Coupled fixed point results on metric spaces defined by binary operations

Abstract

In parallel with the various generalizations of the Banach fixed point theorem in metric spaces, this theory is also transported to some different types of spaces including ultra metric spaces, fuzzy metric spaces, uniform spaces, partial metric spaces, $b$-metric spaces etc. In this context, first we define a binary normed operation on nonnegative real numbers and give some examples. Then we recall the concept of $T$-metric space and some important and fundamental properties of it. A $T$-metric space is a $3$-tuple $(X, T, \diamond)$, where $X$ is a nonempty set, $\diamond$ is a binary normed operation and $T$ is a $T$-metric on $X$. Since the triangular inequality of $T$-metric depends on a binary operation, which includes the sum as a special case, a $T$-metric space is a real generalization of ordinary metric space. As main results, we present three coupled fixed point theorems for bivariate mappings satisfying some certain contractive inequalities on a complete $T$-metric space. It is easily seen that not only existence but also uniqueness of coupled fixed point guaranteed in these theorems. Also, we provide some suitable examples that illustrate our results.