Abstract nonautonomous difference inclusions in locally convex spaces

TL;DR

This work develops a general theory for almost periodic type solutions of abstract nonautonomous difference inclusions in locally convex spaces. It introduces generalized almost periodic concepts in locally convex spaces and establishes existence and uniqueness results for first-order nonautonomous difference equations via a convolution-type representation $x(k)=f(k-1)+\sum_{j=-\infty}^{k-2}[A(k-1)\cdots A(j+1)]f(j)$ under seminorm-based bounds $\kappa(A(k)x)\le c^{\kappa}(k)\kappa(x)$ and summability of the products of $c^{\kappa}$. Under Bohr almost periodic forcing $f$ and (approximately) almost periodic operator coefficients $A(k)$, the paper proves the existence of unique almost periodic, Besicovitch, and $(\omega,c)$-periodic solutions, with extensions to multivalued linear operators via $C$-regularized resolvent frameworks. The higher-order case is handled by reducing to first-order systems and applying the same results, with illustrative examples in $E_{l}$-type spaces and $C_{0}(\Omega)$ (including Dirichlet Laplacian settings) and discussions of limitations for $p\ge3$ in the Besov-like settings, highlighting practical applicability to discrete Poisson and wave-type problems in locally convex spaces.

Abstract

In this paper, we consider abstract nonautonomous difference inclusions in locally convex spaces with integer order differences. We particularly analyze the existence and uniqueness of almost periodic type solutions to abstract nonautonomous difference inclusions. Our results seem to be completely new even in the Banach space setting.

Theorems & Definitions (15)

• Definition 1.1
• Lemma 1.2
• Theorem 2.1
• proof
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Example 2.5
• Theorem 2.6
• Theorem 2.7