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Split equality fixed point problem for quasi-pseudo-contractive mappings with applications

Fixed Point Theory and Applications20152015:208

https://doi.org/10.1186/s13663-015-0458-3

Received: 1 September 2015

Accepted: 5 November 2015

Published: 17 November 2015

Abstract

In this paper, we consider a split equality fixed point problem for quasi-pseudo-contractive mappings which includes split feasibility problem, split equality problem, split fixed point problem etc., as special cases. A unified framework for the study of this kind of problems and operators is provided. The results presented in the paper extend and improve many recent results.

Keywords

split equality fixed point problemquasi-pseudo-contractive mappingquasi-nonexpansive mappingdirected mappingdemicontractive mapping

MSC

49J4049J5247J20

1 Introduction

Let C and Q be nonempty closed and convex subsets of the real Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively. The split feasibility problem \((SFP)\) is formulated as:
$$ \mbox{to find } x^{*}\in C \mbox{ such that } Ax^{*}\in Q, $$
(1.1)
where \(A:H_{1}\rightarrow H_{2}\) is a bounded linear operator. In 1994, Censor and Elfving [1] first introduced the SFP in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. It has been found that the SFP can also be used in various disciplines such as image restoration, computer tomography, and radiation therapy treatment planning [35]. The SFP in an infinite-dimensional real Hilbert space can be found in [2, 4, 610].
Recently, Moudafi [1113] introduced the following split equality feasibility problem (SEFP):
$$ \mbox{to find } x\in C, y\in Q \mbox{ such that } Ax=By, $$
(1.2)
where \(A:H_{1}\rightarrow H_{3}\) and \(B:H_{2}\rightarrow H_{3}\) are two bounded linear operators. Obviously, if \(B=I\) (identity mapping on \(H_{2}\)) and \(H_{3}=H_{2}\), then (1.2) reduces to (1.1). The kind of split equality feasibility problems (1.2) allows asymmetric and partial relations between the variables x and y. The interest is to cover many situations, such as decomposition methods for PDEs, applications in game theory and intensity-modulated radiation therapy.
In order to solve split equality feasibility problem (1.2), Moudafi [11] introduced the following simultaneous iterative method:
$$ \left \{ \textstyle\begin{array}{@{}l} x_{k+1} =P_{C}(x_{k}-\gamma A^{*}(Ax_{k}-By_{k})),\\ y_{k+1} = P_{Q}(y_{k}+\beta B^{*}(Ax_{k+1}-By_{k})), \end{array}\displaystyle \right . $$
(1.3)
and under suitable conditions he proved the weak convergence of the sequence \(\{(x_{n}, y_{n})\}\) to a solution of (1.2) in Hilbert spaces.
In order to avoid using the projection, recently, Moudafi [13] introduced and studied the following problem: Let \(T : H_{1} \to H_{1}\) and \(S : H_{2} \to H_{2}\) be nonlinear operators such that \(\operatorname{Fix}(T) \neq \emptyset\) and \(\operatorname{Fix}(S) \neq \emptyset\), where \(\operatorname{Fix}(T)\) and \(\operatorname{Fix}(S)\) denote the sets of fixed points of T and S, respectively. If \(C = \operatorname{Fix}(T)\) and \(Q = \operatorname{Fix}(S)\), then split equality problem (1.2) reduces to
$$ \mbox{find } x \in \operatorname{Fix}(T) \mbox{ and } y \in \operatorname{Fix}(S) \mbox{ such that } Ax = By, $$
(1.4)
which is called a split equality fixed point problem (in short, SEFPP).

Denote by Γ the solution set of split equality fixed point problem (1.4).

Recently Moudafi [13] proposed the following iterative algorithm for finding a solution of SEFPP (1.4):
$$ \left \{ \textstyle\begin{array}{@{}l} x_{n+1} = T(x_{n} - \gamma_{n} A^{*}(Ax_{n} - By_{n})),\\ y_{n+1} = S(y_{n} + \beta_{n} B^{*}(Ax_{n+1} - By_{n})). \end{array}\displaystyle \right . $$
(1.5)
He also studied the weak convergence of the sequences generated by scheme (1.5) under the condition that T and S are firmly quasi-nonexpansive mappings. Very recently, Che and Li [14] proposed the following iterative algorithm for finding a solution of SEFPP (1.4):
$$ \left \{ \textstyle\begin{array}{@{}l} u_{n} = x_{n} - \gamma_{n} A^{*}(Ax_{n} - By_{n}),\\ x_{n+1} = \alpha_{n} x_{n} + (1 - \alpha_{n})Tu_{n},\\ v_{n} = y_{n} + \gamma_{n} B^{*}(Ax_{n} - By_{n}),\\ y_{n+1} = \alpha_{n} y_{n} + (1 - \alpha_{n})Sv_{n}. \end{array}\displaystyle \right . $$
(1.6)
They also established the weak convergence of the scheme (1.6) under the condition that the operators T and S are quasi-nonexpansive mappings.
The purpose of this paper is two-fold. First, we will consider split equality fixed point problem (1.4) for the class of quasi-pseudo-contractive mappings which is more general than the classes of quasi-nonexpansive mappings, directed mappings, and demicontractive mappings. Second, we modify the iterative scheme (1.6) and propose the following iterative algorithms with weak convergence without using the projection:
$$ \left \{ \textstyle\begin{array}{@{}l} u_{n} = x_{n} - \gamma_{n} A^{*}(Ax_{n} - By_{n}),\\ x_{n+1} = \alpha_{n} x_{n} + (1 - \alpha_{n})((1-\xi)I + \xi T((1-\eta)I + \eta T))u_{n},\\ v_{n} = y_{n} + \gamma_{n} B^{*}(Ax_{n} - By_{n}),\\ y_{n+1} = \alpha_{n} y_{n} + (1 - \alpha_{n})((1-\xi)I + \xi S((1-\eta)I + \eta S))v_{n}. \end{array}\displaystyle \right . $$
(1.7)
Our results provide a unified framework for the study of this kind of problems and this class of operators.

