A new relaxed extragradient-like algorithm for approaching common solutions of generalized mixed equilibrium problems, a more general system of variational inequalities and a fixed point problem

Yifen Ke; Changfeng Ma

doi:10.1186/1687-1812-2013-126

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A new relaxed extragradient-like algorithm for approaching common solutions of generalized mixed equilibrium problems, a more general system of variational inequalities and a fixed point problem

Yifen Ke¹ and
Changfeng Ma¹Email author

Fixed Point Theory and Applications20132013:126

https://doi.org/10.1186/1687-1812-2013-126

Received: 20 August 2012

Accepted: 29 April 2013

Published: 15 May 2013

Abstract

In this paper, we introduce a new iterative algorithm by the relaxed extragradient-like method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of solutions of a more general system of variational inequalities for finite inverse strongly monotone mappings and the set of solutions of a fixed point problem of a strictly pseudocontractive mapping in a Hilbert space. Then we prove strong convergence of the scheme to a common element of the three above described sets.

Keywords

generalized mixed equilibrium problems more general system of variational inequalities fixed point inverse strongly monotone mapping strictly pseudocontractive mapping

1 Introduction

In this paper, we assume that H is a real Hilbert space with the inner product $〈 \cdot, \cdot 〉$ and the induced norm $∥ \cdot ∥$ and C is a nonempty closed convex subset of H. $P_{C}$ denotes the metric projection of H onto C and $F (T)$ denotes the fixed points set of a mapping T. The sequence ${x_{n}}$ converges weakly to x which is denoted by $x_{n} ⇀ x$ .

Let $φ : C \to R$ be a real-valued function and let $A : C \to H$ be a nonlinear mapping. Suppose that $F : C \times C \to R$ is a bifunction.

The generalized mixed equilibrium problem is to find

x \in C

(see, e.g., [1–6]) such that

F (x, y) + φ (y) - φ (x) + 〈 A x, y - x 〉 \geq 0, \forall y \in C .

(1.1)

The set of solutions of (1.1) is denoted by $GMEP (F, φ, A)$ .

If

φ \equiv 0

, then problem (1.1) reduces to the generalized equilibrium problem, which is to find

x \in C

(see, e.g., [7–9]) such that

F (x, y) + 〈 A x, y - x 〉 \geq 0, \forall y \in C .

(1.2)

The set of solutions of (1.2) is denoted by $GEP (F, A)$ .

If

A \equiv 0

, then problem (1.1) reduces to the mixed equilibrium problem, which is to find

x \in C

(see, e.g., [10–13]) such that

F (x, y) + φ (y) - φ (x) \geq 0, \forall y \in C .

(1.3)

The set of solutions of (1.3) is denoted by $MEP (F, φ)$ .

If

F \equiv 0

, then problem (1.1) reduces to the mixed variational inequality of Browder type, which is to find

x \in C

(see, e.g., [3, 14]) such that

φ (y) - φ (x) + 〈 A x, y - x 〉 \geq 0, \forall y \in C .

(1.4)

The set of solutions of (1.4) is denoted by $MVI (C, φ, A)$ .

If

φ \equiv 0

,

A \equiv 0

, then problem (1.1) reduces to the equilibrium problem, which is to find

x \in C

(see, e.g., [15–17]) such that

F (x, y) \geq 0, \forall y \in C .

(1.5)

The set of solutions of (1.5) is denoted by $EP (F)$ .

If

F \equiv 0

,

φ \equiv 0

, then problem (1.1) reduces to the variational inequality, which is to find

x \in C

(see, e.g., [18–29]) such that

〈 A x, y - x 〉 \geq 0, \forall y \in C .

(1.6)

The set of solutions of (1.6) is denoted by $VI (C, A)$ .

If

F \equiv 0

,

A \equiv 0

, then problem (1.1) reduces to the minimized problem, which is to find

x \in C

(see, e.g., [18–28]) such that

φ (y) - φ (x) \geq 0, \forall y \in C .

(1.7)

The set of solutions of (1.7) is denoted by $Argmin (φ)$ .

Let

A, B : C \to H

be two mappings. Ceng et al. [2] considered the following problem of finding

(x^{*}, y^{*}) \in C \times C

such that

{\begin{matrix} 〈 λ A y^{*} + x^{*} - y^{*}, x - x^{*} 〉 \geq 0, \forall x \in C, \\ 〈 μ B x^{*} + y^{*} - x^{*}, x - y^{*} 〉 \geq 0, \forall x \in C, \end{matrix}

(1.8)

which is called a general system of variational inequalities where $λ > 0$ and $μ > 0$ are two constants. In particular, if $A = B$ , $x^{*} = y^{*}$ , then problem (1.8) reduces to the classical variational inequality problem (1.6).

In order to find the common element of the solutions of problem (1.8) and the set of fixed points of one nonexpansive mapping S, Ceng et al. [2] studied the following algorithm: fix

u \in C

,

x_{0} \in C

, and

{\begin{matrix} z_{n} = T_{λ_{n}}^{(Θ, φ)} (x_{n} - λ_{n} F x_{n}), \\ y_{n} = T_{μ_{1}}^{G_{1}} [T_{μ_{2}}^{G_{2}} (z_{n} - μ_{2} B_{2} z_{n}) - μ_{1} B_{1} T_{μ_{2}}^{G_{2}} (z_{n} - μ_{2} B_{2} z_{n})], \\ x_{n + 1} = α_{n} u + β_{n} x_{n} + γ_{n} y_{n} + δ_{n} S y_{n}, n \geq 0 . \end{matrix}

(1.9)

Under appropriate conditions, they obtained one strong convergence theorem.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let

{A_{i}}_{i = 1}^{N} : C \to H

be a family of mappings. Cai and Bu [1] considered the following problem of finding

(x_{1}^{*}, x_{2}^{*}, \dots, x_{N}^{*}) \in C \times C \times \dots \times C

such that

{\begin{matrix} 〈 λ_{N} A_{N} x_{N}^{*} + x_{1}^{*} - x_{N}^{*}, x - x_{1}^{*} 〉 \geq 0, \forall x \in C, \\ 〈 λ_{N - 1} A_{N - 1} x_{N - 1}^{*} + x_{N}^{*} - x_{N - 1}^{*}, x - x_{N}^{*} 〉 \geq 0, \forall x \in C, \\ ⋮ \\ 〈 λ_{2} A_{2} x_{2}^{*} + x_{3}^{*} - x_{2}^{*}, x - x_{3}^{*} 〉 \geq 0, \forall x \in C, \\ 〈 λ_{1} A_{1} x_{1}^{*} + x_{2}^{*} - x_{1}^{*}, x - x_{2}^{*} 〉 \geq 0, \forall x \in C . \end{matrix}

(1.10)

And (1.10) can be rewritten as

{\begin{matrix} 〈 x_{1}^{*} - (I - λ_{N} A_{N}) x_{N}^{*}, x - x_{1}^{*} 〉 \geq 0, \forall x \in C, \\ 〈 x_{N}^{*} - (I - λ_{N - 1} A_{N - 1}) x_{N - 1}^{*}, x - x_{N}^{*} 〉 \geq 0, \forall x \in C, \\ ⋮ \\ 〈 x_{3}^{*} - (I - λ_{2} A_{2}) x_{2}^{*}, x - x_{3}^{*} 〉 \geq 0, \forall x \in C, \\ 〈 x_{2}^{*} - (I - λ_{1} A_{1}) x_{1}^{*}, x - x_{2}^{*} 〉 \geq 0, \forall x \in C, \end{matrix}

(1.11)

which is called a more general system of variational inequalities in Hilbert spaces, where $λ_{i} > 0$ for all $i \in {1, 2, \dots, N}$ . The set of solutions to (1.10) is denoted by Ω. In particular, if $N = 2$ , $A_{1} = B$ , $A_{2} = A$ , $λ_{1} = μ$ , $λ_{2} = λ$ , $x_{1}^{*} = x^{*}$ , $x_{2}^{*} = y^{*}$ , then problem (1.10) reduces to problem (1.8).

In order to find a common element of the solutions of problem (1.10) and the common fixed points of a family of strictly pseudocontractive mappings, Cai and Bu [1] studied the following algorithm: pick any

x_{0} \in H

, set

C_{1} = C

,

x_{1} = P_{C_{1}} x_{0}

, and

{\begin{matrix} u_{n} = T_{r_{M, n}}^{(F_{M}, φ_{M})} (I - r_{M, n} B_{M}) T_{r_{M - 1, n}}^{(F_{M - 1}, φ_{M - 1})} \\ \times (I - r_{M - 1, n} B_{M - 1}) \dots T_{r_{1, n}}^{(F_{1}, φ_{1})} (I - r_{1, n} B_{1}) x_{n}, \\ y_{n} = P_{C} (I - λ_{N} A_{N}) P_{C} (I - λ_{N - 1} A_{N - 1}) \dots P_{C} (I - λ_{2} A_{2}) P_{C} (I - λ_{1} A_{1}) u_{n}, \\ z_{n} = α_{n} y_{n} + (1 - α_{n}) S_{n} y_{n}, \\ C_{n + 1} = {z \in C_{n} : ∥ z_{n} - z ∥ \leq ∥ x_{n} - z ∥}, \\ x_{n + 1} = P_{C_{n + 1}} x_{0}, \forall n \geq 1 . \end{matrix}

(1.12)

Under suitable conditions, they also obtained one strong convergence theorem.

In this paper, motivated and inspired by the above facts, we study a new iterative algorithm by the relaxed extragradient-like method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of solutions of a more general system of variational inequalities for finite inverse strongly monotone mappings and the set of solutions of a fixed point problem of a strictly pseudocontractive mapping in a Hilbert space. Then we prove strong convergence of the scheme to a common element of the three above described sets.

