# Approximation of Common Fixed Points of a Sequence of Nearly Nonexpansive Mappings and Solutions of Variational Inequality Problems

Authors: D. R. Sahu; Shin Min Kang; Vidya Sagar
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theorem extends and improves several corresponding results in the context of nearly nonexpansive mappings.


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## Body

## 1. Introduction

LetC be a nonempty subset of a real Hilbert space H with inner product 〈·,·〉 and norm ∥·∥, respectively. A mapping T:C→H is called the following:(1)
monotone if
(1.1)〈Tx-Ty,x-y〉≥0∀x,y∈C,(2)
η-strongly monotone if there exists a positive real number η such that
(1.2)〈Tx-Ty,x-y〉≥η‖x-y‖2∀x,y∈C,(3)
k-Lipschitzian if there exists a constant k>0 such that
(1.3)‖Tx-Ty‖≤k‖x-y‖∀x,y∈C,(4)
nonexpansive if
(1.4)‖Tx-Ty‖≤‖x-y‖∀x,y∈C,(5)
nearly nonexpansive [1, 2] with respect to a fixed sequence {an} in [0,∞) with an→0 if
(1.5)‖Tnx-Tny‖≤‖x-y‖+an∀x,y∈Candn∈N.In [3], Moudafi proposed viscosity approximation methods of selecting a particular fixed point of a given nonexpansive mapping in Hilbert spaces (see [4] for further developments in both Hilbert and Banach spaces). Let f be a contraction on H. Starting with an arbitrary initial x1∈H, define a sequence {xn} recursively by
(1.6)xn+1=αnf(xn)+(1-αn)Txn∀n∈N,
where {αn} is a sequence in (0,1). It is proved in [4] that under appropriate conditions imposed on {αn}, the sequence {xn} generated by (1.6) strongly converges to the unique solution x*∈C of the variational inequality
(1.7)〈(I-f)x*,x-x*〉≥0∀x∈C,
where C=F(T), the set of fixed points of T.In 2006, Marino and Xu [5] introduced the viscosity iterative method for nonexpansive mappings. Starting with an arbitrary initial x1∈H, define a sequence {xn} recursively by
(1.8)xn+1=αnγf(xn)+(I-αnA)Txn∀n∈N.
They proved that the sequence {xn} generated by (1.8) converges strongly to the unique solution of the variational inequality
(1.9)〈(A-γf)x*,x-x*〉≥0∀x∈C,
which is the optimality condition for the minimization problem
(1.10)min⁡x∈C12〈Ax,x〉-h(x),
where h is a potential function for γf (i.e., h′(x)=γf(x) for all x∈H), and A is a strongly positive bounded linear operator on H; that is, there is a constant γ¯>0 such that (1.11)〈Ax,x〉≥γ¯‖x‖2∀x∈H.
The applications of the iterative method (1.8) have been studied by some researchers (see [6, 7]).Also, Wang [8, 9] and Wang and Hu [10] introduced the iterative method for nonexpansive mappings.Recently, Tian [11] proposed an implicit and an explicit schemes on combining the iterative methods of Marino and Xu [5] and Yamada [12]. He also proved the strong convergence of these two schemes to a fixed point of a nonexpansive mapping T defined on a real Hilbert space under suitable conditions.More recently, Ceng et al. [13] introduced an implicit and an explicit schemes using the properties of projection for finding the fixed points of a nonexpansive mapping defined on the closed convex subset of a real Hilbert space. They also proved the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping which is also a solution of a variational inequality defined on the set of fixed points.Aoyama et al. [14] proved strong convergence of an iterative scheme for a sequence of nonexpansive mappings as follows.Theorem 1.1.
LetX be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and C be a nonempty closed convex subset of X. Let {Tn} be a sequence of nonexpansive mappings from C into itself such that ⋂n=1∞‍F(Tn)≠∅. Let T be a mapping from C into itself defined by Tx=limn→∞Tnx for all x∈C and F(T)=⋂n=1∞‍F(Tn). Let {xn} be a sequence in C generated by the following iterative process:
(1.12)x1=x∈C,xn+1=αnx+(1-αn)Tnxn∀n∈N,
where {αn} is a sequence in [0,1] satisfying the following conditions:(a)
limn→∞αn=0 and ∑n=1∞‍αn=∞;(b)
either∑n=1∞‍|αn+1-αn|<∞ or αn∈(0,1] and limn→∞αn+1/αn=1;(c)
∑n=1∞‍sup{∥Tnz-Tn+1z∥:z∈B}<∞ for any bounded subset B of C.
Then, the sequence{xn} converges strongly to Qx, where Q is the sunny nonexpansive retraction from X onto F(T).LetC be a nonempty subset of a real Hilbert space H. Let 𝒯:={Tn} be a sequence of mappings from C into itself. We denote by F(𝒯) the set of common fixed points of the sequence 𝒯, that is, F(𝒯)=⋂n=1∞‍F(Tn). Fix a sequence {an} in [0,∞) with an→0, and let {Tn} be a sequence of mappings from C into H. Then, the sequence {Tn} is called a sequence of nearly nonexpansive mappings [15] with respect to a sequence {an} if
(1.13)‖Tnx-Tny‖≤‖x-y‖+an∀x,y∈C,n∈N.It is obvious that the sequence of nearly nonexpansive mappings is a wider class of sequence of nonexpansive mappings.In this paper, inspired by Aoyama et al. [14], Ceng et al. [13], and Sahu et al. [15], we introduce an explicit iterative scheme and prove a strong convergence theorem for computing an element of F(𝒯), the set of common fixed points of a sequence 𝒯={Tn} of nearly nonexpansive mappings which is also a solution of a variational inequality over F(𝒯). Our result generalizes and improves the results of Ceng et al. [13], Tian [11], and many other related works.


