A remark on asymptotic regularity and fixed point property

Abstract

In this paper, we show that orbital continuity of a pair of non-commuting mappings of a complete metric space is equivalent to fixed point property under the Proinov type condition. Furthermore, we establish a situation in which orbital continuity turns out to be a necessary and sufficient condition for the existence of a common fixed point of a pair of mappings yet the mappings are not necessarily continuous at the common fixed point.

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

“…4. The technique of proof used in the present paper, essentially based on the concepts of graphic contraction and approximate fixed point sequence, could also be nontrivially applied to other classes of self and nonself single-valued mappings in the literature on metric fixed point theory, see [5,[7][8][9][10][11][12]14,20,23,28,31,36,38,39,42,44,46,50,60,61,63] etc. 5.…”

“…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.…”

Asymptotic regularity, fixed points and successive approximations

“…In the case F = {x * }, i.e., when is a Picard operator, the problem was studied in [127] and [21], see also [24] and [80].…”

Convexity and boundedness relaxation for fixed point theorems in modular spaces

“…Moreover, it has been shown that a lot of metric fixed point theorems can be extended to b-metric spaces [11,58], although b-metric, in the general case, is not continuous, lim [6,21] for further details). The notion of b-metric spaces were introduced to reach the generalization of some known fixed point theorems for single valued mappings and correspondences [9,10,15,16].…”