2 Preliminaries

For solving the equilibrium problem, let us assume that the bifunction F satisfies the following conditions:

(A1)

$F (x, x) = 0$ for all $x \in C$ ;
(A2)

F is monotone, i.e., $F (x, y) + F (y, x) \leq 0$ for any $x, y \in C$ ;
(A3)

F is weakly upper semicontinuous, i.e., for each $x, y, z \in C$ ,
$\underset{t \to 0}{lim sup} F (x + t (z - x), y) \leq F (x, y);$
(A4)

$F (x, \cdot)$ is convex and lower semicontinuous for each $x \in C$ ;
(B1)

For each $x \in H$ and $r > 0$ , there exists a bounded subset $D_{x} \subseteq C$ and $y_{x} \in C$ such that for any $z \in C ∖ D_{x}$ ,
$F (z, y_{x}) + φ (y_{x}) - φ (z) + \frac{1}{r} 〈 y_{x} - z, z - x 〉 < 0;$
(B2)

C is a bounded set.

Let H be a real Hilbert space. It is well known that

{∥ x + y ∥}^{2} = {∥ x ∥}^{2} + {∥ y ∥}^{2} + 2 〈 x, y 〉

(2.1)

and

{∥ x ∥}^{2} - {∥ y ∥}^{2} \leq ∥ x - y ∥ (∥ x ∥ + ∥ y ∥)

(2.2)

for all $x, y \in H$ .

Definition 2.1 Let C be a nonempty closed convex subset of a real Hilbert space H.

(1)

A mapping $T : C \to C$ is said to be nonexpansive if
$∥ T x - T y ∥ \leq ∥ x - y ∥, \forall x, y \in C;$
(2)

A mapping $T : C \to H$ is said to be L-Lipschitzian if there exists $L > 0$ such that
$∥ T x - T y ∥ \leq L ∥ x - y ∥, \forall x, y \in C;$
(3)

A mapping $T : C \to C$ is said to be k-strictly pseudocontractive if there exists a constant $k \in [0, 1)$ such that
${∥ T x - T y ∥}^{2} \leq {∥ x - y ∥}^{2} + k {∥ (I - T) x - (I - T) y ∥}^{2}, \forall x, y \in C .$

(2.3)

It is obvious that $k = 0$ , then the mapping T is nonexpansive;
(4)

A mapping $T : C \to H$ is said to be monotone if
$〈 T x - T y, x - y 〉 \geq 0, \forall x, y \in C;$
(5)

A mapping $T : C \to H$ is said to be α-inverse-strongly monotone if there exists a positive real number α such that
$〈 T x - T y, x - y 〉 \geq α {∥ T x - T y ∥}^{2}, \forall x, y \in C .$

It is obvious that any α-inverse-strongly monotone mapping T is monotone and $\frac{1}{α}$ -Lipschitz continuous.

Definition 2.2

P_{C} : H \to C

is called a metric projection if for every point

x \in H

, there exists a unique nearest point in C, denoted by

P_{C} x

, such that

∥ x - P_{C} x ∥ \leq ∥ x - y ∥, \forall y \in C .

In order to prove our main results in the next section, we recall some lemmas.

Lemma 2.1 [30]

Let C be a nonempty closed convex subset of H and let

T : C \to C

be a k-strictly pseudocontractive mapping, then the following results hold:

(1)

equation (2.3) is equivalent to
$〈 T x - T y, x - y 〉 \leq {∥ x - y ∥}^{2} - \frac{1 - k}{2} {∥ (I - T) x - (I - T) y ∥}^{2}, \forall x, y \in C;$

(2.4)
(2)

T is Lipschitz continuous with a constant $\frac{1 + k}{1 - k}$ , i.e.,
$∥ T x - T y ∥ \leq \frac{1 + k}{1 - k} ∥ x - y ∥, \forall x, y \in C;$

(2.5)
(3)

(Demi-closed principle) $I - T$ is demi-closed on C, that is,
$if x_{n} ⇀ x^{*} \in C and (I - T) x_{n} \to 0, then x^{*} = T x^{*} .$

Lemma 2.2 [1]

Let C be a nonempty closed convex subset of H and let

T : C \to H

be an α-inverse-strongly monotone mapping, then for all

x, y \in C

and

λ > 0

, we have

\begin{array}{rcl} {∥ (I - λ T) x - (I - λ T) y ∥}^{2} & = & {∥ (x - y) - λ (T x - T y) ∥}^{2} \\ = & {∥ x - y ∥}^{2} - 2 λ 〈 T x - T y, x - y 〉 + λ^{2} {∥ T x - T y ∥}^{2} \\ \leq & {∥ x - y ∥}^{2} + λ (λ - 2 α) {∥ T x - T y ∥}^{2} . \end{array}

So, if $0 < λ \leq 2 α$ , then $I - λ T$ is a nonexpansive mapping from C to H.

Lemma 2.3 Let C be a nonempty closed convex subset of H and let

P_{C} : H \to C

be a metric projection, then

(1)

${∥ P_{C} x - P_{C} y ∥}^{2} \leq 〈 x - y, P_{C} x - P_{C} y 〉$ , $\forall x, y \in H$ ;
(2)

moreover, $P_{C}$ is a nonexpansive mapping, i.e., $∥ P_{C} x - P_{C} y ∥ \leq ∥ x - y ∥$ , $\forall x, y \in H$ ;
(3)

$〈 x - P_{C} x, y - P_{C} x 〉 \leq 0$ , $\forall x \in H$ , $y \in C$ ;
(4)

${∥ x - y ∥}^{2} \geq {∥ x - P_{C} x ∥}^{2} + {∥ y - P_{C} x ∥}^{2}$ , $\forall x \in H$ , $y \in C$ .

Lemma 2.4 [4]

Let C be a nonempty closed convex subset of H. Assume that

F : C \times C \to R

satisfies (A1)-(A4) and let

φ : C \to R

be a lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For

r > 0

and

x \in H

, define a mapping

T_{r}^{(F, φ)} : H \to C

as follows:

T_{r}^{(F, φ)} (x) = {z \in C : F (z, y) + φ (y) - φ (z) + \frac{1}{r} 〈 y - z, z - x 〉 \geq 0, \forall y \in C}

for all

x \in H

. Then the following hold:

(1)

for each $x \in H$ , $T_{r}^{(F, φ)} (x) \neq \emptyset$ and $T_{r}^{(F, φ)}$ is single-valued;
(2)

$T_{r}^{(F, φ)}$ is firmly nonexpansive, that is, for any $x, y \in H$ ,
${∥ T_{r}^{(F, φ)} (x) - T_{r}^{(F, φ)} (y) ∥}^{2} \leq 〈 T_{r}^{(F, φ)} (x) - T_{r}^{(F, φ)} (y), x - y 〉;$
(3)

$F (T_{r}^{(F, φ)}) = MEP (F, φ)$ ;
(4)

$MEP (F, φ)$ is closed and convex.

Lemma 2.5 [3]

Let C be a nonempty closed convex subset of H. Assume that

F : C \times C \to R

satisfies (A1)-(A4),

B : C \to H

is a continuous monotone mapping and let

φ : C \to R

be a lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For

r > 0

and

x \in H

, define a mapping

K_{r}^{(F, φ)} : H \to C

as follows:

K_{r}^{(F, φ)} (x) = {z \in C : F (z, y) + φ (y) - φ (z) + 〈 B x, y - z 〉 + \frac{1}{r} 〈 y - z, z - x 〉 \geq 0, \forall y \in C}

for all

x \in H

. Then the following hold:

(1)

for each $x \in H$ , $K_{r}^{(F, φ)} (x) \neq \emptyset$ and $K_{r}^{(F, φ)}$ is single-valued;
(2)

$K_{r}^{(F, φ)}$ is firmly nonexpansive, that is, for any $x, y \in H$ ,
${∥ K_{r}^{(F, φ)} (x) - K_{r}^{(F, φ)} (y) ∥}^{2} \leq 〈 K_{r}^{(F, φ)} (x) - K_{r}^{(F, φ)} (y), x - y 〉;$
(3)

$F (K_{r}^{(F, φ)}) = GMEP (F, φ, B)$ ;
(4)

$GMEP (F, φ, B)$ is closed and convex.

Lemma 2.6 Let C be a nonempty closed convex subset of H. Let

{F_{k}}_{k = 1}^{M}

be a family of bifunctions from

C \times C

into ℝ satisfying (A1)-(A4), let

{φ_{k}}_{k = 1}^{M}

be a family of lower semicontinuous functions from C into ℝ, and let

{B_{k}}_{k = 1}^{M}

be a family of

β_{k}

-inverse-strongly monotone mappings from C into H. For

F_{k}

and

φ_{k}

,

k = 1, 2, \dots, M

, assume that either (B1) or (B2) holds. Let

T : C \to H

be a mapping defined by

T (x) = T_{r_{M}}^{(F_{M}, φ_{M})} (I - r_{M} B_{M}) T_{r_{M - 1}}^{(F_{M - 1}, φ_{M - 1})} (I - r_{M - 1} B_{M - 1}) \dots T_{r_{1}}^{(F_{1}, φ_{1})} (I - r_{1} B_{1}) x, \forall x \in C .

Putting

Θ^{0} = I

, where I is an identity mapping,

Θ^{k} = T_{r_{k}}^{(F_{k}, φ_{k})} (I - r_{k} B_{k}) T_{r_{k - 1}}^{(F_{k - 1}, φ_{k - 1})} (I - r_{k - 1} B_{k - 1}) \dots T_{r_{1}}^{(F_{1}, φ_{1})} (I - r_{1} B_{1}), k = 1, 2, \dots, M .

If

x \in ⋂_{k = 1}^{M} GMEP (F_{k}, φ_{k}, B_{k})

and

0 < r_{k} \leq 2 β_{k}

,

k = 1, 2, \dots, M

, then

(1)

$Θ^{k} x = x$ , $k = 1, 2, \dots, M$ ;
(2)

T is nonexpansive.

