Existence of Solutions for Some Nonlinear Problems with Boundary Value Conditions

Abstract

We study the existence of solutions for nonlinear boundary value problems , where denotes the boundary conditions on a compact interval , is a homeomorphism such that , and is a continuous function. All the contemplated boundary value problems are reduced to finding a fixed point for one operator defined on a space of functions, and Schauder fixed point theorem or Leray-Schauder degree is used.

1. Introduction

The purpose of this article is to obtain some existence results for nonlinear boundary value problems of the form where denotes the Dirichlet or mixed boundary conditions on the interval , is a bounded, singular, or classic homeomorphism such that , is a continuous function, and is a positive real number.

Recently, problem (1) in special cases, when is an increasing homeomorphism from to such that and denotes the periodic, Neumann, or Dirichlet boundary conditions, has been investigated by Bereanu and Mawhin in [1].

In [2], the authors have studied problem (1), where and denotes the periodic boundary conditions. They obtained the existence of solutions by means of the Leray-Schauder degree theory. In particular, regular periodic problems with - or -Laplacian on the left hand side were considered by several authors; see, for example, del Pino et al. [3] or Yan [4].

In [5] Benevieri et al. proved an existence result for the periodic boundary value problem: assuming that is a Carathéodory function and is a homeomorphism between and the open ball of with center zero and radius . They used a topological method: the properties of and allowed applying the Leray-Schauder degree. The interest in this class of nonlinear operators is mainly due to the fact that they include the mean curvature operator:

The paper is organized as follows. In Section 2, we introduce some notations and preliminaries, which will be crucial in the proofs of our results. Section 3 is devoted to the study of existence of solutions for the Dirichlet problems with bounded homomorphisms: In particular, Bereanu and Mawhin in [6] proved the existence of at least one solution by means of the Leray-Schauder degree.

Theorem 1 (Bereanu and Mawhin). If the function satisfies the condition the Dirichlet problem has at least one solution.

The main purpose of this section is an extension of the results obtained in the previous theorem. For this, we use topological methods based upon Leray-Schauder degree [7] and more general properties of the function . In Section 4, we use the fixed point theorem of Schauder to show the existence of at least one solution for boundary value problems of the type where (we call it singular). Of course, a solution of (6) is a function of class such that , satisfying the boundary conditions, and the function is continuously differentiable and for all . In Section 5, for boundary conditions and classic homeomorphisms , we investigate the existence of at least one solution using Leray-Schauder degree, where a solution of this problem is any function of class such that is continuously differentiable, which satisfies the boundary conditions and for all . Such problems do not seem to have been studied in the literature. In the present paper generally we follow the ideas of Bereanu and Mawhin [1, 2, 6, 8].

2. Notation and Preliminaries

For fixed , we denote the usual norm in by . Let ) denote the Banach space of continuous functions from into , endowed with the uniform norm , the Banach space of continuously differentiable functions from into , equipped with the usual norm , and the closed subspace of defined by

We introduce the following applications: The Nemytskii operator :  The integration operator :  The following continuous linear applications:

For , we write

The following lemma is an adaptation of a result of [1] to the case of a homeomorphism which is not defined everywhere. We present here the demonstration for better understanding of the development of our research.

Lemma 2. Let . For each , there exists a unique (where denotes the range of ) such thatMoreover, the function is continuous and sends bounded sets into bounded sets.

Proof. Let . We define the continuous application for We now show that the equation has a unique solution . Let be such that that is, It follows that there exists such that Using the injectivity of we deduce that . Let us now show the existence. Because is strictly monotone and , we have that It follows that there exists such that . Consequently for each , (13) has a unique solution. Thus, we define the function such that On the other hand, because , we have that Therefore, the function sends bounded sets into bounded sets.
Finally, we show that is continuous on . Let be a sequence such that in . Since the function sends bounded sets into bounded sets, then is bounded. Hence, is relatively compact. Without loss of generality, passing if necessary to a subsequence, we can assume that where for each we obtain Using the dominated convergence theorem, we deduce that so we have that . Hence, the function is continuous.

The following extended homotopy invariance property of the Leray-Schauder degree can be found in [9].

Proposition 3. Let be a real Banach space, be an open, bounded set, and be a completely continuous operator on such that for each . Then the Leray-Schauder degree is well defined and independent of in , where is the open, bounded (possibly empty) set defined by .

