Iterative methods for variational inequality problems and fixed point problems of a countable family of strict pseudo-contractions in a q-uniformly smooth Banach space

Abstract

In this article, we introduce iterative methods (implicit and explicit) for finding a common fixed point set of a countable family of strict pseudo-contractions, which is a unique solution of some variational inequality. Furthermore, we prove the strong convergence theorems of such iterative scheme in a q-uniformly smooth Banach space which admits a weakly sequentially continuous generalized duality mapping. The results presented in this article extend and generalize the corresponding results announced by Yamada and Ceng et al. from Hilbert spaces to Banach spaces.

Mathematics Subject Classification 2010: 47H09; 47J05; 47J20; 65J15.

1 Introduction

Let X be a real Banach space, and X* be its dual space. Let U = {x ∈ X: ∥x∥ = 1}. A Banach space X is said to be strictly convex if $\frac{∥x+y∥}{2}<1$ for all x, y ∈ X with ∥x∥ = ∥y∥ = 1 and x ≠ y. A Banach space X is called uniformly convex if for each ε > 0 there is a δ > 0 such that for x, y ∈ X with ∥x∥, ∥y∥ ≤ 1 and ∥x - y∥ ≥ ε, ∥x + y∥ ≤ 2(1 - δ) holds. The modulus of convexity of X defined by

$${\delta }_X\left(\epsilon \right)=\mathsf{\text{inf}}\left\{1-∥\frac{1}{2}\left(x+y\right)∥:∥x∥,∥y∥\le 1,∥x-y∥\ge \epsilon \right\},$$

for all ε ∈ [0,2]. X is uniformly convex if δ _x (0) = 0 and δ _x (ε) > 0 for all 0 < ε ≤ 2. It is know that every uniformly convex Banach space is strictly convex and reflexive (see [1]). The norm of X is said to be Gâteaux differentiable if the limit

$$\underset{t\to 0}{\mathsf{\text{lim}}}\frac{∥x+ty∥-∥x∥}{t}$$

exists for each x, y ∈ U. In this case X is smooth. Let ρ _X : [0, ∞) → [0, ∞) be the modulus of smoothness of X defined by

$${\rho }_X\left(\tau \right)=\mathsf{\text{sup}}\left\{\frac{1}{2}\left(∥x+y∥+∥x-y∥\right)-1:x\in U,∥y∥\le \tau \right\}.$$

A Banach space X is said to be uniformly smooth if $\frac{{\rho }_X\left(t\right)}{t}\to 0$ as t → 0. Suppose that q > 1, then X is said to be q-uniformly smooth if there exists c > 0 such that ρ _X (t) ≤ ct^q. It is easy to see that if X is q-uniformly smooth, then q ≤ 2 and X is uniformly smooth. For q > 1, the generalized duality mapping $J_q:X\to 2^{X^{*}}$ is defined by

$$J_q\left(x\right)=\left\{f\in X^{*}:〈x,f〉={∥x∥}^q,∥f∥={∥x∥}^{q-1}\right\},$$

where 〈·,·〉 denotes the duality pairing between X and X*. In particular, J _q = J₂ is called the normalized duality mapping and J _q (x) = ∥x∥^q-2J₂(x) for x ≠ 0. If X := H is a real Hilbert space, then J = I where I is the identity mapping. Further, we have the following properties of the generalized duality mapping J _q :

• (1) J _q (x) = ∥x∥^q-2J₂(x) for all x ∈ X with x ≠ 0.

• (2) J(tx) = t^q-1J _q (x) for all x ∈ X and t ∈ [0, ∞).

• (3) J _q (-x) = -J _q (x) for all x ∈ X.

It is well known that if X is smooth, then J _q is single-valued, which is denoted by j _q (see [1]). The duality mapping J _q from a smooth Banach space X into X* is said to be weakly sequentially continuous generalized duality mapping if for all {x _n } ⊂ X with x _n ⇀ x implies J _q (x _n ) ⇀* J _q (x).

Let C be a nonempty, closed and convex subset of X and T be a self-mapping of C. We denote the fixed points set of the mapping T by Fix(T) = {x ∈ C: Tx = x} and denote → and ⇀ by strong and weak convergence, respectively.

Definition 1.1. A mapping T: C → C is said to be:

• (i) λ-strictly pseudocontractive [2], if for all x, y ∈ C there exists λ > 0 and j _q (x - y) ∈ J _q (x - y) such that

  $$〈Tx-Ty,j_q\left(x-y\right)〉\le {∥x-y∥}^q-\lambda {∥\left(I-T\right)x-\left(I-T\right)y∥}^q,$$

or equivalently

$$〈\left(I-T\right)x-\left(I-T\right)y,j_q\left(x-y\right)〉\ge \lambda {∥\left(I-T\right)x-\left(I-T\right)y∥}^q.$$

• (ii) L-Lipschitzian if for all x, y ∈ C, there exists a constant L > 0 such that

  $$∥Tx-Ty∥\le L∥x-y∥.$$

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 λ-strictly pseudocontractive mapping is $\left(\frac{1+\lambda }{\lambda }\right)$-Lipschitzian (see [3]).

Remark 1.2. Let C be a nonempty subset of a real Hilbert space H and T: C → C be a mapping. Then T is said to be k-strictly pseudocontractive [2], if for all x, y ∈ C, there exists k ∈ [0,1) such that

$${∥Tx-Ty∥}^2\le {∥x-y∥}^2+k{∥\left(I-T\right)x-\left(I-T\right)y∥}^2.$$

(1.1)

It is well known that (1.1) is equivalent to the following:

$$〈Tx-Ty,x-y〉\le ∥x-y∥-\frac{1-k}{2}{∥\left(I-T\right)x-\left(I-T\right)y∥}^2.$$

Let C be a nonempty, closed and convex subset of X and Ψ: C → X be a nonlinear mapping. The variational inequality problem is to find u ∈ C such that

$$〈\mathrm{\Psi }u,j_q\left(v-u\right)〉\ge 0,\forall v\in C,$$

(1.2)

where j _q (v - u) ∈ J _q (v - u). The set of solution of variational inequality problem is denoted by VI(C, Ψ). If X := H is a real Hilbert space, the variational inequality problem reduces to find u ∈ C such that

$$〈\mathrm{\Psi }u,v-u〉\ge 0,\forall v\in C.$$

(1.3)

Applications of variational inequalities span as diverse disciplines as differential equations, time-optimal control, optimization, mathematical programming, mechanics, finance and so on (see, e.g., [4, 5] for more details). 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 can be unified into form (1.2) and (1.3). For more details, we recommend the reader [6–11]. On the author hand, we note that iterative approximation of fixed points of nonexpansive mappings (and of common fixed points of nonexpansive semigroups) have recently been applied to image recovery and signal processing (see, e.g., [12–17]).

A mapping F: C → X is said to be accretive if for all x, y ∈ C there exists j _q (x - y) ∈ J _q (x - y) such that

$$〈Fx-Fy,j_q\left(x-y\right)〉\ge 0.$$

For some η > 0, F: C → X is said to be strongly accretive if for all x, y ∈ C there exists j _q (x -y) ∈ J _q (x - y) such that

$$〈Fx-Fy,j_q\left(x-y\right)〉\ge \eta {∥x-y∥}^q.$$

Remark 1.3. If X := H is a real Hilbert space, accretive and strongly accretive mappings coincide with monotone and strongly monotone mappings, respectively.

