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Abstract and Applied Analysis
Volume 2011, Article ID 545264, 19 pages
http://dx.doi.org/10.1155/2011/545264
Research Article

Existence and Multiplicity of Solutions for a Periodic Hill's Equation with Parametric Dependence and Singularities

1Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2Departamento de Matemáticas, Universidad de Jaén, Campus Las Lagunillas, Ed. B3, 23071 Jaén, Spain

Received 5 July 2010; Revised 27 January 2011; Accepted 24 February 2011

Academic Editor: Pavel Drábek

Copyright © 2011 Alberto Cabada and José Ángel Cid. 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 deal with the existence and multiplicity of solutions for the periodic boundary value problem , where is a positive parameter. The function is allowed to be singular, and the related Green's function is nonnegative and can vanish at some points.

1. Introduction and Preliminaries

In the recent paper [1], the authors obtain existence, multiplicity, and nonexistence results for the periodic problem depending on the parameter . Although not explicitly mentioned in [1], we point out the important fact that the related Green’s function of (1.1) is strictly negative for all .

The aim of this paper is to give complementary results to those of [1] for the case of a nonnegative related Green's function. In particular, we will deal with problem assuming that its Green’s function is nonnegative (for instance, if , this means ). Moreover, in order to give wider applicable results, we will also allow to be singular at (the reader may have in mind the model , for some ).

We note that analogous arguments have been developed in [2] for the fourth-order discrete equation

The main tool used in this paper is Krasnoselskii's fixed point theorem in a cone, which is a classical tool extensively used in the related literature (see, for instance, [1, 35] and references therein). We will use cones of the form where is a fixed constant and is a functional satisfying (i) for all , (ii) for all and .

In particular, in Section 2, we use the standard choice , and in Section 3, we use , which has been recently introduced in [3].

We say that the linear problem is nonresonant when its unique solution is the trivial one. It is well known that if (1.5) is nonresonant, then the nonhomogeneous problem always has a unique solution which, moreover, can be written as where is Green's function related to (1.5). Thus, defining for each the operator given by we have that is a solution of problem (1.2) if and only if .

Throughout the paper, we will use the following notation:

For , we denote by its positive part, and for , we denote by its conjugate, that is . Moreover, for an essentially bounded function , we define and for , we will define

The following section is devoted to prove the existence, multiplicity, and nonexistence of solutions of problem (1.2) by assuming that the related Green’s function is strictly positive, whereas in Section 3, we turn out to the case, where the related Green's function is non negative. We point out that in the recent paper [6], the existence of solution for problem (1.2) with a sign-changing Green's function is studied, but only considering a regular and .

2. Positive Green's Function

In this section, we assume the following hypotheses: or , problem (1.5) is nonresonant and the corresponding Green's function is strictly positive on ,, for a.e. , and , is continuous,.

Notice that condition allows to be singular at . In particular, is satisfied when , (the case is called a weak singularity, while is called an strong singularity).

On the other hand, it is well known that for constant , condition is equivalent to . For a time-dependent and nonnegative potential , Torres gave a sharp -criterium [5] based on an antimaximum principle obtained in a previous work by Torres and Zhang [7]. That criterium has been extended in [8] for sign-changing potentials with strictly positive average. The obtained result is the following.

Proposition 2.1 (see [8, Theorem  3.2]). Define where is the usual Gamma function.
Assume that for some , , and moreover, Then, for all .

In [9], by studying antimaximum principles for the semilinear equation the previous result has been extended to the potentials with nonnegative average as follows.

Lemma 2.2 (see [9, Theorem  3.4 and Remark  3.7]). Assume that for some , , and moreover, Then, for a. e. .

Zhang constructs in [10] some examples of potentials for which the related Green's function is strictly positive, but (2.2) does not hold. In consequence, the best Sobolev constant is not an optimal estimate to ensure the positiveness of Green's function. For optimal conditions in order to get maximum or antimaximum principles, expressed using eigenvalues, Green's functions, or rotation numbers, the reader is referred to the recent work of Zhang [11].

Example 2.3. By Proposition 2.1, Hill's equation with the periodic boundary conditions satisfies , provided that , and moreover, where is given by (2.1). So, for each , the condition is fulfilled if
In particular, it is known that and (since the maximum of is attained at , see [10, Example  4.4]). The graphic of is showed in Figure 1.

545264.fig.001
Figure 1: Graphic of .

From , it follows that , and we define the cone where In both cases, , and for , we define

Next, we give sufficient conditions for the solvability of problem (1.2).

Theorem 2.4. Assume that conditions , , , , and are fulfilled. Then, for each and , the operator given by (1.9) is well defined and completely continuous.
Moreover, if either(i) for any with and for any with , or(ii) for any with and for any with , then has a fixed point in , which is a positive solution of problem (1.2).

Proof. Note that if , then for all , so , and then is well defined. Standard arguments show that and that is completely continuous. Then, from Krasnoselskii's fixed point theorem (see [12, p. 148]), it follows the existence of a fixed point for in which is, by the definition of , a positive solution of problem (1.2).

Before proving the existence and multiplicity results for problem (1.2), we need some technical lemmas proved in the next subsection.

