Abstract
We first introduce and analyze one multistep iterative algorithm by hybrid shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: the generalized mixed equilibrium problem, finitely many variational inclusions, the minimization problem for a convex and continuously Fréchet differentiable functional, and the fixed-point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another multistep iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions.
1. Introduction
Let be a nonempty closed convex subset of a real Hilbert space and the metric projection of onto . Let be a nonlinear mapping on . We denote by Fix the set of fixed points of and by the set of all real numbers. A mapping is called strongly positive on if there exists a constant such that A mapping is called -Lipschitz continuous if there exists a constant such that In particular, if , then is called a nonexpansive mapping; if , then is called a contraction.
Let be a nonlinear mapping on . We consider the following variational inequality problem (VIP): find a point such that The solution set of VIP (3) is denoted by . We remark that VIP (3) was first discussed by Lions [1].
Let be a real-valued function, a nonlinear mapping, and a bifunction. In 2008, Peng and Yao [2] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that We denote the set of solutions of GMEP (4) by GMEP . The GMEP (4) is very general which includes, as special cases, the generalized equilibrium problem [3], the mixed equilibrium problem [4], and the equilibrium problem [5, 6].
In [2], it is assumed that is a bifunction satisfying conditions (H1)–(H4) and is a lower semicontinuous and convex function with restriction (H5), where (H1) for all ; (H2) is monotone; that is, for any ; (H3) is upper hemicontinuous; that is, for each , (H4) is convex and lower semicontinuous for each ; (H5) for each and , there exists a bounded subset and such that for any ,
Given a positive number . Let be the solution set of the auxiliary mixed equilibrium problem; that is, for each , In particular, whenever , is rewritten as . Further, if additionally, then is rewritten as .
Let be two bifunctions satisfying conditions (H1)–(H4), and let be two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP): find such that where and are two constants. It is introduced and studied in [7]. Whenever , the SGEP reduces to a system of variational inequalities, which is considered and studied in [8]. It is worth to mention that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games. In 2010, Ceng and Yao [7] transformed the SGEP into a fixed-point problem of the mapping . Here, we denote the fixed point set of by SGEP.
Let be an infinite family of nonexpansive mappings on and a sequence of nonnegative numbers in . For any , define a mapping on as follows: Such a mapping is called the -mapping generated by and .
Let be a contraction and a strongly positive bounded linear operator on . Assume that is a lower semicontinuous and convex functional, such that satisfy conditions (H1)–(H4) and that are inverse-strongly monotone. Very recently, Ceng et al. [9] introduced the following hybrid extragradient-like iterative algorithm: for finding a common solution of (4), (8), and the fixed point problem of an infinite family of nonexpansive mappings on , where , , and are given. The authors proved the strong convergence of the sequence to a point under some suitable conditions. This point also solves the following optimization problem: where is the potential function of .
Let be a single-valued mapping of into and a multivalued mapping with . Consider the following variational inclusion: find a point such that We denote by the solution set of the variational inclusion (11).
Let be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing over the constraint set which was studied in [10–13]. We denote by the set of minimizers of CMP (12).
Let be a nonempty subset of a normed space . A mapping is called uniformly Lipschitzian if there exists a constant such that Recently, Kim and Xu [14] introduced the concept of asymptotically -strict pseudocontractive mappings in a Hilbert space as below.
Definition 1. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping with sequence if there exists a constant and a sequence in with such that
It is important to note that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with . Subsequently, Sahu et al. [15] considered the concept of asymptotically -strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.
Definition 2. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence if there exist a constant and a sequence in with such that
Put . Then , , and there holds the relation
In this paper, we first introduce and analyze one multistep iterative algorithm by hybrid shrinking projection method for finding a solution of the SGEP (8) with constraints of several problems: the GMEP (4), the CMP (12), finitely many variational inclusions, and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another multistep iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literatures.
2. Preliminaries
Throughout this paper, we assume that is a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of . We write to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . Moreover, we use to denote the weak -limit set of the sequence ; that is,
Recall that a mapping is called(i)monotone if (ii)-strongly monotone if there exists a constant such that (iii)-inverse-strongly monotone if there exists a constant such that
It is obvious that if is -inverse-strongly monotone, then is monotone and -Lipschitz continuous.
The metric (or nearest point) projection from onto 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 3. For given and :(i);(ii);(iii).
