Applied Mathematics and Computation 219 (2013) 5261–5267

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Multiplicity of solutions of a two point boundary value problemfor a fourth-order equation

Alberto Cabada a,1, Stepan Tersian b,⇑a Departamento de Análise Matemática, Facultade de Matemáticas, Universitade de Santiago de Compostela, Santiago de Compostela, Spainb Department of Mathematical Analysis, University of Ruse, 7017 Ruse, Bulgaria

a r t i c l e i n f o a b s t r a c t

Keywords:Three critical point theoremTwo-points BVPFourth-order equation

0096-3003/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.amc.2012.11.066

⇑ Corresponding author.E-mail addresses: alberto.cabada@usc.es (A. Caba

1 Partially supported by FEDER and Ministerio de E

We study the existence of multiple solutions for semi linear fourth-order differential equa-tion describing elastic deflections. The proof of the main result is based on a three criticalpoint theorem.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

A multiplicity result for critical points of C1 functionals on finite-dimensional spaces was proved by Ricceri [13] asfollows:

Theorem 1. Let X be a finite-dimensional real Hilbert space, and let J : X ! R be a C1-function such that

lim infjjxjj!þ1

JðxÞjjxjj2

P 0:

Moreover, let x0 2 X and r; s 2 R with 0 < r < s, be such that

infx2X

JðxÞ 6 Jðx0Þ 6 inffJðxÞ : r 6 jjx� x0jj 6 sg:

Then, there exists k� > 0 such that the equation

xþ k�J0ðxÞ ¼ x0

has at least three solutions.The last theorem was extended to a wider class of functionals in finite-dimensional spaces in [4, Theorem 3]. The proof of

Theorem 1 is based on earlier results of Ricceri on the well-posedness of some minimization problems. Cabada and Iannizz-otto in [3] extended Theorem 1 removing the assumption on the finite dimension of space X and condition on annulusaround point x0. Their given result is the following.

Theorem 2. Let ðX; jj � jjÞ be a uniformly convex Banach space with strictly convex dual space, J 2 C1ðXÞ be a functional withcompact derivative, x0; x1 2 X; p; r 2 R be such that p > 1; r > 0 and jjx1 � x0jj < r. Let the following conditions be satisfied:

(i) lim inf jjxjj!1JðxÞjjxjjp P 0;

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da), sterzian@uni-ruse.bg (S. Tersian).ducación y Ciencia, Spain, MTM 2010-15314.

5262 A. Cabada, S. Tersian / Applied Mathematics and Computation 219 (2013) 5261–5267

(ii) infx2XJðxÞ < inffJðxÞ : jjx� x0jj 6 rg;(iii) Jðx1Þ 6 inffJðxÞ : jjx� x0jj ¼ rg.

Then, there exists a nonempty open set A # �0;þ1½ such that for all k 2 A the functional x #jjx�x0 jjp

p þ kJðxÞ has at least threecritical points in X.

As a consequence of this result we can deduce the following one.

Theorem 3. Suppose that the conditions (ii) and (iii) of Theorem 2 are trivially satisfied and the functional J is bounded from belowon X, i.e.

(i1) 9C 2 R; such that JðxÞP C; 8 x 2 X.Then, the assertion of Theorem 2 is still valid.

Proof. Let us consider the functional

JCðxÞ ¼jjx� xjjp

pþ kðJðxÞ � CÞ:

Since

infx2XðJðxÞ � CÞ ¼ inf

x2XðJðxÞÞ � C

and

inffJðxÞ � C : jjx� x0jj 6 rg ¼ inffJðxÞ : jjx� x0jj 6 rg � C;

the conditions (ii) and (iii) of Theorem 2 are satisfied for the functional JC .Clearly JCðxÞP 0 and therefore (i) is also satisfied for JC . Then, by Theorem 2 there exists a set AC # �0;þ1½ such that for all

k 2 AC the functional JC has at least three critical points, which are also critical points of the functional J. h

Theorem 2 improves Theorem 1 since the space X is supposed to be infinite dimensional and the geometric conditions aremore general and easy to verify in the different applications. Another three critical points theorems were proved and pre-sented in [5,7,12,11,8,1,6,14].

