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Abstract and Applied Analysis
Volume 2013, Article ID 575869, 11 pages
http://dx.doi.org/10.1155/2013/575869
Research Article

Fixed Points for -Graphic Contractions with Application to Integral Equations

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran

Received 18 August 2013; Accepted 1 September 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 N. Hussain 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 aim of this paper is to define modified weak --contractive mappings and to establish fixed point results for such mappings defined on partial metric spaces using the notion of triangular -admissibility. As an application, we prove new fixed point results for graphic weak -contractive mappings. Moreover, some examples and an application to integral equation are given here to illustrate the usability of the obtained results.

1. Introduction and Preliminaries

The concept of partial metric space was introduced by Matthews [1] in 1994. Partial metric space is a generalized metric space in which each object does not necessarily have to have a zero distance from itself. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle [2, 3]. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions on partial metric spaces (e.g., [410]). For a recent survey on the existence of fixed points in different spaces with generalized distance functions, the reader may check [1118]. We start by recalling some definitions and properties of partial metric spaces.

Definition 1. A partial metric on a nonempty set is a function such that, for all ,(p1); (p2); (p3); (p4).
A partial metric space is a pair such that is nonempty set and is a partial metric on .

From the above definition, if , then . But if , may not be in general. A trivial example of a partial metric space is the pair , where is defined as . For more examples of partial metric spaces, we refer to [6, 7].

Each partial metric on generates a topology on which has as a base the family of open -balls , where for all and . A sequence in converges to a point , with respect to , if and only if . A sequence in is called Cauchy sequence if exists and is finite. A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

A sequence in is called -Cauchy sequence if . A partial metric space is said to be -complete if every -Cauchy sequence in converges, with respect to , to a point such that .

The partial metric space , where denotes the set of rational numbers and the partial metric is given by , provides an example of a 0-complete partial metric space which is not complete.

If is a partial metric on , then the function given by is a metric on . A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if . Some more properties of partial metric spaces can be found in [3, 7, 19].

Samet et al. [20] defined the notion of -admissible mappings and proved the following result.

Definition 2. Let be a self-mapping on and a function. One says that is an -admissible mapping if

Denote with the family of nondecreasing functions such that for all , where is the th iterate of .

Theorem 3. Let be a complete metric space and be -admissible mapping. Assume that where . Also, suppose that the following assertions hold:(i)there exists such that ;(ii)either is continuous or for any sequence in with for all and as , one has for all .
Then, has a fixed point.

Very recently, Salimi et al. [21] modified the notions of --contractive mappings and -admissible mappings as follows.

Definition 4 (see [21]). Let be a self-mapping on and two functions. One says that is an -admissible mapping with respect to if Note that if we take , then this definition reduces to Definition 2. Also, if we take, , then we say that is an -subadmissible mapping.

The following result properly contains Theorem 3 and Theorems 2.3 and 2.4 of [22].

Theorem 5 (see [21]). Let be a complete metric space and be -admissible mapping with respect to . Assume that where and Also, suppose that the following assertions hold:(i)there exists such that ;(ii)either is continuous or for any sequence in with for all and as , one has for all . Then, has a fixed point.

In fact, Banach contraction principle and Theorem 5 hold for the following example, but Theorem 3 does not hold.

Example 6 (see [21]). Let be endowed with the usual metric for all , and let be defined by . Also, define, by and by .

In this paper, we define modified weak --contractive mappings and establish fixed point results for such mappings defined on ordinary as well as ordered partial metric spaces using the notion of triangular -admissibility. As an application, we prove new fixed point results for graphic weak -contractive mappings. Moreover, some examples and an application to integral equation are given here to illustrate the usability of the obtained results.

2. Modified Weak --Contractions

Recently, Karapınar et al. [23] introduced the notion of triangular -admissible mapping as follows.

Definition 7 (see [23]). Let and . One says that is a triangular -admissible mapping if(T1) implies ;(T2) imply .

Lemma 8 (see [23]). Let be a triangular -admissible mapping. Assume that there exists such that . Define sequence by . Then,

Motivated by Samet et al. [20] and Salimi et al. [21], we introduce the following mapping.

Definition 9. Let be a partial metric space and , two mappings. If there exists an upper semicontinuous from the right nondecreasing function with for all such that where Then, we say that is a modified weak --contractive mapping.

