Journal of Mathematical Physics, Analysis, Geometry 2019, vol. 15, No 3, pp. 336-353   https://doi.org/10.15407/mag15.03.336     ( to contents , go back )

### 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.

Mathematics Subject Classification 2010: 39A06.
Key words: difference equation, locally convex space, Banach space, locally nilpotent operator.