# Fixed-Point Approximations of Generalized Nonexpansive Mappings via Generalized M-Iteration Process in Hyperbolic Spaces

Authors: Preeyalak Chuadchawna; Ali Farajzadeh; Anchalee Kaewcharoen
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

In this paper, we propose the generalized M-iteration process for approximating the fixed points from Banach spaces to hyperbolic spaces. Using our new iteration process, we proveΔ-convergence and strong convergence theorems for the class of mappings satisfying the condition Cλ and the condition E which is the generalization of Suzuki generalized nonexpansive mappings in the setting of hyperbolic spaces. Moreover, a numerical example is given to present the capability of our iteration process and the solution of the integral equation is also illustrated using our main result.


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## Body

## 1. Introduction

The concept of nonexpansive mappings can be defined in many general setting of metric spaces. Suzuki [1] introduced the concept of generalized nonexpansive single-valued mappings which are called Suzuki generalized nonexpansive mappings or the condition C and received some fixed-point results and convergence results for such mappings in Banach spaces. Also, the new conditions for single-valued mappings are defined by García-Falset et al. [2], called the condition Cλ and the condition E which are weaker than a nonexpansive mapping and stronger than a quasi-nonexpansive mapping. Moreover, the class of mappings satisfying the condition Cλ and the condition E is larger than the class of mappings satisfying the condition C.LetT be a self-mapping on a nonempty subset D of a Banach space X and ωn and σn are real sequences in 0,1 for all n≥0. Mann [3] iteration process was determined by the following method:(1)x0∈D,xn+1=1−ωnxn+ωnTxn.The following is called the S-iteration process defined by Agarwal et al. [4]:(2)x0∈D,yn=1−σnxn+σnTxn,xn+1=1−ωnTxn+ωnTyn.Next, Gursoy et al. [5] introduced the Picard-S-iteration process. By providing an example, they presented that Picard-S-iteration process converges faster than all Mann, Picard, Noor, Ishikawa, CR, SP, S, S∗, Abbas, and Normal-S involving Two-step Mann iteration processes. The Picard-S-iteration process is defined as(3)x0∈D,zn=1−σnxn+σnTxn,yn=1−ωnTxn+ωnTzn,xn+1=Tyn.Recently in 2018, Ullah et al. [6] introduced a new iterative process called the M-iteration process defined as follows:(4)x0∈D,zn=1−ωnxn+ωnTxn,yn=Tzn,xn+1=Tyn.They proved that M-iteration process can be used to approximate the fixed point of Suzuki generalized nonexpansive mappings and obtain weak convergence and strong convergence results on Banach spaces. They also presented that the M-iteration process converges faster than Picard-S-iteration and S-iteration processes by providing an instance.Motivated by above, we introduce a generalized M-iteration process and use our iterative scheme for proving someΔ-convergence and strong convergence theorems for mappings satisfying the condition Cλ and the condition E which is the generalization of Suzuki generalized nonexpansive mappings (the condition (C)) in the setting of hyperbolic spaces. Numerically, we give an example of a mapping satisfying the condition Cλ and the condition E, which does not satisfy the condition C. Finally, we compare the speed of convergence of our generalized M-iteration process with M-iteration process of Ullah et al. [6].


