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Hybrid iterative algorithms for two families of finite maximal monotone mappings

Abstract

In this paper, we introduce and analyze a new general hybrid iterative algorithm for finding a common element of the set of common zeros of two families of finite maximal monotone mappings, the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping in a real Hilbert space. Our algorithm is based on four well-known methods: Mann’s iteration method, composite method, outer-approximation method and extragradient method. We prove the strong convergence theorem for the proposed algorithm. The results presented in this paper extend and improve the corresponding results of Wei and Tan (Fixed Point Theory Appl. 2014:77, 2014). Some special cases are also discussed.

Introduction

Let H be a real Hilbert space and C be a nonempty, closed, and convex subset of H. A mapping \(T:C \to C\) is said to be monotone if

$$\langle Tx - Ty,x - y \rangle\ge0,\quad \forall x,y \in C. $$

A multi-valued mapping \(T:H \to2^{H}\) is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone mapping. Lots of researches are focused on the maximal monotone mapping due to its importance.

In 1976, to solve the inclusion problem \(0 \in Ax\), Rockafellar [1] introduced the following proximal point method:

$$x_{0} \in H, \quad x_{n + 1} = J_{r_{n}}x_{n},\quad n = 0,1,2, \ldots, $$

where \(J_{r_{n}} = ( I + r_{n}A )^{ - 1}\) and \(A:H \to 2^{H}\) is a maximal monotone mapping. It is shown that the iterative sequence \(\{ x_{n} \}\) converges weakly to a zero of A under some appropriate conditions. The strong convergence of the sequence has been extensively discussed by Zegeye and Shahzad [2] and Hu and Liu [3] in Banach spaces.

In 2014, Wei and Tan [4] introduced the following Mann-type composite viscosity iterative scheme for finding the common zeros of two families of finite maximal monotone mappings. In particular, they proved the following theorem.

Theorem 1.1

([4], Theorem 2.2)

Let H be a real Hilbert space, C be a nonempty closed and convex subset of H, \(A_{i},B_{j}\ ( i = 1,2, \ldots,k;j = 1,2, \ldots,l ):C \to C\) be two families of m-accretive mappings. Suppose

$$S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}} = J_{r_{n}}^{A_{k}}J_{r_{n}}^{A_{k - 1}} \cdots J_{r_{n}}^{A_{1}},\qquad T_{r_{n}}=a_{0}I+a_{1}J_{r_{n}}^{B_{1}}+a_{2}J_{r_{n}}^{B_{2}}+\cdots+a_{l} J_{r_{n}}^{B_{l}}, $$

with \(J_{r_{n}}^{A_{i}} = ( I + r_{n}A_{i} )^{ - 1}\), \(J_{r_{n}}^{B_{j}} = ( I + r_{n}B_{j} )^{ - 1}\), \(\sum_{m = 0}^{l} a_{m} = 1\) and \(a_{m} \in ( 0,1 )\). The sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\), and \(\{ u_{n} \}\) are generated by

$$\left \{ \textstyle\begin{array}{@{}l} x_{1} \in C, \\ y_{n} = \beta_{n}f ( x_{n} ) + ( 1 - \beta_{n} )S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}x_{n}, \\ u_{n} = v_{n}f ( y_{n} ) + ( 1 - v_{n} )T_{r_{n}}y_{n}, \\ x_{n + 1} = \alpha_{n}f ( u_{n} ) + ( 1 - \alpha_{n} )u_{n}, \end{array}\displaystyle \right . $$

for every \(n = 0,1,2, \ldots\) , where \(f:C \to C\) is a contraction. If \(D = ( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 )\) is nonempty, \(\{ \alpha_{n} \}\), \(\{ \beta_{n} \}\), and \(\{ v_{n} \}\) are three sequences in \(( 0,1 )\) and \(\{ r_{n} \} \subset ( 0, + \infty )\) satisfy the following conditions:

  1. (i)

    \(\sum_{n = 1}^{\infty} \vert \alpha_{n + 1} - \alpha_{n} \vert < + \infty\), and \(\alpha_{n} \to0\) as \(n\to\infty\);

  2. (ii)

    \(\sum_{n = 1}^{\infty} \beta_{n} = + \infty\), \(\sum_{n = 1}^{\infty} \vert \beta_{n + 1} - \beta_{n} \vert < + \infty\), and \(\beta_{n} \to0\) as \(n \to\infty\);

  3. (iii)

    \(\sum_{n = 1}^{\infty} \vert v_{n + 1} - v_{n} \vert < + \infty\), and \(v_{n} \to0\) as \(n \to\infty\);

  4. (iv)

    \(\sum_{n = 1}^{\infty} \vert r_{n + 1} - r_{n} \vert < + \infty\), and \(r_{n} \to r^{*} \ge\varepsilon> 0\) as \(n \to\infty\).

Then \(\{ x_{n} \}\) converges strongly to a point \(p_{0} \in D\), which is the unique solution of the following variational inequality:

$$\bigl\langle f ( p_{0} ) - p_{0},p_{0} - q \bigr\rangle \ge0, \quad\forall q \in D. $$

Remark 1.1

Actually, the m-accretive mapping in a Hilbert space defined in Wei and Tan [4] is a maximal monotone mapping.

Theorem 1.1 gives rise naturally to the question we concerned: the real sequences \(\{ \alpha_{n} \}\), \(\{ \beta_{n} \}\), \(\{ v_{n} \}\), and \(\{ r_{n} \}\) are satisfying the conditions (i), (ii), (iii), and (iv). When is the restrictions of the real sequences relaxed? The purpose of our paper is to give an affirmative answer to the question. Moreover, a new algorithm and an extensive problem are considered.

On the other hand, the variational inequality problem is to find \(u \in C\) such that

$$\langle Au,v - u \rangle\ge0,\quad \forall v \in C, $$

where \(A:C \to H\) is a nonlinear mapping. The set of solutions of the variational inequality problem is denoted by \(VI ( C,A )\). The variational inequality problem was first discussed by Lions [5]. For finding a solution of the variational inequality problem in Euclidean space \(R^{n}\), Korpelevich [6] introduced the following extragradient method:

$$\left \{ \textstyle\begin{array}{@{}l} x_{0} = x \in C, \\ \overline{x}_{n} = P_{C} ( x_{n} - \lambda Ax_{n} ), \\ x_{n + 1} = P_{C} ( x_{n} - \lambda A\overline{x}_{n} ), \end{array}\displaystyle \right . $$

for every \(n = 0, 1, 2, \ldots\) where C is a nonempty, closed, and convex subset of \(R^{n}\), \(A:C \to R^{n}\) is a monotone and ρ-Lipschitz-continuous mapping, \(\lambda\in ( 0,\frac{1}{\rho} )\). She showed that if \(VI ( C,A )\) is nonempty, then the generated sequences \(\{ x_{n} \}\) and \(\{ \overline{x}_{n} \}\) converge to the same point \(z \in VI ( C,A )\). The extragradient iterative process was successfully generalized and extended not only to Euclidean but also to Hilbert and Banach spaces; see, e.g., the recent references of [79].

Furthermore, Iiduka and Takahashi [10] introduced the following outer-approximation method:

$$\left \{ \textstyle\begin{array}{@{}l} x_{0} = x \in C, \\ y_{n} = \alpha_{n}x_{n} + ( 1 - \alpha_{n} )SP_{C} ( x_{n} -\lambda_{n}Ax_{n} ), \\ C_{n} = \{ z \in C:\Vert y_{n} - z \Vert \le \Vert x_{n} - z \Vert \}, \\ Q_{n} = \{ z \in C: \langle x_{n} - z, x - x_{n} \rangle \ge0 \}, \\ x_{n + 1} = P_{C_{n} \cap Q_{n}}x, \end{array}\displaystyle \right . $$

for every \(n = 0, 1, 2, \ldots\) , where \(A:C \to H\) is an ρ-inverse strongly monotone mapping, \(S:C \to C\) is a nonexpansive mapping, \(0 < a \le\lambda_{n} \le b < 2\rho\) and \(0 \le\alpha_{n} \le c < 1\). They showed that if \(F ( S ) \cap VI ( C,A )\) is nonempty, then the generated sequence \(\{ x_{n} \}\) converges to \(P_{F ( S ) \cap VI ( C,A )}x\). The outer-approximation method was originally introduced by Haugazeau in 1968 and was successfully generalized and extended in recent papers [1114].

In this paper, inspired and motivated by the above work, we introduce the following general hybrid iterative algorithm, which is based on four well-known methods: Mann’s iteration method, the composite method, the outer-approximation method, and the extra-gradient method.

Algorithm 1.1

Let H be a real Hilbert space, C be a nonempty, closed, and convex subset of H, \(A:C \to H\) be a monotone and ρ-Lipschitz-continuous mapping, \(S:C \to C\) be a nonexpansive mapping, \(A_{i},B_{j}\ ( i = 1,2, \ldots,k;j = 1,2, \ldots,l ):C \to C\) be two families of finite maximal monotone mappings. Suppose

$$S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}} = J_{r_{n}}^{A_{k}}J_{r_{n}}^{A_{k - 1}} \cdots J_{r_{n}}^{A_{1}},\qquad T_{r_{n}} = a_{0}I + a_{1}J_{r_{n}}^{B_{1}} + a_{2}J_{r_{n}}^{B_{2}} + \cdots+ a_{l}J_{r_{n}}^{B_{l}}, $$

with \(J_{r_{n}}^{A_{i}} = ( I + r_{n}A_{i} )^{ - 1}\), \(J_{r_{n}}^{B_{j}} = ( I + r_{n}B_{j} )^{ - 1}\), \(\sum_{m = 0}^{l} a_{m} = 1\) and \(a_{m} \in ( 0,1 )\). The sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\), \(\{ z_{n} \}\), and \(\{ w_{n} \}\) are generated by

$$\left \{ \textstyle\begin{array}{@{}l} x_{0} = x \in C, \\ y_{n} = P_{C} ( x_{n} - \lambda_{n}Ax_{n} ), \\ z_{n} = \alpha_{n}x_{n} + ( 1 - \alpha_{n} )SP_{C} ( x_{n} - \lambda_{n}Ay_{n} ), \\ w_{n} = ( 1 - \beta_{n} - \gamma_{n} )z_{n} + \beta_{n}S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} + \gamma_{n}T_{r_{n}}z_{n}, \\ C_{n} = \{ z \in C:\Vert w_{n} - z \Vert \le \Vert x_{n} - z \Vert \}, \\ Q_{n} = \{ z \in C: \langle x_{n} - z, x - x_{n} \rangle \ge 0 \}, \\ x_{n + 1} = P_{C_{n} \cap Q_{n}}x, \end{array}\displaystyle \right . $$

for every \(n = 0, 1, 2, \ldots\) .