2 Preliminaries

In this section, we collect some definitions, notations, and conclusions, which will be needed in proving our main results.

Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and \(T : C \to C\) be a nonlinear mapping.

Definition 2.1

\(T : C \to C\) is said to be:
  1. (i)

    Nonexpansive if \(\|Tx -Ty\| \le\|x - y\|\) \(\forall x, y \in C\).

     
  2. (ii)
    Quasi-nonexpansive if \(\operatorname{Fix}(T) \neq \emptyset\) and
    $$\bigl\| Tx - x^{*}\bigr\| \le\bigl\| x - x^{*}\bigr\| \quad\forall x \in C \mbox{ and } x^{*} \in \operatorname{Fix}(T). $$
     
  3. (iii)
    Firmly nonexpansive if
    $$\|Tx -Ty\|^{2} \le\|x - y\|^{2} - \bigl\| (I -T)x - (I-T)y \bigr\| ^{2} \quad \forall x, y \in C. $$
     
  4. (iv)
    Firmly quasi-nonexpansive if \(\operatorname{Fix}(T) \neq \emptyset\) and
    $$\bigl\| Tx - x^{*}\bigr\| ^{2} \le\bigl\| x - x^{*}\bigr\| ^{2} - \bigl\| (I- T)x \bigr\| ^{2} \quad \forall x \in C \mbox{ and } x^{*} \in \operatorname{Fix}(T). $$
     
  5. (v)
    Strictly pseudo-contractive if there exists \(k \in[0, 1)\) such that
    $$\|Tx - Ty\|^{2} \le\|x - y\|^{2} + k\bigl\| (I - T)x - (I - T)y \bigr\| ^{2} \quad \forall x, y \in C. $$
     
  6. (vi)

    Directed if \(\operatorname{Fix}(T) \neq \emptyset\) and \(\langle Tx - x^{*}, Tx -x \rangle\le0\) \(\forall x \in C\) and \(x^{*} \in \operatorname{Fix}(T)\).

     
  7. (vii)
    Demicontractive if \(\operatorname{Fix}(T) \neq \emptyset\) and there exists \(k \in[0, 1)\) such that
    $$\bigl\| Tx - x^{*}\bigr\| ^{2} \le\bigl\| x - x^{*}\bigr\| ^{2} + k \|Tx - x \|^{2} \quad \forall x \in C \mbox{ and } x^{*} \in \operatorname{Fix}(T). $$
     

Remark 2.2

As pointed out by Bauschke and Combettes [15], \(T : C \to C\) is directed if and only if
$$\bigl\| Tx - x^{*}\bigr\| ^{2} \le\bigl\| x - x^{*}\bigr\| ^{2} - \|Tx - x \|^{2} \quad \forall x \in C \mbox{ and } x^{*} \in F(T). $$
That is to say that the class of directed mappings coincides with that of firmly quasi-nonexpansive mappings.

Remark 2.3

From the above definitions, we note that the class of demicontractive mappings is fundamental; it includes many kinds of nonlinear mappings such as the directed mappings, the quasi-nonexpansive mappings, and the strictly pseudo-contractive mappings with fixed points as special cases.

Definition 2.4

An operator \(T : C \to C\) is said to be pseudo-contractive if
$$\langle Tx - Ty, x - y\rangle\le\|x - y\|^{2} \quad\forall x, y \in C. $$
The interest of pseudo-contractive operators lies in their connection with monotone mappings, namely, T is a pseudo-contraction if and only if \(I - T\) is a monotone mapping. It is well known that T is pseudo-contractive if and only if
$$\|Tx - Ty\|^{2} \le\|x - y\|^{2} + \bigl\| (I - T)x - (I - T)y \bigr\| ^{2} \quad \forall x, y \in C. $$

Definition 2.5

An operator \(T : C \to C\) is said to be quasi-pseudo-contractive if \(\operatorname{Fix}(T) \neq \emptyset\) and
$$\bigl\| Tx - x^{*}\bigr\| ^{2} \le\bigl\| x - x^{*}\bigr\| ^{2} + \|T x - x \|^{2} \quad \forall x \in C \mbox{ and } x^{*} \in F(T). $$
It is obvious that the class of quasi-pseudo-contractive mappings includes the class of demicontractive mappings.

Definition 2.6

(1) A mapping \(T : C \to C\) is said to be demiclosed at 0 if, for any sequence \(\{x_{n}\} \subset C\) which converges weakly to x and with \(\|x_{n} - T(x_{n})\| \to0\), \(T(x) = x\).

(2) A mapping \(T : H \to H\) is said to be semi-compact if, for any bounded sequence \(\{x_{n}\} \subset H\) with \(\|x_{n} - Tx_{n}\| \to0\), there exists a subsequence \(\{x_{n_{i}}\} \subset\{x_{n}\}\) such that \(\{x_{n_{i}}\}\) converges strongly to some point \(x \in H\).

Lemma 2.7

Let H be a real Hilbert space. For any \(x, y \in H\), the following conclusions hold:
$$\begin{aligned}& \bigl\| tx + (1 - t)y\bigr\| ^{2} = t\|x\|^{2} + (1 - t)\|y \|^{2} - t(1 - t)\|x - y\|^{2}, \quad t \in[0; 1]; \end{aligned}$$
(2.1)
$$\begin{aligned}& \|x + y\|^{2} \le\|x\|^{2} + 2 \langle y, x + y \rangle. \end{aligned}$$
(2.2)
Recall that a Banach space X is said to satisfy Opial’s condition, if for any sequence \(\{x_{n}\}\) in X which converges weakly to \(x^{*}\),
$$\limsup_{n\to\infty} \bigl\| x_{n} - x^{*}\bigr\| < \limsup _{n \to\infty} \|x_{n} - y\| \quad \forall y \in X \mbox{ with } y \neq x^{*}. $$
It is well known that every Hilbert space satisfies the Opial condition.

