Some fixed point theorems for the KF-iteration method in hyperbolic metric spaces

The aim of this thesis is to prove the weak w^2-stability and data dependence theorems for contraction mappings and some Δ-convergence and strong convergence theorems for generalized (α,β)-nonexpansive type 1 mappings using the KF-iteration method in hyperbolic metric spaces given by Kohlenbach and to generalize some results that exist in the literature. In addition, it was aimed to show that the KF-iteration method converges to the fixed point of the mapping faster than other iteration methods by comparing some iteration methods in the literature with the KF-iteration method by giving a numerical example for generalized (α,β)-nonexpansive type 1 mappings in hyperbolic metric spaces. 1. To achieve this goal, firstly, by examining the following studies, the KF-iteration was restated in a suitable form for the structure of the hyperbolic metric space given by Kohlenbach. a. In 2022, in the paper written by Ullah, Ahmad and Khan, and in the paper written by Temir and Korkut, the iteration method, which converges to the fixed point faster than some iteration methods in the literature and was called the KF-iteration method by Ullah, Ahmad and Khan, was defined as follows: {■(x_1∈C,@■(z_n=T((1-β_n ) x_n+β_n Tx_n ),@■(y_n=Tz_n,@x_(n+1)=T((1-α_n )Tx_n+α_n 〖Ty〗_n ),∀n≥1,)))┤ where C is a nonempty convex subset of a Banach space X, T is a self-mapping on C, and {α_n },{β_n } are two real sequences in [0,1]. b. Let (X,d) be a metric space and W:X×X×[0,1]→X be a mapping. The mapping W is said to be a convex structure on X if for all x,y,z∈X and α∈[0,1], d(z,W(x,y,α))≤(1-α)d(z,x)+αd(z,y) and a metric space (X,d) together with the convex structure W is called a convex metric space which is denoted by (X,d,W). In 2005, Kohlenbach defined the concept of hyperbolic metric space by adding the following three conditions to Takahashi's definition of convex metric space. Let (X,d) be a metric space and W:X×X×[0,1]→X be a mapping. Then (X,d,W) will be the hyperbolic metric space if the following conditions are satisfied: (i) d(z,W(x,y,α))≤(1-α)d(z,x)+αd(z,y), (ii) d(W(x,y,α),W(x,y,β))=|α-β|d(x,y), (iii) W(x,y,α)=W(y,x,1-α), (iv) d(W(x,z,α),W(y,w,α))≤(1-α)d(x,y)+αd(z,w) for all x,y,z,w∈X and α,β∈[0,1]. The KF-iteration was modified according to hyperbolic metric space in the following way: Let X be a hyperbolic metric space, C be a nonempty convex subset of a hyperbolic metric space X and T: C→C be a mapping. The KF-iteration is defined by {■(x_1∈C,@■(z_n=T(W(x_n,Tx_n,β_n )),@■(y_n=Tz_n,@x_(n+1)=T(W(Tx_n,〖Ty〗_n,α_n )),∀n≥1,)))┤ where {α_n },{β_n }∈[0,1]. 2. Then, by examining the definitions given below, the weak w^2-stability and data dependency theorems for contraction mappings were proved. a. Let (X,d) be a metric space. A mapping T:X→X is said to be a contraction if there exists a constant k∈[0,1) such that for all x,y∈X d(Tx,Ty)≤kd(x,y). b. Let (X,d) be a metric space, T:X→X be a mapping and {x_n}⊂X be an iterative sequence defined by {■(x_1∈X@x_(n+1)=f(T,x_n ),∀n≥1,)┤ where f is a function. Suppose that {x_n} converges strongly to p∈F(T). If, for any equivalent sequence {y_n}⊂X of {x_n}, (lim)┬(n→∞) d(y_(n+1), f(T,y_n ))=0⇒(lim)┬(n→∞) y_n=p then, the iterative sequence {x_n} is said to be weak w^2-stable with respect to T. c. Let (X,d) be a metric space and T,T ̃:X→X be two operators. T ̃ is called an approximate operator of T if d(Tx,T ̃x )≤ε for all x∈X and for a fixed ε>0. After it is shown that the sequence obtained from an iteration method is convergent to the fixed point of the mapping used, it can be shown that the new sequence obtained using the approximation operator for this iteration method is also convergent to the fixed point of the approximation operator. In such a case, the questions of how close the fixed points of both mappings are to each other and how to calculate this distance bring up the concept of data dependency. 3. At the same time, the Δ-convergence and strong convergence theorems for generalized (α,β)-nonexpansive type 1 mappings were proved in line with the following information. a. In 2021, two more general classes of nonexpansive mappings "generalized (α,β)-nonexpansive type 1 and type 2 mappings'' were introduced by Akutsah and Narain. From these classes of mappings, the class of generalized (α,β)-nonexpansive type 1 mappings include many mapping classes such as the generalized α-nonexpansive, mean nonexpansive and mappings satisfying the ( ) condition. (i) Let C be a nonempty subset of a metric space (X,d). A mapping T:C→C is said to be generalized (α,β)-nonexpansive type 1 if there exist α,β,λ∈[0,1) with α≤β and α+β<1 such that for all x,y∈C, λd(x,Tx)≤d(x,y)⇒ d(Tx,Ty)≤αd(Tx,y)+βd(Ty,x)+(1-(α+β))d(x,y) (ii) Let C be a nonempty subset of a metric space (X,d). A mapping T:C→C is said to be generalized (α,β)-nonexpansive type 2 if there exist α,β,λ∈[0,1) with α+β<1 such that for all x,y∈C, λd(x,Tx)≤d(x,y)⇒d(Tx,Ty)≤max{P(x,y),Q(x,y)}, where P(x,y)= αd(Tx,y)+βd(Ty,x)+(1-(α+β))d(x,y) and Q(x,y)= αd(Tx,x)+βd(Ty,y)+(1-(α+β))d(x,y). b. Let (X,d) be a metric space, x∈X and {x_n} be a sequence in X. (i) A sequence {x_n} is called a convergent sequence (to x) if, for every ε>0, there exists n_0=n_0 (ε)∈N such that 𝑑(x_n, x)<ε, for all n≥n_0. We write x_n→ x (n→∞) or lim_(n→∞) x_n=x. (ii) A sequence {x_n} in X is said to Δ-converge to a point x∈X if x is the unique asymptotic center of every subsequence {u_n} of {x_n}. In this case, we write Δ-〖lim〗_(n→∞) x_n=x and call as Δ-limit of {x_n}. 4. Finally, an example of generalized (α,β)-nonexpansive type 1 mappings was given as follows: Let X=R with the usual metric and C=[0,∞). Define a mapping T:C→C by Tx={■(0 if&x∈[0,6/5),@5x/12 if&x∈[6/5,∞).)┤ It was shown that the mapping T is a generalized (5/12,6/12)-nonexpansive type 1 mapping with λ=1/3 in addition, the convergence speeds of other iteration methods in the literature to the fixed point were compared with the KF-iteration method using the MATLAB program. By the researchers, the generalized (α,β)-nonexpansive type 2 mappings in hyperbolic metric spaces can be studied using similar approaches in this thesis and some numerical examples for this class of mappings in hyperbolic metric spaces can also be investigated.