- Research Article
- Open access
- Published:
The Over-Relaxed
-Proximal Point Algorithm for General Nonlinear Mixed Set-Valued Inclusion Framework
Fixed Point Theory and Applications volume 2011, Article number: 840978 (2011)
Abstract
The purpose of this paper is (1) a general nonlinear mixed set-valued inclusion framework for the over-relaxed -proximal point algorithm based on the (
,
)-accretive mapping is introduced, and (2) it is applied to the approximation solvability of a general class of inclusions problems using the generalized resolvent operator technique due to Lan-Cho-Verma, and the convergence of iterative sequences generated by the algorithm is discussed in
-uniformly smooth Banach spaces. The results presented in the paper improve and extend some known results in the literature.
1. Introduction
In recent years, various set-valued variational inclusion frameworks, which have wide applications to many fields including, for example, mechanics, physics, optimization and control, nonlinear programming, economics, and engineering sciences have been intensively studied by Ding and Luo [1], Verma [2], Huang [3], Fang and Huang [4], Fang et al. [5], Lan et al. [6], Zhang et al. [7], respectively. Recently, Verma [8] has intended to develop a general inclusion framework for the over-relaxed -proximal point algorithm [9] based on the
-maximal monotonicity. In 2007-2008, Li [10, 11] has studied the algorithm for a new class of generalized nonlinear fuzzy set-valued variational inclusions involving
-monotone mappings and an existence theorem of solutions for the variational inclusions, and a new iterative algorithm [12] for a new class of general nonlinear fuzzy mulitvalued quasivariational inclusions involving
-monotone mappings in Hilbert spaces, and discussed a new perturbed Ishikawa iterative algorithm for nonlinear mixed set-valued quasivariational inclusions involving
-accretive mappings, the stability [13] and the convergence of the iterative sequences in
-uniformly smooth Banach spaces by using the resolvent operator technique due to Lan et al. [6].
Inspired and motivated by recent research work in this field, in this paper, a general nonlinear mixed set-valued inclusion framework for the over-relaxed -proximal point algorithm based on the
-accretive mapping is introduced, which is applied to the approximation solvability of a general class of inclusions problems by the generalized resolvent operator technique, and the convergence of iterative sequences generated by the algorithm is discussed in
-uniformly smooth Banach spaces. For more literature, we recommend to the reader [1–17].
2. Preliminaries
Let be a real Banach space with dual space
, and let
be the dual pair between
and
, let
denote the family of all the nonempty subsets of
, and let
denote the family of all nonempty closed bounded subsets of
. The generalized duality mapping
is single-valued if
is strictly convex [14], or
is uniformly smooth space. In what follows we always denote the single-valued generalized duality mapping by
in real uniformly smooth Banach space
unless otherwise stated. We consider the following general nonlinear mixed set-valued inclusion problem with
-accretive mappings (GNMSVIP).
Finding such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ1_HTML.gif)
where ,
be single-valued mappings;
be an
-accretive set-valued mapping. A special case of problem (2.1) is the following:
if is a Hilbert space,
is the zero operator in
, and
, then problem (2.1) becomes the inclusion problem
with a
-maximal monotone mapping
, which was studied by Verma [8].
Definition 2.1.
Let be a real Banach space with dual space
, and let
be the dual pair between
and
. Let
and
be single-valued mappings. A set-valued mapping
is said to be
(i)strongly
-accretive, if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ2_HTML.gif)
(ii)-relaxed
-accretive, if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ3_HTML.gif)
(iii)-cocoercive, if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ4_HTML.gif)
(iv)-accretive, if
is
-relaxed
-accretive and
for every
.
Based on the literature [6], we can define the resolvent operator as follows.
Definition 2.2.
Let be a single-valued mapping,
be a strictly
-accretive single-valued mapping and
be an
-accretive set-valued mapping. The resolvent operator
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ5_HTML.gif)
where is a constant.
