizing theirabstract result
Here we are tempted to use the results obtained in Maand Wang [18] to solve the bifurcation problem for BVP (3)it is however obvious that the nonlinear reaction R ni 119906 rarr
ℎ(sin 119906) isin R precludes our application of such results In thenext section we will analyze the bifurcation problem for BVP(3) in an elementary way
Advances in Mathematical Physics 3
3 Proof of the Main Results
In this section we propose two lemmas and then proveTheorem 4 based on them Various calculus theorems areemployed in our proofs and the elementary equality
(119891minus1)1015840
(119891 (119905)) =1
1198911015840(119905)
(14)
is also used repeatedlyFor the sake of convenience we write
119867(119909) = 1198702
(119909) = int
1199090+119909
1199090
ℎ (119905) d119905 (0 le 119909 le 119897) (15)
The function 119867 is strictly increasing on [0 119897] 119867(0) = 0and1198671015840(119909) = ℎ(119909
0+ 119909)119867 is 2120587-periodic because
119867(119909 + 2119897) minus 119867 (119909) = int
119909+2119897
119909
ℎ (119905) d119905 = int119897
minus119897
ℎ (1199090+ 119905) d119905 = 0
(16)
due to Remark 2
Lemma 5 The function 119870 is strictly increasing and differen-tiable on [0 119897] with derivative
1198701015840
(119909) =
1198671015840(119909)
(2radic119867 (119909))
0 lt 119909 le 119897
radicℎ1015840(1199090)
2119909 = 0
(17)
Moreover
1198701015840(119870minus1(119910)) =
1
2119910 (119867minus1)1015840
(1199102)
(0 lt 119910 lt radic119867 (119897)) (18)
Proof Since 119870 = radic119867 and 119867 is positive on (0 119897] the value1198701015840(119909) with 119909 isin (0 119897] can be derived directly from (15) and
thus half of (17) is obtained Consequently
1198701015840
(0) = lim119909darr0
1198701015840
(119909) = lim119909darr0
1198671015840(119909)
2radic119867 (119909)
(19)
Noting that1198671015840 ge 0 on [0 119897] LrsquoHospital rule shows that
(1198701015840
(0))2
= lim119909darr0
(1198671015840(119909))2
4119867 (119909)= lim119909darr0
21198671015840(119909)119867
10158401015840(119909)
41198671015840(119909)
=ℎ1015840(1199090)
2
(20)
Thus the other half of (17) is obtained A simple compu-tation using (17) and (14) gives (18)
Lemma 6 Define
119901 (119905) = int
119905
0
1
radic119867 (119905) minus 119867 (119910)
d119910 (0 lt 119905 le 119897) (21)
For 0 lt 119905 lt 119897 the integral above converges and the function 119901is strictly increasing on (0 119897] Besides
lim119905darr0
119901 (119905) =120587
radic2ℎ1015840(1199090)
(22)
Proof Changing of variable 119910 = 119867minus1(119906119867(119905)) in the integralwe get
int
119905
0
1
radic119867 (119905) minus 119867 (119910)
d119910 = int1
0
radic119867(119905)
radic1 minus 119906
sdot (119867minus1)1015840
(119906 (119867 (119905))) d119906
(0 lt 119905 le 119897)
(23)
By the dominated convergence theorem and (18) the inte-grand of the right side above equals
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
(24)
and thus
119901 (119905) = int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
d119906 (25)
The function 1(1198701015840 ∘ 119870minus1) is strictly increasing on (0radic119867(119897)] because 119870minus1 is strictly increasing and 1198701015840 is strictlydecreasing by our assumption (6) Hence value (24) increasesas 119905 does Note that radic119906(119867(119905)) le radic119867(119905) for 119906 isin [0 1] If119905 isin (0 119897) then
119901 (119905) = int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
d119906
le int
1
0
1
2radic119906 (1 minus 119906)
sdot1
1198701015840(119870minus1(radic119867 (119905)))
d119906
=120587
21198701015840(119870minus1(radic119867 (119905)))
lt infin
(26)
Hence 119901 is real valued and (25) implies that 119901 is strictlyincreasing on (0 119897] Finally equalities (25) (17) and the dom-inated convergence theorem show that
lim119905darr0
119901 (119905)
= lim119905darr0
int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
d119906
= int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(0))
d119906
=120587
2radicℎ1015840(1199090)
(27)
The proof is complete
Proof of Theorem 4 Assume 120593 is a nonconstant solution of(3) The set 119909 isin [119886 119887] 1205931015840(119909) = 0 is open and nonempty in[119886 119887] and thus is a disjoint union of open intervals (provided[119886 119909) and (119909 119887] for 119909 isin (119886 119887) are viewed as ldquoopen intervalsrdquo
4 Advances in Mathematical Physics
in [119886 119887]) Let (1198860 1198870) be such an open interval Then 120593 is
nonconstant on [1198860 1198870] and 1205931015840(119886
0) = 120593
1015840(1198870) We will show
that 1198870minus 1198860gt 120587radic|120582|ℎ
1015840(1199090) which is equivalent to (8)
Since differentiable functions have intermediate valueproperty and 1205931015840(119909) = 0 on (119886
0 1198870) we may assume 1205931015840 gt 0 on
(1198860 1198870) without loss of generality So 120593 is strictly increasing
on [1198860 1198870] with inverse function 119892 = 120593
minus1 defined on[120593(1198860) 120593(1198870)] A simple computation using (14) and the chain
rule for differentiation shows that
119889 ((11198921015840(119910))2
)
119889119910
=
119889(((119892minus1)1015840
(119892 (119910)))
2
)
119889119910
= 2 (119892minus1)1015840
(119892 (119910)) sdot (119892minus1)10158401015840
(119892 (119910)) sdot 1198921015840(119910)
= 2 (119892minus1)10158401015840
(119892 (119910))
(28)
Condition (3) means
(119892minus1)10158401015840
(119892 (119910)) + 120582ℎ (119910) = 0 (120593 (1198860) le 119910 le 120593 (119887
0)) (29)
lim119910darr119892(120593(1198860))
1198921015840(119910) = lim
119910uarr119892(120593(1198870))
1198921015840(119910) = +infin (30)
Integrating both sides of (29) and using (28) we get that
1
2 (1198921015840(119910))2+ 120582119867 (119910) = 119862
1(120593 (1198860) lt 119910 lt 120593 (119887
0)) (31)
with constant1198621 We assume 120582 gt 0without loss of generality
Then
1198921015840(119910) =
1
radic2120582
sdot1
radic1198622minus 119867 (119910)
(120593 (1198860) lt 119910 lt 120593 (119887
0)) (32)
with 1198622= 1198621120582 Consequently
119867(119910) lt 1198622
(120593 (1198860) lt 119910 lt 120593 (119887
0)) (33)
As the properties of 119867 stated before the function 119910 997891rarr
119867(119910) is with period 2119897 decreases on [minus119897 0] and increases on[0 119897] Equalities (30) and (32) yield that
119867(119910) 997888rarr 1198622
(119910 darr 120593 (1198860) or119910 uarr 120593 (119887
0)) (34)
This together with (33) and the properties of 119867 shows thatthere exists 119905 isin (0 119897] and 119911 isin Z such that 119862
2= 119867(119909
0+ 119905)
120593(1198860) = 1199090+ 2119897119911 minus 119905 and 120593(119887
0) = 1199090+ 2119897119911 + 119905 Consequently
1198870minus 1198860= 119892 (120593 (119887
0)) minus 119892 (120593 (119886
0))
= 119892 (120593 (1198870) minus 2119897119911) minus 119892 (120593 (119886
0) minus 2119897119911)
= 119892 (1199090+ 119905) minus 119892 (119909
0minus 119905)
= int
119905
minus119905
1198921015840(1199090+ 119910) d119910
(35)
Since119867 is an even function it follows from (32) that
1198870minus 1198860=
1
radic2120582
int
119905
minus119905
1
radic119867 (119905) minus 119867 (119910)
d119910
= radic2
120582int
119905
0
1
radic119867 (119905) minus 119867 (119910)
d119910
= radic2
120582119901 (119905)
(36)
Note that 119901 is strictly increasing on (0 119897] (see Lemma 6)Hence
119887 minus 119886 ge 1198870minus 1198860= radic
2
120582119901 (119905) gt lim
119905darr0
radic2
120582119901 (119905) =
120587
radic120582ℎ1015840(1199090)
(37)
which implies (8) and thus half of Theorem 4 is provedFor the other half we assume without loss of generality
that 120582 gt 0 and assume (8) holds that is
119887 minus 119886 ge120587
radic120582ℎ1015840(1199090)
(38)
And we show that (2) has a nonconstant solutionFirst it follows from (27) that there exits 119905
0gt 0 such that
radic2
120582119901 (1199050) lt 119887 minus 119886 (39)
Let
1198871= 119886 + radic
2
120582119901 (1199050) lt 119887 (40)
Define a continuous function1198920on [minus119905
0 1199050] by119892(minus119905
0) = 119886
and
1198921015840
0(119910) =
1
radic2120582
sdot1
radic119867 (1199050) minus 119867 (119910)
(minus1199050lt 119910 lt 119905
0) (41)
Definition (21) yields
119892 (1199050) = 119886 + int
1199050
minus1199050
1198921015840
0(119905) d119905
= 119886 + 21
radic2120582
sdot int
1199050
0
1
radic119867(1199050) minus 119867 (119910)
d119905
= 119886 + radic2
120582119901 (1199050)
= 1198871
(42)
Besides
1
2 (1198921015840
0(119910))2+ 120582119867 (119910) = 120582119867 (119905
0) (minus119905
0lt 119910 lt 119905
0) (43)
Advances in Mathematical Physics 5
Differentiating both sides of (45) and using (28) we get
(119892minus1
0)10158401015840
(1198920(119910)) + 120582ℎ (119910) = 0 (minus119905