Proof (1) Since

{B_{k}}_{k = 1}^{M}

is a family of

β_{k}

-inverse-strongly monotone mappings from C into H, so they are continuous monotone mappings. Observe that

\begin{aligned} T_{r_{k}}^{(F_{k}, φ_{k})} (I - r_{k} B_{k}) x \\ = {z \in C : F_{k} (z, y) + φ_{k} (y) - φ_{k} (z) + \frac{1}{r_{k}} 〈 y - z, z - (I - r_{k} B_{k}) x 〉 \geq 0, \forall y \in C} \\ = {z \in C : F_{k} (z, y) + φ_{k} (y) - φ_{k} (z) + 〈 B_{k} x, y - z 〉 + \frac{1}{r_{k}} 〈 y - z, z - x 〉 \geq 0, \forall y \in C} \\ = K_{r_{k}}^{(F_{k}, φ_{k})} (x) . \end{aligned}

By Lemma 2.5, we know that if

x \in ⋂_{k = 1}^{M} GMEP (F_{k}, φ_{k}, B_{k})

then x is the fixed point of the mapping

K_{r_{k}}^{(F_{k}, φ_{k})}

,

k = 1, 2, \dots, M

, so we have

x = K_{r_{k}}^{(F_{k}, φ_{k})} (x) = T_{r_{k}}^{(F_{k}, φ_{k})} (I - r_{k} B_{k}) x,

(2.6)

which implies that x is a fixed point of the mapping

T_{r_{k}}^{(F_{k}, φ_{k})} (I - r_{k} B_{k})

. Therefore we get

Θ^{k} x = x, k = 1, 2, \dots, M .

(2) Since

T_{r}^{(F, φ)}

is firmly nonexpansive, then it is obvious that

T_{r}^{(F, φ)}

is nonexpansive. And from Lemma 2.2, we have

\begin{array}{rcl} ∥ T (x) - T (y) ∥ & = & ∥ Θ^{M} x - Θ^{M} y ∥ \\ = & ∥ T_{r_{M}}^{(F_{M}, φ_{M})} (I - r_{M} B_{M}) Θ^{M - 1} x - T_{r_{M}}^{(F_{M}, φ_{M})} (I - r_{M} B_{M}) Θ^{M - 1} y ∥ \\ \leq & ∥ (I - r_{M} B_{M}) Θ^{M - 1} x - (I - r_{M} B_{M}) Θ^{M - 1} y ∥ \\ \leq & ∥ Θ^{M - 1} x - Θ^{M - 1} y ∥ \leq \dots \leq ∥ Θ^{0} x - Θ^{0} y ∥ \\ = & ∥ x - y ∥, \end{array}

which implies T is nonexpansive. □

Lemma 2.7 [1]

Let C be a nonempty closed convex subset of H. Let

A_{i}

be

α_{i}

-inverse-strongly monotone from C into H, respectively, where

i \in {1, 2, \dots, N}

. Let

G : C \to C

be a mapping defined by

G (x) = P_{C} (I - λ_{N} A_{N}) P_{C} (I - λ_{N - 1} A_{N - 1}) \dots P_{C} (I - λ_{2} A_{2}) P_{C} (I - λ_{1} A_{1}) x, \forall x \in C .

If $0 < λ_{i} \leq 2 α_{i}$ , $i = 1, 2, \dots, N$ , then G is nonexpansive.

Proof Put

Ω^{i} = P_{C} (I - λ_{i} A_{i}) P_{C} (I - λ_{i - 1} A_{i - 1}) \dots P_{C} (I - λ_{1} A_{1})

,

i = 1, 2, \dots, N

, and

Ω^{0} = I

, where I is an identity mapping. Since

P_{C}

is nonexpansive and from Lemma 2.2, we have

\begin{array}{rcl} ∥ G (x) - G (y) ∥ & = & ∥ Ω^{N} x - Ω^{N} y ∥ \\ = & ∥ P_{C} (I - λ_{N} A_{N}) Ω^{N - 1} x - P_{C} (I - λ_{N} A_{N}) Ω^{N - 1} y ∥ \\ \leq & ∥ (I - λ_{N} A_{N}) Ω^{N - 1} x - (I - λ_{N} A_{N}) Ω^{N - 1} y ∥ \\ \leq & ∥ Ω^{N - 1} x - Ω^{N - 1} y ∥ \leq \dots \leq ∥ Ω^{0} x - Ω^{0} y ∥ \\ \leq & ∥ x - y ∥, \end{array}

which implies G is nonexpansive. □

Lemma 2.8 Let C be a nonempty closed convex subset of H. Let

A_{i} : C \to H

be a nonlinear mapping, where

i = 1, 2, \dots, N

. For given

x_{i}^{*} \in C

,

i = 1, 2, \dots, N

,

(x_{1}^{*}, x_{2}^{*}, \dots, x_{N}^{*})

is a solution of problem (1.10) if and only if

x_{1}^{*} = P_{C} (I - λ_{N} A_{N}) x_{N}^{*}, x_{i}^{*} = P_{C} (I - λ_{i - 1} A_{i - 1}) x_{i - 1}^{*}, i = 2, 3, \dots, N,

(2.7)

that is,

x_{1}^{*} = P_{C} (I - λ_{N} A_{N}) P_{C} (I - λ_{N - 1} A_{N - 1}) \dots P_{C} (I - λ_{2} A_{2}) P_{C} (I - λ_{1} A_{1}) x_{1}^{*} .

Proof (⟸) From Lemma 2.3(3), it is obvious that (2.7) is the solution of problem (1.10).

(⟹) Since

\begin{aligned} 〈 λ_{N} A_{N} x_{N}^{*} + x_{1}^{*} - x_{N}^{*}, x - x_{1}^{*} 〉 \geq 0, \forall x \in C \\ \Rightarrow 〈 x_{1}^{*} - (I - λ_{N} A_{N}) x_{N}^{*}, x - x_{1}^{*} 〉 \geq 0, \forall x \in C \\ \Rightarrow 〈 (I - λ_{N} A_{N}) x_{N}^{*} - x_{1}^{*}, (I - λ_{N} A_{N}) x_{N}^{*} - x_{1}^{*} - (I - λ_{N} A_{N}) x_{N}^{*} + x 〉 \leq 0, \forall x \in C \\ \Rightarrow {∥ (I - λ_{N} A_{N}) x_{N}^{*} - x_{1}^{*} ∥}^{2} \leq 〈 (I - λ_{N} A_{N}) x_{N}^{*} - x_{1}^{*}, (I - λ_{N} A_{N}) x_{N}^{*} - x 〉, \forall x \in C \\ \Rightarrow ∥ (I - λ_{N} A_{N}) x_{N}^{*} - x_{1}^{*} ∥ \leq ∥ (I - λ_{N} A_{N}) x_{N}^{*} - x ∥, \forall x \in C \\ \Rightarrow x_{1}^{*} = P_{C} (I - λ_{N} A_{N}) x_{N}^{*} . \end{aligned}

Similarly, we get

x_{i}^{*} = P_{C} (I - λ_{i - 1} A_{i - 1}) x_{i - 1}^{*}, i = 2, 3, \dots, N .

Therefore we have

x_{1}^{*} = P_{C} (I - λ_{N} A_{N}) P_{C} (I - λ_{N - 1} A_{N - 1}) \dots P_{C} (I - λ_{2} A_{2}) P_{C} (I - λ_{1} A_{1}) x_{1}^{*},

which completes the proof. □

From Lemma 2.8, we know that $x_{1}^{*} = G (x_{1}^{*})$ , that is, $x_{1}^{*}$ is a fixed point of the mapping G, where G is defined by Lemma 2.7. Moreover, if we find the fixed point $x_{1}^{*}$ , it is easy to solve the other points by (2.7).

Lemma 2.9 [31]

Let ${x_{n}}$ and ${y_{n}}$ be bounded sequences in a Banach space X and let ${β_{n}}$ be a sequence in $[0, 1]$ with $0 < {lim inf}_{n \to \infty} β_{n} \leq {lim sup}_{n \to \infty} β_{n} < 1$ . Suppose that $x_{n + 1} = β_{n} x_{n} + (1 - β_{n}) y_{n}$ for all integers $n \geq 0$ and ${lim sup}_{n \to \infty} (∥ y_{n + 1} - y_{n} ∥ - ∥ x_{n + 1} - x_{n} ∥) \leq 0$ . Then ${lim}_{n \to \infty} ∥ y_{n} - x_{n} ∥ = 0$ .

Lemma 2.10 [30]

Let $T : C \to C$ be a nonexpansive mapping with $F (T) \neq \emptyset$ . If $x_{n} ⇀ x^{*}$ and $(I - T) x_{n} \to y^{*}$ , then $(I - T) x^{*} = y^{*}$ .

Lemma 2.11 [30]

Assume that

{α_{n}}

is a sequence of nonnegative real numbers such that

α_{n + 1} \leq (1 - γ_{n}) α_{n} + δ_{n}, \forall n \geq 1,

where

{γ_{n}}

is a sequence in

(0, 1)

and

{δ_{n}}

is a sequence such that

(1)

$\sum_{n = 1}^{\infty} γ_{n} = \infty$ ;
(2)

${lim sup}_{n \to \infty} \frac{δ_{n}}{γ_{n}} \leq 0$ or $\sum_{n = 1}^{\infty} | δ_{n} | < \infty$ .

Then ${lim}_{n \to \infty} α_{n} = 0$ .

3 Main results

In this section, we state and verify our main results. We have the following theorem.

