Abstract
In this paper, we introduce two iterative algorithms for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions.
MSC:49J30, 47H09, 47J20, 49M05.
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1 Introduction
Let H be a real Hilbert space with inner product and norm , C be a nonempty closed convex subset of H and be the metric projection of H onto C. Let be a nonlinear mapping on C. We denote by the set of fixed points of S and by R the set of all real numbers. A mapping V is called strongly positive on H if there exists a constant such that
A mapping is called L-Lipschitz-continuous if there exists a constant such that
In particular, if then S is called a nonexpansive mapping; if then A is called a contraction.
Let be a real-valued function, be a nonlinear mapping and be a bifunction. We consider the generalized mixed equilibrium problem (GMEP) [1] of finding such that
We denote the set of solutions of GMEP (1.1) by . The GMEP (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied in, e.g., [2–8].
Throughout this paper, it is assumed as in [1] that is a bifunction satisfying conditions (H1)-(H4) and is a lower semicontinuous and convex function with restriction (H5), where
-
(H1) for all ;
-
(H2) Θ is monotone, i.e., for any ;
-
(H3) Θ is upper-hemicontinuous, i.e., for each ,
-
(H4) is convex and lower semicontinuous for each ;
-
(H5) for each and , there exist a bounded subset and such that for any ,
Let be two bifunctions, and be two nonlinear mappings. Consider the system of generalized equilibrium problems (SGEP): find such that
where and are two positive constants.
Let be an infinite family of nonexpansive self-mappings on C and be a sequence of nonnegative numbers in . For any , define a self-mapping on H as follows:
Such a mapping is called the W-mapping generated by and .
Let be a contraction and V be a strongly positive bounded linear operator on H. Assume that is a lower semicontinuous and convex functional, that satisfy conditions (H1)-(H4), and that are inverse-strongly monotone. Very recently, motivated by Yao et al. [3], Cai and Bu [4] introduced the following hybrid extragradient-like iterative algorithm:
for finding a common solution of GMEP (1.1), SGEP (1.2), and the fixed point problem of an infinite family of nonexpansive mappings on H, where , , and are given. The authors proved the strong convergence of the sequence generated by the hybrid iterative algorithm (1.4) to a point under some suitable conditions, where is the fixed point set of the mapping . This point also solves the following optimization problem:
where is the potential function of γf.
Let B be a single-valued mapping of C into H and R be a set-valued mapping with . Consider the following variational inclusion: find a point such that
We denote by the solution set of the variational inclusion (1.5). In particular, if , then . If , then problem (1.5) becomes the inclusion problem introduced by Rockafellar [9]. It is known that problem (1.5) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, etc. Let a set-valued mapping be maximal monotone. We define the resolvent operator associated with R and λ as follows:
where λ is a positive number.
In 1998, Huang [10] studied problem (1.5) in the case where R is maximal monotone and B is strongly monotone and Lipschitz-continuous with . Subsequently, Zeng et al. [11] further studied problem (1.5) in the case which is more general than Huang’s [10]. Moreover, the authors [11] obtained the same strong convergence conclusion as in Huang’s result [10]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [5, 12–17] and the references therein.
In 2011, for the case where , Yao et al. [5] introduced and analyzed an iterative algorithms for finding a common element of the set of solutions of the GMEP (1.1), the set of solutions of the variational inclusion (1.5) for maximal monotone and inverse-strongly monotone mappings and the set of fixed points of a countable family of nonexpansive mappings on H.
Recently, Kim and Xu [18] introduced the concept of asymptotically κ-strict pseudocontractive mappings in a Hilbert space.
Definition 1.1 Let C be a nonempty subset of a Hilbert space H. A mapping is said to be an asymptotically κ-strict pseudocontractive mapping with sequence if there exist a constant and a sequence in with such that
Subsequently, Sahu et al. [19] considered the concept of asymptotically κ-strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.