Our algorithm is first used for finding the common zeros of two families of finite maximal monotone mappings. By using Algorithm 1.1, we will find a common element of the set of common zeros of two families of finite maximal monotone mappings, the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. We will prove the strong convergence theorem for the proposed algorithm, which extends and improves the corresponding results in the early and recent literature; see, e.g., [3, 4, 6, 10, 14].

Preliminaries

Throughout our paper, let H be a real Hilbert space with the inner product \(\langle\cdot, \cdot \rangle\) and norm \(\Vert \cdot \Vert \), C be a nonempty, closed, and convex subset of H, and I be the identity mapping on H. We write \(x_{n}\stackrel{w}{\longrightarrow} x\) to indicate that the sequence \(\{ x_{n} \}\) converges weakly to x and \(x_{n} \to x\) to indicate that the sequence \(\{ x_{n} \}\) converges strongly to x. For every point \(x \in H\), there exists an unique nearest point in C, denoted by \(P_{C}x\), such that

$$\Vert x - P_{C}x \Vert \le \Vert x - y \Vert ,\quad \forall y \in C. $$

\(P_{C}\) is said to be the metric projection of H onto C. A mapping \(T:C \to C\) is said to be ρ-Lipschitz-continuous if

$$\Vert Tx - Ty \Vert \le\rho \Vert x - y \Vert ,\quad \forall x,y \in C. $$

\(T:C \to C\) is said to be nonexpansive if

$$\Vert Tx - Ty \Vert \le \Vert x - y \Vert , \quad\forall x,y \in C. $$

Obviously, the 1-Lipschitz-continuous mapping is a nonexpansive mapping. It is well known that \(P_{C}\) is nonexpansive.

We use \(F ( T )\) to denote the set of fixed points of T, that is, \(F ( T ) = \{ x \in C:Tx = x \}\). We use \(T^{ - 1}0\) to denote the set of zeros of T, that is, \(T^{ - 1}0 = \{ x \in C:Tx = 0 \}\). We use \(J_{r}^{T} \) (\(r > 0\)) to denote the resolvent operator of T, that is, \(J_{r}^{T} = ( I + rT )^{ - 1}\). As is well known, \(J_{r}^{T}\) is nonexpansive and \(F ( J_{r}^{T} ) = T^{ - 1}0\).

It is well known that a monotone mapping T is maximal if and only if for \(( x,f ) \in H \times H\), \(\langle x - y,f - g \rangle\ge0\) for every \(( y,g ) \in \operatorname{Graph} ( T )\) implies \(f \in Tx\). Next we provide an example to illustrate the concept of maximal monotone mapping. Let \(A:C \to H\) be a monotone, ρ-Lipschitz-continuous mapping and \(N_{C}v\) be the normal cone to C at \(v \in C\), i.e., \(N_{C}v = \{ \omega\in H: \langle v - u, w \rangle\ge0, \forall u \in C \}\). Define

$$Tv = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} Av + N_{C}v,& \mbox{if }v \in C, \\ \phi, &\mbox{if }v \notin C, \end{array}\displaystyle \right . $$

it is well known that in this case T is maximal monotone and \(0 \in Tv\) if and only if \(v \in VI ( C,A )\); see [15]. At the same time, it is well known that H satisfies Opial’s condition [16], i.e., for any sequence \(\{ x_{n} \}\) with \(x_{n}\stackrel{w}{\longrightarrow}x\), the inequality

$$\liminf_{n \to\infty} \Vert x_{n} - x \Vert < \liminf _{n \to \infty} \Vert x_{n} - y \Vert $$

holds for every \(y \in H\) with \(y \ne x\). H also admits Kadec-Klee property, i.e., sequential weak convergence on the unit sphere coincides with norm convergence; see [17].

In order to prove our main results, we need the following lemmas.

Lemma 2.1

[18]

For \(\forall x \in H\) and \(\forall y \in C\), \(P_{C}\) is characterized by the properties:

  1. (i)

    \(\langle x - P_{C}x, P_{C}x - y \rangle\ge0\);

  2. (ii)

    \(\Vert P_{C}x - y \Vert ^{2} + \Vert x - P_{C}x \Vert ^{2} \le \Vert x - y \Vert ^{2}\).

Lemma 2.2

[4]

For \(\forall x \in H\), \(\forall y \in A^{ - 1}0\) and \(r > 0\),

$$\bigl\Vert J_{r}^{A}x - y \bigr\Vert ^{2} + \bigl\Vert x - J_{r}^{A}x \bigr\Vert ^{2} \le \Vert x - y \Vert ^{2}. $$

Lemma 2.3

[4]

Let H be a real Hilbert space, C be a nonempty, closed, and convex subset of H, \(A_{i},B_{j}\ ( i = 1,2, \ldots ,k;j = 1,2, \ldots,l ):C \to C\) be two families of finite maximal monotone mappings such that \(( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 )\) is nonempty. Suppose

$$S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}} = J_{r_{n}}^{A_{k}}J_{r_{n}}^{A_{k - 1}} \cdots J_{r_{n}}^{A_{1}},\qquad T_{r_{n}} = a_{0}I + a_{1}J_{r_{n}}^{B_{1}} + a_{2}J_{r_{n}}^{B_{2}} + \cdots+ a_{l}J_{r_{n}}^{B_{l}}, $$

with \(J_{r_{n}}^{A_{i}} = ( I + r_{n}A_{i} )^{ - 1}\), \(J_{r_{n}}^{B_{j}} = ( I + r_{n}B_{j} )^{ - 1}\), \(\sum_{m = 0}^{l} a_{m} = 1\), \(a_{m} \in ( 0,1 )\) and \(r_{n} > 0\). Then \(S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}:C \to C\) and \(T_{r_{n}}:C \to C\) are nonexpansive.

Lemma 2.4

[4]

Let H, C, \(A_{i}\), \(B_{j}\), \(S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}\), and \(T_{r_{n}}\) be the same as those in Lemma  2.3, suppose \(( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 )\) is nonempty, then \(F ( S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}} ) = \bigcap_{i = 1}^{k}A_{i}^{ - 1}0\) and \(F ( T_{r_{n}} ) = \bigcap_{j = 1}^{l}B_{j}^{ - 1}0\).

Strong convergence theorems

Theorem 3.1

Let H be a real Hilbert space, C be a nonempty, closed, and convex subset of H, \(A:C \to H\) be a monotone and ρ-Lipschitz-continuous mapping, \(S:C \to C\) be a nonexpansive mapping, \(A_{i},B_{j}\ ( i = 1,2, \ldots,k;j = 1,2, \ldots,l ):C \to C\) be two families of finite maximal monotone mappings such that \(D = ( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 ) \cap F ( S ) \cap VI ( C,A )\) is nonempty. Suppose

$$S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}} = J_{r_{n}}^{A_{k}}J_{r_{n}}^{A_{k - 1}} \cdots J_{r_{n}}^{A_{1}}, \qquad T_{r_{n}} = a_{0}I + a_{1}J_{r_{n}}^{B_{1}} + a_{2}J_{r_{n}}^{B_{2}} + \cdots+ a_{l}J_{r_{n}}^{B_{l}}, $$

with \(J_{r_{n}}^{A_{i}} = ( I + r_{n}A_{i} )^{ - 1}\), \(J_{r_{n}}^{B_{j}} = ( I + r_{n}B_{j} )^{ - 1}\), \(\sum_{m = 0}^{l} a_{m} = 1\) and \(a_{m} \in ( 0,1 )\).

If \(\{ \alpha_{n} \}\), \(\{ \beta_{n} \}\), \(\{ \gamma_{n} \}\), \(\{ \lambda_{n} \}\), and \(\{ r_{n} \}\) satisfy the following conditions:

  1. (i)

    \(0 \le\alpha_{n} \le b < 1\);

  2. (ii)

    \(0 < c \le\beta_{n} \le1\);

  3. (iii)

    \(0 < d \le\gamma_{n} \le1\), \(\beta_{n} + \gamma_{n} \le1\);

  4. (iv)

    \(0 < p \le\lambda_{n} \le q < \frac{1}{\rho}\);

  5. (v)

    \(0 < \eta\le r_{n} < + \infty\),

where b, c, d, p, q and η are constants, then the sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\), \(\{ z_{n} \}\), and \(\{ w_{n} \}\) generated by Algorithm 1.1 converge strongly to \(P_{D}x\).

Proof

We will split the proof into five steps.

Step 1. \(D = ( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 ) \cap F ( S ) \cap VI ( C,A ) \subset C_{n} \cap Q_{n}\).

First, we show \(D \subset C_{n}\).

