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Threestep Mann iterations for a general system of variational inequalities and an infinite family of nonexpansive mappings in Banach spaces
Journal of Inequalities and Applications volume 2013, Article number: 539 (2013)
Abstract
In this paper, let X be a uniformly convex Banach space which either is uniformly smooth or has a weakly continuous duality map. We introduce and consider threestep Mann iterations for finding a common solution of a general system of variational inequalities (GSVI) and a fixed point problem (FPP) of an infinite family of nonexpansive mappings in X. Here threestep Mann iterations are based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. We prove the strong convergence of this method to a common solution of the GSVI and the FPP, which solves a variational inequality on their common solution set. We also give a weak convergence theorem for threestep Mann iterations involving the GSVI and the FPP in a Hilbert space. The results presented in this paper improve, extend, supplement and develop the corresponding results announced in the earlier and very recent literature.
MSC:49J30, 47H09, 47J20.
1 Introduction
Let X be a real Banach space whose dual space is denoted by {X}^{\ast}. The normalized duality mapping J:X\to {2}^{{X}^{\ast}} is defined by
where \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing. It is an immediate consequence of the HahnBanach theorem that J(x) is nonempty for each x\in X. Let U=\{x\in X:\parallel x\parallel =1\} denote the unite sphere of X. A Banach space X is said to be uniformly convex if for each \u03f5\in (0,2], there exists \delta >0 such that for all x,y\in U,
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space X is said to be smooth if the limit
exists for all x,y\in U; in this case, X is also said to have a Gateaux differentiable norm. X is said to have a uniformly Gateaux differentiable norm if for each y\in U, the limit is attained uniformly for x\in U. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for x,y\in U. The norm of X is said to be the Frechet differential if for each x\in U, this limit is attained uniformly for y\in U.
Let C be a nonempty closed convex subset of X, and let T:C\to C be a nonlinear mapping. Denote by Fix(T) the set of fixed points of T, i.e., Fix(T)=\{x\in C:Tx=x\}. Recall that T is nonexpansive if \parallel TxTy\parallel \le \parallel xy\parallel for all x,y\in C. A mapping f:C\to C is said to be a contraction on C if there exists a constant ρ in (0,1) such that \parallel f(x)f(y)\parallel \le \rho \parallel xy\parallel for all x,y\in C. A mapping A:C\to X is said to be accretive if for each x,y\in C there exists j(xy)\in J(xy) such that \u3008AxAy,j(xy)\u3009\ge 0.
Recently, Yao et al. [1] combined the viscosity approximation method and the Mann iteration method and gave the following hybrid viscosity approximation method.
Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X, let T:C\to C be a nonexpansive mapping such that Fix(T)\ne \mathrm{\varnothing} and f\in {\Xi}_{C} with a contractive coefficient \rho \in (0,1), where {\Xi}_{C} is the collection of all contractive selfmappings on C. For an arbitrary {x}_{0}\in C, define \{{x}_{n}\} in the following way:
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are two sequences in (0,1). They proved under certain control conditions on the sequences \{{\alpha}_{n}\} and \{{\beta}_{n}\} that \{{x}_{n}\} converges strongly to a fixed point of T. Subsequently, Ceng and Yao [2] under the convergence of no parameter sequences to zero proved that the sequence \{{x}_{n}\} generated by (YCY) converges strongly to a fixed point of T. Such a result includes [[1], Theorem 1] as a special case.
Theorem 1.1 (see [[2], Theorem 3.1])
Let C be a nonempty closed convex subset of a uniformly smooth Banach space X. Let T:C\to C be a nonexpansive mapping with Fix(T)\ne \mathrm{\varnothing} and f\in {\Xi}_{C} with a contractive coefficient \rho \in (0,1). Given sequences \{{\alpha}_{n}\} and \{{\beta}_{n}\} in [0,1], the following control conditions are satisfied:

(i)
0\le {\beta}_{n}\le 1\rho, \mathrm{\forall}n\ge {n}_{0} for some integer {n}_{0}\ge 0;

(ii)
{\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty};

(iii)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;

(iv)
{lim}_{n\to \mathrm{\infty}}(\frac{{\beta}_{n+1}}{1(1{\beta}_{n+1}){\alpha}_{n+1}}\frac{{\beta}_{n}}{1(1{\beta}_{n}){\alpha}_{n}})=0.
For an arbitrary {x}_{0}\in C, let \{{x}_{n}\} be generated by (YCY). Then
where q\in Fix(T) solves the VIP
Let C be a nonempty closed convex subset of a real Banach space X, and f\in {\Xi}_{C} with a contractive coefficient \rho \in (0,1). Let {\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}} be an infinite family of nonexpansive selfmappings on C, and let {\{{\lambda}_{n}\}}_{n=0}^{\mathrm{\infty}} be a sequence of nonnegative numbers in [0,1]. For any n\ge 0, define a selfmapping {W}_{n} on C as follows:
Such a mapping {W}_{n} is called the Wmapping generated by {T}_{n},{T}_{n1},\dots ,{T}_{0} and {\lambda}_{n},{\lambda}_{n1},\dots ,{\lambda}_{0}; see [3].
In 2012, Ceng et al. [4] introduced and analyzed the following hybrid viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in a strictly convex and reflexive Banach space, which either is uniformly smooth or has a weakly continuous duality map {J}_{\phi} with gauge φ.
Theorem 1.2 (see [[4], Theorem 3.3])
Let C be a nonempty closed convex subset of a reflexive and strictly convex Banach space X. Assume, in addition, that X either is uniformly smooth or has a weakly continuous duality map {J}_{\phi} with gauge φ. Let {\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}} be an infinite family of nonexpansive selfmappings on C such that the common fixed point set F:={\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n})\ne \mathrm{\varnothing} and f\in {\Xi}_{C} with a contractive coefficient \rho \in (0,1). Given sequences \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\lambda}_{n}\} in [0,1], the following conditions are satisfied:

(i)
0\le {\beta}_{n}\le 1\rho, \mathrm{\forall}n\ge {n}_{0} for some {n}_{0}\ge 0, and {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty};

(ii)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;

(iii)
{lim}_{n\to \mathrm{\infty}}(\frac{{\beta}_{n+1}}{1(1{\beta}_{n+1}){\alpha}_{n+1}}\frac{{\beta}_{n}}{1(1{\beta}_{n}){\alpha}_{n}})=0;

(iv)
0<{\lambda}_{n}\le b<1, \mathrm{\forall}n\ge 0 for some constant b\in (0,1).
For an arbitrary {x}_{0}\in C, let \{{x}_{n}\} be generated by
where {W}_{n} is the Wmapping generated by {T}_{n},{T}_{n1},\dots ,{T}_{0} and {\lambda}_{n},{\lambda}_{n1},\dots ,{\lambda}_{0}. Then
In this case,

(i)
if X is uniformly smooth, then q\in F solves the VIP
\u3008qf(q),J(qp)\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F; 
(ii)
if X has a weakly continuous duality map {J}_{\phi} with gauge φ, then q\in F solves the VIP
\u3008qf(q),{J}_{\phi}(qp)\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F.
On the other hand, Cai and Bu [5] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space X, which involves finding ({x}^{\ast},{y}^{\ast})\in C\times C such that
where C is a nonempty, closed and convex subset of X, {B}_{1},{B}_{2}:C\to X are two nonlinear mappings and {\mu}_{1} and {\mu}_{2} are two positive constants. Here, the set of solutions of GSVI (1.1) is denoted by GSVI(C,{B}_{1},{B}_{2}). In particular, if X=H, a real Hilbert space, then GSVI (1.1) reduces to the following GSVI of finding ({x}^{\ast},{y}^{\ast})\in C\times C such that
which is studied in Ceng et al. [6]. The set of solutions of problem (1.2) is still denoted by GSVI(C,{B}_{1},{B}_{2}). In particular, if {B}_{1}={B}_{2}=A, then problem (1.2) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [7]. Further, if {x}^{\ast}={y}^{\ast} additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding {x}^{\ast}\in C such that
The solution set of VIP (1.3) is denoted by VI(C,A). Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [8]. This alternative formulation has been used to suggest and analyze the projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous.
In 1976, Korpelevich [9] proposed an iterative algorithm for solving VIP (1.3) in Euclidean space {\mathbf{R}}^{n}:
with \tau >0 a given number, which is known as the extragradient method (see also [10, 11]). The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention from many authors, who improved it in various ways; see, e.g., [5, 8, 12–29] and references therein, to name but a few.
In particular, whenever X is still a real smooth Banach space, {B}_{1}={B}_{2}=A and {x}^{\ast}={y}^{\ast}, then GSVI (1.1) reduces to the variational inequality problem (VIP) of finding {x}^{\ast}\in C such that
which was considered by Aoyama et al. [30]. Note that VIP (1.5) is connected with the fixed point problem for a nonlinear mapping (see, e.g., [31]), the problem of finding a zero point of a nonlinear operator (see, e.g., [32]) and so on. It is clear that VIP (1.5) extends VIP (1.3) from Hilbert spaces to Banach spaces.
In order to find a solution of VIP (1.5), Aoyama et al. [30] introduced the following Mann iterative scheme for an accretive operator A:
where {\Pi}_{C} is a sunny nonexpansive retraction from X onto C. Then they proved a weak convergence theorem. For related work, please see [33] and the references therein.
Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (1.1) which contains VIP (1.5) as a special case. Very recently, Cai and Bu [5] constructed an iterative algorithm for solving GSVI (1.1) and a fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2uniformly smooth Banach space X.
Lemma 1.1 (see [34])
Let X be a 2uniformly smooth Banach space. Then
where κ is the 2uniformly smooth constant of X and J is the normalized duality mapping from X into {X}^{\ast}.
Theorem 1.3 (see [[5], Theorem 3.1])
Let C be a nonempty closed convex subset of a uniformly convex and 2uniformly smooth Banach space X. Let {\Pi}_{C} be a sunny nonexpansive retraction from X onto C. Let the mapping {B}_{i}:C\to X be {\beta}_{i}inversestrongly accretive with 0<{\mu}_{i}<\frac{{\beta}_{i}}{{\kappa}^{2}} for i=1,2. Let f be a contraction of C into itself with a coefficient \rho \in (0,1). Let {\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}} be an infinite family of nonexpansive mappings of C into itself such that F={\bigcap}_{i=1}^{\mathrm{\infty}}Fix({T}_{i})\cap \Omega \ne \mathrm{\varnothing}, where Ω is a fixed point set of the mapping G:={\Pi}_{C}(I{\mu}_{1}{B}_{1}){\Pi}_{C}(I{\mu}_{2}{B}_{2}). For arbitrarily given {x}_{1}\in C, let \{{x}_{n}\} be a sequence generated by
Suppose that \{{\alpha}_{n}\} and \{{\beta}_{n}\} are two sequences in (0,1) satisfying the following conditions:

(i)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty};

(ii)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1.
Assume that {\sum}_{n=1}^{\mathrm{\infty}}{sup}_{x\in D}\parallel {T}_{n+1}x{T}_{n}x\parallel <\mathrm{\infty} for any bounded subset D of C, and let T be a mapping of C into X defined by Tx={lim}_{n\to \mathrm{\infty}}{T}_{n}x for all x\in C and suppose that Fix(T)={\bigcap}_{n=1}^{\mathrm{\infty}}Fix({T}_{n}). Then \{{x}_{n}\} converges strongly to q\in F, which solves the following VIP:
For the convenience of implementing the argument techniques in [6], the authors [5] used the following inequality in a real smooth and uniform convex Banach space X.
Proposition 1.1 (see [35])
Let X be a real smooth and uniform convex Banach space, and let r>0. Then there exists a strictly increasing, continuous and convex function g:[0,2r]\to \mathbf{R}, g(0)=0 such that
where {B}_{r}=\{x\in X:\parallel x\parallel \le r\}.
In this paper, let X be a uniformly convex Banach space which either is uniformly smooth or has a weakly continuous duality map. Let C be a nonempty closed convex subset of X, {\Pi}_{C} be a sunny nonexpansive retraction from X onto C and f\in {\Xi}_{C} with a contractive coefficient \rho \in (0,1). Motivated and inspired by the research going on in this area, we introduce and analyze threestep Mann iterations for finding a common solution of GSVI (1.1) and a fixed point problem (FPP) of an infinite family of nonexpansive selfmappings on C. Here, threestep Mann iterations are based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. We prove the strong convergence of this method to a common solution of GSVI (1.1) and the FPP, which solves a variational inequality on their common solution set. We also give a weak convergence theorem for threestep Mann iterations involving GSVI (1.2) and the FPP in the case of X=H, a Hilbert space. The results presented in this paper improve, extend, supplement and develop the corresponding results announced in the earlier and very recent literature; see, e.g., [2, 4–6, 29].
2 Preliminaries
Let X be a real Banach space. We define a function \rho :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) called the modulus of smoothness of X as follows:
It is known that X is uniformly smooth if and only if {lim}_{\tau \to 0}\rho (\tau )/\tau =0. Let q be a fixed real number with 1<q\le 2. Then a Banach space X is said to be quniformly smooth if there exists a constant c>0 such that \rho (\tau )\le c{\tau}^{q} for all \tau >0. As pointed out in [36], no Banach space is quniformly smooth for q>2. In addition, it is also known that J is singlevalued if and only if X is smooth, whereas if X is uniformly smooth, then the mapping J is normtonorm uniformly continuous on bounded subsets of X. If X has a uniformly Gateaux differentiable norm, then the duality mapping J is normtoweak^{∗} uniformly continuous on bounded subsets of X. We use the notation ⇀ to indicate the weak convergence and → to indicate the strong convergence.
Let C be a nonempty closed convex subset of X. Recall that a mapping A:C\to X is said to be

(i)
αstrongly accretive if for each x,y\in C, there exists j(xy)\in J(xy) such that
\u3008AxAy,j(xy)\u3009\ge \alpha {\parallel xy\parallel}^{2}for some \alpha \in (0,1);

(ii)
βinversestronglyaccretive if for each x,y\in C, there exists j(xy)\in J(xy) such that
\u3008AxAy,j(xy)\u3009\ge \beta {\parallel AxAy\parallel}^{2}for some \beta >0;

(iii)
λstrictly pseudocontractive [37] (see also [38]) if for each x,y\in C, there exists j(xy)\in J(xy) such that
\u3008AxAy,j(xy)\u3009\le {\parallel xy\parallel}^{2}\lambda {\parallel xy(AxAy)\parallel}^{2}for some \lambda \in (0,1).
It is worth emphasizing that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, e.g., [17, 39, 40].
We list some lemmas that will be used in the sequel. Lemma 2.1 can be found in [41]. Lemma 2.2 is an immediate consequence of the subdifferential inequality of the function \frac{1}{2}{\parallel \cdot \parallel}^{2}.
Lemma 2.1 Let \{{s}_{n}\} be a sequence of nonnegative real numbers satisfying the condition
where \{{\mu}_{n}\} and \{{\nu}_{n}\} are sequences of real numbers such that

(i)
\{{\mu}_{n}\}\subset [0,1] and {\sum}_{n=0}^{\mathrm{\infty}}{\mu}_{n}=\mathrm{\infty}, or equivalently,
\prod _{n=0}^{\mathrm{\infty}}(1{\mu}_{n}):=\underset{n\to \mathrm{\infty}}{lim}\prod _{k=0}^{n}(1{\mu}_{k})=0; 
(ii)
{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\nu}_{n}\le 0, or {\sum}_{n=0}^{\mathrm{\infty}}{\mu}_{n}{\nu}_{n}<\mathrm{\infty}.
Then {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{s}_{n}=0.
Lemma 2.2 (see [42])
Let X be a real Banach space and J be the normalized duality map on X. Then, for any given x,y\in X, the following inequality holds:
Let D be a subset of C, and let Π be a mapping of C into D. Then Π is said to be sunny if
whenever \Pi (x)+t(x\Pi (x))\in C for x\in C and t\ge 0. A mapping Π of C into itself is called a retraction if {\Pi}^{2}=\Pi. If a mapping Π of C into itself is a retraction, then \Pi (z)=z for every z\in R(\Pi ), where R(\Pi ) is the range of Π. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D.
The following lemma concerns the sunny nonexpansive retraction.
Lemma 2.3 (see [43])
Let C be a nonempty closed convex subset of a real smooth Banach space X. Let D be a nonempty subset of C. Let Π be a retraction of C onto D. Then the following are equivalent:

(i)
Π is sunny and nonexpansive;

(ii)
{\parallel \Pi (x)\Pi (y)\parallel}^{2}\le \u3008xy,J(\Pi (x)\Pi (y))\u3009, \mathrm{\forall}x,y\in C;

(iii)
\u3008x\Pi (x),J(y\Pi (x))\u3009\le 0, \mathrm{\forall}x\in C, y\in D.
It is well known that if X=H, a Hilbert space, then a sunny nonexpansive retraction {\Pi}_{C} is coincident with the metric projection from X onto C; that is, {\Pi}_{C}={P}_{C}. If C is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space X and if T:C\to C is a nonexpansive mapping with the fixed point set Fix(T)\ne \mathrm{\varnothing}, then the set Fix(T) is a sunny nonexpansive retract of C.
Lemma 2.4 Let C be a nonempty closed convex subset of a smooth Banach space X. Let {\Pi}_{C} be a sunny nonexpansive retraction from X onto C, and let {B}_{1},{B}_{2}:C\to X be nonlinear mappings. For given {x}^{\ast},{y}^{\ast}\in C, ({x}^{\ast},{y}^{\ast}) is a solution of GSVI (1.1) if and only if {x}^{\ast}={\Pi}_{C}({y}^{\ast}{\mu}_{1}{B}_{1}{y}^{\ast}), where {y}^{\ast}={\Pi}_{C}({x}^{\ast}{\mu}_{2}{B}_{2}{x}^{\ast}).
Proof We can rewrite GSVI (1.1) as
which is obviously equivalent to
because of Lemma 2.3. This completes the proof. □
In terms of Lemma 2.4, we observe that
which implies that {x}^{\ast} is a fixed point of the mapping G. Throughout this paper, the set of fixed points of the mapping G is denoted by Ω.
Lemma 2.5 (see [44])
Let X be a uniformly smooth Banach space, C be a nonempty closed convex subset of X, T:C\to C be a nonexpansive mapping with Fix(T)\ne \mathrm{\varnothing}, and f\in {\Xi}_{C}. Then the net \{{x}_{t}\} defined by {x}_{t}=tf({x}_{t})+(1t)T{x}_{t}, \mathrm{\forall}t\in (0,1), converges strongly to a point in Fix(T). If we define a mapping Q:{\Xi}_{C}\to Fix(T) by Q(f):=s{lim}_{t\to 0}{x}_{t}, \mathrm{\forall}f\in {\Xi}_{C}, then Q(f) solves the VIP
In particular, if f=u\in C is a constant, then the map u\mapsto Q(u) is reduced to the sunny nonexpansive retraction of Reich type from C onto Fix(T), i.e.,
Recall that a gauge is a continuous strictly increasing function \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) such that \phi (0)=0 and \phi (t)\to \mathrm{\infty} as t\to \mathrm{\infty}. Associated to the gauge φ is the duality map {J}_{\phi}:X\to {2}^{{X}^{\ast}} defined by
We say that a Banach space X has a weakly continuous duality map if there exists a gauge φ for which the duality map {J}_{\phi} is singlevalued and weaktoweak^{∗} sequentially continuous. It is known that {l}^{p} has a weakly continuous duality map with gauge \phi (t)={t}^{p1} for all 1<p<\mathrm{\infty}. Set
Then {J}_{\phi}(x)=\partial \Phi (\parallel x\parallel ) for all x\in X, where ∂ denotes the subdifferential in the sense of convex analysis; see [42] for more details.
The first part of the following lemma is an immediate consequence of the subdifferential inequality, and the proof of the second part can be found in [45].
Lemma 2.6 Assume that X has a weakly continuous duality map {J}_{\phi} with gauge φ.

