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Hybrid methods for common solutions in Hilbert spaces with applications
Journal of Inequalities and Applications volume 2014, Article number: 183 (2014)
Abstract
In this paper, hybrid methods are investigated for treating common solutions of nonlinear problems. A strong convergence theorem is established in the framework of real Hilbert spaces.
1 Introduction and preliminaries
Common solutions to variational inclusion, equilibrium and fixed point problems have been recently extensively investigated based on iterative methods; see [1–33] and the references therein. The motivation for this subject is mainly to its possible applications to mathematical modeling of concrete complex problems, which use more than one constraint. The aim of this paper is to investigate a common solution of variational inclusion, equilibrium and fixed point problems. The organization of this paper is as follows. In Section 1, we provide some necessary preliminaries. In Section 2, a hybrid method is introduced and analyzed. Strong convergence theorems are established in the framework of Hilbert spaces. In Section 3, applications of the main results are discussed.
In what follows, we always assume that H is a real Hilbert space with the inner product and the norm . Let C be a nonempty, closed, and convex subset of H and let be the metric projection from H onto C. Let be a mapping. stands for the fixed point set of S; that is, .
Recall that S is said to be contractive iff there exists a constant such that
If , then S is said to be nonexpansive. Let be a mapping. If C is nonempty closed and convex, then the fixed point set of S is nonempty.
Recall that A is said to be monotone iff
Recall that A is said to be strongly monotone iff there exists a constant such that
For such a case, A is also said to be α-strongly monotone. Recall that A is said to be inverse-strongly monotone iff there exists a constant such that
For such a case, A is also said to be α-inverse-strongly monotone.
Recall that a set-valued mapping is said to be monotone iff, for all , , and imply . M is maximal iff the graph of R is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping M is maximal if and only if, for any , , for all implies . For a maximal monotone operator M on H, and , we may define the single-valued resolvent , where denote the domain of M. It is well known that is firmly nonexpansive, and , where , and .
Let be a inverse-strongly monotone mapping, and let F be a bifunction of into ℝ, where ℝ denotes the set of real numbers. We consider the following generalized equilibrium problem.
In this paper, the set of such an is denoted by .
To study the equilibrium problems (1.1), we may assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each ,
(A4) for each , is convex and weakly lower semicontinuous.
In order to prove our main results, we also need the following lemmas.
Lemma 1.1 [34]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(1)
;
-
(2)
or .
Then .
Lemma 1.2 [35]
Let be a bifunction satisfying (A1)-(A4). Then, for any and , there exists such that
Define a mapping as follows:
then the following conclusions hold:
-
(1)
is single-valued;
-
(2)
is firmly nonexpansive, i.e., for any ,
-
(3)
;
-
(4)
is closed and convex.
Let be a family of infinitely nonexpansive mappings and be a nonnegative real sequence with , . For define a mapping as follows:
Such a mapping is nonexpansive from C to C and it is called a W-mapping generated by and .
Lemma 1.3 [36]
Let be a family of infinitely nonexpansive mappings with a nonempty common fixed point set and let be a real sequence such that , where l is some real number, . Then
-
(1)
is nonexpansive and , for each ;
-
(2)
for each and for each positive integer k, the limit exists;
-
(3)
the mapping defined by
(1.3)
is a nonexpansive mapping satisfying and it is called the W-mapping generated by and .
Lemma 1.4 [27]
Let be a family of infinitely nonexpansive mappings with a nonempty common fixed point set and let be a real sequence such that , . If K is any bounded subset of C, then
Throughout this paper, we always assume that , .
Lemma 1.5 [37]
Let be a mapping and let be a maximal monotone operator. Then .
Lemma 1.6 [38]
Let and be bounded sequences in H and let be a sequence in with . Suppose that for all and
Then .
Lemma 1.7 [39]
Let a Lipschitz monotone mapping and let be the normal cone to C at ; that is, . Define
Then D is maximal monotone and if and only if .
2 Main results
Theorem 2.1 Let C be a nonempty closed convex subset of a Hilbert space H and F a bifunction from to ℝ which satisfies (A1)-(A4). Let be a -inverse-strongly monotone mapping, be a -inverse-strongly monotone mapping, be a -inverse-strongly monotone mapping, a maximal monotone operator such that and a maximal monotone operator such that . Let be a family of infinitely nonexpansive mappings. Assume that . Let and be a sequence generated by
where u is a fixed element in C, is such that
is the sequence generated in (1.2), , , and are sequences in such that for each and , , and are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
-
(a)
, , ;
-
(b)
and ;
-
(c)
;
-
(d)
.
Then the sequence converges strongly to , where .
