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Convergence of a regularization algorithm for nonexpansive and monotone operators in Hilbert spaces
Fixed Point Theory and Applications volume 2014, Article number: 180 (2014)
Abstract
Variational inequality, fixed point and generalized equilibrium problems are investigated via a regularization algorithm. It is proved that the sequence generated in the regularization algorithm converges strongly to a common solution of the three problems in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding ones announced by many authors.
1 Introduction and preliminaries
In this paper, we always assume that H is a real Hilbert space with inner product and induced norm for . Let C be a nonempty, closed, and convex subset of H.
Let be a mapping. 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 the classical variational inequality is to find an such that
In this paper, we always use to denote the solution set of (1.1) and use denote the metric projection from H onto C. It is well known that is a solution of (1.1) iff x is a fixed point of the mapping , where is a constant, I stands for the identity mapping. If A is strongly monotone and Lipschitz continuous, the existence and uniqueness of solutions of equilibrium (1.1) is guaranteed by the Banach contraction principle.
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 .
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
For such a case, S is also said to be α-contractive. We know that the mapping enjoys a unique fixed point and Picard’s algorithm can be employed to approximate its unique fixed point.
Recall that S is said to be nonexpansive iff
If C is a closed, bounded and convex subset of H, then is not empty; see [1].
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 ; see [2] and the references therein.
Let be a 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 is denoted by , i.e.,
If , the zero mapping, then the problem (1.3) is reduced to the following equilibrium problem [3]:
In this paper, the set of such an is denoted by .
If , then the problem (1.3) is reduced to the classical variational inequality (1.1).
To study equilibrium problems (1.3) and (1.4), 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 semi-continuous.
Many important problems have reformulations which require finding solutions of equilibriums (1.3) and (1.4), for instance, image recovery, inverse problems, network allocation, transportation problems and optimization problems; see [3–11] and the references therein. For solving solutions of equilibriums (1.3) and (1.4), regularization methods recently have been extensively studied; see [11–28] and the references therein.
In this paper, motivated and inspired by the research going on in this direction, we study the variational inequality (1.1), and the fixed point and equilibrium problem (1.3) based on a regularization algorithm. It is proved that the sequence generated in the regularization algorithm converges strongly to a common solutions of the three problems in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results in Chang et al. [11], Takahashi and Takahashi [13] and Hao [29].
The following lemmas play an important role in our paper.
Lemma 1.1 [3]
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.
Lemma 1.2 [30]
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 1.3 [2]
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
(1.5)
is a nonexpansive mapping satisfying and it is called the W-mapping generated by and .
Lemma 1.4 [31]
Let and be bounded sequences in H and let be a sequence in with . Suppose that for all and
Then .
Lemma 1.5 [11]
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 1.6 [10]
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 H. Let F be a bifunction from to ℝ which satisfies (A1)-(A4) and let be a κ-contraction. Let be an α-inverse-strongly monotone mapping and let be a β-inverse-strongly monotone mapping. Let be a τ-inverse-strongly monotone mapping. Let be a family of infinitely nonexpansive mappings. Assume that is not empty. Let , , and be sequences in such that . Let , , and be positive number sequences. Let and let be a sequence generated by
where is such that , , and is the sequence generated in (1.5). Assume that the following restrictions hold:
-
(a)
and ,
-
(b)
and ,
-
(c)
and ,
-
(d)
, ,
-
(e)
,
where a, , , b, , and are real constants. Then converges strongly to , which solves uniquely the following variational inequality:
Proof Since A is inverse-strongly monotone, we see from restriction (b) that
This shows that is nonexpansive. In the same way, we find that and are nonexpansive. Note that can be re-written as . Let . It follows that
Putting , we see that .
This implies that
This yields the result that the sequence is bounded, and so are , , and . Without loss of generality, we can assume that there exists a bounded set such that . Since , we find that
and
Let in (2.2) and in (2.3). By adding up these two inequalities, we obtain
This implies that
It follows that
where is an appropriate constant such that
It follows from (2.4) that
Hence, we have
where . This implies from (2.5) that
where K is the bounded subset of C defined above. Let . It follows that
By use of (2.6), we find that
This implies that
It follows from restrictions (a)-(e) that
This implies from Lemma 1.4 that . It follows that
Since A is inverse-strongly monotone, we find that
It follows that
Hence, we have
By use of the restrictions (a), (d), and (e), we obtain from (2.7)
Since the metric projection is firmly nonexpansive, we find that
Hence, we have
This further implies that
which yields
By use of restrictions (b), (d), and (e), we find from (2.7) that
Since B is inverse-strongly monotone, we find that
It follows that
Hence, we have
By use of the restrictions (c), (d), and (e), we obtain from (2.7)
Since the metric projection is firmly nonexpansive, we find that
Hence, we have
This further implies that
which yields
By use of restrictions (c), (d), and (e), we find from (2.7) that
Since T is inverse-strongly monotone, we find that
This implies that
In view of the restrictions (a), (d), and (e), we see from (2.7) that
Since is firmly nonexpansive, we find that
This in turn implies that
It follows that
By use of restrictions (a), (d), and (e), we see from (2.7) and (2.12) that
Next, we prove that
where . 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 . Since
In view of the restrictions (d) and (e), we obtain from (2.7)
Note that
In view of (2.8), (2.11), (2.13), and (2.14), we find that
Suppose the contrary, , i.e., . Since , we find from Opial’s condition [32] that
On the other hand, we have
In view of Lemma 1.5, we obtain from (2.15) . It follows that . Thus one derives a contradiction. Thus, we have .