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let

{F_{k}}_{k = 1}^{M}

be a family of bifunctions from

C \times C

into ℝ satisfying (A1)-(A4), let

{φ_{k}}_{k = 1}^{M} : C \to R

be a family of lower semicontinuous and convex functions and let

{B_{k}}_{k = 1}^{M}

be a family of

β_{k}

-inverse-strongly monotone mappings from C into H. Let

A_{i}

be

α_{i}

-inverse-strongly monotone from C into H, respectively, where

i \in {1, 2, \dots, N}

. Let S be a δ-strict pseudocontractive mapping from C into itself such that

F = [⋂_{k = 1}^{M} GMEP (F_{k}, φ_{k}, B_{k})] \cap F (G) \cap F (S) \neq \emptyset

, where G is defined by Lemma 2.7. For

F_{k}

and

φ_{k}

,

k = 1, 2, \dots, M

, assume that either (B1) or (B2) holds. Pick any

x_{0} \in C

, let

{x_{n}} \subset C

be a sequence generated by

{\begin{matrix} z_{n} = T_{r_{M, n}}^{(F_{M}, φ_{M})} (I - r_{M, n} B_{M}) T_{r_{M - 1, n}}^{(F_{M - 1}, φ_{M - 1})} \\ \times (I - r_{M - 1, n} B_{M - 1}) \dots T_{r_{1, n}}^{(F_{1}, φ_{1})} (I - r_{1, n} B_{1}) x_{n}, \\ y_{n} = P_{C} (I - λ_{N} A_{N}) P_{C} (I - λ_{N - 1} A_{N - 1}) \dots P_{C} (I - λ_{2} A_{2}) P_{C} (I - λ_{1} A_{1}) z_{n}, \\ x_{n + 1} = a_{n} x_{0} + b_{n} x_{n} + c_{n} y_{n} + d_{n} S y_{n}, \forall n \geq 0, \end{matrix}

(3.1)

where

λ_{i} \in (0, 2 α_{i})

,

i = 1, 2, \dots, N

,

δ \in (0, 1)

.

{a_{n}}, {b_{n}}, {c_{n}}, {d_{n}} \subset [0, 1]

satisfy the following conditions:

(i)

$a_{n} + b_{n} + c_{n} + d_{n} = 1$ and $(c_{n} + d_{n}) δ \leq c_{n}$ for all $n \geq 0$ ;
(ii)

${lim}_{n \to \infty} a_{n} = 0$ and $\sum_{n = 0}^{\infty} a_{n} = \infty$ ;
(iii)

$0 < {lim inf}_{n \to \infty} b_{n} \leq {lim sup}_{n \to \infty} b_{n} < 1$ and ${lim inf}_{n \to \infty} d_{n} > 0$ ;
(iv)

${lim}_{n \to \infty} (\frac{c_{n + 1}}{1 - b_{n + 1}} - \frac{c_{n}}{1 - b_{n}}) = 0$ ;
(v)

$0 < {lim inf}_{n \to \infty} r_{k, n} \leq {lim sup}_{n \to \infty} r_{k, n} < 2 β_{k}$ , $k = 1, 2, \dots, M$ .

Then ${x_{n}} \subset C$ converges strongly to $P_{F} x_{0}$ .

Proof Putting

\begin{array}{rcl} Θ_{n}^{k} & = & T_{r_{k, n}}^{(F_{k}, φ_{k})} (I - r_{k, n} B_{k}) T_{r_{k - 1, n}}^{(F_{k - 1}, φ_{k - 1})} (I - r_{k - 1, n} B_{k - 1}) \dots T_{r_{1, n}}^{(F_{1}, φ_{1})} (I - r_{1, n} B_{1}), \\ \forall k \in {1, \dots, M}, n \in N, \end{array}

and

Ω^{i} = P_{C} (I - λ_{i} A_{i}) P_{C} (I - λ_{i - 1} A_{i - 1}) \dots P_{C} (I - λ_{2} A_{2}) P_{C} (I - λ_{1} A_{1}), \forall i \in {1, 2, \dots, N},

$Θ_{n}^{0} = Ω^{0} = I$ , where I is the identity mapping on H. Then we have that $z_{n} = Θ_{n}^{M} x_{n}$ and $y_{n} = Ω^{N} z_{n}$ . From Lemma 2.6 and Lemma 2.7, it can be seen easily that $Θ_{n}^{k}$ and $Ω^{i}$ are nonexpansive, where $k \in {1, 2, \dots, M}$ , $i \in {1, 2, \dots, N}$ . We divide the proof into six steps.

Step 1. Firstly, we show that ${x_{n}}$ is bounded.

Indeed, take

p \in F

arbitrarily. Since

p = Θ_{n}^{k} p = S p

,

\forall k \in {1, 2, \dots, M}

,

\forall n \in N

. By Lemma 2.6, we have

∥ z_{n} - p ∥ = ∥ Θ_{n}^{M} x_{n} - Θ_{n}^{M} p ∥ \leq ∥ x_{n} - p ∥ .

(3.2)

It follows from Lemma 2.7 and (3.2) that

∥ y_{n} - p ∥ = ∥ Ω^{N} z_{n} - Ω^{N} p ∥ \leq ∥ z_{n} - p ∥ \leq ∥ x_{n} - p ∥ .

(3.3)

Furthermore, from (3.1), we have

\begin{array}{rcl} ∥ x_{n + 1} - p ∥ & = & ∥ a_{n} x_{0} + b_{n} x_{n} + c_{n} y_{n} + d_{n} S y_{n} - p ∥ \\ = & ∥ a_{n} (x_{0} - p) + b_{n} (x_{n} - p) + c_{n} (y_{n} - p) + d_{n} (S y_{n} - p) ∥ \\ \leq & a_{n} ∥ x_{0} - p ∥ + b_{n} ∥ x_{n} - p ∥ + ∥ c_{n} (y_{n} - p) + d_{n} (S y_{n} - p) ∥ . \end{array}

(3.4)

Since

(c_{n} + d_{n}) δ \leq c_{n}

, (2.3) and (2.4), we have

\begin{aligned} {∥ c_{n} (y_{n} - p) + d_{n} (S y_{n} - p) ∥}^{2} \\ = c_{n}^{2} {∥ y_{n} - p ∥}^{2} + d_{n}^{2} {∥ S y_{n} - p ∥}^{2} + 2 c_{n} d_{n} 〈 S y_{n} - p, y_{n} - p 〉 \\ \leq c_{n}^{2} {∥ y_{n} - p ∥}^{2} + d_{n}^{2} [{∥ y_{n} - p ∥}^{2} + δ {∥ y_{n} - S y_{n} ∥}^{2}] \\ + 2 c_{n} d_{n} [{∥ y_{n} - p ∥}^{2} - \frac{1 - δ}{2} {∥ y_{n} - S y_{n} ∥}^{2}] \\ = {(c_{n} + d_{n})}^{2} {∥ y_{n} - p ∥}^{2} + [d_{n}^{2} δ - (1 - δ) c_{n} d_{n}] {∥ y_{n} - S y_{n} ∥}^{2} \\ = {(c_{n} + d_{n})}^{2} {∥ y_{n} - p ∥}^{2} + d_{n} [(c_{n} + d_{n}) δ - c_{n}] {∥ y_{n} - S y_{n} ∥}^{2} \\ \leq {(c_{n} + d_{n})}^{2} {∥ y_{n} - p ∥}^{2}, \end{aligned}

which implies that

∥ c_{n} (y_{n} - p) + d_{n} (S y_{n} - p) ∥ \leq (c_{n} + d_{n}) ∥ y_{n} - p ∥ .

(3.5)

From (3.2)-(3.5) it follows that

\begin{array}{rcl} ∥ x_{n + 1} - p ∥ & \leq & a_{n} ∥ x_{0} - p ∥ + b_{n} ∥ x_{n} - p ∥ + ∥ c_{n} (y_{n} - p) + d_{n} (S y_{n} - p) ∥ \\ \leq & a_{n} ∥ x_{0} - p ∥ + b_{n} ∥ x_{n} - p ∥ + (c_{n} + d_{n}) ∥ y_{n} - p ∥ \\ \leq & a_{n} ∥ x_{0} - p ∥ + b_{n} ∥ x_{n} - p ∥ + (c_{n} + d_{n}) ∥ x_{n} - p ∥ \\ = & a_{n} ∥ x_{0} - p ∥ + (1 - a_{n}) ∥ x_{n} - p ∥ . \end{array}

So, we have

∥ x_{n + 1} - p ∥ \leq max {∥ x_{0} - p ∥, ∥ x_{n} - p ∥}, \forall n \geq 0 .

By induction, we obtain that

∥ x_{n} - p ∥ \leq ∥ x_{0} - p ∥, \forall n \geq 0 .

Hence, ${x_{n}}$ is bounded. Consequently, we deduce immediately that ${z_{n}}$ , ${y_{n}}$ , ${S y_{n}}$ are bounded.

Step 2. Next, we prove that ${lim}_{n \to \infty} ∥ x_{n + 1} - x_{n} ∥ = 0$ .

Indeed, define

x_{n + 1} = b_{n} x_{n} + (1 - b_{n}) w_{n}

for all

n \geq 0

. It follows that

\begin{array}{rcl} w_{n + 1} - w_{n} & = & \frac{x_{n + 2} - b_{n + 1} x_{n + 1}}{1 - b_{n + 1}} - \frac{x_{n + 1} - b_{n} x_{n}}{1 - b_{n}} \\ = & \frac{a_{n + 1} x_{0} + c_{n + 1} y_{n + 1} + d_{n + 1} S y_{n + 1}}{1 - b_{n + 1}} - \frac{a_{n} x_{0} + c_{n} y_{n} + d_{n} S y_{n}}{1 - b_{n}} \\ = & \frac{a_{n + 1} x_{0}}{1 - b_{n + 1}} - \frac{a_{n} x_{0}}{1 - b_{n}} + \frac{c_{n + 1} (y_{n + 1} - y_{n}) + d_{n + 1} (S y_{n + 1} - S y_{n})}{1 - b_{n + 1}} \\ + (\frac{c_{n + 1}}{1 - b_{n + 1}} - \frac{c_{n}}{1 - b_{n}}) y_{n} + (\frac{d_{n + 1}}{1 - b_{n + 1}} - \frac{d_{n}}{1 - b_{n}}) S y_{n} . \end{array}

(3.6)