In this paper, we apply Theorem 2 to a two-point boundary value problem for fourth-order differential equation describ-ing elastic deflections of a rod with clamped ends. As a model problem we consider the following one for the nonlinearfourth-order equation

ðPkÞ :

uð4ÞðxÞ þ kf ðx;uðxÞÞ ¼ 0; 0 < x < 1;uð0Þ ¼ u0ð0Þ ¼ u00ð1Þ ¼ 0;u000ð1Þ ¼ kgðuð1ÞÞ;

8><>:

where the functions f : ½0;1� � R! R and g : R! R are continuous and k P 0 is a real parameter.Assume that the following conditions on the functions f and g are fulfilled:

ðf1Þ f : ½0;1� � R! R is a continuous function;ðf2Þ There exist k2 > k1 > 0, such that t f ðx; tÞP 0 for every x 2 ½0;1� and for all t, such that jtj 6 k1 or jtjP k2;ðg1Þ g : R! R is a continuous function;ðg2Þ t g(t) P 0 for all t, such that jtj 6 k1 or jtjP k2;ðhÞ There exist c1 < �k1 or c1 > k1 and a > b > 0 such that:

Z c1

0f ðx; tÞdt 6 �a; 8 x 2 ½0;1� and

Z c1

0gðtÞdt 6 b:

Our main result is the following

Theorem 4. Assume that the conditions ðf1Þ; ðf2Þ; ðg1Þ; ðg2Þ and ðhÞ on the functions f and g are satisfied. Then, there exists a setA # �0;þ1½ such that for each k 2 A, the problem Pkð Þ has at least two nontrivial solutions.

Theorem 4 is proved in Section 3. Further comments and examples are also given. The proof is based on the variationalapproach presented in Section 2. We refer to the reader for similar problems, obtained by topological and variational meth-ods, given in [2,10,9,8] and references therein.

2. Preliminaries

In this section we give some definitions and auxiliary results for the variational formulation and treatment of the problemðPkÞ.

A. Cabada, S. Tersian / Applied Mathematics and Computation 219 (2013) 5261–5267 5263

Let Xp; p P 2 be the space

Xp :¼ fu 2W2;pð0;1Þ : uð0Þ ¼ u0ð0Þ ¼ 0g;

with the norm

jjujjpp :¼Z 1

0ju00ðtÞjpdt:

Here W2;pð0;1Þ is the Sobolev space of functions u : ½0;1� ! R such that u and its generalized derivative u0 are absolutelycontinuous functions and u00 2 Lpð0;1Þ. If p ¼ 2 we denote

X :¼ X2 ¼ fu 2W2;2ð0;1Þ ¼ H2ð0;1Þ : uð0Þ ¼ u0ð0Þ ¼ 0g;

which is a Hilbert space with inner product

hu;vi :¼Z 1

0u00ðtÞv 00ðtÞdt

and norm

jjujj2 :¼Z 1

0ðu00ðtÞÞ2dt:

By a standard way one can prove the following

Lemma 5. The following assertions are fulfilled:

(a) The norm

jjujjpp ¼Z 1

0ju00ðtÞjpdt

is equivalent to the usual norm in Xp

jjujjpXp¼Z 1

0ju00ðtÞjp þ ju0ðtÞjp þ juðtÞjp� �

dt:

(b) The inclusion Xp � C1ð½0;1�Þ is compact.

To study the problem ðPkÞ, we consider the functional I : X ! R defined by

IðuÞ :¼ jjujj2

2þ kJðuÞ;

where

JðuÞ :¼Z 1

0Fðx;uðxÞÞdxþ Gðuð1ÞÞ

and

Fðx; tÞ ¼Z t

0f ðx; sÞds; GðtÞ ¼

Z t

0gðsÞds:

The functional I : X ! R is Fréchet differentiable and, for any u;v 2 X, the following identity holds

hI0ðuÞ;vi ¼Z 1

0u00ðxÞv 00ðxÞdxþ k

Z 1

0f ðx;uðxÞÞvðxÞdxþ gðuð1ÞÞvð1Þ

� �:

Following the framework of Tersian and Chaparova [15] and Yang et al. [16], we arrive at the standard regularity property.

Lemma 6. If u is a critical point of I in X, then u is a classical solution of the problem ðPkÞ.

3. Proof of the main result and comments

This section is devoted to proof the main result of this paper. Moreover, we present an example in which we apply thegeneral result to a particular case of problem ðPkÞ.

5264 A. Cabada, S. Tersian / Applied Mathematics and Computation 219 (2013) 5261–5267

Proof of Theorem 4. By Lemma 5 for p ¼ 2 and the continuity of functions f and g we deduce that the derivative of thefunctional J is compact.

