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The common solution for a generalized equilibrium problem, a variational inequality problem and a hierarchical fixed point problem
Journal of Inequalities and Applications volume 2015, Article number: 53 (2015)
Abstract
The present paper aims to deal with a new iterative method to find a common solution of a generalized equilibrium problem, a variational inequality problem and a hierarchical fixed point problem for a sequence of nearly nonexpansive mappings. It is proved that the proposed method converges strongly to a common solution of above problems under some assumptions. The results here improve and extend some recent corresponding results by many other authors.
1 Introduction
Let H be a real Hilbert space whose inner product and norm are denoted by \(\langle\cdot,\cdot\rangle\) and \(\Vert \cdot \Vert \), respectively, C be a nonempty, closed, and convex subset of H. It is well known that for any \(x\in H\), there exists a unique point \(y_{0}\in C\) such that
Here, \(y_{0}\) is denoted by \(P_{C}x\), where \(P_{C}\) is called the metric projection of H onto C.
Let us recall some kinds of nonlinear mappings as follows, which are needed in the next sections. A mapping \(T:C\rightarrow H\) is called L-Lipschitzian if there exists a constant \(L>0\) such that \(\Vert Tx-Ty\Vert \leq L\Vert x-y\Vert \), \(\forall x,y\in C\). In particular, if \(L\in {}[0,1)\), then T is said to be a contraction; if \(L=1\), then T is called a nonexpansive mapping. Let us fix a sequence \(\{a_{n}\}\) in \([0,\infty ) \) with \(a_{n}\rightarrow0\). If the inequality \(\Vert T^{n}x-T^{n}y\Vert \leq \Vert x-y\Vert +a_{n}\) holds for all \(x,y\in C\) and \(n\geq1\), then T is said to be nearly nonexpansive [1, 2] with respect to \(\{a_{n}\}\). Let \(\{ T_{n} \} \) be a sequence of mappings from C into H. Then the sequence \(\{ T_{n} \} \) is called a sequence of nearly nonexpansive mappings [3, 4] with respect to a sequence \(\{a_{n}\}\) if
It is obvious that the sequence of nearly nonexpansive mappings is a wider class of sequence of nonexpansive mappings. A mapping \(A:C\rightarrow H\) is called α-inverse strongly monotone if there exists a positive real number \(\alpha>0\) such that
and a mapping \(F:C\rightarrow H\) is called η-strongly monotone if there exists a constant \(\eta\geq0\) such that
In particular, if \(\eta=0\), then F is said to be monotone.
Let \(G:C\times C\rightarrow \mathbb{R} \) be a bifunction and B be a nonlinear mapping. The generalized equilibrium problem, denoted by GEP, is to find a point \(x\in C\) such that
for all \(y\in C\), and the solution of the problem (1.2) is denoted by \(\operatorname {GEP}( G ) \), i.e.,
If \(B=0\), then the GEP is reduced to equilibrium problem, denoted by EP, which is to find a point \(x\in C\) such that
for all \(y\in C\). The set of solutions of EP is denoted by \(\operatorname {EP}(G)\). In the case of \(G=0\), then GEP is equivalent to find a \(x\in C\) such that
for all \(y\in C\). The problem (1.3) is called variational inequality problem, denoted by \(VI ( C,B ) \), and the solution of \(VI ( C,B ) \) is denoted by Ω, i.e.,
The generalized equilibrium problem includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity, the Nash equilibrium problem in noncooperative games, the vector optimization problem, etc. Hence, the existence of solutions of generalized equilibrium problems has been extensively studied by many authors in the literature (see, e.g., [5–9]).
Let \(S:C\rightarrow H\) be a nonexpansive mapping. The following problem is called a hierarchical fixed point problem: Finding \(x^{\ast}\in \operatorname {Fix}(T)\) such that
where \(\operatorname {Fix}( T ) \) is the set of fixed points of T, i.e., \(\operatorname {Fix}( T ) = \{ x\in C:Tx=x \} \). The problem (1.4) is equivalent to the following fixed point problem: Finding an \(x^{\ast }\in C\) that satisfies \(x^{\ast}=P_{\operatorname {Fix}(T)}Sx^{\ast}\). Since \(\operatorname {Fix}(T)\) is closed and convex, the metric projection \(P_{\operatorname {Fix}(T)}\) is well defined.
It is well known that the hierarchical fixed point problem (1.4) links with some monotone variational inequalities and convex programming problems; see [10–15]. Therefore, there exist various methods to solve the hierarchical fixed point problem; see Yao and Liou in [16], Xu in [17], Marino and Xu in [18] and Bnouhachem and Noor in [19].
