Existence and Approximation of Fixed Points of Enriched Contractions and Enriched φ-Contractions

Abstract

We obtain existence and uniqueness fixed point theorems as well as approximation results for some classes of mappings defined by symmetric contractive type conditions in a convex metric space in the sense of Takahashi. By using a new approach, i.e., the technique of enrichment of contractive type mappings, we obtain general results which extend the well known Banach contraction mapping principle from metric spaces as well as other corresponding results for enriched mappings defined on Banach spaces. To indicate the relevance of our new results, we present some important particular cases and future directions of research.

Enhancing Ant-Based Algorithms for Medical Image Edge Detection by Admissible Perturbations of Demicontractive Mappings

“…(1) We show, both analytically and empirically, that one can improve ant-based algorithms for image edge detection by using admissible perturbations of demicontractive mappings as test functions. For another possible way to find more selective test functions that can improve ant-based algorithms for image edge detection, we refer to the classes of the so-called enriched mappings-which are in fact admissible perturbations of various contractive operators-which were very recently studied by the authors of [26][27][28][29][30][31][32][33][34].…”

Some fixed point theorems on equivalent metric spaces

“…One of the significant results from the several new contractive conditions which have been developed in an attempt to obtain more refined fixed point results is the concept of (l, L)-Berinde contractions. Some related fixed point results can be found in [7,18,31] etc. For all these results it is important to give some examples which involve the fixed point theory as a must-have tool in the study of the solutions of differential and integral equations, see for example [8,12,13,15,18].…”

Fixed point theorems for enriched Ćirić-Reich-Rus contractions in Banach spaces and convex metric spaces

“…but the reverse is not valid: there are various examples of convex metric spaces which cannot be embedded in any Banach space (see [60], Example 1; [1], Examples 1 and 2; [2], [29], [42] etc.). The next lemmas present some fundamental properties of a convex metric space in the sense of Definition 3.5 (see [60,2] for more details and [23] for their proofs). Let (X, d, W ) be a convex metric space and T : X → X be a self mapping.…”

“…On the other hand, in the recent papers [12]- [14], [17]- [19], [22] and [23], the authors used the technique of enriching contractive type mappings in order to generalize, in the setting of a Hilbert space, Banach space or convex metric space, well known and important classes of contractive type mappings from the metric fixed point theory.…”