0le 119910 le 119905
0) (44)
From (43) it follows that
lim119910darrminus1199050
1198921015840
0(119910) = lim
119910uarr1199050
1198921015840(119910) = +infin (45)
Define 1205930= 119892minus1
0 Then (44) and (45) say that
12059310158401015840
0(119909) + 120582ℎ (120593
0(119909)) = 0 (119886 lt 119909 lt 119887
1)
1205931015840
0(119886) = 120593
1015840
0(1198871) = 0
(46)
From (46) and assumption (H1) we see that
lim119909uarr1198871
12059310158401015840
0(119909) = lim
119909uarr1198871
minus 120582ℎ (1205930(119909)) = lim
119909uarr1199050
minus 120582ℎ (119910) = ℎ (1199050) = 0
(47)
and similarly lim119909darr11988612059310158401015840
0(119909) = 0 By defining 120593
0(119909) = 120593
0(1198871)
when 1198871lt 119909 le 119887 the function120593
0is extended and thus defined
on [119886 119887] such that
12059310158401015840
0(119909) + 120582ℎ (120593
0(119909)) = 0 (119886 le 119909 le 119887)
1205931015840
0(119886) = 120593
1015840
0(119887) = 0
(48)
The proof is complete
Example 7 (BVP (2) revisited) Again we are concerned withBVP (2) namely
12059310158401015840+ 120582 sin120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(49)
The problem has nonconstant solutions if and only if (119887 minus119886)2|120582| lt 120587
2 Indeed we see that
sin (119909 + 120587) = minus sin119909 (119909 isin R) (50)
and for some 1199090isin R
0 lt sin119909 = minus sin (minus119909) (0 lt 119909 lt 120587)
ℎ1015840
(0) gt 0
(51)
119909 997891997888rarr radicint
119909
0
sin 119905 d119905 is a concave function on [0 120587] (52)
Note that (52) is trivial since
radicint
119909
0
sin 119905 d119905 = radic1 minus cos119909 = radic2 cos 1199092
(0 le 119909 le 120587) (53)
Remark 8 In this paper we have solved a class of bifurcationproblems for Neumann boundary value problems for semi-linear elliptic equations namely
12059310158401015840+ 120582ℎ ∘ 120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(54)
the governing equation occurring in this boundary valueproblem generalizes the classical beam equation in the sensethat the nonlinear interaction assumes the form R ni 119906 997891rarr
ℎ(sin 119906) isin R instead of R ni 119906 997891rarr sin 119906 isin R What is moreimportant is that the bifurcation problem for the classicalbeam equation can be solved using abstract bifurcationtheorems in nonlinear analysis while the generalized beamequations can not be We provided a unified approach tounderstand this class of problems Indeed our method isquite general and very elementary
It is worthwhile to mention that bifurcation problemsassociated with beam equations other than the type (3) havebeen extensively studied see [19ndash22] and the profound refer-ences cited therein The approaches frequently used in theliterature are quite different from ours and have as founda-tions much advanced complicated knowledge in functionalanalysis
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees for theirvaluable suggestions
References
[1] L Nirenberg Topics in Nonlinear Functional Analysis vol 6of Courant Lecture Notes in Mathematics New York UniversityCourant Institute of Mathematical Sciences New York NYUSA 2001 Chapter 6 by E Zehnder Notes by R A ArtinoRevised reprint of the 1974 original
[2] T Ma and S Wang ldquoBifurcation of nonlinear equations ISteady state bifurcationrdquoMethods and Applications of Analysisvol 11 no 2 pp 155ndash178 2004
[3] T Ma and S Wang ldquoBifurcation of nonlinear equations IIDynamic bifurcationrdquo Methods and Applications of Analysisvol 11 no 2 pp 179ndash209 2004
[4] M A Krasnoselskii Topological Methods in the Theory ofNonlinear Integral Equations translated byAHArmstrong andedited by J Burlak Pergamon Press New York NY USA 1964
[5] S N Chow and J K Hale Methods of Bifurcation Theory vol251 of rundlehren der Mathematischen Wissenschaften [Funda-mental Principles of Mathematical Science] Springer New YorkNY USA 1982
[6] D Henry Geometric Theory of Semilinear Parabolic EquationsLecture Notes in Mathematics Springer New York NY USA1981
[7] T Kato Perturbation Theory for Linear Operators Classicsin Mathematics reprint of the 1980 edition Springer BerlinGermany 1995
[8] T Ma and S Wang ldquoStructure of 2D incompressible flows withthe DIRichlet boundary conditionsrdquo Discrete and ContinuousDynamical Systems Series B vol 1 no 1 pp 29ndash41 2001
[9] T Ma and S Wang ldquoStructural classification and stability ofdivergence-free vector fieldsrdquo Physica D vol 171 no 1-2 pp107ndash126 2002
6 Advances in Mathematical Physics
[10] T Ma and S Wang ldquoAttractor bifurcation theory and its appli-cations to Rayleigh-Benard convectionrdquo Communications onPure and Applied Analysis vol 2 no 4 pp 591ndash599 2003
[11] T Ma and S Wang ldquoDynamic bifurcation and stability in theRayleighBenard convectionrdquo Communications in MathematicalSciences vol 2 no 2 pp 159ndash183 2004
[12] T Ma and S Wang Bifurcation Theory and Applications vol53 of World Scientific Series on Nonlinear Science Series AMonographs and Treatises World Scientific Publishing Co PteLtd Hackensack NJ Hackensack NJ USA 2005
[13] T Ma and S Wang Geometric Theory of Incompressible Flowswith Applications to Fluid Dynamics Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 2005
[14] H Berestycki ldquoOn some nonlinear Sturm-Liouville problemsrdquoJournal of Differential Equations vol 26 no 3 pp 375ndash390 1977
[15] G Birkhoff A Source Book in Classical Analysis Harvard Uni-versity Press Cambridge Mass USA 1973
[16] P H Rabinowitz ldquoNonlinear Sturm-Liouville problems forsecond order ordinary differential equationsrdquo Communicationson Pure and Applied Mathematics vol 23 pp 939ndash961 1970
[17] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005
[18] T Ma and S Wang Stability and Bifurcation Problems for Non-linear Evolution Equations Science Press Beijing China 2007(Chinese)
[19] M A Abdul Hussain ldquoBifurcation solutions of elastic beamsequation with small perturbationrdquo International Journal ofMathematical Analysis vol 3 no 17ndash20 pp 879ndash888 2009
[20] J Berkovits ldquoOn the bifurcation of large amplitude solutionsfor a system of wave and beam equationsrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 52 no 1 pp 343ndash354 2003
[21] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[22] C Polymilis C Skokos G Kollias G Servizi and G TurchettildquoBifurcations of beam-beam like mapsrdquo Journal of Physics AMathematical and General vol 33 no 5 pp 1055ndash1064 2000
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 Advances in Mathematical Physics
points the number of solutions to (3) may change Howeververy little further work has been done to determine whetherthe number of solutions changes at these points In this paperwe give such a criterion for a class of nonlinear problems
Theorem 4 Let 119886 119887 120582 isin R with 119886 lt 119887 Assume (H1)-(H2)Then
plusmn1205872
((119887 minus 119886)2ℎ1015840(1199090))
(7)
are two bifurcation points for BVP (3) Besides (3) has non-constant solutions if and only if
1205872lt |120582| (119887 minus 119886)
2ℎ1015840(1199090) (8)
The proof of a bifurcation assertion of a nonlinear equa-tion often has as ingredients such topological arguments asKrasnoselskiirsquos and Rabinowitzrsquos theorems on bifurcationThese arguments usually have the assumption that the alge-braic multiplicity of the associated linear eigenvalue problemis odd see [1ndash3] and the references therein Since then severalauthors have also attempted to remove such oddness assump-tion see [1 2 4] In particular Ma and Wang [2] developedan elaborate algorithm to prove steady state bifurcationassertions concerning nonlinear equations this algorithmdoes not assume the oddness of the algebraic multiplicitySee [5ndash13] for more studies on bifurcation problems Ourapproach to prove Theorem 4 does not assume such paritycondition on the algebraic multiplicity
As a matter of fact BVP (2) is a special case of Sturm-Liouville problem or boundary value problems for ellipticpartial differential equations Therefore BVP (2) possibly indisguise has been studied extensively in the literature for theexistence of solutions satisfying certain prescribed propertiesfor qualitative properties of solutions and so on see [14ndash17]and the profound references cited therein
The remainder of this paper is organized as follows InSection 2 we introduce some nonlinear functional analysisand formulate the problem in a formal way and in Section 3we give the proof of Theorem 4
2 The Existing Bifurcation Results for BVP (2)
In this section we mainly give a brief review of the existingresults in the literature concerning bifurcation problems forBVP (2) which can be viewed as archetypes of bifurcationproblems for BVP (3) Indeed bifurcation problems for BVP(2) have been often provided as illustration examples to testthe proposed abstract bifurcation-problems-solving methodin the literature see [1 12 13 18]
