Abstract
Let $(X,d)$ be a metric space. A map $T:X \mapsto X$ is said to be a $(\delta,L)$ weak contraction [1] if there exists $\delta \in (0,1)$ and $L\geq 0$ such that the following inequality holds for all $x,y \in X$:
$d(Tx,Ty)\leq \delta d (x,y)+Ld(y,Tx)$
On the other hand, the idea of convex contractions appeared in [2] and [3]. In the first part of this paper, motivated by [1]-[3], we introduce a concept of convex $(\delta,L)$ weak contraction, and obtain a fixed point theorem associated with this mapping. In the second part of this paper, we consider the map is a non-self map, and obtain a best proximity point theorem. Finally, we leave the reader with some open problems.
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3. Clement Boateng Ampadu, A new proof of the convex contraction mapping theorem in metric spaces, Internat. J. Math. Arch., to appear. https://drive.google.com/file/d/0BwtkpMtWoUlEV0d4QUhnaVlqOHc/view
4. Clement Ampadu, Fixed Point Theory for Higher-Order Mappings, lulu.com, 2016. ISBN: 5800118959925
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