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Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces
Fixed Point Theory and Applications volumeÂ 2015, ArticleÂ number:Â 200 (2015)
Abstract
In this paper, we propose a new iterative sequence for solving common problems which consist of split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems of the generated sequence \(\{x_{n}\}\) by the shrinking projection method. Our results improve and extend the previous results given in the literature.
1 Introduction
Throughout this paper, let \(\mathbb{R}\) and \(\mathbb{N}\) denote the set of all real numbers and the set of all positive integers, respectively. Let H be a real Hilbert space and C be a nonempty closed convex subset of H.
A mapping \(T:C\times C\to\mathbb{R}\) is said to be asymptotically nonexpansive if there exists a sequence \(\{k_{n}\}\subset[1,\infty)\) with \(\lim_{n\to\infty}k_{n}=1\) such that
for all \(x, y\in C\). It is easy to see that, if \(k_{n}\equiv1\), then T is said to be nonexpansive. We denote the set of fixed point of T by \(F(T)\), that is, \(F(T)=\{x\in C:Tx=x\}\). There are many iterative methods for solving a fixed point problem corresponding to an asymptotically nonexpansive mapping (see also [1â€“3]).
Recall that a Hilbert space H satisfies Opialâ€™s condition [4], that is, for any subsequence \(\{x_{n}\}\subset H\) with \(x_{n}\rightharpoonup x\), the following inequality
holds for all \(y\in H\) with \(y\neq x\). Furthermore, a Hilbert space H has a KadecKlee property, i.e., \(x_{n}\rightharpoonup x\) and \(\Vert x_{n}\Vert \to \Vert x\Vert \) imply \(x_{n}\to x\). In fact, from
we can conclude that a Hilbert space has a KadecKlee property.
In 1994, Blum and Oettli [5] introduced the equilibrium problem which is to find \(x\in C\) such that
They denoted the solution set of problem (1.1) as \(EP(F)\). Since the wellknown problems were variational problems, complementary problems, fixed point problems, saddle point problems and other problems proposed from the equilibrium problem, it has become the most attractive topic for many mathematicians [6â€“8]. They have widely spread its applications to other applied disciplines including physics, chemistry, economics and engineering (see, for example, [9â€“12]).
In 1997, Combettes and Hirstoaga [13] proposed an iterative method for solving problem (1.1) by the assumption that \(EP(F)\neq \emptyset\). Moreover, there are many new iteratively generated sequences for solving this problem together with fixed point problems (see [14â€“17]).
Later, the socalled split equilibrium problem was introduced (shortly, \(SEP\)). Let \(H_{1}\), \(H_{2}\) be two real Hilbert spaces. Let C, Q be closed convex subsets of \(H_{1}\) and \(H_{2}\), respectively, and let \(A:H_{1}\to H_{2}\) be a bounded linear operator. Further, let \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to\mathbb{R}\) be two bifunctions. The \(SEP\) is to find the element \(x^{*}\in C\) such that
and such that
The solution sets of problems (1.2) and (1.3) are symbolized by \(EP(F_{1})\) and \(EP(F_{2})\), respectively. Therefore, we denote \(\Omega=\{v\in C: v\in EP(F_{1}) \mbox{ such that } Av\in EP(F_{2})\}\) as the solution set of \(SEP\).
Clearly, the \(SEP\) contains two equilibrium problems, that is, we find out the solution of one equilibrium problem, i.e., its image under a given bounded linear operator, must be the solution of another equilibrium problem. In order to find a common solution of equilibrium problems, it has been mostly considered in the same spaces. However, we normally found that, in the reallife problems, it may be considered in different spaces. That is how the \(SEP\) works very well for this case (see, for example, [18]). Moreover, the split variational inequality problem (shortly, \(SVIP\)) is its special case, which is to find \(x^{*}\in C\) such that
and corresponding to
where \(f:H_{1}\to H_{1}\) and \(g:H_{1}\to H_{2}\) are nonlinear mappings and \(A:H_{1}\to H_{2}\) is a bounded linear operator (see [19]).