In particular Ma and Wang [18] proposed an abstractmethod which generalizes slightly the previous one obtainedby Nirenberg [1] In presenting their method the authorsfixed two Banach spaces 119883 and 119884 for which 119883 embedscontinuously and densely into119884The abstract problemwhichthey were concerned with reads
119871120582119906 + 119866
120582(119906) = 0 (9)
where 119871120582 119883 rarr 119884 120582 isin R is a family of bounded linear
operators and 119866120582 119883 rarr 119884 is a family of continuous map-
pings They assumed the following
(H3) 119871120582is in the form 119871
120582= 119860 + 119861
120582with 119860 as a linear
topological isomorphism of 119883 onto 119884 and 119861120582as
compact linear operators hence the spectrum of 119871120582
consists of the exactly countably many eigenvalues120573119896(120582) (listed by algebraic multiplicities) of 119871
120582 there
exists 1205820for which
1205731(1205820) = 120573119895(1205820) 120573
119895(1205820) = 0 forall119895 gt 1
120573119895(1205820) lt 0 if 120582 lt 120582
0
1205731(1205820) = 0 if 120582 = 120582
0
120573119895(1205820) gt 0 if 120582 gt 120582
0
(10)
(H4) For any 120576 gt 0 there exists a 120575 gt 0 such that
1003817100381710038171003817119866120582 (119906)1003817100381710038171003817119884lt 120576 forall119906 isin 119883 with 119906
119883lt 120575 forall120582 isin R (11)
119866120582is analytic in the sense that
119866120582= 1198660
120582+
infin
sum
119899=1
119866119899
120582(119906 119906 119906⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119899 copies) (12)
119866119899
120582is a continuous symmetric 119899-form on119883The precise problem with which they are concerned is
whether there is a 1205820isin R given in such a way that if 119906
120582= 0
with 120582 in a neighborhood of 1205820is a collection of solutions to
BVP (9) then
119906120582997888rarr 0 in 119883 as 120582 997888rarr 120582
0 (13)
If there exists a 1205820which satisfies the above requirements
then 1205820is called a bifurcation point for nonlinear problem
(9) also problem (9) is said to bifurcate from 1205820
Concerning (9) they proved the followingAssume (H3)-(H4) Then 120582
0is a candidate bifurcation
point of the nonlinear problem (9)The proof of the above theorem provided in [18] utilizes
such complicated methods as Lyapunov-Schmidt reductionmethod Morse index theory and so forth
Ma and Wang [18] used the above theorem to obtainthe bifurcation results for BVP (2) Indeed they wrote firstly119883 = 119906 isin 119867
2(0 1) 119906
1015840(0) = 119906
1015840(1) 119884 = 119871
2(0 1) 119871
120582119906 =
(11988921198891199092)119906 minus 120582119906 and 119866
120582(119906) = 120582 sin 119906 minus 120582119906 thereby recasting
BVP (2) into one of the forms (9) and secondly they solvedthis new bifurcation problem for BVP (2) by utilizing theirabstract result
Here we are tempted to use the results obtained in Maand Wang [18] to solve the bifurcation problem for BVP (3)it is however obvious that the nonlinear reaction R ni 119906 rarr
ℎ(sin 119906) isin R precludes our application of such results In thenext section we will analyze the bifurcation problem for BVP(3) in an elementary way
Advances in Mathematical Physics 3
3 Proof of the Main Results
In this section we propose two lemmas and then proveTheorem 4 based on them Various calculus theorems areemployed in our proofs and the elementary equality
(119891minus1)1015840
(119891 (119905)) =1
1198911015840(119905)
(14)
is also used repeatedlyFor the sake of convenience we write
119867(119909) = 1198702
(119909) = int
1199090+119909
1199090
ℎ (119905) d119905 (0 le 119909 le 119897) (15)
The function 119867 is strictly increasing on [0 119897] 119867(0) = 0and1198671015840(119909) = ℎ(119909
0+ 119909)119867 is 2120587-periodic because
119867(119909 + 2119897) minus 119867 (119909) = int
119909+2119897
119909
ℎ (119905) d119905 = int119897
minus119897
ℎ (1199090+ 119905) d119905 = 0
(16)
due to Remark 2
Lemma 5 The function 119870 is strictly increasing and differen-tiable on [0 119897] with derivative
1198701015840
(119909) =
1198671015840(119909)
(2radic119867 (119909))
0 lt 119909 le 119897
radicℎ1015840(1199090)
2119909 = 0
(17)
Moreover
1198701015840(119870minus1(119910)) =
1
2119910 (119867minus1)1015840
(1199102)
(0 lt 119910 lt radic119867 (119897)) (18)
Proof Since 119870 = radic119867 and 119867 is positive on (0 119897] the value1198701015840(119909) with 119909 isin (0 119897] can be derived directly from (15) and
thus half of (17) is obtained Consequently
1198701015840
(0) = lim119909darr0
1198701015840
(119909) = lim119909darr0
1198671015840(119909)
2radic119867 (119909)
(19)
Noting that1198671015840 ge 0 on [0 119897] LrsquoHospital rule shows that
(1198701015840
(0))2
= lim119909darr0
(1198671015840(119909))2
4119867 (119909)= lim119909darr0
21198671015840(119909)119867
10158401015840(119909)
41198671015840(119909)
=ℎ1015840(1199090)
2
(20)
Thus the other half of (17) is obtained A simple compu-tation using (17) and (14) gives (18)
Lemma 6 Define
119901 (119905) = int
119905
0
1
radic119867 (119905) minus 119867 (119910)
d119910 (0 lt 119905 le 119897) (21)
For 0 lt 119905 lt 119897 the integral above converges and the function 119901is strictly increasing on (0 119897] Besides
lim119905darr0
119901 (119905) =120587
radic2ℎ1015840(1199090)
(22)
Proof Changing of variable 119910 = 119867minus1(119906119867(119905)) in the integralwe get
int
119905
0
1
radic119867 (119905) minus 119867 (119910)
d119910 = int1
0
radic119867(119905)
radic1 minus 119906
sdot (119867minus1)1015840
(119906 (119867 (119905))) d119906
(0 lt 119905 le 119897)
(23)
By the dominated convergence theorem and (18) the inte-grand of the right side above equals
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
(24)
and thus
119901 (119905) = int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
d119906 (25)
The function 1(1198701015840 ∘ 119870minus1) is strictly increasing on (0radic119867(119897)] because 119870minus1 is strictly increasing and 1198701015840 is strictlydecreasing by our assumption (6) Hence value (24) increasesas 119905 does Note that radic119906(119867(119905)) le radic119867(119905) for 119906 isin [0 1] If119905 isin (0 119897) then
119901 (119905) = int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
d119906
le int
1
0
1
2radic119906 (1 minus 119906)
sdot1
1198701015840(119870minus1(radic119867 (119905)))
d119906
=120587
21198701015840(119870minus1(radic119867 (119905)))
lt infin
(26)
Hence 119901 is real valued and (25) implies that 119901 is strictlyincreasing on (0 119897] Finally equalities (25) (17) and the dom-inated convergence theorem show that
lim119905darr0
119901 (119905)
= lim119905darr0
int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
d119906
= int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(0))
d119906
=120587
2radicℎ1015840(1199090)
(27)
The proof is complete
Proof of Theorem 4 Assume 120593 is a nonconstant solution of(3) The set 119909 isin [119886 119887] 1205931015840(119909) = 0 is open and nonempty in[119886 119887] and thus is a disjoint union of open intervals (provided[119886 119909) and (119909 119887] for 119909 isin (119886 119887) are viewed as ldquoopen intervalsrdquo
4 Advances in Mathematical Physics
in [119886 119887]) Let (1198860 1198870) be such an open interval Then 120593 is
nonconstant on [1198860 1198870] and 1205931015840(119886
0) = 120593
1015840(1198870) We will show
that 1198870minus 1198860gt 120587radic|120582|ℎ
1015840(1199090) which is equivalent to (8)
Since differentiable functions have intermediate valueproperty and 1205931015840(119909) = 0 on (119886
0 1198870) we may assume 1205931015840 gt 0 on
(1198860 1198870) without loss of generality So 120593 is strictly increasing
on [1198860 1198870] with inverse function 119892 = 120593
minus1 defined on[120593(1198860) 120593(1198870)] A simple computation using (14) and the chain
rule for differentiation shows that
119889 ((11198921015840(119910))2
)
119889119910
=
119889(((119892minus1)1015840
(119892 (119910)))
2
)
119889119910
= 2 (119892minus1)1015840
(119892 (119910)) sdot (119892minus1)10158401015840
(119892 (119910)) sdot 1198921015840(119910)
= 2 (119892minus1)10158401015840
(119892 (119910))
(28)
Condition (3) means
(119892minus1)10158401015840
(119892 (119910)) + 120582ℎ (119910) = 0 (120593 (1198860) le 119910 le 120593 (119887
0)) (29)
lim119910darr119892(120593(1198860))
1198921015840(119910) = lim
119910uarr119892(120593(1198870))
1198921015840(119910) = +infin (30)
Integrating both sides of (29) and using (28) we get that
1
2 (1198921015840(119910))2+ 120582119867 (119910) = 119862
1(120593 (1198860) lt 119910 lt 120593 (119887
0)) (31)
with constant1198621 We assume 120582 gt 0without loss of generality
Then
1198921015840(119910) =
1
radic2120582
sdot1