Definition 1.2 Let C be a nonempty subset of a Hilbert space H. A mapping is said to be an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence if there exist a constant and a sequence in with such that
Put . Then (), (), and (1.6) reduce to the relation
Whenever for all in (1.7), then S is an asymptotically κ-strict pseudocontractive mapping with sequence . The authors [19] derived the weak and strong convergence of the modified Mann iteration processes for an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence . More precisely, they first established one weak convergence theorem for the following iterative scheme:
where , , and ; and then obtained another strong convergence theorem for the following iterative scheme:
where , , and .
Inspired by the above facts, we in this paper introduce two iterative algorithms for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions. The results presented in this paper are the supplement, extension, improvement, and generalization of the previously known results in this area.
2 Preliminaries
Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by and , respectively. Let C be a nonempty closed convex subset of H. We write to indicate that the sequence converges weakly to x and to indicate that the sequence converges strongly to x. Moreover, we use to denote the weak ω-limit set of the sequence , i.e.,
Definition 2.1 A mapping is called
-
(i)
monotone if
-
(ii)
η-strongly monotone if there exists a constant such that
-
(iii)
ζ-inverse-strongly monotone if there exists a constant such that
It is easy to see that the projection is 1-inverse-strongly monotone (in short, 1-ism). Inverse-strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.
Definition 2.2 A differentiable function is called:
-
(i)
convex, if
where is the Frechet derivative of K at x;
-
(ii)
strongly convex, if there exists a constant such that
It is easy to see that if is a differentiable strongly convex function with constant then is strongly monotone with constant .
The metric (or nearest point) projection from H onto C is the mapping which assigns to each point the unique point satisfying the property
Some important properties of projections are gathered in the following proposition.
Proposition 2.1 For given and :
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, . (This implies that is nonexpansive and monotone.)
By using the technique of [20], we can readily obtain the following elementary result.
Proposition 2.2 (see [[6], Lemma 1 and Proposition 1])
Let C be a nonempty closed convex subset of a real Hilbert space H and let be a lower semicontinuous and convex function. Let be a bifunction satisfying the conditions (H1)-(H4). Assume that
-
(i)
is strongly convex with constant and the function is weakly upper semicontinuous for each ;
-
(ii)
for each and , there exist a bounded subset and such that for any ,
Then the following hold:
-
(a)
for each , ;
-
(b)
is single-valued;
-
(c)
is nonexpansive if is Lipschitz-continuous with constant and
where for ;
-
(d)
for all and
-
(e)
;
-
(f)
is closed and convex.
In particular, whenever is a bifunction satisfying the conditions (H1)-(H4) and , , then, for any ,
( is firmly nonexpansive) and
In this case, is rewritten as . If, in addition, , then is rewritten as (see [[21], Lemma 2.1] for more details).
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma 2.1 Let X be a real inner product space. Then we have the following inequality:
Lemma 2.2 Let H be a real Hilbert space. Then the following hold:
-
(a)
for all ;
-
(b)
for all and with ;
-
(c)
If is a sequence in H such that , it follows that
Lemma 2.3 ([[19], Lemma 2.5])
Let H be a real Hilbert space. Given a nonempty closed convex subset of H and points and given also a real number , the set
is convex (and closed).
Lemma 2.4 ([[19], Lemma 2.6])
Let C be a nonempty subset of a Hilbert space H and be an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence . Then
for all and .
Lemma 2.5 ([[19], Lemma 2.7])
Let C be a nonempty subset of a Hilbert space H and be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in C such that and as . Then as .
Lemma 2.6 (Demiclosedness principle [[19], Proposition 3.1])
Let C be a nonempty closed convex subset of a Hilbert space H and be a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence . Then is demiclosed at zero in the sense that if is a sequence in C such that and , then .
Lemma 2.7 ([[19], Proposition 3.2])
Let C be a nonempty closed convex subset of a Hilbert space H and be a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence such that . Then is closed and convex.
Remark 2.1 Lemmas 2.6 and 2.7 give some basic properties of an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence . Moreover, Lemma 2.6 extends the demiclosedness principles studied for certain classes of nonlinear mappings; see [19] for more details.
Lemma 2.8 ([[22], p.80])
Let , , and be sequences of nonnegative real numbers satisfying the inequality
If and , then exists. If, in addition, has a subsequence which converges to zero, then .
