On \(l^ p\) solutions of discrete Volterra equations
Sufficient conditions are obtained to ensure the solution of \(u(n)=\sum^{n-1}_{i=0}f(n,i,u(i))+g(n)\), a discrete analogue of a nonlinear Volterra integral equation, is in the space \(\ell_ \omega^ p(N,B)\), the space of functions from \(N\) to \(B\) which satisfy \(\sup_{n\geq 0}(\omega^{-1}(n)\sum^{n-1}_{i=0}\| x(i)\|^ p)<\infty\) for some \(\omega:N\to[0,\infty)\) with \(\omega(n)\neq 0\), \(N=\{0,1,2,3,\dots\}\), \((B,\|\cdot\|)\) a Banach space and fixed \(p\), \(1\leq p<\infty\). \((f:N\times N\times B\to B\), \(g:N\to B\), and \(u:N\to B)\). Similar results are obtained for discrete operator equations.
scientific article; zbMATH DE number 92520