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

Abstract

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. 2010 Mathematics Subject Classification. 34C27, 35B15, 46E30. Key words and phrases. Lebesgue spaces with variable exponents, Stepanov almost automorphy with variable exponents, asymptotical Stepanov almost automorphy with variable exponents, abstract Volterra integro-differential inclusions, abstract (multi-term) fractional differential equations. The second named author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.