Some new existence results for boundary value problems involving ?-Caputo fractional derivative

Abstract

This paper concerns the boundary value problem for a fractional differential equation involving a generalized Caputo fractional derivative in b?metric spaces. The used fractional operator is given by the kernel k(t, s) = ?(t) ? ?(s) and the derivative operator 1/??(t) d/dt . Some existence results are obtained based on fixed point theorem of ?-??Graghty contraction type mapping. In the end, we provide some illustrative examples to justify the acquired results.