(i)
For all x,y\in X, the following inequality holds:
\Phi (\parallel x+y\parallel )\le \Phi (\parallel x\parallel )+\u3008y,{J}_{\phi}(x+y)\u3009. 
(ii)
Assume that a sequence \{{x}_{n}\} in X is weakly convergent to a point x. Then the following identity holds:
\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}\Phi (\parallel {x}_{n}y\parallel )=\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}\Phi (\parallel {x}_{n}x\parallel )+\Phi (\parallel yx\parallel ),\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in X.
Lemma 2.7 ([[31], Theorem 3.1 and its proof])
Let X be a reflexive Banach space and have a weakly continuous duality map {J}_{\phi} with gauge φ, let C be a nonempty closed convex subset of X, let T:C\to C be a nonexpansive mapping with Fix(T)\ne \mathrm{\varnothing}, and let f\in {\Xi}_{C}. Then \{{x}_{t}\} defined by {x}_{t}=tf({x}_{t})+(1t)T{x}_{t}, \mathrm{\forall}t\in (0,1), converges strongly to a point in Fix(T) as t\to {0}^{+}. Define Q:{\Xi}_{C}\to Fix(T) by Q(f):=s{lim}_{t\to {0}^{+}}{x}_{t}. Then Q(f) solves the variational inequality
In particular, if f=u\in C is a constant, then the map u\mapsto Q(u) is reduced to the sunny nonexpansive retraction of Reich type from C onto Fix(T), i.e.,
Recall that X satisfies Opial’s property [46] provided, for each sequence \{{x}_{n}\} in X, the condition {x}_{n}\rightharpoonup x implies
It is known in [46] that each {l}^{p} (1\le p<\mathrm{\infty}) enjoys this property, while {L}^{p} does not unless p=2. It is known in [47] that every separable Banach space can be equivalently renormed so that it satisfies Opial’s property. We denote by {\omega}_{w}({x}_{n}) the weak ωlimit set of \{{x}_{n}\}, i.e.,
Also, recall that in a Hilbert space H, the following equality holds:
Lemma 2.8 (see [48])
Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let {\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}} be a sequence of nonexpansive mappings on C. Suppose that {\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n}) is nonempty. Let \{{\lambda}_{n}\} be a sequence of positive numbers with {\sum}_{n=0}^{\mathrm{\infty}}{\lambda}_{n}=1. Then a mapping S on C defined by Sx={\sum}_{n=0}^{\mathrm{\infty}}{\lambda}_{n}{T}_{n}x for x\in C is defined well, nonexpansive and Fix(S)={\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n}) holds.
Lemma 2.9 (see [30])
Let C be a nonempty closed convex subset of a smooth Banach space X. Let {\Pi}_{C} be a sunny nonexpansive retraction from X onto C, and let A:C\to X be an accretive mapping. Then, for all \lambda >0,
Lemma 2.10 (see [[49], Lemma 3.2])
Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let {\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}} be a sequence of nonexpansive selfmappings on C such that {\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n})\ne \mathrm{\varnothing}, and let {\{{\lambda}_{n}\}}_{n=0}^{\mathrm{\infty}} be a sequence of positive numbers in (0,b] for some b\in (0,1). Then, for every x\in C and k\ge 0, the limit {lim}_{n\to \mathrm{\infty}}{U}_{n,k}x exists.
Using Lemma 2.10, one can define a mapping W:C\to C as follows:
Such W is called the Wmapping generated by the sequences {\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}} and {\{{\lambda}_{n}\}}_{n=0}^{\mathrm{\infty}}. Throughout this paper, we always assume that {\{{\lambda}_{n}\}}_{n=0}^{\mathrm{\infty}} is a sequence of positive numbers in (0,b] for some b\in (0,1).
Lemma 2.11 (see [[49], Lemma 3.3])
Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let {\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}} be a sequence of nonexpansive selfmappings on C such that {\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n})\ne \mathrm{\varnothing}, and let {\{{\lambda}_{n}\}}_{n=0}^{\mathrm{\infty}} be a sequence of positive numbers in (0,b] for some b\in (0,1). Then Fix(W)={\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n}).
Lemma 2.12 (see [[50], Lemma 2])