Consequently, is nonexpansive and monotone.
If is an -inverse-strongly monotone mapping of into , then it is obvious that is -Lipschitz continuous. We also have that if , then is a nonexpansive mapping from to .
Definition 4. A mapping is said to be(a)nonexpansive if (b)firmly nonexpansive if is nonexpansive, or equivalently, if is -inverse strongly monotone (-ism), alternatively, is firmly nonexpansive if and only if can be expressed as where is nonexpansive; projections are firmly nonexpansive.
It can be easily seen that if is nonexpansive, then is monotone.
Definition 5. A mapping is said to be an averaged mapping if it can be written as the average of the identity and a nonexpansive mapping; that is, where and is nonexpansive. More precisely, when the last equality holds, we say that is -averaged. Thus firmly nonexpansive mappings (in particular, projections) are -averaged mappings.
Proposition 6 (see [16]). Let be a given mapping.(i) is nonexpansive if and only if the complement is -ism.(ii)If is -ism, then for , is -ism.(iii) is averaged if and only if the complement is -ism for some . Indeed, for is -averaged if and only if is -ism.
Proposition 7 (see [16]). Let be given operators.(i)If for some and if is averaged and is nonexpansive, then is averaged.(ii) is firmly nonexpansive if and only if the complement is firmly nonexpansive.(iii)If for some and if is firmly nonexpansive and is nonexpansive, then is averaged.(iv)The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then the composite is -averaged, where .(v)If the mappings are averaged and have a common fixed point, then The notation denotes the set of all fixed points of the mapping ; that is, .
By using the technique in [4], we can readily obtain the following elementary result.
Proposition 8 (see [9, Lemma 1 and Proposition 1]). Let be a nonempty closed convex subset of a real Hilbert space 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 exists 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.
Remark 9. In Proposition 6, whenever is a bifunction satisfying the conditions (H1)–(H4) and , , we have for any , ( is firmly nonexpansive) and In this case, is rewritten as . If, in addition, , then is rewritten as .
We need some facts and tools in a real Hilbert space which are listed as lemmas below.
Lemma 10. Let be a real inner product space. Then there holds the following inequality:
Lemma 11. Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 3(i)) implies
Lemma 12 (see [17], demiclosedness principle). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on . Then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here is the identity operator of .
Lemma 13 (see [18, 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 is said to satisfy the Opial condition [17] if for any given sequence which converges weakly to an element , there holds the inequality
It is well known in [17] that every Hilbert space satisfies the Opial condition.
Lemma 14 (see [19, Proposition 3.1]). Let be a nonempty closed convex subset of a real Hilbert space and let be a sequence in . Suppose that where and are sequences of nonnegative real numbers such that and . Then converges strongly in .
Lemma 15 (see [20]). Let be a closed convex subset of a real Hilbert space . Let be a sequence in and . Let . If is such that and satisfies the condition then as .
Lemma 16. Let be a real Hilbert space. Then the following hold:(a) for all ;(b) for all and with ;(c)if is a sequence in such that , it follows that
Assume that is a maximal monotone mapping. Then, for , associated with , the resolvent operator can be defined as
In terms of Huang [21] (see also [22]), there holds the following property for the resolvent operator .
Lemma 17. is single-valued and firmly nonexpansive; that is, Consequently, is nonexpansive and monotone.
Lemma 18 (see [23]). Let be a maximal monotone mapping with . Then for any given is a solution of problem (11) if and only if satisfies
Lemma 19 (see [22]). Let 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 20 (see [23]). Let be a maximal monotone mapping with and a monotone, continuous, and single-valued mapping. Then for each . In this case, is maximal monotone.
Lemma 21 (see [22, Lemma 2.5]). Let be a real Hilbert space. Given a nonempty closed convex subset of and points and given also a real number , the set
is convex (and closed).
Recall that a set-valued mapping is called monotone if for all , and imply
A set-valued mapping is called maximal monotone if is monotone and for each , where is the identity mapping of . We denote by the graph of . It is known that a monotone mapping is maximal if and only if, for for every implies . Let be a monotone, -Lipschitz-continuous mapping and let be the normal cone to at ; that is,
Define
Then, is maximal monotone and if and only if ; see [24].
Lemma 22 (see [15, Lemma 2.6]). Let be a nonempty subset of a Hilbert space and an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then for all and .
