Fixed point under set-valued relation-theoretic nonlinear contractions and application

Abstract

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.

On relation-theoretic F−contractions and applications in F−metric spaces

“…On another point of note, Turinici [22] investigated the order-theoretic fixed point, while Ran and Reurings [18] rediscovered an order-theoretic variant of the Banach contraction principle [4]. Recently, Tomar et al [26] gave a novel response to the open question presented by Rhoades [17] on continuity at a fixed point while proving a fixed point of a set-valued map satisfying relation-theoretic contractions in a partial Pompeiu-Hausdorff metric space.…”

Fixed point, its geometry, and application via ω‐interpolative contraction of Suzuki type mapping

“…In 1930, Caccioppoli 1 enhanced and improved the conclusions of Banach 2 from normed space to metric space. Following him, numerous authors investigated many abstract spaces to generalize the conclusions of Banach 2 and Caccioppoli, 3 for instance, previous works [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and references therein. However, all such efforts were not beneficial, which was the main reason for the new explorations.…”

Fixed Point Theorems for Nonexpansive Mappings under Binary Relations

“…In 2015, Alam and Imdad [17] established a novel version of the Banach contraction principle, employing an amorphous binary relation. In recent years, various metrical fixed theorems were proved under different types of contractivity conditions, employing certain binary relations (e.g., [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]). In such results, the involved contraction conditions remain relatively weaker than the usual contraction conditions, as these are required to hold merely for those elements which are related in the underlying binary relation.…”