Let \{{x}_{n}\} and \{{z}_{n}\} be bounded sequences in a Banach space X, and let \{{\beta}_{n}\} be a sequence of nonnegative numbers in [0,1] with 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1. Suppose that {x}_{n+1}={\beta}_{n}{x}_{n}+(1{\beta}_{n}){z}_{n} for all integers n\ge 0 and {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}(\parallel {z}_{n+1}{z}_{n}\parallel \parallel {x}_{n+1}{x}_{n}\parallel )\le 0. Then {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{z}_{n}\parallel =0.
Lemma 2.13 (see [34])
Given a number r>0. A real Banach space X is uniformly convex if and only if there exists a continuous strictly increasing function g:[0,\mathrm{\infty})\to [0,\mathrm{\infty}), g(0)=0 such that
for all \lambda \in [0,1] and x,y\in X such that \parallel x\parallel \le r and \parallel y\parallel \le r.
We will also use the following elementary lemmas in the sequel.
Lemma 2.14 (see [51])
Let \{{a}_{n}\} and \{{b}_{n}\} be the sequences of nonnegative real numbers such that {\sum}_{n=0}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty} and {a}_{n+1}\le {a}_{n}+{b}_{n} for all n\ge 0. Then {lim}_{n\to \mathrm{\infty}}{a}_{n} exists.
Lemma 2.15 (Demiclosedness principle [42])
Assume that T is a nonexpansive selfmapping of a nonempty closed convex subset C of a Hilbert space H. If T has a fixed point, then IT is demiclosed. That is, whenever {x}_{n}\rightharpoonup x in C and (IT){x}_{n}\to y in H, it follows that (IT)x=y. Here, I is the identity operator of H.
3 Main results
In this section, in order to prove our main results, we will use the following useful lemmas whose proofs will be omitted since they can be proved by standard arguments.
Lemma 3.1 Let C be a nonempty closed convex subset of a smooth Banach space X, and let the mapping {B}_{i}:C\to X be {\zeta}_{i}strictly pseudocontractive and {\eta}_{i}strongly accretive with {\zeta}_{i}+{\eta}_{i}\ge 1 for i=1,2. Then, for {\mu}_{i}\in (0,1], we have
for i=1,2. In particular, if 1\frac{{\zeta}_{i}}{1+{\zeta}_{i}}(1\sqrt{\frac{1{\eta}_{i}}{{\zeta}_{i}}})\le {\mu}_{i}\le 1, then I{\mu}_{i}{B}_{i} is nonexpansive for i=1,2.
Lemma 3.2 Let C be a nonempty closed convex subset of a smooth Banach space X. Let {\Pi}_{C} be a sunny nonexpansive retraction from X onto C, and let the mapping {B}_{i}:C\to X be {\zeta}_{i}strictly pseudocontractive and {\eta}_{i}strongly accretive with {\zeta}_{i}+{\eta}_{i}\ge 1 for i=1,2. Let G:C\to C be the mapping defined by
If 1\frac{{\zeta}_{i}}{1+{\zeta}_{i}}(1\sqrt{\frac{1{\eta}_{i}}{{\zeta}_{i}}})\le {\mu}_{i}\le 1, then G:C\to C is nonexpansive.
We now state and prove the main result of this paper.
Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Assume, in addition, that X either is uniformly smooth or has a weakly continuous duality map {J}_{\phi} with gauge φ. Let {\Pi}_{C} be a sunny nonexpansive retraction from X onto C. Let the mapping {B}_{i}:C\to X be {\zeta}_{i}strictly pseudocontractive and {\eta}_{i}strongly accretive with {\zeta}_{i}+{\eta}_{i}\ge 1 for i=1,2. Let f\in {\Xi}_{C} with a contractive coefficient \rho \in (0,1). Let {\{{\lambda}_{n}\}}_{n=0}^{\mathrm{\infty}} be a sequence of positive numbers in (0,b] for some b\in (0,1) and {\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}} be an infinite family of nonexpansive selfmappings on C such that F:={\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n})\cap \Omega \ne \mathrm{\varnothing}, where Ω is the fixed point set of the mapping G={\Pi}_{C}(I{\mu}_{1}{B}_{1}){\Pi}_{C}(I{\mu}_{2}{B}_{2}) with 1\frac{{\zeta}_{i}}{1+{\zeta}_{i}}(1\sqrt{\frac{1{\eta}_{i}}{{\zeta}_{i}}})\le {\mu}_{i}\le 1 for i=1,2. For an arbitrary {x}_{0}\in C, let \{{x}_{n}\} be generated by
where {W}_{n} is the Wmapping generated by {T}_{n},{T}_{n1},\dots ,{T}_{0} and {\lambda}_{n},{\lambda}_{n1},\dots ,{\lambda}_{0}, and \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\sigma}_{n}\} are sequences in [0,1]. Suppose that the following conditions hold:

(i)
0\le {\beta}_{n}\le 1\rho, \mathrm{\forall}n\ge {n}_{0} for some {n}_{0}\ge 0, and {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty};

(ii)
{lim}_{n\to \mathrm{\infty}}\frac{{\beta}_{n+1}}{1(1{\beta}_{n+1}){\alpha}_{n+1}}\frac{{\beta}_{n}}{1(1{\beta}_{n}){\alpha}_{n}}=0 and {lim}_{n\to \mathrm{\infty}}{\sigma}_{n+1}{\sigma}_{n}=0;

(iii)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;

(iv)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\sigma}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\sigma}_{n}<1.
Then
In this case,

(i)
if X is uniformly smooth, then q\in F solves the VIP
\u3008qf(q),J(qp)\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F; 
(ii)
if X has a weakly continuous duality map {J}_{\phi} with gauge φ, then q\in F solves the VIP
\u3008qf(q),{J}_{\phi}(qp)\u3009\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F.
Proof First of all, let us show that \{{x}_{n}\} is bounded. Indeed, taking an element p\in F={\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n})\cap \Omega arbitrarily, we obtain that Gp=p and p={W}_{n}p for all n\ge 0. By Lemma 3.2, we know that G is nonexpansive. It follows from the nonexpansivity of G and {W}_{n} that
and
By induction, we have
Hence \{{x}_{n}\} is bounded, and so are the sequences \{{y}_{n}\}, \{{z}_{n}\}, \{G{x}_{n}\}, \{{W}_{n}{z}_{n}\} and \{f({x}_{n})\}.
Suppose that {x}_{n}\to q\in F as n\to \mathrm{\infty}. Then q=Gq and q={W}_{n}q for all n\ge 0. From (3.1) it follows that
and
that is, {x}_{n}\to q. Again from (3.1) we obtain that
Conversely, suppose that {\beta}_{n}(f({x}_{n}){x}_{n})\to 0 (n\to \mathrm{\infty}). Put {\gamma}_{n}=(1{\beta}_{n}){\alpha}_{n} for each n\ge 0. Then it follows from conditions (i) and (iii) that
and hence
Define {\stackrel{\u02c6}{z}}_{n} by
Observe that
It follows that
In the meantime, simple calculations show that
which hence yields
Taking into account the nonexpansivity of {T}_{k} and {U}_{n,k}, from (CWY) we have
where {sup}_{n\ge 0}\parallel {T}_{n+1}{z}_{n}{z}_{n}\parallel \le {M}_{0} for some {M}_{0}>0. Thus, from (3.4), (3.5) and (3.6), we get
where {sup}_{n\ge 0}\{\parallel f({x}_{n})\parallel +\parallel {W}_{n}{z}_{n}\parallel +\parallel G{x}_{n}\parallel +\parallel {x}_{n}\parallel +{M}_{0}\}\le M for some M>0. Then it immediately follows that
From condition (ii) and 0<{\lambda}_{k}\le b<1, \mathrm{\forall}k\ge 0, we deduce that
Hence by Lemma 2.12 we have
It follows from (3.2) and (3.3) that
From (3.1) we have
This implies that
Since {x}_{n+1}{x}_{n}\to 0 and {\beta}_{n}(f({x}_{n}){x}_{n})\to 0, we get
Observe that
It follows from condition (iii), (3.7) and (3.8) that
Also, utilizing Lemma 2.13, we obtain from (3.1) that for p\in F
and hence
Thus, we get
From (3.7), conditions (iii), (iv) and the boundedness of \{{x}_{n}\} and \{{y}_{n}\}, it follows that
Utilizing the properties of g, we have
This immediately implies that
Note that
which together with (3.9) and (3.11) implies that
Also, note that
From [[52], Remark 2.2] (see also [[53], Remark 3.1]), we have
It follows that
In terms of (2.3) and Lemma 2.11, W:C\to C is a nonexpansive mapping such that Fix(W)={\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n}). Define a mapping Vx=(1\theta )Wx+\theta Gx, where θ is a constant in (0,1). Then by Lemma 2.8, we have that Fix(V)=Fix(W)\cap Fix(G)=F. Moreover, from (3.10) and (3.12), we get
that is,
In the following, we discuss two cases.
(i) Firstly, suppose that X is uniformly smooth. Let {x}_{t} be the unique fixed point of the contraction mapping {T}_{t} given by
By Lemma 2.5, we can define q:=s{lim}_{t\to {0}^{+}}{x}_{t}, and q\in Fix(V)=F solves the VIP
Let us show that
Note that
Applying Lemma 2.2, we derive
where
The last inequality implies
It follows that
where {M}_{1}>0 is a constant such that {M}_{1}\ge {\parallel {x}_{t}{x}_{n}\parallel}^{2} for all n\ge 0 and small enough t\in (0,1). Taking the lim sup as t\to {0}^{+} in (3.15) and noticing the fact that the two limits are interchangeable due to the fact that the duality map J is uniformly normtonorm continuous on any bounded subset of X, we get (3.14).
Now, let us show that {x}_{n}\to q as n\to \mathrm{\infty}.
Indeed, utilizing Lemma 2.2, we obtain from (3.1) that
and hence
Therefore, applying Lemma 2.1 to (3.17), we conclude from (3.14) and condition (i) that {x}_{n}\to q as n\to \mathrm{\infty}.
(ii) Secondly, suppose that X has a weakly continuous duality map {J}_{\phi} with gauge φ. Let {x}_{t} be the unique fixed point of the contraction mapping {T}_{t} given by
By Lemma 2.7, we can define q:=s{lim}_{t\to {0}^{+}}{x}_{t}, and q\in Fix(V)=F solves the VIP
Let us show that
We take a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} such that
Since X is reflexive and \{{x}_{n}\} is bounded, we may further assume that {x}_{{n}_{k}}\rightharpoonup \overline{x} for some \overline{x}\in C. Since {J}_{\phi} is weakly continuous, utilizing Lemma 2.6(ii), we have
Put \Gamma (x)={lim\hspace{0.17em}sup}_{k\to \mathrm{\infty}}\Phi (\parallel {x}_{{n}_{k}}x\parallel ), \mathrm{\forall}x\in X. It follows that
From (3.13), we have
Furthermore, observe that
Combining (3.21) with (3.22), we obtain
Hence V\overline{x}=\overline{x} and \overline{x}\in Fix(V)=F. Thus, from (3.18) and (3.20), it is easy to see that
Therefore, we deduce that (3.19) holds.
Next, let us show that {x}_{n}\to q as n\to \mathrm{\infty}. Indeed, utilizing Lemma 2.6(i), we obtain from (3.1) that
and hence
Applying Lemma 2.1 to (3.23), we conclude from (3.19) and condition (i) that
which implies that \parallel {x}_{n}q\parallel \to 0 (n\to \mathrm{\infty}), i.e., {x}_{n}\to q (n\to \mathrm{\infty}). This completes the proof. □
Corollary 3.1 The conclusion in Theorem 3.1 still holds, provided the conditions (i)(iv) are replaced by the following:

(i)
0\le {\beta}_{n}\le 1\rho, \mathrm{\forall}n\ge {n}_{0} for some {n}_{0}\ge 0;

(ii)
{lim}_{n\to \mathrm{\infty}}{\beta}_{n}{\beta}_{n+1}=0 and {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty};

(iii)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}{\alpha}_{n+1}=0 and 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;