## 2.   Preliminaries

Throughout this paper, we denote byI the identity operator of H. Also, we denote by → and ⇀ the strong convergence and weak convergence, respectively. The symbol ℕ stands for the set of all natural numbers.LetC be a nonempty closed convex subset of a real Hilbert space H. Then, for any x∈H, there exists a unique nearest point in C, denoted by PC(x), such that
(2.1)‖x-PC(x)‖≤‖x-y‖∀y∈C.
The mapping PC is called the metric projection from H onto C (see [1]).LetC be a nonempty subset of a real Hilbert space H and T1,T2:C→H be two mappings. We denote ℬ(C), the collection of all bounded subsets of C. The deviation between T1 and T2 on B∈ℬ(C), denoted by 𝒟B(T1,T2), is defined by
(2.2)DB(T1,T2)=sup⁡{‖T1x-T2x‖:x∈B}.The following lemmas will be needed to prove our main result.Lemma 2.1 (see [16]).
The metric projection mappingPC is characterized by the following properties:(a)
PC(x)∈C for all x∈H;(b)
〈x-PC(x),PC(x)-y〉≥0 for all x∈H and y∈C;(c)
∥x-y∥2≥∥x-PC(x)∥2+∥y-PC(x)∥2 for all x∈H and y∈C;(d)
〈PC(x)-PC(y),x-y〉≥∥PC(x)-PC(y)∥2 for all x,y∈H.Lemma 2.2 (see [13]).
LetV:C→H be an L-Lipschitzian mapping and F:C→H be a k-Lipschitzian and η-strongly monotone operator. Then, for 0≤γL<μη,
(2.3)〈x-y,(μF-γV)x-(μF-γV)y〉≥(μη-γL)‖x-y‖2∀x,y∈C.
That is, μF-γV is strongly monotone with coefficient μη-γL.Lemma 2.3 (see [12]).
LetC be a nonempty subset of a real Hilbert space H. Suppose that λ∈(0,1) and μ>0. Let F:C→H be a k-Lipschitzian and η-strongly monotone operator on C. Define the mapping G:C→H by
(2.4)Gx=x-λμFx∀x∈C.
Then G is a contraction that provided μ<2η/k2. More precisely, for μ∈(0,2η/k2),
(2.5)‖Gx-Gy‖≤(1-λτ)‖x-y‖∀x,y∈C,
where τ=1-1-μ(2η-μk2)∈(0,1].Lemma 2.4 (see [1]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a nonexpansive mapping. Then I-T is demiclosed at zero; that is, if {xn} is a sequence in C weakly converging to some x∈C and the sequence {(I-T)xn} converges strongly to 0, then x∈F(T).Lemma 2.5 (see [17]).
Assume that{tn} is a sequence of nonnegative real numbers such that
(2.6)tn+1≤(1-αn)tn+αnβn∀n∈N,
where {αn} and {βn} are sequences of nonnegative real numbers which satisfy the following conditions:(a)
{αn}n=1∞⊂(0,1) and ∑n=1∞‍αn=∞;(b)
limsupn→∞βn≤0, or(b’)
∑n=1∞‍αnβn is convergent. Then limn→∞tn=0.Lemma 2.6 (see [18]).
LetC be a nonempty closed convex subset of a real Hilbert space H and λi>0.(i=1,2,3,…,N) such that ∑i=1N‍λi=1. Let T1,T2,T3,…,TN:C→C be nonexpansive mappings such that ⋂i=1N‍F(Ti)≠∅, and let T=∑i=1N‍λiTi. Then T is nonexpansive from C into itself and F(T)=⋂i=1N‍F(Ti).


