Some new fixed point theorems under $$(\mathcal {A},\mathcal {S})$$ ( A , S ) -contractivity conditions

Abstract

In this manuscript we introduce a new class of contractivity conditions for mappings from a metric space into itself endowed with a binary relation (that is not necessarily a partial order). These conditions unify several kinds of contractive operators under some basic axioms, and they lead us to present some results about existence and uniqueness of fixed points that extend and generalize many theorems in the field of fixed point theory. One of the most attractive properties of this new class of operators is that they are not necessarily contractive, that is, there are non-contractive operators for which the presented results are applicable.

Relation-theoretic metrical coincidence theorems

“…Shahzad et al [19] and Roldán-López-de-Hierro and Shahzad [18] used the term "R-nondecreasing" instead of "R-preserving".…”

“…Motivated by the notion of F-invariant subset of X 6 (for the mapping F : X 3 → X) due to Charoensawan [48] and the notion of F-closed subset of X 4 (for the mapping F : X 2 → X) due to Kutbi et al [49], Alam and Imdad [16] introduced the following. Here it can be pointed out that instead of saying that f is R-continuous, Roldán-López-de-Hierro et al [17], Shahzad et al [19] and Roldán-López-de-Hierro and Shahzad [18] said that f is (d, R)-nondecreasingcontinuous. The following notion is a generalization of 'd-self-closedness of partial order relation ( )' defined by Turinici [4].…”

“…Define the constant sequences z 0 n = x and z k n = y, then using (19), we have z 0 n+1 = f z 0 n = x and z k n+1 = f z k n = y ∀ n ∈ N 0 . Put z 1 0 = z 1 , z 2 0 = z 2 , ..., z k−1 0 = z k−1 .…”

“…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.…”

Ample Spectrum Contractions and Related Fixed Point Theorems

“…The above-mentioned classes of contractions have been included in a new family of contractive mappings, called (A, S)-contractions, that extend and unify several results in fixed point theory (see [5]). Theoretical notions introduced in such manuscript were later developed by other researchers (see [6]) even with applications to fuzzy partial differential equations (see [7]) and optimal solutions and applications to nonlinear matrix and integral equations (see [8]).…”