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RANDOM COINCIDENCE POINTS FOR MULTI-VALUED NON-LINEAR CONTRACTIONS IN PARTIALLY ORDERED METRIC SPACES.

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

ABSTRACT

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

1. INTRODUCTION

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)

Equations

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.

Equations

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.

Equations

Theorem 2.11.

Equations

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:

Equations

RERENCES

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[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.
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