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 (X,d) and a convexity mapping W:X2×[0,1]→X which satisfy

d(z,W(x,y,λ))≤(1-λ)d(z,x)+λd(z,y),

d(W(x,y,λ1),W(x,y,λ2))=λ1-λ2d(x,y),

W(x,y,λ)=W(x,y,1-λ),

d(W(x,z,λ),W(y,w,λ))≤(1-λ)d(x,y)+λd(z,w),

for all x,y,z,w∈X and λ,λ1,λ2∈[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:Xk×[0,1]k→X is called a generalized convex structure on X if for each xi∈X and λi∈[0,1](1)dq,Wx1,x2,…,xk;λ1,λ2,…,λk≤∑i=1kλidq,xiholds for q∈X and ∑i=1kλi=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:Xk×[0,1]k→X. A general hyperbolic space is a metric space (X,d) associated with the mapping W and it satisfies the following:

where [0,1]λk=λ1,λ2,…,λk, for each λi∈[0,1] and xi,yi,y∈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 Xk=Rk be endowed with the metric d(x_,y_)=∑ikxi-yi/(1+|xi-yi|) and W(x1,x2,…,xk;λ1,λ2,…,λk)=∑λixi, for x_,y_∈Rk; then, metric d on Rk 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)≤η for all x∈X and η>0.

We shall employ the class of quasi-contractive operator: (2)dTx,Ty≤adx,y+ϵdx,Txfor x,y∈X,ϵ≥0,a∈0,1in [13] to prove the following lemma.

Lemma 3.

Let (X,d) be a metric space and let T:X→X be a map satisfying (2). Then, for all k∈N and ϵ≥0(3)dTkx,Tky≤∑i=1kkiak-iϵidx,Tix+akdx,y,for all x,y∈X and a∈(0,1).

Proof.

Let T be an operator satisfying (2); we claim that Tkx also satisfies (2).

Then, (4)dTkx,Tky≤ϵdx,Tkx+adTk-1x,Tk-1y≤ϵdx,Tkx+aϵdx,Tk-1x+a2dTk-2x,Tk-2y≤⋯≤∑i=1kkiak-iϵidx,Tix+akdx,yfor each ak∈(0,1) and ϵi≥0. Thus, Tkx satisfies (3).

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

Lemma 4 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

Let ann=0∞ be a nonnegative sequence for which there exists n0∈N such that, for all n≥n0, one has the following inequality: (5)an+1≤1-rnan+rntn,where rn∈(0,1), for all n∈N, ∑n=1∞rn=∞, and tn≥0 for n∈N. Then, (6)0≤limn→∞supan≤limn→∞suptn.

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∈FT 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→K be maps satisfying (3), where S is an approximate operator of T. Let xn,un⊂K be two iterative sequences associated with T, respectively, to S given as follows: for x0,u0∈X(7)xn=Wxn-1,Txn,T2xn,…,Tkxn;αn,0,αn,1,αn,2,…,αn,k;∑i=0kαn,i=1,n≥1,(8)un=Wun-1,Sun,S2un,…,Skun;αn,0,αn,1,αn,2,…,αn,k;∑i=0kαn,i=1,n≥1,where αn,i, βn,i are sequences in [0,1], for i=0,1,2,…,k, k∈N with ∑(1-αn,0)=∞.

If p∈FT, q∈FS and η>0, then (9)dp,q≤η1-a2.

Proof.

Let x0,u0∈X, p∈FT, and q∈FS. By using (GW1)–(GW4), (7), (8), and (3), we get (10)dxn,un=dWxn-1,Txn,T2xn,…,Tkxn;αn,0,αn,1,αn,2,…,αn,k,Wun-1,Sun,S2un,…,Skun;αn,0,αn,1,αn,2,…,αn,k≤αn,0dxn-1,un-1+∑i=1kαn,idTixn,Sixn+dSixn,Siun≤αn,0dxn-1,un-1+∑i=1kαn,iη+∑i=1kkiak-iϵidxn,Sixn+akdxn,unwhich implies (11)dxn,un≤αn,01-∑i=1kαn,iaidxn-1,un-1+∑i=1kαn,i1-∑i=1kαn,iaiη+∑i=1kkiak-iϵidxn,SixnThis further implies (12)dxn,un≤αn,01-1-αn,0adxn-1,un-1+1-αn,01-1-αn,0aη+∑i=1kkiak-iϵidxn,Sixn.Let Qn=αn,0/(1-1-αn,0a); then (13)1-Qn=1-αn,0+1-αn,0a1-1-αn,0a≥1-αn,0+1-αn,0a.Hence, we have (14)Qn≤αn,0+1-αn,0a=1-1-a1-αn,0.Using (14) and the fact that 1-a≤1-(1-αn,0)a then (12) becomes (15)dxn,un≤1-1-a1-αn,0dxn-1,un-1+1-a1-αn,01-a2η+∑i=1kkiak-iϵidxn,Sixn.By letting an=d(xn,un), rn=(1-a)(1-αn,0), and tn=1/1-a2η+∑i=1kkiak-iϵidxn,Sixn in (15).

