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Strong convergence theorems of general split equality problems for quasinonexpansive mappings
Journal of Inequalities and Applications volume 2014, Article number: 367 (2014)
Abstract
The purpose of this paper is to introduce and study the general split equality problem and general split equality fixed point problem in the setting of infinitedimensional Hilbert spaces. Under suitable conditions, we prove that the sequences generated by the proposed new algorithm converges strongly to a solution of the general split equality fixed point problem and the general split equality problem for quasinonexpansive mappings in Hilbert spaces. As an application, we shall utilize our results to study the null point problem of maximal monotone operators, the split feasibility problem, and the equality equilibrium problem. The results presented in the paper extend and improve the corresponding results announced by Moudafi et al. (Nonlinear Anal. 79:117121, 2013; Trans. Math. Program. Appl. 1:111, 2013), Eslamian and Latif (Abstr. Appl. Anal. 2013:805104, 2013) and Chen et al. (Fixed Point Theory Appl. 2014:35, 2014), Censor and Elfving (Numer. Algorithms 8:221239, 1994), Censor and Segal (J. Convex Anal. 16:587600, 2009) and some others.
1 Introduction
Let C and Q be nonempty closed convex subsets of real Hilbert spaces {H}_{1} and {H}_{2}, respectively. The split feasibility problem (SFP) is formulated as
where A:{H}_{1}\to {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, and computer tomograph and radiation therapy treatment planning [3–5]. The SFP in an infinitedimensional real Hilbert space can be found in [2, 4, 6–10].
Assuming that the SFP is consistent, it is not hard to see that {x}^{\ast}\in C solves SFP if and only if it solves the fixed point equation
where {P}_{C} and {P}_{Q} are the metric projection from {H}_{1} onto C and from {H}_{2} onto Q, respectively, \gamma >0 is a positive constant and {A}^{\ast} is the adjoint of A.
A popular algorithm to be used to solve SFP (1.1) is due to Byrne’s CQalgorithm [2]:
where \gamma \in (0,2/\lambda ) with λ being the spectral radius of the operator {A}^{\ast}A.
Recently, Moudafi [11] introduced the following split equality problem (SEP):
where A:{H}_{1}\to {H}_{3} and B:{H}_{2}\to {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). This kind of split equality problem (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 the split equality problem (1.2), Moudafi [11] introduced the following relaxed alternating CQalgorithm:
where
and c:{H}_{1}\to R (respectively q:{H}_{2}\to R) is a convex and subdifferentiable function. Under suitable conditions, he proved that the sequence \{{x}_{n}\} defined by (1.4) converges weakly to a solution of the split equality problem (1.2).
Each nonempty closed convex subset of a Hilbert space can be regarded as a set of fixed points of a projection. In [12], Moudafi and AlShemas introduced the following split equality fixed point problem:
where S:{H}_{1}\to {H}_{1} and T:{H}_{2}\to {H}_{2} are two firmly quasinonexpansive mappings, F(S) and F(T) denote the fixed point sets of S and T, respectively.