Let us see that J is bounded from below.From f1ð Þ and f2ð Þ we have that the function F attains its finite minimum at a point of ½0;1� � ½�k2;�k1�, if c1 < 0, or in

½0;1� � ½k1; k2�, if c1 > 0. By g1ð Þ and g2ð Þ it follows that G is bounded from below and then, J is bounded from below too.To verify assumption (ii) of Theorem 2, let us define the function

unðxÞ ¼0; 0 6 x 6 1=n;

c1 cos2ðnpx2 Þ; 1=n 6 x 6 2=n;

c1; 2=n 6 x 6 1;

8><>:

where c1 satisfies assumption ðhÞ.It is clear that un 2 X � C1ð½0;1�Þ; 8 n 2 N.We have

JðunÞ ¼Z 1=n

0Fðx;0Þdxþ

Z 2=n

1=nF x; c1 cos2 npx

2

� �� �dxþ

Z 1

2=nFðx; c1Þdxþ Gðc1Þ: ð1Þ

Since c1 cos2 npx2

� ��� �� 6 jc1j, by f1ð Þ there exists A > 0 such that

F x; c1 cos2 npx2

� �� ���� ��� 6 A:

Then, by (1) and ðhÞ, we have that

JðunÞ 6An� a 1� 2

n

� �þ b ¼ Aþ 2a

nþ b� a:

Moreover, for sufficiently large n, it is clear that ðAþ 2aÞ=n < a� b and therefore JðunÞ < 0 for such n. As consequence,infu2XJðuÞ < 0.

On the other hand, since Fðx; tÞP 0 for all ðx; tÞ 2 ½0;1� � ½�k1; k1� and GðtÞP 0 for jtj 6 k1, we have that

inffJðuÞ : u 2 X; jjujj1 6 k1gP 0:

From the fact that jjujj1 6 jjujj, we deduce that

fu 2 X; jjujj 6 k1g# fu 2 X; jjujj1 6 k1g

and, as a consequence

inffJðuÞ : u 2 X; jjujj 6 k1gP inffJðuÞ : u 2 X; jjujj1 6 k1gP 0:

So, we deduce condition (ii) with r ¼ k1 and x0 ¼ 0.In order to verify (iii) we take r ¼ k1 and x0 ¼ x1 ¼ 0 and deduce that

0 ¼ Jð0Þ 6 inffJðuÞ : u 2 X; jjujj 6 k1g 6 inffJðuÞ : u 2 X; jjujj ¼ k1g:

The result is proved. h

We finish this section by the following example for a particular case of functions f and g.

Example 7. Define the functions f and g as follows: f ðx; tÞ ¼ aðxÞf1ðtÞ where aðxÞ ¼ 2þ sinðpxÞ,

f1ðtÞ ¼

ðt � 2pÞ2 t P 2p;3ðt � pÞðt � 2pÞ; p 6 t 6 2p;sin t; �p 6 t 6 p;�3ðt þ pÞðt þ 2pÞ; �2p 6 t 6 �p;�ðt þ 2pÞ2 t 6 �2p

8>>>>>><>>>>>>:

and

gðtÞ ¼

0 t P 2p;1p2 ðt � pÞðt � 2pÞ; p 6 t 6 2p;sin t; �p 6 t 6 p;� 1

p2 ðt þ pÞðt þ 2pÞ; �2p 6 t 6 �p;0 t 6 �2p:

8>>>>>><>>>>>>:

Taking c1 ¼ �3p we have

A. Cabada, S. Tersian / Applied Mathematics and Computation 219 (2013) 5261–5267 5265

Z c1

0f ðx; tÞdt ¼ aðxÞ

Z �p

0sin tdt � 3

Z �2p

�pðt þ pÞðt þ 2pÞdt �

Z �3p

�2pðt þ 2pÞ2dt

� �¼ aðxÞ 2� p3

6

� �6 2� p3

6� �3:1677

and

Z c1

0gðtÞdt ¼

Z �p

0sin tdt � 1

p2

Z �2p

�pðt þ pÞðt þ 2pÞdt ¼ 2� p

6� 1:4764:

So, assumptions ðf1Þ–ðhÞ are satisfied with k1 ¼ p; k2 ¼ 2p; c1 ¼ �3p;a ¼ 16 p

3 � 2 and b ¼ 2� 16 p. In consequence there exists

a set A # �0;þ1½ such that for each k 2 A, the problem Pkð Þ has at least two nontrivial solutions. The graphs of functions f1ðtÞand gðtÞ for �3p 6 t 6 3p are presented on Figs. 1,2 respectively.