Thus, by Lemma 4, inequality (15) becomes (16)limn→∞dxn,un≤11-a2η+∑i=1kkiak-iϵilimn→∞dxn,Sixn.for (17)0≤dxn,Sixn≤dxn,p+Sip,Sixn≤1+aidxn,p⟶0,asn⟶∞Therefore, (18)dp,q≤η1-a2.

Theorem 6.

Let K⊂(X,d,W) and T,S:K→K be two maps satisfying (3), where S is an approximate operator of T. Let xn,un be two implicit Kirk-Ishikawa iterative sequences associated with T, respectively, to S given as follows: for x0,u0∈X(19)xn=Wyn-1,Txn,T2xn,…,Tkxn;αn,0,αn,1,αn,2,…,αn,k;∑i=0kαn,i=1,yn-1=Wxn-1,Tyn-1,T2yn-1,…,Tsyn-1;βn,0,βn,1,βn,2,…,βn,s;∑i=0sβn,i=1,n≥1,(20)un=Wvn-1,Sun,S2un,…,Skun;αn,0,αn,1,αn,2,…,αn,k;∑i=0kαn,i=1,vn-1=Wun-1,Svn-1,S2vn-1,…,Ssvn-1;βn,0,βn,1,βn,2,…,βn,s;∑i=0sβn,i=1,n≥1,where αn,ik, βn,is are sequences in [0,1], for ik=01k;is=0(1)s; k and s are fixed integers such that k≥s with ∑(1-αn,0)=∞. Assume that p∈FT, q∈FS, and η>0; then (21)dp,q≤2η1-a2.

Proof.

Let x0,u0∈X. By taking xn of (19) and un of (20) using conditions (GW1)–(GW4) and (3), we obtain (22)dxn,un≤αn,01-∑ik=1kαn,ikaikdyn-1,vn-1+∑ik=1kαn,ik1-∑ik=1kαn,ikaikη+∑ik=1kkikak-ikϵikdxn,Sikxn.Similarly, yn-1 of (19) and vn-1 of (20) give (23)dyn-1,vn-1≤βn,01-∑is=1sβn,isaisdxn-1,un-1+∑is=1sβn,is1-∑is=1sβn,isaisη+∑is=1ssisas-isϵisdxn,Sisxn.By combining (22) and (23), we have (24)dxn,un≤αn,01-∑ik=1kαn,ikaikβn,01-∑is=1sβn,isaisdxn-1,un-1+∑is=1sβn,is1-∑is=1sβn,isais×η+∑is=1ssisas-isϵisdxn,Sisxn+∑ik=1kαn,ik1-∑ik=1kαn,ikaikη+∑ik=1kkikak-ikϵikdxn,Sikxn.This is further reduced to (25)dxn,un≤αn,0βn,01-1-αn,0a1-1-βn,0adxn-1,un-1+αn,01-βn,01-1-αn,0a1-1-βn,0a×η+∑is=1ssisas-isϵisdxn,Sisxn+1-αn,01-1-αn,0aη+∑ik=1kkikak-ikϵikdxn,Sikxn.Using the ansatz prescribed in (14), we get (26)dxn,un≤1-1-a1-αn,0dxn-1,un-1+1-a1-αn,01-a2×2η+∑is=1ssisas-isϵisdxn,Sisxn+∑ik=1kkikak-ikϵikdxn,Sikxn.Using the condition of Lemma 4, we conclude that (27)limn→∞dxn,un⟶dp,q≤2η1-a2.This following example is adopted from [14].

Example 7.

Let T:R→R be given by (28)Tx=0ifx∈-∞,2-0.5ifx∈2,∞with the unique fixed point being 0. Then, T is quasi-contractive operator.

Also, consider the map S:R→R, (29)Sx=1if x∈-∞,2-1.5if x∈2,∞with the unique fixed point 1.

Take η to be the distance between the two maps as follows: (30)dSx,Tx≤1,∀x∈R.Let x0=u0=0 be the initial datum, αn,0=βn,0=1-2/n, and αn,i=βn,i=1/n for n≥5, i=1,2. Note that αn,i=βn,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 of the 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∈(0,1). On the other hand, the result will also be valid if we choose T sufficiently close to S.

Number of iterations

Iteration (7)

Iteration (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

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)→0 as η→0, which is quite remarkable. Also observe there is a tie-in between Theorems 5 and 6 in the following order: (31)dp,q≤η1-a2≤2η1-a2.Thus, for any case of k=1,2, we have (32)supdp,q:dp,q≤kη1-a2,for each k.In Example 7 above, η=1 is chosen, but for higher k, it is suitable to choose η=1/k.

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.

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