## 2. Preliminaries

For this research, we discuss on the setting of hyperbolic spaces which was introduced by Kohlenbach [7], containing normed linear spaces and convex subsets and Hadamard manifolds [8], CAT(0) spaces in the sense of Gromov [9], and the Hilbert ball equipped with the hyperbolic metric [8].A hyperbolic space is a tripleX,d,W where X,d is a metric space and W:X2×0,1⟶X such that(W1)dw,Wu,v,ω≤1−ωdw,u+ωdw,v,(W2)dWu,v,ω,Wu,v,σ=ω−σdu,v,(W3)Wu,v,ω=Wv,u,1−ω,(W4)dWu,z,ω,Wv,w,ω≤1−ωdu,v+ωdz,w,for all u,v,w,z∈X and ω,σ∈0,1.First we recall some definitions, lemmas, and propositions that will be used in the next part.A hyperbolic spaceX,d,W is called uniformly convex [10] if for all u,v,z∈X, r>0, and ε∈0,2 there exists δ∈0,1 such that dv,u≤r, dz,u≤r, and dv,z≥εr. Then, we have(5)dWv,z,12,u≤1−δr.A mappingη:0,∞×0,2⟶0,1 which provides δ=ηr,ε for a given r>0 and ε∈0,2 is well known as a modulus of uniform convexity of X. We call η as a monotone if it decreases with r (for a fixed ε), i.e., for any given ε>0 and for any r2≥r1>0, we have ηr2,ε≤ηr1,ε.A nonempty subsetD of a hyperbolic space X is said to be convex if Wu,v,γ∈D for any u,v∈D and γ∈0,1.Ifu,v∈X and γ∈0,1, then we use the notion 1−γu⊕γv for Wu,v,γ. In [10], it is remarked that any normed space X,⋅ is a hyperbolic space, with 1−γu⊕γv:=1−γu+γv. Hence, the class of uniformly convex hyperbolic spaces is a natural generalization of uniformly convex Banach spaces.LetD be a nonempty subset of metric space X. If Tq=q, then q is said to be a fixed point of a mapping T. The set of all fixed points of T is denoted by FT.Definition 1.
(see [11]). A mapping T:D⟶D is said to be(i)
Nonexpansive ifdTu,Tv≤du,v for all u,v∈D;(ii)
Quasi-nonexpansive ifFT≠∅ and dTu,Tq≤du,q for all u∈D and q∈FT.
Suzuki [1] defined the notion of Suzuki generalized nonexpansive mappings which is called the condition C. Such mappings are weaker than nonexpansive mappings and stronger than quasi-nonexpansive mappings.Definition 2.
(see [1]). A mapping T:D⟶D is called to satisfy the condition C if(6)12du,Tu≤du,vimpliesdTu,Tv≤du,v,for all u,v∈D.
Falset et al. [2] introduced the definition of generalized nonexpansive mappings as follows.Definition 3.
(see [2]). A mapping T:D⟶D is called to satisfy the condition Eμ provided that(7)du,Tv≤μdu,Tu+du,v,for allu,v∈D.
We say thatT satisfies the condition E whenever T satisfies the condition Eμ for some μ≥1.Proposition 1.
(see [12]). Let T:D⟶D be a mapping which satisfies the condition C. Then, T satisfies the condition E for some μ=3.There is an example presented that there exists a mapping which satisfies the conditionE, but it fails to satisfy the condition C. Then, the mappings satisfying the condition E are more generalized than mappings satisfying the condition C.Example 1.
(see [13]). Let X=C0,1, and under the supremum norm, consider a nonempty subset D of X defined by(8)D=f∈C0,1:0=f0≤f1=1.
For anyg∈D, associate a function Fg:D⟶D defined by Fght=g°ht=ght. It is easy to verify that Fg satisfies the condition E1 but does not satisfy the condition C.Definition 4.
(see [2]). Let T:D⟶D be a mapping and λ∈0,1. Then, T is said to satisfy the condition Cλ if for all u,v∈D(9)λdu,Tu≤du,vimpliesdTu,Tv≤du,v.
Of course, it follows that the conditionC is the special case for λ=1/2 in the condition Cλ.
On the contrary, the notion of a mapping satisfying the conditionI was introduced by Senter and Dotson [14].Definition 5.
LetT:D⟶D said to satisfy the condition I, if there exists a nondecreasing function f:0,∞⟶0,∞ satisfying f0=0 and ft>0 for all t∈0,∞, such that(10)dv,Tv≥fdv,FTfor allv∈D.
We need the following definition of convergence in hyperbolic spaces [15] which is called Δ-convergence. It plays an essential role in the principle results. In order to define Δ-convergence, we need the following concepts.
Letxn be a bounded sequence in a hyperbolic space X. We can define a function r⋅,xn:X⟶0,∞ by(11)rx,xn=limsupn⟶∞dx,xn,for allx∈X.
An asymptotic radius of a bounded sequencexn with respect to a nonempty subset D of X is determined and denoted by(12)rDxn=infrx,xn:x∈D.
An asymptotic center of a bounded sequencexn with respect to a nonempty subset D of X is determined and denoted by(13)ACDxn=x∈X:rx,xn≤ry,xn,for ally∈D.
The sequencexn in X is said to Δ-converge to x∈X if x is a unique asymptotic center of wn for every subsequence wn of xn. In this case, we write Δ-limn⟶∞xn=x and call x the Δ-lim of xn.
Next, we review some definitions and lemmas.Lemma 1.
(see [16]). Let X be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity η. Then, every bounded sequence xn in X has a unique asymptotic center with respect to any nonempty closed convex subset D of X.Lemma 2.
(see [17]). Let X be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity η. Let x∈X and αn be a sequence in a,b for some a,b∈0,1. If xn and yn are sequences in X such that limsupn⟶∞dxn,x≤c, limsupn⟶∞dyn,x≤c, and limn⟶∞dWxn,yn,αn,x=c for some c≥0. Then,(14)limn⟶∞dxn,yn=0.Lemma 3.
(see [18]). Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with a monotone modulus of uniform convexity η. Let T:D⟶D be a mapping which satisfies the condition Cλ for some λ∈0,1 and the condition E. Suppose that xn is a bounded sequence in D such that limn⟶∞dxn,Txn=0. Then, T has a fixed point.As a consequence of Theorem 3.2 [18], we receive the following lemma.Lemma 4.
LetD be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with a monotone modulus of uniform convexity η. Let T:D⟶D be a mapping which satisfies the condition Cλ for some λ∈0,1. Then, FT is closed.