3. Dirichlet Problems with Bounded Homeomorphisms

In this section we are interested in Dirichlet boundary value problems of the type where is a homeomorphism, , and is continuous. In order to apply Leray-Schauder degree theory to show the existence of at least one solution of (24), we introduce, for , the family of Dirichlet boundary value problems: LetClearly is an open set in and is nonempty because . Using Lemma 2, we can define the operator by Here with an abuse of notation is understood as the operator defined by . It is clear that is continuous and sends bounded sets into bounded sets.

When the boundary conditions are periodic or Neumann, an operator has been considered by Bereanu and Mawhin [6].

The following lemma plays a pivotal role in studying the solutions of problem (25).

Lemma 4. The operator is well defined and continuous. Moreover, if is such that , then is solution of (25).

Proof. Let . We show that in fact . It is clear thatwhere the continuity of and follows from the continuity of applications and .
On the other hand using Lemma 2, we have Therefore and is well defined. The continuity of follows by the continuity of the operators which compose it .
Now suppose that is such that . It follows from (27) that for all . Differentiating (30), we obtain Applying to both of its members we have thatDifferentiating again, we deduce that for all . Thus, satisfies problem (25). This completes the proof.

Remark 5. Note that the reciprocal of Lemma 4 is not true because we cannot guarantee that for every solution of (25).

In our main result, we need the following lemma to obtain the required a priori bounds for the possible fixed points of .

Lemma 6. Assume that there exist and such that , for all . If is such that , then where .

Proof. Let and be such that . Using Lemma 4, we have that is solution of (25), which implies thatwhere, for all , we obtain On the other hand, using inequality (34) we have thatUsing the integration by parts formula, the boundary conditions, and the fact that , we deduce that Since and is solution of (25), it follows thatand hence On the other hand, since , we getfor all . It follows thatwhich implies that , where . Using again the boundary conditions, we have thatand hence Finally, if , then , so the proof is complete.

Let be such that and consider the set Since the set , then we deduce that is nonempty. Moreover, it is clear that is open and bounded in and . On the other hand using an argument similar to the one introduced in the proof of Lemma   in [6], it is not difficult to see that is well defined and completely continuous and

3.1. Existence Results

In this subsection, we present and prove our main result.

Theorem 7. If satisfies conditions of Lemma 6, then problem (24) has at least one solution.

Proof. Let be the operator given by (27). Using Proposition 3, we deduce thatwhere . Thus, there exists such that , which is a solution for (24).

Remark 8. Note that Theorem 7 is a generalization of Theorem 1.

Corollary 9. Assume that is an increasing homomorphism. Let be such thatfor all and . If is such that , then where .

Proof. Since is an increasing homomorphism we have thatfor all . Using Lemma 6 with for all , we can obtain the conclusion of Corollary 9. The proof is achieved.

Theorem 10. If satisfies conditions of Corollary 9, then problem (24) has at least one solution.

Let us give now an application of Theorem 10 when is unbounded.

Example 11. Consider the Dirichlet problem where .
It is not difficult to verify that is an increasing homeomorphism and is a continuous function such thatSo, we can choose and to see that Corollary 9 holds and so, using Theorem 10, we obtain that (53) has at least one solution.

4. Problems with Singular Homeomorphisms and Three-Point Boundary Conditions

In this section we study the existence of at least one solution for boundary value problems of the type where is a homeomorphism such that and is a continuous function.

In order to transform problem (55) to a fixed point problem we use a similar argument introduced in Lemma 2 for .

Lemma 12. is a solution of (55) if and only if is a fixed point of the operator defined on by

Proof. If is solution of (55), thenfor all . Applying to both members and using the fact that , we deduce thatBy the inversion of in (58), we havewhere . Integrating from 0 to , we have thatBecause , thenUsing an argument similar to the one introduced in Lemma 2, it follows that . Hence,Let be such that . Then for all . Since , therefore, we have that . Differentiating (63), we obtain thatIn particular,Applying to both members and differentiating again, we deduce that for all . This completes the proof.

Lemma 13. The operator is completely continuous.

Proof. Let be a bounded set. Then, if , there exists a constant such that Next, we show that is a compact set. Let be a sequence in , and let be a sequence in such that . Using (67), we have that there exists a constant such that, for all , which implies that Hence the sequence is bounded in . Moreover, for and for all , we have that which implies that is equicontinuous. Thus, by the Arzelà-Ascoli theorem there is a subsequence of , which we call , which is convergent in . Using the fact that is continuous, it follows fromthat the sequence is convergent in . Then, passing to a subsequence if necessary, we obtain that is convergent in . Finally, let be a sequence in . Let be such that Let be a subsequence of such that it converges to . It follows that and converge to . This concludes the proof.