Recall that a Banach space X is said to satisfy the Opial condition [23] if, for any given sequence which converges weakly to an element , we have the inequality
It is well known in [23] that every Hilbert space H satisfies the Opial condition.
Lemma 2.9 (see [[24], Proposition 3.1])
Let C be a nonempty closed convex subset of a real Hilbert space H and let be a sequence in H. Suppose that
where and are sequences of nonnegative real numbers such that and . Then converges strongly in C.
Lemma 2.10 (see [25])
Let C be a closed convex subset of a real Hilbert space H. Let be a sequence in H and . Let . If is such that and satisfies the condition
then as .
Lemma 2.11 (see [[26], Lemma 3.2])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a sequence of nonexpansive self-mappings on C such that and let be a sequence in for some . Then, for every and the limit exists.
Remark 2.2 (see [[27], Remark 3.1])
It can be known from Lemma 2.11 that if D is a nonempty bounded subset of C, then for there exists such that for all
Remark 2.3 (see [[27], Remark 3.2])
Utilizing Lemma 2.11, we define a mapping as follows:
Such a W is called the W-mapping generated by and . Since is nonexpansive, is also nonexpansive. Indeed, observe that for each
If is a bounded sequence in C, then we put . Hence, it is clear from Remark 2.2 that for an arbitrary there exists such that for all
This implies that
Lemma 2.12 (see [[26], Lemma 3.3])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a sequence of nonexpansive self-mappings on C such that , and let be a sequence in for some . Then .
Lemma 2.13 (see [[28], Theorem 10.4 (Demiclosedness Principle)])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be nonexpansive. Then is demiclosed on C. That is, whenever is a sequence in C weakly converging to some and the sequence strongly converges to some y, it follows that . Here I is the identity operator of H.
Recall that a set-valued mapping is called monotone if, for all , , and imply
A set-valued mapping R is called maximal monotone if R is monotone and for each , where I is the identity mapping of H. We denote by the graph of R. It is known that a monotone mapping R is maximal if and only if, for , for every , we have . We illustrate the concept of maximal monotone mapping with the following example.
Let be a monotone, k-Lipschitz-continuous mapping and let be the normal cone to C at , i.e.,
Define
Then T is maximal monotone and if and only if for all (see [9]).
Assume that is a maximal monotone mapping. Let . In terms of Huang [10] (see also [11]), we have the following property for the resolvent operator .
Lemma 2.14 is single-valued and firmly nonexpansive, i.e.,
Consequently, is nonexpansive and monotone.
Lemma 2.15 (see [14])
Let R be a maximal monotone mapping with . Then for any given , is a solution of problem (1.6) if and only if satisfies
Lemma 2.16 (see [11])
Let R be a maximal monotone mapping with and let be a strongly monotone, continuous, and single-valued mapping. Then for each , the equation has a unique solution for .
Lemma 2.17 (see [14])
Let R be a maximal monotone mapping with and be a monotone, continuous and single-valued mapping. Then for each . In this case, is maximal monotone.
Lemma 2.18 (see [29])
Let C be a nonempty closed convex subset of a real Hilbert space H, and be a proper lower semicontinuous differentiable convex function. If is a solution the minimization problem
then
In particular, if solves (OP1), then
3 Strong convergence theorems
In this section, we introduce and analyze an iterative algorithm for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences we will prove strong convergence of the proposed algorithm.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ be a bifunction from to R satisfying (H1)-(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be ζ-inverse-strongly monotone and -inverse-strongly monotone, respectively, where . Let be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let V be a -strongly positive bounded linear operator and be an l-Lipschitzian mapping with . Assume that is nonempty and bounded. Let be the W-mapping defined by (1.4) and , and be three sequences in such that and . Assume that:
-
(i)
is strongly convex with constant and its derivative is Lipschitz-continuous with constant such that the function is weakly upper semicontinuous for each ;
-
(ii)
for each , there exist a bounded subset and such that for any ,
-
(iii)
;
-
(iv)
, , and satisfies
Pick any and set , . Let be a sequence generated by the following algorithm:
where , , and . If is firmly nonexpansive, then the following statements hold:
-
(I)
converges strongly to ;
-
(II)
converges strongly to , which solves the optimization problem
(OP2)provided additionally, where is the potential function of γf.
