# The Implicit Midpoint Procedures for Asymptotically Nonexpansive Mappings

Authors: M. O. Aibinu; S. C. Thakur; S. Moyo
Journal: Journal of Mathematics
 (2020)

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## Abstract

The concept of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings. Implicit midpoint procedures are extremely fundamental for solving equations involving nonlinear operators. This paper studies the convergence analysis of the class of asymptotically nonexpansive mappings by the implicit midpoint iterative procedures. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using a well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. A numerical example is presented to illustrate the convergence result. Under relaxed conditions on the parameters, some algorithms and strong convergence results were derived to obtain some results in the literature as corollaries.


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## Body

## 1. Introduction

LetK be a bounded subset of a Banach space E. A mapping T:K⟶K is called asymptotically nonexpansive if there exists a sequence kn of positive real numbers with kn⟶1 as n⟶∞ for which(1)Tnx−Tny≤knx−y,for allx,y∈Kandn∈ℕ.In 1972, Geobel and Kirk [1] introduced this concept of asymptotically nonexpansive mappings as an important generalization of the class of nonexpansive mappings in which kn=1 for all n [1]. T is said to be uniformly L-Lipschitzian, if there exists a constant L>0 such that(2)Tnx−Tny≤Lx−y,for allx,y∈Kandn∈ℕ.Notice that every asymptotically nonexpansive is uniformlyL-Lipschitzian with a constant L=supn∈ℕkn≥1. The set of fixed point of T,x∈K:Tx=x, will be denoted by FT. A mapping f:K⟶K is called a α-contraction if there exists α∈0,1 such that(3)fx−fy≤αx−y,for allx,y∈K.It is well known that a contractionf on K has a unique fixed point in K.A powerful numerical method for solving ordinary differential equations and differential algebraic equations, which also has a long history, is the implicit midpoint procedure. Akin to the implicit midpoint procedure is the fractal structures of Mandelbrot and Julia sets which have been demonstrated for practical application in quadratic, cubic, and higher degree polynomials. Kang et al. in [2] defined Jungck Noor iteration with s-convexity and established the escape criterions for quadratic, cubic, and nth degree complex polynomials. Auzinger and Frank in [3] studied the structure of the global discretization error for the implicit midpoint and trapezoidal rules applied to nonlinear stiff initial value problems. Bader and Deuflhard in [4] introduced a semi-implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations. The implicit midpoint rule is described as a theoretical foundation of the numerical treatment of problems arising in physical and biological sciences (Kastner-Maresch [5]). The implicit midpoint rule is applied to obtain the periodic solution of a nonlinear time-dependent evolution equation and a Fredholm integral equation (Somali and Davulcua [6]). Consider the ordinary differential equation:(4)x′=ft,x0=x0.A sequencexn is generated by the implicit midpoint rule via the recursion:(5)1hxn+1−xn=fxn+xn+12,n∈ℕ,where h>0 is a stepsize and ℕ is the set of positive integers. For a Lipschitz continuous and sufficiently smooth map f:ℝN⟶ℝN, it is known that, as h⟶0 uniformly over t∈0,t¯ for any fixed t¯>0, the sequence xn generated by (5) converges to the exact solution of (4). By rewriting the function f in the form ft=gt−t, the differential equation (4) becomes x′=gt−t. Consequently, the equilibrium problem associated with the differential equation is transformed to the fixed point problem gt=t [7]. Extension of the implicit midpoint rule to nonexpansive mappings by Alghamdi et al. in [8] generates a recursion sequence:(6)xn+1=1−anxn+anTxn+xn+12,n∈ℕ,where an∈0,1,T:K⟶K is a nonexpansive mapping and K is a closed convex subset of a real Hilbert space H. They proved the weak convergence of (6) under certain conditions on an. Still, in Hilbert space, Xu et al. in [9] used contractions to regularize the implicit midpoint rule (6) and introduced the implicit procedure:(7)xn+1=anfxn+1−anTxn+xn+12,n∈ℕ.They proved a strong convergence theorem for the sequencexn to a fixed point p of T which also solves the variational inequality:(8)I−fp,x−p≥0,∀x∈FT.In 2015, Yao et al. in [7] introduced(9)xn+1=anfxn+bnxn+cnTxn+xn+12,n∈ℕ,which gives a faster approximation compared with (7), where T is a nonexpansive mapping in a Hilbert space and an+bn+cn=1,∀n∈ℕ. Aibinu et al. in [10] compared the rate of convergence of the iteration procedures (6), (7), and (9). Aibinu and Kim in [11] recently introduced a new scheme of the viscosity implicit iterative algorithms for nonexpansive mappings in Banach spaces. Suitable conditions were imposed on the control parameters to prove a strong convergence theorem for the considered iterative sequence. Also, Aibinu and Kim in [12] studied the analytical comparison of schemes (7) and (9) to determine the sequence that converges faster in approximating a fixed point of a nonexpansive mapping. Literature review reveals the following problems, which this paper is devoted to address.Problem 1.
Can one study the implicit iterative procedure (9) for the class of asymptotically nonexpansive mappings which is more general than nonexpansive mappings?Problem 2.
Under what conditions will the main results of Yao et al. in [7] hold for asymptotically nonexpansive mappings in the general Banach spaces?Motivated by the previous works, this paper is devoted for the extension of the previous results in the literature, to a more general space and to study the analogue of algorithm (9) for the class of asymptotically nonexpansive mappings. Therefore, for a Banach space with a uniformly Ga´teaux differentiable norm possessing uniform normal structure and for arbitrary x1∈K, this paper considers the iterative algorithm given by(10)xn+1=anfxn+bnxn+cnTnxn+xn+12,n∈ℕ,where T is an asymptotically nonexpansive mapping and the real sequences an⊂0,1,bn⊂0,1 and cn⊂0,1 are chosen such that an+bn+cn=1∀n∈ℕ. We study the convergence of the important class of asymptotically nonexpansive mappings, which is a generalization of the class of nonexpansive mappings. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using this well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. A numerical example is given to illustrate the convergence result.


