A new method for a semi-positone Hadamard fractional boundary value problem

Abstract

A Hadamard fractional boundary value problem with semi-positone nonlinearity is studied in this paper, and the local existence and uniqueness of solutions are derived by a recent fixed point theorem involving with increasing $\phi -(h,r)$-concave operators defined on ordered spaces. This is an unprecedented approach to solve the semi-positone problems. In practice, this method has a wider range of applicability since it can obtain the local uniqueness of the solutions. Furthermore, we can approximate the unique solution by constructing convergent iterative sequences. In the end, a persuasive example is provided to illustrate that the theoretical results we obtained are applicable and valid.