Proof Since and , we may assume, without loss of generality, that . Since V is a -strongly positive bounded linear operator on H, we know that
Observe that
that is, is positive. It follows that
Put
for all and , and , where I is the identity mapping on H. Then we have that . We divide the rest of the proof into several steps.
Step 1. We show that is well defined. It is obvious that is closed and convex. As the defining inequality in is equivalent to the inequality
by Lemma 2.3 we know that is convex and closed for every .
First of all, we show that for all . Suppose that for some . Take arbitrarily. Since , A is ζ-inverse strongly monotone and , we have
Since , , and is -inverse-strongly monotone, where , , by Lemma 2.14 we deduce that
Combining (3.2) and (3.3), we have
By Lemma 2.2(b), we deduce from (3.1) and (3.4) that
Set . Then, for , by Lemma 2.1 we obtain from (3.1), (3.4), and (3.5)
which hence yields
where , , and (due to and ). Hence . This implies that for all . Therefore, is well defined.
Step 2. We prove that as .
Indeed, let . From and , we obtain
This implies that is bounded and hence , , , and are also bounded. Since and , we have
Therefore exists. From , , by Proposition 2.1(ii) we obtain
which implies
It follows from that and hence
From (3.8) and , we have
Also, utilizing Lemmas 2.1 and 2.2(b) we obtain from (3.1), (3.4), and (3.5)
which leads to
Since , , and , it follows from (3.9) and condition (iii) that
Note that
which yields
So, from (3.9), (3.10), and , we get
Step 3. We prove that , , , and as .
Indeed, taking into consideration that , we may assume, without loss of generality, that . From (3.4) and (3.5) it follows that
Next we prove that
For , we find that
By (3.3), (3.12), and (3.14), we obtain
which implies that
From , , and (3.11), we have
By the firm nonexpansivity of and Lemma 2.2(a), we have
which implies that
Combining (3.12) and (3.16), we have
which implies
From , , (3.11), and (3.15), we know that (3.13) holds.
Next we show that , . It follows from Lemma 2.14 that
Combining (3.12) and (3.17), we have
together with , , implies
From , , and (3.11), we obtain
By Lemma 2.14 and Lemma 2.2(a), we obtain
which implies
Combining (3.12) and (3.19) we get
which implies
From (3.11), (3.18), , and , we have
From (3.20) we get
By (3.13) and (3.21), we have
From (3.8) and (3.22), we have
By (3.11), (3.13), and (3.21), we get
We observe that
From and (3.24), we have
We note that
From (3.23), (3.25), and Lemma 2.4, we obtain
On the other hand, we note that
From (3.25), (3.26), and the uniform continuity of S, we have
In addition, note that
So, from (3.10), (3.24), and Remark 2.3 it follows that
Step 4. we prove that as .
Indeed, since is bounded, there exists a subsequence which converges weakly to some w. From (3.13) and (3.20)-(3.22), we see that , , and , where . Since S is uniformly continuous, by (3.27) we get for any . Hence from Lemma 2.6, we obtain . In the meantime, utilizing Lemma 2.13, we deduce from (3.28) and that (due to Lemma 2.12). Next, we prove that . As a matter of fact, since is -inverse-strongly monotone, is a monotone and Lipschitz-continuous mapping. It follows from Lemma 2.17 that is maximal monotone. Let , i.e., . Again, since , , , we have
that is,
In terms of the monotonicity of , we get
and hence
In particular,
Since (due to (3.20)) and (due to the Lipschitz-continuity of ), we conclude from and , that
It follows from the maximal monotonicity of that , i.e., . Therefore, .
Next, we show that . In fact, from , we know that
From (H2) it follows that
Replacing n by , we have
Put for all and . Then, from (3.29), we have
Since as , we deduce from the Lipschitz-continuity of A and that and as . Further, from the monotonicity of A, we have . So, from (H4), we have the weakly lower semicontinuity of and , then we have
From (H1), (H4), and (3.30) we also have
and hence
Letting , we have, for each ,
This implies that . Therefore,
This shows that . From (3.7) and Lemma 2.10 we infer that as .
