Strong convergence theorems for variational inequalities and fixed points of a countable family of nonexpansive mappings

Abstract

A new general iterative method for finding a common element of the set of solutions of variational inequality and the set of common fixed points of a countable family of nonexpansive mappings is introduced and studied. A strong convergence theorem of the proposed iterative scheme to a common fixed point of a countable family of nonexpansive mappings and a solution of variational inequality of an inverse strongly monotone mapping are established. Moreover, we apply our main result to obtain strong convergence theorems for a countable family of nonexpansive mappings and a strictly pseudocontractive mapping, and a countable family of uniformly k-strictly pseudocontractive mappings and an inverse strongly monotone mapping. Our main results improve and extend the corresponding result obtained by Klin-eam and Suantai (J Inequal Appl 520301, 16 pp, 2009).

Mathematics Subject Classification (2000): 47H09, 47H10

1 Introduction

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. In this paper, we always assume that a bounded linear operator A on H is strongly positive, that is, there is a constant $\bar{\gamma }>0$ such that $⟨Ax,x⟩\ge \bar{\gamma }\left|\right|x|{|}^2$ for all x ∈ H. Recall that a mapping T of H into itself is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y ∈ H. The set of all fixed points of T is denoted by F(T), that is, F(T) = {x ∈ C : x = Tx}. A self-mapping f : H → H is a contraction on H if there is a constant α ∈ [0, 1) such that ||f(x) - f(y) || ≤ α ||x - y|| for all x, y ∈ H.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on H:

$$\underset{x\in F}{min}\frac{1}{2}⟨Ax,x⟩-⟨x,b⟩,$$

(1.1)

where F is the fixed point set of a nonexpansive mapping T on H and b is a given point in H. A mapping B of C into H is called monotone if 〈Bx - By, x - y〉 ≥ 0 for all x, y ∈ C. The variational inequality problem is to find x ∈ C such that 〈Bx, y - x〉 ≥ 0 for all y ∈ C. The set of solutions of the variational inequality is denoted by VI(C, B). A mapping B of C to H is called inverse strongly monotone if there exists a positive real number β such that 〈x - y, Bx - By〉 ≥ β ||Bx - By||² for all x, y ∈ C.

Starting with an arbitrary initial x₀ ∈ H, define a sequence {x _n } recursively by

$$x_{n+1}=\left(I-{\alpha }_{n}A\right)T{x}_n+{\alpha }_{n}bn\ge 0.$$

(1.2)

It is proved by Xu [1] that the sequence {x _n } generated by (1.2) converges strongly to the unique solution of the minimization problem (1.1) provided the sequence {α _n } satisfies certain conditions.

On the other hand, Moudafi [2] introduced the viscosity approximation method for nonexpansive mappings. Let f be a contraction on H. Starting with an arbitrary initial x₀ ∈ H, define a sequence {x _n } recursively by

$$x_{n+1}=\left(1-{\sigma }_n\right)T{x}_n+{\sigma }_{n}f\left(x_n\right)n\ge 0,$$

(1.3)

where {σ _n } is a sequence in (0, 1). It is proved by Moudafi [2] and Xu [3] that under certain appropriate conditions imposed on {σ _n }, the sequence {x _n } generated by (1.3) strongly converges to the unique solution x* in C of the variational inequality

$$⟨\left(I-f\right)x^{*},x-x^{*}⟩\ge 0x\in C.$$

Recently, Marino and Xu [4] combined the iterative method (1.2) with the viscosity approximation method (1.3) and considered the following general iteration process:

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

(1.4)

and proved that if the sequence {α _n } satisfies appropriate conditions, the sequence {x _n } generated by (1.4) converges strongly to the unique solution of the variational inequality

$$⟨\left(A-\gamma f\right)x^{*},x-x^{*}⟩\ge 0x\in C$$

which is the optimality condition for the minimization problem

$$\underset{x\in C}{min}\frac{1}{2}⟨Ax,x⟩-h\left(x\right),$$

where h is a potential function for γ f (i.e., h'(x) = γ f(x) for x ∈ H).

Chen, Zhang and Fan [5] introduced the following iterative process: x₀ ∈ C,

$$x_{n+1}={\alpha }_{n}f\left(x_n\right)+\left(1-{\alpha }_n\right)T{P}_C\left(x_n-{\lambda }_{n}B{x}_n\right),n\ge 0,$$

(1.5)

where {α _n } ⊂ (0, 1) and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < 2β.

They proved that under certain appropriate conditions imposed on {α _n } and {λ _n }, the sequence {x _n } generated by (1.5) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping (say $\bar{x}\in C$), which solves the variational inequality

$$⟨\left(I-f\right)\bar{x},x-\bar{x}⟩\ge 0\forall x\in F\left(T\right)\cap VI\left(C,B\right).$$

Klin-eam and Suantai [6] modify the iterative methods (1.4) and (1.5) by proposing the following general iterative method: x₀ ∈ C,

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

(1.6)

where P _C is the projection of H onto C, f is a contraction, A is a strongly positive linear bounded operator, B is a β-inverse strongly monotone mapping, {α _n } ⊂ (0, 1) and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < 2β. They noted that when A = I and γ = 1, the iterative scheme (1.6) reduced to the iterative scheme (1.5).

Wangkeeree, Petrot and Wangkeeree [7] introduced the following iterative process:

$$\left\{\begin{array}{c}{x}_0=x\in H,\\ y_n={\beta }_n{x}_n+\left(1-{\beta }_n\right)T_n{x}_n,\\ x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)y_n,n\ge 0\end{array}\right.$$

(1.7)

where {α _n }and{β _n } ⊂ (0, 1) and T _n is a countable family of nonexpansive mappings, f is a contraction, and A is a strongly positive linear bounded operator. They proved that under certain appropriate conditions imposed on {α _n }, {β _n } and {T _n }, the sequence {x _n } converges strongly to $\tilde{x}$, which solves the variational inequality:

$$⟨\left(A-\gamma f\right)\tilde{x},\tilde{x}-z⟩\le 0z\in F\left(T\right).$$

In this paper, motivated and inspired by Klin-eam and Suantai [6], we introduced the following iteration to find some solutions of variational inequality and fixed points of countable family of nonexpansive mappings in a Hilbert spaces H: x₀ ∈ C,

$$x_{n+1}=P_C\left({\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)T_n{P}_C\left(x_n-{\lambda }_{n}B{x}_n\right)\right),n\ge 0,$$

(1.8)

where P _C is the projection of H onto C, f is a contraction, A is a strongly positive linear bounded operator, T _n is a countable family of nonexpansive mappings of C into itself, B is a β-inverse strongly monotone mapping, {α _n } ⊂ (0, 1), and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < 2β.

