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Fixed point solutions for variational inequalities in image restoration over quniformly smooth Banach spaces
Journal of Inequalities and Applications volume 2014, Article number: 473 (2014)
Abstract
In this paper, we introduce new implicit and explicit iterative methods for finding a common fixed point set of an infinite family of strict pseudocontractions by the sunny nonexpansive retractions in a real quniformly and uniformly convex Banach space which admits a weakly sequentially continuous generalized duality mapping. Then we prove the strong convergence under mild conditions of the purposed iterative scheme to a common fixed point of an infinite family of strict pseudocontractions which is a solution of some variational inequalities. Furthermore, we apply our results to study some strong convergence theorems in {L}_{p} and {\ell}_{p} spaces with 1<p<\mathrm{\infty}. Our results mainly improve and extend the results announced by Ceng et al. (Comput. Math. Appl. 61:24472455, 2011) and many authors from Hilbert spaces to Banach spaces. Finally, we give some numerical examples for support our main theorem in the end of the paper.
MSC:47H09, 47H10, 47H17, 47J25, 49J40.
1 Introduction
Let {C}_{1},{C}_{2},\dots ,{C}_{n} be nonempty, closed, and convex subsets of a real Hilbert space H such that {\bigcap}_{i=1}^{n}{C}_{i}\ne \mathrm{\varnothing}. The problem of image recovery in a Hilbert space setting by using convex of metric projections {P}_{{C}_{i}}, may be stated as follows: the original unknown image z is known a priori to belong to the intersection of {\{{C}_{i}\}}_{i=1}^{n}; given only the metric projections {P}_{{C}_{i}} of H onto {C}_{i} for i=1,2,\dots ,n recover z by an iterative scheme. Youla and Webb [1] first used iterative methods for applied in image restoration. The problems of image recovery have been studied in a Banach space setting by Kitahara and Takahashi [2] (see also [3, 4]) by using convex combinations of sunny nonexpansive retractions in uniformly convex Banach spaces. On the other hand, Alber [5] studied the problem of image recovery by the products of generalized projections in a uniformly convex and uniformly smooth Banach space whose duality mapping is weakly sequentially continuous (see also [6, 7]). Nakajo et al. [8] and Kimura et al. [9] considered this problem by the sunny nonexpansive retractions and proved convergence of the iterative sequence to a common point of countable nonempty, closed, and convex subsets in a uniformly convex and smooth Banach space, and in a strictly convex, smooth and reflexive Banach space having the KadecKlee property, respectively. Some iterative methods have been studied in problem of image recovery by numerous authors (see [2–5, 10–12]).
The problems of image recovery are connected with the convex feasibility problem, convex minimization problems, multipleset split feasibility problems, common fixed point problems, and variational inequalities. In particular, variational inequality theory has been studied widely in several branches of pure and applied sciences. This field is dynamics and is experiencing an explosive growth in both theory and applications. Indeed, applications of the variational inequalities span as diverse disciplines as differential equations, timeoptimal control, optimization, mathematical programming, mechanics, finance, and so on. Note that most of the variational problems, including minimization or maximization of functions, variational inequality problems, quasivariational inequality problems, decision and management sciences, and engineering sciences problems. Recently, some iterative methods have been developed for solving the fixed point problems and variational inequality problems in quniformly smooth Banach spaces by numerous authors (see [13–24]).
Let A be a strongly positive bounded linear operator on H, that is, there exists a constant \overline{\gamma}>0 such that
Remark 1.1 From the definition of operator A, we note that a strongly positive bounded linear operator A is a \parallel A\parallelLipschitzian and ηstrongly monotone operator.
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:
where C is the fixed point set of a nonexpansive mapping T on H and u is a given point in H.
In 2006, Marino and Xu [25] introduced and considered the following a general iterative method:
where A is a strongly positive bounded linear operator on a real Hilbert space H. They proved that if the sequence \{{\alpha}_{n}\} satisfies appropriate conditions, then the sequence \{{x}_{n}\} generated by (1.3) converges strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem
where C is the fixed point set of a nonexpansive mapping T and h is a potential function for γf (i.e., {h}^{\prime}(x)=\gamma f(x) for all x\in H).
On the other hand, Yamada [26] introduced a hybrid steepest descent method for a nonexpansive mapping T as follows:
where F is a κLipschitzian and ηstrongly monotone operator on a real Hilbert space H with constants \kappa ,\eta >0 and 0<\mu <\frac{2\eta}{{\kappa}^{2}}. He proved that if \{{\lambda}_{n}\} satisfy the appropriate conditions, then the sequence \{{x}_{n}\} generated by (1.6) converges strongly to the unique solution of the variational inequality
Tian [27] combined the iterative method (1.3) with the Yamada method (1.6) and considered a general iterative method for a nonexpansive mapping T on a real Hilbert space H as follows:
Then he proved that the sequence \{{x}_{n}\} generated by (1.8) converges strongly to the unique solution of variational inequality
In 2011, Ceng et al. [28] combined the iterative method (1.3) with Tian’s method (1.8) and consider the following a general composite iterative method:
where A is a strongly positive bounded linear operator on H with coefficient \overline{\gamma}\in (1,2), and \{{\alpha}_{n}\}\subset (0,1) and \{{\beta}_{n}\}\subset (0,1] satisfy appropriate conditions. Then they proved that the sequence \{{x}_{n}\} generated by (1.10) converges strongly to the unique solution {x}^{\ast}\in C of the variational inequality
where C=Fix(T).
In this paper, motivated by the above facts, we introduce new implicit and explicit iterative methods for finding a common fixed point set of an infinite family of strict pseudocontractions by the sunny nonexpansive retractions in a real quniformly and uniformly convex Banach space X which admits a weakly sequentially continuous generalized duality mapping. Consequently, we prove the strong convergence under mild conditions of the purposed iterative scheme to a common fixed point of an infinite family of strict pseudocontractions of nonempty, closed, and convex subsets of X which is a solution of some variational inequalities. Furthermore, we apply our results to the study of some strong convergence theorems in {L}_{p} and {\ell}_{p} spaces with 1<p<\mathrm{\infty}. Our results extend the main result of Ceng et al. [28] in several aspects and the work of many authors from Hilbert spaces to Banach spaces. Finally, we give some numerical examples to support our main theorem in the end of the paper.
2 Preliminaries
Throughout this paper, we denote by X and {X}^{\ast} a real Banach space and the dual space of X, respectively. Let q>1 be a real number. The generalized duality mapping {J}_{q}:X\to {2}^{{X}^{\ast}} is defined by
where \u3008\cdot ,\cdot \u3009 denotes the duality pairing between X and {X}^{\ast}. In particular, {J}_{q}={J}_{2} is called the normalized duality mapping and {J}_{q}(x)={\parallel x\parallel}^{q2}{J}_{2}(x) for x\ne 0. If X:=H is a real Hilbert space, then J=I, where I is the identity mapping. It is well known that if X is smooth, then {J}_{q} is singlevalued, which is denoted by {j}_{q} (see [29]).
A Banach space X is said to be strictly convex if \frac{\parallel x+y\parallel}{2}<1 for all x,y\in X with \parallel x\parallel =\parallel y\parallel =1 and x\ne y. A Banach space X is said to be uniformly convex if, for each \u03f5>0, there exists \delta >0 such that for x,y\in X with \parallel x\parallel ,\parallel y\parallel \le 1 and \parallel xy\parallel \ge \u03f5, \frac{\parallel x+y\parallel}{2}\le 1\delta holds. Let S(X)=\{x\in X:\parallel x\parallel =1\}. The norm of X is said to be Gâteaux differentiable (or X is said to be smooth) if the limit
exists for each x,y\in S(X). The norm of X is said to be uniformly Gâteaux differentiable, if, for each y\in S(X), the limit is attained uniformly for x\in S(X).
Let {\rho}_{X}:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) be the modulus of smoothness of X defined by
A Banach space X is said to be uniformly smooth if \frac{{\rho}_{X}(t)}{t}\to 0 as t\to 0. Suppose that q>1, then X is said to be quniformly smooth if there exists c>0 such that {\rho}_{X}(t)\le c{t}^{q} for all t>0. It is shown in [30] (see also [31]) that there is no Banach space which is quniformly smooth with q>2. If X is quniformly smooth, then X is uniformly smooth. It is well known that each uniformly convex Banach space X is reflexive and strictly convex and every uniformly smooth Banach space X is a reflexive Banach space with uniformly Gâteaux differentiable norm (see [29]). Typical examples of both uniformly convex and uniformly smooth Banach spaces are {L}_{p}, where p>1. More precisely, {L}_{p} is min\{p,2\}uniformly smooth for every p>1.
Let C be a nonempty, closed, and convex subset of X and T be a selfmapping on C. We denote the fixed points set of the mapping T by Fix(T)=\{x\in C:Tx=x\}.
Definition 2.1 A mapping T:C\to C is said to be:

(i)
λstrictly pseudocontractive [32] if, for all x,y\in C, there exist \lambda >0 and {j}_{q}(xy)\in {J}_{q}(xy) such that
\u3008TxTy,{j}_{q}(xy)\u3009\le {\parallel xy\parallel}^{q}\lambda {\parallel (IT)x(IT)y\parallel}^{q},(2.1)or equivalently
\u3008(IT)x(IT)y,{j}_{q}(xy)\u3009\ge \lambda {\parallel (IT)x(IT)y\parallel}^{q}.(2.2) 
(ii)
LLipschitzian if, for all x,y\in C, there exists a constant L>0 such that
\parallel TxTy\parallel \le L\parallel xy\parallel .
If 0<L<1, then T is a contraction and if L=1, then T is a nonexpansive mapping. By the definition, we know that every λstrictly pseudocontractive mapping is (\frac{1+\lambda}{\lambda})Lipschitzian (see [33]).
Remark 2.2 Let C be a nonempty subset of a real Hilbert space H and T:C\to C be a mapping. Then T is said to be kstrictly pseudocontractive [32] if, for all x,y\in C, there exists k\in [0,1) such that
It is well known that (2.3) is equivalent to the following inequality:
A mapping F:C\to X is said to be accretive if, for all x,y\in C, there exists {j}_{q}(xy)\in {J}_{q}(xy) such that
For some \eta >0, F:C\to X is said to be strongly accretive if, for all x,y\in C, there exists {j}_{q}(xy)\in {J}_{q}(xy) such that
Remark 2.3 If X:=H is a real Hilbert space, accretive and strongly accretive mappings coincide with monotone and strongly monotone mappings, respectively.
Let D be a nonempty subset of C. A mapping Q:C\to D is said to be sunny [34] if
whenever Qx+t(xQx)\in C for x\in C and t\ge 0. A mapping Q:C\to D is said to be retraction if Qx=x for all x\in D. Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. It is well known that if X:=H is a real Hilbert space, then a sunny nonexpansive retraction Q is coincident with the metric projection from X onto C.
Lemma 2.4 ([14])
Let C be a closed and convex subset of a smooth Banach space X. Let D be a nonempty subset of C. Let Q:C\to D be a retraction and let j, {j}_{q} be the normalized duality mapping and generalized duality mapping on X, respectively. Then the following are equivalent:

(a)
Q is sunny and nonexpansive.

(b)
{\parallel QxQy\parallel}^{2}\le \u3008xy,j(QxQy)\u3009 for all x,y\in C.

(c)
\u3008xQx,j(yQx)\u3009\le 0 for all x\in C and y\in D.

