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Split equality fixed point problem for quasipseudocontractive mappings with applications
Fixed Point Theory and Applications volume 2015, Article number: 208 (2015)
Abstract
In this paper, we consider a split equality fixed point problem for quasipseudocontractive mappings which includes split feasibility problem, split equality problem, split fixed point problem etc., as special cases. A unified framework for the study of this kind of problems and operators is provided. The results presented in the paper extend and improve many recent results.
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:
where \(A:H_{1}\rightarrow H_{2}\) is a bounded linear operator. In 1994, Censor and Elfving [1] first introduced the SFP in finitedimensional 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 infinitedimensional real Hilbert space can be found in [2, 4, 6–10].
Recently, Moudafi [11–13] introduced the following split equality feasibility problem (SEFP):
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 intensitymodulated radiation therapy.
In order to solve split equality feasibility problem (1.2), Moudafi [11] introduced the following simultaneous iterative method:
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
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):
He also studied the weak convergence of the sequences generated by scheme (1.5) under the condition that T and S are firmly quasinonexpansive mappings. Very recently, Che and Li [14] proposed the following iterative algorithm for finding a solution of SEFPP (1.4):
They also established the weak convergence of the scheme (1.6) under the condition that the operators T and S are quasinonexpansive mappings.
The purpose of this paper is twofold. First, we will consider split equality fixed point problem (1.4) for the class of quasipseudocontractive mappings which is more general than the classes of quasinonexpansive 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:
Our results provide a unified framework for the study of this kind of problems and this class of operators.
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:

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

(ii)
Quasinonexpansive if \(\operatorname{Fix}(T) \neq \emptyset\) and
$$\bigl\ Tx  x^{*}\bigr\ \le\bigl\ x  x^{*}\bigr\ \quad\forall x \in C \mbox{ and } x^{*} \in \operatorname{Fix}(T). $$ 
(iii)
Firmly nonexpansive if
$$\Tx Ty\^{2} \le\x  y\^{2}  \bigl\ (I T)x  (IT)y \bigr\ ^{2} \quad \forall x, y \in C. $$ 
(iv)
Firmly quasinonexpansive if \(\operatorname{Fix}(T) \neq \emptyset\) and
$$\bigl\ Tx  x^{*}\bigr\ ^{2} \le\bigl\ x  x^{*}\bigr\ ^{2}  \bigl\ (I T)x \bigr\ ^{2} \quad \forall x \in C \mbox{ and } x^{*} \in \operatorname{Fix}(T). $$ 
(v)
Strictly pseudocontractive if there exists \(k \in[0, 1)\) such that
$$\Tx  Ty\^{2} \le\x  y\^{2} + k\bigl\ (I  T)x  (I  T)y \bigr\ ^{2} \quad \forall x, y \in C. $$ 
(vi)
Directed if \(\operatorname{Fix}(T) \neq \emptyset\) and \(\langle Tx  x^{*}, Tx x \rangle\le0\) \(\forall x \in C\) and \(x^{*} \in \operatorname{Fix}(T)\).

(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
That is to say that the class of directed mappings coincides with that of firmly quasinonexpansive 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 quasinonexpansive mappings, and the strictly pseudocontractive mappings with fixed points as special cases.
Definition 2.4
An operator \(T : C \to C\) is said to be pseudocontractive if
The interest of pseudocontractive operators lies in their connection with monotone mappings, namely, T is a pseudocontraction if and only if \(I  T\) is a monotone mapping. It is well known that T is pseudocontractive if and only if
Definition 2.5
An operator \(T : C \to C\) is said to be quasipseudocontractive if \(\operatorname{Fix}(T) \neq \emptyset\) and
It is obvious that the class of quasipseudocontractive 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 semicompact 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:
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 \(x^{*}\),
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
where \(\{\gamma_{n}\}\) is a sequence in \((0, 1)\) and \(\{\delta_{n}\}\) is a sequence such that

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

(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 LLipschitzian mapping with \(L \ge1\). Denote
If \(0 < \xi< \eta < \frac{1}{1 + \sqrt{1 + L^{2}}}\), then the following conclusions hold:

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

(2)
If T is demiclosed at 0, then K is also demiclosed at 0.

(3)
In addition, if \(T : H \to H\) is quasipseudocontractive, then the mapping K is quasinonexpansive, 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
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 LLipschizian, we have
Simplifying it, we have
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]). □
Main results
Throughout this section, we assume that:

(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)
\(T : H_{1} \to H_{1}\) and \(S : H_{2} \to H_{2}\) are two LLipschitzian and quasipseudocontractive 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:
In the sequel we use Γ to denote the set of solutions of (3.1), that is,
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
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:

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

(ii)
\(0 < a < \xi_{n} < \eta_{n} < b < \frac{1}{1 + \sqrt{1 + L^{2}}} \forall n \ge1\).
Then the following conclusions hold:

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

(II)
In addition, if S, T are also semicompact, 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
Similarly, from (3.3)(c), we have
Put
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)
Both K and G are quasinonexpansive;

(2)
\(\operatorname{Fix}(K) = \operatorname{Fix}(T)\) and \(\operatorname{Fix}(G) = \operatorname{Fix}(S)\);