2 Preliminaries

Let H be a real Hilbert space with inner product 〈·,·〉 and norm || · ||, and let C be a closed convex subset of H. We write x _n ⇀ x to indicate that the sequence {x _n } converges weakly to x, and x _n → x implies that {x _n } converges strongly to x. For every point x ∈ H, there exists a unique nearest point in C, denoted by P _C x, such that ||x - P _C x|| ≤ ||x - y|| for all y ∈ C and P _C x is called the metric projection of H onto C. We know that P _C is a nonexpansive mapping of H onto C. It is also known that P _C satisfies 〈x - y, P _C x - P _C y〉 ≥ ||P _C x - P _C y||² for every x, y ∈ H. Moreover, P _C x is characterized by the properties: P _C x ∈ C and 〈x - P _C x, P _C x - y〉 ≥ 0 for all y ∈ C. In the context of the variational inequality problem, this implies that

$$u\in VI\left(C,A\right)\iff u=P_C\left(u-\lambda Au\right),\forall \lambda >0.$$

A set-valued mapping T : H → 2^H is called monotone if for all x, y ∈ H, f ∈ Tx and g ∈ Ty imply 〈x - y, f - g〉 ≥ 0. A monotone mapping T : H → 2^H is maximal if the graph G(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f ) ∈ H × H, 〈x - y, f - g〉 ≥ 0 for every (y, g) ∈ G(T) implies f ∈ Tx. Let A be an inverse strongly monotone mapping of C into H, and let N _C v be the normal cone to C at v ∈ C, i.e., N _C v = {w ∈ H : 〈v - u, w〉 ≥ 0, ∀u ∈ C}, and define

$$Tv=\left\{\begin{array}{cc}\hfill Av+N_{C}v,\hfill & \hfill v\in C,\hfill \\ \hfill \varnothing ,\hfill & \hfill v\notin C.\hfill \end{array}\right.$$

Then, T is maximal monotone and 0 ∈ Tv if and only if v ∈ V I(C, A).

Lemma 2.1 Let C be a closed convex subset of a real Hilbert space H. Given x ∈ H and y ∈ C, then

(i) y = P _C x if and only if the inequality 〈x - y, y - z〉 ≥ 0 for all z ∈ C,

(ii) P _C is nonexpansive,

(iii) 〈x - y, P _C x - P _C y〉 ≥ ||P _C x - P _C y||² for all x, y ∈ H,

(iv) 〈x - P _C x, P _C x - y〉 ≥ 0 for all x ∈ H and y ∈ C.

Lemma 2.2 [4] Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient $\bar{\gamma }>0$ and 0 < ρ ≤ ||A|| ^-1, then $\left|\right|I-\rho A\left|\right|\le 1-\rho \bar{\gamma }$.

Lemma 2.3 [8] Assume {a _n } is a sequence of nonnegative real numbers such that

$$a_{n+1}\le \left(1-{\gamma }_n\right)a_n+{\delta }_n,n\ge 0$$

where {γ _n } ⊂ (0, 1) and {δ _n } is a sequence in ℝ such that

(i) ${\sum }_{n=1}^{\infty }{\gamma }_n=\infty$,

(ii) lim sup_n→∞δ _n /γ _n ≤ 0 or ${\sum }_{n=1}^{\infty }\left|{\delta }_n\right|<\infty$.

Then, lim _n→∞a _n = 0.

Lemma 2.4 [9] Let C be a closed convex subset of a real Hilbert space H, and let T : C → C be a nonexpansive mapping such that F(T) ≠ ∅. If a sequence {x _n } in C such that x _n ⇀ z and x _n - Tx _n → 0, then z = Tz.

To deal with a family of mappings, the following conditions are introduced: Let C be a subset of a real Banach space E, and let ${\left\{T_n\right\}}_{n=1}^{\infty }$ be a family of mappings of C such that ${\cap }_{n=1}^{\infty }F\left(T_n\right)\ne \varnothing$. Then, {T _n } is said to satisfy the AKTT-condition [10] if for each bounded subset B of C,

$$\sum _{n=1}^{\infty }sup\left\{\left|\right|T_{n+1}z-T_{n}z\left|\right|:z\in B\right\}<\infty .$$

Lemma 2.5 [10] Let C be a nonempty and closed subset of a Banach space E and let {T _n } be a family of mappings of C into itself which satisfies the AKTT-condition. Then, for each x ∈ C, {T _n x} converges strongly to a point in C. Moreover, let the mapping T be defined by

$$Tx=\underset{n\to \infty }{lim}{T}_{n}x\forall x\in C.$$

Then, for each bounded subset B of C,

$$\underset{n\to \infty }{\lim \sup }\left\{\right||Tz-T_{n}z||\text{‷}:z\in B\}=0.$$

In the sequel, we will write ({T _n }, T ) satisfies the AKTT-condition if {T _n } satisfies the AKTT-condition, and T is defined by Lemma 2.5 with $F\left(T\right)={\cap }_{n=1}^{\infty }F\left(T_n\right)$.

3 Main results

In this section, we prove a strong convergence theorem for a countable family of nonexpansive mappings.

Theorem 3.1 Let C be a closed convex subset of a real Hilbert space H, and let B : C → H be a β-inverse strongly monotone mapping, also let A be a strongly positive linear bounded operator of H into itself with coefficient $\bar{\gamma }>0$ such that ||A|| = 1 and let f : C → C be a contraction with coefficient α(0 < α < 1). Assume that $0<\gamma <\bar{\gamma }∕\alpha$. Let {T _n } be a countable family of nonexpansive mappings from a subset C into itself with $F={\cap }_{n=1}^{\infty }F\left(T_n\right)\cap VI\left(C,B\right)\ne \varnothing$. Suppose {x _n } is the sequence generated by the following algorithm: x₀ ∈ C,

$$x_{n+1}=P_C\left({\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)T_n{P}_C\left(x_n-{\lambda }_{n}B{x}_n\right)\right)$$

for all n = 0, 1, 2, ..., where {α _n } ⊂ (0, 1) and {λ _n } ⊂ (0, 2β ). If {α _n } and {λ _n } are chosen so that λ _n ∈ [a, b] for some a, b with 0 < a < b < 2β ,

$$\begin{array}{cc}\hfill \left(C1\right)\underset{n\to 0}{lim}{\alpha }_n=0;\hfill & \hfill \left(C2\right)\sum _{n=1}^{\infty }{\alpha }_n=\infty ;\hfill \\ \hfill \left(C3\right)\sum _{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_n\right|<\infty ;\hfill & \hfill \left(C4\right)\sum _{n=1}^{\infty }\left|{\lambda }_{n+1}-{\lambda }_n\right|<\infty .\hfill \end{array}$$

Suppose that ({T _n }, T ) satisfies the AKTT-condition. Then, {x _n } converges strongly to q ∈ F, where q = P _F (γ f + I - A)(q) which solves the following variational inequality:

$$⟨\left(\gamma f-A\right)q,p-q⟩\le 0\forall p\in F.$$

Proof. First, we show that the sequence {x _n } is bounded. Consider the mapping I -λ _n B. Since B is a β-inverse strongly monotone mapping, we have that for all x, y ∈ C,

$$\begin{array}{lll}\hfill \left|\right|\left(I-{\lambda }_{n}B\right)x-\left(I-{\lambda }_{n}B\right)y|{|}^2& =\left|\right|\left(x-y\right)-{\lambda }_n\left(Bx-By\right)|{|}^2& \hfill \text{(1)}\\ =\left|\right|x-y|{|}^2-2{\lambda }_n⟨x-y,Bx-By⟩+{\lambda }_n^2||Bx-By|{|}^2& \hfill \text{(2)}\\ \le \left|\right|x-y|{|}^2+{\lambda }_n\left({\lambda }_n-2\beta \right)||Bx-By|{|}^2.& \hfill \text{(3)}\\ \hfill \text{(4) }\end{array}$$

For 0 < λ _n < 2β, implies that ||k(I - λ _n B)x - (I- λ _n B)y||² ≤ ||x - y||².