To solve the split equality fixed point problem (1.5) for firmly quasinonexpansive mappings, Moudafi et al. [11–13] proposed the following iteration algorithm:
Very recently, Eslamian and Latif [14] and Chen et al. [15] introduced and studied some kinds of general split feasibility problem and split equality problem in real Hilbert spaces, and under suitable conditions some strong convergence theorems are proved.
Motivated by the above works, the purpose of this paper is to introduce the following general split equality fixed point problem:
and the general split equality problem:
For solving the GSEFP (1.7) and GSEP (1.8), in Sections 3 and 4, we propose an algorithm for finding the solutions of the general split equality fixed point problem and general split equality problem in a Hilbert space. We establish the strong convergence of the proposed algorithms to a solution of GSEFP and GSEP. As applications, in Section 5 we utilize our results to study the split feasibility problem, the null point problem of maximal monotone operators, and the equality equilibrium problem.
2 Preliminaries
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. In the sequel, denote by F(T) the set of fixed points of a mapping T and by {x}_{n}\to {x}^{\ast} and {x}_{n}\rightharpoonup {x}^{\ast}, the strong convergence and weak convergence of a sequence \{{x}_{n}\} to a point {x}^{\ast}, respectively.
Recall that a mapping T:H\to H is said to be nonexpansive, if \parallel TxTy\parallel \le \parallel xy\parallel, \mathrm{\forall}x,y\in H. A typical example of a nonexpansive mapping is the metric projection {P}_{C} from H onto C\subseteq H defined by \parallel x{P}_{C}x\parallel ={inf}_{y\in C}\parallel xy\parallel. The metric projection {P}_{C} is firmly nonexpansive, i.e.,
and it can be characterized by the fact that
Definition 2.1 A mapping T:H\to H is said to be quasinonexpansive, if F(T)\ne \mathrm{\varnothing}, and
Lemma 2.2 [16]
Let H be a real Hilbert space, and \{{x}_{n}\} be a sequence in H. Then, for any given sequence \{{\lambda}_{n}\} of positive numbers with {\sum}_{i=1}^{\mathrm{\infty}}{\lambda}_{n}=1 such that for any positive integers i, j with i<j, the following holds:
Lemma 2.3 [17]
Let H be a real Hilbert space. For any x,y\in H, the following inequality holds:
Lemma 2.4 [18]
Let \{{t}_{n}\} be a sequence of real numbers. If there exists a subsequence \{{n}_{i}\} of \{n\} such that {t}_{{n}_{i}}<{t}_{{n}_{i}+1} for all i\ge 1, then there exists a nondecreasing sequence \{\tau (n)\} with \tau (n)\to \mathrm{\infty} such that for all (sufficiently large) positive integer numbers n, the following holds:
In fact,
Definition 2.5 (Demiclosedness principle)
Let C be a nonempty closed convex subset of a real Hilbert space H, and T:C\to C be a mapping with F(T)\ne \mathrm{\varnothing}. Then IT is said to be demiclosed at zero, if for any sequence \{{x}_{n}\}\subset C with {x}_{n}\rightharpoonup x and \parallel {x}_{n}T{x}_{n}\parallel \to 0, then x=Tx.
Remark 2.6 It is well known that if T:C\to C is a nonexpansive mapping, then IT is demiclosed at zero.
Lemma 2.7 Let \{{a}_{n}\}, \{{b}_{n}\}, and \{{c}_{n}\} be sequences of positive real numbers satisfying {a}_{n+1}\le (1{b}_{n}){a}_{n}+{c}_{n} for all n\ge 1. If the following conditions are satisfied:

(1)
{b}_{n}\in (0,1) and {\sum}_{n=1}^{\mathrm{\infty}}{b}_{n}=\mathrm{\infty},

(2)
{\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}, or {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\frac{{c}_{n}}{{b}_{n}}\le 0,
then {lim}_{n\to \mathrm{\infty}}{a}_{n}=0.
3 Strong convergence theorem for general split equality fixed point problem
Throughout this section we always assume that

(1)
{H}_{1}, {H}_{2}, {H}_{3} are real Hilbert spaces;

(2)
{\{{S}_{i}\}}_{i=1}^{\mathrm{\infty}}:{H}_{1}\to {H}_{1} and {\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}}:{H}_{2}\to {H}_{2} are two families of onetoone and quasinonexpansive mappings;

(3)
A:{H}_{1}\to {H}_{3} and B:{H}_{2}\to {H}_{3} are two bounded linear operators;

(4)
f=\left[\begin{array}{c}{f}_{1}\\ {f}_{2}\end{array}\right], where {f}_{i}, i=1,2 is a kcontractive mapping on {H}_{i} with k\in (0,1);

(5)
C:={\bigcap}_{i=1}^{\mathrm{\infty}}F({S}_{i}), Q:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}), Γ is the set of solutions of GSEFP (1.7),
P=\left[\begin{array}{c}{P}_{C}\\ {P}_{Q}\end{array}\right],\phantom{\rule{2em}{0ex}}{K}_{i}=\left[\begin{array}{c}{S}_{i}\\ {T}_{i}\end{array}\right],\phantom{\rule{2em}{0ex}}G=[\begin{array}{cc}A& B\end{array}],\phantom{\rule{2em}{0ex}}{G}^{\ast}G=\left[\begin{array}{cc}{A}^{\ast}A& {A}^{\ast}B\\ {B}^{\ast}A& {B}^{\ast}B\end{array}\right]; 
(6)
for any given {w}_{0}\in {H}_{1}\times {H}_{2}, the iterative sequence \{{w}_{n}\}\subset {H}_{1}\times {H}_{2} is generated by
{w}_{n+1}=P[{\alpha}_{n}{w}_{n}+{\beta}_{n}f({w}_{n})+\sum _{i=1}^{\mathrm{\infty}}{\gamma}_{n,i}\left({K}_{i}(I{\lambda}_{n,i}{G}^{\ast}G){w}_{n}\right)],\phantom{\rule{1em}{0ex}}n\ge 0,(3.1)where \{{\alpha}_{n}\}, \{{\beta}_{n}\}, \{{\gamma}_{n,i}\} are the sequences of nonnegative numbers with
{\alpha}_{n}+{\beta}_{n}+\sum _{i=1}^{\mathrm{\infty}}{\gamma}_{n,i}=1\phantom{\rule{1em}{0ex}}\text{for each}n\ge 0.
We are now in a position to give the following main result.
Lemma 3.1 Let {w}^{\ast}=({x}^{\ast},{y}^{\ast}) be a point in C\times Q, i.e., {x}^{\ast}\in C={\bigcap}_{i=1}^{\mathrm{\infty}}F({S}_{i}) and {y}^{\ast}\in Q={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}). Then the following statements are equivalent:

(i)
{w}^{\ast} is a solution to GSEFP (1.7);

(ii)
{w}^{\ast}={K}_{i}({w}^{\ast}) for each i\ge 1 and G({w}^{\ast})=0;

(iii)
for each i\ge 1 and for each \lambda >0, {w}^{\ast} solves the fixed point equations:
{w}^{\ast}={K}_{i}{w}^{\ast}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{w}^{\ast}={K}_{i}(I\lambda {G}^{\ast}G){w}^{\ast}.(3.2)
Proof (i) ⇒ (ii). If {w}^{\ast}\in C\times Q is a solution to GSEFP (1.7), then for each i\ge 1, {w}^{\ast}={K}_{i}{w}^{\ast}, and A{x}^{\ast}=B{y}^{\ast}. This implies that for each i\ge 1, {w}^{\ast}={K}_{i}{w}^{\ast}, and

(ii)
⇒ (iii). If {w}^{\ast}={K}_{i}({w}^{\ast}), \mathrm{\forall}i\ge 1 and G({w}^{\ast})=0, it is easy to see that (3.2) holds.

(iii)
⇒ (i). From (3.2), for each i\ge 1 we have {K}_{i}{w}^{\ast}={K}_{i}(I\lambda {G}^{\ast}G){w}^{\ast}. Since {S}_{i} and {T}_{i} both are onetoone, so is {K}_{i}. Hence we have \parallel {w}^{\ast}(I\lambda {G}^{\ast}G){w}^{\ast}\parallel =0, for any \lambda >0. This implies that {G}^{\ast}G({w}^{\ast})=0, and so
0=\u3008{G}^{\ast}G{w}^{\ast},{w}^{\ast}\u3009=\u3008G{w}^{\ast},G{w}^{\ast}\u3009={\parallel G{w}^{\ast}\parallel}^{2},
i.e., G({w}^{\ast})=A{x}^{\ast}B{y}^{\ast}=0.
This completes the proof of Lemma 3.1. □
Lemma 3.2 If \lambda \in (0,\frac{2}{L}), where L={\parallel G\parallel}^{2}, then (I\lambda {G}^{\ast}G):{H}_{1}\times {H}_{2}\to {H}_{1}\times {H}_{2} is a nonexpansive mapping.
Proof In fact for any w,u\in {H}_{1}\times {H}_{2}, we have
This completes the proof. □
Theorem 3.3 Let {H}_{1}, {H}_{2}, {H}_{3}, \{{S}_{i}\}, \{{T}_{i}\}, A, B, f, C, Q, Γ, P, G, {K}_{i}, {G}^{\ast}G satisfy the above conditions (1)(5). Let \{{w}_{n}\} be the sequence defined by (3.1). If the solution set Γ of GSEFP (1.7) is nonempty and the following conditions are satisfied:

(i)
{\alpha}_{n}+{\beta}_{n}+{\sum}_{i=1}^{\mathrm{\infty}}{\gamma}_{n,i}=1, for each n\ge 0;

(ii)
{lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0, and {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty};

(iii)
{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}{\gamma}_{n,i}>0 for each i\ge 1;

(iv)
\{{\lambda}_{n,i}\}\subset (0,\frac{2}{L}) for each i\ge 1, where L={\parallel G\parallel}^{2};

(v)
for each i\ge 1, the mapping I{K}_{i}(I{\lambda}_{n,i}{G}^{\ast}G) is demiclosed at zero,
then the sequence \{{w}_{n}\} converges strongly to {w}^{\ast}={P}_{\mathrm{\Gamma}}f({w}^{\ast}) which is a solution of GSEFP (1.7).
Proof (I) First we prove that the sequence \{{w}_{n}\} is bounded.
In fact, for any given z\in \mathrm{\Gamma}, it follows from Lemma 3.1 that
By the assumptions and Lemma 3.2, for each \lambda \in (0,\frac{2}{L}), (I\lambda {G}^{\ast}G):{H}_{1}\times {H}_{2}\to {H}_{1}\times {H}_{2} is nonexpansive, and for each i\ge 1, {K}_{i}=\left[\begin{array}{c}{S}_{i}\\ {T}_{i}\end{array}\right] is quasinonexpansive, hence we have
By induction, we can prove that
This shows that \{{w}_{n}\} is bounded, and so is \{f({w}_{n})\}.

(II)
Now we prove that the following inequality holds:
\begin{array}{r}{\alpha}_{n}{\gamma}_{n,i}{\parallel {w}_{n}{K}_{i}(I{\lambda}_{n,i}{G}^{\ast}G){w}_{n}\parallel}^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel {w}_{n}z\parallel}^{2}{\parallel {w}_{n+1}z\parallel}^{2}+{\beta}_{n}{\parallel f({w}_{n})z\parallel}^{2}\phantom{\rule{1em}{0ex}}\text{for each}i\ge 1.\end{array}(3.3)
Indeed, it follows from (3.1) and Lemma 2.2 that for each i\ge 1
This implies that for each i\ge 1
Inequality (3.3) is proved.
It is easy to see that the solution set Γ of GSEFP (1.7) is a nonempty closed and convex subset in C\times Q, hence the metric projection {P}_{\mathrm{\Gamma}} is well defined. In addition, since {P}_{\mathrm{\Gamma}}f:{H}_{1}\times {H}_{2}\to {H}_{1}\times {H}_{2} is a contractive mapping, there exists a {w}^{\ast}\in \mathrm{\Gamma} such that