(d)
\u3008xQx,{j}_{q}(yQx)\u3009\le 0 for all x\in C and y\in D.
Lemma 2.5 ([35])
Suppose that q>1. Then the following inequality holds:
for arbitrary positive real numbers a, b.
In a real quniformly smooth Banach space, Xu [36] proved the following important inequality:
Lemma 2.6 ([36])
Let X be a real quniformly smooth Banach space. Then the following inequality holds:
for all x,y\in X and for some {C}_{q}>0.
Remark 2.7 The constant {C}_{q} satisfying (2.4) is called the best quniform smoothness constant.
Lemma 2.8 ([21])
Let C be a nonempty and convex subset of a real quniformly smooth Banach space X and T:C\to C be a λstrict pseudocontraction. For \gamma \in (0,1), define Sx=(1\gamma )x+\gamma Tx. Then, as \gamma \in (0,\nu ), \nu =min\{1,{(\frac{q\lambda}{{C}_{q}})}^{\frac{1}{q1}}\}, S:C\to C is nonexpansive and Fix(S)=Fix(T), where {C}_{q} is the best quniform smoothness constant.
Definition 2.9 ([37])
Let C be a nonempty, closed, and convex subset of a real quniformly smooth Banach space X. Let {T}_{n,k}={\theta}_{n,k}{S}_{k}+(1{\theta}_{n,k})I, where {S}_{k}:C\to C is {\lambda}_{k}strict pseudocontraction and \{{t}_{n}\} be a nonnegative real sequence with 0\le {t}_{n}\le 1, \mathrm{\forall}n\in \mathbb{N}. For n\ge 1, define a mapping {W}_{n}:C\to C as follows:
Such a mapping {W}_{n} is called the Wmapping generated by {T}_{n,n},{T}_{n,n1},\dots ,{T}_{n,1} and {t}_{n},{t}_{n1},\dots ,{t}_{1}.
Throughout this paper, we will assume that \{{\theta}_{n,k}\} satisfies the following conditions:
(H1) {\theta}_{n,k}\in (0,\nu ], \nu =min\{1,{(\frac{q\overline{\lambda}}{{C}_{q}})}^{\frac{1}{q1}}\} with \overline{\lambda}=inf{\lambda}_{k}>0, \mathrm{\forall}n,k\in \mathbb{N};
(H2) {\theta}_{n+1,k}{\theta}_{n,k}\le {a}_{n}, \mathrm{\forall}n\in \mathbb{N} and 1\le k\le n with {\sum}_{n=1}^{\mathrm{\infty}}{a}_{n}<\mathrm{\infty};
The hypothesis (H2) secures the existence of {lim}_{n\to \mathrm{\infty}}{\theta}_{n,k}, \mathrm{\forall}k\in \mathbb{N}. Set {\theta}_{1,k}:={lim}_{n\to \mathrm{\infty}}{\theta}_{n,k}, \mathrm{\forall}n\in \mathbb{N}. Furthermore, we assume
(H3) {\theta}_{1,k}>0, \mathrm{\forall}k\in \mathbb{N}.
It is obvious that {\theta}_{1,k} satisfies (H1). Using condition (H3), from {T}_{n,k}={\theta}_{n,k}{S}_{k}+(1{\theta}_{n,k})I, we define mappings {T}_{1,k}x:={lim}_{n\to \mathrm{\infty}}{T}_{n,k}x={\theta}_{1,k}{S}_{k}x+(1{\theta}_{1,k})x, \mathrm{\forall}x\in C.
Lemma 2.10 ([37])
Let C be a nonempty, closed, and convex subset of a real quniformly smooth and strictly convex Banach space X. Let {T}_{n,i}={\theta}_{n,i}{S}_{i}+(1{\theta}_{n,i})I, where {S}_{i}:C\to C (i=1,2,\dots) is {\lambda}_{i}strict pseudocontraction with {\bigcap}_{n=1}^{\mathrm{\infty}}Fix({S}_{n})\ne \mathrm{\varnothing} and inf{\lambda}_{i}>0. Let {t}_{1},{t}_{2},\dots be nonnegative real numbers such that 0<{t}_{n}\le b<1, \mathrm{\forall}n\ge 1. Assume the sequence \{{\theta}_{n,k}\} satisfies (H1)(H3). Then

(1)
{W}_{n} is nonexpansive and Fix({W}_{n})={\bigcap}_{n=1}^{\mathrm{\infty}}Fix({S}_{n}) for each n\ge 1;

(2)
for each x\in C and for each positive integer k, the limit {lim}_{n\to \mathrm{\infty}}{U}_{n,k} exists;