Finally, assume additionally that . Note that V is a -strongly positive bounded linear operator and is an l-Lipschitzian mapping with . It is clear that
Hence we deduce that is -strongly monotone. In the meantime, it is easy to see that is -Lipschitzian with constant . Thus, there exists a unique solution p in Ω to the VIP
Equivalently, solves (OP2) (due to Lemma 2.18). Consequently, we deduce from (3.9) and () that
Furthermore, by Lemma 2.1 we conclude from (3.1), (3.4), and (3.5) that
which hence yields
Since , , and , we infer from (3.31) and that as
That is, . This completes the proof. □
Corollary 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from to R satisfying (H1)-(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be ζ-inverse strongly monotone and -inverse-strongly monotone, respectively, for . Let be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let V be a -strongly positive bounded linear operator and be an l-Lipschitzian mapping with . Assume that is nonempty and bounded. Let be the W-mapping defined by (1.4) and , , and be three sequences in such that and . Assume that:
-
(i)
is strongly convex with constant and its derivative is Lipschitz-continuous with constant such that the function is weakly upper semicontinuous for each ;
-
(ii)
for each , there exist a bounded subset and such that for any ,
-
(iii)
;
-
(iv)
for , and satisfies
Pick any and set , . Let be a sequence generated by the following algorithm:
where , , and . If is firmly nonexpansive, then the following statements hold:
-
(I)
converges strongly to ;
-
(II)
converges strongly to , which solves the optimization problem
(OP3)provided additionally, where is the potential function of γf.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from to R satisfying (H1)-(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be ζ-inverse strongly monotone and ξ-inverse-strongly monotone, respectively. Let be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let V be a -strongly positive bounded linear operator and be an l-Lipschitzian mapping with . Assume that is nonempty and bounded. Let be the W-mapping defined by (1.4) and , , and be three sequences in such that and . Assume that:
-
(i)
is strongly convex with constant and its derivative is Lipschitz-continuous with constant such that the function is weakly upper semicontinuous for each ;
-
(ii)
for each , there exist a bounded subset and such that for any ,
-
(iii)
;
-
(iv)
, and satisfies
Pick any and set , . Let be a sequence generated by the following algorithm:
where , , and . If is firmly nonexpansive, then the following statements hold:
-
(I)
converges strongly to ;
-
(II)
converges strongly to , which solves the optimization problem
(OP4)provided additionally, where is the potential function of γf.
4 Weak convergence theorems
In this section, we introduce and analyze another iterative algorithm for finding common solutions of a finite family of variational inclusions for maximal monotone and inverse-strongly monotone mappings with the constraints of two problems: a generalized mixed equilibrium problem and a common fixed point problem of an infinite family of nonexpansive mappings and an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Under mild conditions imposed on the parameter sequences we will prove weak convergence of the proposed algorithm.
Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ be a bifunction from to R satisfying (H1)-(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be ζ-inverse-strongly monotone and -inverse-strongly monotone, respectively, where . Let be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some with sequences and . Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let V be a -strongly positive bounded linear operator and be an l-Lipschitzian mapping with . Assume that is nonempty. Let be the W-mapping defined by (1.4) and , and be three sequences in such that . Assume that:
-
(i)
is strongly convex with constant and its derivative is Lipschitz-continuous with constant such that the function is weakly upper semicontinuous for each ;
-
(ii)
for each , there exist a bounded subset and such that for any ,
-
(iii)
and ;
-
(iv)
, , and satisfies
Pick any and let be a sequence generated by the following algorithm:
Then converges weakly to provided is firmly nonexpansive.
Proof First, let us show that exists for any . Put
for all , , and , where I is the identity mapping on H. Then we see that . Take arbitrarily. Similarly to the proof of Theorem 3.1, we obtain
By Lemma 2.2(b) we get
Repeating the same arguments as in the proof of Theorem 3.1 we have
Then by Lemma 2.1 we deduce from (4.2), (4.3), (4.8), and that
which hence yields
where (due to and ). By Lemma 2.8, we see from that exists. Thus is bounded and so are the sequences , , and .
