Open Access

Strong and weak convergence theorems for split equality generalized mixed equilibrium problem

Fixed Point Theory and Applications20162016:101

DOI: 10.1186/s13663-016-0592-6

Received: 24 February 2016

Accepted: 25 November 2016

Published: 1 December 2016

Abstract

In this paper, we consider split equality generalized mixed equilibrium problem, which is more general than many problems such as split feasibility problem, split equality problem, split equilibrium problem, and so on. We propose a new modified algorithm to obtain strong and weak convergence theorems for split equality generalized mixed equilibrium problem for nonexpansive mappings in Hilbert spaces. Also, we give some applications to other problems. Our results extend some results in the literature.

Keywords

split equality generalized mixed equilibrium problem nonexpansive mappings fixed point demicompactness

MSC

47H09 47J25

1 Introduction

Let H be a real Hilbert space with inner product \(\langle \cdot,\cdot \rangle\) and norm \(\Vert \cdot \Vert \), C be a nonempty closed convex subset of H, and \(F: C \times C \rightarrow \mathbb{R}\) be a bifunction, where \(\mathbb{R}\) is the set of all real numbers. The scalar-valued equilibrium problem is finding a point \(x \in C\) such that
$$ F ( x, y ) \geq0 $$
(1.1)
for all \(y \in C\). The equilibrium problem has been extensively studied, beginning with Blum and Oettli [1]. In 2014, Ahmad and Rahaman [2] introduced the generalized vector equilibrium problem to find a point \(x \in C\) such that
$$F \bigl( \lambda x + ( 1- \lambda ) z, y \bigr) \nsubseteq- C \backslash \{ 0 \} $$
for all \(y, z \in C\) and \(\lambda \in ( 0,1 ]\), where \(F: C \times C \rightarrow 2^{H}\) is a set-valued mapping such that \(F ( \lambda x + ( 1- \lambda ) z, x ) \supseteq \{ 0 \}\). In the scalar case, the generalized equilibrium problem takes the form of finding \(x \in C\) such that
$$ F \bigl( \lambda x + ( 1- \lambda ) z, y \bigr) \geq0 $$
(1.2)
for all \(y, z \in C\) and \(\lambda \in ( 0,1 ]\) under the condition \(F ( \lambda x + ( 1- \lambda ) z, x ) =0\). If \(\lambda =1\), then problem (1.2) is reduced to problem (1.1).
Let \(T: C \rightarrow C\) be a continuous mapping, and \(\phi: C \rightarrow \mathbb{R}\) be a mapping. Very recently, Rahaman et al. [3] considered the generalized mixed equilibrium problem of finding a point \(x \in C\) such that
$$ F \bigl( \lambda x + ( 1- \lambda ) z, y \bigr) + \langle Tx, y - x \rangle + \phi ( y ) - \phi ( x ) \geq0 $$
(1.3)
for all \(y, z \in C\) and \(\lambda \in ( 0,1 ]\). The set of solutions of problem (1.3) is denoted by \(\operatorname{GMEP} ( F, T, \phi )\).
Let \(H_{1}\), \(H_{2}\), and \(H_{3}\) be real Hilbert spaces, and C and Q be nonempty closed and convex subsets of \(H_{1}\) and \(H_{2}\), respectively. Let \(F: C \times C \rightarrow \mathbb{R}\) and \(G: Q \times Q \rightarrow \mathbb{R}\) be two bifunctions, \(T: C \rightarrow C\) and \(S: Q \rightarrow Q\) be nonlinear mappings, \(\phi: C \rightarrow \mathbb{R}\cup \{ +\infty \}\) and \(\varphi: Q \rightarrow \mathbb{R}\cup \{ +\infty \}\) be proper lower semicontinuous and convex mappings such that \(C \cap \operatorname{dom}\phi \neq\emptyset\) and \(Q \cap \operatorname{dom} \varphi \neq\emptyset\), and \(A: H_{1} \rightarrow H_{3}\) and \(B: H_{2} \rightarrow H_{3}\) be bounded linear mappings. In 2015, Rahaman et al. [3] introduced the following split equality generalized mixed equilibrium problem (SEGMEP): find \(x^{*} \in C\) and \(y^{*} \in Q\) such that
$$\begin{aligned}& F \bigl( \lambda_{1} x^{*} + ( 1- \lambda_{1} ) b, x \bigr) + \bigl\langle T x^{*}, x - x^{*} \bigr\rangle + \phi ( x ) - \phi \bigl( x^{*} \bigr) \geq0, \\& G \bigl( \lambda_{2} y^{*} + ( 1- \lambda_{2} ) c, y \bigr) + \bigl\langle S y^{*}, y - y^{*} \bigr\rangle + \varphi ( y ) - \varphi \bigl( y^{*} \bigr) \geq0, \quad \mbox{and} \\& A x^{*} = B y^{*} \end{aligned}$$
(1.4)
for all \(x, b \in C\), \(y, c \in Q\), and \(\lambda_{1}, \lambda_{2} \in ( 0,1 ]\). The solution set of problem (1.4) is denoted by \(\operatorname{SEGMEP} ( F, G, T, S, \phi, \varphi )\). This problem is a generalization of all the following problems.
  1. 1.
    If \(T = S =0\) and \(\lambda_{1} = \lambda_{2} =1\), then the split equality generalized mixed equilibrium problem (SEGMEP) is reduced to the split equality mixed equilibrium problem (SEMEP) introduced by Ma et al. [4]: find \(x^{*} \in C\) and \(y^{*} \in Q\) such that
    $$\begin{aligned}& F \bigl( x^{*}, x \bigr) + \phi ( x ) - \phi \bigl( x^{*} \bigr) \geq0,\quad \forall x \in C, \\& G \bigl( y^{*}, y \bigr) + \varphi ( y ) - \varphi \bigl( y^{*} \bigr) \geq0,\quad \forall y \in Q,\quad \mbox{and} \\& A x^{*} = B y^{*}. \end{aligned}$$
    (1.5)
     
  2. 2.
    If \(T = S = \phi = \varphi =0\), \(B = I\), \(H_{2} = H_{3}\), and \(\lambda_{1} = \lambda_{2} =1\), then problem (1.4) is reduced to the split equilibrium problem (SEqP) introduced by He [5]: find \(x^{*} \in C\) such that
    $$ \begin{aligned} &F \bigl( x^{*}, x \bigr) \geq0,\quad \forall x \in C, \quad \mbox{and} \\ &A x^{*} = y^{*} \in Q \quad \mbox{solves} \quad G \bigl( y^{*}, y \bigr) \geq0,\quad \forall y \in Q. \end{aligned} $$
    (1.6)
     
  3. 3.
    If \(T = S = \phi = \varphi =0\) and \(\lambda_{1} = \lambda_{2} =1\), then problem (1.4) is reduced to the split equality equilibrium problem (SEEP) of finding \(x^{*} \in C\) and \(y^{*} \in Q\) such that
    $$\begin{aligned}& F \bigl( x^{*}, x \bigr) \geq0,\quad \forall x \in C, \\& G \bigl( y^{*}, y \bigr) \geq0,\quad \forall y \in Q,\quad \mbox{and} \\& A x^{*} = B y^{*}. \end{aligned}$$
    (1.7)
     
  4. 4.
    If \(F = G = T = S =0\), then problem (1.4) is reduced to the split equality convex minimization problem (SECMP) of finding \(x^{*} \in C\) and \(y^{*} \in Q\) such that
    $$ \begin{aligned} &\phi ( x ) \geq \phi \bigl( x^{*} \bigr),\quad \forall x \in C,\qquad \varphi ( y ) \geq \varphi \bigl( y^{*} \bigr), \quad \forall y \in Q, \quad \mbox{and} \\ &A x^{*} = B y^{*}. \end{aligned} $$
    (1.8)
     
  5. 5.
    If \(F = G = T = S =0\), \(B = I\), and \(H_{2} = H_{3}\), then problem (1.4) is reduced to the split convex minimization problem (SCMP) of finding \(x^{*} \in C\) such that
    $$ \begin{aligned} &\phi ( x ) \geq \phi \bigl( x^{*} \bigr),\quad \forall x \in C,\qquad \varphi ( y ) \geq \varphi \bigl( y^{*} \bigr), \quad \forall y \in Q, \quad \mbox{and} \\ &A x^{*} = y^{*} \in Q. \end{aligned} $$
    (1.9)
     
  6. 6.
    If \(F = G = \phi = \varphi = T = S =0\), then problem (1.4) is reduced to the split equality problem (SEP) of finding \(x^{*} \in C\) and \(y^{*} \in Q\) such that
    $$ A x^{*} = B y^{*}. $$
    (1.10)
     
  7. 7.
    If \(F = G = \phi = \varphi = T = S =0\), \(B = I\), and \(H_{2} = H_{3}\), then problem (1.4) is reduced to the split feasibility problem (SFP) of finding \(x^{*} \in C\) such that
    $$ A x^{*} \in Q. $$
    (1.11)
    This problem was introduced by Censor and Elfving [6].
     

Since these kinds of problems are related implicitly or explicitly to many areas, such as engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games, computer tomograph, radiation therapy treatment planing, physics, inverse problems that arise from phase retrievals and in medical image reconstruction, and so on, it is very important in mathematics. So, many authors have proposed some algorithms to solve such problems; see, for instance, [714]. We further give some of them.

