Asymptotic regularity, fixed points and successive approximations

Abstract

Let (M,d) be a metric space. In this paper we survey some of the most relevant results which relate the three concepts involved in the title: a) the asymptotic regularity; b) the existence (and uniqueness) of fixed points and c) the convergence of the sequence of successive approximations to the fixed point(s), for a given operator f : M ? M or for two operators f,g : M ? M connected to each other in some sense.

"Approximating fixed points of demicontractive mappings via the quasi-nonexpansive case"

“…Remark 4.5. Note that a condition of the form (4.9) is usually referred as retraction-displacement condition, see the recent papers [22], [23] and [63], where the authors studied the fixed point equation x = T x in terms of a retraction-displacement condition and have given various examples, corresponding to Picard, Krasnoselskii, Mann and Halpern iterative algorithms. A simpler form of Condition I, called Condition II, has been also introduced in [64] and corresponds to the particular case f (t) = αt, with α > 0 a real number, in Condition I.…”

Alternative proofs of some classical metric fixed point theorems by using approximate fixed point sequences

“…5. There exists another important technique for proving metric fixed point theorems which is based on the property of asymptotic regularity of the mappings, see [14,[29][30][31], and which is naturally closely related but independent to the technique emphasized in the current paper, in view of Theorem 3.1 in [13], which shows that, for a nonempty set X and a mapping T : X → X , the following statements are equivalent: (a) there exists a complete metric on X with respect to which T is a continuous graphic contraction; (b) Fi x (T ) = ∅ and there exists a metric on X with respect to which T is asymptotically regular.…”

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

“…The obtained results are important generalizations of the corresponding results for enriched contractions and enriched Kannan mappings in Banach spaces and also for enriched contractions in convex metric spaces, respectively. For other possible directions of research we refer to [3], [8], [9], [11], [16], [20], [21], [31], [43], [48], [53]- [59], [62],... 4. Most of the authors of the recent papers that bear in their title the names of the three mathematicians Ćirić, Reich and Rus to which we owe the important class of contractions we studied in this paper are using the alphabetical order " Ćirić-Reich-Rus".…”