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Abstract and Applied Analysis
Volume 2010, Article ID 439137, 13 pages
http://dx.doi.org/10.1155/2010/439137
Research Article

Positive Fixed Point of Strict Set Contraction Operators on Ordered Banach Spaces and Applications

1School of Applied Science, Beijing Information Science & Technology University, Beijing 100192, China
2Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
3Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received 4 February 2010; Revised 8 May 2010; Accepted 7 July 2010

Academic Editor: Ferhan M. Atici

Copyright © 2010 Meiqiang Feng et al. 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

The fixed point theorem of cone expansion and compression of norm type for a strict set contraction operator is generalized by replacing the norms with a convex functional satisfying certain conditions. We then show how to apply our theorem to prove the existence of a positive solution to a second-order differential equation with integral boundary conditions in an ordered Banach space. An example is worked out to demonstrate the main results.

1. Introduction

The theory of integral and differential equations in Banach spaces, as two new branches of nonlinear functional analysis, has developed for no more than forty years, but it has extensive applications in such domains as the critical point theory, the theory of partial differential equations, and eigenvalue problems. For an introduction of the basic theory of integral and differential equations in Banach spaces, see Guo et al. [1], Guo and Lakshmikantham [2], Lakshmikantham and Leela [3], and Demling [4], and the references therein. In recent years, the theory of integral and differential equations in Banach spaces has become an important area of investigation in both pure and applied mathematics (see, for instance, [518] and references cited therein).

On the other hand, the theory of fixed point is an important tool to study various boundary value problems of ordinary differential equations, difference differential equations, and dynamic equations on time scales. An overview of such results can be found in Guo et al. [1], in Guo and Lakshmikantham [2], and in Demling [4]. The Krasnoselskii's fixed point theorem concerning cone compression and expansion of norm type is worth mentioing here as follows (see [1, 2, 4]).

Theorem 1.1. Let and be two bounded open sets in Banach space , such that and . Let be a cone in and let operator be completely continuous. Suppose that one of the following two conditions is satisfied: (a) and ; (b) and .Then, has at least one fixed point in .

To generalize Theorem 1.1, one may consider the weakening of one or more of the following hypotheses: (i) the operator , (ii) the norm.

In [19], Sun generalized Theorem 1.1 for completely continuous operator to strict set contraction operator and obtain the following results.

Theorem 1.2. Let be a cone of Banach space and with Suppose that is a strict set contraction such that one of the following two conditions is satisfied: (a)(b)Then, has a fixed point .

Recently, in [20], Anderson and Avery generalized the fixed point theorem of cone expansion and compression of norm type by replacing the norms with two functionals satisfying certain conditions to produce a fixed point theorem of cone expansion and compression of functional type. In [21], Guo and Ge extended Krasnoselskii's fixed point theorem by choosing two functionals that satisfy certain conditions which are used in place of the norm. In [22], Zhang and Sun generalized the classical Krasnoselskii's fixed point theorem concerning cone compression and expansion of norm type. The interesting point is that they took place norm by convex functional.

In the past few years, we also notice a class of boundary value problems with integral boundary conditions appeared in heat conduction, chemical engineering, underground water flow, thermo elasticity, and plasma physics. Such problems include two, three, multi point and nonlocal boundary value problems as special cases and attracted the attention of Gallardo [23], Karakostas and Tsamatos [24], Lomtatidze and Malaguti [25], and others included in the references therein. On the other hand, we refer the reader to papers by Ahmad et al. [26], Feng et al. [27], Boucherif [28], Infante and Webb [29], Kang et al. [30], Ma [31], Webb [32], Webb and Infante [33, 34], Yang [35], Zhang et al. [3638], and Chang et al. [39] for other recent results on nonlinear boundary value problems with integral boundary conditions.

