On Fixed Points That Belong to the Zero Set of a Certain Function

Let $$T: X\rightarrow X$$ beZero set a given mapping. The set $${\text {Fix}}(T)$$ is said to be $$\varphi $$ -admissible $$\varphi $$ -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 setZero set of $$\varphi $$ , i.e., $$Z_\varphi =\{x\in X: \varphi (x)=0\}$$ . In this chapter, we present the class of extended simulation functionsExtended simulation function 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 functionsExtended simulation function, for a new class of mappings $$T: X\rightarrow X$$ , with respect to a lower semi-continuousLower 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 metricPartial metric spaces are derived, including Matthews fixed point theoremMatthews 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].