Journal of Function Spaces

Volume 2014 (2014), Article ID 384958, 9 pages

http://dx.doi.org/10.1155/2014/384958

Research Article

Existence of Positive Solutions for Some Superlinear Fourth-Order Boundary Value Problems

Imed Bachar¹ and Habib Mâagli²

¹Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

²Department of Mathematics, College of Sciences and Arts, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia

Received 30 May 2014; Accepted 12 July 2014; Published 11 August 2014

Academic Editor: Józef Banaś

Copyright © 2014 Imed Bachar and Habib Mâagli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We are concerned with the following superlinear fourth-order equation , where are nonnegative constants such that and is a nonnegative continuous function that is required to satisfy some appropriate conditions related to a class satisfying suitable integrability condition. Our purpose is to prove the existence, uniqueness, and global behavior of a classical positive solution to the above problem by using a method based on estimates on the Green function and perturbation arguments.

1. Introduction

A natural motivation for studying higher order BVPs lies in their applications. It is well-known (see for instance [1]) that the deformation of an elastic beam in equilibrium state, whose both ends clamped, can be described by fourth-order BVP

subject to the boundary conditions

When the nonlinearity is nonnegative such problems have been extensively investigated by many researchers, and various forms of the equation and boundary condition have been discussed; see, for example, [1–15] and references therein.

In particular, in 2005, Ma and Tisdel [1] studied the existence of positive solutions to the sublinear BVP

where and is continuous which may be singular at both ends and and satisfying some integrable conditions. Using the method of lower and upper solutions for fourth-order boundary value problems, they have given some necessary and sufficient conditions for the existence of regular positive solutions to the boundary value problem (3).

In [8], the authors used fixed-point index results, to prove the existence of a positive solution for the boundary value problem

where is nondecreasing continuous function allowed to be superlinear and is continuous which may be singular at both ends and satisfying some adequate conditions.

Recently, in [4], the author considered problem (3), with and is a nonnegative continuous function on satisfying some hypotheses related to Karamata regular variation theory. Using the Schauder fixed-point theorem, he established the existence and uniqueness of a positive solution to (3).

Here, by using a method based on estimates on the Green function and perturbation arguments, we show that, for negative nonlinearity , problem (1) has a unique positive classical solution subject to some boundary conditions. More precisely, we are concerned with the following superlinear fourth order problem

where are nonnegative constants such that . Our goal is to answer the questions of existence, uniqueness, and global behavior of a classical positive solution to problem (5), where the nonnegative nonlinear term is required to satisfy some appropriate conditions related to the following class .

Definition 1. A Borel measurable function in belongs to the class if satisfies the following condition:

We will often refer in this paper to

the unique solution of the problem

Observe that, for ,

where .

Also we denote by the Green function of the operator , with the conditions , which can be explicitly given by

where and (see [2]).

The outline of the paper is as follows. In Section 2, we give some sharp estimates on Green’s function , including the following 3G-inequality: for each ,

where .

In particular, we derive from this 3G-inequality that, for each , one has

In Section 3, our purpose is to study the superlinear fourth-order problem (5). The nonlinearity is required to satisfy a combination of the following assumptions.

 (H ₁) is a nonnegative continuous function in . (H ₂)There exists a nonnegative function with such that for each , the map is nondecreasing on . (H ₃)For each , the function is nondecreasing on .

We will first exploit the 3G-inequality to prove that the inverse of fourth-order operators that are perturbed by a zero-order term are positivity preserving. That is, if is a positive measurable function and is a nonnegative function belonging to the class with , then the following problem

has a positive solution. It turns out to prove that problem (13) admits a positive Green function .

Based on the construction of this Green function and by using perturbation arguments, we prove the following.

Theorem 2. Assume -, then problem (5) has a positive solution in satisfying

where is a constant in .

Moreover, if hypothesis is also satisfied, then the solution to problem (5) satisfying (14) is unique.

Corollary 3. Let be a nonnegative function in such that the map is nondecreasing on . Let be a nonnegative continuous function on such that the function belongs to the class . Then for , the following problem

has a unique positive solution in satisfying

Observe that in Theorem 2, we obtain a positive classical solution to problem (5) whose behavior is not affected by the perturbed term. That is, it behaves like the solution of the homogeneous problem (8). As typical example of nonlinearity satisfying , we quote , where ,    is a positive continuous function on such that

and , with . Note that by using (9) and (17) the function belongs to the class .

As usual, let be the set of Borel measurable functions in and let be the set of nonnegative ones.

We define the kernel by

Let . We define the kernel by

Remark 4 (see [1]). Let such that the function is continuous and integrable on , then is the unique solution in of

2. Estimates on the Green’s Function

From the explicit expression of Green’s function (10) we derive the following.

Proposition 5 (see [4, 7]). For , one has

Using (22), we deduce the following.

Corollary 6. For , one has

(i)

(ii)For , the function is continuous on if and only if the integral converges.

Theorem 7 (3G inequality). For each , one has

where .

Proof. To prove the inequality, we denote by and we claim that is a quasi-metric, that is, for each ,

By symmetry, we may assume that .

Using (21), we deduce that

To show (25), we separate the proof into three cases.

Case 1   . In this case, one has

Case 2   . We obtain

Case 3   . One has

This completes the proof.

In the sequel, for any , we recall that

and we denote by

Proposition 8. Let be a function in ; then

(i)

(ii)For , one has

(iii)For , one has

In particular, for , one has

Proof. Let be a function in .

 (i) Using (24) and (23), one has, for ,

 Hence

 (ii) Since, for each , one has , then we deduce by Fatou’s lemma and (12), that

 which implies that, for ,

 (iii) Similarly, we prove inequality (34) by observing that

Inequality (35) follows from Proposition 8 (ii)-(iii) and the fact that . This completes the proof.

