In this paper, the parabolic evolution equation u (t) + A(t)u(t) = f (t) in a reflexive real Banach space is considered. Assuming strong monotonicity, pseudo almost automorphy and other appropriate conditions of the operators A(t) and Stepanov-like pseudo almost automorphy of the forced term f (t), we obtain the Stepanov-like pseudo almost automorphy of the solution to the evolution equation by using the almost automorphic component equation method. This paper extends a known result in the case where A(·) and f are almost automorphic in certain senses. Finally, a concrete example is given to illustrate our results.MSC: Primary 34G20; secondary 43A60
This paper is devoted to generalizing the notion of almost periodic functions on time scales. We introduce a new class of almost periodic time scales called Hausdorff almost periodic time scales by using the Hausdorff distance and propose a more general notion of almost periodic functions on these new time scales. Then we explore some properties of Hausdorff almost periodic time scales and prove that the family of almost periodic functions on Hausdorff almost periodic time scales is a Banach space. Especially, our analysis also indicates that a function on a Hausdorff almost periodic time scale is almost periodic if and only if its affine extension is Bohr almost periodic on the real numbers R. As an application, we establish the existence of almost periodic solutions for a single species model on Hausdorff almost periodic time scales.
MSC: 34N05; 11K70
The shunting inhibitory cellular neural networks with continuously distributed delays and pseudo almost periodic coefficients are considered. First, we make a generalization of the Halanay inequality, and then establish some sufficient conditions for the existence and asymptotical stability of pseudo almost periodic solutions. Finally, a numerical simulation is presented to illustrate the theoretical results.
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