In 2015, Ma et al. [4] introduced the following simultaneous iterative algorithm to obtain weak and strong convergence theorems for (SEMEP):
$$ \left \{ \textstyle\begin{array}{l} F ( u_{n}, u ) + \phi ( u ) - \phi ( u_{n} ) + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0, \\ G ( v_{n}, v ) + \varphi ( v ) - \varphi ( v_{n} ) + \frac{1}{r_{n}} \langle v - v_{n}, v_{n} - y_{n} \rangle \geq0, \\ x_{n +1} = \alpha_{n} u_{n} + ( 1- \alpha_{n} ) T ( u_{n} - \gamma_{n} A^{*} ( A u_{n} - B v_{n} ) ), \\ y_{n +1} = \alpha_{n} u_{n} + ( 1- \alpha_{n} ) S ( v_{n} + \gamma_{n} B^{*} ( A u_{n} - B v_{n} ) ) \end{array}\displaystyle \right . $$
(1.12)
for all \(u \in C\), \(v \in Q\), where \(n \geq1\), and \(T: H_{1} \rightarrow H_{1}\) and \(S: H_{2} \rightarrow H_{2}\) are nonexpansive mappings. In the same year, Rahaman et al. [3] gave the following method as a generalization of algorithm (1.12):
$$ \left \{ \textstyle\begin{array}{l} F ( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u ) + \phi ( u ) - \phi ( u_{n} ) \\ \quad {}+ \langle T u_{n}, u - u_{n} \rangle + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0, \\ G ( \lambda_{2} v_{n} + ( 1- \lambda_{2} ) c, v ) + \varphi ( v ) - \varphi ( v_{n} ) \\ \quad {}+ \langle S v_{n}, v - v_{n} \rangle + \frac{1}{r_{n}} \langle v - v_{n}, v_{n} - y_{n} \rangle \geq0, \\ x_{n +1} = ( 1- \alpha_{n} ) ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} Q ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) \end{array}\displaystyle \right . $$
(1.13)
for all \(u, b \in C\), \(v, c \in Q\), where \(n \geq1\), and \(P: H_{1} \rightarrow H_{1}\) and \(Q: H_{2} \rightarrow H_{2}\) are two demicontractive mappings. They also proved that the sequence \(\{ ( x_{n}, y_{n} ) \}\) generated by algorithm (1.13) converges weakly and strongly to the solution of the split equality generalized mixed equilibrium problem (1.4) under some suitable conditions.

In this paper, inspired by algorithm (1.13), we introduce a modified algorithm to obtain weak and strong convergence results for the split equality generalized mixed equilibrium problem. Also, we give some corollaries and applications for the split equality problem, the split feasibility problem, the split equality mixed convex differentiable optimization problem, the split equality convex minimization problem, and the split equality mixed equilibrium problem. Our results extend some correspoing results of many authors.

2 Preliminaries

Throughout this paper, we use the symbols → and for the strong and weak convergence, respectively. Now, we recall some definitions, lemmas, and properties, which we need in the proof of our main theorem.

Let T be a mapping on a Hilbert space H. The set of fixed points of T is denoted by \(F ( T )\), that is, \(F ( T ) = \{ x \in H: Tx = x \}\). Let C be a nonempty closed convex subset of H. A mapping \(T: C \rightarrow C\) it is said to be
  1. (i)
    a nonexpansive mapping if
    $$\Vert Tx - Ty \Vert \leq \Vert x - y \Vert ,\quad \forall x, y \in C; $$
     
  2. (ii)
    a firmly nonexpansive mapping if
    $$\Vert Tx - Ty \Vert ^{2} \leq \langle Tx - Ty, x - y \rangle,\quad \forall x, y \in C. $$
     

Lemma 1

[15]

Let C be a nonempty closed convex subset of a uniformly convex Banach space X, and \(T:C\rightarrow C\) be a nonexpansive mapping with \(F(T)\neq\emptyset\). Then \(F(T)\) is closed and convex.

Lemma 2

[16]

Let C be a nonempty closed convex subset of a real Hilbert space H, and T be a nonexpansive self-mapping on C. If \(F ( T ) \neq\emptyset\), then \(I-T\) is demiclosed, that is, if \(\{ x_{n} \}\) is a sequence in C weakly converging to some \(x\in C\) and the sequence \(\{ ( I-T ) x_{n} \}\) strongly converges to some y, then \(( I-T ) x=y\). Here, I is the identity operator of H.

Recall that T is said to be demicompact if every bounded sequence \(\{ x_{n} \}\) in C such that \(\{ ( I - T ) x_{n} \}\) converges strongly contains a strongly convergent subsequence.

To solve a generalized mixed equilibrium problem for a bifunction \(F: C \times C \rightarrow \mathbb{R}\) and mappings \(T: C \rightarrow C\) and \(\phi: C \rightarrow \mathbb{R}\), let us assume that the following conditions are satisfied:
  1. A1.

    \(F ( \lambda x + ( 1- \lambda ) b, x ) =0\), \(\forall x \in C\);

     
  2. A2.
    F is monotone, that is, for all \(x, y \in C\),
    $$F \bigl( \lambda x + ( 1- \lambda ) b, y \bigr) + F \bigl( \lambda y + ( 1- \lambda ) b, x \bigr) \leq0; $$
     
  3. A3.
    T is monotone, that is, for all \(x, y \in C\),
    $$\langle Tx - Ty, x - y \rangle \geq0; $$
     
  4. A4.

    \(\forall x \in C\), \(y \mapsto F ( \lambda x + ( 1- \lambda ) b, y )\) is convex and lower semicontinuous;

     
  5. A5.

    F is hemicontinuous in the first argument;

     
  6. A6.

    T is weakly upper semicontinuous;

     
  7. A7.
    For all \(x \in C\), \(\lambda \in ( 0,1 ]\), and \(r >0\), there exist a bounded subset \(D \subseteq C\) and \(a \in C\) such that, for all \(z \in C \backslash D\) and \(b \in C\),
    $$- F \bigl( \lambda a + ( 1- \lambda ) b, z \bigr) + \langle Tz, a - z \rangle + \phi ( a ) - \phi ( z ) + \frac{1}{r} \langle a - z, z - x \rangle < 0. $$
     

Lemma 3

[3]

Let C be a nonempty closed convex subset of a Hilbert space \(H_{1}\). Suppose that the bifunction \(F:C\times C \rightarrow \mathbb{R}\) and the mapping \(T:C\rightarrow C\) satisfy conditions (A1)-(A7). Let \(\phi:C \rightarrow \mathbb{R}\cup\{+\infty\}\) be a proper lower semicontinuous and convex mapping such that \(C\cap \operatorname{dom}\phi=\emptyset\). For \(r>0\), \(\lambda_{1} \in ( 0,1 ]\), and \(x\in H\), let \(J_{r}^{F,T}: H_{1} \rightarrow C\) be the resolvent operator of F and T defined by
$$\begin{aligned} J_{r}^{F, T} ( x ) =& \biggl\{ z \in C: F \bigl( \lambda_{1} z + ( 1- \lambda_{1} ) b, y \bigr) + \langle Tz, y - z \rangle \\ &{}+ \phi ( y ) - \phi ( z ) + \frac{1}{r} \langle y - z, z - x \rangle \geq0, \forall y, b \in C \biggr\} . \end{aligned}$$
(2.1)
Then:
  1. (i)

    For each \(x \in H_{1}\), \(J_{r}^{F, T} ( x ) \neq\emptyset\);

     
  2. (ii)

    \(J_{r}^{F, T}\) is single valued;

     
  3. (iii)
    \(J_{r}^{F, T}\) is firmly nonexpansive, that is,
    $$\bigl\Vert J_{r}^{F, T} ( x ) - J_{r}^{F, T} ( y ) \bigr\Vert ^{2} \leq \bigl\langle J_{r}^{F, T} ( x ) - J_{r}^{F, T} ( y ), x - y \bigr\rangle ,\quad \forall x, y \in H_{1}; $$
     
  4. (iv)

    \(F ( J_{r}^{F, T} ) = \operatorname{GMEP} ( F, T, \phi )\), and it is closed and convex.

     
Let the bifunction \(G: Q \times Q \rightarrow \mathbb{R}\) and the mapping \(S: Q \rightarrow Q\) satisfy conditions (A1)-(A7). Let \(\varphi: C \rightarrow \mathbb{R}\cup\{+\infty\}\) be a proper lower semicontinuous and convex mapping such that \(Q \cap \operatorname{dom} \varphi =\emptyset\). For \(s >0\), \(\lambda_{2} \in ( 0,1 ]\), and \(u \in H_{2}\), let \(J_{s}^{G, S}: H_{2} \rightarrow Q\) be the resolvent operator of G and S defined by
$$\begin{aligned} J_{s}^{G, S} ( u ) =& \biggl\{ v \in Q: G \bigl( \lambda_{2} v + ( 1- \lambda_{2} ) c, w \bigr) + \langle Sv, w - v \rangle \\ &{}+ \varphi ( w ) - \varphi ( v ) + \frac{1}{s} \langle w - v, v - u \rangle \geq0, \forall w, c \in Q \biggr\} . \end{aligned}$$
(2.2)
Then, clearly, \(J_{s}^{G, S}\) satisfies (i)-(iv) of Lemma 3, and \(F ( J_{s}^{G, S} ) = \operatorname{GMEP} ( G, S, \varphi )\).

