An application of Krasnoselskii fixed point theorem to the asymptotic behavior of solutions to difference equations in Banach spaces
Let \(X\) be a Banach space, let \(f:X\to X\) be a Lipschitz mapping and let \(g:X\to X\) be a compact operator. Under some additional assumptions, for each \(x\in X\), there exists a solution \((x_n)\in l_\infty(X)\) to the difference equation \[ \Delta x=\sum^\infty_{i=0} a^i_n f(x_{n+i}) +\sum^n_{i=0} b^i_n g(x_{n+i}) +y_n, \] which is asymptotically equal to \(x\). The proof of this is based on the Krasnoselskii fixed point theorem. Next this result is applied to obtain a similar result for a second order difference equation \[ \Delta(q_n \Delta x_n)+ r_nf(x_n)+ s_ng(x_n)+ z_n=0. \]
scientific article; zbMATH DE number 1497303