Lemma 2.8

Let \(\{a_{n}\}\) be a sequence of nonnegative real numbers such that
$$a_{n+1} \le(1 - \gamma_{n})a_{n} + \delta_{n} \quad \forall n \ge1, $$
where \(\{\gamma_{n}\}\) is a sequence in \((0, 1)\) and \(\{\delta_{n}\}\) is a sequence such that
  1. (1)

    \(\sum_{n=1}^{\infty}\gamma_{n} = \infty\);

     
  2. (2)

    \(\limsup_{n \to\infty} \frac{\delta_{n}}{\gamma_{n}} \le0 \) or \(\sum_{n=1}^{\infty}|\delta_{n}| < \infty\).

     
Then \(\lim_{n\to\infty} a_{n} = 0\).

Lemma 2.9

Let H be a real Hilbert space and \(T : H \to H\) be a L-Lipschitzian mapping with \(L \ge1\). Denote
$$ K: = (1 - \xi)I + \xi T\bigl((1 - \eta)I + \eta T\bigr). $$
(2.3)
If \(0 < \xi< \eta < \frac{1}{1 + \sqrt{1 + L^{2}}}\), then the following conclusions hold:
  1. (1)

    \(\operatorname{Fix}(T) = \operatorname{Fix}(T((1 - \eta)I + \eta T)) = \operatorname{Fix}(K)\).

     
  2. (2)

    If T is demiclosed at 0, then K is also demiclosed at 0.

     
  3. (3)
    In addition, if \(T : H \to H\) is quasi-pseudo-contractive, then the mapping K is quasi-nonexpansive, that is,
    $$\bigl\| Kx - u^{*}\bigr\| \le\bigl\| x - u^{*}\bigr\| \quad\forall x \in H \textit{ and } u^{*} \in \operatorname{Fix}(T) = \operatorname{Fix}(K). $$
     

Proof

(1) If \(x^{*} \in \operatorname{Fix}(T)\), it is obvious that \(x^{*} \in \operatorname{Fix}(T((1 - \eta)I + \eta T))\).

Conversely, if \(x^{*} \in \operatorname{Fix}(T((1 - \eta)I + \eta T))\), i.e., \(x^{*} = T((1 - \eta)x^{*} + \eta Tx^{*})\), letting \(U = (1 - \eta)I + \eta T\), then \(TU x^{*} = x^{*}\). Put \(Ux^{*} = y^{*}\). Then \(Ty^{*} = x^{*}\). Now we prove that \(x^{*} = y^{*}\). In fact, we have
$$\begin{aligned} \bigl\| x^{*} - y^{*}\bigr\| & = \bigl\| x^{*} - Ux^{*}\bigr\| = \bigl\| x^{*} - \bigl((1 - \eta)I + \eta T\bigr)x^{*}\bigr\| \\ & = \eta\bigl\| x^{*} - Tx^{*}\bigr\| = \eta\bigl\| Ty^{*} - Tx^{*}\bigr\| \le L \eta\bigl\| y^{*} - x^{*}\bigr\| . \end{aligned}$$
Since \(0 < L\eta< 1\), we have \(x^{*} = y^{*}\), i.e., \(x^{*} \in \operatorname{Fix}(T)\). This shows that \(\operatorname{Fix}(T) = \operatorname{Fix}(T((1 - \eta)I + \eta T))\).

It is obvious that \(x \in \operatorname{Fix}(K)\) if and only if \(x \in \operatorname{Fix}(T((1 - \eta )I + \eta T))\).

The conclusion (1) is proved.

(2) For any sequence \(\{x_{n}\} \subset H\) satisfying \(x_{n} \rightharpoonup x^{*}\) and \(\|x_{n} - Kx_{n}\| \to0\). Next we show that \(x^{*} \in \operatorname{Fix}(K)\). From conclusion (1), we only need to prove that \(x^{*} \in \operatorname{Fix}(T)\). In fact, since T is L-Lipschizian, we have
$$\begin{aligned} \|x_{n} - Tx_{n}\| & \le\bigl\| x_{n} - T\bigl((1 - \eta)I + \eta T\bigr)x_{n} \bigr\| + \bigl\| T\bigl((1 - \eta)I + \eta T \bigr)x_{n} - Tx_{n}\bigr\| \\ & \le\frac{1}{\xi}\bigl\| x_{n} - (1-\xi)x_{n} - \xi T\bigl((1 - \eta)I + \eta T\bigr)x_{n}\bigr\| + L\eta\|x_{n} - Tx_{n}\| \\ & = \frac{1}{\xi}\|x_{n} - K x_{n}\| + L\eta \|x_{n} - Tx_{n}\|. \end{aligned}$$
Simplifying it, we have
$$ \|x_{n} - Tx_{n}\| \le\frac{1}{\xi(1- L\eta)}\|x_{n} - Kx_{n} \| \to0. $$
(2.4)
Since T is demiclosed at 0, we have \(x^{*} \in F(T) = F(K)\). The conclusion (2) is proved.

The conclusion (3) is obvious (see also [16]). □

3 Main results

Throughout this section, we assume that:
  1. (1)

    \(H_{1}\), \(H_{2}\), and \(H_{3}\) are three real Hilbert spaces. \(A : H_{1} \to H_{3}\) and \(B : H_{2} \to H_{3}\) are two bounded linear operators with their adjoints \(A^{*}\) and \(B^{*}\), respectively;

     
  2. (2)

    \(T : H_{1} \to H_{1}\) and \(S : H_{2} \to H_{2}\) are two L-Lipschitzian and quasi-pseudo-contractive mappings with \(L \ge1\), \(\operatorname{Fix}(T) \neq \emptyset\), and \(\operatorname{Fix}(S) \neq \emptyset\).

     

In the sequel, we denote the strong convergence and weak convergence of a sequence \(\{x_{n}\}\) to a point \(x\in H\) by \(x_{n} \to x\) and \(x_{n} \rightharpoonup x\), respectively.