The next result is based on Schauder’s fixed point theorem.

Theorem 14. Let be continuous. Then (55) has at least one solution.

Proof. Let . Then whereMoreover,Hence,Because the operator is completely continuous and bounded, we can use Schauder’s fixed point theorem to deduce the existence of at least one fixed point. This, in turn, implies that problem (55) has at least one solution. The proof is complete.

5. Problems with Classic Homeomorphisms and Three-Point Boundary Conditions

We finally consider boundary value problems of the form where is a homeomorphism such that and is a continuous function. We remember that a solution of this problem is any function of class such that is continuously differentiable, satisfying the boundary conditions and for all .

Let us consider the operator

Analogously to Section 3, here is understood as the operator defined for It is clear that is continuous and sends bounded sets into bounded sets.

Lemma 15. is a solution of (77) if and only if is a fixed point of the operator .

Proof. Let , and we have the following equivalences:

Remark 16. Note that .

Using an argument similar to the one introduced in Lemma 13, it is easy to see that is completely continuous.

In order to apply Leray-Schauder degree to the operator , we introduced a family of problems depending on parameter . We remember that to each continuous function we associate its Nemytskii operator defined byFor , we consider the family of boundary value problems: Notice that (81) coincides with (77) for . So, for each , the operator associated with (81) by Lemma 15 is the operator , where is defined on byUsing the same arguments as in the proof of Lemma 13 we show that the operator is completely continuous. Moreover, using the same reasoning as above, system (81) (see Lemma 15) is equivalent to the problem

5.1. Existence Results

In this subsection, we present and prove our main results. These results are inspired by works of Bereanu and Mawhin [6] and Manásevich and Mawhin [10]. We denote by the Brouwer degree and by the Leray-Schauder degree and define the mapping by

Theorem 17. Assume that is an open bounded set in such that the following conditions hold: (1)For each problem  has no solution on .(2)The equation  has no solution on , where we consider the natural identification of with related functions in .(3)The Brouwer degreeThen problem (77) has a solution.

Proof. Let . If is a solution of (85), then ; hence is a solution of problem (81). On the other hand, for , if is a solution of (81) and because we have ; then is a solution of (85). It follows that, for , problems (81) and (85) have the same solutions. We assume that, for , (81) does not have a solution on since otherwise we are done with proof. It follows that (81) has no solutions for . If , then (81) is equivalent to the problem and thus, if is a solution of (89), we must have Moreover, is a function of the form . Thus, by (90) which, together with hypothesis (2), implies that . Thus we have proved that (81) has no solution in for all . Then we have that, for each , the Leray-Schauder degree is well defined, and by the homotopy invariance, one hasOn the other hand, we have that But the range of the mapping is contained in the subspace of related functions, isomorphic to . Thus, using a reduction property of Leray-Schauder degree [7, 11], Then, , where denotes the unit operator. Hence, there exists such that , which is a solution for (77).

The following result gives a priori bounds for the possible solutions of (85) and adapts a technique introduced by Ward Jr. [12].

Theorem 18. Assume that satisfies the following conditions: (1)There exists such that  for all .(2)There exists such that, for all , If is such that is solution of (85), then where

Proof. Let be such that is a solution of (85). Then, for all , Using hypothesis (2), we have that It follows that there exists such that and which implies that where Hence, where and . It follows that where . Because is such that we have that and hence . This proves the theorem.

Now we show the existence of at least one solution for problem (77) by means of Leray-Schauder degree.

Theorem 19. Let be continuous and let it satisfy conditions (1) and (2) of Theorem 18. Assume that the following conditions hold for some : (1)The equation  has no solution on , where we consider the natural identification of with related functions in .(2)The Brouwer degree Then problem (77) has a solution.

Proof. Let be such that is a solution of (85). Using Theorem 18, we have where . Thus the conditions of Theorem 17 are satisfied with , where is the open ball in center and radius . This concludes the proof.

Let us give now an application of Theorem 19.

Example 20. Let us consider the problem Let and . If we suppose that and , then On the other hand, if we choose and for all , we have that the equation has no solution on . Then we have that the Brouwer degree is well defined and, by the properties of that degree, we have that where is a regular value of and is the Jacobian of at . So, using Theorem 19, we obtain that the boundary value problem (111) has at least one solution.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this article.

Acknowledgments

This research was supported by CAPES and CNPq/Brazil. The author would like to thank Dr. Pierluigi Benevieri for his kind advice and for the constructive revision of this paper.