Observe that

\begin{aligned} {∥ c_{n + 1} (y_{n + 1} - y_{n}) + d_{n + 1} (S y_{n + 1} - S y_{n}) ∥}^{2} \\ = c_{n + 1}^{2} {∥ y_{n + 1} - y_{n} ∥}^{2} + d_{n + 1}^{2} {∥ S y_{n + 1} - S y_{n} ∥}^{2} + 2 c_{n + 1} d_{n + 1} 〈 S y_{n + 1} - S y_{n}, y_{n + 1} - y_{n} 〉 \\ \leq c_{n + 1}^{2} {∥ y_{n + 1} - y_{n} ∥}^{2} + d_{n + 1}^{2} [{∥ y_{n + 1} - y_{n} ∥}^{2} + δ {∥ (y_{n + 1} - S y_{n + 1}) - (y_{n} - S y_{n}) ∥}^{2}] \\ + 2 c_{n + 1} d_{n + 1} [{∥ y_{n + 1} - y_{n} ∥}^{2} - \frac{1 - δ}{2} {∥ (y_{n + 1} - S y_{n + 1}) - (y_{n} - S y_{n}) ∥}^{2}] \\ = {(c_{n + 1} + d_{n + 1})}^{2} {∥ y_{n + 1} - y_{n} ∥}^{2} + [d_{n + 1}^{2} δ - (1 - δ) c_{n + 1} d_{n + 1}] {∥ (y_{n + 1} - S y_{n + 1}) - (y_{n} - S y_{n}) ∥}^{2} \\ = {(c_{n + 1} + d_{n + 1})}^{2} {∥ y_{n + 1} - y_{n} ∥}^{2} + d_{n + 1} [(c_{n + 1} + d_{n + 1}) δ - c_{n + 1}] \\ \times {∥ (y_{n + 1} - S y_{n + 1}) - (y_{n} - S y_{n}) ∥}^{2} \\ \leq {(c_{n + 1} + d_{n + 1})}^{2} {∥ y_{n + 1} - y_{n} ∥}^{2}, \end{aligned}

which implies that

∥ c_{n + 1} (y_{n + 1} - y_{n}) + d_{n + 1} (S y_{n + 1} - S y_{n}) ∥ \leq (c_{n + 1} + d_{n + 1}) ∥ y_{n + 1} - y_{n} ∥ .

(3.7)

Since

∥ z_{n + 1} - z_{n} ∥ = ∥ Θ_{n}^{M} x_{n + 1} - Θ_{n}^{M} x_{n} ∥ \leq ∥ x_{n + 1} - x_{n} ∥,

(3.8)

then we have

∥ y_{n + 1} - y_{n} ∥ = ∥ Ω^{N} z_{n + 1} - Ω^{N} z_{n} ∥ \leq ∥ z_{n + 1} - z_{n} ∥ \leq ∥ x_{n + 1} - x_{n} ∥ .

(3.9)

Hence it follows from (3.6), (3.7), (3.9) and

\frac{c_{n + 1} + d_{n + 1}}{1 - b_{n + 1}} = \frac{1 - a_{n + 1} - b_{n + 1}}{1 - b_{n + 1}} < 1

that

\begin{aligned} ∥ w_{n + 1} - w_{n} ∥ \leq & (\frac{a_{n + 1}}{1 - b_{n + 1}} + \frac{a_{n}}{1 - b_{n}}) ∥ x_{0} ∥ + \frac{∥ c_{n + 1} (y_{n + 1} - y_{n}) + d_{n + 1} (S y_{n + 1} - S y_{n}) ∥}{1 - b_{n + 1}} \\ + | \frac{c_{n + 1}}{1 - b_{n + 1}} - \frac{c_{n}}{1 - b_{n}} | ∥ y_{n} ∥ + | \frac{d_{n + 1}}{1 - b_{n + 1}} - \frac{d_{n}}{1 - b_{n}} | ∥ S y_{n} ∥ \\ \leq & (\frac{a_{n + 1}}{1 - b_{n + 1}} + \frac{a_{n}}{1 - b_{n}}) ∥ x_{0} ∥ + \frac{c_{n + 1} + d_{n + 1}}{1 - b_{n + 1}} ∥ x_{n + 1} - x_{n} ∥ \\ + | \frac{c_{n + 1}}{1 - b_{n + 1}} - \frac{c_{n}}{1 - b_{n}} | ∥ y_{n} ∥ + | \frac{a_{n + 1} + c_{n + 1}}{1 - b_{n + 1}} - \frac{a_{n} + c_{n}}{1 - b_{n}} | ∥ S y_{n} ∥ \\ \leq & \frac{a_{n + 1}}{1 - b_{n + 1}} (∥ x_{0} ∥ + ∥ S y_{n} ∥) + \frac{a_{n}}{1 - b_{n}} (∥ x_{0} ∥ + ∥ S y_{n} ∥) + ∥ x_{n + 1} - x_{n} ∥ \\ + | \frac{c_{n + 1}}{1 - b_{n + 1}} - \frac{c_{n}}{1 - b_{n}} | (∥ y_{n} ∥ + ∥ S y_{n} ∥) . \end{aligned}

(3.10)

Consequently, it follows from (3.10), conditions (ii), (iv) and

{y_{n}}

,

{S y_{n}}

are bounded that

\begin{aligned} \underset{n \to \infty}{lim sup} (∥ w_{n + 1} - w_{n} ∥ - ∥ x_{n + 1} - x_{n} ∥) \\ \leq \underset{n \to \infty}{lim sup} {\frac{a_{n + 1}}{1 - b_{n + 1}} (∥ x_{0} ∥ + ∥ S y_{n} ∥) + \frac{a_{n}}{1 - b_{n}} (∥ x_{0} ∥ + ∥ S y_{n} ∥) \\ + | \frac{c_{n + 1}}{1 - b_{n + 1}} - \frac{c_{n}}{1 - b_{n}} | (∥ y_{n} ∥ + ∥ S y_{n} ∥)} = 0 . \end{aligned}

Hence, by Lemma 2.9, we get

{lim}_{n \to \infty} ∥ w_{n} - x_{n} ∥ = 0

. Thus, from condition (iii), we have

lim_{n \to \infty} ∥ x_{n + 1} - x_{n} ∥ = lim_{n \to \infty} (1 - b_{n}) ∥ w_{n} - x_{n} ∥ = 0 .

(3.11)

Step 3. We show that ${lim}_{n \to \infty} ∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥ = 0$ , $k = 1, 2, \dots, M$ and ${lim}_{n \to \infty} ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥ = 0$ , $i = 1, 2, \dots, N$ .

It follows from Lemma 2.6 that

\begin{aligned} {∥ z_{n} - p ∥}^{2} & = {∥ Θ_{n}^{M} x_{n} - Θ_{n}^{M} p ∥}^{2} \leq {∥ Θ_{n}^{k} x_{n} - Θ_{n}^{k} p ∥}^{2} \\ = {∥ T_{r_{k, n}}^{(F_{k}, φ_{k})} (I - r_{k, n} B_{k}) Θ_{n}^{k - 1} x_{n} - T_{r_{k, n}}^{(F_{k}, φ_{k})} (I - r_{k, n} B_{k}) Θ_{n}^{k - 1} p ∥}^{2} \\ \leq {∥ (I - r_{k, n} B_{k}) Θ_{n}^{k - 1} x_{n} - (I - r_{k, n} B_{k}) Θ_{n}^{k - 1} p ∥}^{2} \\ \leq {∥ Θ_{n}^{k - 1} x_{n} - p ∥}^{2} + r_{k, n} (r_{k, n} - 2 β_{k}) {∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥}^{2} \\ \leq {∥ x_{n} - p ∥}^{2} + r_{k, n} (r_{k, n} - 2 β_{k}) {∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥}^{2} . \end{aligned}

(3.12)

By Lemma 2.3 and Lemma 2.2, we have

\begin{array}{rcl} {∥ y_{n} - p ∥}^{2} & = & {∥ P_{C} (I - λ_{N} A_{N}) Ω^{N - 1} z_{n} - P_{C} (I - λ_{N} A_{N}) Ω^{N - 1} p ∥}^{2} \\ \leq & {∥ (I - λ_{N} A_{N}) Ω^{N - 1} z_{n} - (I - λ_{N} A_{N}) Ω^{N - 1} p ∥}^{2} \\ \leq & {∥ Ω^{N - 1} z_{n} - Ω^{N - 1} p ∥}^{2} + λ_{N} (λ_{N} - 2 α_{N}) {∥ A_{N} Ω^{N - 1} z_{n} - A_{N} Ω^{N - 1} p ∥}^{2} . \end{array}

By induction, we get

{∥ y_{n} - p ∥}^{2} \leq {∥ z_{n} - p ∥}^{2} + \sum_{i = 1}^{N} λ_{i} (λ_{i} - 2 α_{i}) {∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥}^{2} .

(3.13)

From condition (i) and (3.5), we get

\begin{aligned} {∥ x_{n + 1} - p ∥}^{2} \\ = 〈 a_{n} (x_{0} - p) + b_{n} (x_{n} - p) + c_{n} (y_{n} - p) + d_{n} (S y_{n} - p), x_{n + 1} - p 〉 \\ = a_{n} 〈 x_{0} - p, x_{n + 1} - p 〉 + b_{n} 〈 x_{n} - p, x_{n + 1} - p 〉 + 〈 c_{n} (y_{n} - p) + d_{n} (S y_{n} - p), x_{n + 1} - p 〉 \\ \leq a_{n} 〈 x_{0} - p, x_{n + 1} - p 〉 + b_{n} ∥ x_{n} - p ∥ ∥ x_{n + 1} - p ∥ \\ + ∥ c_{n} (y_{n} - p) + d_{n} (S y_{n} - p) ∥ ∥ x_{n + 1} - p ∥ \\ \leq a_{n} 〈 x_{0} - p, x_{n + 1} - p 〉 + b_{n} ∥ x_{n} - p ∥ ∥ x_{n + 1} - p ∥ + (c_{n} + d_{n}) ∥ y_{n} - p ∥ ∥ x_{n + 1} - p ∥ \\ \leq a_{n} 〈 x_{0} - p, x_{n + 1} - p 〉 + \frac{b_{n}}{2} ({∥ x_{n} - p ∥}^{2} + {∥ x_{n + 1} - p ∥}^{2}) \\ + \frac{c_{n} + d_{n}}{2} ({∥ y_{n} - p ∥}^{2} + {∥ x_{n + 1} - p ∥}^{2}), \end{aligned}

that is,

{∥ x_{n + 1} - p ∥}^{2} \leq \frac{2 a_{n}}{1 + a_{n}} 〈 x_{0} - p, x_{n + 1} - p 〉 + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} + \frac{c_{n} + d_{n}}{1 + a_{n}} {∥ y_{n} - p ∥}^{2} .