Now, we give some iteration schemes which are related with the problems (1.2), (1.3), and (1.4). In 2011, Ceng et al. [25] investigated the following iterative method:
where F is a L-Lipschitzian and η-strongly monotone operator with constants \(L,\eta>0\) andV is a γ-Lipschitzian (possibly non-self-)mapping with constant \(\gamma\geq0\) such that \(0<\mu <\frac {2\eta}{L^{2}}\) and \(0\leq\rho\gamma<1-\sqrt{1-\mu( 2\eta-\mu L^{2} ) }\). They proved that under some approximate assumptions on the operators and parameters, the sequence \(\{x_{n}\}\) generated by (1.5) converges strongly to the unique solution of the variational inequality
Recently, in 2013, Sahu et al. [26] introduced the following iterative process for the sequence of nearly nonexpansive mappings \(\{ T_{n} \} \) defined by (1.1):
where f is a contraction and \(\{ S_{n} \} \) is a sequence of nonexpansive mappings from C into itself. They proved that the sequence \(\{x_{n}\}\) generated by (1.7) converges strongly to the unique solution of the following variational inequality:
In the same year, Bnouhachem and Noor [19] introduced a new iterative scheme to find a common solution of a variational inequality, a generalized equilibrium problem and a hierarchical fixed point problem. Their scheme is as follows:
where \(V_{i}=k_{i}I+ ( 1-k_{i} ) T_{i}\), \(0\leq k_{i}<1\), \(\{ T_{i} \} _{i=1}^{\infty}:C\rightarrow C\) is a countable family of \(k_{i}\)-strict pseudo-contraction mappings, A and B are inverse strongly monotone mappings. They proved that the sequence \(\{ x_{n} \} \) generated by (1.8) converges strongly to a point \(z\in P_{\Omega \cap \operatorname {GEP}( G ) \cap \operatorname {Fix}( T ) }f ( z ) \) which is the unique solution of the following variational inequality:
where \(\operatorname {Fix}( T ) =\bigcap_{i=1}^{\infty} \operatorname {Fix}( T_{i} ) \).
In 2014, Bnouhachem and Chen [20] introduced the following iterative method:
where \(D,A:C\rightarrow H\) are inverse strongly monotone mappings, \(F_{1}:C\times C\rightarrow \mathbb{R} \) is a bifunction, \(\varphi:C\rightarrow \mathbb{R} \) is a proper lower semicontinuous and convex function, \(S,T:C\rightarrow C\) are nonexpansive mappings, \(F:C\rightarrow C\) is Lipschitzian and a strongly monotone mapping and \(U:C\rightarrow C\) is a Lipschitzian mapping. The authors proved the strong convergence of the sequence generated by (1.9) to a common solution of a variational inequality, a generalized mixed equilibrium problem, and a hierarchical fixed point problem.
In addition to all these papers, similar problems are considered in several papers; see, e.g., [21–24].
In this paper, motivated by the above works and by the recent work going in this direction, we introduce an iterative projection method and prove a strong convergence theorem based on this method for computing an approximate element of the common set of solution of a generalized equilibrium problem, a variational inequality problem and a fixed point problem for a sequence of nearly nonexpansive mappings defined by (1.1). The proposed method improves and extends many known results; see, e.g., [4, 11, 25, 27, 28] and the references therein.
2 Preliminaries
Let \(\{ x_{n} \} \) be a sequence in a Hilbert space H and \(x\in H \). Throughout this paper, \(x_{n}\rightarrow x\) denotes the strong convergence of \(\{ x_{n} \} \) to x and \(x_{n}\rightharpoonup x\) denotes the weak convergence. Let C be a nonempty subset of a real Hilbert space H. For solving an equilibrium problem for a bifunction \(G:C\times C\rightarrow \mathbb{R} \), let us assume that G satisfies the following conditions:
-
(A1)
\(G ( x,x ) =0\), \(\forall x\in C\),
-
(A2)
G is monotone, i.e. \(G ( x,y ) +G ( y,x ) \leq0\), \(\forall x,y\in C\),
-
(A3)
\(\forall x,y,z\in C\),
$$ \lim_{t\rightarrow0^{+}}G \bigl( tz+ ( 1-t ) x,y \bigr) \leq G ( x,y ), $$ -
(A4)
\(\forall x\in C\), \(y\longmapsto G ( x,y ) \) is convex and lower semicontinuous.
Lemma 1
[29]
Let C be a nonempty, closed, and convex subset of H, and let G be a bifunction from \(C\times C\) into ℝ satisfying (A1)-(A4). Let \(r>0\) and \(x\in H\). Then there exists \(z\in C\) such that
for all \(x\in C\).
Lemma 2
[30]
Suppose that \(G:C\times C\rightarrow \mathbb{R}\) satisfies (A1)-(A4). For \(r>0\) and \(x\in H\), define a mapping \(T_{r}:H\rightarrow C\) as follows:
for all \(z\in H\). Then the following hold:
-
(1)
\(T_{r}\) is single valued,
-
(2)
\(T_{r}\) is firmly nonexpansive i.e.
$$ \Vert T_{r}x-T_{r}y\Vert ^{2}\leq\langle T_{r}x-T_{r}y,x-y \rangle,\quad\forall x,y\in H, $$ -
(3)
\(\operatorname {Fix}( T_{r} ) =\operatorname {EP}( G ) \),
-
(4)
\(\operatorname {EP}( G ) \) is closed and convex.