## 2. Preliminaries

LetE be a real Banach space with dual E∗ and denote the norm on E by .. The normalized duality mapping J:E⟶2E∗ is defined as(11)Jx=f∈E∗:x,f=xf,x=f,where .,. is the duality pairing between E and E∗. Let BE denotes the unit ball of E. The modulus of convexity of E is defined as follows δEε=inf1−x+y/2:x,y∈BE,x−y≥ε,0≤ε≤2. E is uniformly convex if and only if δEε>0 for every ε∈0,2. E is said to be smooth (or Ga´teaux differentiable) if the limit,(12)limt⟶0+x+ty−xt,exists for each x,y∈BE. E is said to have uniformly Ga^teaux differentiable norm if, for each y∈BE, the limit is attained uniformly for x∈BE and uniformly smooth if it is smooth and the limit is attained uniformly for each x,y∈BE. Recall that if E is smooth, then J is single-valued and onto if E is reflexive. Furthermore, the normalized duality mapping J is uniformly continuous on bounded subsets of E from the strong topology of E to the weak-star topology of E∗ if E is a Banach space with a uniformly Ga^teaux differentiable norm.LetK be a nonempty bounded closed convex subset of a Banach space E, and let the diameter of K be defined by dK=supx−y:x,y∈K. For each x∈K, let rx,K=supx−y:y∈K and let rK=infrx,K:x∈K, and the Chebyshev radius of K is relative to itself. The normal structure coefficient of E (see [13]) is defined as the number:(13)NE=infdKrK:Kbounded closed convex subset ofEwithdK>0.A spaceE, such that NE>1, is said to have uniformly normal structure. Recall that a space with uniformly normal structure is reflexive and that all uniformly convex or uniformly smooth Banach spaces have uniform normal structure [14, 15]. If E is a reflexive Banach space with modulus of convexity δE, then NE≥1−δE1−1. It was proved that if the space E is uniformly convex; then, every asymptotically nonexpansive self-mapping T of K has a fixed point [1]. Also, it has been proved that if liminfρt/t<1/2, then NE>1, where ρt=sup1/2x+ty+x−ty−1,x,y∈BE for t≥0 [16, 17].Recall that a functionf:X⟶R∪+∞ is said to be weakly lower semicontinuous at x0∈X if whenever xn is a sequence in X such that xn converges weakly to x0; then,(14)fx0≤liminfn⟶∞fxn.According to Jung and Kim in [18], let μ be a mean on positive integers ℕ. A mean μ on ℕ is a continuous linear functional on l∞ satisfying μ=1=μ1. It is called a Banach limit if μnan=μnan+1 for every a=a1,a2,…∈l∞.The following lemmas are needed in the sequel.Lemma 1.
(see [19]). Let K be a nonempty closed convex subset of a Banach space E with a uniformly Ga^teaux differentiable norm, let xn be a bounded sequence of E, and let μn be a mean on ℕ. Let x∈K. Then,(15)μnxn−x2=minz∈Kμnxn−z2,if and only if μnz−x,Jxn−x≤0 for all z∈K, where J is the duality mapping of E.Lemma 2.
(see [20]). Let σn be a sequence of nonnegative real numbers satisfying the property:(16)σn+1=1−γnσn+γnβn,n>0,where γn⊂0,1 and βn⊂ℝ such that(i)
∑n=1∞γn=∞(ii)
limsupn⟶∞βn≤0
Then,σn converges to zero, as n⟶∞.Lemma 3.
(see [21]). Let E be a Banach space with uniform normal structure, K a nonempty closed convex and bounded subset of E, and T:K⟶K an asymptotically nonexpansive mapping. Then, T has a fixed point.Lemma 4.
(see [22]). Let E be a real Banach space and J the normalized duality map on E. Then, for any given x,y∈E, the following inequality holds:(17)x+y2≤x2+2y,jx+y,∀jx+y∈Jx+y.Lemma 5.
(see [23, 24]). Let η be a real number and x0,x1,…∈l∞ such that μnxn≤η for all Banach limits. If limsupn⟶∞xn+1−xn≤0, then limsupn⟶∞xn≤η.Lemma 6.
(see [15]). Suppose E is a Banach space with uniformly normal structure, K is a nonempty bounded subset of E, and T:K⟶K is a uniformly k-Lipschitzian mapping with k<NE1/2. Suppose also there exists a nonempty, bounded, closed, and convex subset G of K with the following property:(18)P:p∈Gimpliesωwp∈G,where ωwp is the weak ω-limit set of T at p, that is, the set(19)x∈E:x=weak−limj⟶∞Tnjpfor somenj⟶∞.
Then,T has a fixed point in G.


