Abstract and Applied Analysis

Volume 2013 (2013), Article ID 406727, 5 pages

http://dx.doi.org/10.1155/2013/406727

Research Article

Positive Fixed Points for Semipositone Operators in Ordered Banach Spaces and Applications

Zengqin Zhao and Xinsheng Du

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Received 23 January 2013; Accepted 7 April 2013

Academic Editor: Kunquan Lan

Copyright © 2013 Zengqin Zhao and Xinsheng Du. 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 theory of semipositone integral equations and semipositone ordinary differential equations has been emerging as an important area of investigation in recent years, but the research on semipositone operators in abstract spaces is yet rare. By employing a well-known fixed point index theorem and combining it with a translation substitution, we study the existence of positive fixed points for a semipositone operator in ordered Banach space. Lastly, we apply the results to Hammerstein integral equations of polynomial type.

1. Introduction

Existence of fixed points for positive operators have been studied by many authors; see [1–9] and their references. The theory of semipositone integral equations and semipositone ordinary differential equations has been emerging as an important area of investigation in recent years (see [10–17]). But the research on semipositone operators in abstract spaces are yet rare up to now.

Inspired by a number of semipositone problems for integral equations and ordinary differential equations, we study the existence of positive fixed points for semipositone operators in ordered Banach spaces. Then the results are applied to Hammerstein integral equations of polynomial type.

Let be a real Banach space with the norm , a cone of , and “” the partial ordering defined by , denoting the zero element of , , .

Recall that cone is said to be normal if there exists a positive constant such that implies , the smallest is called the normal constant of . An element is called the least upper bound (i.e., supremum) of set , if it satisfies two conditions: (i) for any ; (ii) , implies . We denote the least upper bound of by , that is, .

Definition 1. Cone is said to be minihedral if exists for each pair of elements . For any in we define .

Definition 2 (see [1, 3]). Let be real Banach spaces, cones of , ,  , and . Then we say is -convex if and only if for all .

Definition 3. Let be real Banach spaces, cones of , and . , . is said to be nondecreasing if implies ; is said to be positive if for any ; is said to be semipositone if (i) there exists an element such that and (ii) there exists an element such that for any .

In order to prove the main results, we need the following lemma which is obtained in [18].

Lemma 4. Let be a real Banach space and a bounded open subset of , with , and is a completely continuous operator, where is a cone in .

(i)Suppose that , for all , , then the fixed point index .(ii)Suppose that , for all , then .

The research on ordered Banach spaces, cones, fixed point index, and the above lemma can be seen in [18, 19].

2. Main Results and Their Proofs

Theorem 5. Let be Banach space, cones, and . Suppose that operator can be expressed as , where the cone and the operator and satisfy the following conditions:

(H1) when is normal and minihedral, is normal;(H2) when is continuous, there exist , , a nondecreasing -convex operator , , and a bounded functional such that

(H3) when is linear completely continuous, there exists such that

(H4) when there exists a positive number such that

with , is the normal constant of . Then has a fixed point .

Proof. For in (H2) and in (H3), we define that

Clearly, is a normal cone of . Since the cone is minihedral, makes sense. By (H4) and (4), we know that

From the condition (H3) and (4), we know that , and hence and

By (7), we have , using (H2) we know that

That is, . This and (2) and (5) imply , for all . Hence,

Suppose that is a bounded set of , is a positive number satisfying , for all . By (7) and normality of , we obtain that

Therefore, (H2) implies that , . Since is normal, the order interval is a bounded set of ; therefore, is a bounded set of . This together with (9), continuity of , and the completely continuity of , we obtain that map into and is completely continuous.

For the in (H4), we let . By (7) we know that

Therefore, from (H2) we obtain that

where is as in (H4).

We prove that

Assume there exist and , such that . Using (12) we have

hence

which contradicts the condition (3), thus (13) holds. By Lemma 4 we know

Take such that , and set

where as in (3), is the normal constant of . In the following, we prove

Assume there exists such that . Using (6), we have , thus it is obtained that

by (17). From (17) we know , thus . This and (H3), (4), and (19) imply

By convexity of we know

By (17) we know , hence (20) and (21) imply

This together with (5) and the condition (H2) imply

This and (23) imply

therefore,

which contradicts (17), thus (18) holds. Using Lemma 4 we have

By (16) and (26) and additivity of fixed point indexes we know that

Thus, has a fixed point on . Hence,

Let . From (6) and we know , then . This together with (4) and (28) imply , so that is a positive fixed point of .

3. Corollary and Applications

From Theorem 5 we obtain the following corollary.

Corollary 6. Suppose that conditions (H1), (H2), and (H3) hold, and in addition assume the following.

(H5) For any , there exists a positive number such that .

Then there exists a small enough such that has a positive solution for any .

Proof. For any fixed , by (H5), we can all take , such that

hence (H4) holds. We take that

Then for , the conditions in Theorem 5 are satisfied. Thus, has a positive fixed point, that is, has a positive solution, and the proof is complete.

We consider the integral equation

where is a bounded closed domain in and , , , , is nonnegative continuous on .

Theorem 7. Suppose that among there exists such that , and there exist nontrivial nonnegative functions , , and a positive number such that

Then (31) has a nontrivial nonnegative solution in .

Proof. Let the Banach space with the sup norm ,

with , ,

Then is normal minihedral, the normal constant . is a cone of , , . is nondecreasing -convex operator, and . is continuous; .

It is known easily that

thus exits in (33) and

By (33), (43), and we have

therefore

From (33), (39), and (44) we know easily that there exists such that . From (37)–(46), we obtain that

Equations (32) and (42) imply that , and hence

By (42), (32), (34), and (37), we obtain that

By (41) we have . This and (34) and (42) get that

From (35) and (36) we know that (H1) is satisfied. By (47) and (48) we obtain that (H2) and (H3) are satisfied. Equations (49) and (50) imply that (H4) is satisfied. Therefore, using Theorem 5, the integral equation (31) has a positive solution in .

Acknowledgments

The authors are very grateful to the referee for his or her valuable suggestions. This research was supported by the National Natural Science Foundation of China (10871116), the Doctoral Program Foundation of Education Ministry of China (20103705120002), Shandong Provincial Natural Science Foundation, China (ZR2012AM006), and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.

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