Let \(T_{1},T_{2}:C\rightarrow H\) be two mappings. We denote \(\mathcal{B} ( C ) \), the collection of all bounded subsets of C. The deviation between \(T_{1}\) and \(T_{2}\) on \(B\in \mathcal{B} ( C ) \), denoted by \(\mathfrak{D}_{B} ( T_{1},T_{2} ) \), is defined by
The following lemmas will be used in the next section.
Lemma 3
[3]
Let C be a nonempty, closed, and bounded subset of a Banach space X and \(\{T_{n}\}\) be a sequence of nearly nonexpansive self-mappings on C with a sequence \(\{a_{n}\}\) such that \(\mathfrak{D} _{C} ( T_{n},T_{n+1} ) <\infty\). Then, for each \(x\in C\), \(\{T_{n}x\}\) converges strongly to some point of C. Moreover, if T is a mapping from C into itself defined by \(Tz=\lim_{n\rightarrow\infty }T_{n}z \) for all \(z\in C\), then T is nonexpansive and \(\lim _{n\rightarrow \infty}\mathfrak{D}_{C} ( T_{n},T ) =0\).
Lemma 4
[25]
Let \(V:C\rightarrow H\) be a γ-Lipschitzian mapping with a constant \(\gamma\geq0\) and let \(F:C\rightarrow H\) be a L-Lipschitzian and η-strongly monotone operator with constants \(L,\eta >0\). Then for \(0\leq\rho\gamma<\mu\eta\),
That is, \(\mu F-\rho V\) is strongly monotone with coefficient \(\mu\eta -\rho\gamma\).
Lemma 5
[31]
Let C be a nonempty subset of a real Hilbert space H. Suppose that \(\lambda\in( 0,1 ) \) and \(\mu>0\). Let \(F:C\rightarrow H\) be a L-Lipschitzian and η-strongly monotone operator on C. Define the mapping \(G:C\rightarrow H\) by
Then G is a contraction that provided \(\mu<\frac{2\eta}{L^{2}}\). More precisely, for \(\mu\in( 0,\frac{2\eta}{L^{2}} ) \),
where \(\nu=1-\sqrt{1-\mu( 2\eta-\mu L^{2} ) }\).
Lemma 6
[32]
Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and T be a nonexpansive self-mapping on C. If \(\operatorname {Fix}( T ) \neq\emptyset\), then \(I-T\) is demiclosed; that is whenever \(\{ x_{n} \} \) is a sequence in C weakly converging to some \(x\in C\) and the sequence \(\{ ( I-T ) x_{n} \} \) strongly converges to some y, it follows that \(( I-T ) x=y\). Here I is the identity operator of H.
Lemma 7
[33]
Assume that \(\{ x_{n} \} \) is a sequence of nonnegative real numbers satisfying the conditions
where \(\{ \alpha_{n} \} \) and \(\{ \beta_{n} \} \) are sequences of real numbers such that
Then \(\lim_{n\rightarrow\infty}x_{n}=0\).
3 Main results
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(A,B:C\rightarrow H\) be α, θ-inverse strongly monotone mappings, respectively. Let \(G:C\times C\rightarrow \mathbb{R} \) be a bifunction satisfying assumptions (A1)-(A4), \(S:C\rightarrow H\) be a nonexpansive mapping and \(\{ T_{n} \} \) be a sequence of nearly nonexpansive mappings with the sequence \(\{ a_{n} \} \) such that \(\mathcal {F}:=\operatorname {Fix}( T ) \cap\Omega\cap \operatorname {GEP}( G ) \neq \emptyset\) where \(Tx=\lim_{n\rightarrow\infty}T_{n}x\) for all \(x\in C\) and \(\operatorname {Fix}( T ) =\bigcap_{n=1}^{\infty} \operatorname {Fix}( T_{n} ) \). It is clear that the mapping T is nonexpansive. Let \(V:C\rightarrow H\) be a γ-Lipschitzian mapping, \(F:C\rightarrow H\) be a L-Lipschitzian and η-strongly monotone operator such that these coefficients satisfy \(0<\mu<\frac{2\eta}{L^{2}}\), \(0\leq\rho\gamma<\nu\), where \(\nu =1-\sqrt{1-\mu( 2\eta-\mu L^{2} ) }\). For an arbitrarily initial value \(x_{1}\), define the sequence \(\{ x_{n} \} \) in C generated by
where \(\{ \lambda_{n} \} \subset( 0,2\alpha) \), \(\{ r_{n} \} \subset( 0,2\theta) \), \(\{ \alpha _{n} \} \) and \(\{ \beta_{n} \} \) are sequences in \([ 0,1] \).