Definition 10. Let be a partial metric space. Let and . One says that is an -continuous function on if for given and in ,

Example 11. Let and . Assume that and defined by Clearly, is not continuous, but is an -continuous on . Indeed, if as and , then and so .

Theorem 12. Let be a 0-complete partial metric space and a modified weak --contractive and triangular -admissible mapping. Suppose that the following assertions hold:(i)there exists such that ;(ii) is an -continuous function on . Then, has a fixed point.

Proof. Let such that . Define a sequence in by for all . Then, by Lemma 8, we have If for some , then is a fixed point for , and the result is proved. Hence, we suppose that for all . Now, by (8) and (12) with , , we get On the other hand, and then, . So, from (13), we have Now, if for some , then which is a contradiction. Hence, for all , we have
This implies that the sequence is decreasing, and so, by (17), there is such that Now, we show that must be equal to .
In fact, if , then we get which is a contradiction. Hence, We prove that is a 0-Cauchy sequence. Suppose, to the contrary, that is not a 0-Cauchy sequence. Then, there is and sequences and such that, for all positive integers , Now, for all , we have Taking the limit as in the above inequality and using (20), we get Again, from taking the limit as and by (20) and (23), we deduce Also, since then by taking limit as in the last inequality and applying (20) and (23), we deduce Similarly, Then, by (8) and (12), we obtain where Taking limit supremum as in the above inequality and applying (20), (23), (25), (27), and (28), we get which is a contradiction. Hence, is a 0-Cauchy sequence.
Since is orbitally -continuous on , as and , then we have So, is a fixed point of .

Theorem 13. Let be a 0-complete partial metric space and a modified weak --contractive and -admissible mapping. Suppose that the following assertions hold:(i)there exists such that ;(ii) for all and if is a sequence in such that for all and as , then for all . Then, has a fixed point.

Proof. Condition (ii) implies property (T2) in definition of triangular -admissible map. Indeed, if and , then applying (ii) to defined by we get for , and hence, . Thus, as in Theorem 12, we obtain a 0-Cauchy sequence such that as . Since, for all and as , then from (ii), we have for all . Then, from (8), we get where By taking limit supremum as in the above inequality, we have which is a contradiction. Hence, . That is, .

Example 14. Let be endowed with the partial metric for all , and let be defined by Define and by
Clearly, is a 0-complete partial metric space. We show that is a triangular -admissible mapping. Let , if , then or . On the other hand, for all , we have . It follows that . Also, if , and then . That is, . Hence, the assertion holds. In reason of the above arguments, .
Now, if is a sequence in such that for all and as , then , and hence, . This implies that for all .
Let . Then, . We get That is, Hence, all conditions of Theorem 13 hold and is a fixed point of .

Corollary 15. Let be a 0-complete partial metric space and be such that for some . Assume that hold for all . Also, suppose that one of the following assertions holds:(i) is triangular -admissible and -continuous on ;(ii) is -admissible, for all and if is a sequence in such that for all and as , then for all . Then, has a fixed point.

Corollary 16. Let be a 0-complete partial metric space and be such that for some . Assume that hold for all where . Also, suppose that one of the following assertions holds:(i) is triangular -admissible and -continuous on ;(ii) is -admissible, for all and if is a sequence in such that for all and as , then for all . Then, has a fixed point.

Proof. Let . Then, . Hence, by (44), we have Thus, . Hence, conditions of Corollary 15 hold, and has a fixed point.

Similarly, we have the following corollary.

Corollary 17. Let be a 0-complete partial metric space and be such that for some . Assume that hold for all where . Also, suppose that one of the following assertions holds:(i) is triangular -admissible and -continuous on ;(ii) is -admissible, for all and if is a sequence in such that for all and as , then for all .
Then, has a fixed point.

Corollary 18 (Matthews [1]). Let be a 0-complete partial metric space and a given mapping satisfying for all , where . Then, has a unique fixed point.

3. Fixed Point Results in Partially Ordered Partial Metric Spaces

Fixed point theorems for monotone operators in partially ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [24, 25] and references therein).

Theorem 19 (see [24]). Let be a partially ordered complete metric space and a continuous increasing self-mapping such that for some . Assume that hold for all with , where . Then, has a fixed point.

