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Ishikawa-hybrid proximal point algorithm for NSVI system
Fixed Point Theory and Applications volume 2012, Article number: 195 (2012)
Abstract
A nonlinear set-valued inclusions system framework for an Ishikawa-hybrid proximal point algorithm is developed and studied using the notion of an -accretive mapping. Convergence analysis for the algorithm of solving the nonlinear set-valued inclusions system and existence analysis of solution for the system are explored along with some results on the resolvent operator corresponding to the -accretive mapping in a Banach space. The result that the sequence generated by the algorithm converges linearly to a solution of the system with the convergence rate is proved.
MSC:49J40, 47H06.
1 Introduction
The nonlinear set-valued inclusions system, which was introduced and studied by Hassouni and Moudafi [1], is a useful and important extension of the variational inequality and variational inclusions system. In recent years, various variational inclusions systems and nonlinear set-valued inclusions systems have been intensively studied. For example, Kassay and Kolumbán [2], Chen, Deng and Tan [3], Yan, Fang and Huang [4], Fang, Huang and Thompson [5], Jin [6], Verma [7], Li, Xu and Jin [8], Kang, Cho and Liu [9]et al. introduced and studied various set-valued variational inclusions systems. For the past few years, many existence results and iterative algorithms for various variational inclusions systems have been studied. For details, please see [1–28] and the references therein.
Example 1.1 In 2001, Chen, Deng and Tan [3] have studied the problem associated with the following system of variational inequalities, which is finding (H, Hilbert space) such that
where is a proper, convex, lower semicontinuous functional and denotes the subdifferential operator of ().
Example 1.2 Let X be a real q-uniformly smooth Banach space, and be four single-valued mappings. Find such that
which is studied by Jin in [6].
Inspired and motivated by Examples 1.1-1.2 and recent research work in this field (see [7, 8]), in this paper, we will introduce and discuss the problem associated with the following class of new nonlinear set-valued inclusions systems (NSVI Systems), which is finding for any such that , , and
where X is a real q-uniformly smooth Banach space, , , and are single-valued mappings; is a set-valued -accretive mapping and is a set-valued -accretive mapping, and are two set-valued mappings.
If , , , , and , then the problem (3) reduces to Example 1.1. If , is a proper, convex, lower semicontinuous functional and denotes the subdifferential operator of (, Hilbert space, and ), then the problem (3) changes to Example 1.2.
If X is a real q-uniformly smooth Banach space, and , and , then the problem (3) reduces to the problem associated with the following variational inclusions:
For any , find and such that
which is developed by Li in 2010 [8].
The main purpose of this paper is to introduce and study a generalized nonlinear set-valued inclusions system framework for an Ishikawa-hybrid proximal point algorithm using the notion of -accretive due to Lan-Cho-Verma [10] in a Banach space, to analyse convergence for the algorithm of solving the system and existence of a solution for the system and to prove the result that sequence generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions system with the convergence rate .
2 Preliminaries
Let X be a real q-uniformly smooth Banach space with a dual space , be the dual pair between X and , denote the family of all the nonempty subsets of X, and denote the family of all nonempty closed bounded subsets of X. The generalized duality mapping is defined by
where is a constant. Let us recall the following results and concepts.
Definition 2.1 A single-valued mapping is said to be τ-Lipschitz continuous if there exists a constant such that
Definition 2.2 A single-valued mapping is said to be
-
(i)
accretive if
-
(ii)
strictly accretive if A is accretive and if and only if , ;
-
(iii)
r-strongly η-accretive if there exists a constant such that
-
(iv)
γ-Lipschitz continuous if there exists a constant such that
-
(v)
Let be a single-valued mapping. A is said to be -relaxed cocoercive with respect to f if for any , there exist two constants such that
Definition 2.3 A set-valued mapping is said to be
-
(i)
D-Lipschitz continuous if there exists a constant such that
where is the Hausdorff metric on .
