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An algorithm for treating asymptotically strict pseudocontractions and monotone operators
Fixed Point Theory and Applications volume 2014, Article number: 52 (2014)
Abstract
In this paper, an algorithm for treating asymptotically κ-strict pseudocontractions and monotone operators is proposed. Convergence analysis of the algorithm is investigated in the framework of Hilbert spaces.
1 Introduction-preliminaries
In this paper, we are concerned with the problem of finding a common element in the intersection , where denotes the fixed point set of the mapping T and denotes the zero point set of the sum of the operator A and the operator B.
The motivation for the common element problem is mainly due to its possible applications to mathematical modeling of concrete complex problems. The common element problems include mini-max problems, complementarity problems, equilibrium problems, common fixed point problems and variational inequalities as special cases; see, for example, [1–35] and the references therein.
Throughout the article, we always assume that H is a real Hilbert space with the inner product and the norm , respectively. Let C be a nonempty closed convex subset of H, and let be the metric projection from H onto C.
Let be a mapping. stands for the zero point set of A; that is, . Recall that A is said to be monotone iff
A is said to be α-strongly monotone iff there exists a constant such that
A is said to be α-inverse-strongly monotone iff there exists a constant such that
It is not hard to see that α-inverse-strongly monotone mappings are Lipschitz continuous. Indeed, we have
This shows that .
Recall that the classical variational inequality, denoted by , is to find such that
One can see that the variational inequality (1.1) is equivalent to a fixed point problem of the mapping , where I is the identity and r is some positive real number. The element is a solution of the variational inequality (1.1) iff satisfies the equation . This alternative equivalent formulation has played a significant role in the studies of variational inequalities and related optimization problems.
A multivalued operator with the domain and the range is said to be monotone if for , , and , we have . A monotone operator B is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. Let I denote the identity operator on H and be a maximal monotone operator. Then we can define, for each , a nonexpansive single-valued mapping by . It is called the resolvent of B. We know that for all and is firmly nonexpansive.
Let be a mapping. In this paper, we use to denote the fixed point set of T; that is, . Recall that T is said to be nonexpansive iff
T is said to be asymptotically nonexpansive iff there exists a sequence such that
T is said to be a κ-strict pseudocontraction iff there exists a constant such that
Note that the class of κ-strict pseudocontractions strictly includes the class of nonexpansive mappings as a special case. That is, T is nonexpansive iff the coefficient . T is said to be an asymptotically κ-strict pseudocontraction iff there exist a constant and a sequence in such that
Note that the class of asymptotically κ-strict pseudocontractions strictly includes the class of asymptotically nonexpansive mappings as a special case. That is, T is asymptotically nonexpansive iff the coefficient .
In [24], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator based on the following iterative algorithm:
where is a sequence in , is a positive real number sequence, is maximal monotone and . It is proved that the sequence generated in (1.2) converges strongly to some provided that the control sequence satisfies some restrictions. Further, using this result, they also investigated the case that , where is a proper lower semicontinuous convex function. Convergence theorems are established in the framework of real Hilbert spaces; for more details, see [24].
Recently, Takahashi et al. studied zero point problems of the sum of two monotone mappings and fixed point problems of a nonexpansive mapping based on the following iterative algorithm:
where and are real number sequences in , is a positive sequence, is a nonexpansive mapping and is an inverse-strongly monotone mapping. It is proved that the sequence generated in (1.3) converges strongly to some provided that the control sequence satisfies some restrictions; for more details, see [2].
Motivated by the above results, we investigate fixed point problems of asymptotically strict pseudocontractions and zero point problems of the sum of two monotone mappings.
In order to state our main results, we need the following tools.
Recall that a space is said to satisfy Opial’s condition [36] if, for any sequence with , where ⇀ denotes the weak convergence, the inequality
holds for every with . Indeed, the above inequality is equivalent to the following:
Lemma 1.1 [2]
Let and be bounded sequences in a Banach space X, and let be a sequence in with . Suppose that for all integers and
Then .
Lemma 1.2 [37]
Let C be a nonempty, closed and convex subset of H, let be a mapping, and let be a maximal monotone operator. Then .
Lemma 1.3 [38]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(i)
;
-
(ii)
or .
Then .
Lemma 1.4 [39]
Let C be a nonempty, closed and convex subset of H. Let be an asymptotically strict pseudocontraction. Then T is Lipschitz continuous and is demiclosed at zero.
2 Main results
Theorem 2.1 Let C be a nonempty closed convex subset of H. Let be an asymptotically κ-strict pseudocontraction. Let be an α-inverse-strongly monotone mapping, and let B be a maximal monotone operator on H. Assume that . Let , and be real number sequences in . Let , where is a positive real number sequence. Let be a sequence in C generated by: is chosen arbitrarily and
Assume that the sequences , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
, ;
-
(c)
;
-
(d)
, ,
where a, b, c, d and e are some real numbers. If T is asymptotically regular, then the sequence converges strongly to some point , where .