Lemma 4

Opial’s lemma [17]

Let H be a real Hilbert space, and \(\{ \mu_{n} \}\) be a sequence in H such that there exists a nonempty set \(W\subset H\) satisfying the following conditions:
  1. (i)

    for every \(\mu \in W\), \(\lim_{n \rightarrow\infty} \Vert \mu_{n} - \mu \Vert \) exists;

     
  2. (ii)

    any weak cluster point of the sequence \(\{ \mu_{n} \}\) belongs to W.

     

Then there exists \(w^{*} \in W\) such that \(\{ \mu_{n} \}\) converges weakly to \(w^{*}\).

Lemma 5

[18]

Let H be a real Hilbert space. Then, we have
$$\Vert x - y \Vert ^{2} = \Vert x \Vert ^{2} - \Vert y \Vert ^{2} -2 \langle x - y, y \rangle $$
and
$$\bigl\Vert \lambda x + ( 1- \lambda ) y \bigr\Vert ^{2} = \lambda \Vert x \Vert ^{2} + ( 1- \lambda ) \Vert y \Vert ^{2} - \lambda ( 1- \lambda ) \Vert x - y \Vert ^{2} $$
for all \(x, y \in H\) and \(\lambda \in [ 0,1 ]\).

3 Main results

Now, we give a new modified iterative algorithm to solve the split equality generalized mixed equilibrium problem. Moreover, we prove strong and weak convergence theorems for nonexpansive mappings in Hilbert spaces. Throughout this section, we always assume that:
  1. B1.

    \(H_{1}\), \(H_{2}\), and \(H_{3}\) are real Hilbert spaces, and \(C \subseteq H_{1}\) and \(Q \subseteq H_{2}\) are nonempty closed convex subsets;

     
  2. B2.

    \(F: C \times C \rightarrow \mathbb{R}\) and \(G: Q \times Q \rightarrow \mathbb{R}\) are bifunctions satisfying conditions (A1), (A2), (A4), (A5), and (A7);

     
  3. B3.

    \(T: C \rightarrow C\) and \(S: Q \times Q \rightarrow \mathbb{R}\) are mappings satisfying conditions (A3), (A6), and (A7);

     
  4. B4.

    \(\phi: C \rightarrow \mathbb{R}\cup \{ +\infty \}\) and \(\varphi: Q \rightarrow \mathbb{R}\cup \{ +\infty \}\) are proper lower semicontinuous and convex mappings such that \(C \cap \operatorname{dom}\phi \neq\emptyset\) and \(Q \cap \operatorname{dom} \varphi \neq\emptyset\);

     
  5. B5.

    \(P_{1}, P_{2}: H_{1} \rightarrow H_{1}\) and \(P_{3}, P_{4}: H_{2} \rightarrow H_{2}\) are nonexpansive mapping;

     
  6. B6.

    \(A: H_{1} \rightarrow H_{3}\) and \(B: H_{2} \rightarrow H_{3}\) are bounded linear mappings.

     
For an arbitrary initial value \(( x_{1}, y_{1} ) \in C \times Q\), define the sequence \(\{ ( x_{n}, y_{n} ) \}\) in \(C \times Q\) generated by
$$ \left \{ \textstyle\begin{array}{l} F ( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u ) + \phi ( u ) - \phi ( u_{n} ) \\ \quad {}+ \langle T u_{n}, u - u_{n} \rangle + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0, \\ G ( \lambda_{2} v_{n} + ( 1- \lambda_{2} ) c, v ) + \varphi ( v ) - \varphi ( v_{n} ) \\ \quad {}+ \langle S v_{n}, v - v_{n} \rangle + \frac{1}{r_{n}} \langle v - v_{n}, v_{n} - y_{n} \rangle \geq0, \\ x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P_{2} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P_{4} ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) \end{array}\displaystyle \right . $$
(3.1)
for all \(u, b \in C\) and \(v, c \in Q\), where \(n \geq1\), \(\lambda_{1}, \lambda_{2} \in ( 0,1 ]\), and the sequences \(\{ \delta_{n} \}\), \(\{ \alpha_{n} \}\), and \(\{ r_{n} \}\) satisfy the following conditions:
  1. C1.

    \(\{ \delta_{n} \}\) is a positive real sequence such that \(\delta_{n} \in ( \varepsilon, \frac{2}{\lambda_{A} + \lambda_{B}} - \varepsilon )\) for sufficiently small ε, where \(\lambda_{A}\) and \(\lambda_{B}\) are the spectral radii of \(A^{*} A\) and \(B^{*} B\), respectively;

     
  2. C2.

    \(\{ \alpha_{n} \}\) is a sequence in \(( 0,1 )\) such that, for some α, \(\beta \in ( 0,1 )\), \(0< \alpha \leq \alpha_{n} \leq \beta <1\);

     
  3. C3.

    \(\{ r_{n} \} \subset ( 0,\infty )\) is such that \(\liminf_{n \rightarrow\infty} r_{n} >0\) and \(\lim_{n \rightarrow\infty} \vert r_{n +1} - r_{n} \vert =0\).

     

Theorem 1

Let \(H_{1}\), \(H_{2}\), \(H_{3}\), F, G, T, S, \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), ϕ, φ, A, and B satisfy conditions (B1)-(B6). Let \(\{ ( x_{n}, y_{n} ) \}\) be a sequence generated by (3.1). If \(\mathcal{F} := \bigcap_{i=1}^{4} F ( P_{i} ) \cap \operatorname{SEGMEP} ( F,G, P_{i}, \phi, \varphi ) \neq\emptyset\), then:
  1. (i)

    the sequence \(\{ ( x_{n}, y_{n} ) \}\) converges weakly to a solution of problem (1.4);

     
  2. (ii)

    if \(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence \(\{ ( x_{n}, y_{n} ) \}\) converges strongly to a solution of problem (1.4).

     

