An implicitly defined iterative sequence for monotone operators in Banach spaces
© Kohsaka; licensee Springer. 2014
Received: 31 October 2013
Accepted: 22 April 2014
Published: 12 May 2014
Given a monotone operator in a Banach space, we show that an iterative sequence, which is implicitly defined by a fixed point theorem for mappings of firmly nonexpansive type, converges strongly to a minimum norm zero point of the given operator. Applications to a convex minimization problem and a variational inequality problem are also included.
MSC:47H05, 47J25, 47N10.
KeywordsBanach space convex minimization mapping of firmly nonexpansive type fixed point monotone operator proximal point method variational inequality
Many nonlinear problems can be formulated as the problem of finding a zero point of a maximal monotone operator in a Banach space. The proximal point method, which was first introduced by Martinet  and generally studied by Rockafellar , is an iterative method for approximating a solution to this problem.
for all , where is a maximal monotone operator, X is a smooth, strictly convex, and reflexive real Banach space, is the normalized duality mapping, and is a sequence of positive real numbers.
The following result was obtained in : If , then is bounded if and only if is nonempty. Further, if X is uniformly convex, the norm of X is uniformly Gâteaux differentiable, is nonempty, , and J is weakly sequentially continuous, then converges weakly to an element of . This is a generalization of the result due to Rockafellar  in Hilbert spaces. See also [4, 5] for some related results.
for all , where A, X, J, and are the same as in (1.1) and is a sequence of . Under some additional assumptions, we show that is well defined and is strongly convergent to an element of of minimal norm; see Theorem 3.5.
The schemes (1.1) and (1.2) above are similar to each other, though their properties are quite different. In fact, the former fails to converge strongly even in Hilbert spaces , whereas the latter converges strongly in Banach spaces. Further, the former is well defined since for each , whereas the latter is not necessarily well defined. To study the well definedness and the asymptotic behavior of in (1.2), we exploit some techniques in [3, 7].
This paper is organized as follows: In Section 2, we give some definitions, recall some known results, and briefly study the existence of a zero point of a monotone operator. In Section 3, using the results in the previous section, we first obtain a convergence theorem for a monotone operator satisfying a range condition; see Theorem 3.1. Using this result, we show a convergence theorem for a maximal monotone operator; see Theorem 3.5. In Section 4, we apply Theorem 3.5 to a convex minimization problem and a variational inequality problem.
Throughout the present paper, we denote by ℕ the set of all positive integers, ℝ the set of all real numbers, X a smooth, strictly convex, and reflexive real Banach space with dual , the norms of X and , the value of at , the strong convergence of a sequence of X to , the weak convergence of a sequence of X to , the norm closure of , coU the convex hull of , the closed convex hull of , and the unit sphere of X, respectively.
converges uniformly in for all . The space X is said to be uniformly convex if for each , there exists such that whenever and . The space X is said to have the Kadec-Klee property if whenever is a sequence of X such that and . Every uniformly convex Banach space is both strictly convex and reflexive and has the Kadec-Klee property; see [8, 9].
for all .
for all ; see also . If X is a Hilbert space, then is firmly nonexpansive if and only if it is of firmly nonexpansive type.
For an operator , the domain , the range , and the graph of A are defined by , , and , respectively. The operator A is said to be monotone if whenever . It is also said to be maximal monotone if A is monotone and there is no monotone operator such that and . Let C be a nonempty closed convex subset of X and a monotone operator such that . Then the mapping defined by for all is of firmly nonexpansive type and ; see [12, 14]. We know the following lemma.
Lemma 2.1 ()
a sequence of such that , and the mapping defined by for all and . If is a sequence of C such that and , then u is an element of .
We know the following result for mappings of firmly nonexpansive type.
Lemma 2.2 ()
is nonempty if and only if is bounded for some ;
is closed and convex.
Using Lemma 2.2, we can show the following.
Lemma 2.3 Let C be a nonempty closed convex subset of X and a mapping of firmly nonexpansive type. Suppose that , , and is bounded. Then the mapping βT has a unique fixed point.
for all . This implies that S is of firmly nonexpansive type. Since C is convex, , and , we know that S is a mapping of C into itself. Further, since and is bounded, the sequence is bounded for all . Thus Lemma 2.2 implies that is nonempty.