(3)
the mapping W:C\to C defined by
Wx:=\underset{n\to \mathrm{\infty}}{lim}{W}_{n}x=\underset{n\to \mathrm{\infty}}{lim}{U}_{n,1}x,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in C,
is a nonexpansive mapping satisfying Fix(W)={\bigcap}_{n=1}^{\mathrm{\infty}}Fix({S}_{n}) and it is called the Wmapping generated by {S}_{1},{S}_{2},\dots and {t}_{1},{t}_{2},\dots and {\theta}_{n,k}, \mathrm{\forall}n\in \mathbb{N} and 1\le k\le n.
Lemma 2.11 ([37])
Let C be a nonempty, closed, and convex subset of a real quniformly smooth and strictly convex Banach space X. Let {T}_{n,i}={\theta}_{n,i}{S}_{i}+(1{\theta}_{n,i})I, where {S}_{i}:C\to C (i=1,2,\dots) is {\lambda}_{i}strict pseudocontraction with {\bigcap}_{n=1}^{\mathrm{\infty}}Fix({S}_{n})\ne \mathrm{\varnothing} and inf{\lambda}_{i}>0. Let {t}_{1},{t}_{2},\dots be nonnegative real numbers such that 0<{t}_{n}\le b<1, \mathrm{\forall}n\ge 1. Assume the sequence \{{\theta}_{n,k}\} satisfies (H1)(H3). If \{{\omega}_{n}\} is a bounded sequence in C, then
In the following, the notation ⇀ and → denote the weak and strong convergence, respectively. The duality mapping {J}_{q} from a smooth Banach space X into {X}^{\ast} is said to be weakly sequentially continuous generalized duality mapping if, for all \{{x}_{n}\}\subset X, {x}_{n}\rightharpoonup x implies {J}_{q}({x}_{n}){\rightharpoonup}^{\ast}{J}_{q}(x).
A Banach space X is said to be satisfy Opial’s condition [38], that is, for any sequence \{{x}_{n}\} in X, {x}_{n}\rightharpoonup x implies that
By Theorem 3.2.8 in [39], it is well known that if X admits a weakly sequentially continuous generalized duality mapping, then X satisfies Opial’s condition.
Lemma 2.12 ([13])
Let C be a nonempty, closed, and convex subset of a real quniformly smooth Banach space X which admits weakly sequentially continuous generalized duality mapping {j}_{q} from X into {X}^{\ast}. Let T:C\to C be a nonexpansive mapping. Then, for all \{{x}_{n}\}\subset C, if {x}_{n}\rightharpoonup x and {x}_{n}T{x}_{n}\to 0, then x=Tx.
Lemma 2.13 ([40])
Let \{{a}_{n}\}, \{{\mu}_{n}\}, and \{{\delta}_{n}\} be real sequences of nonnegative numbers such that
where {\sigma}_{n}\in (0,1), {\sum}_{n=1}^{\mathrm{\infty}}{\sigma}_{n}=\mathrm{\infty}, {\mu}_{n}=\circ ({\sigma}_{n}) and {\sum}_{n=1}^{\mathrm{\infty}}{\delta}_{n}<\mathrm{\infty}. Then {lim}_{n\to \mathrm{\infty}}{a}_{n}=0.
3 Main results
In order to prove our main result, the following lemma is needed.
Lemma 3.1 Let C be a nonempty, closed, and convex subset of a real quniformly smooth Banach space X with the best quniform smoothness constant {C}_{q}>0. Let F:C\to X be a κLipschitzian and ηstrongly accretive operator with constants \kappa ,\eta >0. Let 0<\mu <{(\frac{q\eta}{{C}_{q}{\kappa}^{q}})}^{\frac{1}{q1}} and \tau =\mu (\eta \frac{{C}_{q}{\mu}^{q1}{\kappa}^{q}}{q}). Then for t\in (0,min\{1,\frac{1}{q\tau}\}), the mapping S:C\to X defined by S:=It\mu F is a contraction with constant 1t\tau.
Proof Since 0<\mu <{(\frac{q\eta}{{C}_{q}{\kappa}^{q}})}^{\frac{1}{q1}} and t\in (0,min\{1,\frac{1}{q\tau}\}). This implies that 1t\tau \in (0,1). From Lemma 2.6, for all x,y\in C, we have
It follows that
Hence, we have S:=It\mu F is a contraction with constant 1t\tau. □
Lemma 3.2 Let C be a nonempty, closed, and convex subset of a real quniformly smooth Banach space X and G:C\to X be a mapping.

(i)
If G is a δstrongly accretive and λstrictly pseudocontractive mapping wit \delta +\lambda >1, then IG is a contraction with constant {L}_{\delta ,\lambda}:={(\frac{1\delta}{\lambda})}^{\frac{1}{q}}.

(ii)
If G is a δstrongly accretive and λstrictly pseudocontractive mapping with \delta +\lambda >1. For a fixed number t\in (0,1), then ItG is a contraction with constant 1(1{L}_{\delta ,\lambda})t.
Proof (i) For all x,y\in C, from (2.2), we have
Observe that
It follows that
Hence, IG is a contraction with constant {L}_{\delta ,\lambda}.
(ii) Since IG is a contraction with constant {L}_{\delta ,\lambda}. For all t\in (0,1), we have
Hence, ItG is a contraction with constant 1(1{L}_{\delta ,\lambda})t. This completes the proof. □
3.1 Implicit iteration scheme
Let C be a nonempty, closed, and convex subset of a real reflexive and quniformly smooth Banach space X which admits a weakly sequentially continuous generalized duality mapping {j}_{q}. Let {Q}_{C} be a sunny nonexpansive retraction from X onto C. Let F:C\to X be a κLipschitzian and ηstrongly accretive operator with constants \kappa ,\eta >0, G:C\to X be a δstrongly accretive and λstrictly pseudocontractive mapping with \delta +\lambda >1, V:C\to X be an LLipschitzian mapping with constant L\ge 0 and T:C\to C be a nonexpansive mapping such that Fix(T)\ne \mathrm{\varnothing}. Let 0<\mu <{(\frac{q\eta}{{C}_{q}{\kappa}^{q}})}^{\frac{1}{q1}} and 0\le \gamma L<\tau, where \tau =\mu (\eta \frac{{C}_{q}{\mu}^{q1}{\kappa}^{q}}{q}). For each \sigma \in (\frac{{L}_{\delta ,\lambda}}{\tau \gamma L},min\{1,\frac{1}{q\tau},\frac{1+{L}_{\delta ,\lambda}}{\tau \gamma L}\}) and t\in (0,1), we define a mapping {S}_{t}:C\to C defined by
It is easy to see immediately that {S}_{t} is a contraction. Indeed, for all x,y\in C, from Lemmas 3.1 and 3.2(ii), we have
where \theta :=\sigma (\tau \gamma L){L}_{\delta ,\lambda}. Since \tau \gamma L>0 and {L}_{\delta ,\lambda}\in (0,1), observe that
It follows that
and
This implies that \theta =\sigma (\tau \gamma L){L}_{\delta ,\lambda}\in (0,1), which together with t\in (0,1) gives
Hence {S}_{t} is a contraction. By the Banach contraction principle, {S}_{t} has a unique fixed point, denote by {x}_{t}, which uniquely solves the fixed point equation
The following proposition summarizes the properties of the net \{{x}_{t}\}.
Proposition 3.3 Let \{{x}_{t}\} be defined by (3.2). Then the following hold:

(i)
\{{x}_{t}\} is bounded for each t\in (0,1);

(ii)
{lim}_{t\to 0}\parallel {x}_{t}T{x}_{t}\parallel =0;

(iii)
\{{x}_{t}\} defines a continuous curve from (0,1) into C.
Proof (i) Take p\in Fix(T), and denote a mapping {S}_{t}:C\to C by
From (3.1), we have
where \theta :=\sigma (\tau \gamma L){L}_{\delta ,\lambda}. It follows that
Hence \{{x}_{t}\} is bounded, so are \{V{x}_{t}\}, \{FT{x}_{t}\}, and \{GT{x}_{t}\}.

