Local expansions and accretive mappings
Let X and Y be complete metric spaces with Y metrically convex, let \(D\subset X\) be open, fix \(u_ 0\in X\), and let \(d(u)=d(u_ 0,u)\) for all \(u\in D\). Let \(f:X\to 2^ Y\) be a closed mapping which maps open subsets of D onto open sets in Y, and suppose f is locally expansive on D in the sense that there exists a continuous nonincreasing function \(c:R^+\to R^+\) with \(\int^{+\infty}c(s)ds=+\infty\) such that each point \(x\in D\) has a neighborhood N for which \(dist(f(u),f(v))\geq c(\max \{d(u),d(v))\})d(u,v)\) for all u,\(v\in N\). Then, given \(y\in Y\), it is shown that \(y\in f(D)\) iff there exists \(x_ 0\in D\) such that for \(x\in X\backslash D\), \(dist(y,f(x_ 0))\leq dist(u,f(x)).\) This theorem is then applied to the study of existence of zeros of (set-valued) locally strongly accretive and \(\Phi\)-accretive mappings in Banach spaces. These results extend earlier work of \textit{W. A. Kirk} and \textit{R. Schöneberg} [J. Math. Anal. Appl. 72, 114-121 (1979; Zbl 0415.54008)]. The proof of the theorem stated above is based on ideas of \textit{F. Browder} [Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Pure Math. 18, pt.2 (1976; Zbl 0327.47022)].
existence of zeros of set-valued locally strongly accretive mappings in Banach spaces
scientific article; zbMATH DE number 3848078