Almost automorphic and asymptotically almost automorphic type functions in Lebesgue spaces with variable exponents $L^{p(x)}$.

The paper introduces and studies the class of (asymptotically) Stepanov almost automorphic functions with variable exponents. Any function belonging this class needs to be (asymptotically) Stepanov almost automorphic. A few relevant applications to abstract Volterra integro-differential inclusions in Banach spaces is presented.