radic1198622minus 119867 (119910)
(120593 (1198860) lt 119910 lt 120593 (119887
0)) (32)
with 1198622= 1198621120582 Consequently
119867(119910) lt 1198622
(120593 (1198860) lt 119910 lt 120593 (119887
0)) (33)
As the properties of 119867 stated before the function 119910 997891rarr
119867(119910) is with period 2119897 decreases on [minus119897 0] and increases on[0 119897] Equalities (30) and (32) yield that
119867(119910) 997888rarr 1198622
(119910 darr 120593 (1198860) or119910 uarr 120593 (119887
0)) (34)
This together with (33) and the properties of 119867 shows thatthere exists 119905 isin (0 119897] and 119911 isin Z such that 119862
2= 119867(119909
0+ 119905)
120593(1198860) = 1199090+ 2119897119911 minus 119905 and 120593(119887
0) = 1199090+ 2119897119911 + 119905 Consequently
1198870minus 1198860= 119892 (120593 (119887
0)) minus 119892 (120593 (119886
0))
= 119892 (120593 (1198870) minus 2119897119911) minus 119892 (120593 (119886
0) minus 2119897119911)
= 119892 (1199090+ 119905) minus 119892 (119909
0minus 119905)
= int
119905
minus119905
1198921015840(1199090+ 119910) d119910
(35)
Since119867 is an even function it follows from (32) that
1198870minus 1198860=
1
radic2120582
int
119905
minus119905
1
radic119867 (119905) minus 119867 (119910)
d119910
= radic2
120582int
119905
0
1
radic119867 (119905) minus 119867 (119910)
d119910
= radic2
120582119901 (119905)
(36)
Note that 119901 is strictly increasing on (0 119897] (see Lemma 6)Hence
119887 minus 119886 ge 1198870minus 1198860= radic
2
120582119901 (119905) gt lim
119905darr0
radic2
120582119901 (119905) =
120587
radic120582ℎ1015840(1199090)
(37)
which implies (8) and thus half of Theorem 4 is provedFor the other half we assume without loss of generality
that 120582 gt 0 and assume (8) holds that is
119887 minus 119886 ge120587
radic120582ℎ1015840(1199090)
(38)
And we show that (2) has a nonconstant solutionFirst it follows from (27) that there exits 119905
0gt 0 such that
radic2
120582119901 (1199050) lt 119887 minus 119886 (39)
Let
1198871= 119886 + radic
2
120582119901 (1199050) lt 119887 (40)
Define a continuous function1198920on [minus119905
0 1199050] by119892(minus119905
0) = 119886
and
1198921015840
0(119910) =
1
radic2120582
sdot1
radic119867 (1199050) minus 119867 (119910)
(minus1199050lt 119910 lt 119905
0) (41)
Definition (21) yields
119892 (1199050) = 119886 + int
1199050
minus1199050
1198921015840
0(119905) d119905
= 119886 + 21
radic2120582
sdot int
1199050
0
1
radic119867(1199050) minus 119867 (119910)
d119905
= 119886 + radic2
120582119901 (1199050)
= 1198871
(42)
Besides
1
2 (1198921015840
0(119910))2+ 120582119867 (119910) = 120582119867 (119905
0) (minus119905
0lt 119910 lt 119905
0) (43)
Advances in Mathematical Physics 5
Differentiating both sides of (45) and using (28) we get
(119892minus1
0)10158401015840
(1198920(119910)) + 120582ℎ (119910) = 0 (minus119905
0le 119910 le 119905
0) (44)
From (43) it follows that
lim119910darrminus1199050
1198921015840
0(119910) = lim
119910uarr1199050
1198921015840(119910) = +infin (45)
Define 1205930= 119892minus1
0 Then (44) and (45) say that
12059310158401015840
0(119909) + 120582ℎ (120593
0(119909)) = 0 (119886 lt 119909 lt 119887
1)
1205931015840
0(119886) = 120593
1015840
0(1198871) = 0
(46)
From (46) and assumption (H1) we see that
lim119909uarr1198871
12059310158401015840
0(119909) = lim
119909uarr1198871
minus 120582ℎ (1205930(119909)) = lim
119909uarr1199050
minus 120582ℎ (119910) = ℎ (1199050) = 0
(47)
and similarly lim119909darr11988612059310158401015840
0(119909) = 0 By defining 120593
0(119909) = 120593
0(1198871)
when 1198871lt 119909 le 119887 the function120593
0is extended and thus defined
on [119886 119887] such that
12059310158401015840
0(119909) + 120582ℎ (120593
0(119909)) = 0 (119886 le 119909 le 119887)
1205931015840
0(119886) = 120593
1015840
0(119887) = 0
(48)
The proof is complete
Example 7 (BVP (2) revisited) Again we are concerned withBVP (2) namely
12059310158401015840+ 120582 sin120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(49)
The problem has nonconstant solutions if and only if (119887 minus119886)2|120582| lt 120587
2 Indeed we see that
sin (119909 + 120587) = minus sin119909 (119909 isin R) (50)
and for some 1199090isin R
0 lt sin119909 = minus sin (minus119909) (0 lt 119909 lt 120587)
ℎ1015840
(0) gt 0
(51)
119909 997891997888rarr radicint
119909
0
sin 119905 d119905 is a concave function on [0 120587] (52)
Note that (52) is trivial since
radicint
119909
0
sin 119905 d119905 = radic1 minus cos119909 = radic2 cos 1199092
(0 le 119909 le 120587) (53)
Remark 8 In this paper we have solved a class of bifurcationproblems for Neumann boundary value problems for semi-linear elliptic equations namely
12059310158401015840+ 120582ℎ ∘ 120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(54)
the governing equation occurring in this boundary valueproblem generalizes the classical beam equation in the sensethat the nonlinear interaction assumes the form R ni 119906 997891rarr
ℎ(sin 119906) isin R instead of R ni 119906 997891rarr sin 119906 isin R What is moreimportant is that the bifurcation problem for the classicalbeam equation can be solved using abstract bifurcationtheorems in nonlinear analysis while the generalized beamequations can not be We provided a unified approach tounderstand this class of problems Indeed our method isquite general and very elementary
It is worthwhile to mention that bifurcation problemsassociated with beam equations other than the type (3) havebeen extensively studied see [19ndash22] and the profound refer-ences cited therein The approaches frequently used in theliterature are quite different from ours and have as founda-tions much advanced complicated knowledge in functionalanalysis
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees for theirvaluable suggestions
References
[1] L Nirenberg Topics in Nonlinear Functional Analysis vol 6of Courant Lecture Notes in Mathematics New York UniversityCourant Institute of Mathematical Sciences New York NYUSA 2001 Chapter 6 by E Zehnder Notes by R A ArtinoRevised reprint of the 1974 original
[2] T Ma and S Wang ldquoBifurcation of nonlinear equations ISteady state bifurcationrdquoMethods and Applications of Analysisvol 11 no 2 pp 155ndash178 2004
[3] T Ma and S Wang ldquoBifurcation of nonlinear equations IIDynamic bifurcationrdquo Methods and Applications of Analysisvol 11 no 2 pp 179ndash209 2004
[4] M A Krasnoselskii Topological Methods in the Theory ofNonlinear Integral Equations translated byAHArmstrong andedited by J Burlak Pergamon Press New York NY USA 1964
[5] S N Chow and J K Hale Methods of Bifurcation Theory vol251 of rundlehren der Mathematischen Wissenschaften [Funda-mental Principles of Mathematical Science] Springer New YorkNY USA 1982
[6] D Henry Geometric Theory of Semilinear Parabolic EquationsLecture Notes in Mathematics Springer New York NY USA1981
[7] T Kato Perturbation Theory for Linear Operators Classicsin Mathematics reprint of the 1980 edition Springer BerlinGermany 1995
[8] T Ma and S Wang ldquoStructure of 2D incompressible flows withthe DIRichlet boundary conditionsrdquo Discrete and ContinuousDynamical Systems Series B vol 1 no 1 pp 29ndash41 2001
[9] T Ma and S Wang ldquoStructural classification and stability ofdivergence-free vector fieldsrdquo Physica D vol 171 no 1-2 pp107ndash126 2002
6 Advances in Mathematical Physics
[10] T Ma and S Wang ldquoAttractor bifurcation theory and its appli-cations to Rayleigh-Benard convectionrdquo Communications onPure and Applied Analysis vol 2 no 4 pp 591ndash599 2003
[11] T Ma and S Wang ldquoDynamic bifurcation and stability in theRayleighBenard convectionrdquo Communications in MathematicalSciences vol 2 no 2 pp 159ndash183 2004
[12] T Ma and S Wang Bifurcation Theory and Applications vol53 of World Scientific Series on Nonlinear Science Series AMonographs and Treatises World Scientific Publishing Co PteLtd Hackensack NJ Hackensack NJ USA 2005
[13] T Ma and S Wang Geometric Theory of Incompressible Flowswith Applications to Fluid Dynamics Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 2005
[14] H Berestycki ldquoOn some nonlinear Sturm-Liouville problemsrdquoJournal of Differential Equations vol 26 no 3 pp 375ndash390 1977
[15] G Birkhoff A Source Book in Classical Analysis Harvard Uni-versity Press Cambridge Mass USA 1973
[16] P H Rabinowitz ldquoNonlinear Sturm-Liouville problems forsecond order ordinary differential equationsrdquo Communicationson Pure and Applied Mathematics vol 23 pp 939ndash961 1970
[17] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005
[18] T Ma and S Wang Stability and Bifurcation Problems for Non-linear Evolution Equations Science Press Beijing China 2007(Chinese)
[19] M A Abdul Hussain ldquoBifurcation solutions of elastic beamsequation with small perturbationrdquo International Journal ofMathematical Analysis vol 3 no 17ndash20 pp 879ndash888 2009