Proof

(i) Let \(( x, y )\in \mathcal{F}\). So, \(x \in F ( P_{1} ) \cap F ( P_{2} )\) and \(y \in F ( P_{3} ) \cap F ( P_{4} )\). It is easy to see from Lemma 3 that
$$ \Vert u_{n} - x \Vert = \bigl\Vert J_{r_{n}}^{F, T} ( x_{n} ) - J_{r_{n}}^{F, T} ( x ) \bigr\Vert \leq \Vert x_{n} - x \Vert $$
(3.2)
and
$$ \Vert v_{n} - y \Vert = \bigl\Vert J_{r_{n}}^{F, T} ( y_{n} ) - J_{r_{n}}^{F, T} ( y ) \bigr\Vert \leq \Vert y_{n} - y \Vert . $$
(3.3)
Since \(P_{i}\), \(i =1,2,3,4\), are nonexpansive mappings and
$$\Vert x - y \Vert ^{2} = \Vert x \Vert ^{2} + \Vert y \Vert ^{2} -2 \langle y, x \rangle $$
for all \(x, y \in H\), we get from Lemma 5 that
$$\begin{aligned} \Vert x_{n +1} - x \Vert ^{2} =& \bigl\Vert ( 1- \alpha_{n} ) P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}+ \alpha_{n} P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - x \bigr\Vert ^{2} \\ =& \bigl\Vert ( 1- \alpha_{n} ) \bigl( P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - x \bigr) \\ &{}+ \alpha_{n} \bigl( P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - x \bigr) \bigr\Vert ^{2} \\ \leq& ( 1- \alpha_{n} ) \bigl\Vert u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) - x \bigr\Vert ^{2} \\ &{}+ \alpha_{n} \bigl\Vert u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) - x \bigr\Vert ^{2} \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{} - P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \\ =& \bigl\Vert u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) - x \bigr\Vert ^{2} \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{} - P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \\ =& \Vert u_{n} - x \Vert ^{2} + \delta_{n}^{2} \bigl\Vert A^{*} ( A u_{n} - B v_{n} ) \bigr\Vert ^{2} \\ &{}-2 \delta_{n} \bigl\langle A^{*} ( A u_{n} - B v_{n} ), u_{n} - x \bigr\rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{} - P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \\ \leq& \Vert x_{n} - x \Vert ^{2} + \delta_{n}^{2} \bigl\Vert A^{*} ( A u_{n} - B v_{n} ) \bigr\Vert ^{2} \\ &{}-2 \delta_{n} \bigl\langle A^{*} ( A u_{n} - B v_{n} ), u_{n} - x \bigr\rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2}. \end{aligned}$$
(3.4)
On the other hand, we have
$$\begin{aligned} \delta_{n}^{2} \bigl\Vert A^{*} ( A u_{n} - B v_{n} ) \bigr\Vert ^{2} =& \delta_{n}^{2} \bigl\langle A u_{n} - B v_{n}, A A^{*} ( A u_{n} - B v_{n} ) \bigr\rangle \\ \leq& \lambda_{A} \delta_{n}^{2} \Vert A u_{n} - B v_{n} \Vert ^{2}. \end{aligned}$$
(3.5)
So, it follows from (3.4) and (3.5) that
$$\begin{aligned} \Vert x_{n +1} - x \Vert ^{2} \leq& \Vert x_{n} - x \Vert ^{2} + \lambda_{A} \delta_{n}^{2} \Vert A u_{n} - B v_{n} \Vert ^{2} \\ &{}-2 \delta_{n} \langle A u_{n} - B v_{n}, A u_{n} - Ax \rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2}. \end{aligned}$$
(3.6)
In a similar way, we get
$$\begin{aligned} \Vert y_{n +1} - y \Vert ^{2} \leq& \Vert y_{n} - y \Vert ^{2} + \lambda_{B} \delta_{n}^{2} \Vert A u_{n} - B v_{n} \Vert ^{2} \\ &{}+2 \delta_{n} \langle A u_{n} - B v_{n}, B v_{n} - By \rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2}. \end{aligned}$$
(3.7)
By adding inequalities (3.6) and (3.7) side by side and using \(Ax = By\), we obtain
$$\begin{aligned}& \Vert x_{n +1} - x \Vert ^{2} + \Vert y_{n +1} - y \Vert ^{2} \\& \quad \leq \Vert x_{n} - x \Vert ^{2} + \Vert y_{n} - y \Vert ^{2} + \delta_{n}^{2} ( \lambda_{A} + \lambda_{B} ) \Vert A u_{n} - B v_{n} \Vert ^{2} \\& \qquad {}-2 \delta_{n} \Vert A u_{n} - B v_{n} \Vert ^{2} - \alpha_{n} ( 1- \alpha_{n} ) \bigl\{ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} + \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \bigr\} \\& \quad = \Vert x_{n} - x \Vert ^{2} + \Vert y_{n} - y \Vert ^{2} - \delta_{n} \bigl( 2- \delta_{n} ( \lambda_{A} + \lambda_{B} ) \bigr) \Vert A u_{n} - B v_{n} \Vert ^{2} \\& \qquad {}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\{ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} + \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \bigr\} . \end{aligned}$$
(3.8)
Let \(\xi_{n} ( x, y ) = \Vert x_{n} - x \Vert ^{2} + \Vert y_{n} - y \Vert ^{2}\). Thus, we have from (3.8) that
$$\begin{aligned} \xi_{n +1} ( x, y ) \leq& \xi_{n} ( x, y ) - \delta_{n} \bigl( 2- \delta_{n} ( \lambda_{A} + \lambda_{B} ) \bigr) \Vert A u_{n} - B v_{n} \Vert ^{2} \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\{ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} + \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \bigr\} . \end{aligned}$$
(3.9)
Since \(\alpha_{n} \in ( 0,1 )\) and \(\delta_{n} \in ( \varepsilon, \frac{2}{\lambda_{A} + \lambda_{B}} - \varepsilon )\), we get \(2- \delta_{n} ( \lambda_{A} + \lambda_{B} ) >0\). So, from (3.9) we obtain
$$\xi_{n +1} ( x, y ) \leq \xi_{n} ( x, y ). $$
Therefore, the sequence \(\{ \xi_{n} ( x, y ) \}\) is nonincreasing and lower bounded by 0. Hence, \(\lim_{n \rightarrow\infty} \xi_{n} ( x, y )\) exists. Let \(\lim_{n \rightarrow\infty} \xi_{n} ( x, y ) = \sigma ( x, y )\). So condition (i) of Lemma 4 is satisfied with \(\mu_{n} = ( x_{n}, y_{n} )\), \(\mu^{*} = ( x, y )\), and \(W =\mathcal{F}\). Since the sequence \(\{ \xi_{n} ( x, y ) \}\) converges to a finite limit, we have from inequality (3.9) that
$$\begin{aligned}& \lim_{n \rightarrow\infty} \Vert A u_{n} - B v_{n} \Vert =0, \end{aligned}$$
(3.10)
$$\begin{aligned}& \lim_{n \rightarrow\infty} \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert =0, \end{aligned}$$
(3.11)
and
$$ \lim_{n \rightarrow\infty} \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) - P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert =0. $$
(3.12)
Moreover, since \(\Vert x_{n} - x \Vert ^{2} \leq \xi_{n} ( x, y )\) and \(\Vert y_{n} - y \Vert ^{2} \leq \xi_{n} ( x, y )\), the sequences \(\{ x_{n} \}\) and \(\{ y_{n} \}\) are bounded, and \(\limsup_{n \rightarrow\infty} \Vert x_{n} - x \Vert \) and \(\limsup_{n \rightarrow\infty} \Vert y_{n} - y \Vert \) exist. Also, it follows from (3.2) and (3.3) that \(\limsup_{n \rightarrow\infty} \Vert u_{n} - x \Vert \) and \(\limsup_{n \rightarrow\infty} \Vert v_{n} - y \Vert \) exist. Let us assume that the sequences \(\{ x_{n} \}\) and \(\{ y_{n} \}\) converge weakly to points \(x^{*}\) and \(y^{*}\), respectively. So, by (3.10), the sequence \(\{ u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \}\) converges weakly to \(x^{*}\), and \(\{ v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \}\) converges weakly to \(y^{*}\). By Lemma 5 we get
$$\begin{aligned} \Vert x_{n +1} - x_{n} \Vert ^{2} =& \Vert x_{n +1} - x - x_{n} + x \Vert ^{2} \\ =& \Vert x_{n +1} - x \Vert ^{2} - \Vert x_{n} - x \Vert ^{2} -2 \langle x_{n +1} - x_{n}, x_{n} - x \rangle \\ =& \Vert x_{n +1} - x \Vert ^{2} - \Vert x_{n} - x \Vert ^{2} \\ &{}-2 \bigl\langle x_{n +1} - x^{*}, x_{n} - x \bigr\rangle +2 \bigl\langle x_{n} - x^{*}, x_{n} - x \bigr\rangle . \end{aligned}$$
Hence, we obtain
$$ \lim_{n \rightarrow\infty} \Vert x_{n +1} - x_{n} \Vert =0 $$
(3.13)
and, similarly,
$$ \lim_{n \rightarrow\infty} \Vert y_{n +1} - y_{n} \Vert =0. $$
(3.14)
By Lemma 3, since \(u_{n} = J_{r_{n}}^{F, T} ( x_{n} )\) and \(u_{n +1} = J_{r_{n +1}}^{F, T} ( x_{n +1} )\), we have that, for all \(u \in C\),
$$\begin{aligned}& F \bigl( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u \bigr) + \langle T u_{n}, u - u_{n} \rangle \\& \quad {}+ \phi ( u ) - \phi ( u_{n} ) + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0 \end{aligned}$$
(3.15)
and
$$\begin{aligned} \begin{aligned}[b] &F \bigl( \lambda_{1} u_{n +1} + ( 1- \lambda_{1} ) b, u \bigr) + \langle T u_{n +1}, u - u_{n +1} \rangle \\ &\quad {}+ \phi ( u ) - \phi ( u_{n +1} ) + \frac{1}{r_{n +1}} \langle u - u_{n +1}, u_{n +1} - x_{n +1} \rangle \geq0. \end{aligned} \end{aligned}$$
(3.16)
Taking \(u = u_{n}\) in (3.16) and \(u = u_{n +1}\) in (3.15) and adding the resulting inequalities side by side, we obtain
$$\begin{aligned} 0 \leq& F \bigl( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u_{n +1} \bigr) + F \bigl( \lambda_{1} u_{n +1} + ( 1- \lambda_{1} ) b, u_{n} \bigr) \\ &{}+ \langle T u_{n}, u_{n +1} - u_{n} \rangle + \langle T u_{n +1}, u_{n} - u_{n +1} \rangle \\ &{}+ \frac{1}{r_{n}} \langle u_{n +1} - u_{n}, u_{n} - x_{n} \rangle + \frac{1}{r_{n +1}} \langle u_{n} - u_{n +1}, u_{n +1} - x_{n +1} \rangle. \end{aligned}$$
Using conditions (A2)-(A3), we have
$$\begin{aligned} 0 \leq& \frac{1}{r_{n +1}} \langle u_{n} - u_{n +1}, u_{n +1} - x_{n +1} \rangle + \frac{1}{r_{n}} \langle u_{n +1} - u_{n}, u_{n} - x_{n} \rangle \\ \leq& \biggl\langle u_{n +1} - u_{n}, \frac{u_{n} - x_{n}}{r_{n}} - \frac{u_{n +1} - x_{n +1}}{r_{n +1}} \biggr\rangle \\ =& \biggl\langle u_{n +1} - u_{n}, u_{n} - u_{n +1} + u_{n +1} - x_{n} - \frac{r_{n}}{r_{n +1}} ( u_{n +1} - x_{n +1} ) \biggr\rangle \\ =& \langle u_{n +1} - u_{n}, u_{n} - u_{n +1} \rangle \\ &{}+ \biggl\langle u_{n +1} - u_{n}, x_{n +1} - x_{n} + \biggl( 1- \frac{r_{n}}{r_{n +1}} \biggr) ( u_{n +1} - x_{n +1} ) \biggr\rangle \\ =&- \Vert u_{n +1} - u_{n} \Vert ^{2} \\ &{}+ \biggl\langle u_{n +1} - u_{n}, x_{n +1} - x_{n} + \biggl( 1- \frac{r_{n}}{r_{n +1}} \biggr) ( u_{n +1} - x_{n +1} ) \biggr\rangle , \end{aligned}$$
which implies that
$$\Vert u_{n +1} - u_{n} \Vert ^{2} \leq \Vert u_{n +1} - u_{n} \Vert \biggl( \Vert x_{n +1} - x_{n} \Vert + \biggl\vert 1- \frac{r_{n}}{r_{n +1}} \biggr\vert \Vert u_{n +1} - x_{n +1} \Vert \biggr). $$
Thus, we get
$$ \Vert u_{n +1} - u_{n} \Vert \leq \Vert x_{n +1} - x_{n} \Vert + \biggl\vert 1- \frac{r_{n}}{r_{n +1}} \biggr\vert \Vert u_{n +1} - x_{n +1} \Vert . $$
(3.17)
Using (3.13) and (C3), from (3.17) we get
$$ \lim_{n \rightarrow\infty} \Vert u_{n +1} - u_{n} \Vert =0. $$
(3.18)
In a similar way, we get
$$ \lim_{n \rightarrow\infty} \Vert v_{n +1} - v_{n} \Vert =0. $$
(3.19)
On the other hand, from (3.6) and (3.7) we get
