## 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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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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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

### Mathematical Subject Classifications

- 47H10
- 54H25