[20] J Berkovits ldquoOn the bifurcation of large amplitude solutionsfor a system of wave and beam equationsrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 52 no 1 pp 343ndash354 2003
[21] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[22] C Polymilis C Skokos G Kollias G Servizi and G TurchettildquoBifurcations of beam-beam like mapsrdquo Journal of Physics AMathematical and General vol 33 no 5 pp 1055ndash1064 2000
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
Advances in Mathematical Physics 3
3 Proof of the Main Results
In this section we propose two lemmas and then proveTheorem 4 based on them Various calculus theorems areemployed in our proofs and the elementary equality
(119891minus1)1015840
(119891 (119905)) =1
1198911015840(119905)
(14)
is also used repeatedlyFor the sake of convenience we write
119867(119909) = 1198702
(119909) = int
1199090+119909
1199090
ℎ (119905) d119905 (0 le 119909 le 119897) (15)
The function 119867 is strictly increasing on [0 119897] 119867(0) = 0and1198671015840(119909) = ℎ(119909
0+ 119909)119867 is 2120587-periodic because
119867(119909 + 2119897) minus 119867 (119909) = int
119909+2119897
119909
ℎ (119905) d119905 = int119897
minus119897
ℎ (1199090+ 119905) d119905 = 0
(16)
due to Remark 2
Lemma 5 The function 119870 is strictly increasing and differen-tiable on [0 119897] with derivative
1198701015840
(119909) =
1198671015840(119909)
(2radic119867 (119909))
0 lt 119909 le 119897
radicℎ1015840(1199090)
2119909 = 0
(17)
Moreover
1198701015840(119870minus1(119910)) =
1
2119910 (119867minus1)1015840
(1199102)
(0 lt 119910 lt radic119867 (119897)) (18)
Proof Since 119870 = radic119867 and 119867 is positive on (0 119897] the value1198701015840(119909) with 119909 isin (0 119897] can be derived directly from (15) and
thus half of (17) is obtained Consequently
1198701015840
(0) = lim119909darr0
1198701015840
(119909) = lim119909darr0
1198671015840(119909)
2radic119867 (119909)
(19)
Noting that1198671015840 ge 0 on [0 119897] LrsquoHospital rule shows that
(1198701015840
(0))2
= lim119909darr0
(1198671015840(119909))2
4119867 (119909)= lim119909darr0
21198671015840(119909)119867
10158401015840(119909)
41198671015840(119909)
=ℎ1015840(1199090)
2
(20)
Thus the other half of (17) is obtained A simple compu-tation using (17) and (14) gives (18)
Lemma 6 Define
119901 (119905) = int
119905
0
1
radic119867 (119905) minus 119867 (119910)
d119910 (0 lt 119905 le 119897) (21)
For 0 lt 119905 lt 119897 the integral above converges and the function 119901is strictly increasing on (0 119897] Besides
lim119905darr0
119901 (119905) =120587
radic2ℎ1015840(1199090)
(22)
Proof Changing of variable 119910 = 119867minus1(119906119867(119905)) in the integralwe get
int
119905
0
1
radic119867 (119905) minus 119867 (119910)
d119910 = int1
0
radic119867(119905)
radic1 minus 119906
sdot (119867minus1)1015840
(119906 (119867 (119905))) d119906
(0 lt 119905 le 119897)
(23)
By the dominated convergence theorem and (18) the inte-grand of the right side above equals
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
(24)
and thus
119901 (119905) = int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
d119906 (25)
The function 1(1198701015840 ∘ 119870minus1) is strictly increasing on (0radic119867(119897)] because 119870minus1 is strictly increasing and 1198701015840 is strictlydecreasing by our assumption (6) Hence value (24) increasesas 119905 does Note that radic119906(119867(119905)) le radic119867(119905) for 119906 isin [0 1] If119905 isin (0 119897) then
119901 (119905) = int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
d119906
le int
1
0
1
2radic119906 (1 minus 119906)
sdot1
1198701015840(119870minus1(radic119867 (119905)))
d119906
=120587
21198701015840(119870minus1(radic119867 (119905)))
lt infin
(26)
Hence 119901 is real valued and (25) implies that 119901 is strictlyincreasing on (0 119897] Finally equalities (25) (17) and the dom-inated convergence theorem show that
lim119905darr0
119901 (119905)
= lim119905darr0
int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(radic119906 (119867 (119905))))
d119906
= int
1
0
1
2radic119906 (1 minus 119906) sdot 1198701015840(119870minus1(0))
d119906
=120587
2radicℎ1015840(1199090)
(27)
The proof is complete
Proof of Theorem 4 Assume 120593 is a nonconstant solution of(3) The set 119909 isin [119886 119887] 1205931015840(119909) = 0 is open and nonempty in[119886 119887] and thus is a disjoint union of open intervals (provided[119886 119909) and (119909 119887] for 119909 isin (119886 119887) are viewed as ldquoopen intervalsrdquo
4 Advances in Mathematical Physics
in [119886 119887]) Let (1198860 1198870) be such an open interval Then 120593 is
nonconstant on [1198860 1198870] and 1205931015840(119886
0) = 120593
1015840(1198870) We will show
that 1198870minus 1198860gt 120587radic|120582|ℎ
1015840(1199090) which is equivalent to (8)
Since differentiable functions have intermediate valueproperty and 1205931015840(119909) = 0 on (119886
0 1198870) we may assume 1205931015840 gt 0 on
(1198860 1198870) without loss of generality So 120593 is strictly increasing
on [1198860 1198870] with inverse function 119892 = 120593
minus1 defined on[120593(1198860) 120593(1198870)] A simple computation using (14) and the chain
rule for differentiation shows that
119889 ((11198921015840(119910))2
)
119889119910
=
119889(((119892minus1)1015840
(119892 (119910)))
2
)
119889119910
= 2 (119892minus1)1015840
(119892 (119910)) sdot (119892minus1)10158401015840
(119892 (119910)) sdot 1198921015840(119910)
= 2 (119892minus1)10158401015840
(119892 (119910))
(28)
Condition (3) means
(119892minus1)10158401015840
(119892 (119910)) + 120582ℎ (119910) = 0 (120593 (1198860) le 119910 le 120593 (119887
0)) (29)
lim119910darr119892(120593(1198860))
1198921015840(119910) = lim
119910uarr119892(120593(1198870))
1198921015840(119910) = +infin (30)
Integrating both sides of (29) and using (28) we get that
1
2 (1198921015840(119910))2+ 120582119867 (119910) = 119862
1(120593 (1198860) lt 119910 lt 120593 (119887
0)) (31)
with constant1198621 We assume 120582 gt 0without loss of generality
Then
1198921015840(119910) =
1
radic2120582
sdot1
radic1198622minus 119867 (119910)
(120593 (1198860) lt 119910 lt 120593 (119887
0)) (32)
with 1198622= 1198621120582 Consequently
119867(119910) lt 1198622
(120593 (1198860) lt 119910 lt 120593 (119887
0)) (33)
As the properties of 119867 stated before the function 119910 997891rarr
119867(119910) is with period 2119897 decreases on [minus119897 0] and increases on[0 119897] Equalities (30) and (32) yield that
119867(119910) 997888rarr 1198622
(119910 darr 120593 (1198860) or119910 uarr 120593 (119887
0)) (34)
This together with (33) and the properties of 119867 shows thatthere exists 119905 isin (0 119897] and 119911 isin Z such that 119862
2= 119867(119909
0+ 119905)
120593(1198860) = 1199090+ 2119897119911 minus 119905 and 120593(119887
0) = 1199090+ 2119897119911 + 119905 Consequently
1198870minus 1198860= 119892 (120593 (119887
0)) minus 119892 (120593 (119886
0))
= 119892 (120593 (1198870) minus 2119897119911) minus 119892 (120593 (119886
0) minus 2119897119911)
= 119892 (1199090+ 119905) minus 119892 (119909
0minus 119905)
= int
119905
minus119905
1198921015840(1199090+ 119910) d119910
(35)
Since119867 is an even function it follows from (32) that
1198870minus 1198860=
1
radic2120582
int
119905
minus119905
1
radic119867 (119905) minus 119867 (119910)
d119910
= radic2
120582int
119905
0
1
radic119867 (119905) minus 119867 (119910)
d119910
= radic2
120582119901 (119905)
(36)
Note that 119901 is strictly increasing on (0 119897] (see Lemma 6)Hence
119887 minus 119886 ge 1198870minus 1198860= radic
2
120582119901 (119905) gt lim
119905darr0
radic2
120582119901 (119905) =
120587
radic120582ℎ1015840(1199090)
(37)
which implies (8) and thus half of Theorem 4 is provedFor the other half we assume without loss of generality
that 120582 gt 0 and assume (8) holds that is
119887 minus 119886 ge120587
radic120582ℎ1015840(1199090)
(38)
And we show that (2) has a nonconstant solutionFirst it follows from (27) that there exits 119905
0gt 0 such that
radic2
120582119901 (1199050) lt 119887 minus 119886 (39)
Let
1198871= 119886 + radic
2
120582119901 (1199050) lt 119887 (40)
Define a continuous function1198920on [minus119905
0 1199050] by119892(minus119905
0) = 119886
and
1198921015840
0(119910) =
1
radic2120582
sdot1
radic119867 (1199050) minus 119867 (119910)
(minus1199050lt 119910 lt 119905
0) (41)
Definition (21) yields
119892 (1199050) = 119886 + int
1199050
minus1199050
1198921015840
0(119905) d119905
= 119886 + 21
radic2120582
sdot int
1199050
0
1
radic119867(1199050) minus 119867 (119910)
d119905
= 119886 + radic2
120582119901 (1199050)
= 1198871
(42)
Besides
1
2 (1198921015840
0(119910))2+ 120582119867 (119910) = 120582119867 (119905
0) (minus119905
0lt 119910 lt 119905
