Fixed points in partially ordered metric space using δ-distance hybrid contraction

Introduction

In pure and applied mathematics, metric spaces are very crucial, as metric fixed point theory has wide applications in almost all fields of quantitative sciences. The simplicity and usefulness of Banach contraction Principle [1] made the fixed point theorem, a very popular tool in establishing the existence of solution for an integral equation in many branches of mathematical analysis. In 1969, Nadler [11] extended the Banach contraction principle to multivalued mappings in complete metric spaces and one of the interesting results on weakened contractive condition was reported by Alber et al. [2] . Later, Ran and Reurings [15] weakened the contraction principle by considering single valued mappings over partially ordered complete metric spaces, and applied their results to matrix equations. Since then, many authors have studied the existence of fixed points in partially ordered sets. (refer [3,10,13,14] ). Hybrid contractive conditions, that is, contractive conditions involving single valued and multivalued maps are the further addition to metric fixed point theory and its applications. For a comprehensive survey on fundamental development of hybrid contractions and historical remarks, refer S. L. Singh et al. [12,16,17] . The purpose of this paper is to establish some coincidence point and common fixed point results for multivalued and single valued mappings with weakened hybrid contractive conditions using the concept of pre-order relations, δ distance and altering distance functions in metric space endowed with partial order. An example is illustrated to prove that the generalizations are real improvements. These Results extend the results of Gregorio et al. [7] , Giniswamy et al. [6] and many recent results in the literature. First, some basic definitions and important results which are needed in the main results will be recalled.

Definition 1.1:

A partially ordered set is a system (𝑋, ≼) where X is a non-empty set and ≼ is a binary relation on X satisfying, for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 1. 𝑥 ≼ 𝑥 (Reflexivity) 2. If 𝑥 ≼ 𝑦 and 𝑦 ≼ 𝑥, then x = y (Antisymmetry) 3. If 𝑥 ≼ 𝑦 and y ≼ z, then 𝑥 ≼ 𝑧 (Transitivity) Definition 1.2. The triple (𝑋, 𝑑, ≼) is called a partially ordered metric space, if (𝑋, ≼) is a partially ordered set together with (𝑋, 𝑑) is a metric space.

Throughout this paper, CB(X) denotes the set of all nonempty, closed and bounded subsets of a metric space X endowed with a partial order. The δ-distance between any two subsets A and B of X is given by

By the above definition one can conclude the following: 1. 𝑑(𝑎, 𝐵) ≤ 𝐷(𝐴, 𝐵) ≤ 𝛿(𝐴, 𝐵) 2. 𝑑(𝐴, 𝑏) ≤ 𝐷(𝐴, 𝐵) ≤ 𝛿(𝐴, 𝐵) 3. 𝛿(𝐴, 𝐵) = 0 if and only if 𝐴 = 𝐵 = {𝑎} 4. 𝐷(𝐴, 𝐵) = 0 if and only if 𝐴 = 𝐵 Definition 1.4: Let (X, d, ⪯) be a partially ordered metric space, g: X →X and T: X→ CB(X). If w = gx ∈ Tx for some x ∈ X, then x is said to be a coincidence point of g and T and w is the point of coincidence of g and T.

Definition 1.5: Let (X, d, ⪯) be a partially ordered metric space, g: X → X and T: X → CB(X). Then a point 𝑥 ∈ 𝑋 is said to be a fixed point of g and T if x = gx ∈ Tx. The first three pre-order relations between nonempty subsets of partially ordered metric space appeared in [4,5,18] respectively and the last three are defined in [7] . Definition 1.6: Let (𝑋, 𝑑, ⪯) be a partially ordered metric space. Let 𝐴, 𝐵 ∈ 𝐶𝐵(𝑋), then the pre-order relations are defined as follows:

Definition 1.7: Let (X, d, ⪯) be a partially ordered metric space, g: X → X and T: X → CB(X). Then the mappings g and T are called weakly compatible if gx ∈ Tx for some x ∈ X implies gT (x) ⊆ Tg(x). The altering distance function is defined as follows and it has been utilized in many works in metric fixed point theory. Definition 1.8: [9] A function 𝜓: [0, ∞) → [0,1)is called an altering distance function if the following properties are satisfied: 1. ψ is monotone increasing and continuous, 2. ψ(t) = 0 if and only if t = 0. Lemma 1.9: [11] . Let {𝐴 𝑛 } be a sequence in 𝐶𝐵(𝑋) and lim 𝑛→∞ 𝐷(𝐴 𝑛 , 𝐴) = 0 for 𝐴 ∈ 𝐶𝐵(𝑋). If 𝑥 𝑛 ∈ 𝐴 𝑛 and lim 𝑛→∞ 𝑑(𝑥 𝑛 , 𝑥) = 0 then 𝑥 ∈ 𝐴. Proposition 1.10: [8] . Let X be a given nonempty set. Assume that 𝑔: 𝑋 → 𝑋 and 𝑇: 𝑋 → 2 𝑋 are weakly compatible mappings. If g and T have a unique point of coincidence 𝑤 = 𝑔𝑥 ∈ 𝑇𝑥, then w is the unique common fixed point of g and T.

Main Results

We now state and prove the first main result. Throughout this paper, Φ denotes the class of functions 𝜙: [0, ∞) → [0,1) such that lim 𝑟→𝑡 + sup 𝜙(𝑟) < 1 for each 𝑡 ∈ [0, ∞) Theorem 2.1: Let (𝑋, 𝑑, ≼) be partially ordered metric space. Let 𝑇: 𝑋 → 𝐶𝐵(𝑋) be a multivalued mapping and g: 𝑋 → 𝑋 be a self map satisfying the following conditions 1. 𝑇(𝑥) ⊆ g𝑋 ∀ 𝑥 ∈ 𝑋 and g𝑋 is complete

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