2.1. Auxiliary Results

Lemma 2.5. Assume that conditions , , , , and are satisfied. Then, for each , there exists such that for every , we have

Proof. Fix , and let with . If then, for all the following inequalities hold: and thus .

Lemma 2.6. Assume that conditions , , , , and are fullfiled. Then, for each , there exists such that for every , we have

Proof. Fix , and let with . If then and thus .

Lemma 2.7. Suppose that conditions , , , and are satisfied and . Then, if , there exists such that for every , we have

Proof. Since for , there exists such that for each .
Fix , and let with . Then, and thus .

Lemma 2.8. Assume that hypothesis , , , , and hold. Then, if , there exists such that for every , we have

Proof. Since for , there exists such that for each .
Fix , and let with . Then, and thus .

Lemma 2.9. Suppose that conditions , , , , and are satisfied. Then, if then, there exists such that for every , we have

Proof. Since for , there exists such that for each . We define .
Fix , and let with . Then, and thus .

Lemma 2.10. Assume that , , , , and are fullfiled. Then, if , there exists such that for every , we have

Proof. Since for , there exists such that for each . We define .
Fix , and let with . Then, and thus .

In the sequel, we study separately the two different cases considered in condition ; that is, or .

2.2. The Case

Theorem 2.11. Assume that conditions , , , and are fulfilled. If, moreover, , the following results hold:(1) there exists such that problem (1.2) has a positive solution if ,(2) if , then problem (1.2) has a positive solution for every ,(3) if , then there exists such that problem (1.2) has two positive solutions if ,(4) if and , then there exists such that problem (1.2) has no positive solutions if .

Proof. Fix . Then, for each and with , we have Part 1. Fix , and take given by Lemma 2.5. Then, from Theorem 2.4 (ii), it follows the existence of a positive solution for problem (1.2) if . Part 2. Fix , and take , where is given by Lemma 2.9. Then, from Theorem 2.4 (ii), it follows the existence of a positive solution for problem (1.2). Part 3.Fix , and take , where and are the given by Lemma 2.5.
Now, fix , and take , where is given by Lemma 2.10. Therefore, from Theorem 2.4, it follows the existence of two positive solutions and for problem (1.2) such that Part 4. Since and , there exists such that for all . Define
If for , there exists a positive solution of problem (1.2), we know that and, as consequence, . Therefore, we deduce the following inequalities: and we attain a contradiction.

Example 2.12. Let us consider the forced Mathieu-Duffing-type equation which fits into expression (1.2) by defining and .
Equation (2.30), with , was studied in [13], where a sufficient condition for the existence of a -periodic solution is given. However, since the proof relies in the application of Schauder’s fixed point theorem in a ball centered at the origin, the trivial solution is not excluded. The existence of a nontrivial solution was later obtained by Torres in [5, Corollary 4.2]. More precisely, Torres proves that if function for a.e. and , then the homogeneous problem () (2.30) has at least two nontrivial one-signed -periodic solutions.
In this paper, as a consequence of Example 2.3 and Theorem 2.11, Part 3, we arrive at the following multiplicity result for the inhomogeneous () equation (2.30) with a not necessarily constant sign function .

Corollary 2.13. If condition (2.8) is satisfied and , then there exists such that (2.30) has at least two positive -periodic solutions, provided that .

2.3. The Case

Theorem 2.14. Assume that conditions , , , and hold. If moreover the following results hold: (1) if or , then there exists such that problem (1.2) has a positive solution if ,(2) if then there exists such that problem (1.2) has a positive solution for every ,(3) if and then problem (1.2) has a positive solution for every ,(4) if and then there exists such that problem (1.2) has two positive solutions if ,(5) if and , then problem (1.2) has a positive solution for every ,(6) if and then there exists such that problem (1.2) has two positive solutions if ,(7) if and , then there exists such that problem (1.2) has no positive solutions if .

Proof of Part 1. Fix and take given by Lemma 2.5. In consequence, for all we have
Now, let be fixed, and choose , where is given by Lemma 2.8 when . In case of , we get , with given by Lemma 2.10. In both situations, we arrive at
Thus, Theorem 2.4 implies the existence of a positive solution for problem (1.2). Part 2.Fix , and take given by Lemma 2.6. Now, for each , take , with given by Lemma 2.9, and apply Theorem 2.4.Part 3. For each , take given by Lemmas 2.8 and 2.9, respectively, and apply Theorem 2.4.Part 4.Fix , and take given by Lemma 2.5. Now, for each , take given by Lemma 2.8 and given by Lemma 2.10. Then, Theorem 2.4 implies the existence of two positive solutions and for problem (1.2) such that Part 5. Use Lemmas 2.7 and 2.10 and Theorem 2.4.Part 6. Use Lemmas 2.6, 2.7, and 2.9 and Theorem 2.4 twice.Part 7. The proof follows the same steps as Part 4 in Theorem 2.11.

Remark 2.15. Theorem 2.14 complements [1, Theorem  2.1], since it provides similar results for the problem , with .

