Nonlinear discrete systems with global boundary conditions. (Q1414234)

The author studies the existence of \(l_\infty-\)solutions of nonlinear boundary value problems of the form \(x(k+1)=f(k,x(k))+h(k)\), \(k=0, 1, 2, \ldots\) subject to the global conditions \[ \sum_{k=0}^{\infty}{g(k,x(k))}=y. \] Here \(f: {\mathbb R}^{n+1} \to {\mathbb R}^{n}\) and \(g: {\mathbb R}^{n+1} \to {\mathbb R}^{p}\) are smooth functions, \(h \in l_\infty\), \(y \in {\mathbb R}^{p}\) and \(p \leq n\). He gives some sufficient conditions that ensure that the solvability of this problems is preserved under perturbations on \(h\) and \(y\). The parameter problem \(x(k+1)=f(\lambda,k,x(k))+h(k)\), \(k=0, 1, 2, \ldots\) together with \[ \sum_{k=0}^{\infty}{g(\lambda, k,x(k))}=0 \] is also studied. In this case existence of \(l_\infty\)-solutions in terms of the parameter \(\lambda\) is proved under suitable conditions on \(f\) and \(g\). The behavior of such solutions is also treated.