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

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DOI: https://doi.org/10.28924/2291-8639-22-2024-124
Published: Jul 29, 2024
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Bashir Nawaz, Abdallah Labsir, Kifayat Ullah, Junaid Ahmad, A New Fixed Point Approach for Approximating Solutions of Fractional Differential Equations in Banach Spaces, Int. J. Anal. Appl., 22 (2024), 124.

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Bashir Nawaz, Abdallah Labsir, Kifayat Ullah, Junaid Ahmad

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.

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