$$\begin{aligned} \begin{aligned}[b] \Vert x_{n +1} - x \Vert ^{2} \leq{}& \Vert u_{n} - x \Vert ^{2} + \delta_{n}^{2} \lambda_{A} \Vert A u_{n} - B v_{n} \Vert ^{2} \\ &{}-2 \delta_{n} \langle A u_{n} - B v_{n}, A u_{n} - Ax \rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \end{aligned} \end{aligned}$$
(3.20)
and
$$\begin{aligned} \Vert y_{n +1} - y \Vert ^{2} \leq& \Vert v_{n} - y \Vert ^{2} + \delta_{n}^{2} \lambda_{B} \Vert A u_{n} - B v_{n} \Vert ^{2} \\ &{}+2 \delta_{n} \langle A u_{n} - B v_{n}, B v_{n} - By \rangle \\ &{}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\Vert P_{2} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2}. \end{aligned}$$
(3.21)
Using \(Ax = By\) and adding inequalities (3.20) and (3.21) side by side, we have
$$\begin{aligned}& \Vert x_{n +1} - x \Vert ^{2} + \Vert y_{n +1} - y \Vert ^{2} \\& \quad \leq \Vert u_{n} - x \Vert ^{2} + \Vert v_{n} - y \Vert ^{2} \\& \qquad {}- \delta_{n} \bigl( 2- \delta_{n} ( \lambda_{A} + \lambda_{B} ) \bigr) \Vert A u_{n} - B v_{n} \Vert ^{2} \\& \qquad {}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\{ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \\& \qquad {}+ \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \bigr\} , \end{aligned}$$
(3.22)
where
$$\begin{aligned} \Vert u_{n} - x \Vert ^{2} =& \bigl\Vert J_{r_{n}}^{F, T} ( x_{n} ) - J_{r_{n}}^{F, T} ( x ) \bigr\Vert ^{2} \leq \langle x_{n} - x, u_{n} - x \rangle \\ =& \frac{1}{2} \bigl\{ \Vert x_{n} - x \Vert ^{2} + \Vert u_{n} - x \Vert ^{2} - \Vert x_{n} - u_{n} \Vert ^{2} \bigr\} \end{aligned}$$
(3.23)
and
$$\begin{aligned} \Vert v_{n} - y \Vert ^{2} =& \bigl\Vert J_{r_{n}}^{G, S} ( y_{n} ) - J_{r_{n}}^{F, T} ( y ) \bigr\Vert ^{2} \leq \langle y_{n} - y, v_{n} - y \rangle \\ =& \frac{1}{2} \bigl\{ \Vert y_{n} - y \Vert ^{2} + \Vert v_{n} - y \Vert ^{2} - \Vert y_{n} - v_{n} \Vert ^{2} \bigr\} . \end{aligned}$$
(3.24)
From (3.22)-(3.24) we conclude that
$$\begin{aligned}& \Vert x_{n} - u_{n} \Vert ^{2} + \Vert y_{n} - v_{n} \Vert ^{2} \\& \quad \leq \Vert x_{n} - x \Vert ^{2} - \Vert x_{n +1} - x \Vert ^{2} + \Vert y_{n} - y \Vert ^{2} - \Vert y_{n +1} - y \Vert ^{2} \\& \qquad {}- \delta_{n} \bigl( 2- \delta_{n} ( \lambda_{A} + \lambda_{B} ) \bigr) \Vert A u_{n} - B v_{n} \Vert ^{2} \\& \qquad {}- \alpha_{n} ( 1- \alpha_{n} ) \bigl\{ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \\& \qquad {}+ \bigl\Vert P_{3} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \\& \qquad {}- P_{4} \bigl( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert ^{2} \bigr\} . \end{aligned}$$
(3.25)
Using (3.10)-(3.14), we have
$$ \lim_{n \rightarrow\infty} \Vert x_{n} - u_{n} \Vert =0 $$
(3.26)
and
$$ \lim_{n \rightarrow\infty} \Vert y_{n} - v_{n} \Vert =0. $$
(3.27)
Hence, \(u_{n} \rightharpoonup x^{*}\) and \(v_{n} \rightharpoonup y^{*}\).
Since \(P_{i}\), \(i =1,2,3,4\), are nonexpansive mappings, we obtain
$$\begin{aligned} \Vert u_{n} - P_{1} u_{n} \Vert =& \Vert u_{n} - x_{n +1} + x_{n +1} - P_{1} u_{n} \Vert \\ \leq& \Vert u_{n} - x_{n +1} \Vert + \Vert x_{n +1} - P_{1} u_{n} \Vert \\ =& \Vert u_{n} - u_{n +1} + u_{n +1} - x_{n +1} \Vert \\ &{}+ \bigl\Vert ( 1- \alpha_{n} ) P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}+ \alpha_{n} P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - P_{1} u_{n} \bigr\Vert \\ \leq& \Vert u_{n} - u_{n +1} \Vert + \Vert u_{n +1} - x_{n +1} \Vert \\ &{}+ \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) - P_{1} u_{n} \bigr\Vert \\ &{}+ \alpha_{n} \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{}- P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert \\ \leq& \Vert u_{n} - u_{n +1} \Vert + \Vert u_{n +1} - x_{n +1} \Vert \\ &{}+ \vert \delta_{n} \vert \bigl\Vert A^{*} \bigr\Vert \Vert A u_{n} - B v_{n} \Vert + \alpha_{n} \bigl\Vert P_{1} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \\ &{} - P_{2} \bigl( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) \bigr) \bigr\Vert . \end{aligned}$$
Using (3.10), (3.11), (3.18), and (3.26), we have
$$ \lim_{n \rightarrow\infty} \Vert u_{n} - P_{1} u_{n} \Vert =0. $$
(3.28)
Similarly, using the same steps as before for \(P_{2}\), \(P_{3}\), and \(P_{4}\), we get
$$ \lim_{n \rightarrow\infty} \Vert u_{n} - P_{2} u_{n} \Vert =0,\qquad \lim_{n \rightarrow\infty} \Vert v_{n} - P_{3} v_{n} \Vert =0,\quad \mbox{and} \quad \lim_{n \rightarrow\infty} \Vert v_{n} - P_{4} v_{n} \Vert =0. $$
(3.29)
Since
$$\begin{aligned} \Vert x_{n} - P_{1} x_{n} \Vert =& \Vert x_{n} - u_{n} + u_{n} - P_{1} u_{n} + P_{1} u_{n} - P_{1} x_{n} \Vert \\ \leq& \Vert x_{n} - u_{n} \Vert + \Vert u_{n} - P_{1} u_{n} \Vert + \Vert u_{n} - x_{n} \Vert \\ =&2 \Vert x_{n} - u_{n} \Vert + \Vert u_{n} - P_{1} u_{n} \Vert , \end{aligned}$$
we have from (3.26) and (3.28) that
$$ \lim_{n \rightarrow\infty} \Vert x_{n} - P_{1} x_{n} \Vert =0. $$
(3.30)
Similarly, we have
$$ \lim_{n \rightarrow\infty} \Vert x_{n} - P_{2} x_{n} \Vert =0,\qquad \lim_{n \rightarrow\infty} \Vert y_{n} - P_{3} y_{n} \Vert =0,\quad \mbox{and} \quad \lim_{n \rightarrow\infty} \Vert y_{n} - P_{4} y_{n} \Vert =0. $$
(3.31)
Since the sequences \(\{ x_{n} \}\) and \(\{ y_{n} \}\) converge weakly to \(x^{*}\) and \(y^{*}\), respectively, and \(( I - P_{i} )\), \(i =1,2,3,4\), are demiclosed at zero, it follows from (3.30) and (3.31) that \(x^{*} \in F ( P_{1} ) \cap F ( P_{2} )\) and \(y^{*} \in F ( P_{3} ) \cap F ( P_{4} )\). On the other hand, it is well known that every Hilbert space satisfies Opial’s condition. So, we have that the weakly subsequential limit of \(\{ ( x_{n}, y_{n} ) \}\) is unique.
Now, we show that \(x^{*} \in \operatorname{GMEP} ( F, T, \phi )\) and \(y^{*} \in \operatorname{GMEP} ( G, S, \varphi )\). Since \(u_{n} = J_{r_{n}}^{F, T} ( x_{n} )\), we have
$$F \bigl( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u \bigr) + \langle T u_{n}, u - u_{n} \rangle + \phi ( u ) - \phi ( u_{n} ) + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0 $$
for all \(b, u \in C\) and \(\lambda \in ( 0,1 ]\). From conditions (A2) and (A3) we obtain
$$\begin{aligned} \phi ( u ) - \phi ( u_{n} ) + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq&- F \bigl( \lambda_{1} u_{n} + ( 1- \lambda_{1} ) b, u \bigr) - \langle T u_{n}, u - u_{n} \rangle \\ \geq& F \bigl( \lambda_{1} u + ( 1- \lambda_{1} ) b, u_{n} \bigr) + \langle Tu, u_{n} - u \rangle, \end{aligned}$$
and hence
$$\phi ( u ) - \phi ( u_{n_{k}} ) + \frac{1}{ r_{n_{k}}} \langle u - u_{n_{k}}, u_{n_{k}} - x_{n_{k}} \rangle \geq F \bigl( \lambda_{1} u + ( 1- \lambda_{1} ) b, u_{n_{k}} \bigr) + \langle Tu, u_{n_{k}} - u \rangle. $$
From (3.26) it is easy to see that \(u_{n_{k}} \rightharpoonup x^{*}\). So, we can write \(\lim_{k \rightarrow\infty} \frac{\Vert u_{n_{k}} - x_{n_{k}} \Vert }{r_{n_{k}}} =0\), and from the lower semicontinuity of ϕ we get
$$ F \bigl( \lambda_{1} u + ( 1- \lambda_{1} ) b, x^{*} \bigr) + \bigl\langle Tu, x^{*} - u \bigr\rangle + \phi \bigl( x^{*} \bigr) - \phi ( u ) \leq0 $$
(3.32)
for all \(b, u \in C\). Set \(u_{t} = tu + ( 1- t ) x^{*}\) for \(t \in ( 0,1 ]\) and \(u \in C\). Since C is a convex set, we have \(u_{t} \in C\). Hence, from (3.32) we have
$$ F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, x^{*} \bigr) + \bigl\langle T u_{t}, x^{*} - u_{t} \bigr\rangle + \phi \bigl( x^{*} \bigr) - \phi ( u_{t} ) \leq0. $$
(3.33)
Using inequality (3.33), the convexity of ϕ, and conditions (A1)-(A4), we get
$$\begin{aligned} 0 =& F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, u_{t} \bigr) + ( 1- t ) \langle T u_{t}, u_{t} - u_{t} \rangle + \phi ( u_{t} ) - \phi ( u_{t} ) \\ \leq& tF \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, u \bigr) + ( 1- t ) F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, x^{*} \bigr) \\ &{}+ t\phi ( u ) + ( 1- t ) \phi \bigl( x^{*} \bigr) - \phi ( u_{t} ) + ( 1- t ) \bigl\langle T u_{t}, u_{t} - x^{*} \bigr\rangle \\ &{}+ ( 1- t ) \bigl\langle T u_{t}, x^{*} - u_{t} \bigr\rangle \\ =& t \bigl\{ F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, u \bigr) + ( 1- t ) \bigl\langle T u_{t}, u - x^{*} \bigr\rangle + \phi ( u ) - \phi ( u_{t} ) \bigr\} \\ &{}+ ( 1- t ) \bigl\{ F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, x^{*} \bigr) + \bigl\langle T u_{t}, x^{*} - u_{t} \bigr\rangle + \phi \bigl( x^{*} \bigr) - \phi ( u_{t} ) \bigr\} \\ \leq& t \bigl\{ F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, u \bigr) + ( 1- t ) \bigl\langle T u_{t}, u - x^{*} \bigr\rangle + \phi ( u ) - \phi ( u_{t} ) \bigr\} , \end{aligned}$$
which implies that
$$F \bigl( \lambda_{1} u_{t} + ( 1- \lambda_{1} ) b, u \bigr) + ( 1- t ) \bigl\langle T u_{t}, u - x^{*} \bigr\rangle + \phi ( u ) - \phi ( u_{t} ) \geq0 $$
for all \(b, u \in C\). From the definition of \(u_{t}\) it is clear that \(u_{t} \rightarrow x^{*}\) as \(t \rightarrow0\). Using conditions (A5) and (A6) and the proper lower semicontinuity of ϕ, we obtain
$$F \bigl( \lambda_{1} x^{*} + ( 1- \lambda_{1} ) b, u \bigr) + ( 1- t ) \bigl\langle T x^{*}, u - x^{*} \bigr\rangle + \phi ( u ) - \phi \bigl( x^{*} \bigr) \geq0 $$
for all \(b, u \in C\), which shows that \(x^{*} \in \operatorname{GMEP} ( F, T, \phi )\). By using similar steps we have that \(y^{*} \in \operatorname{GMEP} ( G, S, \varphi )\).
Since \(A: H_{1} \rightarrow H_{3}\) and \(B: H_{2} \rightarrow H_{3}\) are bounded linear mappings and \(\{ u_{n} \}\) and \(\{ v_{n} \}\) converge weakly to \(x^{*}\) and \(y^{*}\), respectively, for arbitrary \(f \in H_{3}^{*}\), we have
$$f ( A u_{n} ) \rightarrow f \bigl( A x^{*} \bigr). $$
Similarly,
$$f ( B v_{n} ) \rightarrow f \bigl( B y^{*} \bigr). $$
Hence, we get
$$A u_{n} - B v_{n} \rightharpoonup A x^{*} - B y^{*}, $$
which implies that
$$\bigl\Vert A x^{*} - B y^{*} \bigr\Vert \leq \liminf_{n \rightarrow\infty} \Vert A u_{n} - B v_{n} \Vert =0, $$
so that \(A x^{*} = B y^{*}\). So, it follows that \(( x^{*}, y^{*} ) \in \operatorname{SEGMEP} ( F, G, T, S, \phi, \varphi )\). Therefore, \(( x^{*}, y^{*} ) \in \mathcal{F}\).

