research article bifurcation problems for generalized...

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


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


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


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


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)


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)


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


[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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Stochastic AnalysisInternational Journal of


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




(119891minus1)1015840

(119891 (119905)) =1

1198911015840(119905)

(14)


119867(119909) = 1198702

(119909) = int

1199090+119909

1199090

ℎ (119905) d119905 (0 le 119909 le 119897) (15)



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


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)



1198701015840

(0) = lim119909darr0

1198701015840

(119909) = lim119909darr0

1198671015840(119909)

2radic119867 (119909)

(19)


(1198701015840

(0))2

= lim119909darr0

(1198671015840(119909))2

4119867 (119909)= lim119909darr0

21198671015840(119909)119867

10158401015840(119909)

41198671015840(119909)

=ℎ1015840(1199090)

2

(20)


Lemma 6 Define

119901 (119905) = int

119905

0

1

radic119867 (119905) minus 119867 (119910)

d119910 (0 lt 119905 le 119897) (21)


lim119905darr0

119901 (119905) =120587

radic2ℎ1015840(1199090)

(22)


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)


1


(24)

and thus

119901 (119905) = int

1

0

1


d119906 (25)


119901 (119905) = int

1

0

1


d119906

le int

1

0

1

2radic119906 (1 minus 119906)

sdot1

1198701015840(119870minus1(radic119867 (119905)))

d119906

=120587

21198701015840(119870minus1(radic119867 (119905)))

lt infin

(26)


lim119905darr0

119901 (119905)

= lim119905darr0

int

1

0

1


d119906

= int

1

0

1


d119906

=120587

2radicℎ1015840(1199090)

(27)






0) = 120593

1015840(1198870) We will show









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)


1

2 (1198921015840(119910))2+ 120582119867 (119910) = 119862

1(120593 (1198860) lt 119910 lt 120593 (119887

0)) (31)


Then

1198921015840(119910) =

1

radic2120582

sdot1

radic1198622minus 119867 (119910)

(120593 (1198860) lt 119910 lt 120593 (119887

0)) (32)


119867(119910) lt 1198622

(120593 (1198860) lt 119910 lt 120593 (119887

0)) (33)



119867(119910) 997888rarr 1198622

(119910 darr 120593 (1198860) or119910 uarr 120593 (119887

0)) (34)


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)


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)



2

120582119901 (119905) gt lim

119905darr0

radic2

120582119901 (119905) =

120587

radic120582ℎ1015840(1199090)

(37)



119887 minus 119886 ge120587

radic120582ℎ1015840(1199090)

(38)


0gt 0 such that

radic2

120582119901 (1199050) lt 119887 minus 119886 (39)

Let

1198871= 119886 + radic

2

120582119901 (1199050) lt 119887 (40)


0 1199050] by119892(minus119905

0) = 119886

and

1198921015840

0(119910) =

1

radic2120582

sdot1

radic119867 (1199050) minus 119867 (119910)

(minus1199050lt 119910 lt 119905

0) (41)


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)



(119892minus1

0)10158401015840

(1198920(119910)) + 120582ℎ (119910) = 0 (minus119905

0le 119910 le 119905

0) (44)



1198921015840

0(119910) = lim

119910uarr1199050

1198921015840(119910) = +infin (45)

Define 1205930= 119892minus1


12059310158401015840

0(119909) + 120582ℎ (120593

0(119909)) = 0 (119886 lt 119909 lt 119887

1)

1205931015840

0(119886) = 120593

1015840

0(1198871) = 0

(46)


lim119909uarr1198871

12059310158401015840

0(119909) = lim

119909uarr1198871

minus 120582ℎ (1205930(119909)) = lim

119909uarr1199050

minus 120582ℎ (119910) = ℎ (1199050) = 0

(47)


0(119909) = 0 By defining 120593

0(119909) = 120593

0(1198871)



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)



12059310158401015840+ 120582 sin120593 = 0 in (119886 119887)

1205931015840

(119886) = 1205931015840

(119887) = 0

(49)






ℎ1015840

(0) gt 0

(51)

119909 997891997888rarr radicint

119909

0



radicint

119909

0


(0 le 119909 le 120587) (53)