Lemma 23 (see [15, Lemma 2.7]). Let be a nonempty subset of a Hilbert space and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in such that and as . Then as .
Lemma 24 (see [15, Proposition 3.1] demiclosedness principle). Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then is demiclosed at zero in the sense that if is a sequence in such that and , then .
Lemma 25 (see [15, Proposition 3.2]). Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that . Then is closed and convex.
3. Strong Convergence Theorem
In this section, we will introduce and analyze one multistep iterative algorithm by hybrid shrinking projection method for finding a solution of the SGEP (8) with constraints of several problems: the GMEP (4), the CMP (12), finitely many variational inclusions, and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. This iterative algorithm is based on Korpelevich’s extragradient method, strongly positive bounded linear operator approach, viscosity approximation method, averaged mapping approach to the GPA in [16], Mann-type iteration method, and shrinking projection method. The following proposition will play a key role in the proof of the main results in this paper.
Proposition CY (see [7]). Let be two bifunctions satisfying conditions (H1)–(H4) and let be -inverse-strongly monotone for . Let for . Then, is a solution of SGEP (8) if and only if is a fixed point of the mapping defined by , where . Here, we denote by SGEP the fixed point set of .
Theorem 26. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse-strongly monotone, respectively, for and . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a -strongly positive bounded linear operator and an -Lipschitzian mapping with . Assume that is nonempty and bounded, where is defined as in Proposition CY. Let be a sequence in and , and sequences in such that and , and let , , and , . Pick any and set . Let be a sequence generated by the following algorithm: where (here is nonexpansive; for each ), and , . Assume that the following conditions are satisfied:(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) for each , and ();(iv) and .
Assume that is firmly nonexpansive. Then we have(i) converges strongly as () to ;(ii) converges strongly as () to provided that , which is the unique solution in to the VIP Equivalently, .
Proof. Since is -Lipschitzian, it follows that is -ism; see [16]. By Proposition 6(ii) we know that for is -ism. So by Proposition 6(iii) we deduce that is -averaged. Now since the projection is -averaged, it is easy to see from Proposition 7(iv) that the composite is -averaged for . Hence we obtain that for each , is -averaged for each . Therefore, we can write
where is nonexpansive and for each . It is clear that
As , and , we may assume, without loss of generality, that , and for all . Since is a -strongly positive bounded linear operator on , we know that
Taking into account that for all , we have
that is, is positive. It follows that
Put
for all , and , where is the identity mapping on . Then we have .
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 21 we know that is convex for every .
First of all, let us show that for all . Suppose that for some . Take arbitrarily. Since , is -inverse strongly monotone and , we have, for any ,
Since , and is -inverse strongly monotone, where , , by Lemma 17 we deduce that for each ,
Combining (57) and (58), we have
Since is -inverse-strongly monotone for , and for , we deduce that, for any ,
(This shows that is nonexpansive.) Also, from (47), (54), (59), and (60), it follows that
which hence yields
By Lemma 16(b), we deduce from (47) and (62) that
So, from (47) and (63) we get
where and . Hence . This implies that for all . Therefore, is well defined.
Step 2. We prove that , and 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 3(ii) we obtain
which implies
It follows from that , and hence
From (68) and , we have
Since and , we have
which immediately leads to
Also, utilizing Lemmas 10 and 16(b) we obtain from (47), (59), (60), and (63) that
and hence
So, it follows that
Since , and , it follows from (70) and the boundedness of , , , and that
Note that
Hence, it follows from (76) and that
Note that
Thus, we deduce from (72) and (78) that
Since and , we have
which together with (80), yields
Step 3. We prove that , , , and as .
Indeed, from (58), (60), and , it follows that
Next let us show that
For , we find that
Combining (83) and (85), we obtain
which immediately implies that
Since and and are bounded sequences, it follows from (78) that
Furthermore, from the firm nonexpansivity of , we have
which leads to
From (83) and (90), we have
which hence implies that
Since and , , and are bounded sequences, it follows from (78) and (88) that (84) holds.
