Nonlinear Contractions and Caputo Tempered Implicit Fractional Differential Equations in b-Metric Spaces with Infinite Delay

Abstract

This paper deals with some existence and uniqueness results for a class of problems for nonlinear Caputo tempered implicit fractional
differential equations in $b$-Metric spaces with initial condition and infinite delay. The results are based on the ${\omega}-{\psi}$-Geraghty type contraction, the $F$-contraction and the fixed point theory. Furthermore, an two illustrations are presented to demonstrate the plausibility of our results.