This paper aims to study extensively some results concerning continuous dependence for implicit Kirk-Mann and implicit Kirk-Ishikawa iterations. In order to equipoise the formation of these algorithms, we introduce a general hyperbolic space which is no doubt a free associate of some known hyperbolic spaces. The present results are extension of other results and they can be used in many applications.

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 and a convexity mapping which satisfy(W1),(W2),(W3),(W4), for all and .

Due to the rich geometric properties of this space, a large amount of results have been published on hyperbolic spaces such as [24]. 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 be a metric space. A mapping is called a generalized convex structure on if for each and holds for and . The metric space together with a generalized convex structure is called a generalized convex metric space.
By letting and , we retrieve the convex metric space in [5, 6], respectively.
We now give the following definition.

Definition 2. Let be a metric space and . A general hyperbolic space is a metric space associated with the mapping and it satisfies the following: (GW1),(GW2), = ,(GW3) = , ,(GW4), ,where , for each and , .
It is easily seen that Definition 2 is hyperbolic space when .
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 be endowed with the metric and , for ; then, metric on associated with 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 [811]. 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 ) of is used to access the same source as in such a way that for all and .
We shall employ the class of quasi-contractive operator: in [13] to prove the following lemma.

Lemma 3. Let be a metric space and let be a map satisfying (2). Then, for all and for all and .

Proof. Let be an operator satisfying (2); we claim that also satisfies (2).
Then, for each and Thus, satisfies (3).
The converse of Lemma 3 is not true for . Hence, condition (3) is more general than (2).

Lemma 4 (see [14]). Let be a nonnegative sequence for which there exists such that, for all , one has the following inequality: where , for all , , and for . Then,

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 as proved in [12].

Theorem 5. Let be a closed subset of a general hyperbolic space and let be maps satisfying (3), where is an approximate operator of . Let be two iterative sequences associated with , respectively, to given as follows: for where , are sequences in , for , with .
If , and , then

Proof. Let , , and . By using (GW1)–(GW4), (7), (8), and (3), we get which implies This further implies Let ; then Hence, we have Using (14) and the fact that then (12) becomes By letting , , and in (15).
Thus, by Lemma 4, inequality (15) becomes for Therefore,

Theorem 6. Let and be two maps satisfying (3), where is an approximate operator of . Let be two implicit Kirk-Ishikawa iterative sequences associated with , respectively, to given as follows: for where , are sequences in , for ; and are fixed integers such that with . Assume that , , and ; then

Proof. Let . By taking of (19) and of (20) using conditions (GW1)–(GW4) and (3), we obtain Similarly, of (19) and of (20) give By combining (22) and (23), we have This is further reduced to Using the ansatz prescribed in (14), we get Using the condition of Lemma 4, we conclude that This following example is adopted from [14].

Example 7. Let be given by with the unique fixed point being . Then, is quasi-contractive operator.
Also, consider the map , with the unique fixed point .
Take to be the distance between the two maps as follows: Let be the initial datum, , and for , . Note that for .
With the aid of MATLAB program, the computational results for the iterations (7) and (19) of operator are presented in Table 1 with stopping criterion .
In Table 1, both iterations (7) and (19) converge to the same fixed point . This implies that, for each of the iterations, the distance between the fixed point of and the fixed point of is . In fact, this result can also be verified without computing the operator by using Theorem 5 or Theorem 6 for any choice of . On the other hand, the result will also be valid if we choose sufficiently close to .

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 as , which is quite remarkable. Also observe there is a tie-in between Theorems 5 and 6 in the following order: Thus, for any case of , we have In Example 7 above, is chosen, but for higher , it is suitable to choose .


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.