## 3. Main Results

Assumption 1.
LetE be a real Banach space with a uniformly Ga´teaux differentiable norm possessing uniform normal structure and K a nonempty bounded closed convex subset of E. Suppose T:K⟶K is an asymptotically nonexpansive mapping with FT≠∅ and f:K⟶K is a contraction with constant α∈0,1. For arbitrary x1∈K and real sequences an⊂0,1,bn⊂0,1 and cn⊂0,1 satisfying an+bn+cn=1,∀n∈ℕ, this paper considers the iterative scheme xn given by (10). Assume that an satisfies the following conditions:(i)
limn⟶∞an=0(ii)
∑n=1∞an=∞(iii)
limn⟶∞kn2−1/an=0where kn is a sequence of positive real numbers with kn⟶1 as n⟶∞. Then, the sequence xn is well defined. To show this, for arbitrary ω∈K, define the mapping Tω:K⟶K:(20)x↦Tω≔anfω+bnω+cnTnω+x2,n∈ℕ.
Then, forx,y∈K,(21)Tωx−Tωy≤cnTnω+x2−Tnω+y2≤cnkn2x−y.
Using condition (iii), for any given positive numberε,0<ε<1−α, there exists a sufficient large positive integer n0, such that, for any n≥n0, we obtain that(22)kn2−1≤2εan,kn−1≤kn+12kn−1≤kn2−12≤εan.
Therefore, sincekn is a sequence of positive real numbers with kn⟶1 as n⟶∞ and kn−1≤εan for all n≥n0, it is obvious that(23)cnkn=1+kn−1−kn+cnkn≤1+εan−kn1−cn=1+εan−1−cn=1+εan−an+bn=1−an1−ε−bn≤1.
Thus, from (21) and (23), it shows that Tω is a contraction. Therefore, (10) is well defined since every contraction in a Banach space has a fixed point.Firstly, the proof of the following lemmas are given, which are useful in establishing the main result.Lemma 7.
LetE be a Banach space with a uniform normal structure, K a nonempty bounded closed convex subset of E, and T:K⟶K an asymptotically nonexpansive mapping. Suppose f:K⟶K is a contraction with constant α∈0,1 and assume that an satisfies conditions i−iii. For an arbitrary x1∈K, the iterative sequence xn given by (10) is bounded.Proof.
The sequencexn is shown to be bounded.
By Lemma3, FT≠∅. Then, for p∈FT,(24)xn+1−p=anfxn+bnxn+cnTnxn+xn+12−p=anfxn−fp+anfp−p+bnxn−p+cnTnxn+xn+12−p≤anfxn−fp+anfp−p+bnxn−p+cnTnxn+xn+12−p≤anfxn−fp+anfp−p+bnxn−p+cnknxn+xn+12−p≤αanxn−p+anfp−p+bnxn−p+cnkn2xn−p+cnkn2xn+1−p.
Therefore,(25)1−cnkn2xn+1−p≤αan+bn+cnkn2xn−p+anfp−p,which is equivalent to(26)2−cnkn2xn+1−p≤2αan+2bn+cnkn2xn−p+anfp−p.
Observe that(27)2−cnkn=2−1−an+bnkn=2−kn−knan+bn=1−kn−1+knan+bn≥1−εan+knan+bn=1+kn−εan+knbn≥1+1−εan+bn,for sufficiently largen.
Also,(28)2αan+2bn+cnkn=2αan+2bn+1−an+bnkn=2αan+2bn+kn−knan+bn=1+kn−1+2αan−knan+2bn−knbn≤1+εan+2αan−knan+2bn−knbn=1−kn−2α−εan−kn−2bn≤1−1−2α−εan−1−2bn=1−1−2α−εan+bn,for sufficiently largen.