In 2012, He [18] proposed the new algorithm for solving a split equilibrium problem and investigated the convergence behavior in several ways including both weak and strong convergence. Moreover, they gave some examples and mentioned that there exist many SEPs, and the new methods for solving it further need to be explored in the future. Later, in 2013, Kazmi and Rizvi [20] considered the iterative method to compute the common approximate solution of a split equilibrium problem, a variational inequality problem and a fixed point problem for a nonexpansive mapping in the framework of real Hilbert spaces. They generated the sequence iteratively as follows:
for each \(n\geq0\), where \(A:H_{1}\to H_{2}\) is a bounded linear operator, \(D:C\to H_{1}\) is a Ï„inverse strongly monotone mapping, \(F_{1}:C\times C\to\mathbb{R}\), \(F_{2}:Q\times Q\to\mathbb{R}\) are two bifunctions. They found that, under the sufficient conditions of \(r_{n}\), \(\lambda_{n}\), Î³, \(\beta_{n}\) and \(\gamma_{n}\), the generated sequence \(\{ x_{n}\}\) converges strongly to a common solution of all mentioned problems.
Recently, in 2014, Bnouhachem [21] introduced a new iterative method for solving split equilibrium problem and hierarchical fixed point problems by defining the sequence \(\{x_{n}\}\) as follows:
for each \(n\geq0\), where S, T are nonexpansive mappings, \(F:C\to C\) is a kLipschitz mapping and Î·strongly monotone, \(U:C\to C\) is a Ï„Lipschitz mapping. Also, they proved some strong convergence theorems for the proposed iteration under some appropriate conditions.
In this paper, motivated and inspired by the results [18, 20, 21] and the recent works in this field, we introduce the shrinking projection method for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems for the proposed new iterative method. In fact, our results improve and extend the results given by some authors.
2 Preliminaries
In this section, we recall some concepts including the assumption which will be needed for the proof of our main result.
Let H be a Hilbert space and C be a nonempty closed convex subset of H. For each \(x\in H\), there exists a unique nearest point of C, denoted by \(P_{C}x\), such that
for all \(y\in C\). \(P_{C}\) is called the metric projection from H onto C. It is well known that \(P_{C}\) is a firmly nonexpansive mapping from H onto C, that is,
for all \(x,y\in H\). Furthermore, for any \(x\in H\) and \(z\in C\), \(z=P_{C}x\) if and only if
for all \(y\in C\). A mapping \(A:C\to H\) is called Î±inverse strongly monotone if there exists \(\alpha>0\) such that
for all \(x,y\in H\). Moreover, we can investigate that, for each \(\lambda \in(0,2\alpha]\), \(I\lambda A\) is a nonexpansive mapping of C into H (see [22]).
Lemma 2.1
In a Hilbert space H, the following identity holds:
for all \(x,y\in H\) and \(\lambda\in[0,1]\).
Lemma 2.2
[23]
Let T be an asymptotically nonexpansive mapping defined on a bounded closed convex subset C of a Hilbert space H. Assume that \(\{x_{n}\}\) is a sequence in C with the following properties:

(1)
\(x_{n}\rightharpoonup z\);

(2)
\(Tx_{n}x_{n}\rightarrow0\).
Then \(z\in F(T)\).
Assumption 2.3
[24]
Let \(F:C\times C\to\mathbb{R}\) be a bifunction satisfying the following conditions:

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

(A2)
F is monotone, i.e., \(F(x,y)+F(y,x)\leq0\) for all \(x,y\in C\);

(A3)
for each \(x,y,z\in C\), \(\lim_{t\downarrow 0}F(tz+(1t)x,y)\leq F(x,y)\);

(A4)
for each \(x\in C\), \(y\mapsto F(x,y)\) is convex and lower semicontinuous.
