The Convex (\delta,L) Weak Contraction Mapping Theorem and its Non-Self Counterpart in Graphic Language
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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.
1975 ◽
pp. 337-344
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1977 ◽
Vol 17
(3)
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pp. 375-389
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2017 ◽
Vol 9
(2)
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pp. 1
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2012 ◽
Vol 2012
(1)
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