Proof First, we show that the mapping , , and are nonexpansive. Indeed, we find from the restriction (a) that
which implies that the mapping is nonexpansive. In the same way, we find and are also nonexpansive. Put . Fixing , we find
It follows that
This implies that is bounded, and so are , , and . Without loss of generality, we can assume that there exists a bounded set such that . Notice that
and
Let in (2.1) and in (2.2). By adding these two inequalities, we obtain
It follows that
It follows that
where is an appropriate constant such that
Since is firmly nonexpansive, we find that
Combining (2.3) with (2.4) yields
Since is also firmly nonexpansive, we find that
Substituting (2.5) into (2.6), we see that
where is an appropriate constant such that
Since is nonexpansive, we find that
where K is the bounded subset of C defined as above. Substituting (2.7) into (2.8), we find that
Letting
we see that
Hence, we have
Substituting (2.9) into (2.10), we find that
It follows from Lemma 1.4 that
In view of Lemma 1.6, we find that . It follows that
For any , we see that
Since
we find that
Using the restrictions (a) and (b), we obtain
It follows from (2.12) that
Since
we have
which implies that
Using the restrictions (a) and (b), we obtain
Note that
This implies that
Using the restrictions (a) and (b), we see that
Since is firmly nonexpansive, we find that
This in turn implies that
Using the restrictions (a) and (b), we see that
Since is also firmly nonexpansive mapping, we see that
which implies that
It follows that
Using the restrictions (a) and (b), we obtain
and
Note that
Using the restrictions (b) and (c), we obtain
On the other hand, one has
Using (2.6), (2.7), (2.8), and (2.9), we find that
Next, we prove that
To see this, we choose a subsequence of such that
Since is bounded, there exists a subsequence of which converges weakly to w. Without loss of generality, we may assume that . Therefore, we see that . We also have .
Next, we show that . Suppose the contrary, , i.e., . Since , we see from Opial’s condition that
On the other hand, we have
In view of Lemma 1.4, we obtain that . This implies from (2.22) that . This is a contradiction. Thus, we have .
Now, we are in a position to prove that . Notice that . Let . Since is monotone, we find that
This implies that . This implies that , that is, .
Now, we prove that . Notice that . Let . Since is monotone, we find that
This implies that . This implies that , that is, .
Next, we show that . Since , for any , we have
Replacing n by , we find from (A2) that
Putting for any and , we see that . It follows that
In view of the monotonicity of , and the restriction (a), we obtain from (A4) that
From (A1) and (A4), we see that
It follows that
It follows from (A3) that . Hence,
Finally, we show that . Note that
This implies that
Using Lemma 1.1, we find that . This completes the proof. □
3 Applications
In this section, we consider some applications of the main results.
Recall that the classical variational inequality is to find an such that
In this paper, we use to denote the solution set of the inequality. It is well known that is a solution of the inequality iff x is a fixed point of the mapping , where is a constant, I stands for the identity mapping. If A is α-inverse-strongly monotone and , then the mapping is nonexpansive. It follows that is closed and convex.
Let be a proper convex lower semicontinuous function. Then the subdifferential ∂g of g is defined as follows:
From Rockafellar [39], we know that ∂g is maximal monotone. It is not hard to verify that if and only if .
Let be the indicator function of C, i.e.,
Since is a proper lower semicontinuous convex function on H, we see that the subdifferential of is a maximal monotone operator. It is clearly that , , and .
Theorem 3.1 Let C be a nonempty closed convex subset of a Hilbert space H and F a bifunction from to ℝ which satisfies (A1)-(A4). Let be a -inverse-strongly monotone mapping, be a -inverse-strongly monotone mapping, be a -inverse-strongly monotone mapping, and be a family of infinitely nonexpansive mappings. Assume that . Let and be a sequence generated by
where is such that
is the sequence generated in (1.2), , , and are sequences in such that for each and , , and are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
-
(a)
, , ;
-
(b)
and ;
-
(c)
;
-
(d)
.
Then the sequence converges strongly to .
Recall that a mapping is said to be a k-strict pseudo-contraction if there exists a constant such that
Putting , where is a k-strict pseudo-contraction, we find that A is -inverse-strongly monotone.
Next, we consider fixed points of strict pseudo-contractions.
Theorem 3.2 Let C be a nonempty closed convex subset of a Hilbert space H and F a bifunction from to ℝ which satisfies (A1)-(A4). Let be a -strict pseudo-contraction, be a -strict pseudo-contraction, be a δ-inverse-strongly monotone mapping, and be a family of infinitely nonexpansive mappings. Assume that . Let and be a sequence generated by
where is such that
is the sequence generated in (1.2), , , and are sequences in such that for each and , , and are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
-
(a)
, , ;
-
(b)
and ;
-
(c)
;
-
(d)
.
Then the sequence converges strongly to .
Proof Taking , wee see that is a -strict pseudo-contraction with and for . In view of Theorem 3.1, we find the desired conclusion immediately. □
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Sun, L. Hybrid methods for common solutions in Hilbert spaces with applications. J Inequal Appl 2014, 183 (2014). https://doi.org/10.1186/1029-242X-2014-183
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DOI: https://doi.org/10.1186/1029-242X-2014-183