## 3. Main Results

For this section, we construct an iteration process in hyperbolic spaces named as “generalized M-iteration process,” as follows:(15)x0∈D,zn=Wxn,Txn,ωn,yn=WTzn,zn,σn,xn+1=WTyn,yn,γn,for all n≥0, ωn, σn, and γn are real sequences in 0,1.Remark that if we takeσn=γn=0, then we obtain that (15) reduces to (4).The following lemmas play crucial role in proving the main theorems of this section.Throughout in this paper, letX,d,W be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity η and D be a nonempty closed convex subset of X.Lemma 5.
LetD be a nonempty closed convex subset of X and T:D⟶D be a mapping which satisfies the condition Cλ for some λ∈0,1 with FT≠∅. For an arbitrary chosen x0∈D, a sequence xn is defined by (15). Then, limn⟶∞dxn,q exists for each q∈FT.Proof.
Assume thatq∈FT and z∈D. By hypothesis, we obtain that T satisfies the condition Cλ for some λ∈0,1. So(16)λdq,Tq=0≤dq,zimplies thatdTq,Tz≤dq,z.
Thus using (15), we obtain that(17)dzn,q=dWxn,Txn,ωn,q≤1−ωndxn,q+ωndTxn,q=1−ωndxn,q+ωndTxn,Tq≤1−ωndxn,q+ωndxn,q≤dxn,q.
Using (15) together with (17), we get(18)dyn,q=dWTzn,zn,σn,q≤1−σndTzn,q+σndzn,q=1−σndTzn,Tq+σndzn,q≤1−σndzn,q+σndzn,q≤dzn,q≤dxn,q.
Again, using (15) and (18), we obtain that(19)dxn+1,q=dWTyn,yn,γn,q≤1−γndTyn,q+γndyn,q=1−γndTyn,Tq+γndyn,q≤1−γndyn,q+γndyn,q≤dyn,q≤dxn,q.
This presents thatdxn,q is bounded below and decreasing. Then, limn⟶∞dxn,q exists for each q∈FT.Lemma 6.
LetD be a nonempty closed convex subset of X and T:D⟶D be a mapping which satisfies the condition Cλ for some λ∈0,1 and the condition E. For an arbitrary chosen x0∈D, a sequence xn is defined by (15). Then, FT≠∅ if and only if xn is bounded and limn⟶∞dxn,Txn=0.Proof.
Assume thatFT≠∅, and let q∈FT. So, it follows from Lemma 5, xn is bounded. Next, we will indicate that limn⟶∞dxn,Txn=0. Since T satisfies the condition Cλ for some λ∈0,1 and q∈FT, for each n∈ℕ we obtain that(20)λdq,Tq=0≤dq,xn.
Therefore,(21)dTq,Txn≤dq,xn.
From Lemma5, we achieve limn⟶∞dxn,q exists for all q∈FT. Assume that limn⟶∞dxn,q=s≥0. If s=0, then(22)dxn,Txn≤dxn,q+dq,Txn≤dxn,q+dTq,Txn≤2dxn,q.
Letting limit asn⟶∞ on both sides of the above estimate, we have(23)limn⟶∞dxn,Txn=0.
Ifs>0, then(24)dzn,q=dWxn,Txn,ωn,q≤1−ωndxn,q+ωndTxn,q=1−ωndxn,q+ωndTxn,Tq≤1−ωndxn,q+ωndxn,q≤dxn,q.
Taking lim sup asn⟶∞ on the above inequality,(25)limsupn⟶∞dzn,q≤limsupn⟶∞dxn,q=s.
It follows from (18) and (19) that dxn+1,q≤dzn,q. By taking lim inf as n⟶∞ on both sides, we obtain that(26)s≤liminfn⟶∞dxn+1,q≤liminfn⟶∞dzn,q.
The relations (25) and (26) imply that(27)limn⟶∞dzn,q=s.
SincedTxn,q=dTxn,Tq≤dxn,q, we get that(28)limsupn⟶∞dTxn,q≤limsupn⟶∞dxn,q≤s.
It follows from Lemma2 through (27) and (28) that(29)limn⟶∞dxn,Txn=0.
Conversely, suppose thatxn is bounded and limn⟶∞dxn,Txn=0. Then, it follows from Lemma 3 that T has a fixed point, so we have FT which is nonempty.
We now establish theΔ-convergence theorem for generalized nonexpansive mappings in hyperbolic spaces.Theorem 1.
LetD be a nonempty closed convex subset of X and T:D⟶D be a mapping which satisfies the condition Cλ for some λ∈0,1 and the condition E with FT≠∅. Let xn be a sequence defined by (15). Then, xn Δ-converges to a fixed point of T.Proof.
