Discussion on Generalized-(αψ,βφ)-Contractive Mappings via Generalized Altering Distance Function and Related Fixed Point Theorems

Abstract

We extend the notion of ( , )-contractive mapping, a very recent concept by Berzig and Karapinar. This allows us to consider contractive conditions that generalize a wide range of nonexpansive mappings in the setting of metric spaces provided with binary relations that are not necessarily neither partial orders nor preorders. Thus, using this kind of contractive mappings, we show some related fixed point theorems that improve some well known recent results and can be applied in a variety of contexts., ∈ : R ⇒ R .(1)

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

“…Several metrical fixed point theorems under arbitrary binary relations are proved by various authors such as Khan et al [10], Ayari et al [27], Roldán-López-de-Hierro [28], Roldán-López-de-Hierro and Shahzad [29], and Shahzad et al [30], which are generalizations of the relation-theoretic contraction principle due to Alam and Imdad [26]. Here we can point out that arbitrary binary relation is general enough and often does not work for certain contractions, so that various fixed/coincidence point theorems are proved in metric spaces equipped with different types of binary relations, for example, preorder (Roldán-López-de-Hierro and Shahzad [11]), transitive relations (Shahzad et al [31]), finitely transitive relations (Berzig and Karapinar [32], Berzig et al [33]), locally finitely transitive relations (Turinici [34,35]), locally finitely T-transitive relations (Alam et al [36]), and locally T-transitive relations (see Alam and Imdad [37]).…”

“…Lemma 2 ([33,35]) Let (X, d) be a metric space, and let {x n } be a sequence in X. If {x n } is not a Cauchy sequence, then there exist > 0 and two subsequences {x n k } and {x m k…”

Relation-theoretic metrical coincidence theorems

“…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.…”

Coincidence and common fixed point results for β-quasi contractive mappings on metric spaces endowed with binary relation

“…More recently, there have been some attempts to generalize the fixed point results for metric spaces endowed with binary relation. For more details, we refer the reader to [11][12][13][14][15][16][17][18][19]. In this paper, some coincidence point results in complete metric spaces endowed with transitive binary relations are first highlighted.…”