So, the mapping I - λ _n B is nonexpansive.

Put y _n = P _C (x _n - λ _n Bx _n ) for all n ≥ 0. Let u ∈ F. Then u = P _C (u - λ _n Bu).

From P _C is nonexpansive implies that

$$\begin{array}{lll}\hfill \left|\right|y_n-u\left|\right|& =\left|\right|P_C\left(x_n-{\lambda }_{n}B{x}_n\right)-P_C\left(u-{\lambda }_{n}Bu\right)\left|\right|& \hfill \text{(1)}\\ \le \left|\right|\left(x_n-{\lambda }_{n}B{x}_n\right)-\left(u-{\lambda }_{n}Bu\right)\left|\right|& \hfill \text{(2)}\\ =\left|\right|\left(I-{\lambda }_{n}B\right)x_n-\left(I-{\lambda }_{n}B\right)u\left|\right|.& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$$

Since I - λ _n B is nonexpansive, we have that ||y _n - u|| ≤ ||x _n - u||. Then

$$\begin{array}{lll}\hfill \left|\right|x_{n+1}-u\left|\right|& =\left|\right|P_C\left({\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)T_n{y}_n\right)-u\left|\right|& \hfill \text{(1)}\\ \le \left|\right|{\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)T_n{y}_n-u\left|\right|& \hfill \text{(2)}\\ =\left|\right|{\alpha }_n\left(\gamma f\left(x_n\right)-Au\right)+\left(I-{\alpha }_{n}A\right)\left(T_n{y}_n-u\right)\left|\right|.& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$$

Since A is strongly positive linear bounded operator, we have

$$\begin{array}{lll}\hfill \left|\right|x_{n+1}-u\left|\right|& \le {\alpha }_n\left|\right|\gamma f\left(x_n\right)-Au\left|\right|+\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|T_n{y}_n-u\left|\right|& \hfill \text{(1)}\\ \le {\alpha }_n\left|\right|\gamma f\left(x_n\right)-\gamma f\left(u\right)\left|\right|+{\alpha }_n\left|\right|\gamma f\left(u\right)-Au\left|\right|+\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|T_n{y}_n-u\left|\right|.& \hfill \text{(2)}\\ \hfill \text{(3) }\end{array}$$

By contraction of f, we have

$$\begin{array}{lll}\hfill \left|\right|x_{n+1}-u\left|\right|& \le \alpha \gamma {\alpha }_n\left|\right|x_n-u\left|\right|+{\alpha }_n\left|\right|\gamma f\left(u\right)-Au\left|\right|+\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|T_n{y}_n-u\left|\right|& \hfill \text{(1) }\\ =\alpha \gamma {\alpha }_n\left|\right|x_n-u\left|\right|+{\alpha }_n\left|\right|\gamma f\left(u\right)-Au\left|\right|+\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|T_n{y}_n-T_{n}u\left|\right|& \hfill \text{(2) }\\ \le \alpha \gamma {\alpha }_n\left|\right|x_n-u\left|\right|+{\alpha }_n\left|\right|\gamma f\left(u\right)-Au\left|\right|+\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|y_n-u\left|\right|& \hfill \text{(3) }\\ \le \alpha \gamma {\alpha }_n\left|\right|x_n-u\left|\right|+{\alpha }_n\left|\right|\gamma f\left(u\right)-Au\left|\right|+\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|x_n-u\left|\right|& \hfill \text{(4) }\\ \le \left(\alpha \gamma {\alpha }_n+1-{\alpha }_n\bar{\gamma }\right)\left|\right|x_n-u\left|\right|+{\alpha }_n\left|\right|\gamma f\left(u\right)-Au\left|\right|& \hfill \text{(5) }\\ \le \left(1-{\alpha }_n\left(\bar{\gamma }-\alpha \gamma \right)\right)\left|\right|x_n-u\left|\right|+{\alpha }_n\left(\bar{\gamma }-\alpha \gamma \right)\frac{\left|\right|\gamma f\left(u\right)-Au\left|\right|}{\bar{\gamma }-\alpha \gamma }& \hfill \text{(6) }\\ \le max\left\{\left|\right|x_n-u\left|\right|,\frac{\left|\right|\gamma f\left(u\right)-Au\left|\right|}{\bar{\gamma }-\alpha \gamma }\right\}.& \hfill \text{(7) }\\ \hfill \text{(8) }\end{array}$$

It follows from induction that $\left|\right|x_n-u\left|\right|\le max\left\{\left|\right|x_0-u\left|\right|,\frac{\left|\right|\gamma f\left(u\right)-Au\left|\right|}{\bar{\gamma }-\alpha \gamma }\right\}$, n ≥ 0.

Therefore, {x _n } is bounded, so are {y _n }, {T _n y _n }, {Bx _n }, and {f (x _n )}.

Next, we show that ||x_n+1- x _n || → 0 and ||y _n - T _n y _n || → 0 as n → ∞.

Since P _C is nonexpansive, we also have

$$\begin{array}{lll}\hfill \left|\right|y_{n+1}-y_n\left|\right|& =\left|\right|P_C\left(x_{n+1}-{\lambda }_{n+1}B{x}_{n+1}\right)-P_C\left(x_n-{\lambda }_{n}B{x}_n\right)\left|\right|& \hfill \text{(1)}\\ \le \left|\right|x_{n+1}-{\lambda }_{n+1}B{x}_{n+1}-\left(x_n-{\lambda }_{n}B{x}_n\right)\left|\right|& \hfill \text{(2)}\\ \le \left|\right|x_{n+1}-{\lambda }_{n+1}B{x}_{n+1}-\left(x_n-{\lambda }_{n+1}B{x}_n\right)\left|\right|+|{\lambda }_n-{\lambda }_{n+1}|\left|\right|B{x}_n\left|\right|& \hfill \text{(3)}\\ =\left|\right|\left(I-{\lambda }_{n+1}B\right)x_{n+1}-\left(I-{\lambda }_{n+1}B\right)x_n\left|\right|+|{\lambda }_n-{\lambda }_{n+1}|\left|\right|B{x}_n\left|\right|.& \hfill \text{(4) }\\ \hfill \text{(5) }\end{array}$$

Since I - λ _n B is nonexpansive, we have

$$\left|\right|y_{n+1}-y_n\left|\right|\le \left|\right|x_{n+1}-x_n\left|\right|+|{\lambda }_n-{\lambda }_{n+1}|\left|\right|B{x}_n\left|\right|.$$

