Abstract and Applied Analysis

Volume 2013, Article ID 406727, 5 pages

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

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

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.

#### References

- A. J. B. Potter, “Applications of Hilbert's projective metric to certain classes of non-homogeneous operators,”
*The Quarterly Journal of Mathematics. Oxford. Second Series*, vol. 28, no. 109, pp. 93–99, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Zhai and C. Guo, “On $\alpha $-convex operators,”
*Journal of Mathematical Analysis and Applications*, vol. 316, no. 2, pp. 556–565, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Zhao and X. Du, “Fixed points of generalized $e$-concave (generalized $e$-convex) operators and their applications,”
*Journal of Mathematical Analysis and Applications*, vol. 334, no. 2, pp. 1426–1438, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - J. R. L. Webb, “Remarks on ${u}_{0}$-positive operators,”
*Journal of Fixed Point Theory and Applications*, vol. 5, no. 1, pp. 37–45, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Zhao, “Multiple fixed points of a sum operator and applications,”
*Journal of Mathematical Analysis and Applications*, vol. 360, no. 1, pp. 1–6, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhao and X. Chen, “Fixed points of decreasing operators in ordered Banach spaces and applications to nonlinear second order elliptic equations,”
*Computers & Mathematics with Applications*, vol. 58, no. 6, pp. 1223–1229, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-B. Zhai and X.-M. Cao, “Fixed point theorems for $\tau -\phi $-concave operators and applications,”
*Computers & Mathematics with Applications*, vol. 59, no. 1, pp. 532–538, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Zhao, “Existence and uniqueness of fixed points for some mixed monotone operators,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 73, no. 6, pp. 1481–1490, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhao, “Fixed points of $\tau -\phi $-convex operators and applications,”
*Applied Mathematics Letters*, vol. 23, no. 5, pp. 561–566, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - R. P. Agarwal, S. R. Grace, and D. O'Regan, “Existence of positive solutions to semipositone Fredholm integral equations,”
*Funkcialaj Ekvacioj*, vol. 45, no. 2, pp. 223–235, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Q. Lan, “Positive solutions of semi-positone Hammerstein integral equations and applications,”
*Communications on Pure and Applied Analysis*, vol. 6, no. 2, pp. 441–451, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Q. Lan, “Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 12, pp. 5979–5993, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. R. Graef and L. Kong, “Positive solutions for third order semipositone boundary value problems,”
*Applied Mathematics Letters*, vol. 22, no. 8, pp. 1154–1160, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. R. L. Webb and G. Infante, “Semi-positone nonlocal boundary value problems of arbitrary order,”
*Communications on Pure and Applied Analysis*, vol. 9, no. 2, pp. 563–581, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Dogan, J. R. Graef, and L. Kong, “Higher order semipositone multi-point boundary value problems on time scales,”
*Computers & Mathematics with Applications*, vol. 60, no. 1, pp. 23–35, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. R. Anderson and C. Zhai, “Positive solutions to semi-positone second-order three-point problems on time scales,”
*Applied Mathematics and Computation*, vol. 215, no. 10, pp. 3713–3720, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Xian, D. O'Regan, and C. Yanfang, “Structure of positive solution sets of semi-positone singular boundary value problems,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 7-8, pp. 3535–3550, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. J. Guo and V. Lakshmikantham,
*Nonlinear Problems in Abstract Cones*, vol. 5 of*Notes and Reports in Mathematics in Science and Engineering*, Academic Press, Boston, Mass, USA, 1988. View at MathSciNet - K. Deimling,
*Nonlinear Functional Analysis*, Springer, Berlin, Germany, 1985. View at MathSciNet