Let A be a strongly positive bounded linear operator on H, that is, there exists a constant $\bar{\gamma }>0$ such that

$$〈Ax,x〉\ge \bar{\gamma }{∥x∥}^2,\mathsf{\text{for}}\mathsf{\text{all}}x\in H.$$

(1.4)

Remark 1.4. From the definition of operator A, we note that a strongly positive bounded linear operator A is a ∥A∥-Lipschitzian and η-strongly 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:

$${\mathsf{\text{min}}}_{x\in C}\frac{1}{2}〈Ax,x〉-〈x,u〉,$$

(1.5)

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 [18] introduced and considered the following general iterative method:

$$x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)T{x}_n,\forall n\ge 0,$$

(1.6)

where A is a strongly positive bounded linear operator on a real Hilbert space H. They, proved that, if the sequence {α _n } of parameters satisfies appropriate conditions, then the sequence {x _n } generated by (1.6) converges strongly to the unique solution of the variational inequality

$$〈\left(\gamma f-A\right)x^{*},x-x^{*}〉\le 0,\forall x\in \mathsf{\text{Fix}}\left(T\right),$$

(1.7)

which is the optimality condition for the minimization problem

$${\mathsf{\text{min}}}_{x\in C}\frac{1}{2}〈Ax,x〉-h\left(x\right),$$

(1.8)

where C is the fixed point set of a nonexpansive mapping T and h is a potential function for γf (i.e., h'(x) = γf(x) for all x ∈ H).

On the other hand, Yamada [19] introduced a hybrid steepest descent method for a non-expansive mapping T as follows:

$$x_{n+1}=T{x}_n-\mu {\lambda }_{n}F\left(T{x}_n\right),\forall n\ge 0,$$

(1.9)

where F is a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and $0<\mu <\frac{2\eta }{{\kappa }^2}$. He proved that if {λ _n } satisfying appropriate conditions, then the sequence {x _n } generated by (1.9) converges strongly to the unique solution of variational inequality

$$〈F{x}^{*},x-x^{*}〉\ge 0,\forall x\in \mathsf{\text{Fix}}\left(T\right).$$

(1.10)

In 2010, Tian [20] combined the iterative method (1.6) with the Yamada's method (1.9) and considered a general iterative method for a nonexpansive mapping T as follows:

$$x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_n\mu F\right)T{x}_n,\forall n\ge 0.$$

(1.11)

Then he proved that the sequence {x _n } generated by (1.11) converges strongly to the unique solution of variational inequality

$$〈\left(\gamma f-\mu F\right)x^{*},x-x^{*}〉\le 0,\forall x\in \mathsf{\text{Fix}}\left(T\right).$$

(1.12)

Very recently, Ceng et al. [21] introduced implicit and explicit iterative schemes for finding the fixed points of a nonexpansive mapping T on a nonempty, closed and convex subset C in a real Hilbert space H as follows:

$$x_t=P_C\left[t\gamma V{x}_t+\left(I-t\mu F\right)T{x}_t\right]$$

(1.13)

and

$$x_{n+1}=P_C\left[{\alpha }_n\gamma V{x}_n+\left(I-{\alpha }_n\mu F\right)T{x}_n\right],\forall n\ge 0,$$

(1.14)

where V is an L-Lipschitzian mapping with a constant L ≥ 0 and F is a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and $0<\mu <\frac{2\eta }{{\kappa }^2}$. Then they proved that the sequences generated by (1.13) and (1.14) converge strongly to the unique solution of variational inequality

$$〈\left(\mu F-\gamma V\right)x^{*},x^{*}-x〉\ge 0,\forall x\in \mathsf{\text{Fix}}\left(T\right).$$

(1.15)

The following questions naturally arise in connection with above results:

Question 1.5. Can Theorem of Ceng et al. [21] be extend from a real Hilbert space to a general Banach space? such as q-uniformly smooth Banach space.

Question 1.6. Can we extend the iterative method of scheme (1.14) to a general iterative scheme define over the set of fixed points of a countable family of strict pseudo-contractions.

The purpose of this article is to give the affirmative answers to these questions mentioned above, motivated by Yamada [19], Tian [20] and Ceng et al. [21], we introduce a general iterative method for finding a common fixed point set of a countable family of strict pseudo-contractions, which is a unique solution of some variational inequality. Furthermore, we prove the strong convergence theorems of such iterative scheme in a q-uniformly smooth Banach space which admits a weakly sequentially continuous generalized duality mapping. The results presented in this article extend and generalize the corresponding results announced by Yamada [19] and Ceng et al. [21] and many others to Banach spaces.

2 Preliminaries

Let D be a nonempty subset of C. A mapping Q: C → D is said to be sunny if

$$Q\left(Qx+t\left(x-Qx\right)\right)=Qx,$$

whenever Qx + t(x - Qx) ∈ C for x ∈ C and t ≥ 0. A mapping Q: C → D is said to be retraction if Qx = x for all x ∈ 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 retraction Q is said to be orthogonal if for each x, x - Qx is normal to D in the sense of James (see [22]). 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 _c is coincident with the metric projection from X onto C.

The following lemma concerns the sunny nonexpansive retraction.

Lemma 2.1. [23] Let C be a closed and convex subset of a real q-uniformly smooth Banach space X. Let Q: X → C be a retraction. Then, Q is an orthogonal retraction if and only if

$$〈x-Qx,j_q\left(y-Qx\right)〉\le 0,\forall x\in Xandy\in C.$$

Lemma 2.2. [24] Let X be a real q-uniformly smooth Banach space. Then the following inequality holds:

$${∥x+y∥}^q\le {∥x∥}^q+q〈y,J_q\left(x\right)〉+C_q{∥y∥}^q,$$

for all x, y ∈ X and for some C _q > 0.

Lemma 2.3. [25] Suppose that q > 1. Then the following inequality holds:

$$ab\le \frac{1}{q}{a}^q+\left(\frac{q-1}{q}\right)b^{\frac{q}{q-1}}$$

for arbitrary positive real numbers a, b.

Lemma 2.4. [26] Let {s _n } be a sequence of nonnegative real numbers, {a _n } be a sequence of [0,1] with ${\sum }_{n=1}^{\infty }{a}_n=\infty$, {c _n } be a sequence of nonnegative real number with ${\sum }_{n=1}^{\infty }{c}_n<\infty$ and {b _n } be a sequence of real numbers with lim sup_n→∞b _n ≤ 0. Suppose that

$$s_{n+1}=\left(1-a_n\right)s_n+a_n{b}_n+c_n,$$

for all n ∈ ℕ. Then, lim_n→∞s _n = 0.

Definition 2.5. Let {T _n } be a family of mappings from a subset C of a Banach space X into itself with ${\bigcap }_{n=1}^{\infty }F\left(T_n\right)\ne \varnothing$. We say that {T _n } satisfies the AKTT-condition (see [26]) if for each bounded subset B of C,

$$\sum _{n=1}^{\infty }\underset{\omega \in B}{\mathsf{\text{sup}}}∥T_{n+1}\omega -T_n\omega ∥<\infty .$$

(2.1)

Lemma 2.6. [26] Suppose that {T _n } satisfy the AKTT-condition such that

(i) For each x ∈ C, {T _n } is converge strongly to some point in C.

(ii) Let the mapping T: C → C defined by Tx = lim_n→∞, T _n x for all x ∈ C.

Then lim_n→∞sup_ω∈B∥T ω - T _n ω∥ = 0 for each bounded subset B of C.

Lemma 2.7. [27, 28] Let C be a closed and convex subset of a smooth Banach space X. Suppose that ${\left\{T_n\right\}}_{n=1}^{\infty }:C\to X$ is a family of λ-strictly pseudocontractive mappings with ${\bigcap }_{n=1}^{\infty }\mathsf{\text{Fix}}\left(T_n\right)\ne \varnothing$ and ${\left\{{\mu }_m\right\}}_{n=1}^{\infty }$ is a real sequence in (0,1) such that ${\sum }_{n=1}^{\infty }{\mu }_n=1$. Then the following conclusions hold:

(i) A mapping G: C → X defined by $G:={\sum }_{n=1}^{\infty }{\mu }_n{T}_n$ is a λ-strictly pseudocontractive mapping.

