## Existence and Asymptotic Behaviour of Solutions of Differential and Integral Equations in Some Function Spaces

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

**Academic Editor:**JÃ³zef BanaÅ›

#### 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, [1â€“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 Greenâ€™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.â€‰(*H*_{1}) is a nonnegative continuous function in .â€‰(*H*_{2})There exists a nonnegative function with such that for each , the map is nondecreasing on .â€‰(*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 , â€‰â€‰ 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 Greenâ€™s Function

From the explicit expression of Greenâ€™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 **1â€‰â€‰*. In this case, one has
*Case **2â€‰â€‰*. We obtain
*Case **3â€‰â€‰*. 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 .â€‰(i) Using (24) and (23), one has, for ,
â€‰Hence
â€‰(ii) Since, for each , one has , then we deduce by Fatouâ€™s lemma and (12), that
â€‰which implies that, for ,
â€‰(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*â€‰(i)*.
*â€‰*In particular, is well defined in .*â€‰(ii)*â€‰**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