As can be seen, the convergence of the sequence \(\{ x_{n} \} \) generated by (3.1) depends on the choice of the control sequences and mappings. We list the following hypotheses on them:
Now, we need the following lemmas to prove our main theorem.
Lemma 8
Assume that the conditions (C1), (C2) hold and \(p\in \mathcal {F}\). Then the sequences \(\{ x_{n} \} \), \(\{ y_{n} \} \), \(\{ z_{n} \} \), and \(\{ u_{n} \} \) generated by (3.1) are bounded.
Proof
It is easy to see that the mapping \(I-r_{n}B\) is nonexpansive, so the mapping \(I-\lambda_{n}A\) is also nonexpansive. From Lemma 2, we have \(u_{n}=T_{r_{n}} ( x_{n}-r_{n}Bx_{n} ) \). Let \(p\in \mathcal {F}\). So, we get \(p=T_{r_{n}} ( p-r_{n}Bp ) \). Then we obtain
From (3.2), we get
It follows from (3.3) that
Since \(\lim_{n\rightarrow\infty}\frac{\beta_{n}}{\alpha_{n}}=0\), without loss of generality, we can assume that \(\beta_{n}\leq\alpha_{n}\), for all \(n\geq1\). This gives us \(\lim_{n\rightarrow\infty}\beta_{n}=0\).
Let \(t_{n}=\alpha_{n}\rho Vx_{n}+ ( I-\alpha_{n}\mu F ) T_{n}y_{n} \). Then we get
From condition (C2), there exists a constant \(M_{1}>0\) such that
Thus, from (3.6) we have
By induction, we get
Hence, we find that \(\{ x_{n} \} \) is bounded. So, the sequences \(\{ y_{n} \} \), \(\{ z_{n} \} \), and \(\{ u_{n} \} \) are bounded. □
Lemma 9
Assume that (C1)-(C3) hold. Let \(p\in \mathcal {F}\) and \(\{ x_{n} \} \) be the sequence generated by (3.1). Then the follow hold:
-
(i)
\(\lim_{n\rightarrow\infty} \Vert x_{n+1}-x_{n}\Vert =0\).
-
(ii)
\(w_{w} ( x_{n} ) \subset \operatorname {Fix}( T ) \) where \(w_{w} ( x_{n} ) \) is the weak w-limit set of \(\{ x_{n} \} \), i.e., \(w_{w} ( x_{n} ) = \{ x:x_{n_{i}}\rightharpoonup x \} \).
Proof
(i) Since the mappings \(P_{C}\) and \(( I-\lambda_{n}A ) \) are nonexpansive, we get
and so
On the other hand, since \(u_{n}=T_{r_{n}} ( x_{n}-r_{n}Bx_{n} ) \) and \(u_{n-1}=T_{r_{n-1}} ( x_{n-1}-r_{n-1}Bx_{n-1} ) \), we have
and
If we take \(y=u_{n-1}\) and \(y=u_{n}\) in (3.9) and (3.10), respectively, then we get
and
It follows from (3.11), (3.12), and monotonicity of the function G that
The last inequality implies that
From (3.13), we have
Without loss of generality, we can assume that there exists a real number μ such that \(r_{n}>\mu>0\) for all positive integers n. Then we obtain
Then we have
where
Hence, we write
where
From conditions (C2) and (C3), we get
So, it follows from (3.16), (3.17), and Lemma 7 that
(ii) First, we show that \(\lim_{n\rightarrow\infty} \Vert u_{n}-x_{n}\Vert =0\). Since \(p\in \mathcal {F}\), from (3.2) and (3.3), we obtain
Then, from (3.19), we get
It follows from (3.18) and from conditions (C1) and (C2) that \(\lim_{n\rightarrow\infty} \Vert Bx_{n}-Bp\Vert =0\) and \(\lim_{n\rightarrow\infty} \Vert Au_{n}-Ap\Vert =0\).
