Convergence and stability of Fibonacci-Mann iteration for a monotone non-Lipschitzian mapping

In this paper, we prove strong convergence and ∆−convergence of Fibonacci-Mann iteration for a monotone non-Lipschitzianmapping (i.e. nearly asymptotically nonexpansive mapping) in partially ordered hyperbolic metric space. Moreover, we prove stability of Fibonacci-Mann iteration. Further, we construct a numerical example to illustrate results. Our results simultaneously generalize the results of Alfuraidan and Khamsi [Bull. Aust. Math. Soc., 2017, 96, 307–316] and Schu [J. Math. Anal. Appl., 1991, 58, 407–413].


Introduction
Metric fixed point theory is one of the important branch of nonlinear analysis. In 1922, Banach laid the foundation of metric fixed point theory. He gave the first fixed point theorem which guarantee the existence and uniqueness of fixed point and provided a constructive method to find fixed point. After the Banach fixed point theorem several generalization came into picture. The two important extension to partially ordered metric space was given by Ran and Reuring [1] and Nieto and Rodrique López [2]. Ran and Reuring [1] applied their results to solve matrix equation while Nieto and Rodrique López [2] applied to solve differential equation. In 1965, Browder [3,4], Göhde [5] and Kirk [6] independently gave the existence theorem for nonexpansive mapping. The existence theorem of Kirk [6] was slightly more general then the theorem of Browder and Göhde. In 2016, Dehaish and Khamsi [7] proved Browder and Göhde fixed point theorem for monotone nonexpansive mapping. Banach used Picard iteration process to approximate the fixed point for contraction mapping. When we work with slightly weaker mapping, then Picard iteration does not converge. So many iteration like Mann, Ishikawa, Krasnoseleski came into picture to sort out this problem. Schu [8] introduced modified Mann iteration based on the good behaviour of Lipschitz constant associated to the iterates of involve mappings. The modified Mann iteration scheme does not convergent for monotone mapping. Therefore, Alfuraidan and Khamsi [9] introduced Fibonacci-Mann iteration scheme and proved strong and weak convergence in partially ordered Banach space for monotone asymptotically nonexpansive mapping. It always remains a question of attraction to prove the results of linear domain into nonlinear domain. In general metric space, we don't have addition and scalar multiplication so that we can not talk about convexity, weak convergence, duals as compare to Banach space. Hence, we are unable to extend those results of Banach space which required convexity assumptions. Due to availability of a special kind of convex structure, hyperbolic spaces provide a natural platform to study the approximation of fixed point.
In this paper, we prove strong convergence and ∆−convergence results of Fibonacci-Mann iteration scheme for monotone nearly asymptotically nonexpansive mapping in partially ordered hyperbolic metric space. Also, we establish the w 2 −stability of Fibonacci-Mann iteration process. Further, to demonstrate the genuineness of result we construct a new example of monotone nearly asymptotically nonexpansive mapping in hyperbolic metric space.

Preliminaries
A metric space (X, d) along with partial ordering ⪯ is denoted by (X, d, ⪯). Two points x and y in X are comparable whenever x ⪯ y or y ⪯ x. Definition 1. Let (X, d, ⪯) be a partial order metric space. The map T : X → X is said to be monotone or order for any x, y ∈ X.

Definition 2.
[10] A mapping T : (X, d, ⪯) → (X, d, ⪯) is said to be monotone nearly Lipschitzian with respect to an if for each n ∈ N, there exist a constant kn ≥ 0 such that where an ∈ [0, ∞) with an → 0 and for every comparable element x, y ∈ X. The infimum of constants kn for which the last inequality hold is denoted by η(T n ) and called the nearly Lipschitz constant. The monotone nearly Lipschitz mapping T with sequence {(an , η(T n ))} is said to be monotone nearly asymptotically nonexpansive if 1. η(T n ) ≥ 1 for all n ∈ N and 2. lim n→∞ η(T n ) = 1.
A point x ∈ X is said to be fixed point of T whenever T(x) = x and the set of fixed point of T is denoted by F(T).
In 2005, Kohlenbach [11] introduced the following definition of hyperbolic metric space.

Definition 3.
Let (X, d) be a metric space, then (X, d, W) will be the hyperbolic metric space if the function W : for all x, y, z, w ∈ X and α, β ∈ [0, 1].
Note 1. If only condition (i) is satisfied, then (X, d, W) will be convex metric space introduced by Takahashi [12]. We say that a subset C of X is said to be convex if x, y ∈ C implies that W(x, y, α) ∈ C.
Linear example of hyperbolic metric space is Banach space and nonlinear examples are Hadamard manifolds, the Hilbert open unit ball equipped with the hyperbolic metric and the CAT(0) spaces. Now we construct a example of nearly asymptotically nonexpansive mapping in hyperbolic metric space which is not in a real line.
for all x = (x 1 , x 2 ) and y = (y 1 , y 2 ) in X. Now for α ∈ [0, 1], define a function W : Then we can easily verify that (X, d, W) is a hyperbolic metric space. Now, suppose that C = [1, 4] × [1,4] and T : C → C be a mapping defined by is a discontinuous type nearly asymptotically nonexpansive mapping with a 1 = 14 and an = 0 for n ≥ 2 n ∈ N. kn = 1 for all n ∈ N.
The generalization of definition of uniformly convex in metric space was first given by Goebel et al. [14].