(3.14)

So, in terms of (3.12) and (3.13), we have

\begin{array}{rcl} {∥ x_{n + 1} - p ∥}^{2} & \leq & \frac{2 a_{n}}{1 + a_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} \\ + \frac{c_{n} + d_{n}}{1 + a_{n}} {{∥ z_{n} - p ∥}^{2} + \sum_{i = 1}^{N} λ_{i} (λ_{i} - 2 α_{i}) {∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥}^{2}} \\ \leq & \frac{2 a_{n}}{1 + a_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} \\ + \frac{c_{n} + d_{n}}{1 + a_{n}} {{∥ x_{n} - p ∥}^{2} + r_{k, n} (r_{k, n} - 2 β_{k}) {∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥}^{2} \\ + \sum_{i = 1}^{N} λ_{i} (λ_{i} - 2 α_{i}) {∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥}^{2}} \\ \leq & \frac{2 a_{n}}{1 + a_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + \frac{1 - a_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} \\ + \frac{c_{n} + d_{n}}{1 + a_{n}} {r_{k, n} (r_{k, n} - 2 β_{k}) {∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥}^{2} \\ + \sum_{i = 1}^{N} λ_{i} (λ_{i} - 2 α_{i}) {∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥}^{2}} . \end{array}

Therefore,

\begin{aligned} r_{k, n} (2 β_{k} - r_{k, n}) {∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥}^{2} + \sum_{i = 1}^{N} λ_{i} (2 α_{i} - λ_{i}) {∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥}^{2} \\ \leq \frac{2 a_{n}}{c_{n} + d_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + \frac{1 - a_{n}}{c_{n} + d_{n}} ({∥ x_{n} - p ∥}^{2} - {∥ x_{n + 1} - p ∥}^{2}) \\ \leq \frac{2 a_{n}}{c_{n} + d_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + \frac{1 - a_{n}}{c_{n} + d_{n}} ∥ x_{n + 1} - x_{n} ∥ (∥ x_{n + 1} - p ∥ + ∥ x_{n} - p ∥) . \end{aligned}

Since

{lim}_{n \to \infty} a_{n} = 0

,

0 < {lim inf}_{n \to \infty} r_{k, n} \leq {lim sup}_{n \to \infty} r_{k, n} < 2 β_{k}

,

k = 1, 2, \dots, M

,

λ_{i} \in (0, 2 α_{i})

,

i = 1, 2, \dots, N

,

δ \in (0, 1)

,

{lim inf}_{n \to \infty} (c_{n} + d_{n}) > 0

and

{x_{n}}

is bounded, we have

lim_{n \to \infty} ∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥ = 0, k = 1, 2, \dots, M,

(3.15)

and

lim_{n \to \infty} ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥ = 0, i = 1, 2, \dots, N .

(3.16)

Step 4. We prove that ${lim}_{n \to \infty} ∥ S y_{n} - y_{n} ∥ = 0$ .

Indeed, utilizing firmly nonexpansive of

T_{r_{k}}^{(F_{k}, φ_{k})}

and Lemma 2.2, we have

\begin{array}{rcl} {∥ Θ_{n}^{k} x_{n} - Θ_{n}^{k} p ∥}^{2} & = & {∥ T_{r_{k, n}}^{(F_{k}, φ_{k})} (I - r_{k, n} B_{k}) Θ_{n}^{k - 1} x_{n} - T_{r_{k, n}}^{(F_{k}, φ_{k})} (I - r_{k, n} B_{k}) p ∥}^{2} \\ \leq & 〈 (I - r_{k, n} B_{k}) Θ_{n}^{k - 1} x_{n} - (I - r_{k, n} B_{k}) p, Θ_{n}^{k} x_{n} - p 〉 \\ = & \frac{1}{2} ({∥ (I - r_{k, n} B_{k}) Θ_{n}^{k - 1} x_{n} - (I - r_{k, n} B_{k}) p ∥}^{2} + {∥ Θ_{n}^{k} x_{n} - p ∥}^{2} \\ - {∥ (I - r_{k, n} B_{k}) Θ_{n}^{k - 1} x_{n} - (I - r_{k, n} B_{k}) p - (Θ_{n}^{k} x_{n} - p) ∥}^{2}) \\ \leq & \frac{1}{2} ({∥ Θ_{n}^{k - 1} x_{n} - p ∥}^{2} + {∥ Θ_{n}^{k} x_{n} - p ∥}^{2} \\ - {∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} - r_{k, n} (B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p) ∥}^{2}), \end{array}

which implies

\begin{array}{rcl} {∥ Θ_{n}^{k} x_{n} - p ∥}^{2} & \leq & {∥ Θ_{n}^{k - 1} x_{n} - p ∥}^{2} - {∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} - r_{k, n} (B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p) ∥}^{2} \\ = & {∥ Θ_{n}^{k - 1} x_{n} - p ∥}^{2} - {∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} ∥}^{2} - r_{k, n}^{2} {∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥}^{2} \\ + 2 r_{k, n} 〈 Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n}, B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p 〉 \\ \leq & {∥ Θ_{n}^{k - 1} x_{n} - p ∥}^{2} - {∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} ∥}^{2} \\ + 2 r_{k, n} 〈 Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n}, B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p 〉 . \end{array}

(3.17)

From (3.14), (3.3), Lemma 2.6 and (3.17), we have

\begin{array}{rcl} {∥ x_{n + 1} - p ∥}^{2} & \leq & \frac{2 a_{n}}{1 + a_{n}} 〈 x_{0} - p, x_{n + 1} - p 〉 + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} + \frac{c_{n} + d_{n}}{1 + a_{n}} {∥ z_{n} - p ∥}^{2} \\ \leq & \frac{2 a_{n}}{1 + a_{n}} 〈 x_{0} - p, x_{n + 1} - p 〉 + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} + \frac{c_{n} + d_{n}}{1 + a_{n}} ∥ Θ_{n}^{k} x_{n} - Θ_{n}^{k} p ∥ \\ \leq & \frac{2 a_{n}}{1 + a_{n}} 〈 x_{0} - p, x_{n + 1} - p 〉 + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} \\ + \frac{c_{n} + d_{n}}{1 + a_{n}} [{∥ Θ_{n}^{k - 1} x_{n} - p ∥}^{2} - {∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} ∥}^{2} \\ + 2 r_{k, n} 〈 Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n}, B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p 〉] \\ \leq & \frac{2 a_{n}}{1 + a_{n}} 〈 x_{0} - p, x_{n + 1} - p 〉 + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} \\ + \frac{c_{n} + d_{n}}{1 + a_{n}} [{∥ x_{n} - p ∥}^{2} - {∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} ∥}^{2} \\ + 2 r_{k, n} 〈 Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n}, B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p 〉] \\ \leq & \frac{2 a_{n}}{1 + a_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} \\ + \frac{c_{n} + d_{n}}{1 + a_{n}} [{∥ x_{n} - p ∥}^{2} - {∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} ∥}^{2} \\ + 2 r_{k, n} ∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} ∥ ∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥] . \end{array}

It follows that

\begin{aligned} \frac{c_{n} + d_{n}}{1 + a_{n}} {∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} ∥}^{2} \\ \leq \frac{2 a_{n}}{1 + a_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + \frac{1 - a_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} - {∥ x_{n + 1} - p ∥}^{2} \\ + \frac{2 r_{k, n} (c_{n} + d_{n})}{1 + a_{n}} ∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} ∥ ∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥ \\ \leq \frac{2 a_{n}}{1 + a_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + {∥ x_{n} - p ∥}^{2} - {∥ x_{n + 1} - p ∥}^{2} \\ + \frac{2 r_{k, n} (c_{n} + d_{n})}{1 + a_{n}} ∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} ∥ ∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥ \\ \leq 2 a_{n} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + ∥ x_{n + 1} - x_{n} ∥ (∥ x_{n} - p ∥ + ∥ x_{n + 1} - p ∥) \\ + \frac{2 r_{k, n} (c_{n} + d_{n})}{1 + a_{n}} ∥ Θ_{n}^{k - 1} x_{n} - Θ_{n}^{k} x_{n} ∥ ∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥ . \end{aligned}

Since

{lim inf}_{n \to \infty} \frac{c_{n} + d_{n}}{1 + a_{n}} > 0

,

a_{n} \to 0

,

∥ x_{n + 1} - x_{n} ∥ \to 0

and

∥ B_{k} Θ_{n}^{k - 1} x_{n} - B_{k} p ∥ \to 0

, we conclude that

lim_{n \to \infty} ∥ Θ_{n}^{k} x_{n} - Θ_{n}^{k - 1} x_{n} ∥ = 0, k = 1, 2, \dots, M .

