Unified common fixed point theorems in complex valued metric spaces via an implicit relation with applications

Abstract

The purpose of this paper is to prove some common fixed point theorems for two pairs of weakly compatible mappings in complex valued metric spaces satisfying an implicit relation. Several illustrative examples are given which demonstrate the usefulness of our utilized implicit relation. Beside generalizing and improving several well known core results of the existing literature we can deduce several new contractions which have not obtained before in complex valued metric spaces. As an application of our results, we prove the existence and uniqueness of common solution of Hammerstein as well as Urysohn integral equations.

Fejér monotonicity and fixed point theorems with applications to a nonlinear integral equation in complex valued Banach spaces

“…Although complex valued metric spaces form a special class of cone metric spaces (see, e.g. [2], [6]), yet the definition of cone metric spaces rely on the underlying Banach space which is not a division ring. Consequently, rational expressions are not meaningful in cone metric spaces, this means that results involving mappings satisfying rational expressions cannot be generalized to cone metric spaces.…”

“…It is known that in cone metric spaces the underlying metric assumes values in linear spaces where the linear space may be even infinite dimensional, whereas in the case of complex valued metric spaces the metric values belong to the set of complex numbers which is one dimensional vector space over the complex field. This instance is the major motivation for the consideration of complex valued metric spaces independently (see, [6]). Hence, results in this direction cannot be generalized to cone metric spaces, but to complex valued metric spaces.…”

“…Several authors have obtained interesting and applicable results in complex valued metric spaces (see, e.g. [2], [3], [5], [8], [6], [11], [25], [36], [40], [41], [42]).…”

Fixed points for occasionally weakly biased mappings of type (A)

“…According to [24], several classical fixed point theorems and common fixed point theorems have been unified considering a general condition by an implicit relation in ( [22,23]) and in other papers. Motivated by the three cited papers and the next ones [1,2,4,5,11,18,25,26,27] and so on, we introduce the new type of implicit relations.…”

Completion of Complex Valued Dislocated Metric Space

“…Beg et al [8] explored the completion of complex-valued strong b-metric space in their recent paper. Some recent work about fixed point is disscussed in [4], [6], [13], [14], [15], [16] and [9]. A new extension of the double controlled metric-type spaces, called double controlled metric-like spaces is discussed in [25], by considering that the self-distance may not be zero.…”