0) (43)
Advances in Mathematical Physics 5
Differentiating both sides of (45) and using (28) we get
(119892minus1
0)10158401015840
(1198920(119910)) + 120582ℎ (119910) = 0 (minus119905
0le 119910 le 119905
0) (44)
From (43) it follows that
lim119910darrminus1199050
1198921015840
0(119910) = lim
119910uarr1199050
1198921015840(119910) = +infin (45)
Define 1205930= 119892minus1
0 Then (44) and (45) say that
12059310158401015840
0(119909) + 120582ℎ (120593
0(119909)) = 0 (119886 lt 119909 lt 119887
1)
1205931015840
0(119886) = 120593
1015840
0(1198871) = 0
(46)
From (46) and assumption (H1) we see that
lim119909uarr1198871
12059310158401015840
0(119909) = lim
119909uarr1198871
minus 120582ℎ (1205930(119909)) = lim
119909uarr1199050
minus 120582ℎ (119910) = ℎ (1199050) = 0
(47)
and similarly lim119909darr11988612059310158401015840
0(119909) = 0 By defining 120593
0(119909) = 120593
0(1198871)
when 1198871lt 119909 le 119887 the function120593
0is extended and thus defined
on [119886 119887] such that
12059310158401015840
0(119909) + 120582ℎ (120593
0(119909)) = 0 (119886 le 119909 le 119887)
1205931015840
0(119886) = 120593
1015840
0(119887) = 0
(48)
The proof is complete
Example 7 (BVP (2) revisited) Again we are concerned withBVP (2) namely
12059310158401015840+ 120582 sin120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(49)
The problem has nonconstant solutions if and only if (119887 minus119886)2|120582| lt 120587
2 Indeed we see that
sin (119909 + 120587) = minus sin119909 (119909 isin R) (50)
and for some 1199090isin R
0 lt sin119909 = minus sin (minus119909) (0 lt 119909 lt 120587)
ℎ1015840
(0) gt 0
(51)
119909 997891997888rarr radicint
119909
0
sin 119905 d119905 is a concave function on [0 120587] (52)
Note that (52) is trivial since
radicint
119909
0
sin 119905 d119905 = radic1 minus cos119909 = radic2 cos 1199092
(0 le 119909 le 120587) (53)
Remark 8 In this paper we have solved a class of bifurcationproblems for Neumann boundary value problems for semi-linear elliptic equations namely
12059310158401015840+ 120582ℎ ∘ 120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(54)
the governing equation occurring in this boundary valueproblem generalizes the classical beam equation in the sensethat the nonlinear interaction assumes the form R ni 119906 997891rarr
ℎ(sin 119906) isin R instead of R ni 119906 997891rarr sin 119906 isin R What is moreimportant is that the bifurcation problem for the classicalbeam equation can be solved using abstract bifurcationtheorems in nonlinear analysis while the generalized beamequations can not be We provided a unified approach tounderstand this class of problems Indeed our method isquite general and very elementary
It is worthwhile to mention that bifurcation problemsassociated with beam equations other than the type (3) havebeen extensively studied see [19ndash22] and the profound refer-ences cited therein The approaches frequently used in theliterature are quite different from ours and have as founda-tions much advanced complicated knowledge in functionalanalysis
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees for theirvaluable suggestions
References
[1] L Nirenberg Topics in Nonlinear Functional Analysis vol 6of Courant Lecture Notes in Mathematics New York UniversityCourant Institute of Mathematical Sciences New York NYUSA 2001 Chapter 6 by E Zehnder Notes by R A ArtinoRevised reprint of the 1974 original
[2] T Ma and S Wang ldquoBifurcation of nonlinear equations ISteady state bifurcationrdquoMethods and Applications of Analysisvol 11 no 2 pp 155ndash178 2004
[3] T Ma and S Wang ldquoBifurcation of nonlinear equations IIDynamic bifurcationrdquo Methods and Applications of Analysisvol 11 no 2 pp 179ndash209 2004
[4] M A Krasnoselskii Topological Methods in the Theory ofNonlinear Integral Equations translated byAHArmstrong andedited by J Burlak Pergamon Press New York NY USA 1964
[5] S N Chow and J K Hale Methods of Bifurcation Theory vol251 of rundlehren der Mathematischen Wissenschaften [Funda-mental Principles of Mathematical Science] Springer New YorkNY USA 1982
[6] D Henry Geometric Theory of Semilinear Parabolic EquationsLecture Notes in Mathematics Springer New York NY USA1981
[7] T Kato Perturbation Theory for Linear Operators Classicsin Mathematics reprint of the 1980 edition Springer BerlinGermany 1995
[8] T Ma and S Wang ldquoStructure of 2D incompressible flows withthe DIRichlet boundary conditionsrdquo Discrete and ContinuousDynamical Systems Series B vol 1 no 1 pp 29ndash41 2001
[9] T Ma and S Wang ldquoStructural classification and stability ofdivergence-free vector fieldsrdquo Physica D vol 171 no 1-2 pp107ndash126 2002
6 Advances in Mathematical Physics
[10] T Ma and S Wang ldquoAttractor bifurcation theory and its appli-cations to Rayleigh-Benard convectionrdquo Communications onPure and Applied Analysis vol 2 no 4 pp 591ndash599 2003
[11] T Ma and S Wang ldquoDynamic bifurcation and stability in theRayleighBenard convectionrdquo Communications in MathematicalSciences vol 2 no 2 pp 159ndash183 2004
[12] T Ma and S Wang Bifurcation Theory and Applications vol53 of World Scientific Series on Nonlinear Science Series AMonographs and Treatises World Scientific Publishing Co PteLtd Hackensack NJ Hackensack NJ USA 2005
[13] T Ma and S Wang Geometric Theory of Incompressible Flowswith Applications to Fluid Dynamics Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 2005
[14] H Berestycki ldquoOn some nonlinear Sturm-Liouville problemsrdquoJournal of Differential Equations vol 26 no 3 pp 375ndash390 1977
[15] G Birkhoff A Source Book in Classical Analysis Harvard Uni-versity Press Cambridge Mass USA 1973
[16] P H Rabinowitz ldquoNonlinear Sturm-Liouville problems forsecond order ordinary differential equationsrdquo Communicationson Pure and Applied Mathematics vol 23 pp 939ndash961 1970
[17] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005
[18] T Ma and S Wang Stability and Bifurcation Problems for Non-linear Evolution Equations Science Press Beijing China 2007(Chinese)
[19] M A Abdul Hussain ldquoBifurcation solutions of elastic beamsequation with small perturbationrdquo International Journal ofMathematical Analysis vol 3 no 17ndash20 pp 879ndash888 2009
[20] J Berkovits ldquoOn the bifurcation of large amplitude solutionsfor a system of wave and beam equationsrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 52 no 1 pp 343ndash354 2003
[21] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[22] C Polymilis C Skokos G Kollias G Servizi and G TurchettildquoBifurcations of beam-beam like mapsrdquo Journal of Physics AMathematical and General vol 33 no 5 pp 1055ndash1064 2000
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
4 Advances in Mathematical Physics
in [119886 119887]) Let (1198860 1198870) be such an open interval Then 120593 is
nonconstant on [1198860 1198870] and 1205931015840(119886
0) = 120593
1015840(1198870) We will show
that 1198870minus 1198860gt 120587radic|120582|ℎ
1015840(1199090) which is equivalent to (8)
Since differentiable functions have intermediate valueproperty and 1205931015840(119909) = 0 on (119886
0 1198870) we may assume 1205931015840 gt 0 on
(1198860 1198870) without loss of generality So 120593 is strictly increasing
on [1198860 1198870] with inverse function 119892 = 120593
minus1 defined on[120593(1198860) 120593(1198870)] A simple computation using (14) and the chain
rule for differentiation shows that
119889 ((11198921015840(119910))2
)
119889119910
=
119889(((119892minus1)1015840
(119892 (119910)))
2
)
119889119910
= 2 (119892minus1)1015840
(119892 (119910)) sdot (119892minus1)10158401015840
(119892 (119910)) sdot 1198921015840(119910)
= 2 (119892minus1)10158401015840
(119892 (119910))
(28)
Condition (3) means
(119892minus1)10158401015840
(119892 (119910)) + 120582ℎ (119910) = 0 (120593 (1198860) le 119910 le 120593 (119887
0)) (29)
lim119910darr119892(120593(1198860))
1198921015840(119910) = lim
119910uarr119892(120593(1198870))
1198921015840(119910) = +infin (30)
Integrating both sides of (29) and using (28) we get that
1
2 (1198921015840(119910))2+ 120582119867 (119910) = 119862
1(120593 (1198860) lt 119910 lt 120593 (119887
0)) (31)
with constant1198621 We assume 120582 gt 0without loss of generality
Then
1198921015840(119910) =
1
radic2120582
sdot1
radic1198622minus 119867 (119910)
(120593 (1198860) lt 119910 lt 120593 (119887
0)) (32)
with 1198622= 1198621120582 Consequently
119867(119910) lt 1198622
(120593 (1198860) lt 119910 lt 120593 (119887
0)) (33)
As the properties of 119867 stated before the function 119910 997891rarr
119867(119910) is with period 2119897 decreases on [minus119897 0] and increases on[0 119897] Equalities (30) and (32) yield that
119867(119910) 997888rarr 1198622
(119910 darr 120593 (1198860) or119910 uarr 120593 (119887
0)) (34)
This together with (33) and the properties of 119867 shows thatthere exists 119905 isin (0 119897] and 119911 isin Z such that 119862
