Stepanov Almost Periodicity on the $$p$$-Adic Numbers with Applications to Vladimirov Semilinear Equations

This paper develops a Stepanov framework for almost periodicity over $$p$$ -adic fields. We begin by defining Stepanov spaces using $$p$$ -adic balls and the normalized Haar measure on the locally compact abelian additive group $${\mathbb Q}_p$$ . Stepanov almost periodicity is then defined through relatively dense sets of translation numbers. To bridge Stepanov and Bohr almost periodicity, we introduce the class of uniformly regular Stepanov almost periodic functions, which enables the derivation of Bohr almost periodic solutions under weaker assumptions. To illustrate the theory, we establish existence results for translation-difference equations, convolution equations, and semilinear evolution equations involving the Vladimirov pseudodifferential operator. The existence of almost periodic mild solutions is obtained by means of the Schauder fixed-point theorem.