In this paper, we introduce several various classes of (?, c)-almost periodic type functions and their Stepanov generalizations. We also consider the corresponding classes of (?, c)-almost periodic type functions depending on two variables and related composition principles. We provide several illustrative examples and applications to the abstract Volterra integro-differential equations in Banach spaces.

The main aim of this paper is to indicate that the notion of semi--periodicity is equivalent with the notion of -periodicity, provided that is a nonzero complex number whose absolute value is not equal to 1.

In this paper, we introduce and systematically analyze the classes of (pre-)$(\mathcal{B},\rho ,\left(t_k\right))$-piecewise continuous almost periodic functions and (pre-)$(\mathcal{B},\rho ,\left(t_k\right))$-piecewise continuous uniformly recurrent functions with values in complex Banach spaces. We weaken substantially, or remove completely, the assumption that the sequence $\left(t_k\right)$ of possible first kind discontinuities of the function under consideration is a Wexler sequence (in order to achieve these aims, we use certain results about Stepanov almost periodic type functions). We provide many applications in the analysis of the existence and uniqueness of almost periodic type solutions for various classes of the abstract impulsive Volterra integro-differential inclusions.

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within the framework of $\beta$-Banach spaces. Moreover, we examine the $\beta$–Ulam–Hyers stability of the solutions, providing insights into the stability behavior under small perturbations. An illustrative example is presented to demonstrate the practical applicability and effectiveness of the theoretical results obtained.

In this work, we provide several applications of (a, k)-regularized C-resolvent families to the abstract impulsive Volterra integro-differential inclusions. The resolvent operator families under our consideration are subgenerated by multivalued linear operators, which can degenerate in the time variable. The use of regularizing operator C seems to be completely new within the theory of the abstract impulsive Volterra integro-differential equations.