## 3.   Main Result

Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C→H be a k-Lipschitzian and η-strongly monotone operator and V:C→H be an L-Lipschitzian mapping. Let 𝒯={Tn} be a sequence of nearly nonexpansive mappings from C into itself with respect to a sequence {an} such that F(𝒯)≠∅ and T be a mapping from C into itself defined by Tx=limn→∞Tnx for all x∈C. Suppose that F(T)=F(𝒯),0<μ<2η/k2 and 0≤γL<τ, where τ=1-1-μ(2η-μk2). For an arbitrary x1∈C, consider the sequence {xn} in C generated by the following iterative process:
(3.1)x1∈C,xn+1=PC[αnγVxn+(I-αnμF)Tnxn]∀n∈N,
where {αn} is a sequence in (0,1) satisfying the conditions:(a)
limn→∞αn=0 and ∑n=1∞‍αn=∞;(b)
either∑n=1∞‍|αn+1-αn|<∞ or limn→∞αn+1/αn=1;(c)
either∑n=1∞‍𝒟B(Tn,Tn+1)<∞ or limn→∞𝒟B(Tn,Tn+1)/αn+1=0 for each B∈ℬ(C);(d)
limn→∞an/αn=0. Then, the sequence {xn} converges strongly to x~∈F(𝒯), where x~ is the unique solution of variational inequality
(3.2)〈(μF-γV)x~,x~-y〉≤0∀y∈F(T).Proof.
LetT be a mapping from C into itself defined by Tx=limn→∞Tnx for all x∈C. It is clear that T is a nonexpansive mapping. So, we have F(T)≠∅. Now, we proceed with the following steps.
Step  1. ({xn} is bounded). Let z∈F(𝒯). From (3.1), we have
(3.3)‖xn+1-z‖=‖PC[αnγVxn+(I-αnμF)Tnxn]-PC(z)‖≤‖αnγVxn+(I-αnμF)Tnxn-z‖≤‖αn(γVxn-μFz)+(I-αnμF)Tnxn-(I-αnμF)Tnz‖≤αnγL‖xn-z‖+αn‖(γV-μF)z‖+(1-αnτ)(‖xn-z‖+an)≤(1-αn(τ-γL))‖xn-z‖+αn‖(γV-μF)z‖+(1-αnτ)an≤(1-αn(τ-γL))‖xn-z‖+αn‖(γV-μF)z‖+an.
Note that limn→∞an/αn=0, so there exists a constant K>0 such that
(3.4)αn‖(γV-μF)z‖+anαn≤K∀n∈N.
Thus, we have
(3.5)‖xn+1-z‖≤(1-αn(τ-γL))‖xn-z‖+αnK≤max⁡⁡{‖xn-z‖,Kτ-γL}∀n∈N.
Hence, {xn} is bounded. So {Tnxn} and {Vxn} are bounded.
Step  2. (∥xn+1-xn∥→0 as n→∞). From (3.1), we have
(3.6)‖xn+1-xn‖=‖PC[αnγVxn+(I-αnμF)Tnxn]-PC[αn-1γVxn-1+(I-αn-1μF)Tn-1xn-1]‖≤‖[αnγVxn+(I-αnμF)Tnxn]-[αn-1γVxn-1+(I-αn-1μF)Tn-1xn-1]‖≤‖αnγ(Vxn-Vxn-1)+γ(αn-αn-1)Vxn-1+(I-αnμF)Tnxn-(I-αnμF)Tnxn-1+Tnxn-1-Tn-1xn-1+αn-1μFTn-1xn-1-αnμFTnxn-1‖≤αnγL‖xn-xn-1‖+‖γ(αn-αn-1)Vxn-1‖+(1-αnτ)‖Tnxn-Tnxn-1‖+‖Tnxn-1-Tn-1xn-1‖+μ‖αn-1FTn-1xn-1-αnFTnxn-1‖≤(1-αn(τ-γL))‖xn-xn-1‖+DB(Tn,Tn-1)+(1-αnτ)an+‖γ(αn-αn-1)Vxn-1‖+μ‖αn-1(FTn-1xn-1-FTnxn-1)-(αn-αn-1)(FTnxn-1)‖≤(1-αn(τ-γL))‖xn-xn-1‖+DB(Tn,Tn-1)(1+μαn-1k)+M|αn-αn-1|+an,
