The Ran–Reurings fixed point theorem without partial order: A simple proof

Abstract

Abstract. The purpose of this note is to generalize the celebrated Ran and Reurings fixed point theorem to the setting of a space with a binary relation that is only transitive (and not necessarily a partial order) and a relation-complete metric. The arguments presented here are simple and straightforward. It is also shown that extensions by Rakotch and Hu-Kirk of Edelstein's generalization of the Banach contraction principle to local contractions on chainable complete metric spaces derive from the theorem of Ran-Reurings. Keywords and Phrases:Existence and uniqueness of fixed point; contraction and local contraction; transitive relation; monotonic chainability; monotoniccomplete metric. Mathematics Subject Classification: 47H10 Dedicated to Professor Andrzej Granas PreliminariesIn 1961, M. Edelstein [3] extended the Banach contraction principle by establishing that every uniform local contraction f : X −→ X of an ε-chainable complete metric space (X, d) has a unique fixed point. In 1962, E. Rakotch [8] refined Edelstein's result to a local contraction f of a complete metric space containing some rectifiable path (i.e., a path of finite length 1 ) joining a given point x 0 to f (x 0 ).Recall that a metric space (X, d) is said to be ε-chainable for some ε > 0, if ∀x, y ∈ X, ∃{u i } m i=0 a finite sequence in X such that: x = u 0 , u m = y and d(u i−1 , u i ) < ε for all i = 1, . . . , m. * e-mail address: hmechaie@brocku.ca 1 The length of (continuous) path γ : [0, 1] −→ X is l(γ) := sup{L(P ) : P ∈ P[0, 1]} where P[0, 1] is the collection of all finite partitions P = {0 = t 0 < t 1 < . . . < tn = 1} of [0, 1], and L(P ) = n i=1 d(γ(t i−1 ), γ(t i )).

Relation-theoretic contraction principle

“…Finally, we furnish two illustrative examples in support of Theorem 3.1, which do not satisfy the hypotheses of the previous results [10,19,2,17,18,12,11,5,13] but have fixed points. A. Alam and M. Imdad Example 3.3.…”

“…Here, it is fascinating to point out that corresponding theorems contained in [10,19,2,17,18,12,11,5,13] cannot be used in the context of the foregoing examples (i.e., Examples 3.3 and 3.4), which substantiate the utility of Theorem 3.1 over corresponding several noted results. Thus, in all, we have extended all the classical results to an arbitrary binary relation.…”

A novel approach to Banach contraction principle in extended quasi-metric spaces

“…Though this example is given for a metric space endowed with a partial order but the same ideas may be used for metric spaces endowed with a directed graph or a binary relation. In fact, this example will shed some light on a better understanding of the fixed point theorems of Ran and Reurings [15], Nieto and Rodríguez-López [14], Jachymski [10], and Ben-El-Mechaiekh [5] in connection with Theorem 3.3. It is easy to check that (X,d) is an extended quasi-metric space.…”

“…Shortly after, Nieto and Rodriguez-Lopez [14] improved the main result of [15] and used their new extension to solve a lot of practical questions belonging to differential equations theory. Further extensions of these results have also been given by (i) Jachymski [10], who proposed a natural extension of problem setting from partial orders to directed graphs, (ii) Ben-El-Mechaiekh [5], who gave a formulation of Ran-Reurings fixed point theorem over metric spaces endowed with a binary relation, (iii) Turinici [20], who extended the metrical convergence of the ambient space to (abstract) sequential convergence structures.…”

Correction to: Relation-theoretic metrical fixed-point results via w-distance with applications

“…Exploiting the concepts of different kind binary relations such as partial order, strict order, preorder, tolerance, transitive etc. on metric space, many mathematician are doing their research during several years, see for example [4,5,9,[12][13][14]. Very recently, Alam and Imdad [2] presented relation-theoretic metrical fixed point results due to famous Banach contraction principle using an amorphous relation.…”