Mild Solutions for Semi-Linear Fractional Evolution Systems with Nonlinear Integral Boundary Conditions: Existence and Uniqueness

Abstract

Fractional evolution equations, focusing on infinite dimensions, are methods for describing processes with memory and processes with hereditary types. This paper constructs a solvability framework that differs from previous neutral fractional evolution equations, where operator-valued multi-point non-local boundary problems were dependent on non-linear integral non-local boundary problems in abstract Banach spaces. We include the Caputo fractional derivative with order α more than 0 but less than 1. The linear part is assumed to generate a strongly continuous semigroup. The author proves that mild solutions(MS)exist by transforming the problem into an equal integral equation and applying the Krasnoselskii’s Fixed Point Theorem. This also gives a different way of solving the previous paper’s problem with Sadovskii’s approach. Under some conditions that are similar to the Lipschitz condition, the author proves that, by the Banach contraction principle, the Mild solution becomes exclusive. Therefore, the present work provides a different mathematical approach for the neutral nonlinear fractional evolution system, considering the boundary structure, the system architect, the fixed-point technique different from the first one, and Sadovskii’s theorem. For the theoretical results to be True, a sample numeric case is solved in MATLAB to demonstrate that the solutions behave according to the fractional order parameters of the model, and the results show that the approximate solutions converge.The results received strengthen the nonlocal fractional evolution equations and nonlocal interacting fractional evolution equations with respect to their boundary interactions and allow us to identify concrete directions for further analytical and numerical investigations.