(3.18)

Therefore we get

\begin{aligned} ∥ x_{n} - z_{n} ∥ & = ∥ Θ_{n}^{0} x_{n} - Θ_{n}^{M} x_{n} ∥ \\ \leq ∥ Θ_{n}^{0} x_{n} - Θ_{n}^{1} x_{n} ∥ + ∥ Θ_{n}^{1} x_{n} - Θ_{n}^{2} x_{n} ∥ + \dots + ∥ Θ_{n}^{M - 1} x_{n} - Θ_{n}^{M} x_{n} ∥ \to 0 \\ as n \to \infty . \end{aligned}

(3.19)

From Lemma 2.3(1), we obtain

\begin{array}{rcl} {∥ Ω^{N} z_{n} - Ω^{N} p ∥}^{2} & = & {∥ P_{C} (I - λ_{N} A_{N}) Ω^{N - 1} z_{n} - P_{C} (I - λ_{N} A_{N}) Ω^{N - 1} p ∥}^{2} \\ \leq & 〈 (I - λ_{N} A_{N}) Ω^{N - 1} z_{n} - (I - λ_{N} A_{N}) Ω^{N - 1} p, Ω^{N} z_{n} - Ω^{N} p 〉 \\ = & \frac{1}{2} ({∥ (I - λ_{N} A_{N}) Ω^{N - 1} z_{n} - (I - λ_{N} A_{N}) Ω^{N - 1} p ∥}^{2} + {∥ Ω^{N} z_{n} - Ω^{N} p ∥}^{2} \\ - {∥ (I - λ_{N} A_{N}) Ω^{N - 1} z_{n} - (I - λ_{N} A_{N}) Ω^{N - 1} p - (Ω^{N} z_{n} - Ω^{N} p) ∥}^{2}) \\ \leq & \frac{1}{2} ({∥ Ω^{N - 1} z_{n} - Ω^{N - 1} p ∥}^{2} + {∥ Ω^{N} z_{n} - Ω^{N} p ∥}^{2} \\ - ∥ Ω^{N - 1} z_{n} - Ω^{N} z_{n} + Ω^{N} p - Ω^{N - 1} p \\ - λ_{N} (A_{N} Ω^{N - 1} z_{n} - A_{N} Ω^{N - 1} p) ∥^{2}), \end{array}

which implies that

\begin{aligned} {∥ Ω^{N} z_{n} - Ω^{N} p ∥}^{2} \\ \leq {∥ Ω^{N - 1} z_{n} - Ω^{N - 1} p ∥}^{2} \\ - {∥ Ω^{N - 1} z_{n} - Ω^{N} z_{n} + Ω^{N} p - Ω^{N - 1} p - λ_{N} (A_{N} Ω^{N - 1} z_{n} - A_{N} Ω^{N - 1} p) ∥}^{2} \\ = {∥ Ω^{N - 1} z_{n} - Ω^{N - 1} p ∥}^{2} \\ - {∥ Ω^{N - 1} z_{n} - Ω^{N} z_{n} + Ω^{N} p - Ω^{N - 1} p ∥}^{2} - λ_{N}^{2} {∥ A_{N} Ω^{N - 1} z_{n} - A_{N} Ω^{N - 1} p) ∥}^{2} \\ + 2 λ_{N} 〈 Ω^{N - 1} z_{n} - Ω^{N} z_{n} + Ω^{N} p - Ω^{N - 1} p, A_{N} Ω^{N - 1} z_{n} - A_{N} Ω^{N - 1} p 〉 \\ \leq {∥ Ω^{N - 1} z_{n} - Ω^{N - 1} p ∥}^{2} - {∥ Ω^{N - 1} z_{n} - Ω^{N} z_{n} + Ω^{N} p - Ω^{N - 1} p ∥}^{2} \\ + 2 λ_{N} ∥ Ω^{N - 1} z_{n} - Ω^{N} z_{n} + Ω^{N} p - Ω^{N - 1} p ∥ ∥ A_{N} Ω^{N - 1} z_{n} - A_{N} Ω^{N - 1} p ∥ . \end{aligned}

(3.20)

By induction, we have

\begin{array}{rcl} {∥ Ω^{N} z_{n} - Ω^{N} p ∥}^{2} & \leq & {∥ z_{n} - p ∥}^{2} - \sum_{i = 1}^{N} {∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥}^{2} \\ + 2 \sum_{i = 1}^{N} λ_{i} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥ \\ \leq & {∥ x_{n} - p ∥}^{2} - \sum_{i = 1}^{N} {∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{N - 1} p ∥}^{2} \\ + 2 \sum_{i = 1}^{N} λ_{i} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥, \end{array}

that is,

\begin{array}{rcl} {∥ y_{n} - p ∥}^{2} & \leq & {∥ x_{n} - p ∥}^{2} - \sum_{i = 1}^{N} {∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥}^{2} \\ + 2 \sum_{i = 1}^{N} λ_{i} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥ . \end{array}

(3.21)

From (3.14) and (3.21),

\begin{array}{rcl} {∥ x_{n + 1} - p ∥}^{2} & \leq & \frac{2 a_{n}}{1 + a_{n}} 〈 x_{0} - p, x_{n + 1} - p 〉 + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} \\ + \frac{c_{n} + d_{n}}{1 + a_{n}} [{∥ x_{n} - p ∥}^{2} - \sum_{i = 1}^{N} {∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥}^{2} \\ + 2 \sum_{i = 1}^{N} λ_{i} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥] \\ \leq & \frac{2 a_{n}}{1 + a_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} \\ + \frac{c_{n} + d_{n}}{1 + a_{n}} [{∥ x_{n} - p ∥}^{2} - \sum_{i = 1}^{N} {∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥}^{2} \\ + 2 \sum_{i = 1}^{N} λ_{i} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥] . \end{array}

It follows that

\begin{aligned} \frac{c_{n} + d_{n}}{1 + a_{n}} \sum_{i = 1}^{N} {∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥}^{2} \\ \leq \frac{2 a_{n}}{1 + a_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + \frac{1 - a_{n}}{1 + a_{n}} {∥ x_{n} - p ∥}^{2} - {∥ x_{n + 1} - p ∥}^{2} \\ + \frac{2 (c_{n} + d_{n})}{1 + a_{n}} \sum_{i = 1}^{N} λ_{i} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥ \\ \leq \frac{2 a_{n}}{1 + a_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + {∥ x_{n} - p ∥}^{2} - {∥ x_{n + 1} - p ∥}^{2} \\ + \frac{2 (c_{n} + d_{n})}{1 + a_{n}} \sum_{i = 1}^{N} λ_{i} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥ \\ \leq \frac{2 a_{n}}{1 + a_{n}} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + ∥ x_{n + 1} - x_{n} ∥ (∥ x_{n + 1} - p ∥ + ∥ x_{n} - p ∥) \\ + \frac{2 (c_{n} + d_{n})}{1 + a_{n}} \sum_{i = 1}^{N} λ_{i} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥ \\ \leq 2 a_{n} ∥ x_{0} - p ∥ ∥ x_{n + 1} - p ∥ + ∥ x_{n + 1} - x_{n} ∥ (∥ x_{n + 1} - p ∥ + ∥ x_{n} - p ∥) \\ + \frac{2 (c_{n} + d_{n})}{1 + a_{n}} \sum_{i = 1}^{N} λ_{i} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ ∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥ . \end{aligned}

Since

{lim inf}_{n \to \infty} \frac{c_{n} + d_{n}}{1 + a_{n}} > 0

,

a_{n} \to 0

,

∥ x_{n + 1} - x_{n} ∥ \to 0

and

∥ A_{i} Ω^{i - 1} z_{n} - A_{i} Ω^{i - 1} p ∥ \to 0

, we conclude that

lim_{n \to \infty} \sum_{i = 1}^{N} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ = 0 .

(3.22)

Therefore, we get

\begin{array}{rcl} ∥ z_{n} - y_{n} ∥ & = & ∥ Ω^{0} z_{n} - Ω^{N} z_{n} ∥ \\ \leq & \sum_{i = 1}^{N} ∥ Ω^{i - 1} z_{n} - Ω^{i} z_{n} + Ω^{i} p - Ω^{i - 1} p ∥ \to 0 as n \to \infty . \end{array}

(3.23)

Thus from (3.19) and (3.23), we have

lim_{n \to \infty} ∥ x_{n} - y_{n} ∥ = 0 .

(3.24)

Observe that

\begin{array}{rcl} d_{n} ∥ S y_{n} - x_{n} ∥ & = & ∥ d_{n} S y_{n} - d_{n} x_{n} ∥ \\ = & ∥ x_{n + 1} - a_{n} x_{0} - b_{n} x_{n} - c_{n} y_{n} - d_{n} x_{n} ∥ \\ = & ∥ x_{n + 1} - x_{n} + (1 - b_{n} - d_{n}) x_{n} - c_{n} y_{n} - a_{n} x_{0} ∥ \\ = & ∥ x_{n + 1} - x_{n} + a_{n} (x_{n} - x_{0}) + c_{n} (x_{n} - y_{n}) ∥ \\ \leq & ∥ x_{n + 1} - x_{n} ∥ + a_{n} ∥ x_{n} - x_{0} ∥ + c_{n} ∥ x_{n} - y_{n} ∥ . \end{array}

Since

{lim inf}_{n \to \infty} d_{n} > 0

,

∥ x_{n + 1} - x_{n} ∥ \to 0

,

a_{n} \to 0

and

∥ x_{n} - y_{n} ∥ \to 0

, we have

lim_{n \to \infty} ∥ S y_{n} - x_{n} ∥ = 0 .

(3.25)

From (3.24) and (3.25), we conclude that

lim_{n \to \infty} ∥ S y_{n} - y_{n} ∥ = 0 .

(3.26)

Step 5. In this step, we prove that ${lim sup}_{n \to \infty} 〈 x_{0} - \bar{x}, x_{n} - \bar{x} 〉 \leq 0$ , where $\bar{x} = P_{F} x_{0}$ .

Indeed, take a subsequence

{x_{n_{i}}}

of

{x_{n}}

such that

\underset{n \to \infty}{lim sup} 〈 x_{0} - \bar{x}, x_{n} - \bar{x} 〉 = lim_{n \to \infty} 〈 x_{0} - \bar{x}, x_{n_{i}} - \bar{x} 〉 .