(ii)
By definition of \{{x}_{t}\}, we have
\begin{array}{rcl}\parallel {x}_{t}T{x}_{t}\parallel & =& \parallel {Q}_{C}[(ItG)T{x}_{t}+t(T{x}_{t}\sigma (\mu FT{x}_{t}\gamma V{x}_{t}))]{Q}_{C}T{x}_{t}\parallel \\ \le & t\parallel (IG)T{x}_{t}\sigma (\mu FT{x}_{t}\gamma V{x}_{t})\parallel \to 0\phantom{\rule{1em}{0ex}}\text{as}t\to 0.\end{array} 
(iii)
Take t,{t}_{0}\in (0,1) and calculate
\begin{array}{rcl}\parallel {x}_{t}{x}_{{t}_{0}}\parallel & =& \parallel {Q}_{C}[(ItG)T{x}_{t}+t(T{x}_{t}\sigma (\mu FT{x}_{t}\gamma V{x}_{t}))]\\ {Q}_{C}[(I{t}_{0}G)T{x}_{{t}_{0}}+t(T{x}_{{t}_{0}}\sigma (\mu FT{x}_{{t}_{0}}\gamma V{x}_{{t}_{0}}))]\parallel \\ \le & \parallel ({t}_{0}t)GT{x}_{t}+(I{t}_{0}G)(T{x}_{t}T{x}_{{t}_{0}})+(t{t}_{0})[T{x}_{t}\sigma (\mu FT{x}_{t}\gamma V{x}_{t})]\\ +{t}_{0}[T{x}_{t}\sigma (\mu FT{x}_{{t}_{0}}\gamma V{x}_{{t}_{0}})[T{x}_{{t}_{0}}\sigma (\mu FT{x}_{{t}_{0}}\gamma V{x}_{{t}_{0}})]]\parallel \\ =& \parallel ({t}_{0}t)GT{x}_{t}+(I{t}_{0}G)(T{x}_{t}T{x}_{{t}_{0}})+(t{t}_{0})[T{x}_{t}\sigma (\mu FT{x}_{t}\gamma V{x}_{t})]\\ +{t}_{0}[\sigma \gamma (V{x}_{t}V{x}_{{t}_{0}})+(I\sigma \mu F)(T{x}_{t}T{x}_{{t}_{0}})]\parallel \\ \le & t{t}_{0}\parallel GT{x}_{t}\parallel +(1(1{L}_{\delta ,\lambda}))\parallel {x}_{t}{x}_{{t}_{0}}\parallel \\ +t{t}_{0}\parallel T{x}_{t}\sigma (\mu FT{x}_{t}\gamma V{x}_{t})\parallel \\ +{t}_{0}(1\sigma (\tau \gamma L))\parallel {x}_{t}{x}_{{t}_{0}}\parallel .\end{array}
It follows that
Since \{V{x}_{t}\}, \{FT{x}_{t}\}, and \{GT{x}_{t}\} are bounded. Hence \{{x}_{t}\} defines a continuous curve from (0,1) into C. □
Theorem 3.4 Assume that \{{x}_{t}\} is defined by (3.2), then \{{x}_{t}\} converges strongly to {x}^{\ast}\in Fix(T) as t\to 0, where {x}^{\ast} is the unique solution of the variational inequality
Proof We observe that
It follows that
First, we show the uniqueness of solution of the variational inequality. Suppose that \tilde{x},{x}^{\ast}\in Fix(T) are solutions of (3.3), then
and
Adding up (3.6) and (3.7), and from Lemma 3.2(i), we obtain
On the other hand, we observe from (3.5) that
Note that (3.8) implies that {x}^{\ast}=\tilde{x} and the uniqueness is proved. Below, we use \tilde{x} to denote the unique solution of the variational inequality (3.3).
Next, we show that {x}_{t}\to {x}^{\ast} as t\to 0. Set {x}_{t}={Q}_{C}{y}_{t}, where {y}_{t}=(ItG)T{x}_{t}+t(T{x}_{t}\sigma (\mu FT{x}_{t}\gamma V{x}_{t})). Assume that \{{t}_{n}\}\subset (0,1) is a sequence such that {t}_{n}\to 0 as n\to \mathrm{\infty}. Put {x}_{n}:={x}_{{t}_{n}} and {y}_{n}:={y}_{{t}_{n}}. For z\in Fix(T), we note that
By Lemma 2.4, we have
It follows from (3.9) and (3.10) that
which implies that
In particular, we have
By reflexivity of a Banach space X and boundedness of \{{x}_{n}\}, there exists a subsequence \{{x}_{{n}_{i}}\} of \{{x}_{n}\} such that {x}_{{n}_{i}}\rightharpoonup \tilde{x} as i\to \mathrm{\infty}. Since a Banach space X has a weakly sequentially continuous generalized duality mapping and by (3.11), we obtain {x}_{{n}_{i}}\to \tilde{x}. By Proposition 3.3(ii), we have {x}_{{n}_{i}}T{x}_{{n}_{i}}\to 0 as i\to \mathrm{\infty}. Hence, it follows from Lemma 2.12 that \tilde{x}\in Fix(T).
Next, we show that \tilde{x} solves the variational inequality (3.3). We note that
we derive
Since IT is accretive (i.e., \u3008(IT)x(IT)y,{j}_{q}(xy)\u3009\ge 0 for x,y\in C). For all v\in Fix(T), it follows from (3.10) and (3.12) that
where {M}_{1}>0 is an appropriate constant such that {M}_{1}={sup}_{t\in (0,1)}\{\parallel G\parallel {\parallel {x}_{t}v\parallel}^{q1},\sigma \mu \parallel F\parallel {\parallel {x}_{t}v\parallel}^{q1}\}. Now, replacing t in (3.13) with {t}_{n} and taking the limit as n\to \mathrm{\infty}, we notice that {x}_{{t}_{n}}T{x}_{{t}_{n}}\to \tilde{x}T\tilde{x}=0, we obtain
That is, \tilde{x}\in Fix(T) is the solution of the variational inequality (3.3). Consequently, \tilde{x}={x}^{\ast} by uniqueness. In a summary, we have shown that each cluster point of \{{x}_{t}\} is equal to {x}^{\ast}. Therefore {x}_{t}\to {x}^{\ast} as t\to 0. This completes the proof. □
3.2 Explicit iteration scheme