Lemma 2.4
[24]
Let C be a nonempty closed convex subset of a Hilbert space H and \(F:C\times C\to\mathbb{R}\) be a bifunction which satisfies conditions (A1)(A4). For any \(x\in H\) and \(r>0\), define a mapping \(T_{r}^{F}:H\to C\) by
Then \(T_{r}^{F}\) is well defined and the following hold:

(1)
\(T_{r}^{F}\) is singlevalued;

(2)
\(T_{r}^{F}\) is firmly nonexpansive, i.e., for any \(x,y\in H\),
$$\begin{aligned} \bigl\Vert T_{r}^{F}xT_{r}^{F}y \bigr\Vert ^{2}\leq\bigl\langle T_{r}^{F}xT_{r}^{F}y,xy \bigr\rangle ; \end{aligned}$$ 
(3)
\(F(T_{r}^{F})=EP(F)\);

(4)
\(EP(F)\) is closed and convex.
3 Main results
In this section, we prove some strong convergence theorems of an iterative algorithm for solving a split equilibrium together with a fixed point problem revolving an asymptotically nonexpansive mapping in the framework of Hilbert spaces.
Theorem 3.1
Let \(H_{1}\), \(H_{2}\) be two real Hilbert spaces and C, Q be nonempty closed convex subsets of Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively. Let \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to\mathbb{R}\) be two bifunctions satisfying conditions (A1)(A4) and \(F_{2}\) be upper semicontinuous in the first argument. Let \(T:C\to C\) be an asymptotically nonexpansive mapping and \(A:H_{1}\to H_{2}\) be a bounded linear operator. Suppose that \(F(T)\cap\Omega\neq\emptyset\), where \(\Omega=\{v\in C: v\in EP(F_{1})\textit{ such that }Av\in EP(F_{2})\}\), and let \(x_{0}\in C\). For \(C_{1}=C\) and \(x_{1}=P_{C_{1}}x_{0}\), define a sequence \(\{x_{n}\}\) iteratively as follows:
for each \(n\geq1\), where \(\theta_{n}=(1\alpha_{n})(k_{n}^{2}1)\sup\{\Vert x_{n}z\Vert ^{2}:z\in\Omega\}\), \(0\leq\alpha_{n}\leq a<1\) for all \(n\in\mathbb{N}\), \(0< b\leq r_{n}<\infty\), \(\gamma\in(0,1/L)\), L is the spectral radius of the operator \(A^{*}A\) and \(A^{*}\) is the adjoint of A. Then the sequence \(\{x_{n}\}\) generated by (3.1) strongly converges to a point \(z_{0}\in F(T)\cap\Omega\).
Proof
First of all, we investigate that, for each \(n\in\mathbb{N}\), \(A^{*}(IT_{r_{n}}^{F_{2}})A\) is a \(\frac{1}{2L}\)inverse strongly monotone mapping. Since \(T_{r_{n}}^{F_{2}}\) is firmly nonexpansive and \((IT_{r_{n}}^{F_{2}})\) is \(\frac{1}{2}\)inverse strongly monotone, it follows that
for all \(x,y\in H\), from which it can be concluded that \(A^{*}(IT_{r_{n}}^{F_{2}})A\) is a \(\frac{1}{2L}\)inverse strongly monotone mapping. Moreover, we claim that since \(\gamma\in(0,\frac{1}{L})\), \(I\gamma A^{*}(IT_{r_{n}}^{F_{2}})A\) is nonexpansive.