Since

{x_{n}}

is bounded, there exists a subsequence of

{x_{n}}

which converges weakly to

x^{*}

. Without loss of generality, we may assume that

x_{n_{i}} ⇀ x^{*}

. From (3.18) and (3.24), we have

Θ_{n_{i}}^{k} x_{n_{i}} ⇀ x^{*}

,

y_{n_{i}} ⇀ x^{*}

, where

k \in {1, 2, \dots, M}

. From (3.26) and Lemma 2.1, we have

x^{*} = S x^{*}

that is

x^{*} \in F (S)

. Utilizing Lemma 2.7, we known that G is nonexpansive. And from (3.23), we obtain

∥ y_{n_{i}} - G y_{n_{i}} ∥ = ∥ G z_{n_{i}} - G y_{n_{i}} ∥ \leq ∥ z_{n_{i}} - y_{n_{i}} ∥ \to 0 as i \to \infty .

According to Lemma 2.10, we obtain $(I - G) x^{*} = 0$ , that is, $x^{*} \in F (G)$ .

Next we prove that

x^{*} \in ⋂_{k = 1}^{M} GMEP (F_{k}, φ_{k}, B_{k})

. Since

Θ_{n}^{k} x_{n} = T_{r_{k, n}}^{(F_{k}, φ_{k})} (I - r_{k, n} B_{k}) Θ_{n}^{k - 1} x_{n}, n \geq 1, k \in {1, 2, \dots, M} .

For all

y \in C

, we have

\begin{aligned} F_{k} (Θ_{n}^{k} x_{n}, y) + φ_{k} (y) - φ_{k} (Θ_{n}^{k} x_{n}) + 〈 B_{k} Θ_{n}^{k - 1} x_{n}, y - Θ_{n}^{k} x_{n} 〉 \\ + \frac{1}{r_{k, n}} 〈 y - Θ_{n}^{k} x_{n}, Θ_{n}^{k} x_{n} - Θ_{n}^{k - 1} x_{n} 〉 \geq 0 . \end{aligned}

By (A2), we have

\begin{aligned} φ_{k} (y) - φ_{k} (Θ_{n}^{k} x_{n}) + 〈 B_{k} Θ_{n}^{k - 1} x_{n}, y - Θ_{n}^{k} x_{n} 〉 + \frac{1}{r_{k, n}} 〈 y - Θ_{n}^{k} x_{n}, Θ_{n}^{k} x_{n} - Θ_{n}^{k - 1} x_{n} 〉 \\ \geq F_{k} (y, Θ_{n}^{k} x_{n}) . \end{aligned}

Replacing n by

n_{i}

in the above inequality, we have

\begin{aligned} φ_{k} (y) - φ_{k} (Θ_{n_{i}}^{k} x_{n_{i}}) + 〈 B_{k} Θ_{n_{i}}^{k - 1} x_{n_{i}}, y - Θ_{n_{i}}^{k} x_{n_{i}} 〉 + \frac{1}{r_{k, n_{i}}} 〈 y - Θ_{n_{i}}^{k} x_{n_{i}}, Θ_{n_{i}}^{k} x_{n_{i}} - Θ_{n_{i}}^{k - 1} x_{n_{i}} 〉 \\ \geq F_{k} (y, Θ_{n_{i}}^{k} x_{n_{i}}) . \end{aligned}

Let

z_{t} = t y + (1 - t) x^{*}

for all

t \in (0, 1]

and

y \in C

. This implies that

z_{t} \in C

. Then we have

\begin{aligned} 〈 z_{t} - Θ_{n_{i}}^{k} x_{n_{i}}, B_{k} z_{t} 〉 \\ \geq 〈 z_{t} - Θ_{n_{i}}^{k} x_{n_{i}}, B_{k} z_{t} 〉 + φ_{k} (Θ_{n_{i}}^{k} x_{n_{i}}) - φ_{k} (z_{t}) \\ - 〈 B_{k} Θ_{n_{i}}^{k - 1} x_{n_{i}}, z_{t} - Θ_{n_{i}}^{k} x_{n_{i}} 〉 - \frac{1}{r_{k, n_{i}}} 〈 z_{t} - Θ_{n_{i}}^{k} x_{n_{i}}, Θ_{n_{i}}^{k} x_{n_{i}} - Θ_{n_{i}}^{k - 1} x_{n_{i}} 〉 + F_{k} (z_{t}, Θ_{n_{i}}^{k}) \\ = φ_{k} (Θ_{n_{i}}^{k} x_{n_{i}}) - φ_{k} (z_{t}) + 〈 z_{t} - Θ_{n_{i}}^{k} x_{n_{i}}, B_{k} z_{t} - B_{k} Θ_{n_{i}}^{k} x_{n_{i}} 〉 \\ + 〈 z_{t} - Θ_{n_{i}}^{k} x_{n_{i}}, B_{k} Θ_{n_{i}}^{k} x_{n_{i}} - B_{k} Θ_{n_{i}}^{k - 1} x_{n_{i}} 〉 \\ - 〈 z_{t} - Θ_{n_{i}}^{k} x_{n_{i}}, \frac{Θ_{n_{i}}^{k} x_{n_{i}} - Θ_{n_{i}}^{k - 1} x_{n_{i}}}{r_{k, n_{i}}} 〉 + F_{k} (z_{t}, Θ_{n_{i}}^{k} x_{n_{i}}) . \end{aligned}

By (3.18) we have

∥ B_{k} Θ_{n_{i}}^{k} x_{n_{i}} - B_{k} Θ_{n_{i}}^{k - 1} x_{n_{i}} ∥ \to 0

as

i \to \infty

. Furthermore, by the monotonicity of

B_{k}

, we obtain

〈 z_{t} - Θ_{n_{i}}^{k} x_{n_{i}}, B_{k} z_{t} - B_{k} Θ_{n_{i}}^{k} x_{n_{i}} 〉 \geq 0

. Then from (A4), the lower semicontinuity of φ and

\frac{Θ_{n_{i}}^{k} x_{n_{i}} - Θ_{n_{i}}^{k - 1} x_{n_{i}}}{r_{k, n_{i}}} \to 0, Θ_{n_{i}}^{k} x_{n_{i}} ⇀ x^{*},

we obtain that

〈 z_{t} - x^{*}, B_{k} z_{t} 〉 \geq φ_{k} (x^{*}) - φ_{k} (z_{t}) + F_{k} (z_{t}, x^{*}) .

(3.27)

Using (A1), (A4) and (3.27), we have

\begin{array}{rcl} 0 & = & F_{k} (z_{t}, z_{t}) + φ_{k} (z_{t}) - φ_{k} (z_{t}) \\ \leq & t F_{k} (z_{t}, y) + (1 - t) F_{k} (z_{t}, x^{*}) + t φ_{k} (y) + (1 - t) φ_{k} (x^{*}) - t φ_{k} (z_{t}) - (1 - t) φ_{k} (z_{t}) \\ = & t [F_{k} (z_{t}, y) + φ_{k} (y) - φ_{k} (z_{t})] + (1 - t) [F_{k} (z_{t}, x^{*}) + φ_{k} (x^{*}) - φ_{k} (z_{t})] \\ \leq & t [F_{k} (z_{t}, y) + φ_{k} (y) - φ_{k} (z_{t})] + (1 - t) 〈 z_{t} - x^{*}, B_{k} z_{t} 〉 \\ = & t [F_{k} (z_{t}, y) + φ_{k} (y) - φ_{k} (z_{t})] + (1 - t) t 〈 y - x^{*}, B_{k} z_{t} 〉, \end{array}

and hence

0 \leq F_{k} (z_{t}, y) + φ_{k} (y) - φ_{k} (z_{t}) + (1 - t) 〈 y - x^{*}, B_{k} z_{t} 〉 .

Letting

t \to 0

, we have, for each

y \in C

,

0 \leq F_{k} (x^{*}, y) + φ_{k} (y) - φ_{k} (x^{*}) + 〈 y - x^{*}, B_{k} x^{*} 〉 .

This implies that

x^{*} \in GMEP (F_{k}, φ_{k}, B_{k})

. Hence

x^{*} \in ⋂_{k = 1}^{M} GMEP (F_{k}, φ_{k}, B_{k})

. Therefore,

x^{*} \in F = [⋂_{k = 1}^{M} GMEP (F_{k}, φ_{k}, B_{k})] \cap F (G) \cap F (S) .

This together with the property of metric projection implies that

\underset{n \to \infty}{lim sup} 〈 x_{0} - \bar{x}, x_{n} - \bar{x} 〉 = lim_{n \to \infty} 〈 x_{0} - \bar{x}, x_{n_{i}} - \bar{x} 〉 = 〈 x_{0} - \bar{x}, x^{*} - \bar{x} 〉 \leq 0 .

(3.28)

Step 6. Finally, we can easily show that $x_{n} \to \bar{x}$ as $n \to \infty$ .

Indeed, from (3.14) and (3.3), we have

\begin{array}{rcl} {∥ x_{n + 1} - \bar{x} ∥}^{2} & \leq & \frac{2 a_{n}}{1 + a_{n}} 〈 x_{0} - \bar{x}, x_{n + 1} - \bar{x} 〉 + \frac{b_{n}}{1 + a_{n}} {∥ x_{n} - \bar{x} ∥}^{2} + \frac{c_{n} + d_{n}}{1 + a_{n}} {∥ x_{n} - \bar{x} ∥}^{2} \\ = & \frac{2 a_{n}}{1 + a_{n}} 〈 x_{0} - \bar{x}, x_{n + 1} - \bar{x} 〉 + (1 - \frac{2 a_{n}}{1 + a_{n}}) {∥ x_{n} - \bar{x} ∥}^{2} . \end{array}

It is clear that $\sum_{n = 1}^{\infty} \frac{2 a_{n}}{1 + a_{n}} = \infty$ . Hence, applying (3.28) and Lemma 2.11, we obtain immediately that $x_{n} \to \bar{x}$ as $n \to \infty$ . This completes the proof. □

Declarations

Acknowledgements

The project is supported by the National Natural Science Foundation of China (Grant Nos. 11071041, 11201074) and Fujian Natural Science Foundation (Grant No. 2013J01003).