Cauchy problem for fractional non-autonomous evolution equations

TL;DR

This paper establishes the existence of solutions for a class of nonlinear fractional evolution equations in Banach spaces, using advanced fixed point and measure of noncompactness techniques, without requiring uniform continuity of the nonlinearity.

Contribution

It introduces new existence results for fractional non-autonomous evolution equations by removing previous restrictions on nonlinearity continuity and noncompactness measure constants.

Findings

Existence of mild solutions under local growth and noncompactness conditions.

Generalization of previous results by eliminating uniform continuity requirement.

Application to fractional PDEs with Dirichlet boundary conditions.

Abstract

This paper deals with the following Cauchy problem to nonlinear time fractional non-autonomous integro-differential evolution equation of mixed type via measure of noncompactness $$ \left\{\begin{array}{ll} ^CD^{\alpha}_tu(t)+A(t)u(t)= f(t,u(t),(Tu)(t), (Su)(t)),\quad t\in [0,a], \\[12pt] u(0)=A^{-1}(0)u_0 \end{array} \right. $$ in infinite-dimensional Banach space $E$, where $ ^CD^{\alpha}_t$ is the standard Caputo's fractional time derivative of order $0<\alpha\leq 1$, $A(t)$ is a family of closed linear operators defined on a dense domain $D(A)$ in Banach space $E$ into $E$ such that $D(A)$ is independent of $t$, $a>0$ is a constant, $f:[0,a]\times E\times E\times E\rightarrow E$ is a Carath\'{e}odory type function, $u_0\in E$, $T$ and $S$ are Volterra and Fredholm integral operators, respectively. Combining the theory of fractional calculus and evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the condition of uniformly continuity of the nonlinearity is not required, and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted. As samples of applications, we consider the initial value problem to a class of time fractional non-autonomous partial differential equation with homogeneous Dirichlet boundary condition at the end of this paper.