Consequently, (26) becomes(29)xn+1−p≤1−1−2α−εan+bn1+1−εan+bnxn−p+2an1+1−εan+bnfp−p=1−21−α−εan1+1−εan+bnxn−p+21−α−εan1+1−εan+bn11−α−εfp−p≤maxxn−p,11−α−εfp−p⋮≤maxx1−p,11−α−εfp−p.
This implies that the sequencexn is bounded and hence fxn and Tnxn are also bounded.
Obviously,(30)Tnxn=Tnxn−p+p≤Tnxn−p+p≤knxn−p+p⋮≤Rmaxx1−p,11−α−εfp−p+p,where R:=supkn.Lemma 8.
LetE be a real Banach space with a uniformly Ga´teaux differentiable norm possessing uniform normal structure, K a nonempty bounded closed convex subset of E,T:K⟶K an asymptotically nonexpansive mapping with sequence kn⊂1,∞,f:K⟶K a continuous mapping, and xn a bounded sequence in K such that limn⟶∞xn−Tnxn=0. Suppose zt is a path in K defined by zt=tfzt+1−tTnzt,t∈0,1 such that zt⟶z as t⟶0+. Then,(31)limsupn⟶∞fz−z,Jxn−z≤0.Proof.
Fromlimn⟶∞xn−Tnxn=0, let M:=supn⟶∞xn−Tnxn and define tn:=xn−Tnxn/M,n∈ℕ. Clearly,(32)limn⟶∞xn−Tnxntn=0.
Sincezt is given by zt=tfzt+1−tTnzt, we can write(33)ztn−xn=tnfztn−xn+1−tnTnztn−xn.
Sincexn,ztn, and Tnxn are bounded, ztn−xn is bounded. By Lemma 4,(34)ztn−xn2≤1−tn2Tnztn−xn2+2tnfztn−xn,Jztn−xn≤1−tn2Tnztn−Tnxn+Tnxn−xn2+2tnfztn−ztn,Jztn−xn+2tnztn−xn2≤1−tn2knztn−xn+xn−Tnxn2+2tnfztn−ztn,Jztn−xn+2tnztn−xn2≤1−tn2kn2ztn−xn2+xn−Tnxn2knztn−xn+xn−Tnxn+2tnfztn−ztn,Jztn−xn+2tnztn−xn2.
Therefore,(35)fztn−ztn,Jxn−ztn≤kn21−tn2−1−2tn2tnztn−xn2+1−tn2xn−Tnxn2tn2knztn−xn+xn−Tnxn≤kn2−11−2tn+kn2tn22tnztn−xn2+1+tn2xn−Tnxn2tn2knztn−xn+xn−Tnxn.
From (32) and by recalling that kn⟶1 as n⟶∞,(36)limsupn⟶∞fztn−ztn,Jxn−ztn≤0.
Observe that(37)fz−z,Jxn−z=fz−z,Jxn−z−Jxn−ztn+fz−fztn,Jxn−ztn+ztn−z,Jxn−ztn+fztn−ztn,Jxn−ztn.
Noting the hypothesis thatztn⟶z as n⟶∞, coupled with the continuity of f and uniform continuity of the duality mapping J on bounded subsets of E, it gives(38)ztn−z,Jxn−ztn⟶0,n⟶∞,fz−fztn,Jxn−ztn⟶0,n⟶∞,fz−z,Jxn−z−Jxn−ztn⟶0,n⟶∞.
From (36)–(38), it can be deduced that(39)limsupn⟶∞fz−z,Jxn−z≤0.Theorem 1.
LetE be a real Banach space with a uniformly Ga´teaux differentiable norm possessing uniform normal structure, K a nonempty bounded closed convex subset of E, and T:K⟶K an asymptotically nonexpansive mapping with the sequence kn of positive real numbers such that kn⟶1 as n⟶∞,supn∈ℕkn≤NE1/2, and FT≠∅. Suppose f:K⟶K is a contraction with constant α∈0,1. Assume that an satisfies conditions i−iii and in addition (iv) limn⟶∞xn−Tnxn=0
