ANALYSIS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS INVOLVING CAPUTO-FABRIZIO AND ATANGANA-BALEANU-CAPUTO FRACTIONAL DERIVATIVES

Abstract

Fixed point theory is a fundamental method in the solution analysis of mathematical problems arising in nonlinear differential equations for which it may appear extremely difficult to find an explicit solution. This dissertation addresses the significant gap in the qualitative analysis of nonlinear fractional differential equations (FDEs) involving Caputo-Fabrizio (CF) and Atangana-Baleanu-Caputo (ABC) fractional derivatives. The study aims to analyze the existence, uniqueness, and stability of solutions to nonlinear fractional differential equations involving the Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives. To establish the existence and uniqueness of solutions, the work utilizes fixed point theorems such as: Krasnoselskii’s fixed point theorem to show the existence of solutions under suitable mappings in a Banach space, Schauder’s fixed point theorem to address existence results for continuous and compact operators, F−contraction type fixed point theory, which extends the well-known Banach contraction principle, provides a unifying framework for various contractive conditions, enabling broader applicability to fractional equations and the Banach contraction principle to prove uniqueness by demonstrating that the operator satisfies a contraction condition. Also, Generalized Gronwall inequality is applied to demonstrate solution existence for nonlinear systems of time-delayed fractional differential equations involving Atangana-Baleanu-Caputo fractional derivatives. The dissertation further explores the stability of solutions, focusing on Ulam–Hyers stability, a measure of how small perturbations in initial conditions affect the solution, ensuring the reliability of the mathematical model, and asymptotic stability ensures that the system not only resists deviations from its equilibrium state but also returns to this equilibrium as time progresses to infinity. This research contributes significantly to the theory of fractional differential equations, offering novel insights into their solvability and stability. By leveraging advanced fixed point theorems and fractional derivative concepts, the dissertation not only enriches mathematical theory but also sets the stage for future explorations in applied sciences