It follows from Lemma6 that xn is a bounded sequence. Thus, xn has a Δ-convergent subsequence. Now, we are going to show that every Δ-convergent subsequence of xn has a unique Δ-limit in FT. Let w and z be Δ-limits of the subsequence wn and zn of xn, respectively. From Lemma 1, we have ACD,wn=w and ACD,zn=z. By Lemma 6, we obtain that limn⟶∞dwn,Twn=0 and limn⟶∞dzn,Tzn=0. Next, we prove that w and z are fixed points of T and reunique should be are unique. Since T satisfies the condition E, there exists μw≥1 such that(30)dwn,Tw≤μwdwn,Twn+dwn,w.
Letting lim sup asn⟶∞ on both sides of the above inequality, we get(31)raTw,wn=limsupn⟶∞dwn,Tw≤limsupn⟶∞μwdwn,Twn+dwn,w≤limsupn⟶∞dwn,w=raw,wn.
The uniqueness of the asymptotic center impliesTw=w. Thus, w is a fixed point of T. Similarly, we also have z as a fixed point of T. Lastly, we show that w=z. Suppose w≠z, and so by the uniqueness of an asymptotic center, we have(32)limsupn⟶∞dxn,w=limsupn⟶∞dwn,w<limsupn⟶∞dwn,z=limsupn⟶∞dxn,z=limsupn⟶∞dzn,z<limsupn⟶∞dzn,w=limsupn⟶∞dxn,w.
This is a contradiction. Thus,w=z. Then, xn Δ-converges to a fixed point of T.
Next, we present some strong convergence theorems.Theorem 2.
LetD be a nonempty closed convex subset of X and T:D⟶D be a mapping which satisfies the condition Cλ for some λ∈0,1 and the condition E with FT≠∅. Let xn be a sequence defined by (15). Then, xn converges strongly to a fixed point of T if and only if liminfn⟶∞dxn,FT=0.Proof.
Assume thatxn converges strongly to q∈FT. So, limn⟶∞dxn,q=0. Because 0≤dxn,FT≤dxn,q, therefore liminfn⟶∞dxn,FT=0.
Conversely, suppose thatliminfn⟶∞dxn,FT=0. It follows from Lemma 5, and we get dxn+1,q≤dxn,q for all q∈FT. Thus, dxn+1,FT≤dxn,FT. Therefore, limn⟶∞dxn,FT exists. From the assumption of our theorem, liminfn⟶∞dxn,FT=0, so we have limn⟶∞dxn,FT=0. Next, we prove that xn is a Cauchy sequence in D. Let for each ε>0. Since limn⟶∞dxn,FT=0, for any given ε>0, there is n0∈ℕ such that(33)dxn,FT<ε2for alln≥n0.
In particular,infdxn0,q:q∈FT<ε/2. Then, there exists q′∈FT such that dxn0,q′<ε/2.
For anym,n≥n0, we get(34)dxn+m,xn≤dxn+m,q′+dq′,xn≤dxn0,q′+dq′,xn0≤ε2+ε2=ε.
This shows thatxn is a Cauchy sequence in D. Since D is a closed subset of a complete hyperbolic space X, D is complete. Then, xn must converge to a point in D. Let limn⟶∞xn=z∈D. Now, we show that z is a fixed point of T. Since T satisfies the condition E, there exists μ≥1 such that(35)dxn,Tz≤μdxn,Txn+dxn,z.
Lettingn⟶∞, it follows from Lemma 6 that dxn,Txn=0. We have dz,Tz=0. Then, z is a fixed point of T. Hence, xn converges strongly to a point in FT.Theorem 3.
LetD be a nonempty closed convex subset of X and T:D⟶D be a mapping which satisfies the condition Cλ for some λ∈0,1 and the condition E with FT≠∅. If a sequence xn is determined by (15) and T satisfies condition I, then xn converges strongly to a fixed point of T.Proof.
From Lemma5, we have limn⟶∞dxn,q exists for each q∈FT and limn⟶∞dxn,q=s for some s≥0. Let q∈FT. If s=0, we obtain the desired result. Assume that s>0. From Lemma 6, we obtain that limn⟶∞dxn,Txn=0. Since T satisfies condition I, we have(36)limn⟶∞fdxn,FT≤limn⟶∞dxn,Txn=0.
Therefore,(37)limn⟶∞fdxn,FT=0.
Fromf:0,∞⟶0,∞ is a nondecreasing mapping satisfying f0=0 and ft>0 for each t∈0,∞, we have limn⟶∞dxn,FT=0. By applying Theorem 2, we obtain the desired result.