Next, we show that \(F(T)\cap\Omega\subset C_{n+1}\) for all \(n\in\mathbb {N}\). Let \(p\in F(T)\cap\Omega\), i.e., \(T_{r_{n}}^{F_{1}}p=p\) and \((I\gamma A^{*}(IT_{r_{n}}^{F_{2}})A)p=p\). By mathematical induction, we have \(p\in C=C_{1}\) and hence \(F(T)\cap\Omega\subset C_{1}\). Let \(F(T)\cap \Omega\subset C_{k}\) for some \(k\in\mathbb{N}\). It follows that
and
where \(M_{k}=\sup\{\Vert x_{k}z\Vert :z\in\Omega\}\) and \(\theta_{k}=(1\alpha _{k})(k_{k}^{2}1)M_{k}^{2}\). It can be concluded that \(p\in C_{k+1}\) and \(F(T)\cap\Omega\subset C_{k+1}\) and, further, \(F(T)\cap\Omega\subset C_{n}\) for all \(n\in\mathbb{N}\).
Next, we show that \(C_{n}\) is closed and convex for all \(n\in\mathbb{N}\). First, it is obvious that \(C_{1}=C\) is closed and convex. By induction, we suppose that \(C_{k}\) is closed and convex for some \(k\in\mathbb{N}\). Let \(z_{m}\in C_{k+1}\subset C_{k}\) with \(z_{m}\to z\). Since \(C_{k}\) is closed, it follows that \(x\in C_{k}\) and \(\Vert y_{k}z_{m}\Vert ^{2} \leq \Vert z_{m}x_{k}\Vert ^{2}+\theta _{k}\). Then we have
Letting \(m\to\infty\), we have
which means that \(z\in C_{k+1}\). Let \(x,y\in C_{k+1}\subset C_{k}\) and \(z=\alpha x+(1\alpha)y\) for any \(\alpha\in[0,1]\). Since \(C_{k}\) is convex, \(z\in C_{k}\), \(\Vert y_{k}x\Vert ^{2}\leq \Vert xx_{k}\Vert ^{2}+\theta_{k}\) and \(\Vert y_{k}y\Vert ^{2}\leq \Vert xx_{k}\Vert ^{2}+\theta_{k}\) and so
Therefore, \(z\in C_{k+1}\) and hence \(C_{k+1}\) is closed and convex. It is immediately concluded that \(C_{n}\) is closed and convex for all \(n\in \mathbb{N}\), which implies that \(\{x_{n}\}\) is well defined.
Next, from \(x_{n}=P_{C_{n}}x_{0}\), we have
for all \(y\in C_{n}\). Since \(p\in F(T)\cap\Omega\), we have
for all \(p\in F(T)\cap\Omega\), that is, we have
This implies that
for all \(n\in\mathbb{N}\). From \(x_{n}=P_{C_{n}}x_{0}\) and \(x_{n+1}=P_{C_{n+1}}x_{0}\in C_{n+1}\subset C_{n}\), we also have
for all \(n\in\mathbb{N}\), and so we have
Hence we have
that is, \(\Vert x_{n}x_{0}\Vert \leq \Vert x_{0}x_{n+1}\Vert \) for all \(n\in\mathbb{N}\). From (3.4), it follows that \(\{x_{n}\}\) is bounded and \(\lim_{n\to \infty} \Vert x_{n}x_{0}\Vert \) exists.
