Non-Autonomous Fractional Evolution Equations With Non-Instantaneous Impulse Conditions of Order (1,2): A Cauchy Problem

Abstract

The non-instantaneous condition is utilized in our study through the employment of the Cauchy problem in order to contract a system of nonlinear non-autonomous mixed-type integro-differential (ID) fractional evolution equations in infinite-dimensional Banach spaces. We reveal the existence of new mild solutions in the condition that the nonlinear function modifies approximately suitable, measure of non-compactness (MNC) form and local growth form using evolution classes along with fractional calculus (FC) theory as well as the fixed-point theorem with respect to k-set-contractive operator and MNC standard set. Consequently, as an example, we consider a fractional non-autonomous partial differential equation (PDE) with a homogeneous Dirichlet boundary condition and a non-instantaneous impulse condition. The conclusion of mild solution regarding the uniqueness and existence of a mild solution for a system with a probability density function and evolution classes is drawn with respect to the related domains.

Approximate Controllability of Fractional Stochastic Evolution Inclusions with Non-Local Conditions

“…Many recent articles have investigated mild solutions and controllability challenges for various types of differential inclusions; see [7] and the citations therein. We direct the reader to [8][9][10] for one method of solving fractional differential equations in impulsive stochastic functional differential systems with state-dependent delay in Hilbert spaces. In pharmacotherapy, some of the kinetics of evolution processes are not adequately captured by the effect of instantaneous signals.…”