## 4. Numerical Example

First, we present an example of the convergence for a mapping satisfying the conditionCλ and the condition E, but it fails to satisfy the condition C. Next, we give an example to compare the speed of convergence of our generalized M-iteration process with M-iteration process of Ullah et al. [6].Example 2.
LetX=ℝ with metric dx,y=x−y and D=0,1. Define W:X2×0,1⟶X by Wx,y,γ:=γx+1−γy for each x,y∈X and γ∈0,1. Thus, X,d,W is a complete uniformly hyperbolic space with a monotone modulus of uniform convexity and D is a nonempty compact convex subset of X. For a given λ∈0,1, let T:D⟶D defined by(38)Tx=x2,x≠1,1+λ2+λ,x=1.
By [2], it is shown that the mapping T satisfies the condition Cλ but it fails to satisfy the condition Cλ′ whenever 0<λ′<λ. Moreover, T satisfies the condition E for some μ=2+λ/2. For an initial point x0=0.1 and ωn=n/10n+2, σn=1/3n+7, and γn=1/3n+7, we have ωn, σn, and and γn belonging in 0,1. Set the stop parameter to xn−0≤10−15, where 0 is a fixed point of T. By using MATLAB, we compute the convergence of the generalized M-iteration (15). The numerical experiments of all iterations for approximating the fixed point 0 are given in Table 1, and the convergence of an iterative sequence generated by generalized M-iteration process is shown in Figure 1.Table 1
Sequence generated by generalized M-iteration process.
Iterate
Generalized M-iteration process
x
0
0.1
x
1
0.032887490635253
x
2
0.011717192936408
x
3
0.004203808068737
x
4
0.001512138889767
⋮
⋮
x
29
0.000000000000013
x
30
0.000000000000005
x
31
0.000000000000002
x
32
0.000000000000000Figure 1
Convergence of an iterative sequence generated by generalized M-iteration process.Example 3.
Let a mappingT:0,1⟶0,1 be defined by(39)Tx=1−x,x∈0,18,x+78,x∈18,1.
By the proof of [6], it is shown that a mapping T satisfies the condition C. Then, T also satisfies the condition Cλ for some λ∈0,1 and the condition E for some μ=3. For an initial value x0=0.9 and ωn=n/10n+2, σn=1, γn=1, and ωn, σn, and γn belong in 0,1. Set the stop parameter to xn−1≤10−15, where 1 is a fixed point of T. We compare our iteration process (15) with M-iteration process (4). Numerically, we can easily see that the sequence generated by the generalized M-iteration converges faster than the sequence generated by the M-iteration process as given in Table 2 and Figure 2.Table 2
Sequences generated by generalized M-iteration process and M-iteration process.
Iterate
Generalized M-iteration process
M-iteration process
x
0
0.9
0.9
x
1
0.9998046875
0.9984375
x
2
0.999999396006266
0.999977366129557
x
3
0.999999998069622
0.999999674477361
x
4
0.999999999993755
0.999999995330944
x
5
0.999999999999980
0.999999999933126
x
6
1
0.999999999999043
x
7
1
0.999999999999986
x
8
1
1
x
9
1
1
x
10
1
1Figure 2
Convergence of iterative sequences generated by generalized M-iteration process and M-iteration process to the fixed point 1 of a mappingT defined in Example 3.