2= 119867(119909
0+ 119905)
120593(1198860) = 1199090+ 2119897119911 minus 119905 and 120593(119887
0) = 1199090+ 2119897119911 + 119905 Consequently
1198870minus 1198860= 119892 (120593 (119887
0)) minus 119892 (120593 (119886
0))
= 119892 (120593 (1198870) minus 2119897119911) minus 119892 (120593 (119886
0) minus 2119897119911)
= 119892 (1199090+ 119905) minus 119892 (119909
0minus 119905)
= int
119905
minus119905
1198921015840(1199090+ 119910) d119910
(35)
Since119867 is an even function it follows from (32) that
1198870minus 1198860=
1
radic2120582
int
119905
minus119905
1
radic119867 (119905) minus 119867 (119910)
d119910
= radic2
120582int
119905
0
1
radic119867 (119905) minus 119867 (119910)
d119910
= radic2
120582119901 (119905)
(36)
Note that 119901 is strictly increasing on (0 119897] (see Lemma 6)Hence
119887 minus 119886 ge 1198870minus 1198860= radic
2
120582119901 (119905) gt lim
119905darr0
radic2
120582119901 (119905) =
120587
radic120582ℎ1015840(1199090)
(37)
which implies (8) and thus half of Theorem 4 is provedFor the other half we assume without loss of generality
that 120582 gt 0 and assume (8) holds that is
119887 minus 119886 ge120587
radic120582ℎ1015840(1199090)
(38)
And we show that (2) has a nonconstant solutionFirst it follows from (27) that there exits 119905
0gt 0 such that
radic2
120582119901 (1199050) lt 119887 minus 119886 (39)
Let
1198871= 119886 + radic
2
120582119901 (1199050) lt 119887 (40)
Define a continuous function1198920on [minus119905
0 1199050] by119892(minus119905
0) = 119886
and
1198921015840
0(119910) =
1
radic2120582
sdot1
radic119867 (1199050) minus 119867 (119910)
(minus1199050lt 119910 lt 119905
0) (41)
Definition (21) yields
119892 (1199050) = 119886 + int
1199050
minus1199050
1198921015840
0(119905) d119905
= 119886 + 21
radic2120582
sdot int
1199050
0
1
radic119867(1199050) minus 119867 (119910)
d119905
= 119886 + radic2
120582119901 (1199050)
= 1198871
(42)
Besides
1
2 (1198921015840
0(119910))2+ 120582119867 (119910) = 120582119867 (119905
0) (minus119905
0lt 119910 lt 119905
0) (43)
Advances in Mathematical Physics 5
Differentiating both sides of (45) and using (28) we get
(119892minus1
0)10158401015840
(1198920(119910)) + 120582ℎ (119910) = 0 (minus119905
0le 119910 le 119905
0) (44)
From (43) it follows that
lim119910darrminus1199050
1198921015840
0(119910) = lim
119910uarr1199050
1198921015840(119910) = +infin (45)
Define 1205930= 119892minus1
0 Then (44) and (45) say that
12059310158401015840
0(119909) + 120582ℎ (120593
0(119909)) = 0 (119886 lt 119909 lt 119887
1)
1205931015840
0(119886) = 120593
1015840
0(1198871) = 0
(46)
From (46) and assumption (H1) we see that
lim119909uarr1198871
12059310158401015840
0(119909) = lim
119909uarr1198871
minus 120582ℎ (1205930(119909)) = lim
119909uarr1199050
minus 120582ℎ (119910) = ℎ (1199050) = 0
(47)
and similarly lim119909darr11988612059310158401015840
0(119909) = 0 By defining 120593
0(119909) = 120593
0(1198871)
when 1198871lt 119909 le 119887 the function120593
0is extended and thus defined
on [119886 119887] such that
12059310158401015840
0(119909) + 120582ℎ (120593
0(119909)) = 0 (119886 le 119909 le 119887)
1205931015840
0(119886) = 120593
1015840
0(119887) = 0
(48)
The proof is complete
Example 7 (BVP (2) revisited) Again we are concerned withBVP (2) namely
12059310158401015840+ 120582 sin120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(49)
The problem has nonconstant solutions if and only if (119887 minus119886)2|120582| lt 120587
2 Indeed we see that
sin (119909 + 120587) = minus sin119909 (119909 isin R) (50)
and for some 1199090isin R
0 lt sin119909 = minus sin (minus119909) (0 lt 119909 lt 120587)
ℎ1015840
(0) gt 0
(51)
119909 997891997888rarr radicint
119909
0
sin 119905 d119905 is a concave function on [0 120587] (52)
Note that (52) is trivial since
radicint
119909
0
sin 119905 d119905 = radic1 minus cos119909 = radic2 cos 1199092
(0 le 119909 le 120587) (53)
Remark 8 In this paper we have solved a class of bifurcationproblems for Neumann boundary value problems for semi-linear elliptic equations namely
12059310158401015840+ 120582ℎ ∘ 120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(54)
the governing equation occurring in this boundary valueproblem generalizes the classical beam equation in the sensethat the nonlinear interaction assumes the form R ni 119906 997891rarr
ℎ(sin 119906) isin R instead of R ni 119906 997891rarr sin 119906 isin R What is moreimportant is that the bifurcation problem for the classicalbeam equation can be solved using abstract bifurcationtheorems in nonlinear analysis while the generalized beamequations can not be We provided a unified approach tounderstand this class of problems Indeed our method isquite general and very elementary
It is worthwhile to mention that bifurcation problemsassociated with beam equations other than the type (3) havebeen extensively studied see [19ndash22] and the profound refer-ences cited therein The approaches frequently used in theliterature are quite different from ours and have as founda-tions much advanced complicated knowledge in functionalanalysis
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees for theirvaluable suggestions
References
[1] L Nirenberg Topics in Nonlinear Functional Analysis vol 6of Courant Lecture Notes in Mathematics New York UniversityCourant Institute of Mathematical Sciences New York NYUSA 2001 Chapter 6 by E Zehnder Notes by R A ArtinoRevised reprint of the 1974 original
[2] T Ma and S Wang ldquoBifurcation of nonlinear equations ISteady state bifurcationrdquoMethods and Applications of Analysisvol 11 no 2 pp 155ndash178 2004
[3] T Ma and S Wang ldquoBifurcation of nonlinear equations IIDynamic bifurcationrdquo Methods and Applications of Analysisvol 11 no 2 pp 179ndash209 2004
[4] M A Krasnoselskii Topological Methods in the Theory ofNonlinear Integral Equations translated byAHArmstrong andedited by J Burlak Pergamon Press New York NY USA 1964
[5] S N Chow and J K Hale Methods of Bifurcation Theory vol251 of rundlehren der Mathematischen Wissenschaften [Funda-mental Principles of Mathematical Science] Springer New YorkNY USA 1982
[6] D Henry Geometric Theory of Semilinear Parabolic EquationsLecture Notes in Mathematics Springer New York NY USA1981
[7] T Kato Perturbation Theory for Linear Operators Classicsin Mathematics reprint of the 1980 edition Springer BerlinGermany 1995
[8] T Ma and S Wang ldquoStructure of 2D incompressible flows withthe DIRichlet boundary conditionsrdquo Discrete and ContinuousDynamical Systems Series B vol 1 no 1 pp 29ndash41 2001
[9] T Ma and S Wang ldquoStructural classification and stability ofdivergence-free vector fieldsrdquo Physica D vol 171 no 1-2 pp107ndash126 2002
6 Advances in Mathematical Physics
[10] T Ma and S Wang ldquoAttractor bifurcation theory and its appli-cations to Rayleigh-Benard convectionrdquo Communications onPure and Applied Analysis vol 2 no 4 pp 591ndash599 2003
[11] T Ma and S Wang ldquoDynamic bifurcation and stability in theRayleighBenard convectionrdquo Communications in MathematicalSciences vol 2 no 2 pp 159ndash183 2004
[12] T Ma and S Wang Bifurcation Theory and Applications vol53 of World Scientific Series on Nonlinear Science Series AMonographs and Treatises World Scientific Publishing Co PteLtd Hackensack NJ Hackensack NJ USA 2005
[13] T Ma and S Wang Geometric Theory of Incompressible Flowswith Applications to Fluid Dynamics Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 2005
[14] H Berestycki ldquoOn some nonlinear Sturm-Liouville problemsrdquoJournal of Differential Equations vol 26 no 3 pp 375ndash390 1977
[15] G Birkhoff A Source Book in Classical Analysis Harvard Uni-versity Press Cambridge Mass USA 1973
[16] P H Rabinowitz ldquoNonlinear Sturm-Liouville problems forsecond order ordinary differential equationsrdquo Communicationson Pure and Applied Mathematics vol 23 pp 939ndash961 1970
[17] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005
[18] T Ma and S Wang Stability and Bifurcation Problems for Non-linear Evolution Equations Science Press Beijing China 2007(Chinese)
[19] M A Abdul Hussain ldquoBifurcation solutions of elastic beamsequation with small perturbationrdquo International Journal ofMathematical Analysis vol 3 no 17ndash20 pp 879ndash888 2009
[20] J Berkovits ldquoOn the bifurcation of large amplitude solutionsfor a system of wave and beam equationsrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 52 no 1 pp 343ndash354 2003
[21] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[22] C Polymilis C Skokos G Kollias G Servizi and G TurchettildquoBifurcations of beam-beam like mapsrdquo Journal of Physics AMathematical and General vol 33 no 5 pp 1055ndash1064 2000
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
Advances in Mathematical Physics 5
Differentiating both sides of (45) and using (28) we get
