Abstract
We consider the problem of best proximity point in locally convex spaces endowed with a weakly convex digraph. For that, we introduce the notions of nonself G-contraction and G-nonexpansive mappings, and we show that for each seminorm, we have a best proximity point. In addition, we conclude our work with a result showing the existence of best proximity point for every seminorm.
Highlights
Fixed point theorems deal with conditions under which maps have invariant points. e theory itself is a beautiful mixture of analysis, topology, and geometry
Fixed point techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory, and physics
If the fixed point equation of given mapping does not have a solution, it is of interest to find an approximate solution for the fixed point equation
Summary
Fixed point theorems deal with conditions under which maps (single or multivalued) have invariant points. e theory itself is a beautiful mixture of analysis (pure and applied), topology, and geometry. [8] Ling established some properties relating the concepts of normal structure and submeans in a Hausdorff locally convex space and obtained a fixed point theorem for left reversible semigroups of nonexpansive mapping with a compactness of the domain. Vuong in [10] established a fixed point theorem for nonexpansive mappings in a locally convex space with normal structure and the compactness of the domain.
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