Implicit linear nonhomogeneous difference equation in Banach and locally convex spaces

The subjects of this work are the implicit linear difference equations $Ax_{n+1}+Bx_n=g_n$ and $Ax_{n+1}=x_n-f_n, n=0,1,2,\ldots$, 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.