Put \(t_{n} = P_{C} ( x_{n} - \lambda_{n}Ay_{n} )\). Take a fixed \(p\in D\) arbitrarily, then we get \(p \in ( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 )\), \(p \in F ( S )\) and \(p \in VI ( C,A )\). From Lemma 2.1(ii) and the monotonicity of A, we have

$$\begin{aligned} &\Vert t_{n} - p \Vert ^{2} \\ &\quad\le \Vert x_{n} - \lambda_{n}Ay_{n} - p \Vert ^{2} - \Vert x_{n} - \lambda_{n}Ay_{n} - t_{n} \Vert ^{2} \\ &\quad= \Vert x_{n} - p \Vert ^{2} - \Vert x_{n} - t_{n} \Vert ^{2} + 2 \lambda_{n} \langle Ay_{n}, p - t_{n} \rangle \\ &\quad= \Vert x_{n} - p \Vert ^{2} - \Vert x_{n} - t_{n} \Vert ^{2} + 2 \lambda_{n} \bigl( \langle Ay_{n} - Ap, p - y_{n} \rangle+ \langle Ap,p - y_{n} \rangle+ \langle Ay_{n}, y_{n} - t_{n} \rangle \bigr) \\ &\quad\le \Vert x_{n} - p \Vert ^{2} - \Vert x_{n} - t_{n} \Vert ^{2} + 2 \lambda_{n} \langle Ay_{n},y_{n} - t_{n} \rangle \\ &\quad= \Vert x_{n} - p \Vert ^{2} - \bigl( \Vert x_{n} - y_{n} \Vert ^{2} + 2 \langle x_{n} - y_{n},y_{n} - t_{n} \rangle+ \Vert y_{n} - t_{n} \Vert ^{2} \bigr) + 2 \lambda_{n} \langle Ay_{n},y_{n} - t_{n} \rangle \\ &\quad= \Vert x_{n} - p \Vert ^{2} - \bigl( \Vert x_{n} - y_{n} \Vert ^{2} + \Vert y_{n} - t_{n} \Vert ^{2} \bigr) + 2 \langle x_{n} - \lambda_{n}Ay_{n} - y_{n},t_{n} - y_{n} \rangle. \end{aligned}$$
(3.1)

Since \(y_{n} = P_{C} ( x_{n} - \lambda_{n}Ax_{n} )\), A is ρ-Lipschitz-continuous and by Lemma 2.1(i), we have

$$\begin{aligned} & \langle x_{n} - \lambda_{n}Ay_{n} - y_{n},t_{n} - y_{n} \rangle \\ &\quad= \langle x_{n} - \lambda_{n}Ax_{n} - y_{n},t_{n} - y_{n} \rangle+ \langle \lambda_{n}Ax_{n} - \lambda _{n}Ay_{n},t_{n} - y_{n} \rangle \\ &\quad\le \langle\lambda_{n}Ax_{n} - \lambda_{n}Ay_{n},t_{n} - y_{n} \rangle \\ &\quad\le\lambda_{n}\rho \Vert x_{n} - y_{n} \Vert \Vert t_{n} - y_{n} \Vert . \end{aligned}$$
(3.2)

Substituting (3.2) into (3.1) and by condition (iv), we have

$$\begin{aligned} &\Vert t_{n} - p \Vert ^{2} \\ &\quad\le \Vert x_{n} - p \Vert ^{2} - \bigl( \Vert x_{n} - y_{n} \Vert ^{2} + \Vert y_{n} - t_{n} \Vert ^{2} \bigr) + 2 \lambda_{n}\rho \Vert x_{n} - y_{n} \Vert \Vert t_{n} - y_{n} \Vert \\ &\quad\le \Vert x_{n} - p \Vert ^{2} - \bigl( \Vert x_{n} - y_{n} \Vert ^{2} + \Vert y_{n} - t_{n} \Vert ^{2} \bigr) + \bigl( \lambda_{n}^{2}\rho^{2}\Vert x_{n} - y_{n} \Vert ^{2} + \Vert t_{n} - y_{n} \Vert ^{2} \bigr) \\ &\quad\le \Vert x_{n} - p \Vert ^{2} + \bigl( \lambda_{n}^{2}\rho^{2} - 1 \bigr)\Vert x_{n} - y_{n} \Vert ^{2} \\ &\quad\le \Vert x_{n} - p \Vert ^{2}. \end{aligned}$$
(3.3)

From (3.3), Algorithm 1.1, the convexity of \(\Vert \cdot \Vert ^{2}\) and the nonexpansiveness of S, we have

$$\begin{aligned} \Vert z_{n} - p \Vert ^{2} &= \bigl\Vert \alpha_{n}x_{n} + ( 1 - \alpha_{n} )St_{n} - p \bigr\Vert ^{2} \\ &\le\alpha_{n}\Vert x_{n} - p \Vert ^{2} + ( 1 - \alpha_{n} )\Vert St_{n} - p \Vert ^{2} \\ &\le\alpha_{n}\Vert x_{n} - p \Vert ^{2} + ( 1 - \alpha_{n} )\Vert t_{n} - p \Vert ^{2} \\ &\le \Vert x_{n} - p \Vert ^{2} + ( 1 - \alpha_{n} ) \bigl( \lambda_{n}^{2} \rho^{2} - 1 \bigr)\Vert x_{n} - y_{n} \Vert ^{2} \\ &\le \Vert x_{n} - p \Vert ^{2}. \end{aligned}$$
(3.4)

By (3.4), Algorithm 1.1, Lemma 2.3, and Lemma 2.4, we have

$$\begin{aligned} \Vert w_{n} - p \Vert ^{2} &= \bigl\Vert ( 1 - \beta_{n} - \gamma_{n} )z_{n} + \beta_{n}S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} + \gamma_{n}T_{r_{n}}z_{n} - p \bigr\Vert ^{2} \\ &\le ( 1 - \beta_{n} - \gamma_{n} )\Vert z_{n} - p \Vert ^{2} + \beta_{n}\bigl\Vert S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - p \bigr\Vert ^{2} + \gamma_{n}\Vert T_{r_{n}}z_{n} - p \Vert ^{2} \\ &\le ( 1 - \beta_{n} - \gamma_{n} )\Vert z_{n} - p \Vert ^{2} + \beta_{n}\Vert z_{n} - p \Vert ^{2} + \gamma_{n}\Vert z_{n} - p \Vert ^{2} \\ &= \Vert z_{n} - p \Vert ^{2} \\ &\le \Vert x_{n} - p \Vert ^{2} + ( 1 - \alpha_{n} ) \bigl( \lambda_{n}^{2} \rho^{2} - 1 \bigr)\Vert x_{n} - y_{n} \Vert ^{2} \\ &\le \Vert x_{n} - p \Vert ^{2}. \end{aligned}$$
(3.5)

By virtue of the definition of \(C_{n}\) and (3.5), we have \(p \in C_{n}\). So, \(D \subset C_{n}\), for every \(n = 0, 1, 2, \ldots\) .

Next, through the mathematical induction method, we will prove that \(\{ x_{n} \}\) is well defined and \(D \subset C_{n} \cap Q_{n}\), \(n = 0, 1, 2, \ldots \) . For \(n = 0\), \(x_{0} = x \in C\), \(Q_{0} = C\). Hence \(D \subset C_{0} \cap Q_{0}\). Suppose that \(\{ x_{n} \}\) is given and \(D \subset C_{k} \cap Q_{k}\) for some \(k \in N\). Because D is nonempty, \(C_{k} \cap Q_{k}\) is nonempty. It is obvious that \(C_{n}\) is closed and \(Q_{n}\) is closed and convex. As \(C_{n} = \{ z \in C:\Vert z_{n} - x_{n} \Vert ^{2} + 2 \langle z_{n} - x_{n}, x_{n} - z \rangle\le0 \}\), we also have \(C_{n}\) is convex, for every \(n = 0, 1, 2, \ldots \) . Thus, \(C_{k} \cap Q_{k}\) is a nonempty closed convex subset of C, so there exists an unique element \(x_{k + 1} \in C_{k} \cap Q_{k}\) such that \(x_{k + 1} = P_{C_{k} \cap Q_{k}}x\). It is obvious that

$$\langle x_{k + 1} - z,x - x_{k + 1} \rangle\ge0,\quad \forall z \in C_{k} \cap Q_{k}. $$

Since \(D \subset C_{k} \cap Q_{k}\), we have

$$\langle x_{k + 1} - z,x - x_{k + 1} \rangle\ge0,\quad \forall z \in D. $$

That is, \(z \in Q_{k + 1}\). Hence \(D \subset Q_{k + 1}\). Therefore, we get \(D \subset C_{k + 1} \cap Q_{k + 1}\). Thus, \(D \subset C_{n} \cap Q_{n}\), for every \(n = 0, 1, 2, \ldots \) .

Step 2. \(\{ x_{n} \}\), \(\{ y_{n} \}\), \(\{ z_{n} \}\), \(\{ t_{n} \}\) and \(\{ w_{n} \}\) are all bounded.

Let \(p_{0} = P_{D}x\), then \(p_{0} \in D \subset C_{n} \cap Q_{n}\) from step 1. From \(x_{n + 1} = P_{C_{n} \cap Q_{n}}x\) and the definition of the metric projection, we have

$$ \Vert x_{n + 1} - x \Vert \le \Vert p_{0} - x \Vert , $$
(3.6)

for every \(n = 0, 1, 2, \ldots \) . Therefore, \(\{ x_{n} \}\) is bounded. By virtue of (3.3), (3.4) and (3.5), we also obtain \(\{ t_{n} \}\), \(\{ z_{n} \}\) and \(\{ w_{n} \}\) are also bounded.