Our object is to solve the following split equality fixed point problem:
$$ \mbox{to find } x^{*} \in \operatorname{Fix}(T), y^{*} \in F(S) \mbox{ such that } Ax^{*} = By^{*}. $$
(3.1)
In the sequel we use Γ to denote the set of solutions of (3.1), that is,
$$ \Gamma= \bigl\{ \bigl(x^{*}, y^{*}\bigr) \in \operatorname{Fix}(T) \times \operatorname{Fix}(S) \mbox{ such that } Ax^{*} = By^{*}\bigr\} , $$
(3.2)
and we assume that \(\Gamma\neq \emptyset\).

Now, we present our algorithm for finding \((x^{*}, y^{*}) \in\Gamma\).

Algorithm 3.1

Initialization: Choose \(\{\alpha_{n}\} \subset(0, 1)\). Take arbitrary \(x_{0} \in H_{1}\), \(y_{0} \in H_{2}\).

Iterative steps: For a given current \(x_{n} \in H_{1}\), \(y_{n} \in H_{2}\) compute
$$ \left \{ \textstyle\begin{array}{@{}l} (\mathrm{a})\quad u_{n} = x_{n} - \gamma_{n} A^{*}(Ax_{n} - By_{n}),\\ (\mathrm{b})\quad x_{n+1} = \alpha_{n} x_{n} + (1 - \alpha_{n})((1 - \xi_{n})I + \xi _{n} T((1 - \eta_{n})I + \eta_{n} T))u_{n},\\ (\mathrm{c})\quad v_{n} = y_{n} + \gamma_{n} B^{*}(Ax_{n} - By_{n}),\\ (\mathrm{d})\quad y_{n+1} = \alpha_{n} y_{n} + (1 - \alpha_{n})((1 - \xi_{n})I + \xi_{n} S((1 - \eta_{n})I + \eta_{n} S))v_{n}. \end{array}\displaystyle \right . $$
(3.3)

Theorem 3.2

Let \(H_{1}\), \(H_{2}\), \(H_{3}\), A, B, S, T, Γ, \(\{x_{n}\}\) and \(\{y_{n}\}\) be the same as above. If T and S are demiclosed at 0 and the following conditions are satisfied:
  1. (i)

    \(\gamma_{n} \in(0, \min(\frac{1}{\|A\|^{2}}, \frac {1}{\|B\|^{2}})) \) \(\forall n \ge1\);

     
  2. (ii)

    \(0 < a < \xi_{n} < \eta_{n} < b < \frac{1}{1 + \sqrt{1 + L^{2}}} \forall n \ge1\).

     
Then the following conclusions hold:
  1. (I)

    the sequence \((\{x_{n}, y_{n}\})\) generated by (3.3) converges weakly to a solution of problem (3.1);

     
  2. (II)

    In addition, if S, T are also semi-compact, then \((\{x_{n}, y_{n}\})\) converges strongly to a solution of problem (3.1).

     

Proof

First we prove the conclusion (I).

For any given \((p, q) \in\Gamma\), then \(p \in \operatorname{Fix}(T)\), \(q \in \operatorname{Fix}(S)\) and \(Ap = Bq\). From (3.3)(a), we have
$$\begin{aligned} \|u_{n} -p \|^{2} &= \bigl\| x_{n} - \gamma_{n} A^{*}(Ax_{n} - By_{n}) - p\bigr\| ^{2} \\ & = \|x_{n} - p\|^{2} + \gamma_{n}^{2} \bigl\| A^{*}(Ax_{n} - By_{n})\bigr\| ^{2} - 2\gamma_{n} \bigl\langle x_{n} - p, A^{*}(Ax_{n} - By_{n})\bigr\rangle \\ & \le\|x_{n} - p\|^{2} + \gamma_{n}^{2} \|A\|^{2}\|Ax_{n} - By_{n}\|^{2} - 2 \gamma_{n} \langle A x_{n} - Ap, Ax_{n} - By_{n}\rangle. \end{aligned}$$
(3.4)
Similarly, from (3.3)(c), we have
$$ \|v_{n} -q \|^{2} \le\|y_{n} - q\|^{2} + \gamma_{n}^{2} \|B\|^{2}\|Ax_{n} - By_{n}\|^{2} + 2\gamma_{n} \langle B y_{n} - Bq, Ax_{n} - By_{n}\rangle. $$
(3.5)
Put
$$\begin{aligned} &K: = (1 - \xi_{n})I + \xi_{n} T\bigl((1 - \eta_{n})I + \eta_{n}T\bigr), \\ &G: = (1 - \xi_{n})I + \xi_{n} S\bigl((1 - \eta_{n})I + \eta_{n} S\bigr). \end{aligned}$$
By the assumptions of Theorem 3.2, condition (ii) and Lemma 2.9, we know that the mappings K and G have the following properties:
  1. (1)

    Both K and G are quasi-nonexpansive;

     
  2. (2)

    \(\operatorname{Fix}(K) = \operatorname{Fix}(T)\) and \(\operatorname{Fix}(G) = \operatorname{Fix}(S)\);

     
  3. (3)

    K and G demiclosed at 0.

