Stepanov-like pseudo-almost automorphic solutions of infinite class in the alpha-norm under the light of measure theory

In this article, we study weighted Stepanov-like pseudo-almost automorphic functions with infinite delay using measure theory. We present a new concept of weighted ergodic functions, which is more general than the classical one. Then, we establish many interesting results on the space of such functions. We also study the existence and uniqueness of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>ν</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> (μ ,ν ) -Stepanov-like pseudo-almost automorphic solutions of infinite class in the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> α -norm for some partial functional differential equations in a Banach space with unbounded delay, using the spectral decomposition of the phase space developed by Adimy and his co-authors. An example is given to illustrate this work.