12059310158401015840+ 120582ℎ ∘ 120593 = 0 in (119886 119887)

1205931015840

(119886) = 1205931015840

(119887) = 0

(54)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(119891minus1)1015840

(119891 (119905)) =1

1198911015840(119905)

(14)


119867(119909) = 1198702

(119909) = int

1199090+119909

1199090

ℎ (119905) d119905 (0 le 119909 le 119897) (15)



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


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)



1198701015840

(0) = lim119909darr0

1198701015840

(119909) = lim119909darr0

1198671015840(119909)

2radic119867 (119909)

(19)


(1198701015840

(0))2

= lim119909darr0

(1198671015840(119909))2

4119867 (119909)= lim119909darr0

21198671015840(119909)119867

10158401015840(119909)

41198671015840(119909)

=ℎ1015840(1199090)

2

(20)


Lemma 6 Define

119901 (119905) = int

119905

0

1

radic119867 (119905) minus 119867 (119910)

d119910 (0 lt 119905 le 119897) (21)


lim119905darr0

119901 (119905) =120587

radic2ℎ1015840(1199090)

(22)


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)


1


(24)

and thus

119901 (119905) = int

1

0

1


d119906 (25)


119901 (119905) = int

1

0

1


d119906

le int

1

0

1

2radic119906 (1 minus 119906)

sdot1

1198701015840(119870minus1(radic119867 (119905)))

d119906

=120587

21198701015840(119870minus1(radic119867 (119905)))

lt infin

(26)


lim119905darr0

119901 (119905)

= lim119905darr0

int

1

0

1


d119906

= int

1

0

1


d119906

=120587

2radicℎ1015840(1199090)

(27)






0) = 120593

1015840(1198870) We will show









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)


1

2 (1198921015840(119910))2+ 120582119867 (119910) = 119862

1(120593 (1198860) lt 119910 lt 120593 (119887

0)) (31)


Then

1198921015840(119910) =

1

radic2120582

sdot1

radic1198622minus 119867 (119910)

(120593 (1198860) lt 119910 lt 120593 (119887

0)) (32)


119867(119910) lt 1198622

(120593 (1198860) lt 119910 lt 120593 (119887

0)) (33)



119867(119910) 997888rarr 1198622

(119910 darr 120593 (1198860) or119910 uarr 120593 (119887

0)) (34)


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)


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)



2

120582119901 (119905) gt lim

119905darr0

radic2

120582119901 (119905) =

120587

radic120582ℎ1015840(1199090)

(37)



119887 minus 119886 ge120587

radic120582ℎ1015840(1199090)

(38)


0gt 0 such that

radic2

120582119901 (1199050) lt 119887 minus 119886 (39)

Let

1198871= 119886 + radic

2

120582119901 (1199050) lt 119887 (40)


0 1199050] by119892(minus119905

0) = 119886

and

1198921015840

0(119910) =

1

radic2120582

sdot1

radic119867 (1199050) minus 119867 (119910)

(minus1199050lt 119910 lt 119905

0) (41)


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)



(119892minus1

0)10158401015840

(1198920(119910)) + 120582ℎ (119910) = 0 (minus119905

0le 119910 le 119905

0) (44)



1198921015840

0(119910) = lim

119910uarr1199050

1198921015840(119910) = +infin (45)

Define 1205930= 119892minus1


12059310158401015840

0(119909) + 120582ℎ (120593

0(119909)) = 0 (119886 lt 119909 lt 119887

1)

1205931015840

0(119886) = 120593

1015840

0(1198871) = 0

(46)


lim119909uarr1198871

12059310158401015840

0(119909) = lim

119909uarr1198871

minus 120582ℎ (1205930(119909)) = lim

119909uarr1199050

minus 120582ℎ (119910) = ℎ (1199050) = 0

(47)


0(119909) = 0 By defining 120593

0(119909) = 120593

0(1198871)



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)



12059310158401015840+ 120582 sin120593 = 0 in (119886 119887)

1205931015840

(119886) = 1205931015840

(119887) = 0

(49)






ℎ1015840

(0) gt 0

(51)

119909 997891997888rarr radicint

119909

0



radicint

119909

0


(0 le 119909 le 120587) (53)