Theorem 3.5 Let C be a nonempty, closed, and convex subset of a real quniformly smooth and uniformly convex Banach space X which admits a weakly sequentially continuous generalized duality mapping {j}_{q}. Let {Q}_{C} be a sunny nonexpansive retraction such that X onto C. Let F:C\to X be a κLipschitzian and ηstrongly accretive operator with constants \kappa ,\eta >0, G:C\to X be a δstrongly accretive and λstrictly pseudocontractive mapping with \delta +\lambda >1, V:C\to X be an LLipschitzian mapping with constant L\ge 0. Let {\{{S}_{i}\}}_{i=1}^{\mathrm{\infty}} be an infinite family of {\lambda}_{i}strictly pseudocontractive mapping from C into itself such that \mathcal{F}:={\bigcap}_{i=1}^{\mathrm{\infty}}Fix({S}_{i})\ne \mathrm{\varnothing}. For given {x}_{1}\in C, define the sequence \{{x}_{n}\} by
where \{{\alpha}_{n}\} is a sequence in (0,1) which satisfies the following conditions:
(C1) {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty};
(C2) {\alpha}_{n+1}{\alpha}_{n}\le \circ ({\alpha}_{n})+{\sigma}_{n} with {\sum}_{n=1}^{\mathrm{\infty}}{\sigma}_{n}<\mathrm{\infty}.
Suppose in addition that \{{\theta}_{n,k}\} satisfies (H1)(H3). Then the sequence \{{x}_{n}\} defined by (3.14) converges strongly to {x}^{\ast}\in \mathcal{F} as n\to \mathrm{\infty}, where {x}^{\ast} is the unique solution of the variational inequality
Proof From the condition (C1), we may assume, without loss of generality, that {\alpha}_{n}\le min\{1,\frac{1}{q\tau}\} for all n\in \mathbb{N}. First, we show that \{{x}_{n}\} is bounded. Take p\in \mathcal{F}, and denote a mapping {S}_{n}^{{\alpha}_{n}}:C\to C by
Then we have
From (3.1), we have
where \theta :=\sigma (\tau \gamma L){L}_{\delta ,\lambda}. By induction, we obtain
Hence, \{{x}_{n}\} is bounded, so are \{V{x}_{n}\}, \{F{W}_{n}{x}_{n}\}, and \{G{W}_{n}{x}_{n}\}.
Next, we show that \parallel {x}_{n+1}{x}_{n}\parallel \to 0 as n\to \mathrm{\infty}. Set {S}_{n}^{{\alpha}_{n}}{x}_{n}={Q}_{C}{y}_{n}, where {y}_{n}=(I{\alpha}_{n}G){W}_{n}{x}_{n}+{\alpha}_{n}({W}_{n}{x}_{n}\sigma (\mu F{W}_{n}{x}_{n}\gamma V{x}_{n})). From (2.5), we have
where {M}_{1}={inf}_{i=1,2,\dots}(\frac{1+2{\lambda}_{i}^{q1}}{{\lambda}_{i}^{q1}}){sup}_{n\ge 1}\{\parallel {x}_{n}p\parallel \} with p\in \mathcal{F}.
On the other hand, we note that
Hence, we have
where {M}_{2}={sup}_{n\ge 1}\{\parallel (1+{\alpha}_{n})I{\alpha}_{n+1}G\sigma {\alpha}_{n}\mu F\parallel ,\parallel {W}_{n+1}{x}_{n}G{W}_{n}{x}_{n}\parallel ,\sigma \parallel \mu F{W}_{n+1}{x}_{n}\gamma V{x}_{n}\parallel \}. It follows from (3.1) and (3.16) that
where M=max\{{M}_{1},{M}_{2}\}. Then, by Lemma 2.13, we have
Next, we show that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}W{x}_{n}\parallel =0. Since
From (3.18) and the condition (C1), we obtain
At the same time, observe that
It follows from (3.19) and Lemma 2.11, we have
Next, we show that
where {x}^{\ast} is the same as in Theorem 3.4. Since \{{x}_{n}\} is bounded, there exists a subsequence \{{x}_{{n}_{i}}\} of \{{x}_{n}\} such that
By reflexivity of a Banach space X and boundedness of \{{x}_{n}\}, without loss of generality, we may assume that {x}_{{n}_{i}}\rightharpoonup v as i\to \mathrm{\infty}. It follows from (3.20) and Lemma 2.12 that v\in \mathcal{F}. Since a Banach space X has a weakly sequentially continuous generalized duality mapping, we obtain
Finally, we show that {x}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}. Set {x}_{n+1}={Q}_{C}{y}_{n}, where {y}_{n}=(I{\alpha}_{n}G){W}_{n}{x}_{n}+{\alpha}_{n}({W}_{n}{x}_{n}\sigma (\mu {W}_{n}{x}_{n}\gamma V{x}_{n})). From Lemmas 2.4 and 2.5, we have
which implies that
We can write (3.22) to the formula
where {\tau}_{n}:=(\sigma (\tau \gamma L){L}_{\delta ,\lambda}){\alpha}_{n} and {\xi}_{n}:=\frac{q{\alpha}_{n}}{1+(q1)(\sigma (\tau \gamma L){L}_{\delta ,\lambda})}\u3008{x}^{\ast}G{x}^{\ast}+\sigma (\gamma V{x}^{\ast}\mu F{x}^{\ast}),{j}_{q}({x}_{n+1}{x}^{\ast})\u3009. Put {c}_{n}=max\{0,{\xi}_{n}\}, from (3.21), we have {c}_{n}\to 0 as n\to \mathrm{\infty}. Then we can rewrite (3.23) as
Therefore, by Lemma 2.13, we conclude that {x}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}. This completes the proof. □
4 Some applications
In this section, we will utilize Theorems 3.4 and 3.5 to study some strong convergence theorems in {L}_{p} (or {\ell}_{p}) spaces with 1<p<\mathrm{\infty}. It well known that Hilbert spaces, {L}_{p} (or {\ell}_{p}) spaces with 1<p<\mathrm{\infty} and the Sobolev spaces {W}_{m}^{p} with 1<p<\mathrm{\infty} are quniformly smooth, i.e.,
Furthermore, we have the following properties of {L}_{p} (or {\ell}_{p}) spaces with 1<p<\mathrm{\infty} (see [36, 39]):

