Printer Friendly


Byline: N. Shafqat, N. Yasmin and Z. Akhter


The aim of this work is to derive the results for random coincidence points for multivalued nonlinear contractions in partially ordered metric spaces. We do it from two different approaches, the first is -symmetric property and the second is by using g - mixed monotone property. These results are the random versions of Hussain and Alotabi [Fixed Point Theory and Appl.,2011, 2011:82]. The present theorems extend certain results due to Ciric, Samet and Vetro.

Key Words: Partially ordered set, -symmetric property, mixed g -monotone property, Compatible maps, Couple random coincidence point.

AMS 2010 Subject Classification: Primary 47H10, Secondary 54H25


Random fixed point theorems are stochastic generalization of classical fixed point theorems. Random fixed point theorems for contraction mappings on separable complete metric spaces were first proved by Spacek [23] and Hans [5,6]. The survey article by Bharucha-Ried [1] attereched the attention of several mathematicians (see Zhang and Huang [26], Hans [5,6], Huang [7], Itoh [10], Lin [14], Papageorgiou [16,17], Shahzad and Hussain [21], Shahzad and Latif [22], Tan and Yuan [24] and give wings to this theory. Itoh [10] extended Spacek and Hans's theorem to multivalued contraction mappings. The stochastic version of the well-known Schauder's fixed point theorem was proved by Sehgal and Singh [20], Ciric and Lakshmikantham [4], Zhu and Xiao [27], Hussain et all [9] and Khan et all [11] proved some coupled random fixed point and coupled random coincidence results in partially ordered complete metric spaces.

Ciric et all [3] proved fixed point theorems for single-valued mappings, extended to a coincid ence theorems for a pair of a random operator f : X X and a multi-valued random operator T : X CB( X ) . The aim of this article is to prove a stochastic analog of the Hussain and Alotaibi [8] coupled coincidences for multi- valued contractions in partially ordered metric spaces for a pair of random operators g: X X and a multi- valued random operator T : X CL( X )

2. Preliminaries

Let ( X , d ) be a metric space. We denote by CB( X ) the collection of non-empty closed bounded subsets of X . For A, B CB( X ) and x X , suppose that D(x, A)


Hussain and Alotaibi [8] proved the following theorems for the existence of coupled coincidence for multi-valued nonlinear contractions using two different approaches, first is based on -symmetric property recently studied in [19] and second one is based on mixed g -monotone property studied by Lakishmikantham and C/iric [13].

Theorem 2.9.


Using the concept of commuting maps and mixed g monotone property, Lakshmikantham and C/iric [13] established the existence of coupled coincidence point results to generalize the results of Bhaskar and lakshmikantham [2]. Hussain and Alotaibi [8] proved the following results by using the mixed g -monotone property for compatible maps F and g in partially ordered metric space, where F is the multi-valued mapping.


Theorem 2.11.


For notational convenience, we use the symbol d for the product metric as well as for the metric on X . Now, we prove the following result that provide the existence of a coupled random coincidence point for compatible maps F and g in partially ordered metric spaces, where F is the multi-valued mappi

Theorem 4.5:



[1] A. T. Bharucha-Ried, Fixed point theorem in probabilistic analysis, Bull. Amer. Math. Soc., 82 (1976), 641-645.

[2] T. G. Bhaskar and V. Lakshmikantham, Fixed point theorem in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379-1393.

[3] L. B. C iric , Jeong S.Ume and Sinisan N. Jesic , On random coincidence and fixed points for a pair of multivalued and single valued mappings, J. Inequal. Appl., Article ID 81045, (2006), 1-12.

[4] L. B. C iric , V. Lakshmikantham, Coupled random fixed point theorems for nonlinear contraction in partially ordered metric spaces, Stoch. Anal. Appl., 27 (2009), 1246-1259.

[5] O. HansEt , Reduzierende zufallige Transformation Czechoslovak Mathematical Journa 7 (1957), 154-158.

[6] O. HansEt , Random operation equations, Proceeding of the 4 Berkeley Symposium on Mathematical Statistics and Probability, vol.2, part 1, University of California Press, California, 1961, pp. 185-202.

[7] N. J. Huang, A principal of randomization of coincidence points with applications, Applied Math. Lett., 12 (1999), 107-113.

[8] N. Hussain and A. Alotaibi, Coupled coincidence for multi-valued contractions in partially ordered metric spaces, Fixed Point Theory and Appl., 2011, 2011:82.

[9] N. Hussain, A. Latif and N. Shafqat, Weak contractive inequalities and compatible mixed monotone random operators in ordered metric spaces, J. Inequal. Appl., 2012:257 (2012) 1-20.

[10] S. Ioth, A random fixed point theorem for a multi-valued contraction mapping, Pacific Journal of Mathematics, 68 (1977), n0. 1, 85-90.

[11] K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bulletin de l'Acad'emie Polonaise des Sciences. S'erie des Sciences Math'ematiques Astronomiques et Physiques 13 (1965), 397403.

[12] A. R. Khan, N. Hussain, N. Yasmin and N. Shafqat, Random coincidence point results for weakly increasing functions in partially ordered metric spaces, Bull. Iranian Math. Soc. (accepted)

[13] V. Lakshimikantham and L. B. C iric , Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (20090, 4341- 4349.

[14] T. C. Lin, Random approximations and random fixed point theorems for non-self map, Proc. Amer. Math. Soc., 103 (1988), 1129-1135.

[15] S. Nadler, Multivalued contraction mappings, Pacfic J. Math, 30 (1969), 475-488.

[16] N. S. Papageorgiou, Random fixed point theorems for multifunctions, Mathematica Japonica 29 (1984), no. 1, 93-106.

[17] N. S. Papageorgiou, Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc. 97 (1986), 507-514.

[18] R. T. Rockafellar, Measurable dependence of convex sets and functions in parameters, j. Math. Anal. Appl. 28(1969), 4-25.

[19] B. Samet and C. Vetro, Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces 4268.

[20] V. M. Sehgal and S. P. Singh, On random approximations and a random fixed point theorems for set valued mappings, Proc. Amer. Math. Soc. 95 (1985), 91-94.

[21] N. Shahzad and N. Hussain, Deterministic and random coincidence point results for f -nonexpansive maps, J. Math. Anal. Appl., 323 (2006), 1038-1046.

[22] N. Shahzad and A. Latif, A random coincidence point theorem, J. Math. Anal. Appl., 245 (2000), 633-638. [23] A. Spacek , Zufalilige Gleichungen, Czechoslovak Mathematical Journal 80 (1955), 462-466.

[24] K. K. Tan and X. Z. Yuan, Random fixed point theorems and approximation in cones, J. Math. Anal. Appl., 185 (1994), 378-390.

[25] D. H. Wagner, Survey of measurable selection theorems. SIAM J. Control Optim. 15:859-903, 1977.

[26] S. S. Zhang and N. J. Huang, On the principle of randomization of fixed points for set-valued mappings with applications, Northeastern Mathematical Journal 7 (1991), 486-491.

[27] X. H. Zhu and J. Z. Xiao, Random periodic point and fixed point results for random monotone mappings in ordered Polish spaces, Fixed Point Theory and Appl., 2010, Article ID 723216, 13 pp.
COPYRIGHT 2015 Asianet-Pakistan
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2015 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Publication:Science International
Article Type:Report
Date:Apr 30, 2015

Terms of use | Privacy policy | Copyright © 2019 Farlex, Inc. | Feedback | For webmasters