Fixed point theorems in bimetric space endowed with binary relation and applications

Abstract

Abstract. In this paper, we obtained some fixed point results for continuous mappings satisfying a generalized contractive condition in the setting of two metrics space endowed with a binary relation. Our theorems generalize and extend several known results in the literature. As application, we establish an existence theorem for the solution of a nonlinear first order differential equation.

Relation-theoretic metrical coincidence theorems

“…(22) We prove this fact by the method of mathematical induction. It follows from (20) that (22) holds for n = 0. Suppose that (22) holds for n = r > 0, i.e.,…”

“…In recent years, various fixed and coincidence point theorems are proved in metric spaces equipped with different types of binary relations, e. ., partial order (see Ran and Reurings [1], Nieto and Rodríguez-López [2] and Nieto and Rodríguez-López [3]), preorder (see Turinici [4], Roldán and Karapinar [5], Roldán-López-de-Hierro and Shahzad [6]), transitive relation (see Ben-El-Mechaiekh [7], Shahzad et al [8]), finitely transitive relation (see Berzig and Karapinar [9], Berzig et al [10]), tolerance (see Turinici [11,12]), strict order (see Ghods et al [13]), symmetric closure (see Samet and Turinici [14], Berzig [15]) and arbitrary binary relation (see Alam and Imdad [16], Roldán-López-de-Hierro [17], Roldán-López-de-Hierro and Shahzad [18], Shahzad et al [19], Khan et al [20], Ayari et al [21]). In the present context, the contraction condition remains relatively weaker than usual contraction as it is required to hold merely for those elements which are related in the underlying relation.…”

Observations on relation-theoretic coincidence theorems under Boyd–Wong type nonlinear contractions

“…In last ten decades, the classical Banach contraction principle [3] has been generalized by numerous authors in the different directions by improving the underlying contraction conditions (e.g., [4][5][6]), enhancing the number of involved mappings [4,7], weakening the involved metrical notions [7,8], and enlarging the class of ambient spaces [9][10][11]. In 2004, Ran and Reurings [12] obtained a variant of the classical Banach contraction principle to a complete metric space endowed with partial order relation, which was slightly modified by Nieto and Rodríguez-López [13] in 2005.…”

Some results on implicit contractive conditions in metric spaces endowed with arbitrary binary relations

“…After that, Berzig considered on the coincidence and common fixed point for contractive mappings, which investigated on metric spaces endowed with binary relation. Asgari and Mousavi reconsidered this space for coupled fixed point theorems while Khan et al were constructing the fixed point theorem in bimetric spaces. Recently, Ahmadullah et al proved fixed point theorems through implicit contractive condition on metric spaces used with an arbitrary binary relation, and Ayari et al also proved the main theorem in the same space for showing the existing of coincidence points and common fixed points.…”