One can verify that the variational approach based on Theorem 2 can be also applied to boundary value problem forfourth-order p-Laplacian equations of the type

ðPp;kÞ :

ðupðu00ðxÞÞ00 þ kf ðx; uðxÞÞ ¼ 0; 0 < x < 1;

uð0Þ ¼ u0ð0Þ ¼ u00ð1Þ ¼ 0;ðupðu00ÞÞ

0ð1Þ ¼ kgðuð1ÞÞ;

8><>:

where p > 2 and upðtÞ ¼ tjtjp�2.In this case we consider the functional Ip : Xp ! R

IpðuÞ :¼jjujjpp

pþ kJðuÞ:

The functional Ip : Xp ! R is Fréchet differentiable and its critical points are weak solutions of the problem ðPp;kÞ. Note thata function u 2 Xp is a weak solution of problem ðPp;kÞ iff for any v 2 Xp, the following identity holds

Fig. 1. Graph of the function f1ðtÞ for �3p 6 t 6 3p.

Fig. 2. Graph of the function gðtÞ for �3p 6 t 6 3p.

5266 A. Cabada, S. Tersian / Applied Mathematics and Computation 219 (2013) 5261–5267

Z 1

0upðu00ðxÞÞv 00ðxÞdxþ k

Z 1

0f ðx; uðxÞÞvðxÞdxþ gðuð1ÞÞvð1Þ

� �¼ 0:

By a solution of the problem ðPp;kÞ we mean a function u 2 C1ð½0;1�Þ such that ðupðu00ðxÞÞ00 2 Cð½0;1�Þ and the boundary con-

ditions and the equation are satisfied in ½0;1�. By a standard way it follows that the weak solutions are solutions of theproblem.

Using Theorem 2 and Lemma 5 we arrive at the following existence and multiplicity result.

Theorem 8. Assume that the conditions ðf1Þ; ðf2Þ; ðg1Þ; ðg2Þ and ðhÞ on the functions f and g are satisfied. Then, there exists a setA # �0;þ1½ such that for each k 2 A, the problem Pp;k

� �has at least two nontrivial solutions.

Finally, we point out that the conclusion of Theorem 4 is still valid if we suppose assumptions ðf1Þ–ðhÞ for functions f and gconsidering the problem

ðPk;lÞ :

uð4ÞðxÞ þ kf ðx;uðxÞÞ ¼ 0; 0 < x < 1;uð0Þ ¼ u0ð0Þ ¼ u00ð1Þ ¼ 0;u000ð1Þ ¼ lgðuð1ÞÞ;

8><>:

The corresponding functional for the problem ðPk;lÞ, Il : X ! R, is defined in this case by

IlðuÞ ¼jjujj2

2þ kJlðuÞ;

where

JlðuÞ ¼Z 1

0Fðx;uðxÞÞdxþ l

kGðuð1ÞÞ

and

Fðx; tÞ ¼Z t

0f ðx; sÞds; GðtÞ ¼

Z t

0gðsÞds:

Since condition ðhÞ is fulfilled, it is obvious that, ifR c1

0 gðtÞdt 6 0 then lk

R c10 gðtÞdt 6 0 6 b for all l > 0. On the contrary, ifR c1

0 gðtÞdt > 0, we have, provided that 0 < l 6 k, that the following inequalities are fulfilled:

Z c1

0

lk

gðtÞdt ¼ lk

Z c1

0gðtÞdt 6

Z c1

0gðtÞdt 6 b:

Thus, following the steps of the proof of Theorem 4, we verify that functional Jl satisfies assumptions (i)–(iii) of Theo-rem 2, and so we arrive at the next existence and multiplicity result for problem ðPk;lÞ.

Theorem 9. Assume that the conditions ðf1Þ; ðf2Þ; ðg1Þ; ðg2Þ and ðhÞ on the functions f and g are satisfied. Then, there exists a setA # �0;þ1½ such that for each k 2 A and for every l 2 ð0; k� if

R c10 gðtÞdt > 0 and for all l > 0 if

R c10 gðtÞdt 6 0, the problem

Pk;l� �

has at least two nontrivial solutions.Combining this last result we can deduce the following existence and multiplicity result for the p-Laplacian case.

Theorem 10. Assume that the conditions ðf1Þ; ðf2Þ; ðg1Þ; ðg2Þ and ðhÞ on the functions f and g are satisfied. Then, there exists a setA # �0;þ1½ such that for each k 2 A and for every l 2 ð0; k� if

R c10 gðtÞdt > 0 and for all l > 0 if

R c10 gðtÞdt 6 0, the problem

ðPp;k;lÞ :

ðupðu00ðxÞÞ00 þ kf ðx; uðxÞÞ ¼ 0; 0 < x < 1;

uð0Þ ¼ u0ð0Þ ¼ u00ð1Þ ¼ 0;ðupðu00ÞÞ

0ð1Þ ¼ lgðuð1ÞÞ

8><>:

has at least two nontrivial solutions.

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