(1)
For 2\le p<\mathrm{\infty}, the spaces {L}_{p} (or {\ell}_{p}) are 2uniformly smooth with {C}_{q}={C}_{2}=p1.

(2)
For 1<p\le 2, the spaces {L}_{p} (or {\ell}_{p}) are puniformly smooth with {C}_{q}={C}_{p}=(1+{t}_{p}^{p1}){(1+{t}_{p})}^{1p}, where {t}_{p} is the unique solution of the equation
(p2){t}^{p1}+(p1){t}^{p2}1=0,\phantom{\rule{1em}{0ex}}0<t<1. 
(3)
Every Hilbert spaces are 2uniformly smooth with {C}_{q}={C}_{2}=1.

(4)
Every {L}_{p} (or {\ell}_{p}) spaces with 1<p<\mathrm{\infty} are quniformly smooth and uniformly convex.

(5)
Every {\ell}_{p} spaces with 1<p<\mathrm{\infty} have weakly sequentially continuous generalized duality mappings, but {L}_{p} spaces (1<p<\mathrm{\infty}, p\ne 2) do not have weakly sequentially continuous generalized duality mappings.
Lemma 4.1 Let X:={L}_{p} (or {\ell}_{p}) with 1<p\le 2. Let C be a nonempty, closed, and convex subset of X. Let F:C\to X be a κLipschitzian and ηstrongly accretive operator with constants \kappa ,\eta >0. Let 0<\mu <{(\frac{p\eta}{{D}_{p}{\kappa}^{p}})}^{\frac{1}{p1}} and \tau =\mu (\eta \frac{{D}_{p}{\mu}^{p1}{\kappa}^{p}}{p}). Then for t\in (0,min\{1,\frac{1}{p\tau}\}), the mapping S:C\to X defined by S:=It\mu F is a contraction with constant 1t\tau.
Lemma 4.2 Let X:={L}_{p} (or {\ell}_{p}) with 2\le p<\mathrm{\infty}. Let C be a nonempty, closed, and convex subset of X. Let F:C\to X be a κLipschitzian and ηstrongly accretive operator with constants \kappa ,\eta >0. Let 0<\mu <\frac{2\eta}{(p1){\kappa}^{2}} and \tau =\mu (\eta \frac{(p1)\mu {\kappa}^{2}}{2}). Then for t\in (0,min\{1,\frac{1}{2\tau}\}), the mapping S:C\to X defined by S:=It\mu F is a contraction with constant 1t\tau.
Lemma 4.3 Let X:=H be a real Hilbert space. Let C be a nonempty, closed, and convex subset of X. Let F:C\to X be a κLipschitzian and ηstrongly accretive operator with constants \kappa ,\eta >0. Let 0<\mu <\frac{2\eta}{{\kappa}^{2}} and \tau =\mu (\eta \frac{\mu {\kappa}^{2}}{2}). Then for t\in (0,min\{1,\frac{1}{2\tau}\}), the mapping S:C\to X defined by S:=It\mu F is a contraction with constant 1t\tau.
4.1 Implicit iteration schemes
Theorem 4.4 Let C be a nonempty, closed, and convex subset of an {\ell}_{p} space for 1<p\le 2. Let {Q}_{C}, F, G, V, and T be the same as in Theorem 3.4. Assume that 0<\mu <{(\frac{p\eta}{{D}_{p}{\kappa}^{p}})}^{\frac{1}{p1}} and 0\le \gamma L<\tau, where \tau =\mu (\eta \frac{{D}_{p}{\mu}^{p1}{\kappa}^{p}}{p}). For \sigma \in (\frac{{L}_{\delta ,\lambda}}{\tau \gamma L},min\{1,\frac{1}{p\tau},\frac{1+{L}_{\delta ,\lambda}}{\tau \gamma L}\}) and t\in (0,1), the sequence \{{x}_{t}\} defined by (3.2) converges strongly to {x}^{\ast}\in Fix(T) as t\to 0, where {x}^{\ast} is the unique solution of the variational inequality (3.3).
Theorem 4.5 Let C be a nonempty, closed, and convex subset of an {\ell}_{p} space for 2\le p<\mathrm{\infty}. Let {Q}_{C}, F, G, V, and T be the same as in Theorem 3.4. Assume that 0<\mu <\frac{2\eta}{(p1){\kappa}^{2}} and 0\le \gamma L<\tau, where \tau =\mu (\eta \frac{(p1)\mu {\kappa}^{2}}{2}). For \sigma \in (\frac{{L}_{\delta ,\lambda}}{\tau \gamma L},min\{1,\frac{1}{2\tau},\frac{1+{L}_{\delta ,\lambda}}{\tau \gamma L}\}) and t\in (0,1), the sequence \{{x}_{t}\} defined by (3.2) converges strongly to {x}^{\ast}\in Fix(T) as t\to 0, where {x}^{\ast} is the unique solution of the variational inequality (3.3).
Remark 4.6 If the spaces {L}_{p} has a weakly sequentially continuous generalized duality mappings, then we obtain Theorems 4.4 and 4.5 hold for {L}_{p} spaces with 1<p<\mathrm{\infty}, p\ne 2.
4.2 Explicit iteration schemes
Theorem 4.7 Let C be a nonempty, closed, and convex subset of an {\ell}_{p} space for 1<p\le 2. Let {Q}_{C}, F, G, V, and {W}_{n} be the same as in Theorem 3.5. Let \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in (0,1) which satisfy the conditions (C1) and (C2) in Theorem 3.5 and \{{\theta}_{n,k}\} satisfies (H1)(H3). Then the sequence \{{x}_{n}\} defined by (3.14) converges strongly to {x}^{\ast}\in \mathcal{F} as n\to \mathrm{\infty}, where {x}^{\ast} is the unique solution of the variational inequality (3.15).
