On a fixed point theorem of Krasnoselskii and its applications (Q1175287)

Let \(X\) be a Banach space and let \(G\) be an unbounded open set in \(X\). Let \(\varphi\) be a continuous real-valued function such that \(0<\varphi(r)<r\) for \(r>0\). An operator \(U:\text{Cl}(G)\to X\) is said to be a \(\varphi\)-contraction if \(\| Ux-Uy\|\leq\varphi(\| x- y\|)\) for all \(x\) and \(y\) in Cl\((G)\). The following Krasnoselski type theorem is proved: Let \(G\) be an unbounded convex open set in \(X\) and \(0\in G\). Let \(U:\text{Cl}(G)\to X\) be either a \(\varphi\)-contraction or the restriction to \(\text{Cl}(G)\) of a bounded operator \(U'\) on \(X\) such that \((U')^ p\) is a \(\varphi\)-contraction for some \(p\geq 1\). Let \(F:\) Cl\((G)\to X\) be a bounded-compact operator. Suppose that \(T=U+F\) maps Cl\((G)\) to itself. If \(| T|<\) then \(T\) has a fixed point. This theorem is applied to a proof of existence of periodic solutions of differential equations on Banach space.