Next, we show that \(\Vert x_{n}x_{n+1}\Vert \to0\). From (3.5), we have
Since the limit of \(\{\Vert x_{n}x_{0}\Vert \}\) exists, we have
Thus, by (3.7) and (3.14), we have
as \(n\to\infty\). Furthermore, since \(T_{r_{n}}^{F_{1}}\) is firmly nonexpansive, we have
where \(z_{n}=(I\gamma A^{*}(IT_{r_{n}}^{F_{2}})A)x_{n}\). Moreover,
which leads to
Letting \(\rho_{n}=k_{n}1\). Then it is clear that \(\rho_{n}\to0\) as \(n\to \infty\) and, by (3.9), we exactly have
By (3.8) and \(\rho_{n}\to0\) as \(n\to\infty\), we have
as \(n\to\infty\). Furthermore, since A is linear bounded and so is \(A^{*}\), we can conclude that
Next, we show that \(\u_{n}x_{n}\\to0\). We investigate the following:
Consequently, by (3.12), we can conclude that
Next, we show that \(\T^{n}x_{n}x_{n}\\to0\). We first consider
and since \(x_{n+1}\in C_{n+1}\subset C_{n}\), we have
which means that
Hence,
and so \(\T^{n}u_{n}x_{n}\\to0\). Consider
Therefore, we have \(\T^{n}x_{n}x_{n}\\to0\) as \(n\to\infty\). Putting \(k_{\infty}=\sup\{k_{n}:n\geq1\}<\infty\), we deduce that
Hence we have \(\Tx_{n}x_{n}\\to0\) as \(n\to\infty\). Without loss of generality, since \(\{x_{n}\}\) is bounded, we may assume that \(x_{n}\rightharpoonup x^{*}\). It is easy to see that \(x^{*}\in C_{n}\) for all \(n\geq1\). On the other hand, we have
It follows that
and so
Hence \(\x_{n}\\to\x^{*}\\). Since every Hilbert space has the KadecKlee property, we immediately have \(x_{n}\to x^{*}\).
Finally, we prove that \(x^{*}\in F(T)\cap\Omega\). Since \(x_{n}\to x^{*}\) and \(x_{n}Tx_{n}\to0\) as \(n\to\infty\), consider
We can see that \(\x^{*}Tx^{*}\=0\) and, further, \(x^{*}\in F(T)\). Therefore, we have \(x^{*}\in F(T)\).
Next, we show that \(x^{*}\in\Omega\). By (3.1), \(u_{n}=T_{r_{n}}^{F_{1}}(I\gamma A^{*}(IT_{r_{n}}^{F_{2}}))\), that is,
for all \(y\in C\). From (A2), it follows that
for all \(y\in C\). Since \(\A^{*}(T_{r_{n}}^{F_{2}}I)Ax_{n}\\to0\), \(\ u_{n}x_{n}\\to0\) and \(\x_{n}x^{*}\\to0\) as \(n\to\infty\), we have
for all \(y\in C\). Let \(y_{t}=ty+(1t)x^{*}\) for any \(0< t\leq1\) and \(y\in C\). It means that \(y_{t}\in C\) and hence
and then \(F_{1}(y_{t},y)\geq0\). Letting \(t\to0\), we immediately have \(F_{1}(x^{*},y)\geq0\), i.e., \(x^{*}\in EP(F_{1})\).
Next, we show that \(Ax^{*}\in EP(F_{2})\). Since A is a bounded linear operator and (3.11), we have
as \(n\to\infty\), which yields that \(T_{r_{n}}^{F_{2}}Ax_{n}\to Ax^{*}\). By the definition of \(T_{r_{n}}^{F_{2}}\), we have
for all \(y\in C\). Since \(F_{2}\) is upper semicontinuous in the first argument, taking limâ€‰sup in (3.15), it follows that
for all \(x,y\in C\), from which it can be concluded that \(Ax^{*}\in EP(F_{2})\). Consequently, \(x^{*}\in\Omega\). This completes the proof.â€ƒâ–¡
In TheoremÂ 3.1, if the mapping T is a nonexpansive mapping, then we immediately have the following.