(iv)
{lim}_{n\to \mathrm{\infty}}{\sigma}_{n}{\sigma}_{n+1}=0 and 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\sigma}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\sigma}_{n}<1.
Proof Observe that
Since {lim}_{n\to \mathrm{\infty}}{\beta}_{n}{\beta}_{n+1}=0 and {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}{\alpha}_{n+1}=0, it follows that
Consequently, all the conditions of Theorem 3.1 are satisfied. So, utilizing Theorem 3.1, we obtain the desired result. □
Corollary 3.2 Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Assume, in addition, that X either is uniformly smooth or has a weakly continuous duality map {J}_{\phi} with gauge φ. Let {\Pi}_{C} be a sunny nonexpansive retraction from X onto C. Let the mapping {B}_{i}:C\to X be {\zeta}_{i}strictly pseudocontractive and {\eta}_{i}strongly accretive with {\zeta}_{i}+{\eta}_{i}\ge 1 for i=1,2. Let f\in {\Xi}_{C} with a contractive coefficient \rho \in (0,1). Let {\{{\lambda}_{n}\}}_{n=0}^{\mathrm{\infty}} be a sequence of positive numbers in (0,b] for some b\in (0,1) and {\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}} be an infinite family of nonexpansive selfmappings on C such that F:={\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n})\cap \Omega \ne \mathrm{\varnothing}, where Ω is the fixed point set of the mapping G={\Pi}_{C}(I{\mu}_{1}{B}_{1}){\Pi}_{C}(I{\mu}_{2}{B}_{2}) with 1\frac{{\zeta}_{i}}{1+{\zeta}_{i}}(1\sqrt{\frac{1{\eta}_{i}}{{\zeta}_{i}}})\le {\mu}_{i}\le 1 for i=1,2. Suppose that \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\sigma}_{n}\} are sequences in [0,1] satisfying the following conditions:

(i)
{lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0 and {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty};

(ii)
{lim}_{n\to \mathrm{\infty}}{\sigma}_{n+1}{\sigma}_{n}=0;

(iii)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;

(iv)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\sigma}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\sigma}_{n}<1.
Then, for an arbitrary but fixed {x}_{0}\in C, the sequence \{{x}_{n}\} defined by (3.1) converges strongly to some q\in F. Moreover,

(i)
if X is uniformly smooth, then q\in F solves the VIP
\u3008qf(q),J(qp)\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F; 
(ii)
if X has a weakly continuous duality map {J}_{\phi} with gauge φ, then q\in F solves the VIP
\u3008qf(q),{J}_{\phi}(qp)\u3009\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F.
Proof Repeating the same arguments as those in the proof of Theorem 3.1, we know that \{{x}_{n}\} is bounded, and so are the sequences \{{y}_{n}\}, \{{z}_{n}\}, \{G{x}_{n}\}, \{{W}_{n}{z}_{n}\} and \{f({x}_{n})\}. Since {lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0, it is easy to see that the following hold:

(i)
{\beta}_{n}(f({x}_{n}){x}_{n})\to 0 (n\to \mathrm{\infty});

(ii)
0\le {\beta}_{n}\le 1\rho, \mathrm{\forall}n\ge {n}_{0} for some integer {n}_{0}\ge 0;

(iii)
{lim}_{n\to \mathrm{\infty}}\frac{{\beta}_{n+1}}{1(1{\beta}_{n+1}){\alpha}_{n+1}}\frac{{\beta}_{n}}{1(1{\beta}_{n}){\alpha}_{n}}=0.
Therefore, all the conditions of Theorem 3.1 are satisfied. So, utilizing Theorem 3.1, we derive the desired result. □
To end this paper, we give a weak convergence theorem for threestep Mann iterations (3.1) involving GSVI (1.2) and an infinite family of nonexpansive mappings {T}_{0},{T}_{1},\dots in a Hilbert space H.
Theorem 3.2 Let C be a nonempty closed convex subset of a Hilbert space H. Let the mapping {B}_{i}:C\to H be {\beta}_{i}inverse strongly monotone for i=1,2. Let f\in {\Xi}_{C} with a contractive coefficient \rho \in (0,1). Let {\{{\lambda}_{n}\}}_{n=0}^{\mathrm{\infty}} be a sequence of positive numbers in (0,b] for some b\in (0,1), and let {\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}} be an infinite family of nonexpansive selfmappings on C such that F:={\bigcap}_{n=0}^{\mathrm{\infty}}Fix({T}_{n})\cap \Omega \ne \mathrm{\varnothing}, where Ω is the fixed point set of the mapping G={P}_{C}(I{\mu}_{1}{B}_{1}){P}_{C}(I{\mu}_{2}{B}_{2}) with 0<{\mu}_{i}\le 2{\beta}_{i} for i=1,2. Suppose that \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\sigma}_{n}\} are sequences in [0,1] satisfying the following conditions:

(i)
{\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}<\mathrm{\infty};

(ii)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1;

(iii)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\sigma}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\sigma}_{n}<1.
Then, for an arbitrary but fixed {x}_{0}\in C, the sequence \{{x}_{n}\} defined by (3.1) converges weakly to a point in F.
Proof First of all, by Lemma 1.1, we know that G:C\to C is nonexpansive. Take an arbitrary p\in F. Repeating the same arguments as those in the proof of Theorem 3.1, we know that \{{x}_{n}\} is bounded, and so are the sequences \{{y}_{n}\}, \{{z}_{n}\}, \{G{x}_{n}\}, \{{W}_{n}{z}_{n}\} and \{f({x}_{n})\}.
It follows from (2.2) and (3.1) that
and hence
Since {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}<\mathrm{\infty} and \{f({x}_{n})\} is bounded, we obtain {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}{\parallel f({x}_{n})p\parallel}^{2}<\mathrm{\infty}. Utilizing Lemma 2.14, we conclude that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel exists. Furthermore, it follows from (3.24) that for all e\ge 0,