A New Fixed Point Approach for Approximating Solutions of Fractional Differential Equations in Banach Spaces

Abstract

This contribution targets the solution of fractional differential equations (FDEs) via novel iterative approach and in a new class of nonlinear mappings. Our approach is based on the class of (α, β, γ)-nonexpansive mappings and three-step M-iterative scheme. Under various assumptions, we first carry out some weak and strong convergence results in a setting of a Banach spaces. After this, we carry out an application of one our main result to find approximate solution for a broad class of FDEs. Eventually, we we construct a new example of (α, β, γ)-nonexpansive mappings and show that this new mapping is not continuous on its whole domain and hence it is not nonexpansive. Using this example, we perform a numerical simulation of various iterative scheme including our M-iterative scheme. It has been observed the numerical effectiveness of the M-iterative scheme is high as compared to the other iterative schemes. Accordingly, our main outcome is new/extends some known results of the literature.