We next show that consists of one point. Suppose that . Then it follows from (2.8) that . Since , we obtain . Thus the strict monotonicity of J implies that . □
As a direct consequence of Lemmas 2.2 and 2.3, we obtain the following.
is nonempty, closed, and convex;
- (ii)if and , then there exists a unique such that(2.9)
Proof Let be the mapping defined by for all . Then we know that T is of firmly nonexpansive type and . Hence is bounded for all . On the other hand, we know that . Therefore, part (i) follows from Lemma 2.2. Part (ii) follows from Lemma 2.3. □
3 Strong convergence of an iterative sequence
In this section, we first show the following strong convergence theorem for a monotone operator satisfying a range condition.
For each , there exists a unique such that ;
if X has the Kadec-Klee property, the norm of X is uniformly Gâteaux differentiable, , and , then the sequence converges strongly to .
Part (i) of Theorem 3.1 follows from Corollary 2.4.
The proof of (i) of Theorem 3.1 Let be given and set . Then is monotone and . Thus we know that is bounded and . Therefore, part (ii) of Corollary 2.4 ensures the conclusion. □
Before proving (ii) of Theorem 3.1, we show the following lemma.
for all and ;
for all and .
By (3.2), (3.3), and , the result follows.
for all and . Thus the result follows. □
We next show (ii) of Theorem 3.1.
The proof of (ii) of Theorem 3.1 Suppose that X has the Kadec-Klee property, the norm of X is uniformly Gâteaux differentiable, , and . Set for all . By (i) of Corollary 2.4, the set is nonempty, closed, and convex. Hence is well defined. We denote by P.
Since for all and is bounded, is bounded. By the definition of , we have for all . Thus is also bounded.
Since , Lemma 2.1 shows that u is an element of . It also follows from (3.5) and that .
Thus we obtain . Since X has the Kadec-Klee property, we have . Consequently, it follows from (3.5) that .
Noting that , we have from (2.2) and (3.8) that . Therefore, we conclude that converges strongly to . □
As a direct consequence of Theorem 3.1, we obtain the following corollary.
For each , there exists a unique such that ;
if X has the Kadec-Klee property, the norm of X is uniformly Gâteaux differentiable, and , then the sequence converges strongly to .
for all . Then, by , we know that the following hold:
A is a monotone operator and ;
for all .
Thus the result follows from Theorem 3.1. □
Remark 3.4 In the case when C is bounded, Corollary 3.3 is reduced to the result obtained in .
Using Theorem 3.1, we can also show the following strong convergence theorem for a maximal monotone operator.
Theorem 3.5 Let X be a smooth, strictly convex, and reflexive real Banach space, a maximal monotone operator such that and is bounded, a sequence of positive real numbers, and a sequence of . Then for each , there exists a unique satisfying (1.2). Moreover, if X has the Kadec-Klee property, the norm of X is uniformly Gâteaux differentiable, , and , then the sequence converges strongly to .
Noting that , we obtain the desired result by Theorem 3.1. □
4 Results deduced from Theorem 3.5
In this final section, we study two applications of Theorem 3.5. Throughout this section, we suppose the following:
X is a uniformly convex real Banach space whose norm is uniformly Gâteaux differentiable;
is a sequence of positive real numbers such that ;
is a sequence of such that .
We first study a convex minimization problem. For a function , we denote by argminf or the set of all such that . In the case when for some , we identify argminf with p. The set of all such that is denoted by . We denote by ∂f the subdifferential mapping of f; see [17, 19] for more details.
Moreover, the sequence converges strongly to .
holds for all and . Thus we know that for all . Consequently, Theorem 3.5 implies the conclusion. □
We finally study a variational inequality problem. For a nonempty closed convex subset C of X and an operator , we denote by the set of all such that for all . In the case when for some , we identify with p. The operator B is said to be hemicontinuous if the mapping defined by for all is continuous with respect to the weak* topology in for all .
Moreover, the sequence converges strongly to .
holds for all and . Thus we have for all . Consequently, Theorem 3.5 implies the conclusion. □
The author would like to thank the anonymous referees for carefully reading the original version of the manuscript. The author is supported by Grant-in-Aid for Young Scientists No. 25800094 from the Japan Society for the Promotion of Science.
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