**Reny George 1,2,\*, Zoran D. Mitrovi´c 3,\* and Stojan Radenovi´c <sup>4</sup>**


Received: 16 October 2020; Accepted: 7 November 2020; Published: 9 November 2020

**Abstract:** Common coupled fixed point theorems for generalized T-contractions are proved for a pair of mappings *S* : *X* × *X* → *X* and *g* : *X* → *X* in a *bv*(*s*)-metric space, which generalize, extend, and improve some recent results on coupled fixed points. As an application, we prove an existence and uniqueness theorem for the solution of a system of nonlinear integral equations under some weaker conditions and given a convergence criteria for the unique solution, which has been properly verified by using suitable example.

**Keywords:** common coupled fixed point; *bv*(*s*)-metric space; T-contraction; weakly compatible mapping

## **1. Introduction**

In the last three decades, the definition of a metric space has been altered by many authors to give new and generalized forms of a metric space. In 1989, Bakhtin [1] introduced one such generalization in the form of a b-metric space and in the year 2000 Branciari [2] gave another generalization in the form a rectangular metric space and generalized metric space. Thereafter, using the above two concepts, many generalizations of a metric space appeared in the form of rectangular b-metric space [3], hexagonal b-metric space [4], pentagonal b-metric space [5], etc. The latest such generalization was given by Mitrovi´c and Radenovi´c [6] in which the authors defined a *bv*(*s*)-metric space which is a generalization of all the concepts told above. Some recent fixed point theorems in such generalized metric spaces can be found in [6–9]. In [10–12], one can find some interesting coupled fixed point theorems and their applications proved in some generalized forms of a metric space. In the present note, we have given coupled fixed point results for a pair of generalized *T*-contraction mappings in a *bv*(*s*)-metric space. Our results are new and it extends, generalize, and improve some of the coupled fixed point theorems recently dealt with in [10–12].

In recent years, fixed point theory has been successfully applied in establishing the existence of solution of nonlinear integral equations (see [11–15] ). We have applied one of our results to prove the existence and convergence of a unique solution of a system of nonlinear integral equations using some weaker conditions as compared to those existing in literature.
