Abstract
The main aim of this paper is to consider the classes of quasi-asymptotically almost periodic functions and Stepanov quasi-asymptotically almost periodic functions in Banach spaces. These classes extend the well known classes of asymptotically almost periodic functions, Stepanov asymptotically almost periodic functions and S-asymptotically $$\omega $$ -periodic functions with values in Banach spaces. We investigate the invariance of introduced properties under the action of finite and infinite convolution products, providing also an illustrative application to abstract non-autonomous semilinear differential equations of first order.
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