Abstract
We prove some common fixed point theorems for two pairs of weakly compatible mappings in 2-metric spaces via an implicit relation. As an application to our main result, we derive Bryant's type generalized fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. Our results improve and extend a host of previously known results. Moreover, we study the existence of solutions of a nonlinear integral equation.
Highlights
Introduction and PreliminariesIn 1963, Gahler [1] initiated the concept of 2-metric space as a natural generalization of a metric space
We prove some common fixed point theorems for two pairs of weakly compatible mappings in 2-metric spaces via an implicit relation
The topology induced by 2-metric space is called 2-metric topology which is generated by the set of all open spheres with two centers
Summary
In 1963, Gahler [1] initiated the concept of 2-metric space as a natural generalization of a metric space. Two pairs (A, S) and (B, T) of self-mappings of a 2-metric space (X, d) are said to satisfy the common property (E.A), if there exist two sequences {xn} and {yn} in X such that nl→im∞d (Axn, t, a) = nl→im∞d (Sxn, t, a). Notice that the recent results, contained in Popa et al [22] proved for weakly compatible mappings under the property (E.A), always require the completeness of the underlying subspace for the existence of common fixed point. A pair (A, S) of self-mappings of a 2-metric space (X, d) is said to satisfy the common limit range property with respect to mapping S, denoted by (CLRS), if there exists a sequence {xn} in X such that nl→im∞ d (Axn, t, a) = nl→im∞ d (Sxn, t, a) = 0,.
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