(ii) $\mathsf{\text{Fix}}\left(G\right)={\bigcap }_{n=1}^{\infty }\mathsf{\text{Fix}}\left(T_n\right)$.

Lemma 2.8. [28] Let C be a closed and convex subset of a smooth Banach space X. Suppose that ${\left\{S_k\right\}}_{k=1}^{\infty }:C\to C$ is a countable family of λ-strictly pseudocontractive mappings with ${\bigcap }_{k=1}^{\infty }\mathsf{\text{Fix}}\left(S_k\right)\ne \varnothing$. For all n ∈ ℕ, define T _n : C → C by $T_{n}x:={\sum }_{k=1}^n{\mu }_n^k{S}_{k}x$ for all x ∈ C, where $\left\{{\mu }_n^k\right\}$ is a family of nonnegative numbers satisfying the following conditions:

(i) ${\sum }_{k=1}^n{\mu }_n^k=1$ for all n ∈ ℕ;

(ii) ${\mu }^k:=\underset{n\to \infty }{\mathsf{\text{lim}}}{\mu }_n^k>0$ for all k ∈ ℕ;

(iii) ${\sum }_{n=1}^{\infty }{\sum }_{k=1}^n\left|{\mu }_{n+1}^k-{\mu }_n^k\right|<\infty$.

Then the following hold:

(1) Each T _n is a λ-strictly pseudocontractive mapping.

(2) {T _n } satisfies the AKTT-condition.

(3) If T: C → C is defined by $Tx={\sum }_{k=1}^{\infty }{\mu }^k{S}_{k}x$ for all x ∈ C, then Tx = lim_n→∞T _n x and $\mathsf{\text{Fix}}\left(T\right)={\bigcap }_{n=1}^{\infty }\mathsf{\text{Fix}}\left(T_n\right)={\bigcap }_{k=1}^{\infty }\mathsf{\text{Fix}}\left(S_k\right)$.

3 Main results

In order to prove our main result, the following lemmas are needed.

Lemma 3.1. Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space X. Let F: C → X be a κ-Lipschitzian and η-strongly accretive operator with constants κ, η > 0. Let $0<\mu <{\left(\frac{q\eta }{C_q{\kappa }^q}\right)}^{\frac{1}{q-1}}$ and $\tau =\mu \left(\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}\right)$. Then for $t\in \left(0,\mathsf{\text{min}}\left\{1,\frac{1}{\tau }\right\}\right)$, the mapping S: C → X define by S := (I - tμF) is a contraction with constant 1-tτ.

Proof. Since $0<\mu <{\left(\frac{q\eta }{C_q{\kappa }^q}\right)}^{\frac{1}{q-1}}$ with q > 1 and $t\in \left(0,\mathsf{\text{min}}\left\{1,\frac{1}{\tau }\right\}\right)$. This implies that 1 - tτ ∈ (0,1). From Lemma 2.2, for all x, y ∈ C, we have

$$\begin{array}{ll}\hfill {∥Sx-Sy∥}^q& ={∥\left(I-t\mu F\right)x-\left(I-t\mu F\right)y∥}^q\\ ={∥\left(x-y\right)-t\mu \left(Fx-Fy\right)∥}^q\\ \le {∥x-y∥}^q-qt\mu 〈Fx-Fy,j_q\left(x-y\right)〉+C_q{t}^q{\mu }^q{∥Fx-Fy∥}^q\\ \le {∥x-y∥}^q-qt\mu \eta {∥x-y∥}^q+C_q{t}^q{\mu }^q{\kappa }^q{∥x-y∥}^q\\ \le \left[1-t\mu \left(q\eta -C_q{\mu }^{q-1}{\kappa }^q\right)\right]{∥x-y∥}^q\\ =\left[1-t\mu q\left(\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}\right)\right]{∥x-y∥}^q\\ \le {\left[1-t\mu \left(\eta -\frac{C_q{\mu }^{q-1}{k}^q}{q}\right)\right]}^q{∥x-y∥}^q\\ ={\left(1-t\tau \right)}^q{∥x-y∥}^q.\end{array}$$

It follows that

$$∥Sx-Sy∥\le \left(1-t\tau \right)∥x-y∥.$$

Hence, we have S := (I - tμF) is a contraction with a constant 1 - tτ.

Lemma 3.2. Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space X which admits weakly sequentially continuous generalized duality mapping j _q from X into X*. Let T: C → C be a nonexpansive mapping. Then, for all {x _n } ⊂ C, if x _n ⇀ x and $x_n-T{x}_n\to 0$, then x = Tx.

Proof. From Lemma 2.2, for all x ∈ C, we have

$$\begin{array}{ll}\hfill {∥x-Tx-\left(x_n-T{x}_n\right)∥}^q& ={∥\left(x_n-x\right)+\left(Tx-T{x}_n\right)∥}^q\\ \le {∥x_n-x∥}^q+q〈Tx-T{x}_n,j_q\left(x_n-x\right)〉+C_q{∥Tx-T{x}_n∥}^q\\ \le 〈x_n-x,j_q\left(x_n-x\right)〉+q〈Tx-T{x}_n,j_q\left(x_n-x\right)〉+C_q〈x-x_n,j_q\left(x-x_n\right)〉.\end{array}$$

Taking the limit as n → ∞ in both sides and noting that j _q is weakly sequentially continuous generalized duality mapping. Then, ∥x - Tx∥^q≤ 0, this implies that x = Tx.

3.1 Implicit iteration scheme

Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space X. Let Q _C be a sunny nonexpansive retraction from X onto C. Let F: C → X be a κ-Lipschitzian and η-strongly accretive operator with constants κ, η > 0, V: C → X be an L-Lipschitzian mapping with a constant L ≥ 0 and T: C → C be a nonexpansive mapping such that Fix(T) ≠ ∅. Let $0<\mu <{\left(\frac{q\eta }{C_q{\kappa }^q}\right)}^{\frac{1}{q-1}}$ and 0 ≤ γL < τ, where $\tau =\mu \left(\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}\right)$. For each $t\in \left(0,\mathsf{\text{min}}\left\{1,\frac{1}{\tau }\right\}\right)$, we define the mapping S _t : C → C by

$$S_{t}x:=Q_C\left[t\gamma Vx+\left(I-t\mu F\right)Tx\right],\forall x\in C.$$

It is easy to see that S _t is a contraction. Indeed, from Lemma 3.1, for all x, y ∈ C, we have

$$\begin{array}{ll}\hfill ∥S_{t}x-S_{t}y∥& =∥Q_C[t\gamma Vx+(I-t\mu F\left)Tx\right]-Q_C[t\gamma Vy+(I-t\mu F)Ty]\\ \le ∥\left[t\gamma Vx+\left(I-t\mu F\right)Tx\right]-\left[t\gamma Vy+\left(I-t\mu F\right)Ty\right]∥\\ \le t\gamma ∥Vx-Vy∥+∥\left(I-t\mu F\right)\left(Tx-Ty\right)∥\\ \le t\gamma L∥x-y∥+\left(1-t\tau \right)∥x-y∥\\ =\left(1-\left(\tau -\gamma L\right)t\right)∥x-y∥.\end{array}$$

Hence, S _t has a unique fixed point, denoted by x _t , which uniquely solve the fixed point equation

$$x_t=Q_C\left[t\gamma V{x}_t+\left(I-t\mu F\right)T{x}_t\right]$$

(3.1)

The following proposition summarizes the properties of the net {x _t }.