-
(ii)
β-strongly η-accretive if there exists a constant such that
Definition 2.4 Let and be single-valued mappings. A set-valued mapping is said to be
-
(i)
accretive if
-
(ii)
η-accretive if
-
(iii)
m-relaxed η-accretive, if there exists a constant such that
-
(iv)
A-accretive if M is accretive and for all ;
-
(v)
-accretive if M is m-relaxed η-accretive and for every .
Based on [10], we can define the resolvent operator as follows.
Lemma 2.5 ([10])
Let be a τ-Lipschitz continuous mapping, be an r-strongly η-accretive mapping, and be a set-valued -accretive mapping. Then the generalized resolvent operator is -Lipschitz continuous; that is,
where , .
Remark 2.6 The -accretive mappings are more general than -monotone mappings, A-monotone operators and η-subdifferential operators in a Banach space or a Hilbert space, and the resolvent operators associated with -accretive mappings include as special cases the corresponding resolvent operators associated with them, respectively [3–6, 9, 25].
In the study of characteristic inequalities in q-uniformly smooth Banach spaces X, Xu [14] proved the following result.
Lemma 2.7 ([14])
Let X be a real uniformly smooth Banach space. Then X is q-uniformly smooth if and only if there exists a constant such that for all ,
Lemma 2.8 ([8])
Let be real, for any real , if , then
3 Existence theorem of solutions
Let us study the existence theorem of solutions for the inclusions system (3).
Theorem 3.1 Let X be a Banach space, be two single-valued mappings, be a -Lipschitz continuous mapping and be a -Lipschitz continuous mapping, be a -Lipschitz continuous mapping (), be an -strongly -accretive mapping, be an -strongly -accretive mapping, be a set-valued -accretive mapping and be a set-valued -accretive mapping. Then the following statements are mutually equivalent:
-
(i)
An element is a solution of the problem (3);
-
(ii)
For , and , the following relations hold:
(5)
where is a constant ();
-
(iii)
For , , , and any , the following relations hold:
(6)
where is a constant ();
Proof This directly follows from the definition of , , and the problem (3) for . □
Theorem 3.2 Let X be a q-uniformly smooth Banach space. Let be two single-valued or -Lipschitz continuous mappings, respectively, be a single-valued -Lipschitz continuous mapping (), be two single-valued or -Lipschitz continuous mappings, respectively. Let be single-valued -strongly -accretive, -Lipschitz continuous, -relaxed cocoercive with respect to f, and be single-valued -strongly -accretive, -Lipschitz continuous, -relaxed cocoercive with respect to g. Let be two set-valued or -Lipschitz continuous mappings, respectively. If is a set-valued -accretive mapping and is a set-valued -accretive mapping, and the following condition holds:
where is the same as in Lemma 2.7 and (), then the problem (3) has a solution , , .
Proof Define two mappings as follows:
For elements , if letting
then by (8), Lemma 2.5 and Lemma 2.7, we have
and by -Lipschitz continuity of and -Lipschitz continuity of S, we obtain
Since A is -Lipschitz continuous and -relaxed cocoercive with respect to f, and f is -Lipschitz continuous so that for , , we have
Combining (9), (10) and (11), we can get
where
For elements , , (), if letting
then by using the same method as the one used above,
hold, where
If setting
and , then from (12), (13) and (14), we have , where
where Ψ is called the matrix for nonlinear set-valued inclusions system. By using [16], we have
Letting
It follows from (16), the assumption of the condition (7) and that , , and there exist and , such that
Therefore, the following relations hold for Theorem 3.1(ii)-(iii):
where is a constant (). Thus, by Theorem 3.1, we know that is a solution of the problem (3). This completes the proof. □
4 Ishikawa-hybrid proximal algorithm
In 2008, Verma developed a hybrid version of the Eckstein-Bertsekas [11] proximal point algorithm, introduced the algorithm based on the -maximal monotonicity framework [7] and studied convergence of the algorithm, and so did Li, Xu and Jin in [12]. Based on Theorem 3.1, we develop an Ishikawa-hybrid proximal point algorithm for finding an iterative sequence solving the problem (3) as follows.
