The aims of this paper are (a) to introduce the concept of the 0-complete m-metric spaces, (b) to obtain the results for $m^w$-Caristi mapping using Kirk’s approach, (c) to investigate the problem of non-cooperative equilibrium (abbreviated as NCE) in two- and three-person games in the structure of game theory and find the solution by employing coupled and tripled fixed-point results within the framework of 0-complete m-metric spaces (m-metric spaces, respectively), and (d) to establish some coupled fixed-point results which extend the scope of metric fixed point theory. We provide some examples to support the concepts and results presented in this paper. As an application of our results in this paper, we obtain the existence of a solution for a nonlinear integral equation.

The objectives of this paper are three steps: first, to generalize the idea of complete $ M $-cone metric spaces over Banach algebra; second, to present a new topological structure utilizing the concept introduced by Fernandez et al. (Fixed point results in M-cone metric space over Banach algebra with an application, Filomat, 36 (2022), 5547–5562.); and third, to explore the idea of Banach algebra type relational theoretic contractions and cyclic Banach algebra type contractions in $ M $-cone metric spaces, establishing several fixed point results for these contractions. To illustrate the discussed concepts and results, several examples are provided. As an application, we discuss a solution of the nonlinear integral equation based on the main results.