3. Proofs of Main Results

In this section, we aim at proving Theorem 2 and Corollary 3. So, we need the following preliminaries results.

For a nonnegative function in such that , we define the function on , by

where and

Next, we establish some inequalities on . In particular, we deduce that is well defined.

Lemma 9. Let be a nonnegative function in such that ; then, for each and , one has

 (i).  In particular, is well defined in . (ii)

 where and .(iii) and .(iv).

Proof. (i) The assertion is clear for .

Assume that inequality in (i) holds for some ; then by using (42) and (12), we obtain

Now, since , it follows that is well defined in .

(ii) Using (22) and (42), we obtain (43) by induction.

(iii) Since , then equalities in (iii) are clear for .

Assume that for a given integer and , one has

Using (42) and Fubini-Tonelli theorem, we obtain

On the other hand, by (42) and (45), we deduce that

(iv) Let and . By Lemma 9 (i) and (22), one has

Hence the series converges.

So we deduce by the dominated convergence theorem and Lemma 9 (iii) that

Proposition 10. Let be a nonnegative function in such that . Then the function is continuous on .

Proof. We claim that for , the function is continuous on .

The assertion is clear for .

Now assume that for a given integer , the function is continuous on . By (42), one has

Since the function is continuous on and by Lemma 9 (i) and (22)

we deduce by the dominated convergence theorem that is continuous on .

This proves our claim.

Now by Lemma 9 (i) and (22), one has, for each ,

This implies that the series is uniformly convergent on and therefore the function is continuous on .

The proof is completed.

Lemma 11. Let be a nonnegative function in such that . Then for in , one has

Proof. Since , we deduce from Lemma 9 (i) that

On the other hand, from the expression of , one has

Since the series is convergent, we deduce by (55) and (42) that

That is,

Now from (54) and Lemma 9 (i) (with ), we obtain

This implies that

So it follows that and by (57) and Lemma 9 (i) (with ), one has

In the sequel, for a given nonnegative function such that , we define the operator by

Using (53) and (22), we obtain the following.

Corollary 12. Let be a nonnegative function in such that and ; then the following statements are equivalent.

(i)The function is continuous on .(ii)The integral converges.

Next, we will prove that the kernel satisfies the following resolvent equation.

Lemma 13. Let be a nonnegative function in such that and . Then satisfies the following resolvent equation:

In particular, if , one has

Proof. Let , then by (57) one has

which implies by Fubini-Tonelli theorem that, for ,

On the other hand, by Lemma 9 (iii) and Fubini-Tonelli theorem, we obtain for and

that is,

So we obtain

This completes the proof.

Proposition 14. Let be a nonnegative function in such that and let such that is continuous and integrable on . Then is the unique nonnegative solution in of the perturbed fourth-order equation (13) satisfying

Proof. It is clear by Corollary 12 that the function is continuous on . Using (62) and (22), there exists a nonnegative constant such that

So we deduce that

Hence by using Remark 4, the function satisfies the equation

and by integration inequalities (53), we obtain (69).

It remains to prove the uniqueness. Assume that is another nonnegative solution in of problem (13) satisfying (69).

Since the function is continuous on and by (69) and (70), the function is integrable on ; then it follows by Remark 4 that the function satisfies

From the uniqueness in Remark 4, we deduce that

Hence

Now since by (69), (70), and (35), one has

then by (63), we deduce that . This completes the proof.

Proof of Theorem 2. Let and with and recall that

Since satisfies , then there exists a positive function in such that and for each , the map is nondecreasing on .

Let

and define the operator on by

By (62) and Proposition 8, one has

and by , we obtain

So we claim that is invariant under . Indeed, using (81) and (80), one has for

Next, we will prove that the operator is nondecreasing on . Indeed, let be such that . Since the map is nondecreasing on , for , we obtain

Now, we consider the sequence defined by and , for . Since is invariant under , one has and by the monotonicity of , we deduce that

Hence by dominated convergence theorem and hypotheses -, we conclude that the sequence converges to a function satisfying

That is,

On the other hand, since by (80), one has , then by applying the operator on both sides of the above equality and using (62) and (63), we conclude that satisfies

Next we aim at proving that is a solution of problem (5). To this end, we remark by (81) and (9) that

This implies by Corollary 6 (ii) that the function is continuous on and so by (87), is continuous on .

Now, since by and (88), the function is continuous and integrable on , we conclude by Remark 4 that is the required solution.

It remains to prove that under condition , is the unique solution to problem (5) satisfying (14). Assume that is another nonnegative solution in to problem (5) satisfying (14). Since , we deduce by (88) that

So the function is continuous and integrable on and by Remark 4, we conclude that the function satisfies

From the uniqueness in problem (8), we deduce that

Now let be the function defined on by

Then by ,   and by (87) and (91), one has

On the other hand, by , we remark that and by (80) we deduce that

Hence by (63), we conclude that . This completes the proof.

Proof of Corollary 3. Let ,  , and . It is clear that hypotheses and are satisfied. Since the function belongs to the class , one has for . Moreover, by a simple computation, for and , we obtain

This implies that the function satisfies hypothesis . So the result follows by Theorem 2.

Example 15. Let and with . Let , and be a positive continuous function on such that

Since the function belongs to the class , then, for , the problem

has a unique positive solution in satisfying

Example 16. Let and with . Let ,, and be a positive continuous function on such that

Let . Since the function , then, for , the problem

has a unique positive solution in satisfying

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for their careful reading of the paper. This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.

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