# Continuous Dependence for Two Implicit Kirk-Type Algorithms in General Hyperbolic Spaces.

1. Introduction

In [1], Kohlenbach defined hyperbolic space in his paper titled "Some Logical Metatheorems with Applications in Functional Analysis, Transactions of the American Mathematical Society, Vol. 357, 89-128." He combined a metric space (X, d) and a convexity mapping W: [X.sup.2] x [0,1] [right arrow] X which satisfy

(W1) d(z, W(x,y,[lambda])) [less than or equal to] (1 - [lambda])d(z, x) + [lambda]d(z, y),

(W2) d(W(x,y,[[lambda].sub.1]),W(x,y,[[lambda].sub.2])) = [absolute value of [[lambda].sub.1] - [[lambda].sub.2]]d(x,y),

(W3) W(x, y, [lambda]) = W(x, y, 1 - [lambda]),

(W4) d(W(x,z,[lambda]),W(y,w,[lambda])) [less than or equal to] (1-[lambda])d(x,y) + [lambda]d(z,w),

for all x, y,z,w [member of] X and X, [[lambda].sub.1],[[lambda].sub.2] [member of] [0,1].

Due to the rich geometric properties of this space, a large amount of results have been published on hyperbolic spaces such as [2-4]. It is observed that conditions (W1)-(W4) can only be fulfilled for two or three distinct points. So, to balance up the proportions of the space against the iterative processes in question, we introduce a general notion of the hyperbolic space. Firstly, we define the following.

Definition 1. Let (X, d) be a metric space. A mapping W: [X.sup.k] x [[0,1].sup.k] [right arrow] X is called a generalized convex structure on X if for each [x.sub.i] [member of] X and [[lambda].sub.i] [member of] [0,1]

[mathematical expression not reproducible] (1)

holds for q [member of] X and [[summation].sup.k.sub.i=1] [[lambda].sub.i] = 1. The metric space (X,d) together with a generalized convex structure W is called a generalized convex metric space.

By letting k = 3 and k = 2, we retrieve the convex metric space in [5,6], respectively.

We now give the following definition.

Definition 2. Let (X d) be a metric space and W : [X.sup.k] x [[0,1].sup.k] [right arrow] X. A general hyperbolic space is a metric space (X, d) associated with the mapping W and it satisfies the following:

[mathematical expression not reproducible]

where [[0,1].sup.k.sub.[lambda]] = [[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.k], for each [[lambda].sub.1] [member of] [0,1] and [x.sub.i],[y.sub.i],y [member of] X, i = 1(1)k.

It is easily seen that Definition 2 is hyperbolic space when k = 2.

We note here that every general hyperbolic space is a generalized convex metric space, but the converse in some cases is not necessarily true.

For example, let [X.sup.k] = [R.sup.k] be endowed with the metric [mathematical expression not reproducible]; then, metric d on [R.sup.k] associated with W is a generalized convex metric space but it does not satisfy all the conditions (GW1)-(GW4).

Two hybrid Kirk-type schemes, namely, Kirk-Mann and Kirk-Ishikawa iterations, were first introduced in normed linear space as appeared in [7]. Remarkable results have been investigated to date for more cases of Kirk-type schemes; see [8-11]. Recently in [12], the implicit Kirk-type schemes were introduced in Banach space for a contractive-type operator and it was also remarkable.

However, there are few or no emphases on the data dependence of the Kirk-type schemes. Hence, this paper aims to study closely the continuous contingency of two Kirk-type schemes in [12], namely, implicit Kirk-Mann and implicit Kirk-Ishikawa iterations in a general hyperbolic space. To do this, a certain approximate operator (say S) of T is used to access the same source as T in such a way that d(Tx, Sx) [less than or equal to] q for all x [member of] X and [eta] >0.

We shall employ the class of quasi-contractive operator:

d (Tx, Ty) [less than or equal to] ad (x, y) + [epsilon]d (x, Tx)

for x,y [member of] X, [epsilon] [greater than or equal to] 0, a [member of] (0,1) (2)

in [13] to prove the following lemma.

Lemma 3. Let (X, d) be a metric space and let T : X [right arrow] X be a map satisfying (2). Then, for all k [member of] N and [epsilon] [greater than or equal to] 0

[mathematical expression not reproducible], (3)

for all x, y [member of] X and a [member of] (0,1).

Proof. Let T be an operator satisfying (2); we claim that [T.sup.k]x also satisfies (2).

Then,

[mathematical expression not reproducible] (4)

for each [a.sup.k] [member of](0,1) and [[epsilon].sup.i] [greater than or equal to] 0. Thus, [T.sup.k]x satisfies (3).

The converse of Lemma 3 is not true for k > 1. Hence, condition (3) is more general than (2).