Again from (3.5), conditions (i) and (v), we have

$$\begin{aligned} \Vert x_{n} - y_{n} \Vert ^{2} &\le\frac{1}{ ( 1 - \alpha_{n} ) ( 1 - \lambda_{n}^{2}\rho^{2} )} \bigl( \Vert x_{n} - p \Vert ^{2} - \Vert w_{n} - p \Vert ^{2} \bigr) \\ &\le\frac{1}{ ( 1 - \alpha_{n} ) ( 1 - \lambda_{n}^{2}\rho^{2} )} \bigl( \Vert x_{n} - p \Vert + \Vert w_{n} - p \Vert \bigr) \bigl( \Vert x_{n} - p \Vert - \Vert w_{n} - p \Vert \bigr) \\ &\le\frac{1}{ ( 1 - \alpha_{n} ) ( 1 - \lambda_{n}^{2}\rho^{2} )} \bigl( \Vert x_{n} - p \Vert + \Vert w_{n} - p \Vert \bigr)\Vert x_{n} - w_{n} \Vert \\ &\le\frac{1}{ ( 1 - b ) ( 1 - q^{2}\rho^{2} )} \bigl( \Vert x_{n} - p \Vert + \Vert w_{n} - p \Vert \bigr)\Vert x_{n} - w_{n} \Vert . \end{aligned}$$
(3.7)

So, \(\{ y_{n} \}\) is bounded.

Step 3. \(\lim_{n \to\infty} \Vert x_{n} - y_{n} \Vert = 0\), \(\lim_{n \to\infty} \Vert x_{n} - z_{n} \Vert = 0\), \(\lim_{n \to\infty} \Vert x_{n} - w_{n} \Vert = 0\) and \(\lim_{n \to\infty} \Vert x_{n} - t_{n} \Vert = 0\).

As \(Q_{n} = \{ z \in C: \langle x_{n} - z,x - x_{n} \rangle \ge0 \}\), we have \(\langle x_{n} - z,x - x_{n} \rangle \ge0\) for all \(z \in Q_{n}\) and by the definition of the metric projection, we have \(x_{n} = P_{Q_{n}}x\). Because of \(x_{n + 1} = P_{C_{n} \cap Q_{n}}x \in C_{n} \cap Q_{n} \subset Q_{n}\) and (3.6), we have

$$\Vert x_{n} - x \Vert \le \Vert x_{n + 1} - x \Vert \le \Vert p_{0} - x \Vert , $$

for every \(n = 0, 1, 2, \ldots \) . Therefore there exists \(\lim_{n \to \infty} \Vert x_{n} - x \Vert \). By Lemma 2.1(ii), we have

$$\Vert x_{n + 1} - x_{n} \Vert ^{2} \le \Vert x_{n + 1} - x \Vert ^{2} - \Vert x_{n} - x \Vert ^{2}, $$

this implies that

$$ \lim_{n \to\infty} \Vert x_{n + 1} - x_{n} \Vert = 0. $$
(3.8)

By \(x_{n + 1} \in C_{n}\) and the definition of \(C_{n}\), we have \(\Vert x_{n + 1} - w_{n} \Vert \le \Vert x_{n + 1} - x_{n} \Vert \) and hence

$$ \Vert w_{n} - x_{n} \Vert \le \Vert w_{n} - x_{n + 1} \Vert + \Vert x_{n + 1} - x_{n} \Vert \le2\Vert x_{n + 1} - x_{n} \Vert . $$
(3.9)

From (3.8) and (3.9), we have

$$ \lim_{n \to\infty} \Vert x_{n} - w_{n} \Vert = 0. $$
(3.10)

From conditions (i), (iv), (3.7) and (3.10), we have

$$ \lim_{n \to\infty} \Vert x_{n} - y_{n} \Vert = 0. $$
(3.11)

In (3.3), using another technique, by condition (iv), we get

$$\begin{aligned} \Vert t_{n} - p \Vert ^{2} &\le \Vert x_{n} - p \Vert ^{2} - \bigl( \Vert x_{n} - y_{n} \Vert ^{2} + \Vert y_{n} - t_{n} \Vert ^{2} \bigr) + 2\lambda_{n}\rho \Vert x_{n} - y_{n} \Vert \Vert t_{n} - y_{n} \Vert \\ &\le \Vert x_{n} - p \Vert ^{2} - \bigl( \Vert x_{n} - y_{n} \Vert ^{2} + \Vert y_{n} - t_{n} \Vert ^{2} \bigr) + \bigl( \Vert x_{n} - y_{n} \Vert ^{2} + \lambda_{n}^{2}\rho^{2}\Vert t_{n} - y_{n} \Vert ^{2} \bigr) \\ &= \Vert x_{n} - p \Vert ^{2} + \bigl( \lambda_{n}^{2}\rho^{2} - 1 \bigr)\Vert t_{n} - y_{n} \Vert ^{2}. \end{aligned}$$
(3.12)

From (3.4) and (3.12), we have

$$\begin{aligned} \Vert z_{n} - p \Vert ^{2} &\le \alpha_{n}\Vert x_{n} - p \Vert ^{2} + ( 1 - \alpha_{n} )\Vert t_{n} - p \Vert ^{2} \\ &\le \Vert x_{n} - p \Vert ^{2} + ( 1 - \alpha_{n} ) \bigl( \lambda_{n}^{2} \rho^{2} - 1 \bigr)\Vert t_{n} - y_{n} \Vert ^{2}. \end{aligned}$$
(3.13)

By (3.5) and (3.13), we have

$$ \Vert w_{n} - p \Vert ^{2} \le \Vert z_{n} - p \Vert ^{2} \le \Vert x_{n} - p \Vert ^{2} + ( 1 - \alpha_{n} ) \bigl( \lambda_{n}^{2} \rho^{2} - 1 \bigr)\Vert t_{n} - y_{n} \Vert ^{2}. $$
(3.14)

So, (3.14) implies that

$$ \begin{aligned}[b] \Vert t_{n} - y_{n} \Vert ^{2} &\le\frac{1}{ ( 1 - \alpha_{n} ) ( 1 - \lambda_{n}^{2}\rho^{2} )} \bigl( \Vert x_{n} - p \Vert ^{2} - \Vert w_{n} - p \Vert ^{2} \bigr)\\ &\le\frac{1}{ ( 1 - \alpha_{n} ) ( 1 - \lambda_{n}^{2}\rho^{2} )} \bigl( \Vert x_{n} - p \Vert + \Vert w_{n} - p \Vert \bigr)\Vert x_{n} - w_{n} \Vert . \end{aligned} $$
(3.15)

Since \(\{ x_{n} \}\), \(\{ w_{n} \}\) are bounded, and by conditions (i), (iv), (3.10), and (3.15), we have

$$ \lim_{n \to\infty} \Vert y_{n} - t_{n} \Vert = 0. $$
(3.16)

By \(\Vert x_{n} - t_{n} \Vert \le \Vert x_{n} - y_{n} \Vert + \Vert y_{n} - t_{n} \Vert \), (3.11) and (3.16), we get

$$ \lim_{n \to\infty} \Vert x_{n} - t_{n} \Vert = 0. $$
(3.17)

From (3.4), (3.5), Lemma 2.2, Lemma 2.3 and Lemma 2.4, we have, for \(\forall p \in D\),

$$\begin{aligned} \Vert w_{n} - p \Vert ^{2} \le{}& ( 1 - \beta_{n} - \gamma_{n} )\Vert z_{n} - p \Vert ^{2} + \beta_{n}\bigl\Vert S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - p \bigr\Vert ^{2} + \gamma_{n}\Vert T_{r_{n}}z_{n} - p \Vert ^{2} \\ \le{}& ( 1 - \beta_{n} )\Vert z_{n} - p \Vert ^{2} + \beta_{n}\bigl\Vert S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - p \bigr\Vert ^{2} \\ \le{}& ( 1 - \beta_{n} )\Vert z_{n} - p \Vert ^{2} + \beta_{n} \bigl( \bigl\Vert S_{r_{n}}^{A_{k - 1} \cdots A_{1}}z_{n} - p \bigr\Vert ^{2} \\ & {}- \bigl\Vert S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - S_{r_{n}}^{A_{k - 1} \cdots A_{1}}z_{n} \bigr\Vert ^{2} \bigr) \\ \le{}&\Vert z_{n} - p \Vert ^{2} - \beta_{n}\bigl\Vert S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - S_{r_{n}}^{A_{k - 1} \cdots A_{1}}z_{n} \bigr\Vert ^{2} \\ \le{}&\Vert x_{n} - p \Vert ^{2} - \beta_{n}\bigl\Vert S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - S_{r_{n}}^{A_{k - 1} \cdots A_{1}}z_{n} \bigr\Vert ^{2}. \end{aligned}$$
(3.18)

Then (3.18) implies that

$$\begin{aligned} \bigl\Vert S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - S_{r_{n}}^{A_{k - 1} \cdots A_{1}}z_{n} \bigr\Vert ^{2} &\le \frac{1}{\beta_{n}} \bigl( \Vert x_{n} - p \Vert ^{2} - \Vert w_{n} - p \Vert ^{2} \bigr) \\ &\le\frac{1}{\beta_{n}} \bigl( \Vert x_{n} - p \Vert + \Vert w_{n} - p \Vert \bigr)\Vert x_{n} - w_{n} \Vert . \end{aligned}$$
(3.19)

From (3.10), (3.19), condition (ii), we have

$$ S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - S_{r_{n}}^{A_{k - 1} \cdots A_{1}}z_{n} \to0, \quad n \to\infty. $$
(3.20)