Since \(T_{r_{n}}\) is firmly nonexpansive mapping, we have
Therefore, we get
Then, from (3.3), (3.19), and (3.20), we obtain
The last inequality implies that
Since \(\lim_{n\rightarrow\infty} \Vert Bx_{n}-Bp\Vert =0\) and \(\{ \Vert y_{n}-p\Vert \} \) is a bounded sequence, by using (3.18) and conditions (C1), (C2), we obtain
On the other hand, since a metric projection \(P_{C}\) satisfies
we write
So, we get
By using (3.19) and (3.22), we have
Therefore, we get
Since \(\lim_{n\rightarrow\infty} \Vert Au_{n}-Ap\Vert =0\) and \(\{ \Vert y_{n}-p\Vert \} \) is a bounded sequence, by using (3.18) and conditions (C1), (C2), we obtain
Also, from (3.21) and (3.23), we have
On the other hand, we get
Since \(\lim_{n\rightarrow\infty}\beta_{n}=0\), again from (3.21) and (3.23), we obtain
Now, we show that \(\lim_{n\rightarrow\infty} \Vert x_{n}-Tx_{n}\Vert =0\). Before that we need to show that \(\lim_{n\rightarrow\infty} \Vert x_{n}-T_{n}x_{n}\Vert =0\):
Since \(a_{n}\rightarrow0\), by using (3.18), (3.25), and condition (C1), we obtain
Hence, from (3.26) and condition (C3), we have
Since \(\{ x_{n} \} \) is bounded, there exists a weak convergent subsequence \(\{ x_{n_{k}} \} \) of \(\{ x_{n} \} \). Let \(x_{n_{k}}\rightharpoonup w\) as \(k\rightarrow\infty\). From the Opial condition, we get \(x_{n}\rightharpoonup w\). So, it follows from Lemma 6 that \(w\in \operatorname {Fix}( T ) \). Therefore, \(w_{w} ( x_{n} ) \subset \operatorname {Fix}( T ) \). □
Theorem 1
Assume that (C1)-(C3) hold. Then the sequence \(\{ x_{n} \} \) generated by (3.1) converges strongly to \(x^{\ast }\in \mathcal {F}\), which is the unique solution of the variational inequality
Proof
Since the mapping T is defined by \(Tx=\lim_{n\rightarrow\infty}T_{n}x\) for all \(x\in C\), by Lemma 3, T is a nonexpansive mapping, and \(\operatorname {Fix}( T ) \neq\emptyset\). Moreover, since the operator \(\mu F-\rho V\) is \(( \mu\eta-\rho\gamma) \)-strongly monotone by Lemma 4, we get the uniqueness of the solution of the variational inequality (3.27). Let us denote this solution by \(x^{\ast}\in \operatorname {Fix}( T ) =\mathcal {F}\).
Now, we divide our proof into three steps.
Step 1. From Lemma 8, since \(\{ x_{n} \} \) is bounded, there exists an element w such that \(x_{n}\rightharpoonup w\). First, we show that \(w\in \mathcal {F}=\operatorname {Fix}( T ) \cap\Omega \cap \operatorname {GEP}( G ) \). It follows from Lemma 9 that \(w\in \operatorname {Fix}( T ) =\bigcap_{n=1}^{\infty} \operatorname {Fix}( T_{n} ) \). Next we show that \(w\in\Omega\). Let \(N_{C}v\) be the normal cone to C at \(v\in C\), i.e.,
Let
Then H is maximal monotone mapping. Let \(( v,u ) \in G ( H ) \). Since \(u-Av\in N_{C}v\) and \(z_{n}\in C\), we get
On the other hand, from the definition of \(z_{n}\), we have
and hence,
Therefore, using (3.28), we get
By using (3.21), (3.23), and (3.24), we get \(u_{n_{i}}\rightharpoonup w\) and \(z_{n_{i}}\rightharpoonup w\) for \(i\rightarrow\infty\). Hence, from (3.29) we have
Since H is maximal monotone, we have \(w\in H^{-1}0\) and hence \(w\in \Omega \).
Finally, we show that \(w\in \operatorname {GEP}( G ) \). By using \(u_{n}=T_{r_{n}} ( x_{n}-r_{n}Bx_{n} ) \), we get
Also, from the monotonicity of G, we have
and
Let \(y\in C\) and \(y_{t}=ty+ ( 1-t ) w\), for \(t\in( 0,1 ] \). Then \(y_{t}\in C\). From (3.30), we get
Since B is Lipschitz continuous, using (3.21) we obtain \(\lim_{k\rightarrow\infty} \Vert Bu_{n_{k}}-Bx_{n_{k}}\Vert =0\). It follows from (3.31), \(u_{n_{k}}\rightharpoonup w\) and the monotonicity of B that
Therefore, from assumptions (A1)-(A4) and (3.32), we have
The last inequality implies that
If we take the limit \(t\rightarrow0^{+}\), we get
Hence, we have \(w\in \operatorname {GEP}( G ) \). Thus, we obtain \(w\in \mathcal {F}=\operatorname {Fix}( T ) \cap\Omega\cap \operatorname {GEP}( G ) \).