(3)
K and G demiclosed at 0.
Hence from (3.3)(b) and (2.1) we have
Similarly from (3.3)(c) and (2.1) we have
Adding (3.6) and (3.7) and by virtue of (3.4) and (3.5), we have
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
hence (3.8) can be written as
This implies that \(\{X_{n}(p, q)\}\) is a nonincreasing sequence, hence the limit \(\lim_{n \to\infty}X_{n}(p, q)\) exists. Therefore the following limits exist:
Rewritten (3.10) as
Letting \(n \to\infty\) and taking the limit in (3.12), we have
This together with (3.13) shows that
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 semicontinuity of squared norm, we have
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
Since S, T are semicompact, 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 \(Ax^{*} = By^{*}\). □
Applications
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
We will denote the solution set of (4.1) by \(VI(M,C)\).
A mapping \(M : C \to H_{1}\) is said to be αinversestrongly monotone if there exists a constant \(\alpha> 0\) such that
It is easy to see that if M is αinversestrongly 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):

(A1)
\(h(x,x) = 0\), for all \(x\in C\);

(A2)
h is monotone, i.e., \(h(x,y) + h(y, x) \le0\) for all \(x, y \in C\);

(A3)
\(\limsup_{t\downarrow0} h(tz + (1t)x, y) \le h(x,y)\) for all \(x, y, z \in C \);

(A4)
for each \(x\in C\), \(y \mapsto h(x,y)\) is convex and lower semicontinuous.
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
Proposition 4.1
[17]
The resolvent operator \(R_{\lambda, h}\) of the equilibrium function h has the following properties :

(1)
\(R_{\lambda, h}\) is singlevalued;

(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)
\(R_{\lambda, h}\) 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 αinversestrongly monotone mappings. The socalled split equality variational inequality problem with respect to T and S is to find \(x^{*} \in C\) and \(y^{*} \in Q\) such that
In the sequel we use Ω to denote the solution set of split equality variational inequality problem (4.4), i.e.,
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
and
It follows from Proposition 4.1 that
and so \(R_{\lambda, f}\) and \(R_{\lambda, g}\) both are quasipseudocontractive and 1Lipschitzian. 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:
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
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).
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 semicontinuous 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 socalled split equality convex minimization problem for ϕ and ψ is to find \(x^{*} \in C\), \(y^{*} \in Q\) such that
In the sequel, we denote by Ω the solution set of split equality convex minimization problem (4.9), i.e.,
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:
and
By the same argument as given in Section 4.1, we know that
Therefore the split equality convex minimization problem for ϕ and ψ is equivalent to the following split equality fixed point problem:
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
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).
References
 1.
Censor, Y, Elfving, T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221239 (1994)
 2.
Byrne, C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 18, 441453 (2002)
 3.
Censor, Y, Bortfeld, T, Martin, N, Trofimov, A: A unified approach for inversion problem in intensitymodulated radiation therapy. Phys. Med. Biol. 51, 23532365 (2006)
 4.
Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiplesets split feasibility problem and its applications. Inverse Probl. 21, 20712084 (2005)
 5.
Censor, Y, Motova, A, Segal, A: Perturbed projections and subgradient projections for the multiplesets split feasibility problem. J. Math. Anal. Appl. 327, 12441256 (2007)
 6.
Xu, HK: A variable Krasnosel’skiiMann algorithm and the multiplesets split feasibility problem. Inverse Probl. 22, 20212034 (2006)
 7.
Yang, Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 20, 12611266 (2004)
 8.
Zhao, J, Yang, Q: Several solution methods for the split feasibility problem. Inverse Probl. 21, 17911799 (2005)
 9.
Chang, Ss, Agarwal, RP: Strong convergence theorems of general split equality problems for quasinonexpansive mappings. J. Inequal. Appl. 2014, Article ID 367 (2014)
 10.
Chang, Ss, Wang, L, Tang, YK, Wang, G: Moudafi’s open question and simultaneous iterative algorithm for general split equality variational inclusion problems and general split equality optimization problems. Fixed Point Theory Appl. 2014, Article ID 215 (2014)
 11.
Moudafi, A: A relaxed alternating CQ algorithm for convex feasibility problems. Nonlinear Anal. 79, 117121 (2013)
 12.
Moudafi, A, AlShemas, E: Simultaneous iterative methods forsplit equality problem. Trans. Math. Program. Appl. 1, 111 (2013)
 13.
Moudafi, A: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275283 (2011)
 14.
Che, H, Li, M: A simultaneous iterative method for split equality problems of two finite families of strictly pseudononspreading mappings without prior knowledge of operator norms. Fixed Point Theory Appl. 2015, Article ID 1 (2015)
 15.
Bauschke, HH, Combettes, PL: A weaktostrong convergence principle for Fejermonotone methods in Hilbert spaces. Math. Oper. Res. 26, 248264 (2001)
 16.
Yao, Y, Liou, YC, Yao, JC: Split common fixed point problem for two quasipseudocontractive operators and its algorithm construction. Fixed Point Theory Appl. 2015, Article ID 127 (2015)
 17.
Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63(14), 123145 (1994)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
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MSC
 49J40
 49J52
 47J20
Keywords
 split equality fixed point problem
 quasipseudocontractive mapping
 quasinonexpansive mapping
 directed mapping
 demicontractive mapping