Lemma 4 (see [14]). Let {[a.sub.n]}.sup.[infinity].sub.n=0] be a nonnegative sequence for which there exists [n.sub.0] [member of] N such that, for all n [greater than or equal to] [n.sub.0], one has the following inequality:

[a.sub.n+1] [less than or equal to] (1-[r.sub.n])[a.sub.n] + [r.sub.n][t.sub.n], (5)

where [r.sub.n] [member of] (0,1),for all n [member of] N, [[summation].sup.[infinity].sub.n=1][r.sub.n] = [infinity], and [t.sub.n] [greater than or equal to] 0 for n [member of] N. Then,

[mathematical expression not reproducible]. (6)

2. Main Results

We present the results for implicit Kirk-Mann and implicit Kirk-Ishikawa iterations using condition (3) and noting that both iterations converge strongly to a fixed point p [member of] [F.sub.T] as proved in [12].

Theorem 5. Let K be a closed subset of a general hyperbolic space (X, d, W) and let T,S : K [right arrow] K be maps satisfying (3), where S is an approximate operator of T. Let {[x.sub.n]}, {[u.sub.n]} [subset] K be two iterative sequences associated with T, respectively, to S given as follows: for [x.sub.0],[u.sub.0] [member of] X

[mathematical expression not reproducible], (7)

[mathematical expression not reproducible]. (8)

where [[alpha].sub.n,i], [[beta].sub.n,i] are sequences in [0,1], for i = 0,1,2, ..., k, k [member of] N with [summation](1 - [[alpha].sub.n,0]) = [infinity].

If p [member of] [F.sub.T], q [member of] [F.sub.s] and [eta] > 0, then

d(p,q) [less than or equal to] [eta]/[(1 - a).sup.2]. (9)

Proof. Let [x.sub.0],[u.sub.0] [member of] X, p [member of] [F.sub.T], and q [member of] [F.sub.s]. By using (GW1)-(GW4), (7), (8), and (3), we get

[mathematical expression not reproducible] (10)

which implies

[mathematical expression not reproducible] (11)

This further implies

[mathematical expression not reproducible]. (12)

[mathematical expression not reproducible]. (13)

Hence, we have

[Q.sub.n] [less than or equal to] [[alpha].sub.n,0] + (1- [[alpha].sub.n,0]) a = 1 - (1-a)(1 - [[alpha].sub.n,0]). (14)

Using (14) and the fact that 1- a [less than or equal to] 1-(1- [[alpha].sub.n,0])a then (12) becomes

[mathematical expression not reproducible]. (15)

By letting [mathematical expression not reproducible].

Thus, by Lemma 4, inequality (15) becomes

[mathematical expression not reproducible]. (16)

for

[mathematical expression not reproducible] (17)

Therefore,

[mathematical expression not reproducible]. (18)

Theorem 6. Let K [subset] (X, d, W) and T,S : K [right arrow] K be two maps satisfying (3), where S is an approximate operator of T. Let [[x.sub.n]], {[u.sub.n]} be two implicit Kirk-Ishikawa iterative sequences associated with T, respectively, to S given as follows: for [x.sub.0], [u.sub.0] [member of] X

[mathematical expression not reproducible], (19)

[mathematical expression not reproducible], (20)

where [mathematical expression not reproducible] are sequences in [0,1], for [i.sub.k] = 0(1)k; [i.sub.s] = 0(1)s; k and s are fixed integers such that k [greater than or equal to] s with [summation](1 - [[alpha].sub.n,0]) = [infinity]. Assume that p [member of] [F.sub.T], q [member of] [F.sub.s], and [eta] > 0; then

d(p,q) [less than or equal to] 2[eta]/[(1 - a).sup.2]. (21)

Proof. Let [x.sub.0],[u.sub.0] [member of] X. Bytaking [x.sub.n] of (19) and [u.sub.n] of (20) using conditions (GW1)-(GW4) and (3), we obtain

[mathematical expression not reproducible]. (22)

Similarly, [y.sub.n-1] of (19) and [v.sub.n-1] of (20) give

[mathematical expression not reproducible]. (23)

By combining (22) and (23), we have

[mathematical expression not reproducible]. (24)

This is further reduced to

[mathematical expression not reproducible]. (25)

Using the ansatz prescribed in (14), we get

[mathematical expression not reproducible]. (26)

Using the condition of Lemma 4, we conclude that

[mathematical expression not reproducible]. (27)

This following example is adopted from [14].

Example 7. Let T : R [right arrow] R be given by

[mathematical expression not reproducible] (28)

with the unique fixed point being 0. Then, T is quasi-contractive operator.

Also, consider the map S : R [right arrow] R,

[mathematical expression not reproducible] (29)

with the unique fixed point 1.

Take [eta] to be the distance between the two maps as follows:

d(Sx,Tx) [less than or equal to] 1, [for all]x [member of] R. (30)

Let [x.sub.0] = [u.sub.0] = 0 be the initial datum, [[alpha].sub.n,0] = [[beta].sub.n,0] = 1-2/[square root of (n)], and [[alpha].sub.n,i] = [[beta].sub.n,i] = 1/[square root of (n)] for n [greater than or equal to] 5, i = 1,2. Note that [[alpha].sub.n,i] = [[beta].sub.n,i] = 0 for n= 1(1)4.