for some constant M>0. Thus, using Lemma 2.5, we get ∥xn+1-xn∥→0 as n→∞.
Step  3. We have (∥xn-Txn∥→0 as n→∞). Note that
(3.7)‖xn-Tnxn‖≤‖xn-xn+1‖+‖xn+1-Tnxn‖=‖xn-xn+1‖+‖PC[αnγVxn+(I-αnμF)Tnxn]-PC(Tnxn)‖≤‖xn-xn+1‖+‖αnγVxn+(I-αnμF)Tnxn-Tnxn‖=‖xn-xn+1‖+αn‖γVxn-μFTnxn‖⟶0asn⟶∞.
Since
(3.8)‖xn-Txn‖≤‖xn-Tnxn‖+‖Tnxn-Txn‖≤‖xn-Tnxn‖+DB(Tn,T),
it follows that limn→∞∥xn-Txn∥=0.
Step  4. We have (limsupn→∞〈xn-x~,(γV-μF)x~〉≤0). Let us choose a subsequence {xnk} of {xn} such that
(3.9)limsup⁡n→∞〈xn-x~,(γV-μF)x~〉=lim⁡k→∞〈xnk-x~,(γV-μF)x~〉.
Without loss of generality, we may assume that xnk⇀z∈C. By using Lemma 2.4, we get that z∈F(T). Note that F(T)=F(𝒯), it follows that z∈F(𝒯). Hence from (3.2), we get the following:
(3.10)limsup⁡n→∞〈xn-x~,(γV-μF)x~〉=〈z-x~,(γV-μF)x~〉≤0.
Step  5. We have (xn→x~ as n→∞). Set yn=αnγVxn+(I-αnμF)Tnxn and γn=αn(τ-γL). Noticing that xn+1=PC(yn). From (3.1), we have
(3.11)‖xn+1-x~‖2=〈yn-x~,xn+1-x~〉+〈PC(yn)-yn,PC(yn)-x~〉≤〈yn-x~,xn+1-x~〉=αn〈γVxn-μFx~,xn+1-x~〉+〈(I-αnμF)Tnxn-(I-αnμF)Tnx~,xn+1-x~〉=αnγ〈Vxn-Vx~,xn+1-x~〉+αn〈γVx~-μFx~,xn+1-x~〉+〈(I-αnμF)Tnxn-(I-αnμF)Tnx~,xn+1-x~〉≤αnγL‖xn-x~‖‖xn+1-x~‖+αn〈(γV-μF)x~,xn+1-x~〉+(1-αnτ)(‖xn-x~‖+an)‖xn+1-x~‖=(1-αn(τ-γL))‖xn-x~‖‖xn+1-x~‖+αn〈(γV-μF)x~,xn+1-x~〉+(1-αnτ)an‖xn+1-x~‖≤(1-αn(τ-γL))12(‖xn-x~‖2+‖xn+1-x~‖2)+αn〈(γV-μF)x~,xn+1-x~〉+an‖xn+1-x~‖.
Hence, we have
(3.12)‖xn+1-x~‖2≤1-αn(τ-γL)1+αn(τ-γL)‖xn-x~‖2+2αn1+γn〈(γV-μF)x~,xn+1-x~〉+2an1+γn‖xn+1-x~‖≤(1-αn(τ-γL))‖xn-x~‖2+2αn1+γn〈(γV-μF)x~,xn+1-x~〉+2an1+γn‖xn+1-x~‖=(1-γn)‖xn-x~‖2+γnδn+2an1+γn‖xn+1-x~‖,
where
(3.13)δn=2(1+γn)(τ-γL)〈(γV-μF)x~,xn+1-x~〉.
Noticing that limn→∞an/αn=0, it follows from Lemma 2.5 that limn→∞xn=x~. This completes the proof.
Now, we derive the main result of Ceng et al. ([13], Theorem  3.2) as the following corollary.Corollary 3.2.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C→H be a k-Lipschitzian and η-strongly monotone operator and V:C→H be an L-Lipschitzian mapping. Let T:C→C be a nonexpansive mapping such that F(T)≠∅. Suppose that 0<μ<2η/k2 and 0≤γL<τ, where τ=1-1-μ(2η-μk2). For an arbitrary x1∈C, consider the sequence {xn} generated by the following iterative process:
(3.14)x1∈C,xn+1=PC[αnγVxn+(I-αnμF)Txn]∀n∈N,
where {αn} is a sequence in (0,1) satisfying the conditions (a) and (b) of Theorem 3.1.
Then, the sequence{xn} converges strongly to x~∈F(T), where x~ is the unique solution of the following variational inequality:
(3.15)〈(μF-γV)x~,x~-y〉≤0∀y∈F(T).Again, we derive the result of Tian ([11], Theorem  3.2) as the following corollary.Corollary 3.3.
LetH be a real Hilbert space. Let f be an α-contraction on H and F:H→H be a k-Lipschitzian and η-strongly monotone operator. Let T:H→H be a nonexpansive mapping such that F(T)≠∅. Suppose that 0<μ<2η/k2 and 0<γα<τ, where τ=μ(η-μk2/2). For an arbitrary x1∈H, consider the sequence {xn} generated by the following iterative process:
(3.16)x1∈H,xn+1=αnγf(xn)+(I-αnμF)Txn∀n∈N,
where {αn} is a sequence in (0,1) satisfying the conditions (a) and (b) of Theorem 3.1.
Then, the sequence{xn} converges strongly to x~∈F(T), where x~ is the unique solution of the following variational inequality:
(3.17)〈(μF-γf)x~,x~-y〉≤0∀y∈F(T).The following result obtains immediately from Theorem3.1.Corollary 3.4.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C→H be a k-Lipschitzian and η-strongly monotone operator and V:C→H be an L-Lipschitzian mapping. Let {Tn} be a sequence of nonexpansive mappings from C into itself such that ⋂n=1∞‍F(Tn)≠∅ and T be a mapping from C into itself defined by Tx=limn→∞Tnx for all x∈C. Suppose that 0<μ<2η/k2 and 0≤γL<τ, where τ=1-1-μ(2η-μk2). For an arbitrary x1∈C, consider the sequence {xn} in C generated by the following iterative process:
(3.18)x1∈C,xn+1=PC[αnγVxn+(I-αnμF)Tnxn]∀n∈N,
where {αn} is a sequence in (0,1) satisfying the conditions (a)–(c) of Theorem 3.1.
Then, the sequence{xn} converges strongly to x~∈⋂n=1∞‍F(Tn), where x~ is the unique solution of the following variational inequality:
(3.19)〈(μF-γV)x~,x~-y〉≤0∀y∈⋂n=1∞F(Tn).