Theorem 20. Let be a partially ordered 0-complete partial metric space and an increasing self-mapping such that for some . Assume that hold for all with . Now, if is a continuous mapping on , then has a fixed point.

Proof. Define, by At first we prove that is a triangular -admissible mapping. Let , then . As is an increasing mapping, we have . That is, . Also, let and , then and . So, from transitivity, we have . That is, . Thus, is a triangular -admissible mapping. Also, there exists such that which implies . Let , then . Now, from (49), we have . That is, Hence, all conditions of Theorem 12 are satisfied, and has a fixed point.

Theorem 21 (see [25]). Let be a partially ordered complete metric space and an increasing mapping such that for all with , where . Suppose that the following assertions hold:(i)there exists such that ;(ii)if is a sequence in such that for all and as , then for all . Then, has a fixed point.

Theorem 22. Let be a partially ordered 0-complete partial metric space and an increasing mapping such that for all with . Suppose that the following assertions hold:(i)there exists such that ;(ii)if is a sequence in such that for all and as , then for all . Then, has a fixed point.

Proof. Define as in proof of Theorem 20. Assume for all such that as . Then, for all . Hence, by (ii), we get for all and so for all . Proceeding as in proof of Theorem 20, we can deduce that is a modified weak --contractive and -admissible mapping and there exists such that . Hence, all conditions of Theorem 13 hold, and has a fixed point.

Remark 23. Similarly, we may obtain more fixed point results on ordered partial metric spaces as immediate consequences of Corollaries 1517.

4. Fixed Point Results for Graphic Contractions

Consistent with Jachymski [26], let be a partial metric space, and let denote the diagonal of the Cartesian product . Consider a directed graph such that the set of its vertices coincides with and the set of its edges contains all loops, that is, . We assume that has no parallel edges, so we can identify with the pair . Moreover, we may treat as a weighted graph (see [26]) by assigning to each edge the distance between its vertices. If and are vertices in a graph , then a path in from to of length is a sequence of vertices such that , and for . A graph is connected if there is a path between any two vertices. is weakly connected if is connected (see for details [11, 2628]).

Recently, some results have appeared providing sufficient conditions for a mapping to be a Picard operator if is endowed with a graph. The first result in this direction was given by Jachymski [26].

Definition 24 (see [26]). One says that a mapping is a Banach -contraction or simply -contraction if preserves edges of , that is, and decreases weights of edges of in the following way:

Definition 25 (see [26]). A mapping is called -continuous if given and sequence

Definition 26. Let be a partial metric space endowed with a graph and a self-mapping. If there exists an upper semicontinuous from the right function with for all such that where Then, is called a weak -graphic contractive mapping.

Theorem 27. Let be a 0-complete partial metric space endowed with a graph and a weak -graphic contractive mapping. Suppose that the following assertions hold:(i)there exists such that ;(ii) is -continuous on ;(iii) and imply for all , that is, is a quasi-order [26]. Then, has a fixed point.

Proof. Define by At first we prove that is a triangular -admissible mapping. Let , then . As is a weak -graphic contractive mapping, we have . That is, . Also, let and , then and . So, from (iii), we have . That is, . Thus, is a triangular -admissible mapping. Let be -continuous on . Then, That is, which implies is -continuous on . From (i), there exists such that . That is, .
If , then . Now, since is a weak -graphic contractive mapping, so . That is, Hence, all conditions of Theorem 12 are satisfied, and has a fixed point.

If is a connected graph, then condition (iii) of Theorem 27 is automatically satisfied. Thus, we have the following result.

Corollary 28. Let be a 0-complete partial metric space endowed with a graph and a weak -graphic contractive mapping. Suppose that the following assertions hold:(i)there exists such that ;(ii) is -continuous on ; (iii) is a connected graph. Then, has a fixed point.

Theorem 29. Let be a 0-complete partial metric space endowed with a graph and a weak -graphic contractive mapping. Suppose that the following assertions hold:(i)there exists such that ;(ii)if is a sequence in such that for all and as , then for all . Then, has a fixed point.

Proof. Define, as in proof of Theorem 27. Condition (ii) implies that is a quasi-order, that is, and imply for all (see Remark 3.1 [26]). Let, as and for all . Then, is a sequence in such that for all and as . So, by (ii), we have for all . That is, . All other conditions of Theorem 13, follow similarly as in proof of Theorem 27 and consequently has a fixed point.