So we obtain

$$\begin{array}{l}\left|\right|x_{n+1}-x_n\left|\right|=\left|\right|P_C({\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)T_n{y}_n)-P_C({\alpha }_{n-1}\gamma f(x_{n-1})+(I-{\alpha }_{n-1}A)T_{n-1}{y}_{n-1})\left|\right|\\ \le \left|\right|{\alpha }_n\gamma (f(x_n)-f(x_{n-1}))+\gamma ({\alpha }_n-{\alpha }_{n-1})f(x_{n-1})+(I-{\alpha }_{n}A)(T_n{y}_n-T_{n-1}{y}_{n-1})\\ +({\alpha }_n-{\alpha }_{n-1})A{T}_{n-1}{y}_{n-1}\left|\right|\\ \le {\alpha }_n\alpha \gamma \left|\right|x_n-x_{n-1}\left|\right|+\gamma |{\alpha }_n-{\alpha }_{n-1}|\left|\right|f(x_{n-1})\left|\right|+(1-{\alpha }_n\overline{\gamma })\left|\right|T_n{y}_n-T_{n-1}{y}_{n-1}\left|\right|\\ +|{\alpha }_n-{\alpha }_{n-1}|\left|\right|A{T}_{n-1}{y}_{n-1}\left|\right|\\ \le {\alpha }_n\alpha \gamma \left|\right|x_n-x_{n-1}\left|\right|+\gamma |{\alpha }_n-{\alpha }_{n-1}|\left|\right|f(x_{n-1})\left|\right|+(1-{\alpha }_n\overline{\gamma })(||T_n{y}_n-T_n{y}_{n-1}||\\ +\left|\right|T_n{y}_{n-1}-T_{n-1}{y}_{n-1}\left|\right|)+|{\alpha }_n-{\alpha }_{n-1}\left|\right|\left|A{T}_{n-1}{y}_{n-1}\right||\\ \le {\alpha }_n\alpha \gamma \left|\right|x_n-x_{n-1}\left|\right|+\gamma |{\alpha }_n-{\alpha }_{n-1}|\left|\right|f(x_{n-1})\left|\right|+(1-{\alpha }_n\overline{\gamma })(||y_n-y_{n-1}||\\ +\left|\right|T_n{y}_{n-1}-T_{n-1}{y}_{n-1}\left|\right|)+|{\alpha }_n-{\alpha }_{n-1}\left|\right|\left|A{T}_{n-1}{y}_{n-1}\right||\\ ={\alpha }_n\alpha \gamma \left|\right|x_n-x_{n-1}\left|\right|+\gamma |{\alpha }_n-{\alpha }_{n-1}|\left|\right|f(x_{n-1})\left|\right|+(1-{\alpha }_n\overline{\gamma })\left|\right|y_n-y_{n-1}\left|\right|\\ +(1-{\alpha }_n\overline{\gamma })\left|\right|T_n{y}_{n-1}-T_{n-1}{y}_{n-1}\left|\right|+|{\alpha }_n-{\alpha }_{n-1}|\left|\right|A{T}_{n-1}{y}_{n-1}\left|\right|\\ \le {\alpha }_n\alpha \gamma \left|\right|x_n-x_{n-1}\left|\right|+\gamma |{\alpha }_n-{\alpha }_{n-1}|\left|\right|f(x_{n-1})\left|\right|+(1-{\alpha }_n\overline{\gamma })\left|\right|x_n-x_{n-1}\left|\right|\\ +(1-{\alpha }_n\overline{\gamma })|{\lambda }_{n-1}-{\lambda }_n|\left|\right|B{x}_{n-1}\left|\right|+(1-{\alpha }_n\overline{\gamma })\left|\right|T_n{y}_{n-1}-T_{n-1}{y}_{n-1}\left|\right|\\ +|{\alpha }_n-{\alpha }_{n-1}|\left|\right|A{T}_{n-1}{y}_{n-1}\left|\right|\\ =(1-(\overline{\gamma }-\alpha \gamma ){\alpha }_n)\left|\right|x_n-x_{n-1}\left|\right|+\gamma |{\alpha }_n-{\alpha }_{n-1}|\left|\right|f(x_{n-1})\left|\right|\\ +(1-{\alpha }_n\overline{\gamma })|{\lambda }_{n-1}-{\lambda }_n|\left|\right|B{x}_{n-1}\left|\right|+(1-{\alpha }_n\overline{\gamma })\left|\right|T_n{y}_{n-1}-T_{n-1}{y}_{n-1}\left|\right|\\ +|{\alpha }_n-{\alpha }_{n-1}|\left|\right|A{T}_{n-1}{y}_{n-1}\left|\right|\\ \le (1-(\overline{\gamma }-\alpha \gamma ){\alpha }_n)\left|\right|x_n-x_{n-1}\left|\right|+2L|{\alpha }_n-{\alpha }_{n-1}|+M|{\lambda }_{n-1}-{\lambda }_n|\\ +\underset{y\in \left\{y_n\right\}}{{\displaystyle sup}}\left|\right|T_{n}y-T_{n-1}y\left|\right|,\end{array}$$

where L = max{sup_n∈ℕ||AT_{n- 1}y_{n - 1}||, sup_n∈ℕγ ||f (x_{n - 1})||} and M = sup{||Bx_{n- 1}|| : n∈ℕ}.

Since {T _n } satisfies the AKTT-condition, we get that

$$\sum _{n=1}^{\infty }\underset{y\in \left\{y_n\right\}}{sup}\left|\right|T_{n}y-T_{n-1}y\left|\right|<\infty .$$

From condition (C3), (C4) and by Lemma 2.3, we have ||x_n+1- x _n || → 0.

For u ∈ F and u = P _C (u - λ _n Bu), we have

$$\begin{array}{l}\left|\right|x_{n+1}-u|{|}^2=||P_C({\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)T_n{y}_n)-P_C(u)|{|}^2\\ \le \left|\right|{\alpha }_n(\gamma f(x_n)-Au)+(I-{\alpha }_{n}A)(T_n{y}_n-u)|{|}^2\\ \le {({\alpha }_n||\gamma f(x_n)-Au||+||I-{\alpha }_{n}A|\left|\right||T_n{y}_n-u||)}^2\\ \le {({\alpha }_n||\gamma f(x_n)-Au||+(1-{\alpha }_n\overline{\gamma })||y_n-u||)}^2\\ \le {\alpha }_n\left|\right|\gamma f(x_n)-Au|{|}^2+(1-{\alpha }_n\overline{\gamma })||y_n-u|{|}^2\\ +2{\alpha }_n(1-{\alpha }_n\overline{\gamma })\left|\right|\gamma f(x_n)-Au\left|\right|\left|\right|y_n-u\left|\right|\\ \le {\alpha }_n\left|\right|\gamma f(x_n)-Au|{|}^2+(1-{\alpha }_n\overline{\gamma })||(I-{\lambda }_{n}B)x_n-(I-{\lambda }_{n}B)u|{|}^2\\ +2{\alpha }_n(1-{\alpha }_n\overline{\gamma })\left|\right|\gamma f(x_n)-Au\left|\right|\left|\right|y_n-u\left|\right|\\ \le {\alpha }_n\left|\right|\gamma f(x_n)-Au|{|}^2+(1-{\alpha }_n\overline{\gamma })(\left|\right|x_n-u|{|}^2-2{\lambda }_n〈x_n-u\mathrm{,}B{x}_n-Bu〉\\ +{\lambda }_n^2\left|\right|B{x}_n-Bu|{|}^2)+2{\alpha }_n(1-{\alpha }_n\overline{\gamma })\left|\right|\gamma f(x_n)-Au\left|\right|\left|\right|y_n-u\left|\right|\\ \le {\alpha }_n\left|\right|\gamma f(x_n)-Au|{|}^2+(1-{\alpha }_n\overline{\gamma })(\left|\right|x_n-u|{|}^2-2{\lambda }_n\beta ||B{x}_n-Bu|{|}^2\\ +{\lambda }_n^2\left|\right|B{x}_n-Bu|{|}^2)+2{\alpha }_n(1-{\alpha }_n\overline{\gamma })\left|\right|\gamma f(x_n)-Au\left|\right|\left|\right|y_n-u\left|\right|\\ ={\alpha }_n\left|\right|\gamma f(x_n)-Au|{|}^2+(1-{\alpha }_n\overline{\gamma })\left(\left|\right|x_n-u|{|}^2+{\lambda }_n({\lambda }_n-2\beta )||B{x}_n-Bu|{|}^2\right)\\ +2{\alpha }_n(1-{\alpha }_n\overline{\gamma })\left|\right|\gamma f(x_n)-Au\left|\right|\left|\right|y_n-u\left|\right|\\ \le {\alpha }_n\left|\right|\gamma f(x_n)-Au|{|}^2+||x_n-u|{|}^2+(1-{\alpha }_n\overline{\gamma })b(b-2\beta )\left|\right|B{x}_n-Bu|{|}^2\\ +2{\alpha }_n(1-{\alpha }_n\overline{\gamma })\left|\right|\gamma f(x_n)-Au\left|\right|\left|\right|y_n-u\left|\right|\mathrm{.}\end{array}$$