Again, using a similar technique to (3.18), we have

$$\begin{aligned} \Vert w_{n} - p \Vert ^{2} \le{}& ( 1 - \beta_{n} )\Vert z_{n} - p \Vert ^{2} + \beta_{n}\bigl\Vert S_{r_{n}}^{A_{k - 1} \cdots A_{1}}z_{n} - p \bigr\Vert ^{2} \\ \le{}& ( 1 - \beta_{n} )\Vert z_{n} - p \Vert ^{2} + \beta_{n} \bigl( \bigl\Vert S_{r_{n}}^{A_{k - 2}A_{k - 3} \cdots A_{1}}z_{n} - p \bigr\Vert ^{2} \\ & {} - \bigl\Vert S_{r_{n}}^{A_{k - 1} \cdots A_{1}}z_{n} - S_{r_{n}}^{A_{k - 2}A_{k - 3} \cdots A_{1}}z_{n} \bigr\Vert ^{2} \bigr) \\ \le{}&\Vert z_{n} - p \Vert ^{2} - \beta_{n}\bigl\Vert S_{r_{n}}^{A_{k - 1}A_{k - 2} \cdots A_{1}}z_{n} - S_{r_{n}}^{A_{k - 2}A_{k - 3} \cdots A_{1}}z_{n} \bigr\Vert ^{2} \\ \le{}&\Vert x_{n} - p \Vert ^{2} - \beta_{n}\bigl\Vert S_{r_{n}}^{A_{k - 1}A_{k - 2} \cdots A_{1}}z_{n} - S_{r_{n}}^{A_{k - 2}A_{k - 3} \cdots A_{1}}z_{n} \bigr\Vert ^{2}. \end{aligned}$$

Using the same method in (3.19) and (3.20), we have

$$ S_{r_{n}}^{A_{k - 1}A_{k - 2} \cdots A_{1}}z_{n} - S_{r_{n}}^{A_{k - 2} \cdots A_{1}}z_{n} \to0,\quad n \to\infty. $$
(3.21)

By induction, we have the following results:

$$\begin{aligned}& S_{r_{n}}^{A_{k - 2}A_{k - 3} \cdots A_{1}}z_{n} - S_{r_{n}}^{A_{k - 3} \cdots A_{1}}z_{n} \to0,\quad n \to\infty, \end{aligned}$$
(3.22)
$$\begin{aligned}& \cdots, \\& S_{r_{n}}^{A_{2}A_{1}}z_{n} - S_{r_{n}}^{A_{1}}z_{n} \to0,\quad n \to \infty, \end{aligned}$$
(3.23)
$$\begin{aligned}& S_{r_{n}}^{A_{1}}z_{n} - z_{n} \to0,\quad n \to \infty. \end{aligned}$$
(3.24)

From (3.20)-(3.24), we claim

$$ S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - z_{n} \to0,\quad n \to \infty. $$
(3.25)

By Lemma 2.2, Lemma 2.3, Lemma 2.4, (3.4), and the convexity of \(\Vert \cdot \Vert ^{2}\), \(\forall p \in D\), we have

$$\begin{aligned} \Vert w_{n} - p \Vert ^{2} &\le ( 1 - \beta_{n} - \gamma_{n} )\Vert z_{n} - p \Vert ^{2} + \beta_{n}\bigl\Vert S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - p \bigr\Vert ^{2} + \gamma_{n}\Vert T_{r_{n}}z_{n} - p \Vert ^{2} \\ &\le ( 1 - \gamma_{n} )\Vert z_{n} - p \Vert ^{2} + \gamma_{n}\Vert T_{r_{n}}z_{n} - p \Vert ^{2} \\ &\le ( 1 - \gamma_{n} )\Vert z_{n} - p \Vert ^{2} + \gamma_{n} \Biggl( a_{0}\Vert z_{n} - p \Vert ^{2} + \sum_{j = 1}^{l} a_{j}\bigl\Vert J_{r_{n}}^{B_{j}}z_{n} - p \bigr\Vert ^{2} \Biggr) \\ &\le ( 1 - \gamma_{n} )\Vert z_{n} - p \Vert ^{2} + \gamma_{n} \Biggl( a_{0}\Vert z_{n} - p \Vert ^{2} + \sum_{j = 1}^{l} a_{j} \bigl( \Vert z_{n} - p \Vert ^{2} - \bigl\Vert J_{r_{n}}^{B_{j}}z_{n} - z_{n} \bigr\Vert ^{2} \bigr) \Biggr) \\ &\le \Vert z_{n} - p \Vert ^{2} - \gamma_{n} \sum_{j = 1}^{l} a_{j}\bigl\Vert J_{r_{n}}^{B_{j}}z_{n} - z_{n} \bigr\Vert ^{2} \\ &\le \Vert x_{n} - p \Vert ^{2} - \gamma_{n} \sum_{j = 1}^{l} a_{j}\bigl\Vert J_{r_{n}}^{B_{j}}z_{n} - z_{n} \bigr\Vert ^{2}. \end{aligned}$$
(3.26)

Then (3.26) implies that

$$\begin{aligned} \bigl\Vert J_{r_{n}}^{B_{j}}z_{n} - z_{n} \bigr\Vert ^{2} &\le \frac{1}{\gamma_{n}\sum_{j = 1}^{l} a_{j}} \bigl( \Vert x_{n} - p \Vert ^{2} - \Vert w_{n} - p \Vert ^{2} \bigr) \\ &\le\frac{1}{\gamma_{n}\sum_{j = 1}^{l} a_{j}} \bigl( \Vert x_{n} - p \Vert + \Vert w_{n} - p \Vert \bigr)\Vert x_{n} - w_{n} \Vert . \end{aligned}$$
(3.27)

From the boundedness of \(\{ x_{n} \}\) and \(\{ w_{n} \}\), condition (iii), (3.10) and (3.27), we have

$$ J_{r_{n}}^{B_{j}}z_{n} - z_{n} \to0,\quad n \to \infty. $$
(3.28)

For \(j = 1, 2, \ldots, l\). From (3.28), we have

$$ \lim_{n \to\infty} \Vert T_{r_{n}}z_{n} - z_{n} \Vert = 0. $$
(3.29)

By Algorithm 1.1, we have

$$\Vert w_{n} - z_{n} \Vert = \beta_{n}\bigl\Vert S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} - z_{n} \bigr\Vert + \gamma_{n}\Vert T_{r_{n}}z_{n} - z_{n} \Vert ^{2}. $$

By (3.25) and (3.29), we have

$$ \lim_{n \to\infty} \Vert w_{n} - z_{n} \Vert = 0. $$
(3.30)

By (3.10) and (3.30), we get

$$ \lim_{n \to\infty} \Vert x_{n} - z_{n} \Vert = 0. $$
(3.31)

By (3.17) and (3.31), we get

$$ \lim_{n \to\infty} \Vert z_{n} - t_{n} \Vert = 0. $$
(3.32)

Step 4. \(W ( x_{n} ) \subset D\), where \(W ( x_{n} )\) denotes the set of all the weak limit points of \(\{ x_{n} \}\).

Since \(\{ x_{n} \}\) is bounded, there exists a subsequence of \(\{ x_{n} \}\); for simplicity, we still denote it by \(\{ x_{n} \}\), such that \(x_{n}\stackrel{w}{\longrightarrow}u\) as \(n \to\infty\). In the following, we will prove \(u \in D\).

First, we show \(u \in F ( S )\). Assume \(u \notin F ( S )\), i.e., \(u \ne Su\). Since \(z_{n} = \alpha_{n}x_{n} + ( 1 - \alpha_{n} )St_{n}\), we have

$$\begin{aligned} ( 1 - \alpha_{n} )\Vert St_{n} - t_{n} \Vert &= \bigl\Vert \alpha_{n} ( t_{n} - x_{n} ) + ( z_{n} - t_{n} ) \bigr\Vert \\ &\le\alpha_{n}\Vert t_{n} - x_{n} \Vert + \Vert z_{n} - t_{n} \Vert . \end{aligned}$$
(3.33)

By virtue of condition (i), (3.17), (3.32) and (3.33), we have

$$ \lim_{n \to\infty} \Vert St_{n} - t_{n} \Vert = 0. $$
(3.34)

By (3.17) and \(x_{n}\stackrel{w}{\longrightarrow}u\), we have \(t_{n}\stackrel{w}{\longrightarrow}u\), where \(\{ t_{n} \} \) is a subsequence of \(\{ t_{n} \}\) for simplicity. From (3.34) and Opial’s condition [16], we have

$$\begin{aligned} \liminf_{n \to\infty} \Vert t_{n} - u \Vert &< \liminf_{n \to \infty} \Vert t_{n} - Su \Vert \\ &= \liminf_{n \to\infty} \Vert t_{n} - St_{n} + St_{n} - Su \Vert \\ &\le\liminf_{n \to\infty} \Vert t_{n} - St_{n} \Vert + \liminf_{n \to\infty} \Vert St_{n} - Su \Vert \\ &\le\liminf_{n \to\infty} \Vert t_{n} - u \Vert . \end{aligned}$$

This is a contradiction. So, we obtain \(u \in F ( S )\).

Second, we show \(u \in\bigcap_{i = 1}^{k}A_{i}^{ - 1}0\). From \(x_{n} - z_{n} \to0\) and \(x_{n}\stackrel{w}{\longrightarrow}u\), we have \(z_{n}\stackrel{w}{\longrightarrow}u\), where \(\{ z_{n} \}\) is a subsequence of \(\{ z_{n} \}\) for simplicity. From (3.20)-(3.24), we have \(S_{r_{n}}^{A_{1}}z_{n}\stackrel{w}{\longrightarrow}u\), \(S_{r_{n}}^{A_{2}A_{1}}z_{n}\stackrel{w}{\longrightarrow}u\), \(S_{r_{n}}^{A_{k - 1}A_{k - 2} \cdots A_{1}}z_{n}\stackrel{w}{\longrightarrow}u\) and \(S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n}\stackrel{w}{\longrightarrow}u\).

