Fixed points of monotone nonexpansive mappings with a graph. (English) Zbl 1327.47047
Let \((X,\|\cdot\|)\) be a Banach space and \(\tau\) be a Hausdorff topology on \(X\) which is weaker than the norm topology. In the present paper, the author proves the following.
Theorem 3.1. Let \(X\) be a Banach space which satisfies the \(\tau\)-Opial condition, let \(C\) be a bounded convex \(\tau\)-compact nonempty subset of \(X\) not reduced to one point and \(G\) be a directed graph such that the set of vertices \(V(G)=C\). Assume that the triple \((C,\|\cdot\|, G)\) has the property \((P)\) and the \(G\)-intervals are convex. If \(T: C\to C\) is a \(G\)-monotone nonexpansive mapping with some \(x_0\in C\) such that \((x_0,T_0)\in E(G)\) (\(=\) the set of edges), then \(T\) has a fixed point.
Theorem 3.1. Let \(X\) be a Banach space which satisfies the \(\tau\)-Opial condition, let \(C\) be a bounded convex \(\tau\)-compact nonempty subset of \(X\) not reduced to one point and \(G\) be a directed graph such that the set of vertices \(V(G)=C\). Assume that the triple \((C,\|\cdot\|, G)\) has the property \((P)\) and the \(G\)-intervals are convex. If \(T: C\to C\) is a \(G\)-monotone nonexpansive mapping with some \(x_0\in C\) such that \((x_0,T_0)\in E(G)\) (\(=\) the set of edges), then \(T\) has a fixed point.
Reviewer: Jarosław Górnicki (Rzeszów)
MSC:
| 47H10 | Fixed-point theorems |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H05 | Monotone operators and generalizations |
| 46B20 | Geometry and structure of normed linear spaces |
| 05C20 | Directed graphs (digraphs), tournaments |
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