Then, for an arbitraryx1∈K, the iterative sequence xn given by (10) converges in norm to a fixed point p of T which is the unique solution of variational inequality:(40)I−fp,jx−p≥0,∀x∈FT.Proof.Step 1.
It is shown in this step thatlimn⟶∞xn−Txn=0.
Observe that(41)xn−Txn=xn−xn+1+xn+1−Tnxn+Tnxn−Txn≤xn−xn+1+xn+1−Tnxn+Tnxn−Txn≤xn+1−xn+xn+1−Tnxn+k1xn−Tn−1xn,for some k1∈ℝ+, where ℝ+ denotes the set of positive real numbers. It is required that(42)limn⟶∞xn+1−xn=limn⟶∞xn+1−Tnxn=limn⟶∞xn−Tn−1xn=0.
To evaluatelimn⟶∞xn+1−xn,(43)xn+1−xn≤xn+1−Tnxn+Tnxn−xn=anfxn+bnxn+cnTnxn+xn+12−Tnxn+Tnxn−xn≤anfxn−Tnxn+bnxn−Tnxn+cnTnxn+xn+12−Tnxn+Tnxn−xn≤anfxn−Tnxn+1+bnxn−Tnxn+cnknxn+xn+12−xn=anfxn−Tnxn+1+bnxn−Tnxn+cnkn2xn+1−xn.
LetM>0 be a constant such that(44)M≥supfxn−Tnxn,n∈ℕ.
Then,(45)1−cnkn2xn+1−xn≤anM+1+bnxn−Tnxn,2−cnkn2xn+1−xn≤anM+1+bnxn−Tnxn,1+1−εan+bn2xn+1−xn≤anM+1+bnxn−Tnxn,xn+1−xn≤2an1+1−εan+bnM+21+bn1+1−εan+bnxn−Tnxn2anM+4xn−Tnxn.
Using conditions (i) and (iv),(46)limn⟶∞xn+1−xn=0.
Next, to evaluatelimn⟶∞xn+1−Tnxn,(47)xn+1−Tnxn=anfxn+bnxn+cnTnxn+xn+12−Tnxn≤anfxn−Tnxn+bnxn−Tnxn+cnTnxn+xn+12−Tnxn≤anfxn−Tnxn+bnxn−Tnxn+cnknxn+xn+12−xn=anfxn−Tnxn+bnxn−Tnxn+cnkn2xn+1−xn≤anM+bnxn−Tnxn+cnkn2xn+1−xn⟶0,asn⟶∞.
In a similar manner, one can apply (46) and (47) to obtain that(48)xn−Tn−1xn=xn−Tn−1xn−1+Tn−1xn−1−Tn−1xn≤xn−Tn−1xn−1+Tn−1xn−1−Tn−1xn≤xn−Tn−1xn−1+kn−1xn−xn−1⟶0,asn⟶∞.
Consequently, from (46)–(48), it can be deduced that(49)limn⟶∞xn−Txn=0.Step 2.
It is claimed that(50)limsupn⟶∞fp−p,Jxn−p≤0,where p is a fixed point of T.
Simply by applying Lemma7, it is obtained that(51)limsupn⟶∞fp−p,Jxn−p≤0.
Alternatively, define a functionϕ:K⟶ℝ+ by(52)ϕz=μnxn−z2,for all z∈K. Observe that ϕ is continuous, convex, and ϕz⟶∞ as z⟶∞. Since E is reflexive, ϕ attains its infimum over K. Let z0∈K such that ϕz0=minz∈Kϕz and let(53)G=x∈K:ϕx=minz∈Kϕz.
Then,G is nonempty because z0∈G and it is also closed, convex, and bounded (for more details, see e.g., [15, 19, 25]). Moreover, T has a fixed point in G by Lemma 6. Indeed, let p∈G and ωwp be the weak w-limit set of T at p, that is, the set x∈K:x:=w−limjTnjpfor somenj⟶∞. Then, from the weak-lower-semicontinuity of ϕ and the fact that limn⟶∞xn−Tnxn=0 (condition (iv)), the following estimate is obtained:(54)ϕx≤liminfj⟶∞ϕTnjp≤limsupn⟶∞ϕTnp=limsupn⟶∞μnxn−Tnp2≤limsupn⟶∞μnxn−Tnxn+Tnxn−Tnp2≤limsupn⟶∞μnTnxn−Tnp2≤limsupn⟶∞μnkn2xn−p2≤μnxn−p2=ϕp.
This shows thatG satisfies the property (P). By applying Lemma 6, x=Tx, it follows that G∩FT≠∅. Suppose that p∈G∩FT. Then, Lemma 1 gives that(55)μnx−p,Jxn−p≤0,for all x∈K. In particular,(56)μnfp−p,Jxn−p≤0.