Next we show that , . As a matter of fact, observe that
Combining (60), (83), and (93), we have
which together with , for all , implies that
Since and and are bounded sequences, it follows from (78) that
By Lemma 16(a) and Lemma 17, we obtain
which implies
Combining (60), (83), and (98), we have
So, we conclude that
Since and , , and are bounded, from (78) and (96) we get
From (101) we get
Taking into account that , we conclude from (84) and (102) that
On the other hand, for simplicity, we write , and for all . Then
We now show that ; that is, . As a matter of fact, for , it follows from (59), (60), and (83) that
which immediately yields
Since and and are bounded, from (78) we get
Also, in terms of the firm nonexpansivity of and the -inverse strong monotonicity of for , we obtain from , and (60) that
Thus, we have
Consequently, from (59), (105), and (109) it follows that
which hence leads to
Since and , , , and are bounded sequences, we conclude from (78) and (107) that
Furthermore, from (59), (105), and (110) it follows that
which hence yields
Since and , and are bounded sequences, we conclude from (78) and (107) that
Note that
Hence from (113) and (116) we get
Observe that
Hence, from (76), (103), and (118) we have
It is clear that
where for each . Hence we have
From the boundedness of () and (due to (120)), it follows that
In addition, from (68) and (78), we have
We note that
From (82), (124), and Lemma 22, we obtain
In the meantime, we note that
From (82), (126), and the uniform continuity of , we have
Step 4. We prove that as .
Indeed, since is bounded, there exists a subsequence which converges weakly to some . From (78), (84), (103), and (101), we have that , , and , where . Since is uniformly continuous, by (128) we get for any . Hence from Lemma 24, we obtain . In the meantime, utilizing Lemma 12, we deduce from , , (118), and (123) that and . 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 20 that is maximal monotone. Let ; that is, . Again, since , , , we have
that is,
In terms of the monotonicity of , we get
and hence
In particular,
Since (due to (90)) and (due to the Lipschitz continuity of ), we conclude from and that
It follows from the maximal monotonicity of that ; that is, . Therefore, .
Next, we show that . In fact, from , we know that
From (H2) it follows that
Replacing by , we have
Put for all and . Then, from (137) we have
Since as , we deduce from the Lipschitz continuity of and that and as . Further, from the monotonicity of , we have . So, from (H4), the weakly lower semicontinuity of , and , we have
From (H1), (H4), and (139) we also have
and hence
Letting , we have, for each ,
This implies that . Consequently, . This shows that . From (65) and Lemma 15 we infer that as .
Finally, assume additionally that . It is clear that
So, we know that is -strongly monotone with constant . In the meantime, it is easy to see that is -Lipschitzian with constant . Thus, there exists a unique solution in to the VIP
Equivalently, . Furthermore, from (59), (60), and (83) we get
which hence yields
Since , , and , are bounded, we infer from (146) that
which together with Minty’s lemma [4] implies that
This shows that is a solution in to the VIP (144). Utilizing the uniqueness of solutions in to the VIP (144), we get . This completes the proof.
Corollary 27. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse-strongly monotone, respectively, for and . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a -strongly positive bounded linear operator and an -Lipschitzian mapping with . Assume that is nonempty and bounded where is defined as in Proposition CY. Let be a sequence in and , , and be sequences in such that and , and let , and , . Pick any and set , . Let be a sequence generated by the following algorithm: where (here is nonexpansive, for each ), and , . Assume that the following conditions are satisfied:(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) for each , and ;(iv) and .
Assume that is firmly nonexpansive. Then we have(i) converges strongly as to ;(ii) converges strongly as to provided that , which is the unique solution in to the VIP Equivalently, .
Corollary 28. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse-strongly monotone, respectively, for . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Assume that is nonempty and bounded where is defined as in Proposition CY. Let be a sequence in and and be sequences in such that and , and let and . Pick any and set , . Let be a sequence generated by the following algorithm:
where (here is nonexpansive, for each ) and . Assume that the following conditions are satisfied:(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) for each , and ;(iv) and .
Assume that is firmly nonexpansive. Then we have(i) converges strongly as to ;(ii) converges strongly as to provided that , which is the unique solution in to the VIP
Equivalently, .
Corollary 29. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse-strongly monotone, respectively, for . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping for some with sequence such that . Let be a -strongly positive bounded linear operator and an -Lipschitzian mapping with . Assume that is nonempty and bounded where is defined as in Proposition CY. Let be a sequence in and , and be sequences in such that and , and let and . Pick any and set . Let be a sequence generated by the following algorithm:
where (here is nonexpansive, for each ), and , . Assume that the following conditions are satisfied:(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) for each , and ;(iv) and .