Theorem  3.2(2°) in [26] and Theorem 2.3(2) in [29] are extended to weak -graphic contractive maps defined on a 0-complete partial metric space as follows.

Corollary 30. Let be a 0-complete partial metric space endowed with a graph and a weak -graphic contractive mapping. Suppose that the following assertions hold:(i) there exists such that ;(iis) is a quasi-order and if is a sequence in such that for all and as , then there is a subsequence with for all . Then, has a fixed point.

Proof. Condition (iis) implies that of in Theorem 29 (see Remark 3.1 [26]). Now, the conclusion follows from Theorem 29.

Corollary 31. Let be a 0-complete partial metric space and -chainable for some , that is, given , there is and a sequence such that , and for . Suppose that is a mapping satisfying Then, has a fixed point.

Proof. Consider the graph with and . Then, -chainability of means is connected, and hence, is quasi-order. If , then so , hence is a -contraction. Let be in with , then for sufficiently large , so there is such that . Thus, by Corollary 30, has a fixed point.

Definition 32. Let be a partially ordered partial metric space endowed with a graph and a self-mapping. If there exists an upper semicontinuous from the right function with for all such that, for all , with , where where Then, we say is an ordered weak -graphic contractive mapping.

Theorem 33. Let be a partially ordered 0-complete partial metric space endowed with a graph and an ordered weak -graphic contractive mapping. Suppose that the following assertions hold:(i)there exists such that with ,(ii)either is -continuous in and and imply or; (iii)if is a sequence in such that with for all and as , then with for all . Then, has a fixed point.

Proof. Define by At first, we prove that is a triangular -admissible mapping. Let , then with . As is an ordered weak -graphic contractive mapping, we have where . That is, . Also, let and , then with and with . So from (ii), we have . Also, and implies . Hence, . Thus, is a triangular -admissible mapping. Let be -continuous on . Then, That is, which implies is -continuous on . From (i), there exists such that . That is, .
Let , then with . Now, since is an ordered weak -graphic contractive mapping, then . That is, Hence, all conditions of Theorem 12 (or 13) are satisfied, and has a fixed point.

Remark 34. All our results established above are new even in the setting of complete metric spaces.

5. Application to Existence of Solutions of Integral Equations

Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [30, 31] and references therein). In this section, we apply our result to the existence of a solution of an integral equation. Let be the set of real continuous functions defined on , and let be defined by for all . Then, is a complete metric space. Also, assume this metric space endowed with a graph .

Consider the integral equation and let be defined by We assume that (A) is continuous; (B) is continuous; (C) is continuous; (D)there exists an upper semicontinuous from the right nondecreasing function with for all such that for all , (F) there exist such that ;(G) if is a sequence in such that for all and as , then for all ;(H) assume

Theorem 35. Under assumptions , the integral equation (72) has a solution in .

Proof. Consider the mapping defined by (73). Let . Then, from , we deduce Then, That is, implies Thus, all the hypotheses of Theorem 29 are satisfied, and hence, the mapping has a fixed point, that is, a solution in of the integral equation (72).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first and second authors acknowledge with thanks DSR, KAU, for financial support.

