Fractional non-autonomous evolution equation with nonlocal conditions

The aim of this paper is to discuss the existence of mild solutions for a class of time fractional non-autonomous evolution equations with nonlocal conditions and measure of noncompactness in infinite-dimensional Banach spaces. Combining the theory of fractional calculus and evolution families, the fixed point theorem with respect to k-set-contractive operator and a new estimation technique of the measure of noncompactness, we obtain the new existence results of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and noncompactness measure conditions. Our results generalize and improve some previous results on this topic by deleting the compactness condition on nonlocal function g and extending the study of fractional autonomous evolution equations in recent years to non-autonomous ones. Finally, as samples of applications, we consider a time fractional non-autonomous partial differential equation with homogeneous Dirichlet boundary condition and nonlocal conditions.