Proposition 3.3. Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space X. Let Q _c be a sunny nonexpansive retraction from X onto C. Let F: C → X be a κ-Lipschitzian and η-strongly accretive operator with constants κ, η > 0, V: C → X be an L-Lipschitzian mapping with a constant L ≥ 0 and T: C → C be a nonexpansive mapping such that Fix(T) ≠ ∅. Let $0<\mu <{\left(\frac{q\eta }{C_q{\kappa }^q}\right)}^{\frac{1}{q-1}}$ and 0 ≤ γL < τ, where $\tau =\mu \left(\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}\right)$.

then

(i) {x _t } is bounded for each $t\in \left(0,\mathsf{\text{min}}\left\{1,\frac{1}{\tau }\right\}\right)$.

(ii) ${\mathsf{\text{lim}}}_{t\to 0}∥x_t-T{x}_t∥=0$.

(iii) {x _t } defines a continuous curve from $\left(0,\mathsf{\text{min}}\left\{1,\frac{1}{\tau }\right\}\right)$ into C.

Proof.

• (i) Taking $\bar{x}\in \mathsf{\text{Fix}}\left(T\right)$. Then, we have

  $$\begin{array}{ll}\hfill ∥x_t-\bar{x}∥& =∥Q_C\left[t\gamma V{x}_t+\left(I-t\mu F\right)T{x}_t\right]-Q_C\bar{x}∥\\ \le ∥\left[t\gamma V{x}_t+\left(I-t\mu F\right)T{x}_t\right]-\bar{x}∥\\ =∥t\left(\gamma V{x}_t-\mu F\bar{x}\right)+\left(I-t\mu F\right)\left(T{x}_t-\bar{x}\right)∥\\ \le t∥\gamma V{x}_t-\mu F\bar{x}∥+\left(1-t\tau \right)∥T{x}_t-\bar{x}∥\\ \le t\gamma ∥V{x}_t-V\bar{x}∥+t∥\gamma V\bar{x}-\mu F\bar{x}∥+\left(1-t\tau \right)∥x_t-\bar{x}∥\\ \le \left(1-\left(\tau -\gamma L\right)t\right)∥x_t-\bar{x}∥+t∥\gamma V\bar{x}-\mu F\bar{x}∥.\end{array}$$

It follows that

$$∥x_t-\bar{x}∥\le \frac{∥\gamma V\bar{x}-\mu F\bar{x}∥}{\tau -\gamma L}.$$

Hence, {x _t } is bounded, so are {Vx _t } and {FTx _t }.

• (ii) By definition of {x _t }, we have

  $$\begin{array}{ll}\hfill ∥x_t-T{x}_t∥& =∥Q_C\left[t\gamma V{x}_t+\left(I-t\mu F\right)T{x}_t\right]-Q_{C}T{x}_t∥\\ \le ∥t\gamma V{x}_t+\left(I-t\mu F\right)T{x}_t-T{x}_t∥\\ =t∥\gamma V{x}_t-\mu FT{x}_t∥\to 0\mathsf{\text{as}}t\to 0.\end{array}$$

• (iii) Take t, $t_0\in \left(0,\mathsf{\text{min}}\left[1,\frac{1}{\tau }\right]\right)$. From Lemma 3.1, we have

  $$\begin{array}{ll}\hfill ∥x_t-x_{t_0}∥& =∥Q_C\left[t\gamma V{x}_t+\left(I-t\mu F\right)T{x}_t\right]-Q_C\left[t_0\gamma V{x}_{t_0}+\left(I-t\mu F\right)T{x}_{t_0}\right]∥\\ \le ∥\left[t\gamma V{x}_t+\left(I-t\mu F\right)T{x}_t\right]-\left[t_0\gamma V{x}_{t_0}+\left(I-t\mu F\right)T{x}_{t_0}\right]∥\\ =∥\left(t-t_0\right)\gamma V{x}_t+t_0\gamma \left(V{x}_t-V{x}_{t_0}\right)-\left(t-t_0\right)\mu FT{x}_t+\left(I-t_0\mu F\right)\left(T{x}_t-T{x}_{t_0}\right)∥\\ \le \left(\gamma ∥V{x}_t∥+\mu ∥FT{x}_t∥\right)\left|t-t_0\right|+\left(1-\left(\tau -\gamma L\right)t_0\right)∥x_t-x_{t_0}∥.\end{array}$$

It follows that

$$∥x_t-x_{t_0}∥\le \frac{\left(\gamma ∥V{x}_t∥+\mu ∥FT{x}_t∥\right)}{\left(\tau -\gamma L\right)t_0}\left|t-t_0\right|.$$

Since {Vx _t } and {FTx _t } is bounded. Hence, {x _t } defines a continuous curve from $\left(0,\mathsf{\text{min}}\left\{1,\frac{1}{\tau }\right\}\right)$ into C.

Theorem 3.4. Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space X which admits a weakly sequentially continuous generalized duality mapping j _q from X into X*. Let Q _c be a sunny nonexpansive retraction such that Q _c is an orthogonal from X onto C. Let F: C → X be a κ-Lipschitzian and η-strongly accretive operator with constants κ, η > 0, V: C → X be an L-Lipschitzian mapping with constant L ≥ 0 and T: C → C be a nonexpansive mapping such that Fix(T) ≠ ∅. Let $0<\mu <{\left(\frac{q\eta }{C_q{\kappa }^q}\right)}^{\frac{1}{q-1}}$ and 0 ≤ γL < τ, where $\tau =\mu \left(\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}\right)$. For each $t\in \left(0,\mathsf{\text{min}}\left\{1,\frac{1}{\tau }\right\}\right)$, let {x _t } defined by (3.1), then {x _t } converges strongly to x* ∈ Fix(T) as t → 0, which x* is the unique solution of the variational inequality

$$〈\left(\mu F-\gamma V\right)x^{*},j_q\left(x^{*}-z\right)〉\le 0,\forall z\in \mathsf{\text{Fix}}\left(T\right).$$

(3.2)

Proof. We observe that

$$\begin{array}{ll}\hfill \frac{C_q{\mu }^{q-1}{\kappa }^q}{q}>0& \iff \eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}<\eta \\ \iff \mu \left(\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}\right)<\mu \eta \\ \iff \tau <\mu \eta .\end{array}$$

(3.3)

It follows that

$$0\le \gamma L<\tau <\mu \eta .$$

(3.4)

First, we show the uniqueness of solution of the variational inequality (3.3). Suppose that $\tilde{x},x^{*}\in \mathsf{\text{Fix}}\left(T\right)$ are solutions of (3.3), then

$$〈\left(\mu F-\gamma V\right)x^{*},j_q\left(x^{*}-\tilde{x}\right)〉\le 0$$

(3.5)

and

$$〈\left(\mu F-\gamma V\right)\tilde{x},j_q\left(\tilde{x}-x^{*}\right)〉\le 0.$$

(3.6)

Adding up (3.5) and (3.6), we have

$$\begin{array}{ll}\hfill 0& \ge 〈\left(\mu F-\gamma V\right)x^{*}-\left(\mu F-\gamma V\right)\tilde{x},j_q\left(x^{*}-\tilde{x}\right)〉\\ =\mu 〈F{x}^{*}-F\tilde{x},j_q\left(x^{*}-\tilde{x}\right)〉-\gamma 〈V{x}^{*}-V\tilde{x},j_q\left(x^{*}-\tilde{x}\right)〉\\ \ge \mu \eta {∥x^{*}-\tilde{x}∥}^q-\gamma ∥V{x}^{*}-V\tilde{x}∥{∥x^{*}-\tilde{x}∥}^{q-1}\\ \ge \left(\mu \eta -\gamma L\right){∥x^{*}-\tilde{x}∥}^q.\end{array}$$

Note that (3.4) implies that $x^{*}=\tilde{x}$ and the uniqueness is proves. Below, we use x* to denote the unique solution of the variational inequality (3.3).

