Fixed Points for $\alpha-\beta_E$-Geraghty contractions on $b$-metric spaces and applications to matrix equations

Abstract

In this paper, we introduce the notion of $\alpha-\beta_E$-Geraghty contraction type mappings on $b$-metric spaces and prove the existence and uniqueness of fixed point for such mappings.
 These results are generalizations of the recent results in [Fulga and Proca, Fixed points for $\varphi_E$-Geraghty contractions, Abstract and Applied Analysis, in press].
 We give some examples illustrating the presented results. An application on matrix equations and  numerical algorithms are also provided.