Remark 2.3.
The -accretive mappings are more general than
-monotone mappings and
-accretive mappings in Banach space or Hilbert space, and the resolvent operators associated with
-accretive mappings include as special cases the corresponding resolvent operators associated with
-monotone operators,
-accretive mappings,
-monotone operators,
-subdifferential operators [1–7, 11–13].
Lemma 2.4 (see [6]).
Let be
-Lipschtiz continuous mapping,
be an
-strongly
-accretive mapping, and
be an
-accretive set-valued mapping. Then the generalized resolvent operator
is
-Lipschitz continuous, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ6_HTML.gif)
where .
In the study of characteristic inequalities in -uniformly smooth Banach spaces, Xu [14] proved the following result.
Lemma 2.5.
Let be a real uniformly smooth Banach space. Then
is
-uniformly smooth if and only if there exists a constant
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ7_HTML.gif)
3. The Over-Relaxed
-Proximal Point Algorithm
This section deals with an introduction of a generalized version of the over-relaxed proximal point algorithm and its applications to approximation solvability of the inclusion problem of the form (2.1) based on the -accretive set-valued mapping.
Let be a set-valued mapping, the set
be the graph of
, which is denoted by
for simplicity, This is equivalent to stating that a mapping is any subset
of
, and
. If
is single-valued, we shall still use
to represent the unique y such that
rather than the singleton set
. This interpretation will depend greatly on the context. The inverse
of
is
.
Definition 3.1.
Let be a set-valued mapping. The map
, the inverse of
, is said to be general
-Lipschitz continuous at 0 if, and only if there exist two constants
for any
, a solution
of the inclusion
exist and the
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ8_HTML.gif)
holds.
Lemma 3.2.
Let be a
-uniformly smooth Banach space,
be a
-Lipschtiz continuous mapping,
be an
-strongly
-accretive mapping,
be a
-Lipschtiz continuous mapping, and
be an
-accretive set-valued mapping. If
, and for all
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ9_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ10_HTML.gif)
Proof.
Let be a
-uniformly smooth Banach space,
be a
-Lipschtiz continuous mapping,
be an
-strongly
-accretive mapping, and
be an
-accretive set-valued mapping. Let us set
and
, then by using Definition 2.2, Lemmas 2.4, 2.5, and (3.2), we can have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ11_HTML.gif)
Therefore, (3.3) holds.
Lemma 3.3.
Let be a
-uniformly smooth Banach space,
be a
-Lipschtiz continuous mapping,
be an
-strongly
-accretive and nonexpansive mapping,
be an
-Lipschtiz continuous mapping, and
, and
be an
-accretive set-valued mapping. Then the following statements are mutually equivalent.
(i)An element is a solution of problem (2.1).
(ii)For a , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ12_HTML.gif)
(iii)For a , holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ13_HTML.gif)
where is a constant.
Proof.
This directly follows from definitions of and
.
Lemma 3.4.
Let be a
-uniformly smooth Banach space,
be a
-Lipschtiz continuous mapping,
be an
-strongly
-accretive and nonexpansive mapping,
be an
-Lipschtiz continuous and
-strongly
-accretive mapping, and
, and
be an
-accretive set-valued mapping. If the following conditions holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ14_HTML.gif)
where is the same as in Lemma 2.5, and
. Then the problem (2.1) has a solution
.
Proof.
Define as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ15_HTML.gif)
For elements , if letting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ16_HTML.gif)
then by (3.1) and (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ17_HTML.gif)
By using -strongly
-accretive of
,
-strongly
-accretive of
, and Lemma 2.5, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ18_HTML.gif)
Combining (3.10)-(3.11), by using nonexpansivity of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ19_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ20_HTML.gif)
It follows from (3.7)–(3.12) that has a fixed point in
, that is, there exist a point
such that
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ21_HTML.gif)
This completes the proof.
Based on Lemma 3.3, we can develop a general over-relaxed -proximal point algorithm to approximating solution of problem (2.1) as follows.