     
Hence from (3.3)(b) and (2.1) we have
$$\begin{aligned} \|x_{n+1} - p\|^{2} & = \bigl\| \alpha_{n} x_{n} + (1 - \alpha_{n}) \bigl((1 - \xi_{n})I + \xi _{n} T\bigl((1 - \eta_{n})I + \eta_{n} T\bigr) \bigr)u_{n} - p)\bigr\| ^{2} \\ & = \bigl\| \alpha_{n} (x_{n} -p) + (1 - \alpha_{n}) (Ku_{n} - p)\bigr\| ^{2} \\ & = \alpha_{n} \|x_{n} -p\|^{2} + (1 - \alpha_{n})\|Ku_{n} - p\|^{2} - \alpha_{n} (1 - \alpha_{n})\|Ku_{n} - x_{n}\|^{2} \\ & \le\alpha_{n} \|x_{n} -p\|^{2} + (1 - \alpha_{n})\|u_{n} - p\|^{2} - \alpha_{n} (1 - \alpha_{n})\|Ku_{n} - x_{n}\|^{2}. \end{aligned}$$
(3.6)
Similarly from (3.3)(c) and (2.1) we have
$$ \|y_{n+1} - q\|^{2} \le\alpha_{n} \|y_{n} - q\|^{2} + (1 - \alpha_{n})\| v_{n} - q \|^{2} - \alpha_{n} (1 - \alpha_{n})\|G v_{n} - y_{n}\|^{2}. $$
(3.7)
Adding (3.6) and (3.7) and by virtue of (3.4) and (3.5), we have
$$\begin{aligned} & \|x_{n+1} - p\|^{2} + \|y_{n+1} - q \|^{2} \\ &\quad \le\alpha_{n} \|x_{n} -p\|^{2} + \alpha_{n} \|y_{n} - q\|^{2} + (1 - \alpha _{n})\|u_{n} - p\|^{2} + (1 - \alpha_{n}) \| v_{n} - q\|^{2} \\ &\qquad{} - \alpha_{n} (1 - \alpha_{n})\|Ku_{n} - x_{n}\|^{2} - \alpha_{n} (1 - \alpha _{n})\|G v_{n} - y_{n}\|^{2} \\ &\quad \le\alpha_{n} \|x_{n} -p\|^{2} + (1 - \alpha_{n})\bigl\{ \|x_{n} - p\|^{2} + \gamma _{n}^{2} \|A\|^{2}\|Ax_{n} - By_{n}\|^{2} \\ &\qquad{}- 2\gamma_{n} \langle A x_{n} - Ap, Ax_{n} - By_{n}\rangle\bigr\} \\ &\qquad{} + \alpha_{n} \|y_{n} - q\|^{2} + (1 - \alpha_{n})\bigl\{ \|y_{n} - p\|^{2} + \gamma_{n}^{2} \|B\|^{2}\|Ax_{n} - By_{n}\|^{2} \\ &\qquad{}+ 2\gamma_{n} \langle B y_{n} - Bq, Ax_{n} - By_{n}\rangle\bigr\} \\ &\qquad{} - \alpha_{n} (1 - \alpha_{n})\|Ku_{n} - x_{n}\|^{2} - \alpha_{n} (1 - \alpha _{n})\|G v_{n} - y_{n}\|^{2} \\ &\quad = \|x_{n} -p\|^{2} + \|y_{n} - q \|^{2} + \gamma_{n}^{2}(1 - \alpha_{n}) \bigl\{ \|A\|^{2} + \|B\|^{2}\bigr\} \|Ax_{n} - By_{n}\|^{2} \\ &\qquad{} -(1 - \alpha_{n})2\gamma_{n} \bigl\{ \langle A x_{n} - Ap, Ax_{n} - By_{n}\rangle - \langle B y_{n} - Bq, Ax_{n} - By_{n}\rangle\bigr\} \\ &\qquad{} - \alpha_{n} (1 - \alpha_{n})\bigl\{ \|Ku_{n} - x_{n}\|^{2} + \|G v_{n} - y_{n}\|^{2}\bigr\} \\ &\quad = \|x_{n} -p\|^{2} + \|y_{n} - q \|^{2} + \gamma_{n}^{2}(1 - \alpha_{n}) \bigl\{ \|A\|^{2} + \|B\|^{2}\bigr\} \|Ax_{n} - By_{n}\|^{2} \\ &\qquad{} - (1 - \alpha_{n})2\gamma_{n} \|Ax_{n} - By_{n}\|^{2} - \alpha_{n} (1 - \alpha_{n})\bigl\{ \|Ku_{n} - x_{n}\|^{2} + \|G v_{n} - y_{n}\|^{2}\bigr\} \\ &\qquad(\mbox{since }Ap = Bq) \\ &\quad = \|x_{n} -p\|^{2} + \|y_{n} - q \|^{2} - (1 - \alpha_{n}) \gamma_{n}\bigl( 2 - \gamma _{n} \bigl(\|A\|^{2} + \|B\|^{2}\bigr)\bigr) \|Ax_{n} - By_{n}\|^{2} \\ &\qquad{} - \alpha_{n} (1 - \alpha_{n})\bigl\{ \|Ku_{n} - x_{n}\|^{2} + \|G v_{n} - y_{n}\|^{2}\bigr\} . \end{aligned}$$
(3.8)
Since \(\gamma_{n} \in(0, \min \{\frac{1}{\|A\|^{2}}, \frac{1}{\|A\|^{2}}\} )\), \(\gamma_{n} \|A\|^{2} < 1\) and \(\gamma_{n} \|B\|^{2} < 1\). So \(0 < \gamma_{n} (\|A\|^{2} + \|B\|^{2}) < 2\). This implies that \(\gamma_{n}(2 - \gamma_{n} (\|A\|^{2} + \|B\|^{2})) > 0\).
Putting
$$ X_{n}(p, q)= \|x_{n} - p\|^{2} + \|y_{n} - q\|^{2}, $$
(3.9)
hence (3.8) can be written as
$$\begin{aligned} X_{n+1}(p, q) \le{}& X_{n}(p, q) - (1 - \alpha_{n})\gamma_{n}\bigl(2 - \gamma_{n} \bigl(\|A \|^{2} + \|B\|^{2}\bigr)\bigr)\|Ax_{n} - By_{n}\|^{2} \\ &{}- \alpha_{n} (1- \alpha_{n})\bigl\{ \|Ku_{n} - x_{n}\|^{2} + \|G v_{n} - y_{n} \|^{2}\bigr\} \\ \le{}& X_{n}(p, q). \end{aligned}$$
(3.10)
This implies that \(\{X_{n}(p, q)\}\) is a non-increasing sequence, hence the limit \(\lim_{n \to\infty}X_{n}(p, q)\) exists. Therefore the following limits exist:
$$ \lim_{n \to\infty} \|x_{n} - p\| \quad \mbox{and}\quad \lim _{n \to\infty} \|y_{n} - q\| \quad \forall (p, q) \in\Gamma. $$
(3.11)
Rewritten (3.10) as
$$\begin{aligned} &(1 - \alpha_{n})\gamma_{n}\bigl(2 - \gamma_{n} \bigl(\|A\|^{2} + \|B\|^{2}\bigr)\bigr) \|Ax_{n} - By_{n}\|^{2} \\ &\quad{} + \alpha_{n} (1- \alpha_{n})\bigl\{ \|Ku_{n} - x_{n}\|^{2} + \|G v_{n} - y_{n} \|^{2}\bigr\} \le X_{n}(p, q) - X_{n+1}(p, q). \end{aligned}$$
(3.12)
Letting \(n \to\infty\) and taking the limit in (3.12), we have
$$ \|Ax_{n} - By_{n}\| \to0; \qquad \|Ku_{n} - x_{n}\| \to0; \qquad \|G v_{n} - y_{n}\| \to0. $$
(3.13)
From (3.13) and (3.3) we have
$$ \left \{ \textstyle\begin{array}{@{}l} \lim_{n\to\infty}\|u_{n} - x_{n}\| \to0 \mbox{ and } \lim_{n\to\infty }\|v_{n} - y_{n}\| \to0,\\ \lim_{n\to\infty}\| x_{n+1} - x_{n}\| \\ \quad= \lim_{n\to\infty}(1 - \alpha _{n})\|((1 - \xi_{n})I + \xi_{n} T((1 - \eta_{n})I + \eta_{n} T))u_{n} - x_{n}\|\\ \quad =\lim_{n\to\infty}(1 - \alpha_{n})\|K u_{n} - x_{n}\| = 0,\\ \lim_{n\to\infty}\| y_{n+1} - y_{n}\| \\ \quad= \lim_{n\to\infty}(1 - \beta _{n})\|((1 - \xi_{n})S + \xi_{n} S((1 - \eta_{n})I + \eta_{n} S))y_{n} - y_{n}\|\\ \quad =\lim_{n\to\infty}(1 - \alpha_{n})\|G v_{n} - y_{n}\| = 0. \end{array}\displaystyle \right . $$
(3.14)
This together with (3.13) shows that
$$ \left \{ \textstyle\begin{array}{@{}l} \|K u_{n} - u_{n}\| \le\|K u_{n} - x_{n}\| + \|x_{n} - u_{n}\| \to0;\\ \|G v_{n} - v_{n}\| \le\|G v_{n} - y_{n}\| + \|y_{n} - v_{n}\| \to0. \end{array}\displaystyle \right . $$
(3.15)
Since \(\{x_{n}\}\) and \(\{y_{n}\}\) are bounded sequences, there exist some weakly convergent subsequences, say \(\{x_{n_{i}}\} \subset\{x_{n}\}\) and \(\{y_{n_{i}}\}\subset\{y_{n}\}\) such that \(x_{n_{i}} \rightharpoonup x^{*}\) and \(y_{n_{i}} \rightharpoonup y^{*}\). Since every Hilbert space has the Opial property. The Opial property guarantees that the weakly subsequential limit of \(\{(x_{n}, y_{n})\}\) is unique. Therefore we have \(x_{n} \rightharpoonup x^{*}\) and \(y_{n} \rightharpoonup y^{*}\).