(III)
Now we prove that {w}_{n}\to {w}^{\ast}.
For this purpose, we consider two cases.
Case I. Suppose that the sequence \{\parallel {w}_{n}{w}^{\ast}\parallel \} is monotone. Since \{\parallel {w}_{n}{w}^{\ast}\parallel \} is bounded, \{\parallel {w}_{n}{w}^{\ast}\parallel \} is convergent. Since {w}^{\ast}\in \mathrm{\Gamma}, in (3.3) taking z={w}^{\ast} and letting n\to \mathrm{\infty}, in view of conditions (ii) and (iii), we have
On the other hand, by Lemma 2.3 and (3.1), we have
Simplifying we have
where {\eta}_{n}=\frac{2(1k){\beta}_{n}}{1{\beta}_{n}k}, {\delta}_{n}=\frac{{\beta}_{n}M}{2(1k)}+\frac{1}{1k}\u3008f({w}^{\ast}){w}^{\ast},{w}_{n+1}{w}^{\ast}\u3009, M:={sup}_{n\ge 0}\parallel {w}_{n}{w}^{\ast}\parallel.
By condition (ii), {lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty}, and so {\sum}_{n=1}^{\mathrm{\infty}}{\eta}_{n}=\mathrm{\infty}.
Next we prove that
In fact, since \{{w}_{n}\} is bounded in C\times Q, there exists a subsequence \{{w}_{{n}_{k}}\}\subset \{{w}_{n}\} with {w}_{{n}_{k}}\rightharpoonup {v}^{\ast} (some point in C\times Q), and {\lambda}_{{n}_{k},i}\to {\lambda}_{i}\in (0,\frac{2}{L}) such that
In view of (3.5)
Again by the assumption that for each i\ge 1, the mapping I{K}_{i}(I{\lambda}_{n,i}{G}^{\ast}G) is demiclosed at zero, hence we have
By Lemma 3.1, this implies that {v}^{\ast}\in \mathrm{\Gamma}. In addition, since {w}^{\ast}={P}_{\mathrm{\Gamma}}f({w}^{\ast}), we have
This shows that (3.7) is true. Taking {a}_{n}={\parallel {w}_{n}{w}^{\ast}\parallel}^{2}, {b}_{n}={\eta}_{n}, and {c}_{n}={\delta}_{n}{\eta}_{n} in Lemma 2.7, all conditions in Lemma 2.7 are satisfied. We have {w}_{n}\to {w}^{\ast}.
Case II. If the sequence \{\parallel {w}_{n}{w}^{\ast}\parallel \} is not monotone, by Lemma 2.4, there exists a sequence of positive integers: \{\tau (n)\}, n\ge {n}_{0} (where {n}_{0} is large enough) such that
Clearly \{\tau (n)\} is nondecreasing, \tau (n)\to \mathrm{\infty} as n\to \mathrm{\infty}, and for all n\ge {n}_{0}
Therefore \{\parallel {w}_{\tau (n)}{w}^{\ast}\parallel \} is a nondecreasing sequence. According to Case I, {lim}_{n\to \mathrm{\infty}}\parallel {w}_{\tau (n)}{w}^{\ast}\parallel =0 and {lim}_{n\to \mathrm{\infty}}\parallel {w}_{\tau (n)+1}{w}^{\ast}\parallel =0. Hence we have
This implies that {w}_{n}\to {w}^{\ast} and {w}^{\ast}={P}_{\mathrm{\Gamma}}f({w}^{\ast}) is a solution of GSEFP (1.7).
This completes the proof of Theorem 3.3. □
Remark 3.4 Theorem 3.3 extends and improves the main results in Moudafi et al. [11–13] in the following aspects:

(a)
For the mappings, we extend the mappings from firmly quasinonexpansive mappings to an infinite family of onetoone quasinonexpansive mappings.

(b)
For the algorithms, we propose new iterative algorithms which are different from ones given in [11–13].

(c)
For the convergence, the iterative sequence proposed by our algorithm converges strongly to a solution of GSEFP (1.7). But the iterative sequences proposed in [11–13] are only of weak convergence to a solution of the split equality problem.
4 Strong convergence theorem for general split equality problem
Throughout this section we always assume that

(1)
{H}_{1}, {H}_{2}, {H}_{3} are real Hilbert spaces; {\{{C}_{i}\}}_{i=1}^{\mathrm{\infty}}\subset {H}_{1} and {\{{Q}_{i}\}}_{i=1}^{\mathrm{\infty}}\subset {H}_{2} are two families of nonempty closed and convex subsets with C={\bigcap}_{i=1}^{\mathrm{\infty}}{C}_{i}\ne \mathrm{\varnothing} and Q={\bigcap}_{i=1}^{\mathrm{\infty}}{Q}_{i}\ne \mathrm{\varnothing};

(2)
{P}_{{C}_{i}} (resp. {P}_{{Q}_{i}}) is the metric projection from {H}_{1} onto {C}_{i} (resp. {H}_{2} onto {Q}_{i}), and {P}_{i}:=\left[\begin{array}{c}{P}_{{C}_{i}}\\ {P}_{{Q}_{i}}\end{array}\right], i=1,2,\dots , and P:=\left[\begin{array}{c}{P}_{C}\\ {P}_{Q}\end{array}\right];