Algorithm 3.5.
Let be a
-uniformly smooth Banach space,
be a
-Lipschtiz continuous mapping,
be an
-strongly
-accretive and nonexpansive mapping,
be an
-strongly
-accretive mapping and
-Lipschitz continuous, and
, and
be an
-accretive set-valued mapping. Let
,
and
be three nonnegative sequences such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ22_HTML.gif)
where ,
and each satisfies condition (3.7).
Step 1.
For an arbitrarily chosen initial point , set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ23_HTML.gif)
where the satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ24_HTML.gif)
Step 2.
The sequence is generated by an iterative procedure
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ25_HTML.gif)
and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ26_HTML.gif)
where .
Remark 3.6.
For a suitable choice of the mappings ,
,
,
,
, and space
, then the Algorithm 3.5 can be degenerated to the hybrid proximal point algorithm [16, 17] and the over-relaxed
-proximal point algorithm [8].
Theorem 3.7.
Let be a
-uniformly smooth Banach space. Let
and
be single-valued mappings, and let
be a set-valued mapping and
be the inverse mapping of the mapping
satisfying the following conditions:
(i) is
-Lipschtiz continuous;
(ii) be an
-strongly
-accretive mapping and nonexpansive;
(iii) be an
-Lipschtiz continuous and
-strongly
-accretive mapping;
(iv) be an
-accretive set-valued mapping;
(v)the be
-Lipschitz continuous at 0
;
(vi),
and
be three nonnegative sequences such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ27_HTML.gif)
where ,
and each satisfies condition (3.7),
(vii)let the sequence generated by the general over-relaxed
-proximal point algorithm (3.6) be bounded and
be a solution of problem (2.1), and the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ29_HTML.gif)
hold. Then the sequence converges linearly to a solution
of problem (2.1) with convergence rate
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ30_HTML.gif)
Proof.
Let the be a solution of the Framework (2.1) for the conditions (i)–(iv) and Lemma 3.4. Suppose that the sequence
which generated by the hybrid proximal point Algorithm 3.5 is bounded, from Lemma 3.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ31_HTML.gif)
We infer from Lemma 3.3 that any solution to (2.1) is a fixed point of . First, in the light of Lemma 3.2, we show
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ32_HTML.gif)
where and
.
For , and under the assumptions (including the condition (vii) (3.21)), then
since the
is
-Lipschitz continuous at 0. Indeed, it follows that
from
. Next, by using the condition (iv) and (3.1), and setting
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ33_HTML.gif)
Now applying Lemma 3.3, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ34_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ35_HTML.gif)
where and
.
Next we start the main part of the proof by using the expression
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ36_HTML.gif)
Let us set and
for simple. We begin with estimating
and later using (3.2), the nonexpansivity of
, (3.21) and (3.28) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ37_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ38_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ39_HTML.gif)
and ,
,
, and
.
Since , we have
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ40_HTML.gif)
Next, we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ41_HTML.gif)
This implies from (3.38) and (3.39) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ42_HTML.gif)
Since is an
-strongly
-accretive mapping (and hence,
), this implies from (3.35) that the sequence
converges strongly to
for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ43_HTML.gif)
where ,
, and
.
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ44_HTML.gif)
By (3.22), it follows that from the condition (vi), and the sequence
generated by the hybrid proximal point Algorithm 3.5 converges linearly to a solution
of problem (2.1) with convergence rate
. This completes the proof.
Corollary 3.8.
Let be a Hilbert space
,
be an
-strongly monotone and nonexpansive mapping
,
is a zero operator,
be an
-maximal set-valued monotone.
, and the condition (3.21) hold, the
be
-Lipschitz continuous at 0
. Let
,
and
be the same as in Algorithm 3.5. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ45_HTML.gif)
then the bounded sequence generated by the general over-relaxed
-proximal point algorithm converges linearly to a solution
of problem (2.1) with convergence rate
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F840978/MediaObjects/13663_2010_Article_1432_Equ46_HTML.gif)
and ,
,
.