On the other hand, from (3.14), one gets \(u_{n} \rightharpoonup x^{*}\) and \(v_{n} \rightharpoonup y^{*}\). By (3.15) and the demiclosed property of K and G, we have \(Kx^{*} = x^{*}\) and \(Gy^{*} = y^{*}\). This implies that \(x^{*} \in \operatorname{Fix}(T)\) and \(y^{*} \in \operatorname{Fix}(S)\).

Now we are left to show that \(Ax^{*} = By^{*}\). In fact, since \(Ax_{n} - By_{n} \rightharpoonup Ax^{*} - By^{*}\), by using the weakly lower semi-continuity of squared norm, we have
$$\bigl\| Ax^{*} - By^{*}\bigr\| ^{2} = \liminf_{n \to\infty} \|Ax_{n} - By_{n}\|^{2} = \lim_{n \to\infty} \|Ax_{n} - By_{n}\|^{2} = 0. $$
Thus \(Ax^{*} = By^{*}\). This completes the proof of the conclusion (I).
Now we prove the conclusion (II). In fact, by virtue of (2.4), (3.13), and (3.14), we have
$$ \left \{ \textstyle\begin{array}{@{}l} \|x_{n} - Tx_{n} \| \le\frac{1}{\xi_{n} (1- L\eta_{n})}\|x_{n} - Kx_{n} \| \to 0;\\ \|y_{n} - Sy_{n} \| \le\frac{1}{\xi_{n} (1- L\eta_{n})}\|y_{n} - Gy_{n} \| \to0. \end{array}\displaystyle \right . $$
(3.16)
Since S, T are semi-compact, it follows from (3.16) that there exist subsequences \(\{x_{n_{i}}\} \subset\{x_{n}\}\) and \(\{y_{n_{j}}\} \subset\{ y_{n}\}\) such that \(x_{n_{i}} \to x\) (some point in \(F(T)\)) and \(y_{n_{j}} \to y\) (some point in \(F(S)\)). It follows from (3.11), \(x_{n} \rightharpoonup x^{*}\), and \(y_{n} \rightharpoonup y^{*}\) that \(x_{n} \to x^{*}\) and \(y_{n} \to y^{*}\) and \(Ax^{*} = By^{*}\). □

4 Applications

4.1 Application to the split equality variational inequality problem

Throughout this section, we assume that \(H_{1}\), \(H_{2}\), and \(H_{3}\) are three real Hilbert spaces. C and Q both are nonempty and closed convex subsets of \(H_{1}\) and \(H_{2}\), respectively and assume that \(A : H_{1} \to H_{3}\) and \(B : H_{2} \to H_{3}\) are two bounded linear operator with its adjoint \(A^{*}\) and \(B^{*}\), respectively.

