We familiarize a notion of a fixed circle in a partial metric space, which is not necessarily the same as a circle in a Euclidean space. Next, we establish novel fixed circle theorems and verify these by illustrative examples with geometric interpretation to demonstrate the authenticity of the postulates. Also, we study the geometric properties of the set of non-unique fixed points of a discontinuous self-map in reference to fixed circle problems and responded to an open problem regarding the existence of a maximum number of points for which there exist circles. This paper is concluded by giving an application to activation function to exhibit the feasibility of results, thereby providing a better insight into the analogous explorations.

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.

The aim of the present paper is to obtain an answer to an open problem due to Sastry et al. [5] by using the relationship between the continuity and reciprocal continuity of mappings in the setting of control functions which alter distances.

In this paper we obtain two known solutions of the problem of continuity of contractive mappings at fixed point under alternative set of conditions; these known solutions followed the first solution of Rhoades problem by Pant in 1999. We show that these two solutions characterize completeness and we also compare these with some recent solutions of the Rhoades problem.