With the aid of MATLAB program, the computational results for the iterations (7) and (19) of operator S are presented in Table 1 with stopping criterion 1e - 8.

In Table 1, both iterations (7) and (19) converge to the same fixed point 1. This implies that, for each ofthe iterations, the distance between the fixed point of S and the fixed point of T is 1. In fact, this result can also be verified without computing the operator S by using Theorem 5 or Theorem 6 for any choice of a [member of] (0,1). On the other hand, the result will also be valid if we choose T sufficiently close to S.

3. Concluding Remarks

These results exhibit sufficient conditions under which approximate fixed points depend continuously on parameters. In fact, the above two results show that d(p, q) [right arrow] 0 as [eta] [right arrow] 0, which is quite remarkable. Also observe there is a tie-in between Theorems 5 and 6 in the following order:

d(p,q) [less than or equal to] [eta]/[(1 - a).sup.2] [less than or equal to] 2[eta]/[(1 - a).sup.2]. (31)

Thus, for any case of k = 1,2, we have

sup {d(p,q) : d (p, q) [less than or equal to] k[eta]/[(1 - a).sup.2]}, for each k. (32)

In Example 7 above, [eta] = 1 is chosen, but for higher k, it is suitable to choose [eta] = 1/k.

https://doi.org/10.1155/2017/6570367

Disclosure

The authors agreed to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

Conflicts of Interest

Authors hereby declare that there are no conflicts of interest.

References

[1] U. Kohlenbach, "Some logical metatheorems with applications in functional analysis," Transactions of the American Mathematical Society, vol. 357, no. 1, pp. 89-128, 2005.

[2] A. R. Khan, H. Fukhar-ud-din, and M. A. A. Khan, "An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces," Fixed Point Theory and Applications, vol. 2012, article 54, 12 pages, 2012.

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[4] L. Zhang and X. Wang, "A-convergence for common fixed points of two asymptotically nonexpansive nonself mappings in hyperbolic spaces," International Journal of Mathematical Analysis, vol. 9, no. 5-8, pp. 385-393, 2015.

[5] A. Rafiq and S. Zafar, "On the convergence of implicit Ishikawa iterations with error to a common fixed point of two mappings in convex metric space," General Mathematics, vol. 14, no. 2, pp. 95-108, 2006.

[6] W. Takahashi, "A convexity in metric space and nonexpansive mappings I," Kodai Mathematical Seminar Reports, vol. 22, pp. 142-149, 1970.

[7] M. O. Olatinwo, "Some stability results for two hybrid fixed point iterative algorithms in normed linear space," Matematichki Vesnik, vol. 61, no. 4, pp. 247-256, 2009.

[8] H. Akewe, G. A. Okeke, and A. F. Olayiwola, "Strong convergence and stability of Kirk-multistep-type iterative schemes for contractive-type operators," Fixed Point Theory and Applications, vol. 2014, no. 1, article 45, 2014.

[9] R. Chugh and V. Kumar, "Stability of hybrid fixed point iterative algorithms of Kirk-Noor type in normed linear space for self and nonself operators," International Journal of Contemporary Mathematical Sciences, vol. 7, no. 24, pp. 1165-1184, 2012.

[10] F. Gursoy, V. Karakaya, and B. E. Rhoades, "Data dependence results of new multi-step and S-iterative schemes for contractive-like operators," Fixed Point Theory and Applications, vol. 2013, article 76, 2013:76, 12 pages, 2013.

[11] N. Hussain, R. Chugh, V. Kumar, and A. Rafiq, "On the rate of convergence of Kirk-type iterative schemes," Journal of Applied Mathematics, vol. 2012, Article ID 526503, 22 pages, 2012.

[12] O. T. Wahab and K. Rauf, "On faster implicit hybrid Kirk-multistep schemes for contractive-type operators," International Journal of Analysis, vol. 2016, Article ID 3791506, 10 pages, 2016.

[13] V. Berinde, "A convergence theorem for some mean value fixed point iterations procedures," Demonstratio Mathematica, vol. 38, p. 17784, 2005.

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K. Rauf, (1) O. T. Wahab, (2) and S. M. Alata (1)

(1) Department of Mathematics, University of Ilorin, Ilorin, Nigeria

(2) Department of Statistics and Mathematical Sciences, Kwara State University, Malete, Nigeria

Correspondence should be addressed to K. Rauf; krauf@unilorin.edu.ng

Received 17 January 2017; Accepted 4 May 2017; Published 21 June 2017

Academic Editor: Hichem Ben-El-Mechaiekh
```Table 1

Number of    Iteration    Iteration
iterations      (7)          (19)

5            0.8944272    0.9888544
6            0.9806270    0.9996247
7            0.9952716    0.9999776
8            0.9986151    0.9999981
9            0.9995384    0.9999998
10           0.9998303    1.0000000
11           0.9999326    1.0000000
...             ...          ...
21           0.9999999    1.0000000
22           0.9999999    1.0000000
23           1.0000000    1.0000000
```
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