So, we obtain

$$\begin{array}{c}-\left(1-{\alpha }_n\bar{\gamma }\right)b\left(b-2\beta \right)\left|\right|B{x}_n-Bu|{|}^2\\ \le {\alpha }_n\left|\right|\gamma f\left(x_n\right)-Au|{|}^2+\left(\left|\right|x_n-u\left|\right|+\left|\right|x_{n+1}-u\left|\right|\right)\left(\left|\right|x_n-u\left|\right|-\left|\right|x_{n+1}-u\left|\right|\right)+{\epsilon }_n\\ \le {\alpha }_n\left|\right|\gamma f\left(x_n\right)-Au|{|}^2+{\epsilon }_n+||x_n-x_{n+1}||\left(\left|\right|x_n-u\left|\right|+\left|\right|x_{n+1}-u\left|\right|\right),\end{array}$$

where ${\epsilon }_n=2{\alpha }_n\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|\gamma f\left(x_n\right)-Au\left|\right|\left|\right|y_n-u\left|\right|$.

Since α _n → 0 and ||x_n+1- x _n || → 0, we obtain ||Bx _n - Bu|| → 0 as n → ∞.

Further, by Lemma 2.1, we have

$$\begin{array}{l}\left|\right|y_n-u|{|}^2=||P_C(x_n-{\lambda }_{n}B{x}_n)-P_C(u-{\lambda }_{n}Bu)|{|}^2\\ \le 〈(x_n-{\lambda }_{n}B{x}_n)-(u-{\lambda }_{n}Bu),y_n-u〉\\ =\frac{1}{2}(||(x_n-{\lambda }_{n}B{x}_n)-(u-{\lambda }_{n}Bu)|{|}^2+\left|\right|y_n-u|{|}^2\\ -\left|\right|(x_n-{\lambda }_{n}B{x}_n)-(u-{\lambda }_{n}Bu)-(y_n-u)|{|}^2)\\ \le \frac{1}{2}(||x_n-u|{|}^2+\left|\right|y_n-u|{|}^2-||(x_n-y_n)-{\lambda }_n(B{x}_n-Bu)|{|}^2)\\ \le \left|\right|x_n-u|{|}^2-||x_n-y_n|{|}^2+2{\lambda }_n〈x_n-y_n\mathrm{,}B{x}_n-Bu〉-{\lambda }_n^2\left|\right|B{x}_n-Bu|{|}^2\mathrm{.}\end{array}$$

So, we have

$$\begin{array}{lll}\hfill \left|\right|x_{n+1}-u|{|}^2& =\left|\right|P_C\left({\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)T_n{y}_n\right)-P_C\left(u\right)|{|}^2& \hfill \text{(1)}\\ \le \left|\right|{\alpha }_n\left(\gamma f\left(x_n\right)-Au\right)+\left(I-{\alpha }_{n}A\right)\left(T_n{y}_n-u\right)|{|}^2& \hfill \text{(2)}\\ \le {\left({\alpha }_n\left|\right|\gamma f\left(x_n\right)-Au\left|\right|+\left|\right|I-{\alpha }_{n}A\left|\right|\left|\right|T_n{y}_n-u\left|\right|\right)}^2& \hfill \text{(3)}\\ \le {\left({\alpha }_n\left|\right|\gamma f\left(x_n\right)-Au\left|\right|+\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|y_n-u\left|\right|\right)}^2& \hfill \text{(4)}\\ \le {\alpha }_n\left|\right|\gamma f\left(x_n\right)-Au|{|}^2+\left(1-{\alpha }_n\bar{\gamma }\right)||y_n-u|{|}^2& \hfill \text{(5)}\\ +2{\alpha }_n\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|\gamma f\left(x_n\right)-Au\left|\right|\left|\right|y_n-u\left|\right|& \hfill \text{(6)}\\ \le {\alpha }_n\left|\right|\gamma f\left(x_n\right)-Au|{|}^2+\left(1-{\alpha }_n\overline{\gamma }\right)||x_n-u|{|}^2-\left(1-{\alpha }_n\overline{\gamma }\right)\left|\right|x_n-y_n|{|}^2& \hfill \text{(7)}\\ +2\left(1-{\alpha }_n\bar{\gamma }\right){\lambda }_n⟨x_n-y_n,B{x}_n-Bu⟩-\left(1-{\alpha }_n\bar{\gamma }\right){\lambda }_n^2\left|\right|B{x}_n-Bu|{|}^2& \hfill \text{(8)}\\ +2{\alpha }_n\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|\gamma f\left(x_n\right)-Au\left|\right|\left|\right|y_n-u\left|\right|,& \hfill \text{(9)}\\ \hfill \text{(10) }\end{array}$$

which implies

$$\begin{array}{lll}\hfill \left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|x_n-y_n|{|}^2& \le {\alpha }_n\left|\right|\gamma f\left(x_n\right)-Au|{|}^2+\left(\left|\right|x_n-u\left|\right|+\left|\right|x_{n+1}-u\left|\right|\right)||x_n-x_{n+1}||& \hfill \text{(1)}\\ +2\left(1-{\alpha }_n\bar{\gamma }\right){\lambda }_n⟨x_n-y_n,B{x}_n-Bu⟩-\left(1-{\alpha }_n\bar{\gamma }\right){\lambda }_n^2\left|\right|B{x}_n-Bu|{|}^2& \hfill \text{(2)}\\ +2{\alpha }_n\left(1-{\alpha }_n\bar{\gamma }\right)\left|\right|\gamma f\left(x_n\right)-Au\left|\right|\left|\right|y_n-u\left|\right|.& \hfill \text{(3)}\\ \hfill \text{(4) }\end{array}$$

Since α _n → 0, ||x_n+1- x _n || → 0, and ||Bx _n - Bu|| → 0, we obtain ||x _n - y _n || → 0 as n → ∞.

