Abstract
The existence and multiplicity of positive solutions are established for second-order periodic boundary value problem. Our results are based on the theory of a fixed point index for A-proper semilinear operators defined on cones due to Cremins. Our approach is different in essence from other papers and the main results of this paper are also new.
1. Introduction
In the present paper, we discuss the existence of positive solutions of the periodic boundary value problem (PBVP) for second-order differential equation where is a continuous function. Our purpose here is to provide sufficient conditions for the existence of multiple positive solutions to the periodic boundary value problem (1.1). This will be done by applying the theory of a fixed point index for A-proper semilinear operators defined on cones obtained by Cremins [1].
We are interested in positive solutions of (1.1), because we have been motivated by a problem from the Theory of Nonlinear Elasticity modelling radial oscillations of an elastic spherical membrane made up of a neo-Hookean material and subjected to an internal pressure. Because of wide interests in physics and engineering, second-order periodic boundary value problems have been studied widely in the literature; we refer the reader to [2β30] and references therein. In [6, 7], by using Krasnoselskiiβs fixed point theorem, the existence and multiplicity of positive solutions are established to the periodic boundary value problem on
Agarwal et al. [8] discussed the existence of positive solutions for the second-order differential equation where and are continuous -periodic positive functions and . By employing fixed point index theory in cones, they found sufficient conditions for the existence of at least one positive solution. Recently, Torres [9] and Yao [10] obtained some results on the existence of positive solutions of a general periodic boundary value problem In this case, the problem (1.4) has no Green function. In order to overcome this difficulty, their main technique is to rewrite the original PBVP (1.4) as an equivalent one, so that the Krasnoselskii fixed point theorem on compression and expansion of cones can be applied. Inspired by the above work, the aim of this paper is to consider the existence and multiplicity of positive solutions for the periodic boundary value problem (1.1). The method we used here is different in essence from other papers and the main results of this paper are also new.
This paper is organized as follows. In Section 2, we give some preliminaries and establish several lemmas, and the main theorems are formulated and proved in Section 3. Finally, in Section 4, we give two examples to illustrate our results.
2. Notation and Preliminaries
We start by introducing some basic notation relative to theory of the fixed point index for A-proper semilinear operators defined on cones established by Cremins (see [1]).
Let and be Banach spaces, a linear subspace of , , and sequences of oriented finite-dimensional subspaces such that in for every and dist for every , where and are sequences of continuous linear projections. The projection scheme is then said to be admissible for maps from to .
Definition 2.1 (see [1]). A map is called approximation-proper (A-proper) at a point with respect to if is continuous for each and whenever is bounded with , then there exists a subsequence such that , and . is said to be A-proper on a set if it is A-proper at all points of .
Let be a cone in a finite-dimensional Banach space , and let be open and bounded with . Let be continuous such that on , where and denote the closure and boundary, respectively, of relative . Let be an arbitrary retraction.
The following definition of finite-dimensional index forms the basis of generalized index for A-proper maps .
Definition 2.2 (see [1]). One defines
where the degree is the Brouwer degree and is a ball containing .
Now let be a cone in an infinite-dimensional Banach space with projection scheme such that for every . Let be an arbitrary retraction and an open bounded set such that . Let be such that is A-proper at 0. Write and . Then is a finite dimensional retraction.
Definition 2.3 (see [1]). If on , then one defines
that is, the index is the set of limit points of , where the finite dimensional index is that defined above.
Let be a Fredholm map of index zero, and let be continuous projectors such that ImKer, KerIm and Ker Ker, ImIm. The restriction of to domKer, denoted , is a bijection onto Im with continuous inverse ImdomKer. Since dimβImdimβKer, there exists a continuous bijection Im Ker. Let be a cone in an infinite-dimensional Banach space with projection scheme . If we let , then dom is a linear bijection with bounded inverse. Thus is a cone in the Banach space .
Let be open and bounded with a bounded Fredholm operator of index zero, and dom a bounded continuous nonlinear operator such that is A-proper at 0.
We can now extend the definition of the index to A-proper maps of the form acting on cones.
Definition 2.4 (see [1]). Let be a retraction from to , and assume maps to and on . One defines the fixed point index of over as
where is defined as for each , and the index on the right is that of Definition 2.3.
For convenience, we recall some properties of .
Proposition 2.5 (see [1]). Let be Fredholm of index zero, and let be open and bounded. Assume that maps to , and on . Then one has
(existence property) if , then there exists such that ;(normality property) if , then , where and for every ;(additivity property) if for , where and are disjoint relatively open subsets of , then
with equality if either of indices on the right is a singleton;(homotopy invariance property) if is an A-proper homotopy on for and and for , then is independent of , where .
The following two lemmas will be used in this paper.
Lemma 2.6. If is Fredholm of index zero, is an open bounded set, and , and let be A-proper for . Assume that is bounded and maps to . If there exists , such that for every and all , then .
Proof. Choose a real number such that and define by Trivially, and from (2.5) we obtain Again, by homotopy invariance property in Proposition 2.5, we have However, In fact, if , the existence property in Proposition 2.5 implies that there exists such that Then which contradicts (2.6). So
Remark 2.7. The original condition of [1, Theoremββ5] was given with instead of . The modification is necessary since otherwise it cannot guarantee that .
We assume that there is a continuous bilinear form on such that if and only if for each . This condition implies that if is a basis in , then the linear map defined by , is an isomorphism and that if , then for and for .