References

  1. S. G. Matthews, “Partial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 728, pp. 183–197, Annals of the New York Academy of Sciences, 1994.
  2. L. \'Cirić, B. Samet, H. Aydi, and C. Vetro, “Common fixed points of generalized contractions on partial metric spaces and an application,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2398–2406, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  3. N. Hussain, Z. Kadelburg, S. Radenović, and F. R. Al-Solamy, “Comparison functions and fixed point results in partial metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 605781, 15 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  4. A. Amini-Harandi, “Metric-like spaces, partial metric spaces and fixed points,” Fixed Point Theory and Applications, vol. 2012, article 204, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  5. D. Ilić, V. Pavlović, and V. Rakočević, “Some new extensions of Banach's contraction principle to partial metric space,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1326–1330, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  6. Z. Golubović, Z. Kadelburg, and S. Radenović, “Coupled coincidence points of mappings in ordered partial metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 192581, 18 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. A. Kutbi, J. Ahmad, N. Hussain, and M. Arshad, “Common fixed point results for mappings with rational expressions,” Abstract and Applied Analysis, vol. 2013, Article ID 549518, 11 pages, 2013. View at Publisher · View at Google Scholar
  8. H. K. Nashine, Z. Kadelburg, S. Radenović, and J. K. Kim, “Fixed point theorems under Hardy-Rogers contractive conditions on 0-complete ordered partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 180, 2012. View at Publisher · View at Google Scholar
  9. Z. Kadelburg, H. K. Nashine, and S. Radenović, “Fixed point results under various contractive conditions in partial metric spaces,” Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales A. View at Publisher · View at Google Scholar
  10. W. Long, S. Shukla, S. Radenović, and S. Radojević, “Some coupled coincidence and common fixed point results for hybrid pair of mappings in 0-complete partial metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 145, 2013. View at Publisher · View at Google Scholar
  11. M. Abbas and T. Nazir, “Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph,” Fixed Point Theory and Applications, vol. 2013, article 20, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  12. Lj. \'Cirić, B. Samet, C. Vetro, and M. Abbas, “Fixed point results for weak contractive mappings in ordered -metric spaces,” Fixed Point Theory, vol. 13, no. 1, pp. 59–72, 2012. View at Google Scholar · View at MathSciNet
  13. M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, John Wiley & Sons, New York, NY, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  14. P. Kumam, C. Vetro, and F. Vetro, “Fixed points for weak α-ψ-contactions in partial metric spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 986028, 9 pages, 2013. View at Publisher · View at Google Scholar
  15. A. Shoaib, M. Arshad, and J. Ahmad, “Fixed point results of locally cotractive mappings in ordered quasi partial metric spaces,” The Scientific World Journal, vol. 2013, Article ID 194897, 8 pages, 2013. View at Google Scholar
  16. S. Shukla, S. Radenović, and V. C. Rajić, “Some common fixed point theorems in 0−σ−complete metric-like spaces,” Vietnam Journal of Mathematics, 2013. View at Publisher · View at Google Scholar
  17. H. K. Nashine, Z. Kadelburg, and S. Radenović, “Fixed point theorems via various cyclic contractive conditions in partial metric spaces,” Publications de LfInstitut Mathematique, Nouvelle Serie Tome, vol. 93, no. 107, pp. 69–93, 2013. View at Google Scholar
  18. A. G. Bin Ahmad, Z. M. Fadail, V. C. Rajić, and S. Radenović, “Nonlinear contractions in 0-complete partial metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 451239, 12 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. S. M. Alsulami, N. Hussain, and A. Alotaibi, “Coupled fixed and coincidence points for monotone operators in partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 173, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  20. B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for αψ-contractive type mappings,” Nonlinear Analysis: Theory and Methods, vol. 75, no. 4, pp. 2154–2165, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  21. P. Salimi, A. Latif, and N. Hussain, “Modified αψ-contractive mappings with applications,” Fixed Point Theory and Applications, vol. 2013, article 151, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  22. E. Karapınar and B. Samet, “Generalized (αψ) contractive type mappings and related fixed point theorems with applications,” Abstract and Applied Analysis, vol. 2012, Article ID 793486, 17 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  23. E. Karapınar, P. Kumam, and P. Salimi, “On αψ-Meir-Keeler contractive mappings,” Fixed Point Theory and Applications, vol. 2013, article 94, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  25. J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  26. J. Jachymski, “The contraction principle for mappings on a metric space with a graph,” Proceedings of the American Mathematical Society, vol. 136, no. 4, pp. 1359–1373, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  27. I. Beg, A. R. Butt, and S. Radojević, “The contraction principle for set valued mappings on a metric space with a graph,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1214–1219, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  28. F. Bojor, “Fixed point theorems for Reich type contractions on metric spaces with a graph,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 9, pp. 3895–3901, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  29. F. Bojor, “Fixed point of ϕ-contraction in metric spaces endowed with a graph,” Annals of the University of Craiova. Mathematics and Computer Science Series, vol. 37, no. 4, pp. 85–92, 2010. View at Google Scholar · View at MathSciNet
  30. R. P. Agarwal, N. Hussain, and M. A. Taoudi, “Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations,” Abstract and Applied Analysis, vol. 2012, Article ID 245872, 15 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  31. N. Hussain, A. R. Khan, and R. P. Agarwal, “Krasnosel'skii and Ky Fan type fixed point theorems in ordered Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 3, pp. 475–489, 2010. View at Google Scholar · View at MathSciNet