Finally, we conclude that, for each \(( x^{*}, y^{*} ) \in \mathcal{F}\), \(\lim_{n \rightarrow\infty} ( \Vert x_{n} - x^{*} \Vert + \Vert y_{n} - y^{*} \Vert )\) exists and each weak cluster point of the sequence \(\Vert ( x^{*}, y^{*} ) \Vert \) belongs to \(\mathcal{F}\). Let \(H = H_{1} \times H_{2}\) with norm \(\Vert ( x, y ) \Vert = \sqrt{\Vert x \Vert ^{2} + \Vert y \Vert ^{2}}\), \(W=\mathcal{F}\), \(\mu_{n} = ( x_{n}, y_{n} )\), and \(\mu = ( x^{*}, y^{*} )\). From Lemma 4 we see that there exists \(( \overline{x}, \overline{y} ) \in \mathcal{F}\) such that \(x_{n} \rightharpoonup \overline{x}\) and \(y_{n} \rightharpoonup \overline{y}\). Therefore, the sequence \(\{ ( x_{n}, y_{n} ) \}\) generated by the iterative algorithm (3.1) converges weakly to a solution of problem (1.4) in \(\mathcal{F}\). This completes the proof.

(ii) Now, we prove the strong convergence of the sequence \(\{ ( x_{n}, y_{n} ) \}\) generated by the iterative algorithm (3.1) under the demicompact condition.

Since \(P_{i}\), \(i =1,2,3,4\), are demicompact, \(\{ x_{n} \}\) and \(\{ y_{n} \}\) are bounded sequences, and \(\lim_{n \rightarrow\infty} \Vert x_{n} - P_{1} x_{n} \Vert =0\), \(\lim_{n \rightarrow\infty} \Vert x_{n} - P_{2} x_{n} \Vert =0\), \(\lim_{n \rightarrow\infty} \Vert y_{n} - P_{3} y_{n} \Vert =0\), and \(\lim_{n \rightarrow\infty} \Vert y_{n} - P_{4} y_{n} \Vert =0\), there exist subsequences \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) and \(\{ y_{n_{k}} \}\) of \(\{ y_{n} \}\) such that \(\{ x_{n_{k}} \}\) and \(\{ y_{n_{k}} \}\) converge strongly to some points \(u^{*}\) and \(v^{*}\), respectively. The weak convergence of \(\{ x_{n_{k}} \}\) and \(\{ y_{n_{k}} \}\) to \(x^{*}\) and \(y^{*}\), respectively, implies that \(x^{*} = u^{*}\) and \(y^{*} = v^{*}\). It follows from the demiclosedness of \(P_{i}\) that \(x^{*} \in F ( P_{1} )\cap F ( P_{2} )\) and \(y^{*} \in F ( P_{3} )\cap F ( P_{4} )\). Using similar steps to the previous ones, we get that \(x^{*} \in \operatorname{GMEP} ( F, T, \phi )\) and \(y^{*} \in \operatorname{GMEP} ( G, S, \varphi )\). Thus, we have
$$\bigl\Vert A x^{*} - B y^{*} \bigr\Vert = \lim _{k \rightarrow\infty} \Vert A x_{n_{k}} - B y_{n_{k}} \Vert =0, $$
which implies that \(A x^{*} = B y^{*}\). Hence, \(( x^{*}, y^{*} ) \in \mathcal{F}\). On the other hand, since \(\xi_{n} ( x, y ) = \Vert x_{n} - x \Vert ^{2} + \Vert y_{n} - y \Vert ^{2}\) for \(( x, y )\in \mathcal{F}\), we know that \(\lim_{k \rightarrow\infty} \xi_{n_{k}} ( x^{*}, y^{*} )=0\). From conjecture (i) we see that \(\lim_{n \rightarrow\infty} \xi_{n} ( x^{*}, y^{*} )\) exists; therefore, \(\lim_{n \rightarrow\infty} \xi_{n} ( x^{*}, y^{*} )=0\). So, the iterative scheme (3.1) converges strongly to a solution of problem (1.4). This completes the proof of conjecture (ii). □

Taking \(F = G = T = S = \phi = \varphi =0\) in Theorem 1, we get the following convergence theorem for the split equality problem (1.10).