In [1], Cremins extended a continuation theorem related to that of Mawhin [31] and Petryshyn [32] for semilinear equations to cones; refer to [1, Corollaryββ1] for the details. By Lemma 2.6 and [1, Corollaryββ1], we obtain the following existence theorem of positive solutions to a semilinear equation in cones. It is worth mentioning that the positive or nonnegative solutions of an operator equation were also discussed by a recent paper of OβRegan and Zima [33] and the earlier papers [34β38].
Lemma 2.8. If is Fredholm of index zero, is a cone, and are open bounded sets such that and . Suppose that is A-proper for with bounded. Assume that
and ,
for ,
for ,
,
there exists , such that
Then there exists such that .
Corollary 2.9. Assume all conditions of Lemma 2.8 hold except and assume for each . Then the same conclusion holds.
Proof. We show that implies , that is, , for each . Here . Otherwise, the proof is finished. If , then it follows from that has a solution in , and Corollary 2.9 is proved. If and for some , then and by condition ; that is, , contradicting the fact that . This completes the proof of Corollary 2.9.
The following lemma can be found by of [32, Lemmaββ2].
Lemma 2.10. Suppose either or is compact, then is A-proper for .
3. Main Results
The goal of this section is to apply Lemma 2.8 to discuss the existence and multiplicity of positive solutions for the PBVP (1.1).
Let endowed with the norm and let with the norm and , then is a cone of .
We define then PBVP (1.1) can be written as
It is easy to check that so that is a Fredholm operator of index zero.
Next, define the projections by and by
Furthermore, we define the isomorphism as , where . It is easy to verify that the inverse operator of as , where
For notational convenience, we set or By routine methods of advanced calculus, we get
Now we can state and prove our main results.
Theorem 3.1. Assume that there exist two positive numbers such that
if one of the two conditions(i)(ii)
is satisfied, then the PBVP (1.1) has at least one positive solution satisfying .
Proof. It is easy to see . Without loss of generality, let .
First, we note that , as so defined, is Fredholm of index zero, is compact by Arzela-Ascoli theorem, and thus is A-proper for by of Lemma 2.10.
For each , then by condition
This implies that condition of Lemma 2.8 is satisfied. To apply Lemma 2.8, we should define two open bounded subsets of so that ()β() of Lemma 2.8 hold.
We prove only Case (i). In the same way, we can prove Case (ii).
Let
Clearly, and are bounded and open sets and
Next we show that (i) implies . For this purpose, suppose that there exist and such that then for all . Let , such that . From boundary conditions, we have . Then we have the following two cases. Case 1 (). In this case, . Since boundary condition , we have . So we have . It follows from being continuous in that there exists , such that when . Thus, . Hence,
and is not the maximum on , a contradiction.Case 2 (). In this case, . This gives
which contradicts (i). So for each and , we have . Thus, of Lemma 2.8 is satisfied.
To prove of Lemma 2.8, we define the bilinear form as
It is clear that is continuous and satisfies for every . In fact, for any and , we have , a constant, and there exists such that for each . By , we get
Let , then , so we have by condition (i)
Thus, and of Lemma 2.8 are verified.
Finally, we prove of Lemma 2.8 is satisfied. We may suppose that . Otherwise, the proof is completed.
Let . We claim that
In fact, if not, there exist , such that
Since , operating on both sides of the latter equation by , we obtain
that is,
For any , we have . Then there exists , such that . By condition (i) and ,
in contradiction to (3.19). So (3.16) holds; that is, of Lemma 2.8 is verified.
Thus, all conditions of Lemma 2.8 are satisfied and there exists such that and the assertion follows. Thus, and .
Let be the integer part of . The following result concerns the existence of positive solutions.
Theorem 3.2. Assume that there exist positive numbers such that
,
if one of the two conditions(i), (ii),
is satisfied, then the PBVP (1.1) has at least positive solutions satisfying .
Proof. Modeling the proof of Theorem 3.1, we can prove that if there exist two positive numbers such that then PBVP (1.1) has at least one positive solution satisfying .
By the claim, for every pair of positive numbers , (1.1) has at least positive solutions satisfying .
We have the following existence result for two positive solutions.
Corollary 3.3. Assume that there exist three positive numbers such that , if one of the two conditions(i), (ii)is satisfied, then the PBVP (1.1) has at least two positive solutions satisfying .
We also have the following existence result for three positive solutions.
Corollary 3.4. Assume that there exist four positive numbers such that, if one of the two conditions(i), , and (ii), , ,is satisfied, then the PBVP (1.1) has at least three positive solutions satisfying .
Remark 3.5. From similar arguments and techniques, we can also deal with the following periodic boundary value problem (PBVP) We can also verify that the similar results presented in this paper are valid for PBVP (3.21); we omit the details here.
4. Some Examples
In this section, we give some examples to illustrate the main results of the paper.
Example 4.1. Consider the following second-order periodic boundary value problem (PBVP):
where . In this case, .
Corresponding to the assumptions of Corollary 3.3, we set , and . It is easy to check that the other conditions of Corollary 3.3 are satisfied; hence, PBVP (4.1) has at least two positive solutions satisfying .
Example 4.2. Consider the periodic boundary value problem (PBVP) Now, let ; thus, . Set . Then Corollary 3.4 ensures that there exist at least three positive solutions satisfying .
Acknowledgments
The project is supported financially by the National Science Foundation of china (10971179) and the Natural Science Foundation of Changzhou university (JS201008).