On a cyclic Jungck modified TS-iterative procedure with application examples

Abstract

This article investigates some convergence properties of quasi-cyclic and cyclic Jungck modified TS-iterative schemes in complete metric spaces and Banach spaces. The uniqueness of the best proximity points is investigated. It is basically assumed that one of the self-mappings is asymptotically nonexpansive while the other is asymptotically contractive with several particular cases. Some application examples are also discussed.

Introduction

Many of the existing results on fixed point theory can be formulated either for fixed points and best proximity points of cyclic self-mappings or for fixed points of non-cyclic self-mappings. See, for instance, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [37], [38], [39]. Applications of fixed point theory can be found in the background literature including asymptotic behaviors, stability and related mixed expansive and non-expansive properties the (see, for instance, [18], [20], [25], [26], [27], [28], [29], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44]). Very often, such problems are strongly linked to Fixed Point Theory since stable equilibrium points of dynamic processes are candidates to convergence points of the iterative schemes which are also fixed points of the self-mapping generating the solution trajectory under some contractive or nonexpansive conditions, [34], [35]. On the other hand, Mann iteration processes or their well-known extended versions of Ishikawa and Jungck iterative processes and their various extensions are receiving important research attention in the last years. See, for instance, [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [27], [29] and also references therein. This manuscript is devoted to the study of a class of cyclic Jungck modified TS-iterative scheme which involves two combined self-mappings in the computations one being asymptotically nonexpansive while the other possesses contractive constraints. The existence and uniqueness of best proximity and fixed points of both self-mappings is investigated together with some conditions for such points to be common to both self-mappings. Some application examples are given for the solution sequences of discrete difference equations including an extended Venteŕs theorem.

The real sequences {y_n} and {x_n} are equivalent, being denoted by {y_n} ≈ {x_n}, if both have the same limit.

If T is a self-mapping on some nonempty set C then F(T) denotes the set of fixed points of T:C → C.

Real sequences ${\{x_n\}}_{n\in {\mathbf{N}}_0}$, where N₀ = N ∪ {0}, are simply denoted by {x_n} when no confusion is expected.

If T is a self-mapping on some nonempty union of subsets ${\bigcup }_{i\in \overline{p}}{A}_i$ then BP(T) denotes the set of best proximity points of T in ${\bigcup }_{i\in \overline{p}}{A}_i$.