## Best approximation and fixed points in strong $$M$$-starshaped metric spaces.(English)Zbl 0826.41029

In this paper, the notion of strong $$M$$-starshaped metric spaces have been introduced. For these spaces, four results have been obtained. The first two generalize a result of W. G. Dotson jun. [Proc. Am. Math. Soc. 38, 155-156 (1973; Zbl 0274.47029)] on fixed points of non-expansive maps and the other two extend and subsume several known results on the existence of fixed points of best approximation. Strong $$M$$-starshaped metric spaces: – Let $$X$$ be a metric space, $$M \subset X$$ and $$I = [0,1]$$. $$X$$ is said to be
(i) $$M$$-starshaped if there exists a mapping $$W : X \times M \times I \to X$$ satisfying $d \bigl( x,W(y,q, \lambda) \bigr) \leq \lambda d(x,y) + (1 - \lambda) d(x,q)$ for every $$x,y \in X$$, all $$q \in M$$ and all $$\lambda \in I$$,
(ii) strong $$M$$, starshaped if it is $$M$$-starshaped and $$w$$ satisfies $d \bigl( W(x,q, \lambda),\;W(y,q, \lambda) \bigr) \leq \lambda d(x,y)$ for every $$x,y \in X$$, all $$q \in M$$ and all $$\lambda \in I$$.

### MSC:

 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems

Zbl 0274.47029
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