Next, we show that x _t → x* as t → 0. Setting $y_t=t\gamma V{x}_t+\left(I-t\mu F\right)T{x}_t$, where $t\in \left(0,\mathsf{\text{min}}\left\{1,\frac{1}{\tau }\right\}\right)$. Then, we can rewrite (3.1) as x _t = QCyt. Assume that {t _n } ⊂ (0,1) is a sequence such that t _n → 0 as n → ∞. Putting $x_n:=x_{t_n}$ and $y_n:y_{t_n}$. For z ∈ Fix(T), we note that

$$\begin{array}{ll}\hfill x_n-z& =Q_C{y}_n-y_n+y_n-z\\ =Q_C{y}_n-y_n+t_n\left(\gamma V{x}_n-\mu Fz\right)+\left(I-t_n\mu F\right)\left(T{x}_n-z\right).\end{array}$$

(3.7)

By Lemma 2.1, we have

$$〈Q_C{y}_n-y_n,j_q\left(Q_C{y}_n-z\right)〉\le 0.$$

(3.8)

It follows from (3.7) and (3.8) that

$$\begin{array}{ll}\hfill {∥x_n-z∥}^q=& 〈Q_C{y}_n-y_n,j_q\left(Q_C{y}_n-z\right)〉+〈\left(I-t_n\mu F\right)\left(T{x}_n-z\right),j_q\left(x_n-z\right)〉\\ \hfill +& t_n〈\gamma V{x}_n-\mu Fz,j_q\left(x_n-z\right)〉\\ \hfill \le & \left(1-t_n\tau \right){∥x_n-z∥}^q+t_n〈\gamma V{x}_n-\mu Fz,j_q\left(x_n-z\right)〉.\end{array}$$

Thus, we have

$$\begin{array}{ll}\hfill {∥x_n-z∥}^q& \le \frac{1}{\tau }〈\gamma V{x}_n-\mu Fz,j_q\left(x_n-z\right)〉\\ =\frac{1}{\tau }\left\{\gamma 〈V{x}_n-Vz,j_q\left(x_n-z\right)〉+〈\gamma Vz-\mu Fz,j_q\left(x_n-z\right)〉\right\}\\ \le \frac{1}{\tau }\left\{\gamma L{∥x_n-z∥}^q+〈\gamma Vz-\mu Fz,j_q\left(x_n-z\right)〉\right\},\end{array}$$

which implies that

$${∥x_n-z∥}^q\le \frac{1}{\tau -\gamma L}〈\gamma Vz-\mu Fz,j_q\left(x_n-z\right)〉.$$

In particular, we have

$${∥x_{n_i}-z∥}^q\le \frac{1}{\tau -\gamma L}〈\gamma Vz-\mu Fz,j_q\left(x_{n_i}-z\right)〉.$$

(3.9)

By reflexivity of a Banach space X and boundedness of {x _n }, there exists a subsequence $\left\{x_{n_i}\right\}$ of {x _n } such that $x_{n_i}\rightharpoonup \tilde{x}$ as i → ∞. Since Banach space X has a weakly sequentially continuous generalized duality mapping and by (3.9), we obtain $x_{n_i}\to \tilde{x}$. By Proposition 3.3 (ii), we have $x_n-T{x}_n\to 0$ as n → ∞. Hence, it follows from Lemma 3.2 that $\tilde{x}\in \mathsf{\text{Fix}}\left(T\right)$.

Next, we show that $\tilde{x}$ solves the variational inequality (3.3). We note that

$$x_t=Q_C{y}_t=Q_C{y}_t-y_t+t\gamma V{x}_t+\left(I-t\mu F\right)T{x}_t,$$

we derive that

$$\left(\mu F-\gamma V\right)x_t=\frac{1}{t}\left(Q_C{y}_t-y_t\right)-\frac{1}{t}\left(I-T\right)x_t+\mu \left(F{x}_t-FT{x}_t\right).$$

(3.10)

Since I - T is accretive (i.e., 〈(I - T)x - (I - T)y, j _q (x - y)〉 ≥ 0, for x, y ∈ C). For all z ∈ Fix(T), it follows from (3.10) and Lemma 2.1 that

$$\begin{array}{ll}\hfill 〈\left(\mu F-\gamma V\right)x_t,j_q\left(x_t-z\right)〉& =\frac{1}{t}〈Q_C{y}_t-y_t,j_q\left(Q_C{y}_t-z\right)〉-\frac{1}{t}〈\left(I-T\right)x_t-\left(I-T\right)z,j_q\left(x_t-z\right)〉\\ +\mu 〈F{x}_t-FT{x}_t,j_q\left(x_t-z\right)〉\\ \le \mu 〈F{x}_t-FT{x}_t,j_q\left(x_t-z\right)〉\\ \le \mu ∥F{x}_t-FT{x}_t∥{∥x_t-z∥}^{q-1}\\ \le ∥x_t-T{x}_t∥M,\end{array}$$

(3.11)

where M > 0 is a constant such that M = sup{μκ∥x _t - z∥^q-1}, where $t\in \left(0,\mathsf{\text{min}}\left\{1,\frac{1}{\tau }\right\}\right)$. Now, replacing t in (3.11) with t _n and taking the limit as n → ∞, we noticing that $x_{t_n}-T{x}_{t_n}\to \tilde{x}-T\tilde{x}=0$ for $\tilde{x}\in \mathsf{\text{Fix}}\left(T\right)$, we obtain $〈\left(\mu F-\gamma V\right)\tilde{x},j_q\left(\tilde{x}-z\right)〉\le 0$. That is $\tilde{x}\in \mathsf{\text{Fix}}\left(T\right)$ is the solution of variational inequality (3.3). Consequently, $\tilde{x}=x^{*}$ by uniqueness. Therefore x _t → x* as t → 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 q-uniformly smooth Banach space X which admits a weakly sequentially continuous generalized duality mapping j _q from X into X*. Let Q _c be a sunny nonexpansive retraction such that Q _c is an orthogonal from X onto C. Let F: C → X be a κ-Lipschitzian and η-strongly accretive operator with constants κ, η > 0, V: C → X be an L-Lipschitzian mapping with constant L ≥ 0. Let $0<\mu <{\left(\frac{q\eta }{C_q{\kappa }^q}\right)}^{\frac{1}{q-1}}$ and 0 ≤ γL < τ, where $\tau =\mu \left(\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}\right)$. Let ${\left\{T_n\right\}}_{n=1}^{\infty }:C\to C$ be a family of λ-strict pseudo-contractions with 0 < λ < 1. Define a mapping S _n x := (1 - γ _n )x + γ _n T _n x for all x ∈ C and n ≥ 1. Assume that $\Omega :={\bigcap }_{n=1}^{\infty }\mathsf{\text{Fix}}\left(T_n\right)\ne \varnothing$. Let {x _n } be a sequence defined by x₁ ∈ C and

$$x_{n+1}=Q_C\left[{\alpha }_n\gamma V{x}_n+\left(I-{\alpha }_n\mu F\right)S_n{x}_n\right],\forall n\ge 1,$$

(3.12)

where {α _n } and {γ _n } are sequences in (0,1) which satisfy the following conditions:

(C1) lim_n→∞α _n = 0 and ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$;

(C2) either ${\sum }_{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_n\right|<\infty$ or $\underset{n\to \infty }{\mathsf{\text{lim}}}\frac{{\alpha }_{n+1}}{{\alpha }_n}=1$;

(C3) $0<{\gamma }_n\le \delta ,\delta =\mathsf{\text{min}}\left\{1,{\left(\frac{q\lambda }{C_q}\right)}^{\frac{1}{q-1}}\right\}$ and ${\sum }_{n=1}^{\infty }\left|{\gamma }_{n+1}-{\gamma }_n\right|<\infty$.