Motivated by works mentioned above, we intend in this paper to generalize the fixed point theorem of cone expansion and compression of norm type for strict set contraction operator. The generalization allows the user to choose a convex functional that satisfies certain conditions which are used in place of the norm. In applications to boundary value problems, the functional will typically be maximum of the function over a specific interval. The flexibility of using functionals instead of norms allows the theorem to be used in a wider variety of situations. Our results either improve or generalize the corresponding results due to [1922] and many of others. As an application of our main results, we consider the existence of positive solutions for second-order differential equations with integral boundary conditions in an ordered Banach space. On the other hand, our conditions are weaker than those of [22].

The organization of this paper is as follows. We will introduce some lemmas and notations in the rest of this section. In Section 2, the main results will be stated and proved. In Section 3, as an application of our main results, the existence of positive solutions for a second-order boundary value problem with integral boundary conditions in ordered Banach spaces is considered. Finally, in Section 4, one example is also included to illustrate the main results.

Basic facts about ordered Banach space can be found in [14]. Here we just recall a few of them. The cone in induces a partial order on , that is, if and only if . is said to be normal if there exists a positive constant such that implies . Without loss of generality, suppose, in the present paper, the normal constant .

For a bounded set in Banach space , we denote the Kuratowski measure of noncompactness (see [14], for further understanding). The operator is said to be a -set contraction if is continuous and bounded and there is a constant such that for any bounded ; a -set contraction with is called a strict set contraction.

In the following, denote the Kuratowski's measure of noncompactness by .

For the application in the sequel, we first state the following definition and lemmas which can be found in [1], and some notation.

Definition 1.3. Let be a bounded set of a real Banach space . Let be expressed as the union of a finite number of sets such that the diameter of each set does not exceed , that is, with . Clearly, . is called the Kuratowski's measure of noncompactness.

Definition 1.4. Let be a cone of a real Banach space . If then is a dual cone of cone .

Definition 1.5. Let be a cone of real Banach space . is said to be a convex functional on if for all and .

Definition 1.6. A subset is said to be a retract of if there exists a continuous mapping satisfying

Lemma 1.7. Let , be a bounded set and uniformly continuous and bounded from into ; then

2. Main Results

Lemma 2.1 (see [22]). Let be a cone in a real Banach space . If is a uniformly continuous convex functional with and for , then , is a retract of .

Lemma 2.2. Let be a real Banach space, the norm in , a cone in , and , where is a positive real number. Suppose that is a -set contraction with and is a uniformly continuous convex functional with , , and . If (i) and there exists such that and (ii), hold, then the fixed point index .

Proof. Without loss of generality, we suppose (If , then let . The proof is the same as the following process). Let , then is a strict set contraction. Considering , , . If there exists , such that then which contradicts with (ii). Then by the homotopy invariance property of fixed point index, we have
Let Define . It follows from that From the fact that , we have . In fact, since we have . On the other hand, for combining this with , we have then Let , then by (i) we obtain and , and , where
Let Then, we have and then we obtain that is the strict set contraction. In addition, it is obvious that is uniformly continuous about for all
If there exists such that then which contradicts with (ii). Thus by the homotopy invariance property of fixed point index, we have
Since is a retract of by Lemma 2.1, there exists a retraction satisfying Let then is strict set contraction. From (i) and the definition of , we have Therefore, , that is, , . Then
If then , which implies that has a fixed point in . Thus It is a paradox. The proof is complete.

Lemma 2.3 (see [2]). Let be a cone and a bounded open set in with . Suppose that is condensing and Then .

Lemma 2.4. Let be a cone and a bounded open set in . Suppose that is a -set contraction with and is a uniformly continuous convex functional with and for . If and for then .

Proof. If there exist and such that then Therefore, It is a paradox. From Lemma 2.3, it follows that . The proof is complete.

Theorem 2.5. Let be a bounded open set in such that , and and Suppose that is a -set contraction with and is a uniformly continuous convex functional with and and . If (a), (b) and there exists such that and and , hold, then has at least one fixed point in

Proof. It is easy to obtain the results by Lemmas 2.2 and 2.4. So we omit it.