Theorem 4.8 Let C be a nonempty, closed, and convex subset of an {\ell}_{p} space for 2\le p<\mathrm{\infty}. Let {Q}_{C}, F, G, V, and {W}_{n} be the same as in Theorem 3.5. Let \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in (0,1) which satisfy the conditions (C1) and (C2) in Theorem 3.5 and \{{\theta}_{n,k}\} satisfies (H1)(H3). Then the sequence \{{x}_{n}\} defined by (3.14) converges strongly to {x}^{\ast}\in \mathcal{F} as n\to \mathrm{\infty}, where {x}^{\ast} is the unique solution of the variational inequality (3.15).
Remark 4.9 If the spaces {L}_{p} has a weakly sequentially continuous generalized duality mappings, then we obtain Theorems 4.7 and 4.8 hold for {L}_{p} spaces with 1<p<\mathrm{\infty}, p\ne 2.
5 Numerical examples
In this section, we give a simple example and some numerical experiment result to explain the convergence of the sequence (3.14) as follows:
Example 5.1 Let X=\mathbb{R} and C=[0,\frac{1}{2}]. Let q=2 and {j}_{q}=I. We define a mapping {Q}_{C} as follows:
In terms of Theorem 3.5, set \sigma =\mu =\gamma =1 and {\alpha}_{n}=\frac{1}{n}. Then we see that {\alpha}_{n}=\frac{1}{n} satisfies (C1) and (C2) with {\sigma}_{n}=\frac{1}{{n}^{2}}. Moreover, we define the mappings F, G, and V as follows:
It is easy to observe that F is 1Lipschitzian and \frac{2}{3}strongly accretive, G is 1strongly accretive and λstrictly pseudocontraction for \lambda >0 and V is 1Lipschitzian. For each n\in \mathbb{N}, set {S}_{n}=I. We show that {W}_{n}=I. Since {T}_{n,k}={\theta}_{n,k}{S}_{k}+(1{\theta}_{n,k})I, where {S}_{k} is a {\lambda}_{k}strictly pseudocontractive mapping and \{{\theta}_{n,k}\} satisfies (H1)(H3). It is observe that {T}_{n,k} is a nonexpansive mapping. From (2.5), we have
and we compute (2.5) in a similar way to above, we obtain
Since {S}_{n}=I and {t}_{n}=\alpha, for all n\in \mathbb{N}, we have
Under the above assumptions, (3.14) is simplified as follows:
Since the assumptions of Theorem 3.5 are satisfied in Example 5.1, the sequence (5.1) converges to {x}^{\ast}=0, which is the unique fixed point of {S}_{n}.
Next, we show the numerical results by using MATLAB 7.11.0. We presented numerical comparisons for two cases of iteration process with different initial values, which show the convergence of the sequence (5.1).
When we choose {x}_{1}=0.05 and {x}_{1}=0.1, we see that the iteration process of sequence \{{x}_{n}\} converges to {x}^{\ast}=0 at n=8\text{,}615 and n=28\text{,}946, respectively, as shown in Table 1 and Figures 1 and 2.
From the figures, we can see that \{{x}_{n}\} is a monotone decreasing sequence and converges to 0, but an iterative process with initial value {x}_{1}=0.05 is converges faster than an iterative process with initial value {x}_{1}=0.1.
Remark 5.2 Note that Lemma 3.1 and Lemma 3.2 play an important role in the proof of Theorems 3.4 and 3.5. These are proved in the framework of the more general quniformly smooth Banach space.
Remark 5.3 Our main result extends the main result of Ceng et al. [28] in the following respects:

(1)
An iterative process (1.10) is to extend to a general iterative process defined over the set of fixed points of an infinite family of strict pseudocontractions in a more general quniformly smooth Banach space.

(2)
The self contraction mapping f:H\to H in [[28], Theorem 3.2] is extended to the case of a nonself Lipschitzian mapping V:C\to X on a nonempty, closed, and convex subset C of a real quniformly smooth Banach space X.

(3)
The control condition (C3) in [[28], Theorem 3.2] is removed by weaker than control condition {\alpha}_{n+1}{\alpha}_{n}\le \circ ({\alpha}_{n})+{\sigma}_{n} with {\sum}_{n=1}^{\mathrm{\infty}}{\sigma}_{n}<\mathrm{\infty}.
Furthermore, our method is extended to develop a new iterative method and method of proof is very different from that in Ceng et al. [28] because our method involves the sunny nonexpansive retraction and the infinite family of strict pseudocontractions.
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