Suppose in addition that ${\left\{T_n\right\}}_{n=1}^{\infty }$ satisfies the AKTT-condition. Let T: C → C be the mapping defined by Tx = lim_n→∞T _n x for all x ∈ C and suppose that $\mathsf{\text{Fix}}\left(T\right)={\bigcap }_{n=1}^{\infty }\mathsf{\text{Fix}}\left(T_n\right)$. Then the sequence {x _n } defined by (3.12) converges strongly to x* ∈ Fix(T) as n → ∞, which x* is the unique solution of the variational inequality

$$〈\left(\mu F-\gamma V\right)x^{*},j_q\left(x^{*}-z\right)〉\le 0,\forall z\in \mathsf{\text{Fix}}\left(T\right).$$

(3.13)

Proof. From the condition (C 1), we may assume, without loss of generality, that ${\alpha }_n\le \mathsf{\text{min}}\left\{1,\frac{1}{\tau }\right\}$ for all n ∈ ℕ. First, we show that {x _n } is bounded. From Lemma 2.2 and the condition (C 3), for all x, y ∈ C, we have

$$\begin{array}{ll}\hfill {∥S_{n}x-S_{n}y∥}^q& ={∥\left(1-{\gamma }_n\right)x+{\gamma }_n{T}_{n}x-\left[\left(1-{\gamma }_n\right)y+{\gamma }_n{T}_{n}y\right]∥}^q\\ ={∥x-y-{\gamma }_n\left[x-y-\left(T_{n}x-T_{n}y\right)\right]∥}^q\\ \le {∥x-y∥}^q-q{\gamma }_n〈x-y-\left(T_{n}x-T_{n}y\right),j_q\left(x-y\right)〉+C_q{\gamma }_n^q{∥x-y-\left(T_{n}x-T_{n}y\right)∥}^q\\ ={∥x-y∥}^q-q{\gamma }_n{∥x-y∥}^q+q{\gamma }_n〈T_{n}x-T_{n}y,j_q\left(x-y\right)〉+C_q{\gamma }_n^q{∥x-y-\left(T_{n}x-T_{n}y\right)∥}^q\\ \le {∥x-y∥}^q+q{\gamma }_n\left({∥x-y∥}^q-\lambda {∥x-y-\left(T_{n}x-T_{n}y\right)∥}^q\right)-q{\gamma }_n{∥x-y∥}^q\\ +C_q{\gamma }_n^q{∥x-y-\left(T_{n}x-T_{n}y\right)∥}^q\\ ={∥x-y∥}^q+\left(C_q{\gamma }_n^q-q{\gamma }_n\lambda \right){∥x-y-\left(T_{n}x-T_{n}y\right)∥}^q\\ \le {∥x-y∥}^2.\end{array}$$

It follows that ∥S _n x - S _n y∥ ≤ ∥x - y∥, which implies that S _n is nonexpansive and Fix(T _n ) = Fix(S _n ). Taking $\bar{x}\in \Omega$. Then we have

$$\begin{array}{ll}\hfill ∥x_{n+1}-\bar{x}∥& =∥Q_C\left[{\alpha }_n\gamma V{x}_n+\left(I-{\alpha }_n\mu F\right)S_n{x}_n{x}_n\right]-Q_C\bar{x}∥\\ \le ∥{\alpha }_n\gamma V{x}_n+\left(I-{\alpha }_n\mu F\right)S_n{x}_n{x}_n-\bar{x}∥\\ =∥{\alpha }_n\left(\gamma V{x}_n-\mu F\bar{x}\right)+\left(I-{\alpha }_n\mu F\right)\left(S_n{x}_n-\bar{x}\right)∥\\ \le {\alpha }_n∥\gamma V{x}_n-\mu F\bar{x}∥+\left(1-{\alpha }_n\tau \right)∥S_n{x}_n-\bar{x}∥\\ \le {\alpha }_n\gamma ∥V{x}_n-V\bar{x}∥+{\alpha }_n∥\gamma V\bar{x}-\mu F\bar{x}∥+\left(1-{\alpha }_n\tau \right)∥x_n-\bar{x}∥\\ \le \left(1-\left(\tau -\gamma L\right){\alpha }_n\right)∥x_n-\bar{x}∥+{\alpha }_n∥\gamma V\bar{x}-\mu F\bar{x}∥\\ =\left(1-\left(\tau -\gamma L\right){\alpha }_n\right)∥x_n-\bar{x}∥+\left(\tau -\gamma L\right){\alpha }_n\frac{∥\gamma V\bar{x}-\mu F\bar{x}∥}{\tau -\gamma L}.\end{array}$$

By induction, we have

$∥x_n-\bar{x}∥\le \mathsf{\text{max}}\left\{∥x_1-\bar{x}∥,\frac{∥\gamma V\bar{x}-\mu F\bar{x}∥}{\tau -\gamma L}\right\},\forall n\ge 1.$

Hence, {x _n } is bounded, so are {Vx _n } and {FS _n x _n }.

Next, we show that ∥x_n+1- x _n ∥ → 0 as n → ∞. Since

$$\begin{array}{ll}\hfill ∥S_{n+1}{x}_{n+1}-S_n{x}_n∥& \le ∥S_{n+1}{x}_{n+1}-S_{n+}{x}_n∥+∥S_{n+1}{x}_n-S_n{x}_n∥\\ \le ∥x_{n+1}-x_n∥+∥\left(1-{\gamma }_{n+1}\right)x_n+{\gamma }_{n+1}{T}_{n+1}{x}_n-\left[\left(1-{\gamma }_n\right)x_n+{\gamma }_n{T}_n{x}_n\right]∥\\ =∥x_{n+1}-x_n∥+∥\left({\gamma }_{n+1}-{\gamma }_n\right)\left(T_{n+1}{x}_n-x_n\right)+{\gamma }_n\left(T_{n+1}{x}_n-T_n{x}_n\right)∥\\ \le ∥x_{n+1}-x_n∥+\left|{\gamma }_{n+1}-{\gamma }_n\right|∥T_{n+1}{x}_n-x_n∥+{\gamma }_n∥T_{n+1}{x}_n-T_n{x}_n∥.\end{array}$$

(3.14)

On the other hand, we have

$$\begin{array}{ll}\hfill ∥x_{n+2}-x_{n+1}∥& =∥Q_C\left[{\alpha }_{n+1}\gamma V{x}_{n+1}+\left(I-{\alpha }_{n+1}\mu F\right)S_{n+1}{x}_{n+1}\right]-Q_C\left[{\alpha }_n\gamma V{x}_n+\left(I-{\alpha }_n\mu F\right)S_n{x}_n\right]∥\\ \le ∥\left[{\alpha }_{n+1}\gamma V{x}_{n+1}+\left(I-{\alpha }_{n+1}\mu F\right)S_{n+1}{x}_{n+1}\right]-\left[{\alpha }_n\gamma V{x}_n+\left(I-{\alpha }_n\mu F\right)S_n{x}_n\right]∥\\ =∥{\alpha }_{n+1}\gamma \left(V{x}_{n+1}-V{x}_n\right)+\left({\alpha }_{n+1}-{\alpha }_n\right)\gamma V{x}_n+\left(I-{\alpha }_{n+1}\mu F\right)\left(S_{n+1}{x}_{n+1}-S_n{x}_n\right)\\ +\left({\alpha }_n-{\alpha }_{n+1}\right)\mu F{S}_n{x}_n∥\\ \le {\alpha }_{n+1}\gamma L∥x_{n+1}-x_n∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|\left(\gamma ∥V{x}_n∥+\mu ∥F{S}_n{x}_n∥\right)\\ +\left(1-{\alpha }_{n+1}\tau \right)∥S_{n+1}{x}_{n+1}-S_n{x}_n∥.\end{array}$$

