The notion of triangular inequality plays an important role in determining the structure of distance spaces.In particular, the structure of fuzzy metric spaces depends on the triangular inequality and the concerned t-norm.In most of the fixed point theorems in fuzzy metric spaces both the triangular inequality and the concerned t-norm have a major impact on the proof of fixed point theorems.Inspired by the concept of graphical metric space, it was recently introduced in N. Saleem et al., On Graphical Fuzzy Metric Spaces with Application to Fractional Differential Equations, Fractal and Fract., 6:5 (2022), 238:1-12, the notion of graphical fuzzy metric space and proved some fixed point results.The triangular inequality in such spaces is replaced by a weaker one which is directly associated with the graphical structure affine with the space.In this paper some observations on the recent results of Saleem et al. are made and so the results are revisited.Some related topological properties with some new fixed point results in graphical fuzzy metric spaces are also proved.The results of this paper generalize and extend Banach contraction principle and some other known results in this new setting.Several examples are given which support the claims and illustrate the significance of the new concepts and results.

The purpose of this paper is to explore the split variational inclusion problem in Hilbert spaces.A splitting algorithm is constructed for solving the split variational inclusion with the help of self-adaptive techniques.Convergence analysis of the proposed algorithm is provided under additional conditions.

In this paper we prove new fixed point results in partially ordered bicomplete quasimetric spaces.Our results extends/generalized celebrated results, on the one hand, by Nieto and Rodrguez-Lpez for contraction mappings in partially ordered complete metric spaces and, on the other hand, by Schellekens for contraction mappings in bicomplete quasi-metric spaces.Moreover, it is also shown that neither our assumptions can be weakened nor our results can be deduced from the celebrated Kleene's fixed point theorem.Finally, an application of our results to the asymptotic analysis of recurrence equations is given.

We present a unified approach to describe a possibly wide class of separable Banach spaces which are extremal with respect to the minimal displacement of k-Lipschitz self-maps of the closed unit ball.The prominent member of this class, which plays a central role in our considerations, is the Banach space c 0 of real sequences converging to 0, provided with the maximum norm.Indeed, we show that if a separable Banach space X contains an isomorphic (resp.isometric) copy of c 0 , then X as well as all subspaces of X of finite codimension are extremal (resp.strictly extremal).Our result encompasses and significantly extends a collection of all known examples of separable Banach spaces which are extremal (resp.strictly extremal).

In this paper, we consider a new class of generalized enriched cyclic contraction and establish a related fixed point theorem in the setting of a Banach space.As an application to the fractals, we develop a new iterated function system (IFS) consisting of enriched cyclic (bn, n(t), n(t))contractions.

In this paper, existence of pairs of solutions is obtained for compact potential operators on Hilbert spaces.An application to a second-order boundary value problem is also given as an illustration of our results.

The chances for a mapping taken at random from a given set of mappings to have approximate fixed points are studied in this paper.We start from the discrete case to range more abstract spaces as metric measure spaces.Initial insights for this work are elementary, and some of the observations may already be known.At the same time, they seem to point the way to deeper questions and raise the potential for future study.

Using recent fixed point theory of the author we establish some new maximal element results for general majorized type maps defined on Hausdorff topological vector spaces.Then by constructing appropriate majorized type maps we can then guarantee the existence of equilibria for N -person games.