Algorithm 4.1 Let X be a q-uniformly smooth Banach space. Let be two single-valued or -Lipschitz continuous mappings, respectively, be a single-valued -Lipschitz continuous mapping (), be two single-valued or -Lipschitz continuous mappings, respectively. Let be single-valued -strongly -accretive, -Lipschitz continuous, -relaxed cocoercive with respect to f, and be single-valued -strongly -accretive, -Lipschitz continuous, -relaxed cocoercive with respect to g. Let be two set-valued or -Lipschitz continuous mappings, respectively, be a set-valued -accretive mapping and be a set-valued -accretive mapping. Suppose that , , , and () are ten nonnegative sequences such that
then we can get and , as follows.
Step 1: For arbitrarily chosen initial points , , we choose suitable , , setting
where , satisfy
and
where , satisfy
By using Nadler [15], we can choose suitable , such that
Therefore, we obtain and , and give the next step for generating sequences , , and .
Step 2: From and , , the sequences , , and are generated by the iterative procedure
and
By using Nadler [15], we can choose suitable , such that
for .
Remark 4.2 If we choose some suitable operators A, B, , , F, G, S, T, M, N, f, g and a space X, then Algorithm 4.1 can degenerate to a number of known algorithms for solving the system of variational inequalities and variational inclusions (see [2–6, 8–10, 25]).
5 Convergence of Ishikawa-hybrid proximal Algorithm 4.1
In this section, we prove that generated by Ishikawa-hybrid proximal Algorithm 4.1 converges linearly to a solution of the problem (3) as the convergence rate .
Theorem 5.1 Let X be a q-uniformly smooth Banach space. Let be two single-valued or -Lipschitz continuous mappings, respectively, be a single-valued -Lipschitz continuous mapping (), be two single-two valued or -Lipschitz continuous mappings, respectively. Let be a single-valued -strongly -accretive and -Lipschitz continuous mapping, and let be a single-valued -strongly -accretive and -Lipschitz continuous mapping. Let be two set-valued or -Lipschitz continuous mappings, respectively, A be -relaxed cocoercive with respect to f and B be -relaxed cocoercive with respect to g. Suppose that is a set-valued -accretive mapping and is a set-valued -accretive mapping, and the following conditions hold:
and eight nonnegative sequences , , , and () satisfy the following conditions:
Then the problem (3) has a solution , , and the sequence generated by Ishikawa-hybrid proximal Algorithm 4.1 converges linearly to a solution of the problem (3) as the convergence rate
where is the same as in Lemma 2.5, ().
Proof Let (, ) be the solution of the problem (3), then for any ,
For , we write
It follows from the hypotheses of the mappings A, f, F, S, M, and in Algorithm 4.1 that
that is,
where , and .
From (24)-(27) and (13), we have
By Algorithm 4.1, and , we have
It follows from (26)-(29) that
and
where
For , we write
By using the hypotheses of the mappings B, g, G, T, N, and in Theorem 5.1, and the same method as the one above, we can get
that is,
where , and .
Moreover, we have
for (19) .
It follows from (26) that
Combining (30), (31), (32), (33) and (19), we have
By using (22) and (23), let
where
Let and , then from (33), (34) and (35), we have , where
which is called the matrix for a nonlinear set-valued inclusions system involving -accretive mappings. By using [16], we have
Let
It follows from (21)-(23), Theorem 3.1 and [15] that and there exist and , [17] such that
and the sequence generated by Ishikawa-hybrid proximal Algorithm 4.1 converges linearly to a solution of the problem (3) as the convergence rate
where is the same as in Lemma 2.7, (). This completes the proof. □
Remark 5.2 For a suitable choice of the mappings A, B, , F, G, M, N, S, T, f, g and X, we can obtain several known results in [7, 9, 11, 12] as special cases of Theorem 5.1.
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Li, H.G., Qiu, M. Ishikawa-hybrid proximal point algorithm for NSVI system. Fixed Point Theory Appl 2012, 195 (2012). https://doi.org/10.1186/1687-1812-2012-195
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DOI: https://doi.org/10.1186/1687-1812-2012-195