(119892minus1
0)10158401015840
(1198920(119910)) + 120582ℎ (119910) = 0 (minus119905
0le 119910 le 119905
0) (44)
From (43) it follows that
lim119910darrminus1199050
1198921015840
0(119910) = lim
119910uarr1199050
1198921015840(119910) = +infin (45)
Define 1205930= 119892minus1
0 Then (44) and (45) say that
12059310158401015840
0(119909) + 120582ℎ (120593
0(119909)) = 0 (119886 lt 119909 lt 119887
1)
1205931015840
0(119886) = 120593
1015840
0(1198871) = 0
(46)
From (46) and assumption (H1) we see that
lim119909uarr1198871
12059310158401015840
0(119909) = lim
119909uarr1198871
minus 120582ℎ (1205930(119909)) = lim
119909uarr1199050
minus 120582ℎ (119910) = ℎ (1199050) = 0
(47)
and similarly lim119909darr11988612059310158401015840
0(119909) = 0 By defining 120593
0(119909) = 120593
0(1198871)
when 1198871lt 119909 le 119887 the function120593
0is extended and thus defined
on [119886 119887] such that
12059310158401015840
0(119909) + 120582ℎ (120593
0(119909)) = 0 (119886 le 119909 le 119887)
1205931015840
0(119886) = 120593
1015840
0(119887) = 0
(48)
The proof is complete
Example 7 (BVP (2) revisited) Again we are concerned withBVP (2) namely
12059310158401015840+ 120582 sin120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(49)
The problem has nonconstant solutions if and only if (119887 minus119886)2|120582| lt 120587
2 Indeed we see that
sin (119909 + 120587) = minus sin119909 (119909 isin R) (50)
and for some 1199090isin R
0 lt sin119909 = minus sin (minus119909) (0 lt 119909 lt 120587)
ℎ1015840
(0) gt 0
(51)
119909 997891997888rarr radicint
119909
0
sin 119905 d119905 is a concave function on [0 120587] (52)
Note that (52) is trivial since
radicint
119909
0
sin 119905 d119905 = radic1 minus cos119909 = radic2 cos 1199092
(0 le 119909 le 120587) (53)
Remark 8 In this paper we have solved a class of bifurcationproblems for Neumann boundary value problems for semi-linear elliptic equations namely
12059310158401015840+ 120582ℎ ∘ 120593 = 0 in (119886 119887)
1205931015840
(119886) = 1205931015840
(119887) = 0
(54)
the governing equation occurring in this boundary valueproblem generalizes the classical beam equation in the sensethat the nonlinear interaction assumes the form R ni 119906 997891rarr
ℎ(sin 119906) isin R instead of R ni 119906 997891rarr sin 119906 isin R What is moreimportant is that the bifurcation problem for the classicalbeam equation can be solved using abstract bifurcationtheorems in nonlinear analysis while the generalized beamequations can not be We provided a unified approach tounderstand this class of problems Indeed our method isquite general and very elementary
It is worthwhile to mention that bifurcation problemsassociated with beam equations other than the type (3) havebeen extensively studied see [19ndash22] and the profound refer-ences cited therein The approaches frequently used in theliterature are quite different from ours and have as founda-tions much advanced complicated knowledge in functionalanalysis
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author is grateful to the anonymous referees for theirvaluable suggestions
References
[1] L Nirenberg Topics in Nonlinear Functional Analysis vol 6of Courant Lecture Notes in Mathematics New York UniversityCourant Institute of Mathematical Sciences New York NYUSA 2001 Chapter 6 by E Zehnder Notes by R A ArtinoRevised reprint of the 1974 original
[2] T Ma and S Wang ldquoBifurcation of nonlinear equations ISteady state bifurcationrdquoMethods and Applications of Analysisvol 11 no 2 pp 155ndash178 2004
[3] T Ma and S Wang ldquoBifurcation of nonlinear equations IIDynamic bifurcationrdquo Methods and Applications of Analysisvol 11 no 2 pp 179ndash209 2004
[4] M A Krasnoselskii Topological Methods in the Theory ofNonlinear Integral Equations translated byAHArmstrong andedited by J Burlak Pergamon Press New York NY USA 1964
[5] S N Chow and J K Hale Methods of Bifurcation Theory vol251 of rundlehren der Mathematischen Wissenschaften [Funda-mental Principles of Mathematical Science] Springer New YorkNY USA 1982
[6] D Henry Geometric Theory of Semilinear Parabolic EquationsLecture Notes in Mathematics Springer New York NY USA1981
[7] T Kato Perturbation Theory for Linear Operators Classicsin Mathematics reprint of the 1980 edition Springer BerlinGermany 1995
[8] T Ma and S Wang ldquoStructure of 2D incompressible flows withthe DIRichlet boundary conditionsrdquo Discrete and ContinuousDynamical Systems Series B vol 1 no 1 pp 29ndash41 2001
[9] T Ma and S Wang ldquoStructural classification and stability ofdivergence-free vector fieldsrdquo Physica D vol 171 no 1-2 pp107ndash126 2002
6 Advances in Mathematical Physics
[10] T Ma and S Wang ldquoAttractor bifurcation theory and its appli-cations to Rayleigh-Benard convectionrdquo Communications onPure and Applied Analysis vol 2 no 4 pp 591ndash599 2003
[11] T Ma and S Wang ldquoDynamic bifurcation and stability in theRayleighBenard convectionrdquo Communications in MathematicalSciences vol 2 no 2 pp 159ndash183 2004
[12] T Ma and S Wang Bifurcation Theory and Applications vol53 of World Scientific Series on Nonlinear Science Series AMonographs and Treatises World Scientific Publishing Co PteLtd Hackensack NJ Hackensack NJ USA 2005
[13] T Ma and S Wang Geometric Theory of Incompressible Flowswith Applications to Fluid Dynamics Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 2005
[14] H Berestycki ldquoOn some nonlinear Sturm-Liouville problemsrdquoJournal of Differential Equations vol 26 no 3 pp 375ndash390 1977
[15] G Birkhoff A Source Book in Classical Analysis Harvard Uni-versity Press Cambridge Mass USA 1973
[16] P H Rabinowitz ldquoNonlinear Sturm-Liouville problems forsecond order ordinary differential equationsrdquo Communicationson Pure and Applied Mathematics vol 23 pp 939ndash961 1970
[17] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005
[18] T Ma and S Wang Stability and Bifurcation Problems for Non-linear Evolution Equations Science Press Beijing China 2007(Chinese)
[19] M A Abdul Hussain ldquoBifurcation solutions of elastic beamsequation with small perturbationrdquo International Journal ofMathematical Analysis vol 3 no 17ndash20 pp 879ndash888 2009
[20] J Berkovits ldquoOn the bifurcation of large amplitude solutionsfor a system of wave and beam equationsrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 52 no 1 pp 343ndash354 2003
[21] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[22] C Polymilis C Skokos G Kollias G Servizi and G TurchettildquoBifurcations of beam-beam like mapsrdquo Journal of Physics AMathematical and General vol 33 no 5 pp 1055ndash1064 2000
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 Advances in Mathematical Physics
[10] T Ma and S Wang ldquoAttractor bifurcation theory and its appli-cations to Rayleigh-Benard convectionrdquo Communications onPure and Applied Analysis vol 2 no 4 pp 591ndash599 2003
[11] T Ma and S Wang ldquoDynamic bifurcation and stability in theRayleighBenard convectionrdquo Communications in MathematicalSciences vol 2 no 2 pp 159ndash183 2004
[12] T Ma and S Wang Bifurcation Theory and Applications vol53 of World Scientific Series on Nonlinear Science Series AMonographs and Treatises World Scientific Publishing Co PteLtd Hackensack NJ Hackensack NJ USA 2005
[13] T Ma and S Wang Geometric Theory of Incompressible Flowswith Applications to Fluid Dynamics Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 2005
[14] H Berestycki ldquoOn some nonlinear Sturm-Liouville problemsrdquoJournal of Differential Equations vol 26 no 3 pp 375ndash390 1977
[15] G Birkhoff A Source Book in Classical Analysis Harvard Uni-versity Press Cambridge Mass USA 1973
[16] P H Rabinowitz ldquoNonlinear Sturm-Liouville problems forsecond order ordinary differential equationsrdquo Communicationson Pure and Applied Mathematics vol 23 pp 939ndash961 1970
[17] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005
[18] T Ma and S Wang Stability and Bifurcation Problems for Non-linear Evolution Equations Science Press Beijing China 2007(Chinese)
[19] M A Abdul Hussain ldquoBifurcation solutions of elastic beamsequation with small perturbationrdquo International Journal ofMathematical Analysis vol 3 no 17ndash20 pp 879ndash888 2009
[20] J Berkovits ldquoOn the bifurcation of large amplitude solutionsfor a system of wave and beam equationsrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 52 no 1 pp 343ndash354 2003
[21] S Lenci G Menditto and A M Tarantino ldquoHomoclinic andheteroclinic bifurcations in the non-linear dynamics of a beamresting on an elastic substraterdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 615ndash632 1999
[22] C Polymilis C Skokos G Kollias G Servizi and G TurchettildquoBifurcations of beam-beam like mapsrdquo Journal of Physics AMathematical and General vol 33 no 5 pp 1055ndash1064 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
top related