12059310158401015840+ 120582ℎ ∘ 120593 = 0 in (119886 119887)

1205931015840

(119886) = 1205931015840

(119887) = 0

(54)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0) = 120593

1015840(1198870) We will show









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)


1

2 (1198921015840(119910))2+ 120582119867 (119910) = 119862

1(120593 (1198860) lt 119910 lt 120593 (119887

0)) (31)


Then

1198921015840(119910) =

1

radic2120582

sdot1

radic1198622minus 119867 (119910)

(120593 (1198860) lt 119910 lt 120593 (119887

0)) (32)


119867(119910) lt 1198622

(120593 (1198860) lt 119910 lt 120593 (119887

0)) (33)



119867(119910) 997888rarr 1198622

(119910 darr 120593 (1198860) or119910 uarr 120593 (119887

0)) (34)


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)


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)



2

120582119901 (119905) gt lim

119905darr0

radic2

120582119901 (119905) =

120587

radic120582ℎ1015840(1199090)

(37)



119887 minus 119886 ge120587

radic120582ℎ1015840(1199090)

(38)


0gt 0 such that

radic2

120582119901 (1199050) lt 119887 minus 119886 (39)

Let

1198871= 119886 + radic

2

120582119901 (1199050) lt 119887 (40)


0 1199050] by119892(minus119905

0) = 119886

and

1198921015840

0(119910) =

1

radic2120582

sdot1

radic119867 (1199050) minus 119867 (119910)

(minus1199050lt 119910 lt 119905

0) (41)


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)



(119892minus1

0)10158401015840

(1198920(119910)) + 120582ℎ (119910) = 0 (minus119905

0le 119910 le 119905

0) (44)



1198921015840

0(119910) = lim

119910uarr1199050

1198921015840(119910) = +infin (45)

Define 1205930= 119892minus1


12059310158401015840

0(119909) + 120582ℎ (120593

0(119909)) = 0 (119886 lt 119909 lt 119887

1)

1205931015840

0(119886) = 120593

1015840

0(1198871) = 0

(46)


lim119909uarr1198871

12059310158401015840

0(119909) = lim

119909uarr1198871

minus 120582ℎ (1205930(119909)) = lim

119909uarr1199050

minus 120582ℎ (119910) = ℎ (1199050) = 0

(47)


0(119909) = 0 By defining 120593

0(119909) = 120593

0(1198871)



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)



12059310158401015840+ 120582 sin120593 = 0 in (119886 119887)

1205931015840

(119886) = 1205931015840

(119887) = 0

(49)






ℎ1015840

(0) gt 0

(51)

119909 997891997888rarr radicint

119909

0



radicint

119909

0


(0 le 119909 le 120587) (53)


12059310158401015840+ 120582ℎ ∘ 120593 = 0 in (119886 119887)

1205931015840

(119886) = 1205931015840

(119887) = 0

(54)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119892minus1

0)10158401015840

(1198920(119910)) + 120582ℎ (119910) = 0 (minus119905

0le 119910 le 119905

0) (44)



1198921015840

0(119910) = lim

119910uarr1199050

1198921015840(119910) = +infin (45)

Define 1205930= 119892minus1


12059310158401015840

0(119909) + 120582ℎ (120593

0(119909)) = 0 (119886 lt 119909 lt 119887

1)

1205931015840

0(119886) = 120593

1015840

0(1198871) = 0

(46)


lim119909uarr1198871

12059310158401015840

0(119909) = lim

119909uarr1198871

minus 120582ℎ (1205930(119909)) = lim

119909uarr1199050

minus 120582ℎ (119910) = ℎ (1199050) = 0

(47)


0(119909) = 0 By defining 120593

0(119909) = 120593

0(1198871)



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)



12059310158401015840+ 120582 sin120593 = 0 in (119886 119887)

1205931015840

(119886) = 1205931015840

(119887) = 0

(49)






ℎ1015840

(0) gt 0

(51)

119909 997891997888rarr radicint

119909

0



radicint

119909

0


(0 le 119909 le 120587) (53)


12059310158401015840+ 120582ℎ ∘ 120593 = 0 in (119886 119887)

1205931015840

(119886) = 1205931015840

(119887) = 0

(54)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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