Method of locally linear approximation of nonlinear difference operators by weakly regular operators
The author studies the difference operator \[ (\mathcal{F}\mathbf{x})_n:=x_n+f_n(x_{n-1},\dots,x_{n-p}),\quad n\in\mathbb{Z},\;p\in\mathbb{N}, \] acting in the Banach space \(\ell_{\infty}(\mathbb{Z},E)\) of two-sided sequences \(\mathbf{x}=(x_n)_{n\in\mathbb{\scriptstyle{Z}}}\), \(x_n\in E\), with the norm \[ \|\mathbf{x}\|_{\ell_{\infty}(\mathbb{\scriptstyle{Z}},E)}:=\sup_{n\in\mathbb{\scriptstyle{Z}}}\|x_n\|_E, \] where \(E\) denotes a finite-dimensional Banach space with the norm \(\|\cdot\|_E\). Moreover, it is assumed that the mappings \(f_n:E^p\to E\), \(n\in\mathbb Z\), are continuous and for each \(r\in(0,\infty)\) satisfying the inequality \[ \sup_{n\in\mathbb{\scriptstyle{Z}},\|y_1\|_E\leq r,\dots,\|y_p\|_E\leq r} \|f_n(y_1,\dots,y_p)\|_E<\infty. \] In the main result, see Theorem 1, sufficient conditions under which the equation \(\mathcal{F}\mathbf{x}=\mathbf{h}\) has at least one solution \(\mathbf{x}\in\ell_{\infty}(\mathbb{Z},E)\) for any \(\mathbf{h}\in\ell_{\infty}(\mathbb{Z},E)\), are established. The proof is based on the method of a local linear approximation of \(\mathcal{F}\) by weakly regular operators. Some applications and illustrating examples of the main result are also provided.
scientific article; zbMATH DE number 6157084
scientific article