Definition 4.
Let (X, d, W) be a hyperbolic metric space. We say that X is uniformly convex if for any a ∈ X, for every r > 0, and for each ε > 0 In 1976, Lim [15] introduced the concept of ∆−convergence in metric space.

∆-convergence and strong convergence theorem
In this section, we prove strong and ∆−convegence of Fibonacci-Mann iteration for nearly asymptotically nonexpansive mapping in the setting of uniformly convex hyperbolic metric space.
Theorem 1. Let X be a complete uniformly convex partially ordered hyperbolic metric space and C be a nonempty, convex and closed subset of X and let T : C → C be a monotone nearly asymptotically nonexpansive mapping with sequence {(an , η(T n ))} and F(T) ≠ ϕ such that Proof. Let p ∈ F(T). It follows form Lemma 4 that T f (n) (xn) ⪯ p. Now, It follows from Lemma 3 that lim Thus, we have T f (n) x * = x * , which completes the proof. Thus, for a given ϵ > 0 there exist a K(ϵ) ∈ N such that d(xn , F(T)) < ϵ 2 whenever n > K(ϵ).

Theorem 2. Let X be a complete uniformly convex partially ordered hyperbolic metric space and C be a nonempty, convex and closed subset of X and let T : C → C be a monotone nearly asymptotically nonexpansive mapping with sequence {(an , η(T n ))} and F(T) ≠ ϕ such that
Hence xn is a Cauchy sequence in C. Since C is closed subset of X then lim n→∞ xn = x * ∈ C. Theorem 3. Let X be a complete uniformly convex partially ordered hyperbolic metric space and C be a nonempty, convex and compact subset of X and let T : C → C be a monotone nearly asymptotically nonexpansive mapping with sequence {(an , η(T n ))} and F(T) ≠ ϕ such that ∑︀ ∞ n=1 an < ∞ and ≤ d(xn k , T f (n k ) xn k ) + η(T f (n k ) )d(xn k , p) + η(T f (n k ) )a f (n k ) .
By uniqueness of limit we obtain T f (n k ) p = p. That is p ∈ F(T). Since lim n→∞ d(xn , p) exist for every p ∈ F(T).
Hence xn converges strongly to p ∈ F(T).

Stability result
A fixed point iteration is numerically stable if small perturbation (due to approximation, rounding errors etc.) during computation will produce small changes on the approximate value of the fixed point computed by methods. In 1988, Harder and Hicks [20] gave the formal definition of stability and proved some stability result for Picard, Mann and Kirk fixed point iteration procedures under various contractive conditions. Let us define the stability.

Definition 8.
Let (X, d) be a metric space, T be a self mapping on X and {xn} be an iterative sequence produced by the mapping T such that {︃ where x 1 is an initial approximation and f is a function. Assume that {xn} converges strongly to p ∈ F(T). If for an arbitrary sequence {yn} ⊂ X, then the iterative sequence {xn} is said to be stable with respect to T or simply stable. The following definition of w 2 −stability was given by Timis [22] in 2010. Theorem 4. Let X be a uniformly convex partially ordered hyperbolic metric space and C be a nonempty, convex and closed subset of X and let T : C → C be a monotone nearly asymptotically nonexpansive mapping with sequence {(an , η(T n ))} and F(T) ≠ ϕ such that ∑︀ ∞ n=1 an < ∞ and ∑︀ ∞ n=1 (η(T n ) − 1) < ∞. If sequence {xn} is defined by (2.1) with x 1 ⪯ Tx 1 (or Tx 1 ⪯ x 1 ) where 0 < a ≤ αn , βn ≤ b < 1 and x 1 ∈ C. If p ⪯ x 1 (or x 1 ⪯ p) for some p ∈ F(T) and {yn} be any equivalent sequence of {xn} with xn ⪯ yn ( or yn ⪯ xn), then the iteration process (2.1) is weak w 2 −stable with respect to T.
Suppose that εn → 0 as n → ∞. Then Thus {xn} is weak w 2 −stable with respect to T.

Example
In this section, we construct an example of monotone nearly asymptotically nonexpansive mapping and show the convergence behaviour of Mann and Finbonacci-Mann iteration procedure.
It is observed that when we take tn = (0, .5], both Mann iteration (MI) and Fibonacci-Mann iteration (FMI) converges to fixed point and Fibonacci-Mann iteration converges faster then Mann iteration. But when we take tn = (.5, 1), Fibonacci-Mann iteration converge for all initial value but Mann fails to converge to fixed point for some initial value. Figure 1 and Figure 2 shows the convergence behaviour of Mann iteration and Fibonacci-Mann iteration for tn = .5 and tn = .55 for different initial values.