Positive solutions for two-point conformable fractional differential equations by monotone iterative scheme

Abstract

In this paper, two successively iterative schemes have been provided to show the existence of nontrivial solutions for nonlinear conformable fractional differential equation involving nonlocal boundary condition and a parameter. The iterative sequences begin with some constant. The fractional derivative in this study is based on the newly defined and so called ”conformable fractional derivative”. The corresponding Green’s function that is singular at zero has been derived. Because of this singularity, the fixed point theorem can not be applied directly, thus a sequence of operators that are completely continuous is constructed and uniform convergence of these operators to the underlying operator is shown. Then a fixed point result on the order interval is applied. Nontrivial solutions of the problem and the positive solutions of the problem that are the limit of the iterative sequences constructed has been demonstrated.