  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 inifinite convolution products, providing also an
illustrative application to abstract nonautonomous differential equations of
first order.