(3)
A:{H}_{1}\to {H}_{3} and B:{H}_{2}\to {H}_{3} are two bounded linear operators;

(4)
f, G, {G}^{\ast}G are the same as in Theorem 3.3.
The socalled general split equality problem (GSEP) is
Lemma 4.1 Let {H}_{1}, {H}_{2}, {H}_{3}, P, \{{P}_{i}\}, A, B, f, C, Q, G, {G}^{\ast}G be the same as above. Then a point {w}^{\ast}=({x}^{\ast},{y}^{\ast}) is a solution to GSEP (4.1), if and only if for each i\ge 1 and for each \lambda >0, {w}^{\ast} solves the following fixed point equations:
Proof In fact, a point {w}^{\ast}=({x}^{\ast},{y}^{\ast}) is a solution of GSEP (4.1)
This completes the proof of Lemma 4.1. □
The metric projections {P}_{{C}_{i}} and {P}_{{Q}_{i}} are nonexpansive with F({P}_{{C}_{i}})={C}_{i} and F({P}_{{Q}_{i}})={Q}_{i}, i\ge 1. This implies that the metric projections {P}_{{C}_{i}} and {P}_{{Q}_{i}} all are quasinonexpansive. In addition, by Lemma 3.2, for each i\ge 1 and each \lambda \in (0,\frac{2}{L}), the mapping {P}_{i}(I\lambda {G}^{\ast}G):{H}_{1}\times {H}_{2}\to {C}_{i}\times {Q}_{i} is nonexpansive. By Remark 2.6, for each i\ge 1 and each \lambda \in (0,\frac{2}{L}), the mapping (I{P}_{i}(I\lambda {G}^{\ast}G)) is demiclosed at zero.
Consequently, we have the following.
Theorem 4.2 Let {H}_{1}, {H}_{2}, {H}_{3}, P, \{{P}_{i}\}, A, B, f, C, Q, G, {G}^{\ast}G be the same as above. Let \{{w}_{n}\} be the sequence generated by {w}_{0}\in {H}_{1}\times {H}_{2}
If the solution set {\mathrm{\Gamma}}_{1} of GSEP (4.1) is nonempty and the following conditions are satisfied:

(i)
{\alpha}_{n}+{\beta}_{n}+{\sum}_{i=1}^{\mathrm{\infty}}{\gamma}_{n,i}=1, for each n\ge 0;

(ii)
{lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0, and {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty};

(iii)
{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}{\gamma}_{n,i}>0 for each i\ge 1;

(iv)
\{{\lambda}_{n,i}\}\subset (0,\frac{2}{L}) for each i\ge 1, where L={\parallel G\parallel}^{2},
then the sequence \{{w}_{n}\} defined by (4.3) converges strongly to a solution {w}^{\ast} of GSEP (4.1) and {w}^{\ast}={P}_{{\mathrm{\Gamma}}_{1}}f({w}^{\ast}).
Proof Taking {S}_{i}={P}_{{C}_{i}}, {T}_{i}={P}_{{Q}_{i}}, and {K}_{i}={P}_{i}, i=1,2,\dots in Theorem 3.3, we know that {S}_{i} and {T}_{i} both are nonexpansive with F({S}_{i})={C}_{i} and F({T}_{i})={Q}_{i} and so they are quasinonexpansive mappings, and C={\bigcap}_{i=1}^{\mathrm{\infty}}F({S}_{i}) and Q={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}). Therefore all conditions in Theorem 3.3 are satisfied. The conclusion of Theorem 4.2 can be obtained from Lemma 4.1 and Theorem 3.3 immediately. □
Remark 4.3 Theorem 4.2 extends and improves the corresponding results in Censor and Elfving [1], Moudafi et al. [11, 12], Eslamian and Latif [14], Chen et al. [15], Censor and Segal [19].
5 Applications
In this section we shall utilize the results presented in the paper to give some applications.
5.1 Application to split feasibility problem
Let C\subset {H}_{1} and Q\subset {H}_{2} be two nonempty closed convex subsets and A:{H}_{1}\to {H}_{2} be a bounded linear operator. The socalled split feasibility problem (SFP) [1] is to find
Let {P}_{C} and {P}_{Q} be the metric projection from {H}_{1} onto C and {H}_{2} onto Q, respectively. Thus F({P}_{C})=C and F({P}_{Q})=Q. From Theorem 4.2 we have the following.
Theorem 5.1 Let {H}_{1}, {H}_{2} be two real Hilbert spaces, A:{H}_{1}\to {H}_{2} be a bounded linear operator and I be the identity mapping on {H}_{2}. Let C\subset {H}_{1} and Q\subset {H}_{2} be nonempty closed convex subsets and {P}_{C} and {P}_{Q} are the metric projections from {H}_{1} onto C and {H}_{2} onto Q, respectively. Let \{{w}_{n}\} be the sequence generated by {w}_{0}\in {H}_{1}\times {H}_{2}:
where f is the mapping as given in Theorem 4.2 and
If the solution set {\mathrm{\Gamma}}_{2} of SFP (5.1) is nonempty and the following conditions are satisfied:

(i)
{\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1, for each n\ge 0;

(ii)
{lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0, and {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty};

(iii)
{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}{\gamma}_{n}>0;

(iv)
\{{\lambda}_{n}\}\subset (0,\frac{2}{L}), where L={\parallel U\parallel}^{2},
then the sequence \{{w}_{n}\} defined by (5.2) converges strongly to a solution {w}^{\ast} of SFP (5.1) and {w}^{\ast}={P}_{{\mathrm{\Gamma}}_{2}}f({w}^{\ast}).
Proof In Theorem 4.2 taking {H}_{2}={H}_{3}, B=I, G=U, \{{C}_{i}\}=\{C\}, and \{{Q}_{i}\}=\{Q\}, the conclusions of Theorem 5.1 can be obtained from Theorem 4.2 immediately. □
Remark Theorem 5.1 generalizes and extends the main results of Censor and Elfving [1] and Censor and Segal [19] from weak convergence to strong convergence.
5.2 Application to null point problem of maximal monotone operators
Let {H}_{1}, {H}_{2}, {H}_{3}, A, B, be the same as in Theorem 3.3. Let M:{H}_{1}\to {H}_{1}, and N:{H}_{2}\to {H}_{2} be two strictly maximal monotone operators. It is well known that the associated resolvent mappings {J}_{\mu}^{M}(x):={(I+\mu M)}^{1} and {J}_{\mu}^{N}(x):={(I+\mu N)}^{1} of M and N, respectively, are onetoone nonexpansive mappings, and
Denote S={J}_{\mu}^{M}, T={J}_{\mu}^{N}, C={M}^{1}(0)=F({J}_{\mu}^{M}), and Q={N}^{1}(0)=F({J}_{\mu}^{N}), then the general split equality fixed point problem (1.7) is reduced to the following null point problem related to strictly maximal monotone operators M and N (NPP(M,N)):
From Theorem 3.3 we can obtain the following.
Theorem 5.2 Let {H}_{1}, {H}_{2}, {H}_{3}, A, B, f, G, be the same as in Theorem 3.3. Let C, Q, S, and T be the same as above. Let \{{w}_{n}\} be the sequence generated by {w}_{0}\in {H}_{1}\times {H}_{2}
where P=\left[\begin{array}{c}{P}_{C}\\ {P}_{Q}\end{array}\right], K=\left[\begin{array}{c}S\\ T\end{array}\right]. If the solution set {\mathrm{\Gamma}}_{3} of NPP(M,N) (5.5) is nonempty and the following conditions are satisfied:

(i)
{\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1, for each n\ge 0;

(ii)
{lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0, and {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty};

(iii)
{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}{\gamma}_{n}>0;

