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 [3–5]. The SFP in an infinite-dimensional real Hilbert space can be found in [2, 4, 6–10].

Recently, Moudafi [11–13] 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).