Locally Lipschitz mappings and asymptotic behavior of solutions of difference equations in Banach spaces (Q2757034)

The authors study the asymptotic behavior of the difference equation NEWLINE\[NEWLINE \Delta x_n=\sum_{i=0}^\infty f_n^i(x_{n+1})+y_n NEWLINE\]NEWLINE in a Banach space. They give conditions guaranteeing that the equation has a solution \((x_n)\) asymptotically equal to \(x\in X\), \(X\) being the Banach space (i.e., there exists \(x\in X\) such that \(\|x_n-x\|_X\to 0\) as \(n\to\infty\)). Their criterion improves various results of \textit{J. Popenda} and \textit{E. Schmeidel} [Indian J. Pure Appl. Math. 28, No. 3, 319-327 (1997; Zbl 0877.39007)] and \textit{E. Schmeidel} [Commun. Appl. Nonlinear Anal. 4, No. 2, 87-92 (1997; Zbl 0877.39005), Demonstr. Math. 30, No. 1, 193-197 (1997; Zbl 0880.39010)].