Next, we have

$$\begin{array}{lll}\hfill \left|\right|x_{n+1}-T_n{y}_n\left|\right|& =\left|\right|P_C\left({\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)T_n{y}_n\right)-P_C\left(T_n{y}_n\right)\left|\right|& \hfill \text{(1)}\\ \le \left|\right|{\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)T_n{y}_n-T_n{y}_n\left|\right|& \hfill \text{(2)}\\ ={\alpha }_n\left|\right|\gamma f\left(x_n\right)-A{T}_n{y}_n\left|\right|.& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$$

Since α _n → 0 and {f (x _n )}, {AT _n y _n } are bounded, we have ||x_n+1- T _n y _n || → 0 as n → ∞. Since

$$\left|\right|x_n-T_n{y}_n\left|\right|\le \left|\right|x_n-x_{n+1}\left|\right|+\left|\right|x_{n+1}-T_n{y}_n\left|\right|,$$

it implies that ||x _n - T _n y _n || → 0 as n → ∞. Since

$$\begin{array}{lll}\hfill \left|\right|x_n-T_n{x}_n\left|\right|& \le \left|\right|x_n-T_n{y}_n\left|\right|+\left|\right|T_n{y}_n-T_n{x}_n\left|\right|& \hfill \text{(1)}\\ \le \left|\right|x_n-T_n{y}_n\left|\right|+\left|\right|y_n-x_n\left|\right|,& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

we obtain ||x _n - T _n x _n || → 0 as n → ∞. Moreover, from

$$\left|\right|y_n-T_n{y}_n\left|\right|\le \left|\right|y_n-x_n\left|\right|+\left|\right|x_n-T_n{y}_n\left|\right|,$$

it follows that ||y _n - T _n y _n || → 0 as n → ∞.

By ||y _n - x _n || → 0, ||T _n y _n - x _n || → 0 and Lemma 2.5, we have

$$\begin{array}{lll}\hfill \left|\right|T{x}_n-x_n\left|\right|& \le \left|\right|T{x}_n-T{y}_n\left|\right|+\left|\right|T{y}_n-T_n{y}_n\left|\right|+\left|\right|T_n{y}_n-x_n\left|\right|& \hfill \text{(1)}\\ \le \left|\right|x_n-y_n\left|\right|+sup\left\{\left|\right|T_{n}z-Tz\left|\right|:z\in \left\{y_n\right\}\right\}+\left|\right|T_n{y}_n-x_n\left|\right|.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

Hence, lim_n→∞||Tx _n - x _n || = 0. Observe that P _F (γ f +I - A) is a contraction.

By Lemma 2.2, we have that $\left|\right|I-A\left|\right|\le 1-\bar{\gamma }$, and since $0<\gamma <\bar{\gamma }∕\alpha$, we get

$$\begin{array}{lll}\hfill \left|\right|P_F\left(\gamma f+I-A\right)x-P_F\left(\gamma f+I-A\right)y\left|\right|& \le \left|\right|\left(\gamma f+I-A\right)x-\left(\gamma f+I-A\right)y\left|\right|& \hfill \text{(1) }\\ \le \gamma \left|\right|f\left(x\right)-f\left(y\right)\left|\right|+\left|\right|I-A\left|\right|\left|\right|x-y\left|\right|& \hfill \text{(2) }\\ \le \gamma \alpha \left|\right|x-y\left|\right|+\left(1-\bar{\gamma }\right)\left|\right|x-y\left|\right|& \hfill \text{(3) }\\ =\left(1-\left(\bar{\gamma }-\gamma \alpha \right)\right)\left|\right|x-y\left|\right|.& \hfill \text{(4) }\\ \hfill \text{(5) }\end{array}$$

Then, Banach's contraction mapping principle guarantees that P _F (γ f +I - A) has a unique fixed point, say q ∈ H. That is, q = P _F (γ f + I - A)q. By Lemma 2.1, we obtain

$$⟨\left(\gamma f-A\right)q,x-q⟩\le 0\mathsf{\text{for}}\mathsf{\text{all}}x\in F.$$

(3.1)

Choose a subsequence $\left\{y_{n_k}\right\}$ of {y _n } such that

$$\underset{n\to \infty }{\text{lim}\text{‷}\text{sup}}〈(\gamma f-A)q,T_n{y}_n-q〉=\underset{k\to \infty }{\text{lim}}〈(\gamma f-A)q,T_{n_k}{y}_{n_k}-q〉.$$

As $\left\{y_{n_k}\right\}$ is bounded, there exists a subsequence $\left\{y_{n_{k_i}}\right\}$ of $\left\{y_{n_k}\right\}$ which converges weakly to p. Without loss of generality, we may assume that $y_{n_k}\rightharpoonup p$.

Since ||y _n - T _n y _n || → 0, we obtain $T_{n_k}{y}_{n_k}\rightharpoonup p$. Since ||x _n - Tx _n || → 0, ||x _n - y _n || → 0 and by Lemma 2.4-2.5, we have $p\in {\cap }_{n=1}^{\infty }F\left(T_n\right)$. Let

$$Sv=\left\{\begin{array}{cc}\hfill Bv+N_{C}v,\hfill & \hfill v\in C,\hfill \\ \hfill \varnothing ,\hfill & \hfill v\notin C.\hfill \end{array}\right.$$

where N _C v is normal cone to C at v ∈ C, that is N _C v = {w ∈ H : 〈v - u, w〉 ≥ 0, ∀u ∈ C}. Then S is a maximal monotone. Let (v, w) ∈ G(S). Since w - Bv ∈ N _C v and y _n ∈ C, we have 〈v - y _n , w - Bv〉 ≥ 0. On the other hand, by Lemma 2.1 and from y _n = P _C (x _n - λ _n Bx _n ), we have

$$\begin{array}{lll}\hfill ⟨v-y_n,y_n-\left(x_n-{\lambda }_{n}B{x}_n\right)⟩& \ge 0& \hfill \text{(1)}\\ \hfill ⟨v-y_n,\left(y_n-x_n\right)∕{\lambda }_n+B{x}_n⟩& \ge 0.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

Hence,

$$\begin{array}{lll}\hfill ⟨v-y_{n_k},w⟩& \ge ⟨v-y_{n_k},Bv⟩& \hfill \text{(1)}\\ \ge ⟨v-y_{n_k},Bv⟩-⟨v-y_{n_k},\frac{y_{n_k}-x_{n_k}}{{\lambda }_n}+B{x}_{n_k}⟩& \hfill \text{(2)}\\ =⟨v-y_{n_k},Bv-B{x}_{n_k}-\frac{y_{n_k}-x_{n_k}}{{\lambda }_n}⟩& \hfill \text{(3)}\\ =⟨v-y_{n_k},Bv-B{y}_{n_k}⟩+⟨v-y_{n_k},B{y}_{n_k}-B{x}_{n_k}⟩-⟨v-y_{n_k},\frac{y_{n_k}-x_{n_k}}{{\lambda }_n}⟩& \hfill \text{(4)}\\ \ge ⟨v-y_{n_k},B{y}_{n_k}-B{x}_{n_k}⟩-⟨v-y_{n_k},\frac{y_{n_k}-x_{n_k}}{{\lambda }_n}⟩.& \hfill \text{(5)}\\ \hfill \text{(6) }\end{array}$$

This implies 〈v - p, w〉 ≥ 0. Since S is maximal monotone, we have p ∈ S ^-10 and hence p ∈ V I(C, B). We obtain that p ∈ F. By (3.1), we have 〈(γ f - A)q, p - q〉 ≤ 0. It follows that

$$\underset{n\to \infty }{limsup}⟨\left(\gamma f-A\right)q,T_n{y}_n-q⟩=\underset{k\to \infty }{lim}⟨\left(\gamma f-A\right)q,T_{n_k}{y}_{n_k}-q⟩=⟨\left(\gamma f-A\right)q,p-q⟩\le 0.$$

