# $${S}_{H}$$-metric spaces and fixed-point theorems for multi-valued weak contraction mappings

## Abstract

In this note for every $$S$$-metric space $$(X,S),$$ we define a new $$S$$-metric $${S}_{H}$$ called Hausdorff S-metric on $$CB(X)$$ and show that if $$(X,S)$$ is complete, $$(K\left(X\right), {S}_{H})$$ is complete too, where K(X) is the set of all compact nonempty subsets of $$X$$ and the notion of weak contraction multi-valued mappings on complete metric spaces (Kritsana Neammanee & Annop Kaewkhao, 2011) is generalized to complete $$S$$-metric spaces. This idea is used to establish some fixed-point theorems for weak contractive multi-valued mappings from $$(X,S)$$ into $$(CB\left(X\right),{S}_{H})$$.

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## Acknowledgements

The authors would like to express their appreciation to the referee for his/her valuable comments and suggestions that improved the original version.

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Correspondence to M. Sabbaghan.

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Pourgholam, A., Sabbaghan, M. $${S}_{H}$$-metric spaces and fixed-point theorems for multi-valued weak contraction mappings. Math Sci (2021). https://doi.org/10.1007/s40096-021-00381-w

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### Keywords

• Collage theorem
• Fixed point
• Hausdorff -metric space
• Multi-valued mapping
• Multi-valued Zamfirescu mapping
• Weak contraction

• 47H10
• 54H25
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