Assume that is firmly nonexpansive. Then we have(i) converges strongly as to ;(ii) converges strongly as to provided that , which is the unique solution in to the VIP
Equivalently, .
4. Weak Convergence Theorem
In this section, we will introduce and analyze another multistep iterative algorithm involving no shrinking projection method for finding a solution of the SGEP (8) with constraints of several problems: the GMEP (4), the CMP (12), finitely many variational inclusions, and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove weak convergence theorem for the iterative algorithm under mild assumptions. This iterative algorithm is based on Korpelevich’s extragradient method, strongly positive bounded linear operator approach, viscosity approximation method, averaged mapping approach to the GPA in [16], and Mann-type iteration method.
Theorem 30. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone and -inverse-strongly monotone, respectively, for and . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a -strongly positive bounded linear operator and an -Lipschitzian mapping with . Assume that is nonempty where is defined as in Proposition CY. Let be a sequence in and and sequences in such that and , and let and . Pick any and let be a sequence generated by the following algorithm: where (here is nonexpansive, for each ). Assume that the following conditions are satisfied:(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) for each , and ;(iv) and .Then converges weakly to provided that is firmly nonexpansive.
Proof. Since is -Lipschitzian, it follows that is -ism; see [16]. Repeating the same arguments as in Theorem 26, we can write
where is nonexpansive and for each . It is clear that
As , and , we may assume, without loss of generality, that , and for all . Put
for all , and , where is the identity mapping on . Then we have .
Take a fixed arbitrarily. Let us show the existence of . Indeed, repeating the same arguments as in the proof of Theorem 26, we can obtain that
Utilizing (158) and (168) we obtain
Since , and , by Lemma 13 we have that exists. Thus is bounded and so are the sequences , , and .
Also, utilizing Lemmas 10 and 16(b) we obtain from (158), (164), (165), and (168) that
and hence
So, it follows that
Since , and , it follows from the existence of and the boundedness of , , and that
Note that
Hence, it follows from (171) and that
In the meantime, from (168) and (171) it follows that
which together with leads to
Consequently, from , , , and the existence of , we get
Since , from (179) we have
Note that
Hence from (176) and (180) we have
Repeating the same arguments as those of Step 3 in the proof of Theorem 26, we can obtain that , , , , , and , as .
Since is bounded, there exists a subsequence of which converges weakly to . It is easy to see that , , , and , where . Since is uniformly continuous and as , we get for any . Hence from Lemma 24, we obtain . In the meantime, utilizing Lemma 12, we deduce from , , (118), and (123) that and . Repeating the same arguments as those of Step 4 in the proof of Theorem 26, we can conclude that and . Therefore, . 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 14, we have that converges strongly to some . Since converges weakly to , we have
Therefore we obtain . This completes the proof.
Corollary 31. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse-strongly monotone, respectively, for and . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a -strongly positive bounded linear operator and an -Lipschitzian mapping with . Assume that is nonempty where is defined as in Proposition CY. Let be a sequence in and , , and sequences in such that and , and let , and , . Pick any and let be a sequence generated by the following algorithm: where (here is nonexpansive, for each ). Assume that the following conditions are satisfied:(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) for each , and ;(iv) and .Then converges weakly to provided that is firmly nonexpansive.
Corollary 32. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse-strongly monotone, respectively, for . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Let be a -strongly positive bounded linear operator and an -Lipschitzian mapping with . Assume that is nonempty, where is defined as in Proposition CY. Let be a sequence in and , , and sequences in such that and , and let , and . Pick any and let be a sequence generated by the following algorithm: where (here is nonexpansive, for each ). Assume that the following conditions are satisfied:(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) for each , and ;(iv) and .Then converges weakly to provided that is firmly nonexpansive.
Corollary 33. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse-strongly monotone, respectively, for . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping for some with sequence such that . Let be a -strongly positive bounded linear operator and be an -Lipschitzian mapping with . Assume that is nonempty, where is defined as in Proposition CY. Let be a sequence in and , and sequences in such that and , and let and . Pick any and let be a sequence generated by the following algorithm: where (here is nonexpansive, for each ). Assume that the following conditions are satisfied:(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) for each , and ;(iv) and .Then converges weakly to provided that is firmly nonexpansive.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research 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). This work was supported partly by the National Science Council of the Republic of China.