Fixed point theorems for enriched Kannan mappings in CAT(0) spaces

Abstract

We present enriched Kannan and enriched Bianchini mappings in the framework of unique geodesic spaces. For such mappings, we establish the existence and uniqueness of a fixed point in the setting of CAT(0) spaces and show that an appropriate Krasnoselskij scheme converges with certain rate to the fixed point. We proved some inclusion relations between enriched Kannan mapping and some applicable mappings such as strongly demicontractive mapping. Finally, we give an example in a nonlinear CAT(0) space and perform numerical experiments to support the theoretical results.

On Enriched Suzuki Mappings in Hadamard Spaces

“…Proof. We have, according to (6), that lim n→∞ d(x n , Tx n ) converges, and since, by assumption, T is demicompact, there exists a subsequence {x n k } ⊆ {x n } converging to some point p ∈ C. On the other hand, p is also the ∆-limit of {x n k }, and hence, p ∈ Fix(T). Lastly, the fact that the entire sequence {x n } n≥0 converges to p follows from the inequality d(x n+1 , p) ≤ d(x n , p), n ≥ 0, established above.…”

“…A natural task is to extend these fruitful methods and ideas to other settings, such as geodesic spaces, or, more precisely, to complete CAT(0) spaces also known as Hadamard spaces, which can be seen as important nonlinear generalizations of Hilbert spaces, and provide a suitable setting for nonlinear analysis and optimization problems (see also [5,6]). The fact that this is a suitable setting for developing fixed point theoretic results has been indicated in the pioneering works of Kirk [7,8] (also, for a basic introduction into the subject, we refer the reader to [9]).…”