(3.15)

Substituting (3.14) into (3.15), we obtain

$$\begin{array}{ll}\hfill ∥x_{n+2}-x_{n+1}∥& \le {\alpha }_{n+1}\gamma L∥x_{n+1}-x_n∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|\left(\gamma ∥V{x}_n∥+\mu ∥F{S}_n{x}_n∥\right)\\ +\left(1-{\alpha }_{n+1}\tau \right)\left[∥x_{n+1}-x_n∥+\left|{\gamma }_{n+1}-{\gamma }_n\right|∥T_{n+1}{x}_n-x_n∥+{\gamma }_n∥T_{n+1}{x}_n-T_n{x}_n∥\right]\\ \le \left(1-\left(\tau -\gamma L\right){\alpha }_{n+1}\right)∥x_{n+1}-x_n∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|M_1+\left|{\gamma }_{n+1}-{\gamma }_n\right|M_2\\ +∥T_{n+1}{x}_n-T_n{x}_n∥,\end{array}$$

where M₁ = sup_n≥1{γ∥Vx _n ∥, μ∥FS _n x _n ∥} and M₂ = sup_n≥1{∥T_n+1x _n - x _n ∥}. It follows from the conditions (C 2), (C 3) and Lemma 2.4 that

$${\mathsf{\text{lim}}}_{n\to \infty }∥x_{n+1}-x_n∥=0.$$

(3.16)

Next, we show that ∥x _n - Sx _n ∥ → 0 as n → ∞. For any bounded subset B of C, we observe that

$$\begin{array}{ll}\hfill {\mathsf{\text{sup}}}_{\omega \in B}∥S_{n+1}\omega -S_n\omega ∥& ={\mathsf{\text{sup}}}_{\omega \in B}∥\left(1-{\gamma }_{n+1}\right)\omega +{\gamma }_{n+1}{T}_{n+1}\omega -\left(\left(1-{\gamma }_n\right)\omega +{\gamma }_n{T}_n\omega \right)∥\\ \le \left|{\gamma }_{n+1}-{\gamma }_n\right|{\mathsf{\text{sup}}}_{\omega \in B}∥\omega ∥+{\gamma }_{n+1}{\mathsf{\text{sup}}}_{\omega \in B}∥T_{n+1}\omega -T_n\omega ∥\\ +\left|{\gamma }_{n+1}-{\gamma }_n\right|{\mathsf{\text{sup}}}_{\omega \in B}∥T_n\omega ∥\\ \le \left|{\gamma }_{n+1}-{\gamma }_n\right|M_3+{\mathsf{\text{sup}}}_{\omega \in B}∥T_{n+1}\omega -T_n\omega ∥,\end{array}$$

where M₃ = sup_n≥1{∥ω∥, ∥T _n ω∥}. From the condition (C 3) and {T _n } satisfies the AKTT-condition, then we have

$${\sum }_{n=1}^{\infty }{\mathsf{\text{sup}}}_{\omega \in B}∥S_{n+1}\omega -T_n\omega ∥<\infty ,$$

that is {S _n } satisfies the AKTT-condition, we can define nonexpansive mapping S: C → C by Sx = lim_n→∞S _n x for all x ∈ C. Since {γ _n } is bounded, there exits a subsequence $\left\{{\gamma }_{n_i}\right\}$ of {γ _n } such that ${\gamma }_{n_i}\to \nu$ as i → ∞. It follows that

$$Sx={\mathsf{\text{lim}}}_{i\to \infty }{S}_{n_i}x={\mathsf{\text{lim}}}_{i\to \infty }\left[\left(1-{\gamma }_{n_i}\right)x+{\gamma }_{n_i}{T}_{n_j}x\right]=\left(1-\nu \right)x+\nu Tx,\forall x\in C.$$

That is, Fix(S) = Fix(T). Hence, $\mathsf{\text{Fix}}\left(S\right)={\bigcap }_{n=1}^{\infty }\mathsf{\text{Fix}}\left(T_n\right)={\bigcap }_{n=1}^{\infty }\mathsf{\text{Fix}}\left(S_n\right)=\Omega$. We observe that

$$\begin{array}{ll}\hfill ∥x_n-S_n{x}_n∥& \le ∥x_n-x_{n+1}∥+∥x_{n+1}-S_n{x}_n∥\\ =∥x_n-x_{n+1}∥+∥Q_C\left[{\alpha }_n\gamma V{x}_n+\left(I-{\alpha }_n\mu F\right)S_n{x}_n\right]-Q_C{S}_n{x}_n∥\\ \le ∥x_n-x_{n+1}∥+∥{\alpha }_n\gamma V{x}_n+\left(I-{\alpha }_n\mu F\right)S_n{x}_n-S_n{x}_n∥\\ =∥x_n-x_{n+1}∥+{\alpha }_n∥\gamma V{x}_n-\mu F{S}_n{x}_n∥.\end{array}$$

From the condition (C 1) and (3.16), we have

$${\mathsf{\text{lim}}}_{n\to \infty }∥x_n-S_n{x}_n∥=0.$$

(3.17)

On the other hand, we observe that

$$\begin{array}{ll}\hfill ∥x_n-S{x}_n∥& \le ∥x_n-S_n{x}_n∥+∥S_n{x}_n-S{x}_n∥\\ \le ∥x_n-S_n{x}_n∥+{\mathsf{\text{sup}}}_{\omega \in \left\{x_n\right\}}∥S_n\omega -S\omega ∥,\end{array}$$

which implies by Lemma 2.6 and (3.17) that

$${\mathsf{\text{lim}}}_{n\to \infty }∥x_n-S{x}_n∥=0.$$

(3.18)

Next, we show that

$$\mathsf{\text{lim}}{\mathsf{\text{sup}}}_{n\to \infty }〈\left(\gamma V-\mu F\right)x^{*},j_q\left(x_n-x^{*}\right)〉\le 0,$$

where x* is the same as in Theorem 3.4. To show this, we take a subsequence $\left\{x_{n_i}\right\}$ of {x _n } such that

$$\mathsf{\text{lim}}{\mathsf{\text{sup}}}_{n\to \infty }〈\left(\gamma V-\mu F\right)x^{*},j_q\left(x_n-x^{*}\right)〉=\underset{i\to \infty }{\mathsf{\text{lim}}}〈\left(\gamma V-\mu F\right)x^{*},j_q\left(x_{n_i}-x^{*}\right)〉.$$

By reflexivity of a Banach space X and boundedness of {x _n }, there exists a subsequence $\left\{x_{n_i}\right\}$ of {x _n } such that $x_{n_i}\rightharpoonup z$ as i → ∞. It follows from (3.18) and Lemma 3.2 that z ∈ Ω. Since Banach space X has a weakly sequentially continuous generalized duality mapping, we obtain that

$$\begin{array}{ll}\hfill \mathsf{\text{lim}}{\mathsf{\text{sup}}}_{n\to \infty }〈\left(\gamma V-\mu F\right)x^{*},j_q\left(x_n-x^{*}\right)〉& ={\mathsf{\text{lim}}}_{i\to \infty }〈\left(\gamma V-\mu F\right)x^{*},j_q\left(x_{n_i}-x^{*}\right)〉\\ =〈\left(\gamma V-\mu F\right)x^{*},j_q\left(z-x^{*}\right)〉\le 0.\end{array}$$

