×

Sets in the ranges of nonlinear accretive operators in Banach spaces. (English) Zbl 0830.47048

Summary: Let \(X\) be a real Banach space and \(G\subset X\) open and bounded. Assume that one of the following conditions is satisfied:
(i) \(X^*\) is uniformly convex and \(T: \overline G\to X\) is demicontinuous and accretive;
(ii) \(T: \overline G\to X\) is continuous and accretive;
(iii) \(T: X\supset D(T)\to X\) is \(m\)-accretive and \(\overline G\subset D(T)\).
Assume, further, that \(M\subset X\) is pathwise connected and such that \(M\cap TG\neq \emptyset\) and \(M\cap \overline{T(\partial G)}= \emptyset\). Then \(M\subset \overline{TG}\). If, moreover, case (i) or (ii) holds and \(T\) is of type \((S_1)\), or case (iii) holds and \(T\) is of type \((S_2)\), then \(M\subset TG\). Various results of Morales, Reich and Torrejón, and the author are improved and/or extended.

MSC:

47J25 Iterative procedures involving nonlinear operators
47J05 Equations involving nonlinear operators (general)
47H05 Monotone operators and generalizations
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI EuDML