Sincexn+1−xn⟶0 as n⟶∞ (46), and the duality mapping J is norm to weak-star uniformly continuous on bounded set (earlier mentioned in Section 2); this gives(57)limn⟶∞fp−p,Jxn+1−p−fp−p,Jxn−p=0.
Therefore, the sequencefp−p,Jxn−p satisfies the conditions of Lemma 5. Thus,(58)limsupn⟶∞fp−p,Jxn−p≤0.Step 3.
Lastly, it is shown thatxn⟶p:(59)xn+1−p2=anfxn−p,Jxn+1−p+bnxn−p,Jxn+1−p+cnTnxn+xn+12−p,Jxn+1−p=anfxn−fp,Jxn+1−p+bnxn−p,Jxn+1−p+cnTnxn+xn+12−p,Jxn+1−p+anfp−p,Jxn+1−p≤αanxn−pxn+1−p+bnxn−pxn+1−p+cnknxn+xn+12−pxn+1−p+anfp−p,Jxn+1−p≤αanxn−pxn+1−p+bnxn−pxn+1−p+cnkn2xn−p+xn+1−pxn+1−p+anfp−p,Jxn+1−p=αan+bn+cnkn2xn−pxn+1−p+cnkn2xn+1−p2+anfp−p,Jxn+1−p=2αan+2bn+cnkn2xn−pxn+1−p+cnkn2xn+1−p2+anfp−p,Jxn+1−p≤2αan+2bn+cnkn4xn−p2+xn+1−p2+cnkn2xn+1−p2+anfp−p,Jxn+1−p=2αan+2bn+3cnkn4xn+1−p2+2αan+2bn+cnkn4xn−p2+anfp−p,Jxn+1−p.
Therefore,(60)1−2αan+2bn+3cnkn4xn+1−p2≤2αan+2bn+cnkn4xn−p2+anfp−p,Jxn+1−p,4−2αan+2bn+3cnkn4xn+1−p2≤2αan+2bn+cnkn4xn−p2+anfp−p,Jxn+1−p,xn+1−p2≤2αan+2bn+cnkn4−2αan+2bn+3cnknxn−p2+4an4−2αan+2bn+3cnknfp−p,Jxn+1−p.
Observe that(61)4−2αan+2bn+3cnkn=4−2αan−2bn−3kn1−an−bn=4−3kn−2αan+3knan−2bn+3knbn=1−3kn−1−2αan+3knan−2bn+3knbn≥1−3εan−2αan+3knan−2bn+3knbn=1+3kn−2α−3εan+3kn−2bn≥1+3−2α−3εan+3−2bn=1+3−2α−3εan+bn,for sufficiently largen.
Consequently,(62)xn+1−p2≤1−1−2α−εan+bn1+3−2α−3εan+bnxn−p2+4an1+3−2α−3εan+bnfp−p,Jxn+1−p=1−41−α−εan1+3−2α−3εan+bnxn−p2+41−α−εan1+3−2α−3εan+bn11−α−εfp−p,Jxn+1−p.
By applying Lemma2 to (58) and (62), it is deduced that xn⟶p as n⟶∞,where, respectively,γn:=41−α−εan/1+3−2α−3εan+bn and βn:=1/1−α−εfp−p,Jxn+1−p.Remark 1.
It is generally known that a Hilbert space is a subclass of uniformly convex Banach spaces. Moreover, it is obvious that every nonexpansive mapping is an asymptotically nonexpansive mapping. Consequently, Theorem1 is an improvement and generalization of the main results of Zhao et al. [26], Aibinu et al. [10], Luo et al. [27], and Yao et al. [7] as stated below.Corollary 1.
(see [26]). Let K be a nonempty closed and convex subset of a Hilbert space H, T:K⟶K be an asymptotically nonexpansive mapping with a sequence kn⊂1,+∞,limn⟶∞kn=1 and FT≠∅. Let f be a contraction on K with coefficient c∈0,1. For an arbitrary initial point x1∈K, let xn be the sequence generated by(63)xn+1=anfxn+1−anTnxn+xn+12,n∈ℕ.
Then, the sequencexn converges strongly to p=PFTfp which solves the following variational inequality (8).Proof.