(3.19)

Finally, we show that x _n → x* as n → ∞. Setting y _n = α _n γVx _n +(I-α _n μF)S _n x _n , ∀n ≥ 1. Then, we can rewrite (3.12) as x_n+1= Qcy _n . It follows from Lemmas 2.1 and 2.3 that

$$\begin{array}{ll}\hfill {∥x_{n+1}-x^{*}∥}^q& =〈y_n-x^{*},j_q\left(x_{n+1}-x^{*}\right)〉+〈Q_C{y}_n-y_n,j_q\left(Q_C{y}_n-x^{*}\right)〉\\ \le 〈y_n-x^{*},j_q\left(x_{n+1}-x^{*}\right)〉\\ ={\alpha }_n〈\gamma V{x}_n-\mu F{x}^{*},j_q\left(x_{n+1}-x^{*}\right)〉+〈\left(I-{\alpha }_n\mu F\right)\left(S_n{x}_n-x^{*}\right),j_q\left(x_{n+1}-x^{*}\right)〉\\ ={\alpha }_n〈\gamma V{x}_n-V{x}^{*},j_q\left(x_{n+1}-x^{*}\right)〉+{\alpha }_n〈\gamma V{x}^{*}-\mu F{x}^{*},j_q\left(x_{n+1}-x^{*}\right)〉\\ +〈\left(I-{\alpha }_n\mu F\right)\left(S_n{x}_n-x^{*}\right),j_q\left(x_{n+1}-x^{*}\right)〉\\ \le {\alpha }_n\gamma L∥x_n-x^{*}∥{∥x_{n+1}-x^{*}∥}^{q-1}+{\alpha }_n〈\gamma V{x}^{*}-\mu F{x}^{*},j_q\left(x_{n+1}-x^{*}\right)〉\\ +\left(1-{\alpha }_n\tau \right)∥x_n-x^{*}∥{∥x_{n+1}-x^{*}∥}^{q-1}\\ =\left(1-\left(\tau -\gamma L\right){\alpha }_n\right)∥x_n-x^{*}∥{∥x_{n+1}-x^{*}∥}^{q-1}+{\alpha }_n〈\gamma L{x}^{*}-\mu F{x}^{*},j_q\left(x_{n+1}-x^{*}\right)〉\\ \le \left(1-\left(\tau -\gamma L\right){\alpha }_n\right)\left[\frac{1}{q}{∥x_n-x^{*}∥}^q+\left(\frac{q-1}{q}\right){∥x_{n+1}-x^{*}∥}^q\right]\\ +{\alpha }_n〈\gamma V{x}^{*}-\mu F{x}^{*},j_q\left(x_{n+1}-x^{*}\right)〉,\end{array}$$

which implies that

$$\begin{array}{ll}\hfill {∥x_{n+1}-x^{*}∥}^q& \le \frac{1-\left(\tau -\gamma L\right){\alpha }_n}{1+\left(q-1\right)\left(\tau -\gamma L\right){\alpha }_n}{∥x_n-x^{*}∥}^q+\frac{q{\alpha }_n}{1+\left(q-1\right)\left(\tau -\gamma L\right){\alpha }_n}〈\gamma V{x}^{*}-\mu F{x}^{*},j_q\left(x_{n+1}-x^{*}\right)〉\\ \le \left(1-\left(\tau -\gamma L\right){\alpha }_n\right){∥x_n-x^{*}∥}^q+\frac{q{\alpha }_n}{1+\left(q-1\right)\left(\tau -\gamma L\right){\alpha }_n}〈\gamma V{x}^{*}-\mu F{x}^{*},j_q\left(x_{n+1}-x^{*}\right)〉.\end{array}$$

(3.20)

Put a _n = (τ - γL)α _n and $b_n=\frac{q}{\left(1+\left(q-1\right)\left(\tau -\gamma L\right){\alpha }_n\right)\left(\tau -\gamma L\right)}〈\gamma V{x}^{*}-\mu F{x}^{*},j_q\left(x_{n+1}-x^{*}\right)〉$. Then (3.20) reduces to formula

$${∥x_{n+1}-x^{*}∥}^q\le \left(1-a_n\right){∥x_n-x^{*}∥}^q+a_n{b}_n.$$

It follows from the condition (C 1) and (3.19) that ${\sum }_{n=1}^{\infty }{a}_n=\infty$ and lim sup_n→∞b _n ≤ 0. From Lemma 2.4, we obtain that x _n → x* as n → ∞. This completes the proof.

Remark 3.6. Note that Lemma 3.1 is quite similar to the result of Yamada [19] which is obtained in a real Hilbert space but we extended that result to a real q-uniformly smooth Banach space.

Remark 3.7. Theorems 3.4 and 3.8 extend and generalize the main result of Ceng et al. [21] in the following ways:

• (i) From a real Hilbert space to a real q-uniformly smooth Banach space which admits a weakly sequentially continuous generalized duality mapping.

• (ii) From a nonexpansive mapping to a countable family of a strict pseudo-contractions mapping.

From Lemmas 2.7, 2.8 and Theorem 3.8, we obtain the following result.

Theorem 3.8. Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space X which admits a weakly sequentially continuous generalized duality mapping j _q from X into X*. Let Q _c be a sunny nonexpansive retraction such that Q _c is an orthogonal from X onto C. Let F: C → X be a κ-Lipschitzian and η-strongly accretive operator with constants κ, η > 0, V: C → X be an L-Lipschitzian mapping with a constant L ≥ 0. Let $0<\mu <{\left(\frac{q\eta }{C_q{\kappa }^q}\right)}^{\frac{1}{q-1}}$ and 0 ≤ γL < τ, where $\tau =\mu \left(\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}\right)$. Let ${\left\{S_k\right\}}_{k=1}^{\infty }:C\to C$ be a sequence of λ _k -strict pseudo-contractions such that ${\bigcap }_{k=1}^{\infty }\mathsf{\text{Fix}}\left(S_k\right)\ne \varnothing$ and λ := inf{λ _k : k ∈ ℕ} > 0. Define a mapping $S_{n}x:=\left(1-{\gamma }_n\right)x+{\gamma }_n{\sum }_{k=1}^{\infty }{\mu }_n^k{S}_{k}x$ for all x ∈ C and n ≥ 1. Let {x _n } be a sequence defined by x₁ ∈ C and

$$x_{n+1}=Q_C\left[{\alpha }_n\gamma V{x}_n+\left(I-{\alpha }_n\mu F\right)S_n{x}_n\right],\forall n\ge 1,$$

(3.21)

where {α _n } and {γ _n } are sequences in (0,1) which satisfy the conditions (C 1)-(C 3) of Theorem 3.8 and $\left\{{\mu }_n^k\right\}$ is a sequence which satisfies the conditions (i)-(iii) of Lemma 2.8. Let T: C → C be the mapping defined by $Tx={\sum }_{k=1}^{\infty }{\mu }^k{S}_{k}x$ for all x ∈ C. Then the sequence {x _n } defined by (3.21) converges strongly to $x^{*}\in {\bigcap }_{k=1}^{\infty }\mathsf{\text{Fix}}\left(S_k\right)$ as n → ∞, which x* is the unique solution of the variational inequality

$$〈\left(\mu F-\gamma V\right)x^{*},j_q\left(x^{*}-z\right)〉\le 0,\forall z\in {\bigcap }_{k=1}^{\infty }\mathsf{\text{Fix}}\left(S_k\right).$$

(3.22)