Step 2. We show that \(\limsup_{n\rightarrow\infty} \langle ( \rho V-\mu F ) x^{\ast},x_{n}-x^{\ast} \rangle\leq0\), where \(x^{\ast}\) is the unique solution of variational inequality (3.27). Since the sequence \(\{ x_{n} \} \) is bounded, it has a weak convergent subsequence \(\{ x_{n_{k}} \} \) such that
Let \(x_{n_{k}}\rightharpoonup w\), as \(k\rightarrow\infty\). It follows from Step 1 that \(w\in \mathcal {F}\). Hence
Step 3. Finally, we show that the sequence \(\{ x_{n} \} \) generated by (3.1) converges strongly to the point \(x^{\ast}\). By using the iteration (3.1), we have
Since the metric projection \(P_{C}\) satisfies the inequality
from (3.33), we get
Hence, from Lemma 5, we obtain
The last inequality implies that
and
Since \(\frac{\beta_{n}}{\alpha_{n}}\rightarrow0\) and \(\frac {a_{n}}{\alpha _{n}}\rightarrow0\), we get
So, it follows from Lemma 7 that the sequence \(\{ x_{n} \} \) generated by (3.1) converges strongly to \(x^{\ast}\in \mathcal {F}\) which is the unique solution of variational inequality (3.27). □
Putting \(A=0\) in Theorem 1, we have the following corollary.
Corollary 1
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(B:C\rightarrow H\) be θ-inverse strongly monotone mapping, \(G:C\times C\rightarrow \mathbb{R} \) be a bifunction satisfying assumptions (A1)-(A4), \(S:C\rightarrow H\) be a nonexpansive mapping and \(\{ T_{n} \} \) be a sequence of nearly nonexpansive mappings with the sequence \(\{ a_{n} \} \) such that \(\mathcal {F}:=\operatorname {Fix}( T ) \cap\Omega\cap \operatorname {GEP}( G ) \neq \emptyset\) where \(Tx=\lim_{n\rightarrow\infty}T_{n}x\) for all \(x\in C\) and \(\operatorname {Fix}( T ) =\bigcap_{n=1}^{\infty} \operatorname {Fix}( T_{n} ) \). Let \(V:C\rightarrow H\) be a γ-Lipschitzian mapping, \(F:C\rightarrow H \) be a L-Lipschitzian and η-strongly monotone operator such that these coefficients satisfy \(0<\mu<\frac{2\eta}{L^{2}}\), \(0\leq\rho \gamma <\nu\), where \(\nu=1-\sqrt{1-\mu( 2\eta-\mu L^{2} ) }\). For an arbitrarily initial value \(x_{1}\in C\), consider the sequence \(\{ x_{n} \} \) in C generated by
where \(\{ r_{n} \} \subset( 0,2\theta) \), \(\{ \alpha_{n} \} \) and \(\{ \beta_{n} \} \) are sequences in \([ 0,1 ] \) satisfying the conditions (C1)-(C3) except the condition \(\lim_{n\rightarrow\infty}\frac{\vert\lambda _{n}-\lambda _{n-1}\vert}{\alpha_{n}}=0\). Then the sequence \(\{ x_{n} \} \) generated by (3.34) converges strongly to \(x^{\ast }\in \mathcal {F}\), where \(x^{\ast}\) is the unique solution of variational inequality (3.27).
In Theorem 1, if we take \(A=0\) and \(\beta_{n}=0\) for all \(n\geq1\), then we have the following corollary.
Corollary 2
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(B:C\rightarrow H\) be θ-inverse strongly monotone mapping, \(G:C\times C\rightarrow \mathbb{R} \) be a bifunction satisfying assumptions (A1)-(A4), \(\{ T_{n} \} \) be a sequence of nearly nonexpansive mappings with the sequence \(\{ a_{n} \} \) such that \(\mathcal {F}:=\operatorname {Fix}( T ) \cap\Omega \cap \operatorname {GEP}( G ) \neq\emptyset\) where \(Tx=\lim_{n\rightarrow\infty }T_{n}x\) for all \(x\in C\) and \(\operatorname {Fix}( T ) =\bigcap_{n=1}^{\infty }\operatorname {Fix}( T_{n} ) \). Let \(V:C\rightarrow H\) be a γ-Lipschitzian mapping, \(F:C\rightarrow H\) be a L-Lipschitzian and η-strongly monotone operator such that these coefficients satisfy \(0<\mu <\frac{2\eta}{L^{2}}\), \(0\leq\rho\gamma<\nu\), where \(\nu=1-\sqrt {1-\mu ( 2\eta-\mu L^{2} ) }\). For an arbitrarily initial value \(x_{1}\in C\), consider the sequence \(\{ x_{n} \} \) in C generated by
where \(\{ r_{n} \} \subset( 0,2\theta) \), \(\{ \alpha_{n} \} \) is a sequence in \([ 0,1 ] \) satisfying the conditions (C1)-(C3) except the conditions \(\lim_{n\rightarrow \infty}\frac{\beta_{n}}{\alpha_{n}}=0\), \(\lim_{n\rightarrow\infty}\frac{ \vert\lambda_{n}-\lambda_{n-1}\vert}{\alpha_{n}}=0\) and \(\lim_{n\rightarrow\infty}\frac{\vert\beta_{n}-\beta _{n-1}\vert}{\alpha_{n}}=0\). Then the sequence \(\{ x_{n} \} \) generated by (3.35) converges strongly to \(x^{\ast }\in \bigcap_{n=1}^{\infty} \operatorname {Fix}( T_{n} ) \cap\Omega\cap \operatorname {GEP}( G ) \), where \(x^{\ast}\) is the unique solution of variational inequality (3.27).
