mixed nonlocal boundary value problem for implicit fractional differential equation involving both retarded and advanced arguments

In this paper, we investigate the existence and uniqueness of solutions for nonlinear implicit hilfer-hadamard fractional differential equations involving both retarded and advanced arguments and nonlocal mixed boundary conditions. we also use the banach contraction mapping principle and schaefer’s fixed point theorem to show the existence and uniqueness of solutions. the results obtained here extend the work of benchohra et al. [10] and haoues et al. [18]. an example is also given to illustrate the results.