Approximation theorems and fixed point theorems for 1-set-contraction mappings in abstract cones. (English) Zbl 0943.47038
Statements analogous to the Ky Fan approximation theorem are established for \(1\)-set-contractive, semi-contractive, and LANE mappings of the intersection of a closed ball, annulus, and a sphere with a closed cone, \(P\), in a real Banach space. As an application, some fixed point theorems for the continuous mappings \(f : \{x\in P: \left\|x\right\|\leq R\} \to P\), where \(R>0\), are proved.
Reviewer: Andrei Ronto (Kyïv
MSC:
47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |
47H10 | Fixed-point theorems |
41A50 | Best approximation, Chebyshev systems |