Abstract
AbstractWe prove that every firmly nonexpansive-like mapping from a closed convex subset "Equation missing" of a smooth, strictly convex and reflexive Banach pace into itself has a fixed point if and only if "Equation missing" is bounded. We obtain a necessary and sufficient condition for the existence of solutions of an equilibrium problem and of a variational inequality problem defined in a Banach space.
Highlights
Let C be a subset of a Banach space E
The following statements are equivalent: i Fix T / ∅ for every mapping T : C → C which is of type (Q); ii C is bounded
The following statements are equivalent: i Fix T / ∅ for every mapping T : C → C which is of type (R); ii C is bounded
Summary
Let C be a subset of a Banach space E. The following statements are equivalent: i Fix S / ∅ for every firmly nonexpansive mapping S : C → C; ii C is bounded.
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