## 5. Applications to Integral Equations

In this section, we prove the existence of a solution of a nonlinear quadratic integral equation taken from Allahari et al. [19].LetCI be the set of all continuous functions defined on I=0,1 and d:CI×CI⟶ℝ defined by(40)dx,y=supt∈I|xt−yt|,for allx,y∈CI.We can see thatCI,d is a metric space and a hyperbolic space with the modulus of uniform convexity.LetΓ be the set of functions γ:0,+∞⟶0,+∞ which satisfy the following condition:(i)
γ is nondecreasing and γt≤t for all t∈0,+∞.Consider the nonlinear quadratic equation as follows:(41)xt=ht+ω∫01kt,sfs,xsds,t∈I,ω≥0,where h:I⟶ℝ, f:I×ℝ⟶ℝ, and k:I×I⟶ℝ. From now on, let T:CI⟶CI be defined by(42)Txt=ht+ω∫01kt,sfs,xsdsfort∈I.Suppose that the following conditions hold:(A1)h:I⟶ℝ is continuous;(A2)f:I×ℝ⟶ℝ is continuous, ft,x≥0 and there exist L≥0 and γ∈Γ such that for all t∈I and for all a,b∈ℝ, ft,a−ft,b≤Lγa−b;(A3)k:I×I⟶ℝ is continuous at t∈I for every s∈I and measurable at s∈I for all t∈I such that kt,s≥0 and ∫01kt,sds≤K;(A4)ωKL≤1;(A5) The sequencexn defined as in (15) is bounded and limn⟶∞dxn,Txn=0.Theorem 4.
Under assumptions (A1)–(A5), the integral equation (41) has a solution in CI.Proof.
Letx,y∈CI such that(43)λdx,Tx≤dx,y,for some λ∈0,1. Using (A2), we obtain that(44)Txt−Tyt=ht+ω∫01kt,sfs,xsds−ht−ω∫01kt,sfs,ysds≤ω∫01kt,sfs,xs−fs,ysds≤ω∫01kt,sLγxs−ysds.
Sinceγ is nondecreasing, we have(45)γxs−ys≤γsups∈Ixs−ys=γdx,y.
From (A3), we obtain that(46)Txt−Tyt≤ωKLγdx,y.
Therefore,(47)dTx,Ty=supt∈ITxt−Tyt≤ωKLγdx,y≤dx,y.
This implies thatT satisfies the condition Cλ for λ∈0,1.
Letx,y∈CI. We obtain that(48)xt−Tyt=xt−ht+ω∫01kt,sfs,ysds≤xt−ht+ω∫01kt,sfs,xsds+ht+ω∫01kt,sfs,xsds−ht+ω∫01kt,sfs,ysds=xt−Txt+Txt−Tyt.
This implies that(49)dx,Ty≤dx,Tx+dTx,Ty≤dx,Tx+dx,y≤μdx,Tx+dx,y,for some μ≥1. Then, T satisfies the condition E. By Lemma 3 and (A5), we get that FT≠∅. Therefore, all assumptions in Theorem 1 are satisfied. Hence, xn Δ-converges to a fixed point of T in CI. This implies that the integral equation (41) has a solution in CI.


## 6. Conclusion Remark

Our space is a uniformly convex hyperbolic space which is a natural generalization of a uniformly convex Banach space. Since our mappings are more general than mappings of Ullah et al. [6] and the sequence generated by the generalized M-iteration process (15) converges faster than the sequence generated by the M-iteration process (4), our results improve the results of Ullah et al. [6].



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