On approximate solutions for a class of semilinear fractional-order differential equations in Banach spaces

Abstract

Abstract We apply the topological degree theory for condensing maps to study approximation of solutions to a fractional-order semilinear differential equation in a Banach space. We assume that the linear part of the equation is a closed unbounded generator of a C 0 $C_{0}$ -semigroup. We also suppose that the nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We justify the scheme of semidiscretization of the Cauchy problem for a differential equation of a given type and evaluate the topological index of the solution set. This makes it possible to obtain a result on the approximation of solutions to the problem.

Keywords

• fractional differential equation
• semilinear differential equation
• Cauchy problem
• approximation
• semidiscretization
• index of the solution set