Since \(( I + r_{n}A_{1} )S_{r_{n}}^{A_{1}}z_{n} = z_{n}\), by (3.24) and condition (v), we have

$$A_{1}S_{r_{n}}^{A_{1}}z_{n} = \frac{1}{r_{n}} \bigl( z_{n} - S_{r_{n}}^{A_{1}}z_{n} \bigr) \to0,\quad n \to\infty. $$

So, \(A_{1}u = 0\) and then \(u \in A_{1}^{ - 1}0\).

Since \(( I + r_{n}A_{2} )S_{r_{n}}^{A_{2}A_{1}}z_{n} = S_{r_{n}}^{A_{1}}z_{n}\), and by (3.23), we have

$$A_{2}S_{r_{n}}^{A_{2}A_{1}}z_{n} = \frac{1}{r_{n}} \bigl( S_{r_{n}}^{A_{1}}z_{n} - S_{r_{n}}^{A_{2}A_{1}}z_{n} \bigr) \to 0,\quad n \to\infty. $$

So, \(A_{2}u = 0\) and then \(u \in A_{2}^{ - 1}0\).

By induction, we have

$$A_{k}S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{2}A_{1}}z_{n} = \frac{1}{r_{n}} \bigl( S_{r_{n}}^{A_{k - 1} \cdots A_{2}A_{1}}z_{n} - S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{2}A_{1}}z_{n} \bigr) \to0,\quad n \to\infty. $$

So, \(A_{k}u = 0\) and then \(u \in A_{k}^{ - 1}0\). Thus, \(u \in\bigcap_{i = 1}^{k}A_{i}^{ - 1}0\).

Third, we show \(u \in\bigcap_{j = 1}^{l}B_{j}^{ - 1}0\). In fact, by virtue of (3.28) and condition (v), we have

$$B_{j}J_{r_{n}}^{B_{j}}z_{n} = \frac{1}{r_{n}} \bigl( z_{n} - J_{r_{n}}^{B_{j}}z_{n} \bigr) \to0,\quad n \to\infty. $$

So, \(B_{j}u = 0\), for \(j = 1, 2, \ldots, l\). We can easily see that \(u \in\bigcap_{j = 1}^{l}B_{j}^{ - 1}0\).

Finally, we show \(u \in VI ( C,A )\). Let

$$Tv = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} Av + N_{C}v,& \mbox{if }v \in C, \\ \phi, &\mbox{if }v \notin C, \end{array}\displaystyle \right . $$

where \(N_{C}v\) is the normal cone to C at \(v \in C\). Let \(G ( T )\) be the graph of T and \(( v,\omega ) \in G ( T )\). So, we have \(\omega\in Tv = Av + N_{C}v\) and hence \(\omega- Av \in N_{C}v\). From the definition of the normal cone and \(t_{n} = P_{C} ( x_{n} - \lambda_{n}Ay_{n} ) \in C\), we have

$$ \langle v - t_{n}, \omega- Av \rangle\ge0. $$
(3.35)

From Lemma 2.1(i), we have

$$\langle x_{n} - \lambda_{n}Ay_{n} - t_{n}, t_{n} - v \rangle \ge 0,\quad \forall v \in C, $$

and hence

$$ \biggl\langle v - t_{n}, \frac{t_{n} - x_{n}}{\lambda_{n}} + Ay_{n} \biggr\rangle \ge0. $$
(3.36)

For simplicity, we assume that \(\{ y_{n} \}\) and \(\{ t_{n} \}\) are also subsequences of \(\{ y_{n} \}\) and \(\{ t_{n} \}\) respectively. Because of \(x_{n} - y_{n} \to0\), \(x_{n} - t_{n} \to0\) and \(x_{n}\stackrel{w}{\longrightarrow}u\), we have \(y_{n}\stackrel{w}{\longrightarrow}u\) and \(xt_{n}\stackrel{w}{\longrightarrow}u\). By the monotonicity of A, (3.35), and (3.36), we obtain

$$\begin{aligned} \langle v - t_{n}, \omega \rangle&\ge \langle v - t_{n}, Av \rangle \\ &\ge \langle v - t_{n}, Av \rangle- \biggl\langle v - t_{n}, \frac{t_{n} - x_{n}}{\lambda_{n}} + Ay_{n} \biggr\rangle \\ &= \langle v - t_{n}, Av - At_{n} \rangle+ \langle v - t_{n}, At_{n} - Ay_{n} \rangle- \biggl\langle v - t_{n}, \frac{t_{n} - x_{n}}{\lambda_{n}} \biggr\rangle \\ &\ge \langle v - t_{n}, At_{n} - Ay_{n} \rangle- \biggl\langle v - t_{n}, \frac{t_{n} - x_{n}}{\lambda_{n}} \biggr\rangle . \end{aligned}$$

Hence, we obtain \(\langle v - u, \omega- 0 \rangle\ge0\) as \(n \to\infty\). Since T is maximal monotone, we have \(0 \in Tu\) and so \(u \in VI ( C,A )\).

Simply stated, \(u \in D\). We get \(W ( x_{n} ) \subset D\).

Step 5. \(x_{n} \to P_{D}x\) as \(n \to\infty\).

Let \(p_{0} = P_{D}x\), \(\forall u \in W ( x_{n} )\), from step 4, we have \(u \in D\). Suppose \(x_{n}\stackrel{w}{\longrightarrow}u\) as \(n \to\infty\), where \(\{ x_{n} \}\) is looked as a subsequence of \(\{ x_{n} \}\) for simplicity. From the definition of the metric projection and the weak lower semi-continuity of \(\Vert \cdot \Vert \), we have

$$\Vert p_{0} - x \Vert \le \Vert u - x \Vert \le\liminf _{n \to \infty} \Vert x_{n} - x \Vert \le\limsup _{n \to\infty} \Vert x_{n} - x \Vert \le \Vert p_{0} - x \Vert . $$

So, we obtain

$$\lim_{n \to\infty} \Vert x_{n} - x \Vert = \Vert u - x \Vert . $$

From \(x_{n} - x\stackrel{w}{\longrightarrow}u - x\) and the Kadec-Klee property, we have \(x_{n} - x \to u - x\) and hence \(x_{n} \to u\).

Since \(x_{n} = P_{Q_{n}}x\), \(p_{0} \in D \subset C_{n} \cap Q_{n} \subset Q_{n}\) and Lemma 2.1(i), we have

$$- \Vert p_{0} - x_{n} \Vert ^{2} = \langle p_{0} - x_{n},x_{n} - x \rangle+ \langle p_{0} - x_{n},x - p_{0} \rangle\ge \langle p_{0} - x_{n},x - p_{0} \rangle. $$

As \(n \to\infty\), we get \(- \Vert p_{0} - u \Vert ^{2} \ge \langle p_{0} - u,x - p_{0} \rangle\ge0\). Hence \(u = p_{0}\). This implies that \(x_{n} \to p_{0} = P_{D}x\) as \(n \to\infty\). From step 3, it is easy to see \(y_{n} \to P_{D}x\), \(z_{n} \to P_{D}x\) and \(w_{n} \to P_{D}x\) as \(n \to\infty\).

This completes the proof. □

From Theorem 3.1, we can get some strong convergence theorems.

Theorem 3.2

Let H be a real Hilbert space, C be a nonempty, closed, and convex subset of H, \(A:C \to H\) be a monotone and ρ-Lipschitz-continuous mapping, \(A_{i},B_{j}\ ( i = 1,2, \ldots,k; j = 1,2, \ldots,l ):C \to C\) be two families of finite maximal monotone mappings such that \(D = ( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 ) \cap VI ( C,A )\) is nonempty. Suppose

$$S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}} = J_{r_{n}}^{A_{k}}J_{r_{n}}^{A_{k - 1}} \cdots J_{r_{n}}^{A_{1}},\qquad T_{r_{n}} = a_{0}I + a_{1}J_{r_{n}}^{B_{1}} + a_{2}J_{r_{n}}^{B_{2}} + \cdots+ a_{l}J_{r_{n}}^{B_{l}}, $$

with \(J_{r_{n}}^{A_{i}} = ( I + r_{n}A_{i} )^{ - 1}\), \(J_{r_{n}}^{B_{j}} = ( I + r_{n}B_{j} )^{ - 1}\), \(\sum_{m = 0}^{l} a_{m} = 1\) and \(a_{m} \in ( 0,1 )\).

The sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\), \(\{ z_{n} \}\) and \(\{ w_{n} \}\) are generated by

$$\left \{ \textstyle\begin{array}{@{}l} x_{0} = x \in C, \\ y_{n} = P_{C} ( x_{n} - \lambda_{n}Ax_{n} ), \\ z_{n} = P_{C} ( x_{n} - \lambda_{n}Ay_{n} ), \\ w_{n} = ( 1 - \beta_{n} - \gamma_{n} )z_{n} + \beta_{n}S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} + \gamma_{n}T_{r_{n}}z_{n}, \\ C_{n} = \{ z \in C:\Vert w_{n} - z \Vert \le \Vert x_{n} - z \Vert \}, \\ Q_{n} = \{ z \in C: \langle x_{n} - z, x - x_{n} \rangle \ge 0 \}, \\ x_{n + 1} = P_{C_{n} \cap Q_{n}}x, \end{array}\displaystyle \right . $$

if \(\{ \beta_{n} \}\), \(\{ \gamma_{n} \}\), \(\{ \lambda_{n} \}\) and \(\{ r_{n} \}\) satisfy the following conditions:

  1. (i)

    \(0 < c \le\beta_{n} \le1\);

  2. (ii)

    \(0 < d \le\gamma_{n} \le1\), \(\beta_{n} + \gamma_{n} \le1\);

  3. (iii)

    \(0 < p \le\lambda_{n} \le q < \frac{1}{\rho}\);

  4. (iv)

    \(0 < \eta\le r_{n} < + \infty\),

for every \(n = 0, 1, 2, \ldots \) , where c, d, p, q and η are constants, then the sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\), \(\{ z_{n} \}\), and \(\{ w_{n} \}\) converge strongly to \(P_{D}x\).

