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Published in:
2018 | OriginalPaper | Chapter
7. On Fixed Points That Belong to the Zero Set of a Certain Function
Authors : Praveen Agarwal, Mohamed Jleli, Bessem Samet
Published in: Fixed Point Theory in Metric Spaces
Publisher: Springer Singapore
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Abstract
Let \(T: X\rightarrow X\) be a given mapping. The set \({\text {Fix}}(T)\) is said to be \(\varphi \)-admissible with respect to a certain mapping \(\varphi : X\rightarrow [0,\infty )\), if \(\emptyset \ne \text{ Fix }(T)\subseteq Z_\varphi \), where \(Z_\varphi \) denotes the zero set of \(\varphi \), i.e., \(Z_\varphi =\{x\in X: \varphi (x)=0\}\). In this chapter, we present the class of extended simulation functions recently introduced by Roldán and Samet [13], which is more large than the class of simulation functions, introduced by Khojasteh et al. [8]. We obtain a \(\varphi \)-admissibility result involving extended simulation functions, for a new class of mappings \(T: X\rightarrow X\), with respect to a lower semi-continuous function \(\varphi : X\rightarrow [0,\infty )\), where X is a set equipped with a certain metric d. From the obtained results, some fixed point theorems in partial metric spaces are derived, including Matthews fixed point theorem [9]. Moreover, we answer to three open problems posed by Ioan A. Rus in [16].The main references for this chapter are the papers [7, 13, 17].