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.
Highlights
Fixed point theory is a branch of mathematics that has multiple applications in almost all scientific fields of study
The presented contractions are called ample spectrum contractions because they are an attempt to generalize all known contractions that are defined by contractivity conditions that involve only the terms d( x, y) and d( Tx, Ty)
The following result is useful in order to study when an ample spectrum contraction can have multiple fixed points
Summary
Fixed point theory is a branch of mathematics that has multiple applications in almost all scientific fields of study. In the original definition of (A, S)-contractions, inspired by the previous contributions, the authors established a strict inequality that must be verified for some pairs of points related under a binary relation In this manuscript, we improve such results in several ways: (1) the given family of auxiliary functions is more general;. (2) coherently, the presented contractivity condition is weaker; and (3) the set of points that have to satisfy the contractivity condition is smaller These improvements let us show that the above-commented contractions are particular cases of our study, and new families of contractive maps correspond to this new approach (see [9,10,11]). The presented contractions are called ample spectrum contractions because they are an attempt to generalize all known contractions that are defined by contractivity conditions that involve only the terms d( x, y) and d( Tx, Ty)
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