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In this paper, we investigate several new classes of multi-dimensional almost automorphic type sequences and focus on their applications to various difference equations involving Volterra difference equations. We provide many structural results, illustrative examples and open problems about the notion under consideration. [ABSTRACT FROM AUTHOR]

This paper investigates diverse classes of multidimensional Weyl and Doss ρ -almost periodic functions in a general measure setting. This study establishes the fundamental structural properties of these generalized ρ -almost periodic functions, extending previous classes such as m-almost periodic and (equi-)Weyl-p-almost periodic functions. Notably, a new class of (equi-)Weyl-p-almost periodic functions is introduced, where the exponent p > 0 is general. This paper delves into the abstract Volterra integro-differential inclusions, showcasing the practical implications of the derived results. This work builds upon the extensions made in the realm of Levitan N-almost periodic functions, contributing to the broader understanding of mathematical functions in diverse measure spaces. [ABSTRACT FROM AUTHOR]

In the present work, we concentrate on a certain class of nonlinear difference equations and propose sufficient conditions for the existence of their almost automorphic solutions. In our analysis, we invert an appropriate mapping and obtain the main existence outcomes by utilizing the contraction mapping principle. As the second objective of the manuscript, we reconsider one of the landmark results, namely the Bohr–Neugebauer theorem, in the qualitative theory of dynamical equations, and we investigate the relationship between the existence of almost automorphic solutions and the existence of solutions with a relatively compact range for the proposed difference equation type. Thus, we provide a discrete counterpart of the Bohr–Neugebauer theorem due to the almost automorphy notion under some technical conditions. [ABSTRACT FROM AUTHOR]

In this paper, we aim to study the neutral-type delayed Caputo fractional differential equations of the form C D α x t − g t , x t = f t , x t , t ∈ t 0 , ∞ , t 0 ≥ 0 with order 0 < α < 1 , which can be used to describe the growth processes in real-life sciences at which the present growth depends on not only the past state but also the past growth rate. Our ultimate goal in this study is to concentrate on the convergence of the solutions to a predetermined constant by establishing a linkage between the delayed fractional differential equation and an integral equation. In our analysis, the sufficient conditions for the asymptotic results are obtained due to fixed point theory. The utilization of the contraction mapping principle is a convenient approach in obtaining technical conditions that guarantee the asymptotic constancy of the solutions. [ABSTRACT FROM AUTHOR]

In this study, we focus on a specific type of functional differential equation x' (t) = a(t)x(t) - λf(t; x(t); x (x (t))); t ∊ R, called iterative differential equation, which may depend on a future state. By employing a cone theoretical fixed point theorem, we propose some sufficient conditions for the existence of positive periodic solutions of the iterative functional differential equation. Our approach enables us to obtain some results that are not covered by the existing literature. [ABSTRACT FROM AUTHOR]

Let 핋 be a nonempty, closed, and arbitrary set of real numbers, namely a time scale, and consider the following delay dynamical equation xΔ (t) = a (t) x (t) + f (t, x (ϑ (t))), t ∈ 핋, where ϑ stands for the abstract delay function. The main goal of this study is three-fold: obtaining the existence of an equi-bounded solution, proving the asymptotic stability of the zero solution, and showing the existence of a periodic solution based on new periodicity concept on time scales for the given delayed equation under certain conditions. In our analysis, we propose an alternative variation of parameters formulation by using an auxiliary function to invert a mapping for the utilization of fixed point theory. [ABSTRACT FROM AUTHOR]