Theorem 2.6. Let and a bounded open set in such that Suppose that is a -set contraction with and is a uniformly continuous convex functional with and and . If (a) and there exists such that and and (b)are satisfied, then has at least one fixed point in

Proof. It is easy to obtain the results by Lemmas 2.2 and 2.4. So we omit it.

Remark 2.7. If we let , then is completely continuous. Comparing with Corollary of [22], our conditions are weaker.

Corollary 2.8. Let be a bounded open set in such that , and Suppose that is a -set contraction with and is a uniformly continuous convex functional with and and . If (a), (b) with and hold, then has at least one fixed point in

Proof. It follows by taking .

Corollary 2.9. Let and a bounded open set in such that Suppose that is a -set contraction with and is a uniformly continuous convex functional with and and . If (a) with and (b), hold, then has at least one fixed point in

Proof. It follows by taking .

3. Applications

Throughout the remainder of this paper, we apply the above results to a second-order differential equation in Banach spaces: subject to the following integral boundary conditions: where , is the zero element of , and is nonnegative.

We consider problem (3.1)-(3.2) in , in which . Evidently, is a Banach space with norm for .

To establish the existence of positive solutions in of (3.1)-(3.2), let us list the following assumptions.

?? and for any is uniformly continuous on . Further suppose that is nonnegative, and there exist nonnegative constants with such that where , , and .

It is easy to see that the problem (3.1)-(3.2) has a solution if and only if is a solution of the operator equation where

From (3.6) and (3.7), we can prove that have the following properties.

Proposition 3.1. Assume that holds. Then for we have

Proposition 3.2. For , we have

Proposition 3.3. Let , . Then for all , we have

Proposition 3.4. Assume that holds. Then for , we have where is defined in , and

Proof. By (3.6) and (3.9), we have
On the other hand, noticing , we obtain

Proposition 3.5. Assume that holds. Then for all we have

Proof. By (3.10), we have

We construct a cone by where

It is easy to see that is a cone of .

We will make use of the following lemmas.

Lemma 3.6. Suppose that holds. Then for each is strict set contraction on , that is, there exists a constant such that for any , where .

Proof. By , we know that is uniformly continuous on . Hence, is bounded on . This together with (3.4) and Lemma 1.7 implies that From being uniformly continuous and bounded on , we can obtain that is continuous and bounded from into .
On the other hand, it is clear that and using a similar method as in the proof of Lemma in [40], we can get that
Therefore, where , The proof is complete.

Lemma 3.7. Suppose that holds. Then and is a strict set contraction.

Proof. From (3.5) and (3.15), we obtain Therefore, , that is, .
Next by Lemma 3.6, one can prove that is a strict set contraction. So it is omitted.

Let

Theorem 3.8. Assume that holds and is normal. If there exist with such that for and for where then problem (3.1)-(3.2) has at least one positive solution.

Proof. Let be the cone preserving, strict set contraction that was defined by (3.5).
Let . Then is a uniformly continuous convex functional with and for Let
It is clear that and are open sets in with and If we have which implies that is bounded.
If then .
If then and .
Hence, the proof is finished by Corollary 2.9.

4. Example

Example 4.1. To illustrate how our main results can be used in practice, we present an example. For the convenience of computation, we study a two-point boundary value problem. Now we consider the following boundary value problem: where and Hence , In this case , and . Let ,?? then
Select then we can see that Hence, the conditions of the Theorem 3.8 are satisfied. Then problem (4.1) has at least one positive solution.

Remark 4.2. Example 4.1 implies that there is a large number of functions that satisfy the conditions of Theorem 3.8. In addition, the conditions of Theorem 3.8 are also easy to check.

Acknowledgments

This work is sponsored by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201008430), the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201010772018), the 2010 level of scientific research of improving project (5028123900), the Level of Graduate Education of Improving-Graduate Technology Innovation Project (5028211000), and Beijing Municipal Education Commission (71D0911003). The authors thank the referee for his/her careful reading of the manuscript and useful suggestions.

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