Next, we show that . Let T be the maximal monotone mapping defined by
For any given , we have . Since , by the definition of , we have . Since , we see that . It follows that
Since B is Lipschitz continuous, we see that . Notice that D is maximal monotone and hence . This shows that . In the same way, we find that .
Next, we show that . Since , for any , we find from (A2) that
Putting for any and , we see that . It follows from (2.16) that
In view of the monotonicity of T, and the restriction (a), we obtain from (A4)
From (A1) and (A4), we see that
It follows from (A3) that . This proves that .
Finally, we show that , as . Note that
This implies that
From the restriction (d), we obtain from Lemma 1.2 . This completes the proof. □
Corollary 2.2 Let C be a nonempty closed convex subset of H. Let F be a bifunction from to ℝ which satisfies (A1)-(A4) and let be a κ-contraction. Let be a τ-inverse-strongly monotone mapping. Let be a family of infinitely nonexpansive mappings. Assume that is not empty. Let , , and be sequences in such that . Let be a positive number sequence. Let and let be a sequence generated by
where is such that , , and is the sequence generated in (1.5). Assume that the following restrictions hold:
-
(a)
and ,
-
(b)
, ,
-
(c)
,
where a and b are real constants. Then converges strongly to , which solves uniquely the following variational inequality:
Corollary 2.3 Let C be a nonempty closed convex subset of H. Let F be a bifunction from to ℝ which satisfies (A1)-(A4) and let be a κ-contraction. Let be a β-inverse-strongly monotone mapping. Let be a τ-inverse-strongly monotone mapping. Let be a family of infinitely nonexpansive mappings. Assume that is not empty. Let , , and be sequences in such that . Let and be positive number sequences. Let and let be a sequence generated by
where is such that , , and is the sequence generated in (1.5). Assume that the following restrictions hold:
-
(a)
and ,
-
(b)
and ,
-
(c)
, ,
-
(d)
,
where a, , b, and are real constants. Then converges strongly to , which solves uniquely the following variational inequality:
Corollary 2.4 Let C be a nonempty closed convex subset of H. Let F be a bifunction from to ℝ which satisfies (A1)-(A4) and let be a κ-contraction. Let be an α-inverse-strongly monotone mapping and let be a β-inverse-strongly monotone mapping. Let be a family of infinitely nonexpansive mappings. Assume that is not empty. Let , , and be sequences in such that . Let , , and be positive number sequences. Let and let be a sequence generated by
where is such that , , and is the sequence generated in (1.5). Assume that the following restrictions hold:
-
(a)
and ,
-
(b)
and ,
-
(c)
and ,
-
(d)
, ,
-
(e)
,
where a, , , b, , and are real constants. Then converges strongly to , which solves uniquely the following variational inequality:
Proof In Theorem 2.1, put . Then, for all , we have
Taking with and choosing a sequence of real numbers with , we obtain the desired result by Theorem 2.1. □
Corollary 2.5 Let C be a nonempty closed convex subset of H. Let be a κ-contraction and let be a τ-inverse-strongly monotone mapping. Let be an α-inverse-strongly monotone mapping and let be a β-inverse-strongly monotone mapping. Let be a family of infinitely nonexpansive mappings. Assume that is not empty. Let , , and be sequences in such that . Let , , and be positive number sequences. Let and let be a sequence generated by
where is the sequence generated in (1.5). Assume that the following restrictions hold:
-
(a)
and ,
-
(b)
and ,
-
(c)
and ,
-
(d)
, ,
-
(e)
,
where a, , , b, , and are real constants. Then converges strongly to , which solves uniquely the following variational inequality:
Proof Putting , we find that
is equivalent to
that is, . This completes the proof. □
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Yuan, Q., Zhang, Y. Convergence of a regularization algorithm for nonexpansive and monotone operators in Hilbert spaces. Fixed Point Theory Appl 2014, 180 (2014). https://doi.org/10.1186/1687-1812-2014-180
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DOI: https://doi.org/10.1186/1687-1812-2014-180