## 4. Application

Recall that the so-called problem of image recovery is essentially to find a common element of finitely many nonexpansive retractsC1,C2,…,Cr of C with ⋂i=1r‍Ci≠∅. It is easy to see that every nonexpansive retraction Pi of C onto Ci is a nonexpansive mapping of C into itself. There is no doubt that the problem of image recovery is equivalent to finding a common fixed point of finitely many nonexpansive mappings P1,P2,…,Pr of C into itself. Applying our main result, we obtain the following result which improves a number of results connected to the problem of image recovery.Theorem 4.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C→H be a k-Lipschitzian and η-strongly monotone operator and V:C→H be an L-Lipschitzian mapping. Let λi>0(i=1,2,3,…,N) such that ∑i=1N‍λi=1 and T1,T2,T3,…,TN:C→C be nonexpansive mappings such that ⋂i=1N‍F(Ti)≠∅. Suppose that 0<μ<2η/k2 and 0≤γL<τ, where τ=1-1-μ(2η-μk2). For an arbitrary x1∈C, consider the sequence {xn} in C generated by the following iterative process:(4.1)x1∈C,xn+1=PC[αnγVxn+(I-αnμF)∑i=1N‍λiTixn]∀n∈N,
where {αn} is a sequence in (0,1) satisfying the conditions (a) and (b) of Theorem 3.1.
Then, the sequence{xn} converges strongly to x~∈⋂i=1N‍F(Ti), where x~ is the unique solution of the following variational inequality:
(4.2)〈(μF-γV)x~,x~-y〉≤0∀y∈⋂i=1NF(Ti).Proof.
DefineT=∑i=1N‍λiTi. Then T is nonexpansive mapping from C into itself. Thus, using Lemma 2.6, we get F(T)=⋂i=1N‍F(Ti). Therefore, the proof follows from Corollary 3.2.


## 5.   Numerical Example

For showing the effectiveness and convergence of the sequence generated by the considered iterative scheme, we discuss the following example.Example 5.1.
LetH=ℝ and C=[0,1]. Let T be a self-mapping defined by Tx=1-x for all x∈C. Let F,V:C→H be two mappings defined by Fx=4x and Vx=2x for all x∈C, where F is a k-Lipschitzian and η-strongly monotone, and V is an L-Lipschitzian mapping. We take 0<μ<2η/k2 and 0≤γL<τ, and we have μ=1/4,τ=1 and γ=1/4. Define {αn} in (0,1) by αn=1/n+1. Without loss of generality, we may assume that an=1/n3/2 for all n∈ℕ. For each n∈ℕ, define Tn:C→C by
(5.1)Tnx={1-x,ifx∈[0,1),an,ifx=1.
In [15], it is proved that 𝒯={Tn} is a sequence of nearly nonexpansive mappings from C into itself such that F(𝒯)={1/2} and Tx=limn→∞Tnx for all x∈C, where T is nonexpansive mapping.It can be observed that all the assumptions of Theorem3.1 are satisfied and the sequence {xn} generated by (3.1) converges to a unique solution 1/2 of variational inequality (3.2) over F(𝒯). The graphical presentation of the convergence of the sequence {xn} generated by the iterative scheme (3.1) is given in Figure 1.Figure 1



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