A Study on Non-autonomous Second Order Evolution Equations with Nonlocal Conditions

This article deals with the nonlocal problem to a class of nonlinear non-autonomous second order integro-differential evolution equation of mixed type via measure of noncompactness in infinite-dimensional Banach spaces. Based on the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness combined with the theory of evolution families to investigate the existence of mild solutions for a class of nonlinear non-autonomous second order integro-differential evolution equations with nonlocal condition in infinite-dimensional Banach spaces, we obtained the existence of mild solutions under the weak situation that the nonlinear function satisfy some appropriate growth condition and non-compactness 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. Finally, an example is given to show the applications of the obtained results.