Common Coupled Fixed Point Theorems for Two Hybrid Pairs of Mappings under φ-ψ Contraction

We introduce the concept of (EA) property and occasional w-compatibility for hybrid pair F : X × X → 2X and f : X → X. We also introduce common (EA) property for two hybrid pairs F, G : X → 2X and f, g : X → X. We establish some common coupled fixed point theorems for two hybrid pairs of mappings under φ-ψ contraction on noncomplete metric spaces. An example is also given to validate our results. We improve, extend and generalize several known results. The results of this paper generalize the common fixed point theorems for hybrid pairs of mappings and essentially contain fixed point theorems for hybrid pair of mappings.


Introduction and Preliminaries
Let ( , ) be a metric space and let CB( ) be the set of all nonempty closed bounded subsets of . Let ( , ) denote the distance from to ⊂ and let denote the Hausdorff metric induced by ; that is, (1) The study of fixed points for multivalued contractions and nonexpansive mappings using the Hausdorff metric was initiated by Markin [1]. The existence of fixed points for various multivalued contractive mappings has been studied by many authors under different conditions. The theory of multivalued mappings has application in control theory, convex optimization, differential inclusions, and economics. In 1969, Nadler [2] extended the famous Banach contraction principle [3] from single-valued mapping to multivalued mapping and proved the fixed point theorem for the multivalued contraction. Many authors proved fixed point theorems for hybrid pair of mappings without assuming the continuity of any mapping involved including [4][5][6][7].
In [8], Gnana Bhaskar and Lakshmikantham established some coupled fixed point theorems and applied these results to study the existence and uniqueness of solution for periodic boundary value problems. Luong and Thuan [9] generalized the results of Gnana Bhaskar and Lakshmikantham [8]. Berinde [10] extended the results of Gnana Bhaskar and Lakshmikantham [8] and Luong and Thuan [9]. Lakshmikantham andĆirić [11] proved coupled coincidence and common coupled fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces and extended the results of Gnana Bhaskar and Lakshmikantham [8]. Jain et al. [12] extended and generalized the results of Berinde [10], Gnana Bhaskar and Lakshmikantham [8], Lakshmikantham andĆirić [11], and Luong and Thuan [9].
Recently Samet et al. [16] claimed that most of the coupled fixed point theorems in the setting of single valued mappings on ordered metric spaces are consequences of well-known fixed point theorems.
These concepts were extended by Abbas et al. [17] to multivalued mappings and who obtained coupled coincidence point and common coupled fixed point theorems involving hybrid pair of mappings satisfying generalized contractive conditions in complete metric spaces. Very few authors studied coupled fixed point theorems for hybrid pair of mappings including [17][18][19][20].
Aamri and El Moutawakil [21] defined (EA) property for self-mappings which contained the class of noncompatible mappings. Kamran [22] extended the (EA) property for hybrid pair : → and : → 2 . Liu et al. [23] introduced common (EA) property for hybrid pairs of single and multivalued mappings and gave some new common fixed point theorems under hybrid contractive conditions. Abbas and Rhoades [24] extended the concept of occasionally weakly compatible mappings for hybrid pair : → and : → 2 .
In this paper, we introduce the concept of (EA) property and occasional -compatibility for hybrid pair : × → 2 and : → . We also introduce common (EA) property for two hybrid pairs , : × → 2 and , : → . We establish some common coupled fixed point theorems for two hybrid pairs of mappings under -contraction on noncomplete metric spaces. The -contraction is weaker contraction than the contraction defined in Gnana Bhaskar and Lakshmikantham [8] and Luong and Thuan [9]. We improve, extend, and generalize the results of Berinde [10], Gnana Bhaskar and Lakshmikantham [8], Jain et al. [12], Lakshmikantham andĆirić [11], Liu et al. [23], and Luong and Thuan [9]. The results of this paper generalize the common fixed point theorems for hybrid pairs of mappings and essentially contain fixed point theorems for hybrid pair of mappings.

Main Results
We first define the following.
Consider the sequences Therefore, the pairs { , } and { , } are said to satisfy the common (EA) property.

Put
= and = in Theorem 21, and we get the following result.