TakeE to be a Hilbert space in Theorem 1. Moreover, since an+bn+cn=1,∀n∈ℕ, by taking bn=0 in (10), we have cn=1−an. Therefore, we obtain (63) from (10) by taking bn=0. Hence, the desired result follows from Theorem 1.Corollary 2.
(see [10]). Let E be a uniformly smooth Banach space and K be a nonempty closed convex subset of E. Let T:K⟶K be a nonexpansive mapping with FT≠∅ and f:K⟶K be a c-contraction. Suppose an satisfies iandii and bn satisfies(C1) 0<liminfn⟶∞βn≤limsupn⟶∞βn<1(C2) limn⟶∞βn+1−βn=0
For an arbitraryx1∈K, define the iterative sequence xn by (9). Then, as n⟶∞, the sequence xn converges in norm to a fixed point p of T, where p is the unique solution in FT to the variational inequality (8).Proof.
Every nonexpansive mapping is an asymptotically nonexpansive mapping and all uniformly convex or uniformly smooth Banach spaces have uniform normal structure (see [14, 15]). Therefore, it suffices to show that limn⟶∞xn−Txn=0:(64)xn−Txn≤xn−xn+1+xn+1−Txn=xn−xn+1+anfxn−Txn+bnxn−Txn+cnTxn+xn+12−Txn=xn−xn+1+anfxn−Txn+bnxn−Txn+cnxn+xn+12−xn=xn−xn+1+anfxn−Txn+bnxn−Txn+cn2xn−xn+1.
LetM be a constant such that M>supn∈Nfxn−Txn and b=supn1−bn>0 since bn<1. Then,(65)xn−Txn≤2+cn21−bnxn+1−xn+an1−bnM≤2+cn2bxn+1−xn+anbM⟶0,asn⟶∞.
This completes the proof.Corollary 3.
(see [27]). Let K be a closed convex subset of a uniformly smooth Banach space E. Let T:K⟶K be a nonexpansive mapping with FT≠∅ and f:K⟶K a contraction with coefficient c∈0,1. Let xn be a sequence generated by the viscosity implicit midpoint rule (7), where an is a sequence in 0,1 such that it satisfies conditions iandii and either C3∑n=1∞an−an−1<+∞ or limn⟶∞an/an−1=1. Then, xn converges strongly to a fixed point p of T, which also solves the variational inequality (8).Proof.
Takebn=0 in equation (9). The desired result follows from Corollary (28).Corollary 4.
(see [7]). Let K be a nonempty closed convex subset of a real Hilbert space H. Let T:K⟶K be a nonexpansive mapping with FT≠∅. Suppose f:K⟶K is a c-contraction. For given x0∈K arbitrarily, let the sequence xn be generated by (9). Assume that an satisfies i and ii and bn satisfiesC10<liminfn⟶∞bn≤limsupn⟶∞bn<1C4limn⟶∞bn+1−bn=0
Then, the sequencexn generated by (9) converges strongly to p=PFTQp.Proof.
Observe that a Hilbert space is a subclass of a uniformly smooth Banach space. By takingE to be a Hilbert space in Corollary 2, the result of Yao et al. in [7] is obtained.