Proof

Put \(S = I\) and \(\alpha_{n} = 0\) for all \(n = 0, 1, 2, \ldots \) . By Theorem 3.1, we get the desired results. □

Theorem 3.3

Let H be a real Hilbert space, C be a nonempty, closed, and convex subset of H, \(S:C \to C\) be a nonexpansive mapping, \(A_{i},B_{j}\ ( i = 1,2, \ldots,k;j = 1,2, \ldots,l ):C \to C\) be two families of finite maximal monotone mappings such that \(D = ( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 ) \cap F ( S )\) is nonempty. Suppose

$$S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}} = J_{r_{n}}^{A_{k}}J_{r_{n}}^{A_{k - 1}} \cdots J_{r_{n}}^{A_{1}},\qquad T_{r_{n}} = a_{0}I + a_{1}J_{r_{n}}^{B_{1}} + a_{2}J_{r_{n}}^{B_{2}} + \cdots+ a_{l}J_{r_{n}}^{B_{l}}, $$

with \(J_{r_{n}}^{A_{i}} = ( I + r_{n}A_{i} )^{ - 1}\), \(J_{r_{n}}^{B_{j}} = ( I + r_{n}B_{j} )^{ - 1}\), \(\sum_{m = 0}^{l} a_{m} = 1\) and \(a_{m} \in ( 0,1 )\).

The sequences \(\{ x_{n} \}\), \(\{ z_{n} \}\), and \(\{ w_{n} \}\) are generated by

$$\left \{ \textstyle\begin{array}{@{}l} x_{0} = x \in C, \\ z_{n} = \alpha_{n}x_{n} + ( 1 - \alpha_{n} )SP_{C}x_{n}, \\ w_{n} = ( 1 - \beta_{n} - \gamma_{n} )z_{n} + \beta_{n}S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} + \gamma_{n}T_{r_{n}}z_{n}, \\ C_{n} = \{ z \in C:\Vert w_{n} - z \Vert \le \Vert x_{n} - z \Vert \}, \\ Q_{n} = \{ z \in C: \langle x_{n} - z, x - x_{n} \rangle \ge 0 \}, \\ x_{n + 1} = P_{C_{n} \cap Q_{n}}x, \end{array}\displaystyle \right . $$

if \(\{ \alpha_{n} \}\), \(\{ \beta_{n} \}\), \(\{ \gamma_{n} \}\), and \(\{ r_{n} \}\) satisfy the following conditions:

  1. (i)

    \(0 \le\alpha_{n} \le b < 1\);

  2. (ii)

    \(0 < c \le\beta_{n} \le1\);

  3. (iii)

    \(0 < d \le\gamma_{n} \le1\), \(\beta_{n} + \gamma_{n} \le1\);

  4. (iv)

    \(0 < \eta\le r_{n} < + \infty\),

for every \(n = 0, 1, 2, \ldots \) , where b, c, d and η are constants, then the sequences \(\{ x_{n} \}\), \(\{ z_{n} \}\) and \(\{ w_{n} \}\) converge strongly to \(P_{D}x\).

Proof

Let \(A = 0\) in Theorem 3.1, we obtain the result. □

Theorem 3.4

Let H be a real Hilbert space, C be a nonempty, closed, and convex subset of H, \(A_{i},B_{j}\ ( i = 1,2, \ldots ,k;j = 1,2, \ldots,l ):C \to C\) be two families of finite maximal monotone mappings such that \(D = ( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 )\) is nonempty. Suppose

$$S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}} = J_{r_{n}}^{A_{k}}J_{r_{n}}^{A_{k - 1}} \cdots J_{r_{n}}^{A_{1}},\qquad T_{r_{n}} = a_{0}I + a_{1}J_{r_{n}}^{B_{1}} + a_{2}J_{r_{n}}^{B_{2}} + \cdots+ a_{l}J_{r_{n}}^{B_{l}}, $$

with \(J_{r_{n}}^{A_{i}} = ( I + r_{n}A_{i} )^{ - 1}\), \(J_{r_{n}}^{B_{j}} = ( I + r_{n}B_{j} )^{ - 1}\), \(\sum_{m = 0}^{l} a_{m} = 1\) and \(a_{m} \in ( 0,1 )\). The sequences \(\{ x_{n} \}\) and \(\{ y_{n} \}\) are generated by

$$\left \{ \textstyle\begin{array}{@{}l} x_{0} = x \in C, \\ y_{n} = ( 1 - \beta_{n} - \gamma_{n} )x_{n} + \beta_{n}S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}x_{n} + \gamma_{n}T_{r_{n}}x_{n}, \\ C_{n} = \{ z \in C:\Vert y_{n} - z \Vert \le \Vert x_{n} - z \Vert \}, \\ Q_{n} = \{ z \in C: \langle x_{n} - z, x - x_{n} \rangle \ge 0 \}, \\ x_{n + 1} = P_{C_{n} \cap Q_{n}}x, \end{array}\displaystyle \right . $$

if \(\{ \beta_{n} \}\), \(\{ \gamma_{n} \}\), and \(\{ r_{n} \}\) satisfy the following conditions:

  1. (i)

    \(0 < c \le\beta_{n} \le1\);

  2. (ii)

    \(0 < d \le\gamma_{n} \le1\), \(\beta_{n} + \gamma_{n} \le1\);

  3. (iii)

    \(0 < \eta\le r_{n} < + \infty\),

for every \(n = 0, 1, 2, \ldots \) , where c, d, and η are constants, then the sequences \(\{ x_{n} \}\) and \(\{ y_{n} \}\) converge strongly to \(P_{D}x\).

Proof

Let \(A = 0\), \(S = I\) and \(\alpha_{n} = 0\) for all \(n = 0, 1, 2, \ldots \) in Theorem 3.1, we obtain the result.

If \(A_{i},B_{j}\ ( i = 1,2, \ldots,k;j = 1,2, \ldots,l ):H \to 2^{H}\) are set-valued mappings, the resolvent operator \(J_{r}^{T} \) (\(r > 0 \)) of T, is defined by \(J_{r}^{T}x = \{ z \in H:x \in z + rTz \}= ( I + rT )^{ - 1}x\), \(\forall x \in H\), where I denote the identity mapping on H. As is well known, \(J_{r}^{T}:H \to H\) is a single valued mapping. We have the following strong convergence theorem. □

Theorem 3.5

Let H be a real Hilbert space, \(A:H \to H\) be a monotone and ρ-Lipschitz-continuous mapping, \(S:H \to H\) be a nonexpansive mapping, \(A_{i},B_{j}\ ( i = 1,2, \ldots,k; j = 1,2, \ldots,l ): H \to2^{H}\) be two families of finite set-valued maximal monotone mappings such that \(D = ( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 ) \cap F ( S ) \cap A^{ - 1}0\) is nonempty. Suppose

$$S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}} = J_{r_{n}}^{A_{k}}J_{r_{n}}^{A_{k - 1}} \cdots J_{r_{n}}^{A_{1}},\qquad T_{r_{n}} = a_{0}I + a_{1}J_{r_{n}}^{B_{1}} + a_{2}J_{r_{n}}^{B_{2}} + \cdots+ a_{l}J_{r_{n}}^{B_{l}}, $$

with \(J_{r_{n}}^{A_{i}} = ( I + r_{n}A_{i} )^{ - 1}\), \(J_{r_{n}}^{B_{j}} = ( I + r_{n}B_{j} )^{ - 1}\), \(\sum_{m = 0}^{l} a_{m} = 1\) and \(a_{m} \in ( 0,1 )\).

The sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\) and \(\{ z_{n} \}\) are generated by

$$\left \{ \textstyle\begin{array}{@{}l} x_{0} = x \in H, \\ y_{n} = \alpha_{n}x_{n} + ( 1 - \alpha_{n} )S ( x_{n} - \lambda_{n}A ( x_{n} - \lambda_{n}Ax_{n} ) ), \\ z_{n} = ( 1 - \beta_{n} - \gamma_{n} )y_{n} + \beta_{n}S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}y_{n} + \gamma_{n}T_{r_{n}}y_{n}, \\ C_{n} = \{ z \in C:\Vert z_{n} - z \Vert \le \Vert x_{n} - z \Vert \}, \\ Q_{n} = \{ z \in C: \langle x_{n} - z, x - x_{n} \rangle \ge 0 \}, \\ x_{n + 1} = P_{C_{n} \cap Q_{n}}x, \end{array}\displaystyle \right . $$

if \(\{ \alpha_{n} \}\), \(\{ \beta_{n} \}\), \(\{ \gamma_{n} \}\), \(\{ \lambda_{n} \}\), and \(\{ r_{n} \}\) satisfy the following conditions in Theorem  3.1, then the sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\) and \(\{ z_{n} \}\) converge strongly to \(P_{D}x\).

Proof

As is well known, \(P_{H} = I\) and \(VI ( H,A ) = A^{ - 1}0\). Using the similar arguments to those in the proof of Theorem 3.1, we get the desired result immediately. □

Remark 3.1

Theorems 3.1-3.5 greatly improve and extend the previous work in the following respects:

  1. (1)

    We study the problem of finding a common element of the set of common zeros of two families of finite maximal monotone mappings, the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping, i.e., \(( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 ) \cap F ( S ) \cap VI ( C,A )\). The problems of finding common elements of [7], Theorem 2.2 and [17], Theorems 3.1 and 4.1 are all special cases of our problem.

  2. (2)

    The hybrid iterative Algorithm 1.1 greatly generalizes and extends some corresponding algorithms in [4, 6, 10, 14, 19]. It is first used for finding common zeros of two families of finite maximal monotone mappings. The method of proof is also different from the earlier ones.