Let \(M : C \to H_{1}\) be a mapping. The variational inequality problem for mapping M is to find a point \(x^{*} \in C\) such that
$$ \bigl\langle Mx^{*}, z - x^{*}\bigr\rangle \ge0 \quad \forall z \in C. $$
(4.1)
We will denote the solution set of (4.1) by \(VI(M,C)\).
A mapping \(M : C \to H_{1}\) is said to be α-inverse-strongly monotone if there exists a constant \(\alpha> 0\) such that
$$ \langle Mx - My, x - y\rangle\ge\alpha\|Mx - My\|^{2} \quad \forall x, y \in C. $$
(4.2)
It is easy to see that if M is α-inverse-strongly monotone, then M is \(\frac{1}{\alpha}\)-Lipschitzian.
Setting \(h(x, y) = \langle Mx, y - x\rangle: C \times C \to\mathbb{R}\), it is easy to show that h is an equilibrium function, i.e., it satisfies the following conditions, (A1)-(A4):
  1. (A1)

    \(h(x,x) = 0\), for all \(x\in C\);

     
  2. (A2)

    h is monotone, i.e., \(h(x,y) + h(y, x) \le0\) for all \(x, y \in C\);

     
  3. (A3)

    \(\limsup_{t\downarrow0} h(tz + (1-t)x, y) \le h(x,y)\) for all \(x, y, z \in C \);

     
  4. (A4)

    for each \(x\in C\), \(y \mapsto h(x,y)\) is convex and lower semi-continuous.

     
For given \(\lambda > 0\) and \(x\in H\), the resolvent of the equilibrium function h is the operator \(R_{\lambda, h}: H \to C\) defined by
$$ R_{\lambda, h}(x): = \biggl\{ z\in C: h(z,y)+ \frac{1}{\lambda}\langle y-z, z - x \rangle\geq0, \forall y\in C\biggr\} . $$
(4.3)

Proposition 4.1

[17]

The resolvent operator \(R_{\lambda, h}\) of the equilibrium function h has the following properties :
  1. (1)

    \(R_{\lambda, h}\) is single-valued;

     
  2. (2)

    \(\operatorname{Fix}(R_{\lambda, h}) = VI(M,C)\), where \(VI(M,C)\) is the solution set of variational inequality (4.1) which is a nonempty closed and convex subset of C;

     
  3. (3)

    \(R_{\lambda, h}\) is a firmly nonexpansive mapping. Therefore \(R_{\lambda, h}\) is demiclosed at 0.

     
Let \(T : C \to H_{1}\) and \(S : Q \to H_{2}\) be two α-inverse-strongly monotone mappings. The so-called split equality variational inequality problem with respect to T and S is to find \(x^{*} \in C\) and \(y^{*} \in Q\) such that
$$ \left \{ \textstyle\begin{array}{@{}l} (\mathrm{a})\quad \langle Tx^{*}, u - x^{*}\rangle\ge0\quad \forall u \in C,\\ (\mathrm{b})\quad \langle Sy^{*}, v - y^{*} \rangle\ge0 \quad \forall v \in Q,\\ (\mathrm{c})\quad Ax^{*} = By^{*}. \end{array}\displaystyle \right . $$
(4.4)
In the sequel we use Ω to denote the solution set of split equality variational inequality problem (4.4), i.e.,
$$ \Theta= \bigl\{ \bigl(x^{*}, y^{*}\bigr) \in VI(T,C) \times VI(S,Q): Ax^{*} = By^{*}\bigr\} , $$
(4.5)
where \(VI(T,C)\) (resp. \(VI(S, Q)\)) is the solution set of variational inequality (4.4)(a) (resp. (4.4)(b)).
Denote by \(f(x,y) = \langle Tx, y -x \rangle: C \times C \to\mathbb {R}\) and \(g(u,v) = \langle Su, v - u \rangle: Q \times Q \to\mathbb {R}\). For given \(\lambda> 0\), \(x \in H_{1}\), and \(u \in H_{2}\), let \(R_{\lambda, f}(x)\) and \(R_{\lambda, g}(u)\) be the resolvent operator of the equilibrium function f and g, respectively, which are defined by
$$R_{\lambda, f}(x): = \biggl\{ z\in C: f(z,y)+ \frac{1}{\lambda}\langle y-z, z - x \rangle\geq0, \forall y\in C\biggr\} $$
and
$$R_{\lambda, g}(u): = \biggl\{ z\in Q: g(z,v)+ \frac{1}{\lambda}\langle v-z, z - u \rangle\geq0, \forall v\in Q\biggr\} . $$
It follows from Proposition 4.1 that
$$ \operatorname{Fix}(R_{\lambda, f}) = VI(T,C) \neq \emptyset; \qquad \operatorname{Fix}(R_{\lambda, g}) = VI(S,Q) \neq \emptyset, $$
(4.6)
and so \(R_{\lambda, f}\) and \(R_{\lambda, g}\) both are quasi-pseudo-contractive and 1-Lipschitzian. Therefore the split equality variational inequality problem with respect to T and S (4.4) is equivalent to the following split equality fixed point problem:
$$ \mbox{to find } x^{*} \in \operatorname{Fix}(R_{\lambda, f}), y^{*} \in \operatorname{Fix}(R_{\lambda, g}) \mbox{ such that } Ax^{*} = By^{*}. $$
(4.7)

Since \(R_{\lambda, f}\) and \(R_{\lambda, g}\) are firmly nonexpansive with \(\operatorname{Fix}(R_{\lambda, f}) \neq \emptyset\) and \(\operatorname{Fix}(R_{\lambda, g})\neq \emptyset\), the following theorem can be obtained from Theorem 3.2 immediately.

