Nonlinear Urysohn discrete equations on the infinite interval: a fixed-point approach
The present paper is devoted to the study of the nonlinear-Urysohn discrete equation of the form \[ y(i)= \sum^\infty_{j=0} f\bigl(i,j, y(j) \bigr), \quad i\in\{0,1,2,\dots,\} =N^+, \tag{1} \] where \(y:\mathbb{N}^+ \to \mathbb{R}\). A variety of sufficient conditions for the existence of \(C(\mathbb{N}^+, \mathbb{R})\) solutions to (1) are established by means of some new fixed-point theorems obtained by the same authors in earlier papers. Herein \(C(\mathbb{N}^+, \mathbb{R})\) denotes the class of maps \(w\) continuous on \(\mathbb{N}^+\) (discrete topology), the topology being that of uniform convergence on \(N_m=\{0,1, \dots, m\}\) for every \(m\in \mathbb{N}_0\). The results obtained extend and complement many known results in the literature which can be found in the book by \textit{R. P. Agarwal} [``Difference equations and inequalities'' Marcel Dekker New York (1992; Zbl 0925.39001)] and in the paper by the same authors [Math. Methods Appl. Sci. 22, No. 1, 91-99 (1999 Zbl 0927.47042)].
scientific article; zbMATH DE number 1755249