Corollary 3.2
Let \(H_{1}\), \(H_{2}\) be two real Hilbert spaces and C, Q be nonempty closed convex subsets of Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively. Let \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to\mathbb{R}\) be two bifunctions satisfying conditions (A1)(A4) and \(F_{2}\) be upper semicontinuous in the first argument. Let \(T:C\to C\) be a nonexpansive mapping and \(A:H_{1}\to H_{2}\) be a bounded linear operator. Suppose that \(F(T)\cap\Omega\neq\emptyset\), where \(\Omega=\{v\in C: v\in EP(F_{1})\textit{ such that }Av\in EP(F_{2})\}\), and let \(x_{0}\in C\). For \(C_{1}=C\) and \(x_{1}=P_{C_{1}}x_{0}\), define a sequence \(\{x_{n}\} \) iteratively as follows:
for each \(n\in\mathbb{N}\), where \(M_{n}=\sup\{\x_{n}z\:x\in\Omega\}\) and \(\theta_{n}=(1\alpha_{n})(k_{n}^{2}1)M_{n}^{2}\), \(0\leq\alpha_{n}\leq a<1\) for all \(n\in\mathbb{N}\), \(0< b\leq r_{n}<\infty\), \(\gamma\in(0,1/L)\), L is the spectral radius of the operator \(A^{*}A\) and \(A^{*}\) is the adjoint of A. Then the sequence \(\{x_{n}\}\) generated by (3.16) strongly converges to a point \(z_{0}\in F(T)\cap\Omega\).
If \(H_{1}=H_{2}\), \(C=Q\) and \(A=I\) in TheoremÂ 3.1, then we have the following.
Corollary 3.3
Let H be a real Hilbert space and C be a nonempty closed convex subset of a Hilbert space H. Let \(F_{1},F_{2}:C\times C\to\mathbb{R}\) be bifunctions satisfying conditions (A1)(A4) and \(F_{2}\) be upper semicontinuous in the first argument. Let \(T:C\to C\) be an asymptotically nonexpansive mapping. Suppose that \(F(T)\cap\Omega\neq \emptyset\), where \(\Omega=\{v\in C: v\in EP(F_{1})\cap EP(F_{2})\}\), and let \(x_{0}\in C\). For \(C_{1}=C\) and \(x_{1}=P_{C_{1}}x_{0}\), define a sequence \(\{ x_{n}\}\) iteratively as follows:
for each \(n\in\mathbb{N}\), where \(M_{n}=\sup\{\x_{n}z\:z\in\Omega\}\) and \(\theta_{n}=(1\alpha_{n})(k_{n}^{2}1)(M_{n})^{2}\), \(0\leq\alpha_{n}\leq a<1\) and \(0< b\leq r_{n}<\infty\) for all \(n\in\mathbb{N}\). Then the sequence \(\{x_{n}\}\) generated by (3.17) strongly converges to a point \(z_{0}\in F(T)\cap\Omega\).
4 Applications
4.1 Applications to split variational inequality problems
Firstly, we point out the socalled variational inequality problem (shortly, \(VIP\)), which is to find a point \(x^{*}\in C\) which satisfies the following inequality:
for all \(z\in C\). Its solution set is symbolized by \(VI(A,C)\).
In 2012, Censor et al. [19] proposed the split variational inequality problem (shortly, \(SVIP\)) which is formulated as follows:
and such that
where \(A:C\to C\) is a bounded linear operator. The solution set of split variational inequality problem is denoted by the \(SVIP\).
Setting \(F_{1}(x,y)=\langle f(x),yx\rangle\) and \(F_{2}(x,y)=\langle g(x),yx\rangle\), it is clear that \(F_{1}\), \(F_{2}\) satisfy conditions (A1)(A4), where f and g are \(\eta_{1}\) and \(\eta_{2}\)inverse strongly monotone mappings, respectively. Then, by TheoremÂ 3.1, we get the following.