Finally, we prove x _n → q. By ||y _n - u|| ≤ ||x _n - u|| and Schwarz inequality, we have

$$\begin{array}{l}\left|\right|x_{n+1}-q|{|}^2=||P_C({\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)T_n{y}_n)-P_C(q)|{|}^2\\ \le \left|\right|{\alpha }_n(\gamma f(x_n)-Aq)+(I-{\alpha }_{n}A)(T_n{y}_n-q)|{|}^2\\ \le \left|\right|(I-{\alpha }_{n}A)(T_n{y}_n-q)|{|}^2+{\alpha }_n^2||\gamma f(x_n)-Aq|{|}^2\\ +2{\alpha }_n〈(I-{\alpha }_{n}A)(T_n{y}_n-q),\gamma f(x_n)-Aq〉\\ \le {(1-{\alpha }_n\overline{\gamma })}^2\left|\right|y_n-q|{|}^2+{\alpha }_n^2||\gamma f(x_n)-Aq|{|}^2\\ +2{\alpha }_n〈T_n{y}_n-q\mathrm{,}\gamma f(x_n)-Aq〉-2{\alpha }_n^2〈A(T_n{y}_n-q),\gamma f(x_n)-Aq〉\\ \le {(1-{\alpha }_n\overline{\gamma })}^2\left|\right|x_n-q|{|}^2+{\alpha }_n^2||\gamma f(x_n)-Aq|{|}^2\\ +2{\alpha }_n〈T_n{y}_n-q\mathrm{,}\gamma f(x_n)-\gamma f(q)〉+2{\alpha }_n〈T_n{y}_n-q\mathrm{,}\gamma f(q)-Aq〉\\ -2{\alpha }_n^2〈A(T_n{y}_n-q),\gamma f(x_n)-Aq〉\\ \le {(1-{\alpha }_n\overline{\gamma })}^2\left|\right|x_n-q|{|}^2+{\alpha }_n^2||\gamma f(x_n)-Aq|{|}^2\\ +2{\alpha }_n\left|\right|T_n{y}_n-q\left|\right|\left|\right|\gamma f(x_n)-\gamma f(q)\left|\right|+2{\alpha }_n〈T_n{y}_n-q\mathrm{,}\gamma f(q)-Aq〉\\ -2{\alpha }_n^2〈A(T_n{y}_n-q),\gamma f(x_n)-Aq〉\\ \le {(1-{\alpha }_n\overline{\gamma })}^2\left|\right|x_n-q|{|}^2+{\alpha }_n^2||\gamma f(x_n)-Aq|{|}^2\\ +2\gamma \alpha {\alpha }_n\left|\right|y_n-q\left|\right|\left|\right|x_n-q\left|\right|+2{\alpha }_n〈T_n{y}_n-q\mathrm{,}\gamma f(q)-Aq〉\\ -2{\alpha }_n^2〈A(T_n{y}_n-q),\gamma f(x_n)-Aq〉\\ \le {(1-{\alpha }_n\overline{\gamma })}^2\left|\right|x_n-q|{|}^2+{\alpha }_n^2||\gamma f(x_n)-Aq|{|}^2\\ +2\gamma \alpha {\alpha }_n\left|\right|x_n-q|{|}^2+2{\alpha }_n〈T_n{y}_n-q\mathrm{,}\gamma f(q)-Aq〉\\ -2{\alpha }_n^2〈A(T_n{y}_n-q),\gamma f(x_n)-Aq〉\\ \le ({(1-{\alpha }_n\overline{\gamma })}^2+2\gamma \alpha {\alpha }_n)\left|\right|x_n-q|{|}^2+{\alpha }_n(2〈T_n{y}_n-q\mathrm{,}\gamma f(q)-Aq〉\\ +{\alpha }_n\left|\right|\gamma f(x_n)-Aq|{|}^2+2{\alpha }_n||A(T_n{y}_n-q)|\left|\right||\gamma f(x_n)-Aq||)\\ =(1-2(\overline{\gamma }-\gamma \alpha ){\alpha }_n)\left|\right|x_n-q|{|}^2+{\alpha }_n(2〈T_n{y}_n-q\mathrm{,}\gamma f(q)-Aq〉\\ +{\alpha }_n\left|\right|\gamma f(x_n)-Aq|{|}^2+2{\alpha }_n||A(T_n{y}_n-q)|\left|\right||\gamma f(x_n)-Aq||\\ +{\alpha }_n{\overline{\gamma }}^2\left|\right|x_n-q|{|}^2).\end{array}$$

Since {x _n }, {f (x _n )} and {T _n y _n } are bounded, we can take a constant η > 0 such that

$$\eta \ge \left|\right|\gamma f\left(x_n\right)-Aq|{|}^2+2|\left|A\left(T_n{y}_n-q\right)\right|\left|\right||\gamma f\left(x_n\right)-Aq||+{\bar{\gamma }}^2||x_n-q|{|}^2$$

for all n ≥ 0. It follows that

$$\left|\right|x_{n+1}-q|{|}^2\le \left(1-2\left(\bar{\gamma }-\gamma \alpha \right){\alpha }_n\right)||x_n-q|{|}^2+{\alpha }_n{\beta }_n,$$

(3.2)

where β _n = 2〈T _n y _n - q, γ f(q) - Aq〉 +ηα _n . By lim sup_n→∞〈(γ f - A)q, T _n y _n - q〉 ≤ 0, we get lim sup _n→∞β _n ≤ 0. By Lemma 2.3 and (3.2), we can conclude that x _n → q. This completes the proof. ■

Corollary 3.2 Let C be a closed convex subset of a real Hilbert space H, and let B : C → H be a β-inverse strongly monotone mapping, also let f : C → C be a contraction with coefficient α(0 < α < 1). Let {T _n } be a countable family of nonexpansive mappings from a subset C into itself with $F={\cap }_{n=1}^{\infty }F\left(T_n\right)\cap VI\left(C,B\right)\ne \varnothing$. Suppose {x _n } is the sequence generated by the following algorithm: x₀ ∈ C,

$$x_{n+1}={\alpha }_{n}f\left(x_n\right)+\left(1-{\alpha }_n\right)T_n{P}_C\left(x_n-{\lambda }_{n}B{x}_n\right)$$

for all n = 0, 1, 2, ..., where {α _n } ⊂ (0, 1) and {λ _n } ⊂ (0, 2β ). If {α _n } and {λ _n } are chosen so that λ _n ∈ [a, b] for some a, b with 0 < a < b < 2β,

$$\begin{array}{cc}\hfill \left(C1\right)\underset{n\to 0}{lim}{\alpha }_n=0;\hfill & \hfill \left(C2\right)\sum _{n=1}^{\infty }{\alpha }_n=\infty ;\hfill \\ \hfill \left(C3\right)\sum _{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_n\right|<\infty ;\hfill & \hfill \left(C4\right)\sum _{n=1}^{\infty }\left|{\lambda }_{n+1}-{\lambda }_n\right|<\infty .\hfill \end{array}$$

Suppose that ({T _n }, T ) satisfies the AKTT-condition. Then {x _n } converges strongly to q ∈ F, where q = P _F (γ f + I - A)(q) which solves the following variational inequality:

$$⟨\left(\gamma f-I\right)q,p-q⟩\le 0\forall p\in F.$$

Proof. Taking A = I and γ = 1 in Theorem 3.1, we get the results. ■

4 Applications

In this section, we apply the iterative scheme (1.8) and Theorem 3.1 for finding a common fixed point of countable family of nonexpansive mappings and strictly pseudocontractive mapping and inverse strongly monotone mapping.