(iv)
\{{\lambda}_{n}\}\subset (0,\frac{2}{L}), where L={\parallel G\parallel}^{2},
then the sequence \{{w}_{n}\} defined by (5.6) converges strongly to {w}^{\ast}={P}_{{\mathrm{\Gamma}}_{3}}f({w}^{\ast}), which is a solution of NPP(M,N) (5.5).
Proof Since S={J}_{\mu}^{M} and T={J}_{\mu}^{N} both are onetoone nonexpansive with F(S)\ne \mathrm{\varnothing} and F(T)\ne \mathrm{\varnothing}. Hence they are onetoone quasinonexpansive mappings and IK(I{\lambda}_{n}{G}^{\ast}G) is demiclosed at zero. Therefore all conditions in Theorem 3.3 are satisfied. The conclusions of Theorem 5.2 can be obtained from Theorem 3.3 immediately. □
5.3 Application to equality equilibrium problem
Let D be a nonempty closed and convex subset of a real Hilbert H. A bifunction g:D\times D\to (\mathrm{\infty},+\mathrm{\infty}) is said to be a equilibrium function, if it satisfies the following conditions:
(A1) g(x,x)=0, for all x\in D;
(A2) g is monotone, i.e., g(x,y)+g(y,x)\le 0 for all x,y\in D;
(A3) {lim\hspace{0.17em}sup}_{t\downarrow 0}g(tz+(1t)x,y)\le g(x,y) for all x,y,z\in D;
(A4) for each x\in D, y\mapsto g(x,y) is convex and lower semicontinuous.
The socalled equilibrium problem with respective to the equilibrium functions g and D is
Its solution set is denoted by EP(g,D).
For given \lambda >0 and x\in H, the resolvent of the equilibrium function g is the operator {R}_{\lambda ,g}:H\to D defined by
It is well known that the resolvent {R}_{\lambda ,g} of the equilibrium function g has the following properties [20]:

(1)
{R}_{\lambda ,g} is singlevalued;

(2)
F({R}_{\lambda ,g})=EP(g,D) and F({R}_{\lambda ,g}) is a nonempty closed and convex subset of D;

(3)
{R}_{\lambda ,g} is a nonexpansive mapping, and so it is quasinonexpansive.
Definition 5.3 Let h,j:D\times D\to (\mathrm{\infty},+\mathrm{\infty}) be two equilibrium functions and, for given \lambda >0, let {R}_{\lambda ,h} and {R}_{\lambda ,j} be the resolvents of h and j (defined by (5.8)), respectively. Denote by S={R}_{\lambda ,h}, T={R}_{\lambda ,j}, C:=F({R}_{\lambda ,h}), and Q:=F({R}_{\lambda ,j}). Then the equality equilibrium problem with respective to the equilibrium functions h, j, and D is
where A,B:H\to H are two linear and bounded operators.
The following theorem can be obtained from Theorem 3.3 immediately.
Theorem 5.4 Let H be a real Hilbert space, D be a nonempty and closed convex subset of H. Let G, f be the same as in Theorem 3.3. For given \lambda >0, let h, j, {R}_{\lambda ,h}, {R}_{\lambda ,j}, S, T, C, Q be the same as above. Let \{{w}_{n}\} be the sequence generated by {w}_{0}\in H\times H:
where P=\left[\begin{array}{c}{P}_{C}\\ {P}_{Q}\end{array}\right], K=\left[\begin{array}{c}S\\ T\end{array}\right]. If the solution set {\mathrm{\Gamma}}_{4} of EEP(h,j,D) (5.9) is nonempty and the following conditions are satisfied:

(i)
{\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1, for each n\ge 0;

(ii)
{lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0, and {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty};

(iii)
{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}{\gamma}_{n}>0;

(iv)
\{{\lambda}_{n}\}\subset (0,\frac{2}{L}), where L={\parallel G\parallel}^{2},
then the sequence \{{w}_{n}\} converges strongly to {w}^{\ast}={P}_{{\mathrm{\Gamma}}_{4}}f({w}^{\ast}), which is a solution of EEP(h,j,D) (5.9).
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Chang, Ss., Agarwal, R.P. Strong convergence theorems of general split equality problems for quasinonexpansive mappings. J Inequal Appl 2014, 367 (2014). https://doi.org/10.1186/1029242X2014367
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DOI: https://doi.org/10.1186/1029242X2014367
Keywords
 general split equality problem
 general split equality fixed point problem
 quasinonexpansive mapping
 split feasibility problem