We establish a relation theoretic version of the main result of Aydi et al. [H. Aydi, M. Abbas, C. Vetro, Partial Hausdorff metric and Nadler's fixed point theorem on partial metric space, Topol. Appl. (159), 2012, 3234-3242] and extend the results of Alam and Imdad [A. Alam, M. Imdad, Relation-theoretic contraction priciple, J. Fixed Point Theory Appl., 17(4), 2015, 693-702.] for a set-valued map in a partial Pompeiu-Hausdorff metric space. Numerical examples are presented to validate the theoretical finding and to demonstrate that our results generalize, improve and extend the recent results in different spaces equipped with binary relations to their set-valued variant exploiting weaker conditions. Our results provide a new answer, in the setting of relation theoretic contractions, to the open question posed by Rhoades on continuity at fixed point. We also show that, under the assumption of k-continuity, the solution to the Rhoades' problem given by Bisht and Rakocević characterizes completeness of the metric space. As an application of our main result, we solve an integral inclusion of Fredholm type.
The aim of this paper is to obtain some new fixed point theorems for single valued mappings and also for hybrid pair of mappings. Our results extend and generalize well known results due to Aamri and El Moutawakil, Sintunavarat and Kumam, Kadelburg et al., and Kamran; and also generalize and rectify some recent results due to Bisht [On Existence of common fixed points under Lipschitz-Type Mapping pairs with Applications, Numer.
Recently, there has been a considerable effort to obtain new solutions to the Rhoades' open problem on the existence of contractive mappings that
admit discontinuity at the fixed point. An extended version of this problem is also stated using a geometric approach. In this paper, we obtain new
solutions to this extended version of the Rhoades' open problem. A related problem, the fixed-circle problem (resp. fixed-disc problem) is also
studied. Both of these problems are related to the geometric properties of the fixed point set of a self-mapping on a metric space. Furthermore, a new
result about metric completeness and a short discussion on the activation functions used in the study of neural networks are given. By providing
necessary examples, we show that our obtained results are effective.
In this paper, we establish some fixed point theorems for single valued and multi-valued mappings on a complete metric space. Suzuki's and some other fixed point theorems are generalized by taking a more general contractive condition for single valued mappings. It is also proved that our result characterizes the completeness of the metric space. Further, taking generalized contractive condition, a fixed point theorem is also established for multi-valued mappings.
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