Putting \(A=0\) and \(B=0\), we have the following corollary, which gives us an iterative scheme to find a common solution of an equilibrium problem and a hierarchical fixed point problem.
Corollary 3
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(G:C\times C\rightarrow \mathbb{R} \) be a bifunction satisfying assumptions (A1)-(A4), \(S:C\rightarrow H\) be a nonexpansive mapping and \(\{ T_{n} \} \) be a sequence of nearly nonexpansive mappings with the sequence \(\{ a_{n} \} \) such that \(\mathcal {F}:=\operatorname {Fix}( T ) \cap\Omega\cap \operatorname {GEP}( G ) \neq \emptyset\) where \(Tx=\lim_{n\rightarrow\infty}T_{n}x\) for all \(x\in C\) and \(\operatorname {Fix}( T ) =\bigcap_{n=1}^{\infty} \operatorname {Fix}( T_{n} ) \). Let \(V:C\rightarrow H\) be a γ-Lipschitzian mapping, \(F:C\rightarrow H \) be a L-Lipschitzian and η-strongly monotone operator such that these coefficients satisfy \(0<\mu<\frac{2\eta}{L^{2}}\), \(0\leq\rho \gamma <\nu\), where \(\nu=1-\sqrt{1-\mu( 2\eta-\mu L^{2} ) }\). For an arbitrarily initial value \(x_{1}\), define the sequence \(\{ x_{n} \} \) in C generated by
where \(\{ r_{n} \} \subset( 0,\infty) \), \(\{ \alpha_{n} \} \) and \(\{ \beta_{n} \} \) are sequences in \([ 0,1 ] \) satisfying the conditions (C1)-(C3) except the condition \(\lim_{n\rightarrow\infty}\frac{\vert\lambda _{n}-\lambda _{n-1}\vert}{\alpha_{n}}=0\). Then the sequence \(\{ x_{n} \} \) generated by (3.36) converges strongly to \(x^{\ast }\in \bigcap_{n=1}^{\infty} \operatorname {Fix}( T_{n} ) \cap \operatorname {EP}( G ) \), where \(x^{\ast}\) is the unique solution of variational inequality (3.27).
Corollary 4
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(A,B:C\rightarrow H\) be α, θ-inverse strongly monotone mappings, respectively. \(G:C\times C\rightarrow \mathbb{R} \) be a bifunction satisfying assumptions (A1)-(A4), \(S:C\rightarrow H\) be a nonexpansive mapping and \(\{ T_{n} \} \) be a sequence of nonexpansive mappings such that \(\mathcal {F}:=\operatorname {Fix}( T ) \cap \Omega\cap \operatorname {GEP}( G ) \neq\emptyset\) where \(Tx=\lim_{n\rightarrow \infty}T_{n}x\) for all \(x\in C\) and \(\operatorname {Fix}( T ) =\bigcap_{n=1}^{\infty} \operatorname {Fix}( T_{n} ) \). Let \(V:C\rightarrow H\) be a γ-Lipschitzian mapping, \(F:C\rightarrow H\) be a L-Lipschitzian and η-strongly monotone operator such that these coefficients satisfy \(0<\mu<\frac{2\eta}{L^{2}}\), \(0\leq\rho\gamma<\nu\), where \(\nu =1-\sqrt{1-\mu( 2\eta-\mu L^{2} ) }\). For an arbitrarily initial value \(x_{1}\in C\), consider the sequence \(\{ x_{n} \} \) in C generated by (3.1) where \(\{ \lambda_{n} \} \subset( 0,2\alpha ) \), \(\{ r_{n} \} \subset( 0,2\theta) \), \(\{ \alpha_{n} \} \) and \(\{ \beta_{n} \} \) are sequences in \([ 0,1 ] \) satisfying the conditions (C1)-(C3) of Theorem 1 except the condition \(\lim_{n\rightarrow\infty }\frac {a_{n}}{\alpha_{n}}=0\). Then the sequence \(\{ x_{n} \} \) converges strongly to \(x^{\ast}\in \mathcal {F}\), where \(x^{\ast}\) is the unique solution of variational inequality (3.27).
Remark 1
Our results can be reduced to some corresponding results in the following ways:
-
(1)
In our iterative process (3.35), if we take \(G ( x,y ) =0\) for all \(x,y\in C\), \(B=0\), and \(r_{n}=1\) for all \(n\geq1\), then we derive the iterative process
$$ x_{n+1}=P_{C} \bigl[ \alpha_{n}\rho Vx_{n}+ ( I-\alpha_{n}\mu F ) T_{n}x_{n} \bigr] ,\quad n\geq1, $$which is studied by Sahu et al. [4]. Therefore, Theorem 1 generalizes the main result of Sahu et al. [4, Theorem 3.1]. So, our results extend the corresponding results of Ceng et al. [25] and of many other authors.
