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

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 (alpha, beta, gamma)-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 (alpha, beta, gamma)-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.