Theorem 4.2

Let \(H_{1}\), \(H_{2}\), \(H_{3}\), C, Q, A, B, T, S, \(R_{\lambda, f}\), \(R_{\lambda, g}\), Θ be the same as above and assume that \(\Theta\neq \emptyset\). For given \(x_{0} \in C\), \(y_{0} \in Q\), let \((\{x_{n}\}, \{x_{n}\})\) be the sequence generated by
$$ \left \{ \textstyle\begin{array}{@{}l} u_{n} = x_{n} - \gamma_{n} A^{*}(Ax_{n} - By_{n}),\\ x_{n+1} = R_{\lambda, f}( u_{n}),\\ v_{n} = y_{n} + \gamma_{n} B^{*}(Ax_{n} - By_{n}),\\ y_{n+1} = R_{\lambda, g}(v_{n}). \end{array}\displaystyle \right . $$
(4.8)
If \(\gamma_{n} \in(0, \min(\frac{1}{\|A\|^{2}}, \frac{1}{\|B\|^{2}}))\) \(\forall n \ge1\), then the sequence \((\{x_{n}, y_{n}\})\) generated by (4.8) converges weakly to a solution of split equality variational inequality problem (4.4).

4.2 Application to the split equality convex minimization problem

Let C be a nonempty closed convex subset of \(H_{1}\) and Q be a nonempty closed convex subset of \(H_{2}\). Let \(\phi: C \to\mathbb{R}\) and \(\psi: Q \to\mathbb{R}\) be two proper and convex and lower semi-continuous functions and \(A : H_{1} \to H_{3}\) and \(B : H_{2} \to H_{3}\) be two bounded linear operator with its adjoint \(A^{*}\) and \(B^{*}\), respectively.

The so-called split equality convex minimization problem for ϕ and ψ is to find \(x^{*} \in C\), \(y^{*} \in Q\) such that
$$ \phi\bigl(x^{*}\bigr)= \min_{x \in C}\phi(x), \qquad \psi\bigl(y^{*} \bigr) = \min_{x \in Q}\psi (y), \quad\mbox{and}\quad Ax^{*} = By^{*}. $$
(4.9)
In the sequel, we denote by Ω the solution set of split equality convex minimization problem (4.9), i.e.,
$$\begin{aligned} \Omega ={}& \Bigl\{ (p, q) \in C \times Q \mbox{ such that } \phi\bigl(x^{*}\bigr)= \min_{x \in C}\phi(x), \\ &{}\psi\bigl(y^{*}\bigr) = \min _{x \in Q}\psi(y)\mbox{ and }Ax^{*} = By^{*}\Bigr\} \end{aligned}$$
(4.10)
Let \(j(x, y): = \phi(y) - \psi(x): C \times C \to\mathbb{R}\) and \(k(u, v): = \phi(v) - \psi(u): Q \times Q \to\mathbb{R}\). It is easy to see that j and k both are equilibrium functions satisfying the conditions (A1)-(A4).
For given \(\lambda> 0\), \(x \in H_{1}\) and \(u \in H_{2}\), we define the resolvent operators of j and k as follows:
$$R_{\lambda, j}(x): = \biggl\{ z \in C: j(z,y)+ \frac{1}{\lambda}\langle y-z, z - x \rangle\geq0, \forall y \in C\biggr\} $$
and
$$R_{\lambda, k}(u): = \biggl\{ z\in Q: k(z,v)+ \frac{1}{\lambda}\langle v-z, z - u \rangle\geq0, \forall v\in Q\biggr\} . $$
By the same argument as given in Section 4.1, we know that
$$\begin{aligned}& \operatorname{Fix}(R_{\lambda, j}) = \Bigl\{ x^{*} \in C: \phi\bigl(x^{*}\bigr)= \min_{x \in C}\phi(x)\Bigr\} , \qquad \operatorname{Fix}(R_{\lambda, k}) = \Bigl\{ y^{*} \in Q: \psi\bigl(y^{*}\bigr) = \min_{x \in Q}\psi(y)\Bigr\} . \end{aligned}$$
Therefore the split equality convex minimization problem for ϕ and ψ is equivalent to the following split equality fixed point problem:
$$ \mbox{to find } x^{*} \in \operatorname{Fix}(R_{\lambda, j}), y^{*} \in \operatorname{Fix}(R_{\lambda, k}) \mbox{ such that } Ax^{*} = By^{*}. $$
(4.11)

Since \(R_{\lambda, j}\) and \(R_{\lambda, k}\) both are firmly nonexpansive with \(\operatorname{Fix}(R_{\lambda, f}) \neq \emptyset\) and \(\operatorname{Fix}(R_{\lambda, g})\neq \emptyset\), the following theorem can be obtained from Theorem 3.2 immediately.

Theorem 4.3

Let \(H_{1}\), \(H_{2}\), \(H_{3}\), C, Q, A, B, ϕ, ψ, \(R_{\lambda, j}\), \(R_{\lambda, k}\), Ω be the same as above and assume that \(\Omega\neq \emptyset\). For given \(x_{0} \in C\), \(y_{0} \in Q\), let \((\{x_{n}\}, \{x_{n}\})\) be the sequence generated by
$$ \left \{ \textstyle\begin{array}{@{}l} u_{n} = x_{n} - \gamma_{n} A^{*}(Ax_{n} - By_{n}),\\ x_{n+1} = R_{\lambda, j}( u_{n}),\\ v_{n} = y_{n} + \gamma_{n} B^{*}(Ax_{n} - By_{n}),\\ y_{n+1} = R_{\lambda, k}(v_{n}). \end{array}\displaystyle \right . $$
(4.12)
If \(\gamma_{n} \in(0, \min(\frac{1}{\|A\|^{2}}, \frac{1}{\|B\|^{2}}))\) \(\forall n \ge1\), then the sequence \((\{x_{n}, y_{n}\})\) generated by (4.12) converges weakly to a solution of split equality convex minimization problem (4.9).

Declarations

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Center for General Education, China Medical University, Taichung, Taiwan
(2)
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, P.R. China
(3)
Department of Mathematics, Kunming University, Kunming, China

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