-
(2)
If we take S as a nonexpansive self-mapping on C and \(T_{n}=T\) for all \(n\geq1\) such that T is a nonexpansive mapping in (3.1), then it is clear that our iterative process generalizes the iterative process of Wang and Xu [28]. Hence, Theorem 1 generalizes the main result of Wang and Xu [28, Theorem 3.1]. So, our results extend and improve the corresponding results of [11, 27].
-
(3)
The problem of finding the solution of variational inequality (3.27) is equivalent to finding the solutions of hierarchical fixed point problem
$$ \bigl\langle ( I-S ) x^{\ast},x^{\ast}-x \bigr\rangle \leq0,\quad \forall x\in \mathcal {F}, $$where S= \(I- ( \rho V-\mu F ) \).
Example 1
Let \(H=\mathbb{R} \) and \(C= [ 0,1 ] \). Let \(G:C\times C\rightarrow \mathbb{R} \), \(G ( x,y ) =y^{2}+xy-2x^{2}\), \(S=I\), \(A:C\rightarrow H\), \(Ax=2x\), \(B:C\rightarrow H\), \(Bx=3x-1\), \(Vx=4x+2\), \(Fx=5x\), and
for all \(x\in C\). It is clear that \(G ( x,y ) \) is a bifunction satisfying the assumptions (A1)-(A4), S is nonexpansive mapping, A is \(\frac{1}{4}\)-inverse strongly monotone mapping, B is \(\frac{1}{6}\)-inverse strongly monotone mapping, V is γ-Lipschitzian mapping with \(\gamma=4\), F is L-Lipschitzian and η-strongly monotone operator with \(L=\eta=5\) and \(\{ T_{n} \} \) is a sequence of nearly nonexpansive mappings with respect to the sequence \(a_{n}=\frac {1}{2n^{2}-1}\). Define sequences \(\{ \alpha_{n} \} \) and \(\{ \beta _{n} \} \) in \([ 0,1 ] \) by \(\alpha_{n}=\frac{1}{n}\) and \(\beta_{n}=\frac{1}{n^{2}+2}\) for all \(n\geq1\) and take \(\mu=\rho =\frac{1}{5}\), \(\nu=1\), \(r_{n}=\frac{1}{n+3}\), and \(\lambda_{n}=\frac {1}{n+2}\). It is easy to see that all conditions of Theorem 1 are satisfied. First, we find the sequence \(\{ u_{n} \} \) which satisfies the following generalized equilibrium problem for all \(y\in C\):
For all \(n\geq1\), we get
Put \(K ( y ) =y^{2}r_{n}+y ( u_{n}r+3x_{n}r_{n}+u_{n}-r_{n}-x_{n} ) -2u_{n}^{2}r_{n}-3x_{n}u_{n}r_{n}+u_{n}r_{n}-u_{n}^{2}+u_{n}x_{n}\). Then K is a quadratic function of y with coefficients \(a=r_{n}\), \(b=u_{n}r_{n}+3x_{n}r_{n}+u_{n}-r_{n}-x_{n}\), and \(c=-2u_{n}^{2}r_{n}-3x_{n}u_{n}r_{n}+u_{n}r_{n}-u_{n}^{2}+u_{n}x_{n}\). Next, we compute the discriminant Δ of K as follows:
We know that \(K ( y ) \geq0\) for all \(y\in C= [ 0,1 ] \). If it has most one solution in \([ 0,1 ] \), so \(\Delta\leq0\) and hence \(u_{n}=\frac{r_{n}+x_{n} ( 1-3r_{n} ) }{1+3r_{n}}=\frac{1+nx_{n}}{n+6}\). By using this equation, the sequence \(\{ x_{n} \} \) generated by the iterative scheme (3.1) becomes
for all \(n\geq1\), and it converges strongly to \(x^{\ast}=0.5\) which is the unique common fixed point of the sequence \(\{ T_{n} \} \) and the unique solution of the variational inequality (1.6) over \(\bigcap_{n=1}^{\infty} \operatorname {Fix}( T_{n} ) \). Some of the values of the iterative scheme (3.37) for the different initial values \(x_{1}=0.1\), \(x_{1}=0.4\), and \(x_{1}=0.7\) are as in Table 1.
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Karahan, I., Secer, A., Ozdemir, M. et al. The common solution for a generalized equilibrium problem, a variational inequality problem and a hierarchical fixed point problem. J Inequal Appl 2015, 53 (2015). https://doi.org/10.1186/s13660-015-0567-x
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DOI: https://doi.org/10.1186/s13660-015-0567-x