  3. (3)

    All parameter sequences \(\{ \alpha_{n} \}\), \(\{ \beta_{n} \}\), \(\{ \gamma_{n} \}\), \(\{ \lambda_{n} \}\) and \(\{ r_{n} \}\) satisfy weaker restrictions in our theorems than those in the theorems of [3, 4]. For example, \(0 \le\alpha_{n} \le b < 1\), neither \(\sum_{n = 1}^{\infty} \vert \alpha_{n + 1} - \alpha_{n} \vert < + \infty\), nor \(\alpha_{n} \to0\) as \(n \to\infty\).

Applications

In this section, we will give several examples from the practice with numerical analysis with their new algorithms.

Let H be a real Hilbert space, \(\varphi:H \to R \cup \{ + \infty \}\) be a proper, convex, lower semicontinuous functional. The subdifferentiable operator of φ, denoted by \(\partial\varphi: H \to2^{H}\), is defined at \(x \in H\) by

$$\partial\varphi ( x ) = \bigl\{ u \in H: \varphi ( y ) \ge \varphi ( x ) + \langle y - x,u \rangle, \forall y \in H \bigr\} . $$

For each \(x \in H\), \(\partial\varphi ( x )\) is called the subgradient of φ at x. Using different methods, Rockafellar [20] and Alves and Svaiter [21] proved that the subdifferentiable operator is a maximal monotone mapping, respectively. Thus, from Theorem 3.5, we get the following result immediately.

Theorem 4.1

Let H be a real Hilbert space, \(A:H \to H\) be a monotone and ρ-Lipschitz-continuous mapping, \(S:H \to H\) be a nonexpansive mapping, \(\varphi_{1i},\varphi_{2j}\ ( i = 1,2, \ldots,k; j = 1,2, \ldots,l ): H \to R \cup \{ + \infty \}\) be two finite families of proper, convex, lower semicontinuous functionals, \(\partial \varphi_{1i}\) and \(\partial\varphi_{2j}\) be their subdifferentiable operators, respectively, such that \(D = ( \bigcap_{i = 1}^{k}\partial\varphi_{1i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}\partial\varphi_{2j}^{ - 1}0 ) \cap A^{ - 1}0\) is nonempty. Suppose

$$S_{r_{n}}^{\partial\varphi_{1k}\partial\varphi_{1,k - 1} \cdots\partial\varphi_{11}} = J_{r_{n}}^{\partial\varphi_{1k}}J_{r_{n}}^{\partial\varphi_{1,k - 1}} \cdots J_{r_{n}}^{\partial\varphi_{11}},\qquad T_{r_{n}} = a_{0}I + a_{1}J_{r_{n}}^{\partial\varphi_{21}} + a_{2}J_{r_{n}}^{\partial\varphi_{22}} + \cdots+ a_{l}J_{r_{n}}^{\partial \varphi_{2l}}, $$

with \(J_{r_{n}}^{\partial\varphi_{i}} = ( I + r_{n}\partial\varphi_{i} )^{ - 1}\), \(\sum_{m = 0}^{l} a_{m} = 1\) and \(a_{m} \in ( 0,1 )\). The sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\) and \(\{ z_{n} \}\) are generated by

$$\left \{ \textstyle\begin{array}{@{}l} x_{0} = x \in H, \\ y_{n} = \alpha_{n}x_{n} + ( 1 - \alpha_{n} )S ( x_{n} - \lambda_{n}A ( x_{n} - \lambda_{n}Ax_{n} ) ), \\ z_{n} = ( 1 - \beta_{n} - \gamma_{n} )y_{n} + \beta_{n}S_{r_{n}}^{\partial\varphi_{1k}\partial\varphi_{1,k - 1} \cdots \partial\varphi_{11}}y_{n} + \gamma_{n}T_{r_{n}}y_{n}, \\ C_{n} = \{ z \in C:\Vert z_{n} - z \Vert \le \Vert x_{n} - z \Vert \}, \\ Q_{n} = \{ z \in C: \langle x_{n} - z, x - x_{n} \rangle \ge 0 \}, \\ x_{n + 1} = P_{C_{n} \cap Q_{n}}x, \end{array}\displaystyle \right . $$

if \(\{ \alpha_{n} \}\), \(\{ \beta_{n} \}\), \(\{ \gamma_{n} \}\), \(\{ \lambda_{n} \}\), and \(\{ r_{n} \}\) satisfy the conditions in Theorem  3.1, then the sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\), and \(\{ z_{n} \}\) converge strongly to \(P_{D}x\).

We also know a mapping \(T:C \to C\) is called pseudocontractive if

$$\Vert Tx - Ty \Vert ^{2} \le \Vert x - y \Vert ^{2} + \bigl\Vert ( I - T )x - ( I - T )y \bigr\Vert ^{2}, \quad\forall x,y \in C. $$

It is equivalent to the following definition:

$$\langle Tx - Ty,x - y \rangle\le \Vert x - y \Vert ^{2},\quad \forall x,y \in C. $$

If T is a pseudocontractive, ρ-Lipschitz-continuous mapping, then \(A = I - T\) is monotone and \(( \rho+ 1 )\)-Lipschitz-continuous mapping, and \(F ( T ) = VI ( C,A )\), more details see [14, 19]. So, by Theorem 3.1, we have the following result immediately.

Theorem 4.2

Let H be a real Hilbert space, C be a nonempty, closed, and convex subset of H, \(T:C \to C\) be a pseudocontractive and ρ-Lipschitz-continuous mapping, \(S:C \to C\) be a nonexpansive mapping, \(A_{i},B_{j}\ ( i = 1,2, \ldots,k;j = 1,2, \ldots,l ):C \to C\) be two families of finite maximal monotone mappings such that \(D = ( \bigcap_{i = 1}^{k}A_{i}^{ - 1}0 ) \cap ( \bigcap_{j = 1}^{l}B_{j}^{ - 1}0 ) \cap F ( S ) \cap F ( T )\) is nonempty. Suppose

$$S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}} = J_{r_{n}}^{A_{k}}J_{r_{n}}^{A_{k - 1}} \cdots J_{r_{n}}^{A_{1}},\qquad T_{r_{n}} = a_{0}I + a_{1}J_{r_{n}}^{B_{1}} + a_{2}J_{r_{n}}^{B_{2}} + \cdots+ a_{l}J_{r_{n}}^{B_{l}}, $$

with \(J_{r_{n}}^{A_{i}} = ( I + r_{n}A_{i} )^{ - 1}\), \(J_{r_{n}}^{B_{j}} = ( I + r_{n}B_{j} )^{ - 1}\), \(\sum_{m = 0}^{l} a_{m} = 1\) and \(a_{m} \in ( 0,1 )\).

The sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\), \(\{ z_{n} \}\), and \(\{ w_{n} \}\) are generated by

$$\left \{ \textstyle\begin{array}{@{}l} x_{0} = x \in C, \\ y_{n} = P_{C} ( x_{n} - \lambda_{n} ( x_{n} - Tx_{n} ) ), \\ z_{n} = \alpha_{n}x_{n} + ( 1 - \alpha_{n} )SP_{C} ( x_{n} - \lambda_{n} ( y_{n} - Ty_{n} ) ), \\ w_{n} = ( 1 - \beta_{n} - \gamma_{n} )z_{n} + \beta_{n}S_{r_{n}}^{A_{k}A_{k - 1} \cdots A_{1}}z_{n} + \gamma_{n}T_{r_{n}}z_{n}, \\ C_{n} = \{ z \in C:\Vert w_{n} - z \Vert \le \Vert x_{n} - z \Vert \}, \\ Q_{n} = \{ z \in C: \langle x_{n} - z, x - x_{n} \rangle \ge 0 \}, \\ x_{n + 1} = P_{C_{n} \cap Q_{n}}x, \end{array}\displaystyle \right . $$

for every \(n = 0, 1, 2, \ldots \) . If \(\{ \alpha_{n} \}\), \(\{ \beta_{n} \}\), \(\{ \gamma_{n} \}\), \(\{ \lambda_{n} \}\), and \(\{ r_{n} \}\) satisfy the following conditions:

  1. (i)

    \(0 \le\alpha_{n} \le b < 1\);

  2. (ii)

    \(0 < c \le\beta_{n} \le1\);

  3. (iii)

    \(0 < d \le\gamma_{n} \le1\), \(\beta_{n} + \gamma_{n} \le1\);

  4. (iv)

    \(0 < p \le\lambda_{n} \le q < \frac{1}{\rho+ 1}\);

  5. (v)

    \(0 < \eta\le r_{n} < + \infty\),

where b, c, d, p, q and η are constants, then the sequences \(\{ x_{n} \}\), \(\{ y_{n} \}\), \(\{ z_{n} \}\), and \(\{ w_{n} \}\) converge strongly to \(P_{D}x\).

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Acknowledgements

This work was supported by the Innovation Program of Shanghai Municipal Education Commission (15ZZ068), the Ph.D. Program Foundation of Ministry of Education of China (20123127110002) and the Program for Shanghai Outstanding Academic Leaders in Shanghai City (15XD1503100).

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Correspondence to Lu-Chuan Ceng.

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Qiu, YQ., Ceng, LC., Chen, JZ. et al. Hybrid iterative algorithms for two families of finite maximal monotone mappings. Fixed Point Theory Appl 2015, 180 (2015). https://doi.org/10.1186/s13663-015-0428-9

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  • DOI: https://doi.org/10.1186/s13663-015-0428-9

MSC

  • 47H05
  • 47H10
  • 47J25
  • 49J40

Keywords

  • hybrid iterative algorithms
  • maximal monotone mappings
  • fixed points
  • strong convergence