\((\hat{\alpha}-\bar{\psi}) \text -{ Geraghty Contraction }\) on Generalized Metric Spaces with Simulation Function

Abstract

In this paper, we introduce the new definitions and fixed-point theorems for \((\hat{\alpha}-\hat{\psi})\)-Geraghty contraction with an aid of simulation function \(\zeta:[0, \infty) \times[0, \infty) \rightarrow \mathbb{R}\) in generalized metric space satisfying the following condition:if \(\exists \hat{\beta} \in \mathcal{F}\) such that for all \(r, s \in \mathfrak{X}\), then we have\(\zeta[\hat{\alpha}(r, s)(d(\mathcal{P} r, \mathcal{P} \mathcal{s})), \hat{\beta}(\hat{\psi}(d(r, s))) \hat{\psi}(d(r, s))] \geq 0\), where \(\hat{\psi} \in \hat{\Psi}\) and \((\mathfrak{X}, d)\) is a generalized metric space. An example is also given to support our results.