Also, utilizing Lemmas 2.1 and 2.2(b) we obtain from (4.2), (4.3), and (4.8)
which leads to
Since , , and , it follows from the existence of and condition (iii) that
Note that
which yields
So, from (4.11) and , we get
In the meantime, we conclude from (4.2), (4.3), (4.8), and (4.10) that
which, together with , implies that
Consequently, from , , , condition (iii), and the existence of , we get
Since , from (4.13) we have
Combining (4.4), (4.8), and (4.10), we have
which implies
From condition (iv), , , , and the existence of , we get
Combining (4.5), (4.8), and (4.10), we have
which implies
From (4.15), , , , and the existence of , we obtain
Combining (4.6), (4.8), and (4.10), we have
which implies
From , , , , , and the existence of , we obtain
Combining (4.7), (4.8), and (4.10), we get
which implies
From (4.17), , , , and the existence of , we obtain
By (4.18), we have
From (4.16) and (4.19), we have
By (4.14) and (4.20), we obtain
which, together with (4.12) and (4.21), implies that
On the other hand, we observe that
By (4.20) and (4.22), we have
We note that
From (4.13), (4.23), Lemma 2.4, and the uniform continuity of S, we obtain
In addition, note that
So, from (4.11), (4.14), and Remark 2.3 it follows that
Since is bounded, there exists a subsequence of which converges weakly to w. From (4.20) and (4.21), we have and . From (4.24) and the uniform continuity of S, we have for any . So, from Lemma 2.6, we have . In the meantime, by (4.25) and Lemma 2.13, we get (due to Lemma 2.12). Utilizing similar arguments to those in the proof of Theorem 3.1, we can derive . Consequently, . This shows that .
Next let us show that is a single-point set. As a matter of fact, let be another subsequence of such that . Then we get . If , from the Opial condition, we have
This attains a contradiction. So we have . Put . Since , we have . By Lemma 2.9, we see that converges strongly to some . Since converges weakly to w, we have
Therefore we obtain . This completes the proof. □
Corollary 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from to R satisfying (H1)-(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be ζ-inverse-strongly monotone and -inverse-strongly monotone, respectively, for . Let be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some with sequences and . Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let V be a -strongly positive bounded linear operator and be an l-Lipschitzian mapping with . Assume that is nonempty. Let be the W-mapping defined by (1.4) and , , and be three sequences in such that . Assume that:
-
(i)
is strongly convex with constant and its derivative is Lipschitz-continuous with constant such that the function is weakly upper semicontinuous for each ;
-
(ii)
for each , there exist a bounded subset and such that for any ,
-
(iii)
and ;
-
(iv)
for , and satisfies
Pick any and let be a sequence generated by the following algorithm:
Then converges weakly to provided is firmly nonexpansive.
Corollary 4.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from to R satisfying (H1)-(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be ζ-inverse-strongly monotone and ξ-inverse-strongly monotone, respectively. Let be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense for some with sequences and . Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let V be a -strongly positive bounded linear operator and be an l-Lipschitzian mapping with . Assume that is nonempty. Let be the W-mapping defined by (1.4) and , , and be three sequences in such that . Assume that:
-
(i)
is strongly convex with constant and its derivative is Lipschitz-continuous with constant such that the function is weakly upper semicontinuous for each ;
-
(ii)
for each , there exist a bounded subset and such that for any ,
-
(iii)
and ;
-
(iv)
, and satisfies
Pick any and let be a sequence generated by the following algorithm:
Then converges weakly to provided is firmly nonexpansive.
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Acknowledgements
In this research, first author was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). The second author and third author were supported partly by the National Science Council of the Republic of China.
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Ceng, LC., Chen, CM. & Pang, CT. On a finite family of variational inclusions with the constraints of generalized mixed equilibrium and fixed point problems. J Inequal Appl 2014, 462 (2014). https://doi.org/10.1186/1029-242X-2014-462
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DOI: https://doi.org/10.1186/1029-242X-2014-462