Theorem 4.1
Let \(H_{1}\), \(H_{2}\) be two real Hilbert spaces and C, Q be nonempty closed convex subsets of Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively. Let f and g be \(\eta_{1}\) and \(\eta_{2}\)inverse strongly monotone mappings, respectively. Let \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to\mathbb{R}\) be two bifunctions satisfying conditions (A1)(A4), which are defined by f and g, and \(F_{2}\) be upper semicontinuous in the first argument. Let \(T:C\to C\) be an asymptotically nonexpansive mapping and \(A:H_{1}\to H_{2}\) be a bounded linear operator. Suppose that \(F(T)\cap\Omega\neq\emptyset\), where \(\Omega=\{v\in C: v\in EP(F_{1})\textit{ such that }Av\in EP(F_{2})\}\), and let \(x_{0}\in C\). For \(C_{1}=C\) and \(x_{1}=P_{C_{1}}x_{0}\), define a sequence \(\{x_{n}\}\) iteratively as follows:
for each \(n\in\mathbb{N}\), where \(M_{n}=\sup\{\x_{n}z\:z\in\Omega\}\) and \(\theta_{n}=(1\alpha_{n})(k_{n}^{2}1)M_{n}^{2}\), \(0\leq\alpha_{n}\leq a<1\) for all \(n\in\mathbb{N}\), \(0< b\leq r_{n}<\infty\), \(\gamma\in(0,1/L)\), L is the spectral radius of the operator \(A^{*}A\) and \(A^{*}\) is the adjoint of A. Then the sequence \(\{x_{n}\}\) generated by (4.1) strongly converges to a point \(z_{0}\in F(T)\cap\Omega\).
Proof
The desired result can be proved directly through TheoremÂ 3.1.â€ƒâ–¡
4.2 Applications to split optimization problems
In this section, we mention applications to the split optimization problem, which is to find \(x^{*}\in C\) such that
for all \(y\in Q\). We symbolize Î“ for the solution set of the split optimization problem.
Let \(f:C\to\mathbb{R}\) and \(g:Q\to\mathbb{R}\) be two functions satisfying the following assumption:

(1)
for each \(x,y\in C\), \(f(tx+(1t)y)\leq f(y)\), and for each \(u,v\in Q\), \(g(tu+(1t)v)\leq g(v)\);

(2)
\(f(x)\) is concave and upper semicontinuous for all \(x\in C\) and \(g(u)\) is concave and upper semicontinuous for all \(u\in Q\).
Let \(F_{1}(x,y)=f(x)f(y)\) for all \(x,y\in C\) and \(F_{2}(u,v)=g(u)g(v)\) for all \(u,v\in Q\). If f and g satisfy conditions (1) and (2), then it is clear that \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to \mathbb{R}\) are two bifunctions satisfying conditions (A1)(A4). Therefore, by TheoremÂ 3.1, we have the following.
Theorem 4.2
Let \(H_{1}\), \(H_{2}\) be two real Hilbert spaces and C, Q be nonempty closed convex subsets of Hilbert spaces \(H_{1}\) and \(H_{2}\), respectively. Let \(f:C\to\mathbb{R}\) and \(g:Q\to\mathbb{R}\) be two functions satisfying conditions (1) and (2). Let \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to\mathbb{R}\) be two bifunctions satisfying conditions (A1)(A4) and \(F_{2}\) be upper semicontinuous in the first argument. Let \(A:H_{1}\to H_{2}\) be a bounded linear operator. Suppose that \(F(T)\cap\Gamma\neq\emptyset\) and let \(x_{0}\in C\). For \(C_{1}=C\) and \(x_{1}=P_{C_{1}}x_{0}\), define a sequence \(\{x_{n}\}\) iteratively as follows:
for each \(n\in\mathbb{N}\), where \(0\leq\alpha_{n}\leq a<1\), \(0< b\leq r_{n}<\infty\), and \(\gamma\in(0,1/L)\), L is the spectral radius of the operator \(A^{*}A\) and \(A^{*}\) is the adjoint of A. Then the sequence \(\{x_{n}\}\) generated by (4.3) strongly converges to a point \(z_{0}\in F(T)\cap\Gamma\).
Proof
The desired result can be proved directly through TheoremÂ 3.1.â€ƒâ–¡
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Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (1813036HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2014R1A2A2A01002100).
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Witthayarat, U., Abdou, A.A.N. & Cho, Y.J. Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl 2015, 200 (2015). https://doi.org/10.1186/s1366301504485
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DOI: https://doi.org/10.1186/s1366301504485