Implicit Linear Nonhomogeneous Difference
Equation in Banach and Locally Convex
Spaces

S.L. Gefter

School of Mathematics and Computer Science, V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine
E-mail: gefter@karazin.ua

A.L. Piven

School of Mathematics and Computer Science, V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine
E-mail: aleksei.piven@karazin.ua

Received April 16, 2018, revised November 15, 2018.

The paper is dedicated to the 80th anniversary of Anatoliy Georgievich Rutkas

Abstract

The subjects of this work are the implicit linear difference equations $Ax_{n+1}+Bx_n=g_n$ and $Ax_{n+1}=x_n-f_n,\quad n=0,1,2,\ldots$, where $A$ and $B$ are continuous operators acting
in certain locally convex spaces. The existence and uniqueness conditions, along with explicit formulas, are obtained for solutions of these equations. As an application of the general theory produced this way, the equation $Ax_{n+1}=x_n-f_n$ in the space $\mathbb{R}^{\infty}$ of finite sequences and in the space
$\mathbb{R}^M$, where $M$ is an arbitrary set, has been studied.