This is Theorem 3.2 in [8], and if, in addition, ,
then we can have the Proposition 2 in [9].
References
Ding XP, Luo CL: Perturbed proximal point algorithms for general quasi-variational-like inclusions. Journal of Computational and Applied Mathematics 2000,113(1–2):153–165. 10.1016/S0377-0427(99)00250-2
Verma RU: Approximation-solvability of a class of A-monotone variational inclusion problems. Journal KSIAM 2004,8(1):55–66.
Huang N-J: Nonlinear implicit quasi-variational inclusions involving generalized m -accretive mappings. Archives of Inequalities and Applications 2004,2(4):413–425.
Fang Y-P, Huang N-J: H -accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces. Applied Mathematics Letters 2004,17(6):647–653. 10.1016/S0893-9659(04)90099-7
Fang Y-P, Huang N-J, Thompson HB: A new system of variational inclusions with ()-monotone operators in Hilbert spaces. Computers & Mathematics with Applications 2005,49(2–3):365–374. 10.1016/j.camwa.2004.04.037
Lan H-Y, Cho YJ, Verma RU: Nonlinear relaxed cocoercive variational inclusions involving ()-accretive mappings in Banach spaces. Computers & Mathematics with Applications 2006,51(9–10):1529–1538. 10.1016/j.camwa.2005.11.036
Zhang Q-B, Ding X-P, Cheng C-Z: Resolvent operator technique for generalized implicit variational-like inclusion in Banach space. Journal of Mathematical Analysis and Applications 2010,361(2):283–292. 10.1016/j.jmaa.2006.01.090
Verma RU: A general framework for the over-relaxed A -proximal point algorithm and applications to inclusion problems. Applied Mathematics Letters 2009,22(5):698–703. 10.1016/j.aml.2008.05.001
Pennanen T: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Mathematics of Operations Research 2002,27(1):170–191. 10.1287/moor.27.1.170.331
Li H-G: Iterative algorithm for a new class of generalized nonlinear fuzzy set-variational inclusions involving ()-monotone mappings. Advances in Nonlinear Variational Inequalities 2007,10(1):89–100.
Li HG: Approximate algorithm of solutions for general nonlinear fuzzy multivalued quasi-variational inclusions with ()-monotone mappings. Nonlinear Functional Analysis and Applications 2008,13(2):277–289.
Li H-G: Perturbed Ishikawa iterative algorithm and stability for nonlinear mixed quasi-variational inclusions involving ()-accretive mappings. Advances in Nonlinear Variational Inequalities 2008,11(1):41–50.
Lan H-Y: On multivalued nonlinear variational inclusion problems with ()-accretive mappings in Banach spaces. Journal of Inequalities and Applications 2006, 2006:-12.
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis: Theory, Methods & Applications 1991,16(12):1127–1138. 10.1016/0362-546X(91)90200-K
Weng X: Fixed point iteration for local strictly pseudo-contractive mapping. Proceedings of the American Mathematical Society 1991,113(3):727–731. 10.1090/S0002-9939-1991-1086345-8
Verma RU: A hybrid proximal point algorithm based on the ()-maximal monotonicity framework. Applied Mathematics Letters 2008,21(2):142–147. 10.1016/j.aml.2007.02.017
Li HG, Xu AJ, Jin MM: An Ishikawa-hybrid proximal point algorithm for nonlinear set-valued inclusions problem based on ()-accretive framework. Fixed Point Theory and Applications 2010, 2010:-12.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Pan, X., Li, H. & Xu, A. The Over-Relaxed -Proximal Point Algorithm for General Nonlinear Mixed Set-Valued Inclusion Framework.
Fixed Point Theory Appl 2011, 840978 (2011). https://doi.org/10.1155/2011/840978
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2011/840978