Best approximation and fixed points in strong \(M\)-starshaped metric spaces (Q1895295)

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 \textit{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 \(\lambda \in I\).