This paper is mainly focused upon the pseudo S -asymptotically Bloch type periodicity and its applications. Firstly, a new notion of pseudo S -asymptotically Bloch type periodic functions is introduced, and some fundamental properties on pseudo S -asymptotically Bloch type periodic functions are established. Then, the notion and properties of weighted pseudo S -asymptotically Bloch type periodic functions are similarly presented. Finally, the obtained results are applied to investigate the existence and uniqueness of pseudo S -asymptotically Bloch type periodic mild solutions for some semi-linear evolution equations in Banach spaces.

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This paper is mainly concerned with the existence of pseudo Sasymptotically Bloch type periodic solutions to damped evolution equations in Banach spaces. Some existence results for classical Cauchy conditions and nonlocal Cauchy conditions are established through properties of pseudo Sasymptotically Bloch type periodic functions and regularized families. The obtained results show that for each pseudo S-asymptotically Bloch type periodic input forcing disturbance, the output mild solutions to reference equations remain pseudo S-asymptotically Bloch type periodic.

In this paper, we mainly introduce some new notions of generalized Bloch type periodic functions namely pseudo Bloch type periodic functions and weighted pseudo Bloch type periodic functions. A Bloch type periodic function may not be Bloch type periodic under certain small perturbations while it can be quasi Bloch type periodic in sense of generalized Bloch type periodic functions. We firstly show the completeness of spaces of generalized Bloch type periodic functions and establish some further properties such as composition and convolution theorems of such functions. We then apply these results to investigate existence results for generalized Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces. The obtained results show that for each generalized Bloch type periodic input forcing disturbance, the output mild solutions to reference evolution equations remain generalized Bloch type periodic.