Proof First, we show that the mapping is nonexpansive. Indeed, we have
It follows from Restriction (a) that is nonexpansive. Put and fix . It follows from Lemma 1.2 that
In view of Restriction (d), we find that
Substituting (2.1) into (2.2), we obtain that
Putting , we find that for all . Indeed, it is clear that . Suppose that for some positive integer m. It follows that
This finds that is bounded. Putting , we find that
On the other hand, we have
Substituting (2.3) into (2.4), we find that
Put . Since B is monotone, we find that
It follows that . This yields that . This combines with (2.5) to yield that
On the other hand, we have
Substituting (2.6) into (2.7), we find that
It follows from Restrictions (a), (c) and (d) that
It follows from Lemma 1.1 that . Since , we find that . Notice that
Since the norm is convex, we see from (2.2) and (2.8) that
This yields that
In view of Restrictions (a), (b) and (c), we obtain that
Notice that
It follows that
This yields that
It follows from (2.2) that
We therefore obtain that
In view of Restrictions (a), (b) and (c), we find from (2.10) that
Next, we show that , where . To show it, we can choose a subsequence of such that
Since is bounded, we can choose a subsequence of which converges weakly to some point x. We may assume, without loss of generality, that converges weakly to x. Since , we find that . Since B is monotone, we get, for any , that . Replacing n by and letting , we obtain from (2.12) that . This means , that is, . Hence we get . Next, we show that . Notice that
In view of Restriction (a), we find from (2.12) that . Note that
It follows from (2.8) that
Note that
It follows that . This implies from Restriction (d) and (2.13) that . Since T is uniformly L-Lipschitz continuous, we can obtain that . In view of Lemma 1.4, we find that . This implies that
On the other hand, we have
From Lemma 1.3, we find that . This completes the proof. □
If T is asymptotically nonexpansive, then we find the following result.
Corollary 2.2 Let C be a nonempty closed convex subset of H. Let be an asymptotically nonexpansive mapping. Let be an α-inverse-strongly monotone mapping, and let B be a maximal monotone operator on H. Assume that . Let and be real number sequences in . Let , where is a positive real number sequence. Let be a sequence in C generated by: is chosen arbitrarily and
Assume that the sequences , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
, ;
-
(c)
,
where a, b, c and d are some real numbers. If T is asymptotically regular, then the sequence converges strongly to some point , where .
3 Applications
In this section, we shall consider equilibrium problems and variational inequalities.
Let F be a bifunction of into ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem:
In this work, we use to denote the solution set of the equilibrium problem.
To study the equilibrium problems, 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.
Putting for every , we see that the equilibrium problem is reduced to the variational inequality (1.1).
The following lemma can be found in [40].
Lemma 3.1 Let C be a nonempty closed convex subset of H, and let be a bifunction satisfying (A1)-(A4). Then, for any and , there exists such that
Further, define
for all and . Then the following hold:
-
(a)
is single-valued;
-
(b)
is firmly nonexpansive, i.e., for any ,
-
(c)
;
-
(d)
is closed and convex.
Lemma 3.2 [2]
Let C be a nonempty closed convex subset of a real Hilbert space H, let F be a bifunction from to ℝ which satisfies (A1)-(A4), and let be a multivalued mapping of H into itself defined by
Then is a maximal monotone operator with the domain , and
where is defined as in (3.1).
The following result is not derived based on Theorem 2.1 and Lemma 3.2.
Theorem 3.3 Let C be a nonempty closed convex subset of H. Let be an asymptotically κ-strict pseudocontraction. Let F be a bifunction from to ℝ which satisfies (A1)-(A4). Assume that . Let , and be real number sequences in . Let be a sequence in C generated by: is chosen arbitrarily and
Assume that the sequences , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
, ;
-
(c)
;
-
(d)
, ,
where a, b, c, d and e are some real numbers. If T is asymptotically regular, then the sequence converges strongly to some point , where .
If T is asymptotically nonexpansive, then Theorem 3.3 is reduced to the following.
Corollary 3.4 Let C be a nonempty closed convex subset of H. Let be an asymptotically nonexpansive mapping. Let F be a bifunction from to ℝ which satisfies (A1)-(A4). Assume that . Let and be real number sequences in . Let be a sequence in C generated by: is chosen arbitrarily and
Assume that the sequences , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
, ;
-
(c)
,
where a, b, c and d are some real numbers. If T is asymptotically regular, then the sequence converges strongly to some point , where .
Let H be a Hilbert space and be a proper convex lower semicontinuous function. Then the subdifferential ∂f of f is defined as follows:
From Rockafellar [41], we find that ∂f is maximal monotone. It is easy 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.
Lemma 3.5 [2]
Let C be a nonempty closed convex subset of a real Hilbert space H, the metric projection from H onto C, the subdifferential of , where is defined above and . Then
Now, we consider a variation inequality problem.
Theorem 3.6 Let C be a nonempty closed convex subset of H. Let be an asymptotically κ-strict pseudocontraction. Let be an α-inverse-strongly monotone mapping. Assume that . Let , and be real number sequences in . Let be a sequence in C generated by: is chosen arbitrarily and
Assume that the sequences , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
, ;
-
(c)
;
-
(d)
, ,
where a, b, c, d and e are some real numbers. If T is asymptotically regular, then the sequence converges strongly to some point , where .
Proof Put . Next, we show that . Notice that
From Lemma 3.5, we can conclude the desired conclusion immediately. □
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Zhang, M. An algorithm for treating asymptotically strict pseudocontractions and monotone operators. Fixed Point Theory Appl 2014, 52 (2014). https://doi.org/10.1186/1687-1812-2014-52
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DOI: https://doi.org/10.1186/1687-1812-2014-52