Corollary 1

Let \(H_{1}\), \(H_{2}\), \(H_{3}\), \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), A, and B satisfy conditions (B1), (B5), and (B6). For an arbitrary initial value \(( x_{1}, y_{1} ) \in C\times Q\), define the sequence \(\{ ( x_{n}, y_{n} ) \}\) in \(C\times Q\) generated by
$$\left \{ \textstyle\begin{array}{l} x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( x_{n} - \delta_{n} A^{*} ( A x_{n} - B y_{n} ) ) + \alpha_{n} P_{2} ( x_{n} - \delta_{n} A^{*} ( A x_{n} - B y_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( y_{n} + \delta_{n} B^{*} ( A x_{n} - B y_{n} ) ) + \alpha_{n} P_{4} ( y_{n} + \delta_{n} B^{*} ( A x_{n} - B y_{n} ) ), \end{array}\displaystyle \right . $$
where \(n \geq1\), and the sequences \(\{ \delta_{n} \}\) and \(\{ \alpha_{n} \}\) satisfy conditions (C1) and (C2), respectively. If \(\mathcal{F}:= \bigcap_{i =1}^{4} F ( P_{i} ) \cap \operatorname{SEP} \neq\emptyset\), then:
  1. (i)

    The sequence \(\{ ( x_{n}, y_{n} ) \}\) converges weakly to a solution of problem (1.10);

     
  2. (ii)

    If \(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence \(\{ ( x_{n}, y_{n} ) \}\) converges strongly to a solution of problem (1.10).

     

Taking \(B = I\) and \(H_{2} = H_{3}\) in Corollary 1, we obtain the following convergence theorem for the split feasibility problem (1.11).

Corollary 2

Let \(H_{1}\), \(H_{2}\), \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), and A satisfy conditions (B1), (B5), and (B6) with \(A: H_{1} \rightarrow H_{2}\). For an arbitrary initial value \(( x_{1}, y_{1} ) \in C\times Q\), define the sequence \(\{ ( x_{n}, y_{n} ) \}\) in \(C\times Q\) generated by
$$\left \{ \textstyle\begin{array}{l} x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( x_{n} - \delta_{n} A^{*} ( A x_{n} - y_{n} ) ) + \alpha_{n} P_{2} ( x_{n} - \delta_{n} A^{*} ( A x_{n} - y_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( y_{n} + \delta_{n} ( A x_{n} - y_{n} ) ) + \alpha_{n} P_{4} ( y_{n} + \delta_{n} ( A x_{n} - y_{n} ) ), \quad n \geq1, \end{array}\displaystyle \right . $$
where \(\{ \delta_{n} \}\) is a positive real sequence such that \(\delta_{n} \in ( \varepsilon, \frac{1}{\lambda_{A}} - \varepsilon )\) for sufficiently small ε, where \(\lambda_{A}\) denotes the spectral radius of \(A^{*} A\), and \(\{ \alpha_{n} \}\) satisfy condition (C2). If \(\mathcal{F}:= \bigcap_{i =1}^{4} F ( P_{i} ) \cap \operatorname{SFP} \neq\emptyset\), then:
  1. (i)

    The sequence \(\{ ( x_{n}, y_{n} ) \}\) converges weakly to a solution of problem (1.11);

     
  2. (ii)

    If \(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence \(\{ ( x_{n}, y_{n} ) \}\) converges strongly to a solution of problem (1.11).

     

4 Applications

Let C be a nonempty closed convex subset of a real Hilbert space H, and \(\psi: C \rightarrow C\) be a convex and differentiable mapping. It is known that the convex differentiable minimization problem is to find \(x^{*} \in C\) such that
$$ \min_{x \in C} \psi ( x ) = \psi \bigl( x^{*} \bigr). $$
(4.1)
Also, it is well known that a point \(x^{*}\) is a solution of problem (4.1) if and only if
$$ \bigl\langle \nabla \psi \bigl( x^{*} \bigr), y - x^{*} \bigr\rangle \geq0 $$
(4.2)
for all \(y \in C\). Problem (4.2) is called the classical variational inequality problem. If we get \(F ( x^{*}, y ) = \langle \nabla \psi ( x^{*} ), y - x^{*} \rangle\), then the equilibrium problem (1.1) and the variational inequality problem (4.2) have the same solution.
In 2015, Rahaman et al. [3] introduced the split equality mixed convex differentiable optimization problem of finding \(x^{*} \in C\) and \(y^{*} \in Q\) such that
$$\begin{aligned}& \bigl\langle \nabla \psi \bigl( x^{*} \bigr), x - x^{*} \bigr\rangle + \bigl\langle T \bigl( x^{*} \bigr), x - x^{*} \bigr\rangle + \phi ( x ) - \phi \bigl( x^{*} \bigr) \geq0,\quad \forall x \in C, \\& \bigl\langle \nabla \sigma \bigl( y^{*} \bigr), y - y^{*} \bigr\rangle + \bigl\langle S \bigl( y^{*} \bigr), y - y^{*} \bigr\rangle + \varphi ( y ) - \varphi \bigl( y^{*} \bigr) \geq0,\quad \forall y \in Q, \quad \mbox{and} \\& A x^{*} = B y^{*}, \end{aligned}$$
(4.3)
where \(\psi: C \rightarrow H_{1}\) and \(\sigma: Q \rightarrow H_{2}\) are convex differentiable mappings. The set of solutions of the split equality mixed convex differentiable optimization problem (4.3) is denoted by \(\operatorname{SEMCDOP} ( \psi, \sigma, T, S, \phi, \varphi )\). If \(T =0\), then this problem is reduced to the split equality mixed variational inequality problem introduced by Ma et al. [4] in 2015. Also, if \(B = I\) and \(H_{2} = H_{3}\), then problem (4.3) is reduced to the split mixed convex differentiable optimization problem of finding \(x^{*} \in C\) such that
$$\bigl\langle \nabla \psi \bigl( x^{*} \bigr), x - x^{*} \bigr\rangle + \bigl\langle T \bigl( x^{*} \bigr), x - x^{*} \bigr\rangle + \phi ( x ) - \phi \bigl( x^{*} \bigr) \geq0,\quad \forall x \in C, $$
and such that \(A x^{*} = y^{*} \in Q\) solves
$$ \bigl\langle \nabla \sigma \bigl( y^{*} \bigr), y - y^{*} \bigr\rangle + \bigl\langle S \bigl( y^{*} \bigr), y - y^{*} \bigr\rangle + \varphi ( y ) - \varphi \bigl( y^{*} \bigr) \geq0,\quad \forall y \in Q. $$
(4.4)
The solution set of this problem is denoted by \(\operatorname{SMCDOP} ( \psi, \sigma, T, S, \phi, \varphi )\).

Since the gradients ψ and σ are monotone mappings, if \(F ( x^{*}, y ) = \langle \nabla \psi ( x^{*} ), x - x^{*} \rangle\), \(G ( x, y^{*} ) = \langle \nabla \sigma ( y^{*} ), y - y^{*} \rangle\), and \(\lambda_{1} = \lambda_{2} =1\), then F and G satisfy condition (B2). So, we can give the following result.

Theorem 2

Let \(H_{1}\), \(H_{2}\), \(H_{3}\), T, S, ϕ, φ, \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), A, and B satisfy conditions (B1)-(B6) except (B2). Suppose that the mappings \(\psi:C\rightarrow H_{1}\) and \(\sigma:Q\rightarrow H_{2}\) are convex and differentiable mappings. For an arbitrary initial value \(( x_{1}, y_{1} ) \in C\times Q\), define the sequence \(\{ ( x_{n}, y_{n} ) \}\) in \(C\times Q\) generated by
$$\left \{ \textstyle\begin{array}{l} \langle \nabla \psi ( u_{n} ), u - u_{n} \rangle + \phi ( u ) - \phi ( u_{n} ) + \langle T u_{n}, u - u_{n} \rangle + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0, \\ \langle \nabla \sigma ( v_{n} ), v - v_{n} \rangle + \varphi ( v ) - \varphi ( v_{n} ) + \langle S v_{n}, v - v_{n} \rangle + \frac{1}{r_{n}} \langle v - v_{n}, v_{n} - y_{n} \rangle \geq0, \\ x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P_{2} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P_{4} ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) \end{array}\displaystyle \right . $$
for all \(u \in C\) and \(v \in Q\), where \(n \geq1\), and the sequences \(\{ \delta_{n} \}\), \(\{ \alpha_{n} \}\), and \(\{ r_{n} \}\) satisfy conditions (C1)-(C3), respectively. If \(\mathcal{F}:= \bigcap_{i =1}^{4} F ( P_{i} ) \cap \operatorname{SEMCDOP} ( \psi, \sigma, T, S, \phi, \varphi )\neq\emptyset\), then:
  1. (i)

    The sequence \(\{ ( x_{n}, y_{n} ) \}\) converges weakly to a solution of problem (4.3);

     
  2. (ii)

    If \(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence \(\{ ( x_{n}, y_{n} ) \}\) converges strongly to a solution of problem (4.3).

