**1. Introduction**

Fixed point theory is a branch of mathematics that has multiple applications in almost all scientific fields of study. Mainly, it is used to prove the existence (and, in many cases, also uniqueness) of solutions of grea<sup>t</sup> variety of equations arising in theoretical and practical disciplines: matrix equations, differential equations, integral equations, etc. One of its best advantage is the fact that it permits us to deal with linear and nonlinear problems, which makes this discipline into an essential part of nonlinear analysis.

Although it was not the first result in this line of research, Banach contractive mapping principle is widely considered the pioneering statement. Any new result in this area must generalize such principle. There are many directions in which it has been extended and improved: by using weaker contractivity conditions, more general families of auxiliary functions, by involving a partial order, by considering abstract metric spaces, etc.

In recent times, Khojasteh et al. [1] introduced a new class of auxiliary functions, called *simulation functions*, that let us consider a family of contractivity conditions that only involve two arguments: the distance between two points (*d*(*<sup>x</sup>*, *y*)) and the distance between their corresponding images (*d*(*Tx*, *Ty*)) under the considered operator. This work quickly attracted the attention of several researchers because of its potential applications (see, for instance, the work of Roldán López de Hierro et al. [2], who slightly modified the original definition, and those of Roldán López de Hierro and Shahzad [3,4], who presented *R-functions* as extensions of simulation functions).

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]). However, 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 not only the above-commented contractions are particular cases of our study, but also new families of contractive maps correspond to this new approach (see [9–11]). 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*(*<sup>x</sup>*, *y*) and *d*(*Tx*, *Ty*).