Example 2.16. Consider as a model the problem where , and . When and , (2.34) is the Brillouin-beam focusing equation which has been widely studied in the literature (see [5, 10, 14] and references therein). Now, we have the following:

Corollary 2.17. Assume condition (2.8). Then, the following results are satisfied:(i) if , then problem (2.34) has a positive solution for every .(ii) if and , then there exists such that problem (2.34) has a positive solution for every , and there exists such that the problem has no positive solution for .(iii) if and , then there exists such that problem (2.34) has two positive solutions for every .

Proof. Condition (2.8) implies that condition is satisfied. Now, to prove (i), (ii) or (iii) it is enough to apply Theorem 2.14 Part 3, Part 1 and Part 7 or Part 4, respectively.

3. Nonnegative Green's Function

In this section instead of conditions and , we assume Problem (1.5) is nonresonant, the corresponding Green's function is nonnegative on , and , is continuous, and for all .

Notice that is the unique solution of the problem and then asks for this solution to be strictly positive. On the other hand, assumption allows us to consider only regular problems. We will discuss to singular problems in Section 3.1 by means of a truncation technique.

For constant , condition is equivalent to . For nonconstant , condition is satisfied provided that Lemma 2.2 holds.

On the other hand, under condition , it is allowed that so can be equal to 0, and thus, the arguments used in the previous section do not work. So, by assuming that , let us define where if or if . As far as we know, the cone was introduced in [3].

Clearly, , and for , we define

Next, we prove the following result similar to Theorem 2.4.

Theorem 3.1. Assume that , and hold. Then, for each and , the operator given by (1.9) is well defined and completely continuous.
Moreover, if either (i) for any with and for any with , or (ii) for any with and for any with , then has a fixed point in , which is a nonnegative solution of problem (1.2).

Proof. If and assuming (the case being analogous), we obtain Thus, , and it is standard to show that is completely continuous. In consequence, from Krasnoselskii's fixed point theorem (see [12, p.148]), it follows the existence of a fixed point for in which it is, by the definition of , a non negative solution of problem (1.2).

Now, we are going to give sufficient conditions to obtain or . The combination of the next lemmas with Theorem 3.1 will allow us to prove existence and multiplicity results for problem (1.2).

Lemma 3.2. Suppose that the conditions , and are satisfied. Then, for each , there exists such that for every , we have

Proof. Fix , and let with . If then and thus .

Lemma 3.3. Assume that , , and are satisfied. Then, there exists such that for each , we have

Proof. Fix , and let with . Then, and thus .

Lemma 3.4. Let , and be fulfilled. Then, if , there exists such that for every , we have

Proof. Define . Clearly, is a nondecreasing function on ; moreover, since , it is obvious that
Therefore, we have that for , there exists such that for each .
Define , fix , and let with . Then and thus .

Theorem 3.5. Assume , and . The following results hold:(1) if . then there exists such that problem (1.2) has a nonnegative solution if , (2) if and , then problem (1.2) has a nonnegative solution for every .

Proof. The first assertion is a direct consequence of Lemmas 3.2 and 3.3. The second part follows from Lemmas 3.3 and 3.4

Now, we will impose a strong condition on function by assuming that is strictly positive on the whole interval. , for a.e. .

Lemma 3.6. Assume that conditions , and are satisfied. Then, if , there exists such that for every , we have

Proof. Since for , there exists such that for each .
Fix , and let with . Then, and thus .

Now, we are in a position to present the main result of this section.

Theorem 3.7. Suppose that conditions , and are fulfilled. The following assertions are satisfied:(1) if , then there exists such that problem (1.2) has a nonnegative solution if , (2) if and , then problem (1.2) has a nonnegative solution for every , (3) if and , then there exists such that problem (1.2) has no nonnegative solutions if .

Proof. The first assertion is a direct consequence of Lemmas 3.2 and 3.6. The second part follows from Lemmas 3.4 and 3.6.
To prove Part 3, by using that and , we know that there exists such that for all .
By defining we have that if there is any for which there exists a nonnegative solution of problem (1.2), then . So, we arrive at the following contradiction:

3.1. Applications to Singular Equations

Despite the fact that in the previous results we deal with regular functions, it is possible to apply some of them to the singular equation by means of a truncation technique.

To this end, we will consider a function that satisfies is a continuous function such that .

Theorem 3.8. Assume that and conditions , , , and hold.
Then, problem (3.18) has a positive solution for every .

Proof. Let , and define the function
From , it follows that satisfies condition , and . Moreover, implies that . As consequence, Theorem 3.5, Part 2, implies that the modified problem has a nonnegative solution for all . Such function is given by the expression The nonnegativeness of functions , and implies that the solution for all . Therefore, is a positive solution of problem (3.18).

Remark 3.9. Theorem 3.8 is an alternative result to those obtained in [15, 16] by means of Schauder's fixed point theorem.

Example 3.10. Let us consider the repulsive singular differential equation Since satisfies , we can apply Theorem 3.8 to obtain the following.

Corollary 3.11. Assume that , , and hold. If then, (3.22) has a positive -periodic solution for every .

Acknowledgment

This work was partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, project no. MTM2010-15314.

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