A mapping T : C → C is called strictly pseudocontractive if there exists k with 0 ≤ k < 1 such that

$$\left|\right|Tx-Ty|{|}^2\le ||x-y|{|}^2+k\left|\right|\left(I-T\right)x-\left(I-T\right)y|{|}^2\forall x,y\in C.$$

If k = 0, then T is nonexpansive. Put B = I - T, where T : C → C is a strictly pseudocontractive mapping with k. Then, B is ((1 - k)/2)-inverse strongly monotone and B ^-1 (0) = F(T). Hence, for all x, y ∈ C,

$$\left|\right|\left(I-B\right)x-\left(I-B\right)y|{|}^2\le ||x-y|{|}^2+k\left|\right|Bx-By|{|}^2.$$

Conversely, since H is a real Hilbert space, we have

$$\left|\right|\left(I-B\right)x-\left(I-B\right)y|{|}^2\le ||x-y|{|}^2+\left|\right|Bx-By|{|}^2-2⟨x-y,Bx-By⟩.$$

Thus, we have

$$⟨x-y,Bx-By⟩\ge \frac{1-k}{2}\left|\right|Bx-By|{|}^2.$$

Theorem 4.1 Let C be a closed convex subset of a real Hilbert space H, and let A be a strongly positive linear bounded operator of H into itself with coefficient $\bar{\gamma }>0$ such that ||A|| = 1 and let f : C → C be a contraction with coefficient α(0 < α < 1). Assume that $0<\gamma <\bar{\gamma }∕\alpha$. Let {T _n } be a family of nonexpansive mappings of C into itself and let S be a strictly pseudocontractive mapping of C into itself with β such that $F={\cap }_{n=1}^{\infty }F\left(T_n\right)\cap F\left(S\right)\ne \varnothing$. Suppose {x _n } is a sequence generated by the following algorithm: x₀ ∈ C,

$$x_{n+1}=P_C\left({\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)T_n\left(\left(1-{\lambda }_n\right)x_n-{\lambda }_{n}S{x}_n\right)\right)$$

for all n = 0, 1, 2, ..., where {α _n } ⊂ [0, 1) and {λ _n } ⊂ [0, 1 - β). If {α _n } and {λ _n } are chosen so that λ _n ∈ [a, b] for some a, b with 0 < a < b < 1 - β,

$$\begin{array}{cc}\hfill \left(C1\right)\underset{n\to 0}{lim}{\alpha }_n=0;\hfill & \hfill \left(C2\right)\sum _{n=1}^{\infty }{\alpha }_n=\infty ;\hfill \\ \hfill \left(C3\right)\sum _{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_n\right|<\infty ;\hfill & \hfill \left(C4\right)\sum _{n=1}^{\infty }\left|{\lambda }_{n+1}-{\lambda }_n\right|<\infty .\hfill \end{array}$$

Suppose that ({T _n }, T) satisfies the AKTT-condition. Then, {x _n } converges strongly to q ∈ F, such that

$$⟨\left(\gamma f-A\right)q,p-q⟩\le 0\forall p\in F.$$

Proof. Put B = I - S, then B is ((1 - k)/2)-inverse strongly monotone and F(S) = V I(C, B) and P _C (x _n - λ _n Bx _n ) = (1 - λ _n )x _n +λ _n Sx _n . Therefore, by Theorem 3.1, the conclusion follows. ■

Lemma 4.2 [9] Let T : C → H be a k-strictly pseudocontractive, then

(i) the fixed point set F(T) of T is closed convex so that the projection P_F(T)is well defined;

(ii) define a mapping S : C → H by

$$Sx=\mu x+\left(1-\mu \right)Tx,x\in C.$$

(4.1)

If μ ∈ [k, 1), then S is a nonexpansive mapping such that F(T) = F(S).

A family of mappings ${\left\{T_n:C\to H\right\}}_{n=1}^{\infty }$ is called a family of uniformly k-strictly pseudocontractions, if there exists a constant k ∈ [0, 1) such that $\left|\right|T_{n}x-T_{n}y|{|}^2\le ||x-y|{|}^2+k\left|\right|\left(I-T_n\right)x-\left(I-T_n\right)y|{|}^2\forall x,y\in C,\forall n\ge 1.$

Let {T _n : C → C} be a countable family of uniformly k-strictly pseudocontractions. Let ${\left\{S_n:C\to C\right\}}_{n=1}^{\infty }$ be the sequence of mappings defined by (4.1), i.e.,

$$S_{n}x=\mu x+\left(1-\mu \right)T_{n}x,x\in C,\forall n\ge 1\mathsf{\text{with}}\mu \in \left[k,1\right).$$

Corollary 4.3 Let C be a closed convex subset of a real Hilbert space H, and let B : C → H be a β-inverse strongly monotone mapping, also let A be a strongly positive linear bounded operator of H into itself with coefficient $\bar{\gamma }>0$ such that ||A|| = 1 and let f : C → C be a contraction with coefficient α(0 < α < 1). Assume that $0<\gamma <\bar{\gamma }∕\alpha$. Let {T _n } be a countable family of uniformly k-strictly pseudocontractions from a subset C into itself with $F={\cap }_{n=1}^{\infty }F\left(T_n\right)\cap VI\left(C,B\right)\ne \varnothing$. Suppose {x _n } is the sequence generated by the following algorithm: x₀ ∈ C,

$$x_{n+1}=P_C\left({\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S_n{P}_C\left(x_n-{\lambda }_{n}B{x}_n\right)\right)$$

for all n = 0, 1, 2, ..., where {α _n } ⊂ (0, 1) and {λ _n } ⊂ (0, 2β). If {α _n } and {λ _n } are chosen so that λ _n ∈ [a, b] for some a, b with 0 < a < b < 2β,

$$\begin{array}{cc}\hfill \left(C1\right)\underset{n\to 0}{lim}{\alpha }_n=0;\hfill & \hfill \left(C2\right)\sum _{n=1}^{\infty }{\alpha }_n=\infty ;\hfill \\ \hfill \left(C3\right)\sum _{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_n\right|<\infty ;\hfill & \hfill \left(C4\right)\sum _{n=1}^{\infty }\left|{\lambda }_{n+1}-{\lambda }_n\right|<\infty .\hfill \end{array}$$

Then, {x _n } converges strongly to q ∈ F, where q = P _F (γ f + I - A)(q) which solves the following variational inequality:

$$⟨\left(\gamma f-A\right)q,p-q⟩\le 0\forall p\in F.$$

Proof. Let {T _n } be a countable family of uniformly k-strictly pseudo-contractions from a subset C into itself. Set S _n = μI + (1 - μ)T _n where μ ∈ [k, 1). By Lemma 4.2, we have S _n is nonexpansive and F (S _n ) = F (T _n ). Therefore, by Theorem 3.1, the conclusion follows. ■