## 4. Numerical Example

Numerical example is given in this section to illustrate the convergence of the sequence of iteration in the main theorem.Example 1.
LetK be an orthogonal subspace of ℝ2 with the norm u=∑i=12ui2 and the inner product u,v=∑i=12uivi for u=u1,u2 and v=v1,v2. Define a mapping T:K⟶K by(66)Tu=u1,u2,ifΠi=12ui<0,−u1,−u2,ifΠi=12ui≥0,for each u=u1,u2∈K. Then,(67)Tnu−Tnv2=u−−1nv2=u2+v2=u−v2,or(68)Tnu−Tnv2=−1nu−v2=u2+v2=u−v2,for any u,v∈K (Zhang et al. [28]). Take kn⊂1,∞ to be kn:=1+1/2n and observe that Tnu−Tnv≤knu−v for any u,v∈K. Thus, T is an asymptotically nonexpansive mapping on K and FT=0,0∪u1,u2. We define f:ℝ⟶ℝ by fx=1/2x. Let an=1/n,bn=n−1/nn+1, and cn=n−1/n+1. It is obvious that the sequences an,bn, and cn satisfy conditions (i)–(iii) in Assumption 1, and according to (67) or (68), limn⟶∞un−Tnun=u−u=0. The table and figures are given to display the numerical results and convergence of the our sequence of iteration to fixed points of T. The computations are carried out by using MATLAB. Table 1 exhibits numerical values for the nth iterations as xn converges to a fixed point. Figure 1 displays the convergence of xn to the fixed point p=1,−1 with x1=0,1/3. Convergence of xn to the fixed point p=1,−1 with x1=0,1/3 is shown in Figure 2. Graph of xn+1−p as xn converges to the fixed point p=−1,1 with x1=−2,1 is presented in Figure 3.Table 1
Numerical values for thenth iterations as xn converges to a fixed point.
Iteration (n)xn+1p=1,−1,x1=0,1/3p=0,0,x1=1/2,1p=−1,1,x1=−2,110.51520.22360.648520.00080.00030.001030.00030.00010.000440.00020.00010.000250.00010.00010.000260.00010.00000.000170.00010.00000.000180.00010.00000.000190.00010.00000.0001100.00000.00000.0001110.00000.00000.0000120.00000.00000.0000130.00000.00000.0000140.00000.00000.0000150.00000.00000.0000160.00000.00000.0000170.00000.00000.0000180.00000.00000.0000190.00000.00000.0000200.00000.00000.0000Figure 1
Convergence ofxn to the fixed point p=1,−1 with x1=0,1/3.Figure 2
Convergence ofxn to the fixed point p=0,0 with x1=1/2,1.Figure 3
Graph ofxn+1−p as xn converges to the fixed point p=−1,1 with x1=−2,1.


## 5. Conclusion

This paper considered the implicit midpoint rule of asymptotically nonexpansive mappings using the viscosity technique. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established by using this well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. Moreover, the previous results have been extended from the Hilbert space to a Banach space with a uniformly Ga´teaux differentiable norm possessing uniform normal structure. A numerical example is given to show the convergence of the sequence of iteration in the main theorem. The result presented in this paper plays crucial theoretical base of the numerical analysis for nonlinear problems which are often encountered in physical and biological sciences. The future direction for this paper is to determine for the class of asymptotically nonexpansive mappings and the conditions for the convergence of Jungck Noor iteration with s-convexity, defined by Kang et al. [2] and to compare the rate of convergence with the implicit midpoint procedure.



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