Ample Spectrum Contractions and Related Fixed Point Theorems

Abstract

Simulation functions were introduced by Khojasteh et al. as a method to extend several classes of fixed point theorems by a simple condition. After that, many researchers have amplified the knowledge of such kind of contractions in several ways. R-functions, ( R , S ) -contractions and ( A , S ) -contractions can be considered as approaches in this direction. A common characteristic of the previous kind of contractive maps is the fact that they are defined by a strict inequality. In this manuscript, we show the advantages of replacing such inequality with a weaker one, involving a family of more general auxiliary functions. As a consequence of our study, we show that not only the above-commented contractions are particular cases, but also another classes of contractive maps correspond to this new point of view.

Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces

“…After the approaches due to Menger [8] (statistical metric spaces), Kaleva and Seikkala [9], Schweizer and Sklar [10] (probabilistic metric spaces), Kramosil and Michálek [11] (fuzzy metric spaces), and Roldán López de Hierro et al [12,13] (fuzzy spaces), among others, taking into account its potential applications in several fields of study, George and Veeramani introduced in [14] a wide class of fuzzy metric spaces that have enjoyed great success because they are particularly easy to use and interpret. Furthermore, this category of fuzzy metrics has been demonstrated to be special according to the needs of the theory of fixed points (see, for instance, [15][16][17][18][19][20][21][22][23] in several contexts). Some interrelationships among these fuzzy metric structures can be found in [24].…”

Fixed point theory in the setting of $(\alpha,\beta,\psi,\phi)$-interpolative contractions

“…Proof The existence of fixed points follows from Theorem 2 in [15] using item (b) and the uniqueness follows from Theorem 3 in [15] because we assume condition (B 2 ).…”

“…In [15], Roldán López de Hierro and Shahzad introduced a great class of contractions that generalized many previous kinds of contractions with a particular property: they only used the terms d (u, v) and d (Tu, Tv) on their contractivity conditions. In the following definitions, S represents a binary relation on X and S * is given by uS * v when u, v ∈ X and u = v .…”

“…then {a k } → 0. Although it was not explicitly enunciated in [15], we state here the following result which is a simple consequence of the main theorems given in [15]. The trivial binary relation S X on X is given by xS X y for all x, y ∈ X. Theorem 1 ([15]) Each ample spectrum contraction w.r.t.…”

“…More recently, Khojasteh et al [10] introduced the notion of simulation function, which was later subtly modified by Roldán López de Hierro et al in [11]. After a series of successive research papers deepening this area (see [12][13][14]), this last author and Shahzad presented the notion of ample spectrum contraction in [15] with a double aim: to analyze the essential properties, from an abstract point of view, that any contraction that may arise in the future must satisfy and generalize as many results as possible in the field of fixed point theory.…”