On a JH-operators pair of type (A) with applications to integral equations
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Journal of Fixed Point Theory and Applications
On a JH-operators pair of type (A) with applications to integral equations Mujahid Abbas, Wasfi Shatanawi, Sadia Farooq and Zoran D. Mitrovi´c Abstract. The aim of this paper is to introduce a new class of noncommuting mappings called JH-operator pairs of type (A). We obtain the unique point of coincidence point in generalized metric spaces and the common fixed point results of such pairs of mappings in the setting of metric spaces. Some examples are also presented to support the concepts and results proved herein. We prove the existence of solution of nonlinear integral equations as an application of our result. In the end, we give any open question. Mathematics Subject Classification. Primary 47H10, Secondary 55M20. Keywords. Coincidence point, point of coincidence, common fixed point, weakly compatible mapping, JH-operator pair, JH-operator pair of type (R), JH-operator pair of type (A), generalized metric space, metric space.
1. Introduction and preliminaries Let (M, d) be a metric space and s, h : M → M . A point m in M is called a fixed point of s if d(m, sm) = 0. An operator equation of the form s(m) = h(m) = m is called common fixed point equation, whereas an operator equation s(m) = h(m) is called a coincidence point equation. A point m ∈ M which solves a common fixed point equation (coincidence point equation) is called common fixed point (coincidence point) of s and h. A point w ∈ M is called a point of coincidence of s and h if there exists a point m ∈ M such that w = h(m) = s(m). Let C(s, h) and P C(s, h) denote the set of all coincidence points and the set of all point of coincidence of mapping s and h, respectively. A study of conditions sufficient to guarantee the existence of solution of common fixed point equations has its due significance in fixed point theory. There is a multitude of papers which deals with conditions imposed on mappings s and h to obtain the existence of solutions of common fixed point equations, for some results see [2–7,9,11–38]. 0123456789().: V,-vol
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Jungck [17] obtained common fixed point of two mappings where commutativity of the mappings was required. Sessa [30] coined the term weakly commuting maps which is a weaker version of commutativity condition. Jungck [18] generalized the notion of weak commutativity by introducing compatible maps and then weakly compatible maps. Suppose (M, d) is a metric space and s and h are noncommuting selfmappings on M , the pair (s, h) is said to be (1) [15] JH-operators pair if and only if d(w, m) ≤ diam(P C(s, h)), whenever s(m) = h(m) = w ∈ P C(s, h), where diam(P C(s, h)) denotes the diameter of P C(s, h). (2) [36] generalized JH-operators pair if and only if for some n ∈ N, the following holds d(w, m) ≤ [diam(P C(s, h))]n , whenever s(m) = h(m) = w ∈ P C(s, h). (3) [12] P D-operator pair if there is a point m in M such that m ∈ C(s, h) and d(shm, hsm) ≤ diam(P C(s, h)). (4) [24] JH-operators pair of type (R) if P C(s, h) = ∅ and lim d(mn , t) ≤ diam (P C(s, h)) ,
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