     

In Theorem 2, if we take \(B = I\) and \(H_{2} = H_{3}\), then we get the following result.

Corollary 3

Let \(H_{1}\), \(H_{2}\), T, S, ϕ, φ, \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), and A satisfy conditions (B1)-(B6) except (B2) with \(A: H_{1} \rightarrow H_{2}\). Suppose that the mappings \(\psi:C\rightarrow H_{1}\) and \(\sigma:Q\rightarrow H_{2}\) are convex and differentiable mappings. For an arbitrary initial value \(( x_{1}, y_{1} ) \in C\times Q\), define the sequence \(\{ ( x_{n}, y_{n} ) \}\) in \(C\times Q\) generated by
$$\left \{ \textstyle\begin{array}{l} \langle \nabla \psi ( u_{n} ), u - u_{n} \rangle + \phi ( u ) - \phi ( u_{n} ) + \langle T u_{n}, u - u_{n} \rangle + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0, \\ \langle \nabla \sigma ( v_{n} ), v - v_{n} \rangle + \varphi ( v ) - \varphi ( v_{n} ) + \langle S v_{n}, v - v_{n} \rangle + \frac{1}{r_{n}} \langle v - v_{n}, v_{n} - y_{n} \rangle \geq0, \\ x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - v_{n} ) ) + \alpha_{n} P_{2} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - v_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( v_{n} + \delta_{n} ( A u_{n} - v_{n} ) ) + \alpha_{n} P_{4} ( v_{n} + \delta_{n} ( A u_{n} - v_{n} ) ) \end{array}\displaystyle \right . $$
for all \(u \in C\) and \(v \in Q\), where \(n \geq1\), \(\{ \delta_{n} \}\) is a positive real sequences such that \(\delta_{n} \in ( \varepsilon, \frac{1}{\lambda_{A}} - \varepsilon )\) for sufficiently small ε, where \(\lambda_{A}\) denotes the spectral radius of \(A^{*} A\), and the sequences \(\{ \alpha_{n} \}\) and \(\{ r_{n} \}\) satisfy conditions (C2) and (C3), respectively. If \(\mathcal{F}:= \bigcap_{i =1}^{4} F ( P_{i} ) \cap \operatorname{SMCDOP} ( \psi, \sigma, T, S, \phi, \varphi )\neq\emptyset\), then:
  1. (i)

    The sequence \(\{ ( x_{n}, y_{n} ) \}\) converges weakly to a solution of problem (4.4);

     
  2. (ii)

    If \(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence \(\{ ( x_{n}, y_{n} ) \}\) converges strongly to a solution of problem (4.4).

     

In Theorem 1, if we take \(F = G = T = S =0\), then we have the following result for the split equality convex minimization problem (1.8).

Theorem 3

Let \(H_{1}\), \(H_{2}\), \(H_{3}\), ϕ, φ, \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), A, and B satisfy conditions (B1), (B4), (B5), and (B6). For an arbitrary initial value \(( x_{1}, y_{1} ) \in C\times Q\), define the sequence \(\{ ( x_{n}, y_{n} ) \}\) in \(C\times Q\) generated by
$$\left \{ \textstyle\begin{array}{l} \phi ( u ) - \phi ( u_{n} ) + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0, \\ \varphi ( v ) - \varphi ( v_{n} ) + \frac{1}{r_{n}} \langle v - v_{n}, v_{n} - y_{n} \rangle \geq0, \\ x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P_{2} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P_{4} ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) \end{array}\displaystyle \right . $$
for all \(u \in C\) and \(v \in Q\), where \(n \geq1\), and the sequences \(\{ \delta_{n} \}\), \(\{ \alpha_{n} \}\), and \(\{ r_{n} \}\) satisfy conditions (C1)-(C3), respectively. If \(\mathcal{F}:= \bigcap_{i =1}^{4} F ( P_{i} ) \cap \operatorname{SECMP} ( \phi, \varphi )\neq\emptyset\), then:
  1. (i)

    The sequence \(\{ ( x_{n}, y_{n} ) \}\) converges weakly to a solution of problem (1.8);

     
  2. (ii)

    If \(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence \(\{ ( x_{n}, y_{n} ) \}\) converges strongly to a solution of problem (1.8).

     

If we take \(B = I\) and \(H_{2} = H_{3}\) in Theorem 3, then we get the following result for the split convex minimization problem (1.9).

Corollary 4

Let \(H_{1}\), \(H_{2}\), \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), ϕ, φ, and A satisfy conditions (B1), (B4), (B5), and (B6) with \(A: H_{1} \rightarrow H_{2}\). For an arbitrary initial value \(( x_{1}, y_{1} ) \in C\times Q\), define the sequence \(\{ ( x_{n}, y_{n} ) \}\) in \(C\times Q\) generated by
$$\left \{ \textstyle\begin{array}{l} \phi ( u ) - \phi ( u_{n} ) + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0, \\ \varphi ( v ) - \varphi ( v_{n} ) + \frac{1}{r_{n}} \langle v - v_{n}, v_{n} - y_{n} \rangle \geq0, \\ x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - v_{n} ) ) + \alpha_{n} P_{2} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - v_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( v_{n} + \delta_{n} ( A u_{n} - v_{n} ) ) + \alpha_{n} P_{4} ( v_{n} + \delta_{n} ( A u_{n} - v_{n} ) ) \end{array}\displaystyle \right . $$
for all \(u \in C\) and \(v \in Q\), where \(n \geq1\), \(\{ \delta_{n} \}\) is a positive real sequences such that \(\delta_{n} \in ( \varepsilon, \frac{1}{\lambda_{A}} - \varepsilon )\) for sufficiently small ε, where \(\lambda_{A}\) denotes the spectral radius of \(A^{*} A\), and the sequences \(\{ \alpha_{n} \}\) and \(\{ r_{n} \}\) satisfy conditions (C2) and (C3), respectively. If \(\mathcal{F}:= \bigcap_{i =1}^{4} F ( P_{i} ) \cap \operatorname{SCMP} ( \phi, \varphi )\neq\emptyset\), then:
  1. (i)

    The sequence \(\{ ( x_{n}, y_{n} ) \}\) converges weakly to a solution of problem (1.9);

     
  2. (ii)

    If \(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence \(\{ ( x_{n}, y_{n} ) \}\) converges strongly to a solution of problem (1.9).

     

In Theorem 1, if we take \(T = S =0\) and \(\lambda_{1} = \lambda_{2} =1\), then we have the following convergence result for the split equality mixed equilibrium problem (1.5).

Theorem 4

Let \(H_{1}\), \(H_{2}\), \(H_{3}\), F, G, ϕ, φ, \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), A, and B satisfy conditions (B1)-(B6) except (B3). For an arbitrary initial value \(( x_{1}, y_{1} ) \in C\times Q\), define the sequence \(\{ ( x_{n}, y_{n} ) \}\) in \(C\times Q\) generated by
$$\left \{ \textstyle\begin{array}{l} F ( u_{n}, u ) + \phi ( u ) - \phi ( u_{n} ) + \frac{1}{r_{n}} \langle u - u_{n}, u_{n} - x_{n} \rangle \geq0, \\ G ( v_{n}, v ) + \varphi ( v ) - \varphi ( v_{n} ) + \frac{1}{r_{n}} \langle v - v_{n}, v_{n} - y_{n} \rangle \geq0, \\ x_{n +1} = ( 1- \alpha_{n} ) P_{1} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P_{2} ( u_{n} - \delta_{n} A^{*} ( A u_{n} - B v_{n} ) ), \\ y_{n +1} = ( 1- \alpha_{n} ) P_{3} ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) + \alpha_{n} P_{4} ( v_{n} + \delta_{n} B^{*} ( A u_{n} - B v_{n} ) ) \end{array}\displaystyle \right . $$
for all \(u \in C\) and \(v \in Q\), where \(n \geq1\), and the sequences \(\{ \delta_{n} \}\), \(\{ \alpha_{n} \}\), and \(\{ r_{n} \}\) satisfy conditions (C1)-(C3), respectively. If \(\mathcal{F}:= \bigcap_{i =1}^{4} F ( P_{i} ) \cap \operatorname{SEMEP} ( F, G, \phi, \varphi ) \neq\emptyset\), then:
  1. (i)

    The sequence \(\{ ( x_{n}, y_{n} ) \}\) converges weakly to a solution of problem (1.5);

     
  2. (ii)

    If \(P_{i}\), \(i =1,2,3,4\), are demicompact, then the sequence \(\{ ( x_{n}, y_{n} ) \}\) converges strongly to a solution of problem (1.5).

     

Declarations

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)
Department of Mathematics, Faculty of Science, Erzurum Technical University

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