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An IshikawaHybrid Proximal Point Algorithm for Nonlinear SetValued Inclusions Problem Based on Accretive Framework
Fixed Point Theory and Applications volumeÂ 2010, ArticleÂ number:Â 501293 (2010)
Abstract
A general nonlinear framework for an Ishikawahybrid proximal point algorithm using the notion of accretive is developed. Convergence analysis for the algorithm of solving a nonlinear setvalued inclusions problem and existence analysis of solution for the nonlinear setvalued inclusions problem are explored along with some results on the resolvent operator corresponding to accretive mapping due to LanChoVerma in Banach space. The result that sequence generated by the algorithm converges linearly to a solution of the nonlinear setvalued inclusions problem with the convergence rate is proved.
1. Introduction
The setvalued inclusions problem, which was introduced and studied by Di Bella [1], Huang et al. [2], and Jeong [3], is a useful extension of the mathematics analysis. And the variational inclusion(inequality) is an important context in the setvalued inclusions problem. It provides us with a unified, natural, novel, innovative, and general technique to study a wide class of problems arising in different branches of mathematical and engineering sciences. Various variational inclusions have been intensively studied in recent years. Ding and Luo[4], Verma [5], Huang [6], Fang and Huang [7], Lan et al. [8], Fang et al. [9], and Zhang et al. [10] introduced the concepts of subdifferential operators, maximal monotone operators, monotone operators, monotone operators, monotone operators, accretive mappings, monotone operators, and defined resolvent operators associated with them, respectively. Moreover, by using the resolvent operator technique, many authors constructed some approximation algorithms for some nonlinear variational inclusions in Hilbert spaces or Banach spaces. Recently, Verma has developed a hybrid version of the Eckstein and Bertsekas [11] proximal point algorithm, introduced the algorithm based on the maximal monotonicity framework [12], and studied convergence of the algorithm.
On the other hand, in 2008, Li [13] studied the existence of solutions and the stability of perturbed Ishikawa iterative algorithm for nonlinear mixed quasivariational inclusions involving accretive mappings in Banach spaces by using the resolvent operator technique in [14].
Inspired and motivated by recent research work in this field, in this paper, a general nonlinear framework for a Ishikawahybrid proximal point algorithm using the notion of accretive is developed. Convergence analysis for the algorithm of solving a nonlinear setvalued inclusions problem and existence analysis of solution for the nonlinear setvalued inclusions problem are explored along with some results on the resolvent operator corresponding to accretive mapping due to Lan et al. in Banach space. The result that sequence generated by the algorithm converges linearly to a solution of the nonlinear setvalued inclusions problem as the convergence rate is proved.
2. Preliminaries
Let be a real Banach space with dual space and and let the dual pair between and , denote the family of all the nonempty subsets of and the family of all nonempty closed bounded subsets of . The generalized duality mapping is defined by
where is a constant.
The modulus of smoothness of is the function defined by
A Banach space is called uniformly smooth if
is called uniformly smooth if there exists a constant such that
Remark 2.1.
In particular, is the usual normalized duality mapping, and (for all ). If is strictly convex [15], or is uniformly smooth Banach space, then is single valued. In what follows we always denote the singlevalued generalized duality mapping by in real uniformly smooth Banach space unless otherwise stated.
Let ; be singlevalued mappings. Let be a setvalued accretive mapping. We consider nonlinear setvalued mixed variational inclusions problem withaccretive mappings (NSVMVIP).
For any , finding , such that
Remark 2.2.
A special case of problem (2.5) is the following.
(i)If is a Hilbert space, is the zero operator in , is the identity operator in , and , then problem (2.5) becomes the parametric usual variational inclusion with a maximal monotone mapping , which was studied by Verma [12].
(ii)If is a real Banach space, is the identity operator in , and , then problem (2.5) becomes the parametric usual variational inclusion with a accretive mapping, which was studied by Li [13].
It is easy to see that a number of known special classes of variational inclusions and variational inequalities in the problem (2.5) are studied (see [2, 7, 12â€“14]).
Let us recall the following results and concepts.
Definition 2.3.
A singlevalued mapping is said to be Lipschitz continuous if there exists a constant such that
Definition 2.4.
A singlevalued mapping is said to be
(i)accretive if
(ii)strictly accretive, if is accretive and if and only if â€‰â€‰â€‰â€‰,
(iii)strongly accretive if there exists a constant such that
(iv)Lipschitz continuous if there exists a constant such that
Definition 2.5.
A singlevalued mapping is said to be
(i)Lipschitz continuous if there exist constants such that
(ii)relaxed cocoercive with respect to in the first argument if there exist constants , and such that
where are singlevalued mappings.
Definition 2.6.
Let , and let be singlevalued mappings. A setvalued mapping is said to be
(i)accretive if
(ii)accretive if
(iii)strongly accretive if there exists a constant such that
(iv)relaxed accretiveif there exists a constant such that
(v)accretive, if is accretive and for all
(vi)accretive if is relaxed accretive and for all .
Based on the literature [8], we can define the resolvent operator as follows.
Definition 2.7 (see [8]).
Let be a singlevalued mapping, a strictly accretive singlevalued mapping and a accretive mapping. The resolvent operator is defined by
where is a constant.
Remark 2.8.
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 [3â€“14, 16, 17].
Lemma 2.9 (see [8]).
Let be Lipschtiz continuous mapping, be an strongly accretive mapping, and an accretive mapping. Then the generalized resolvent operator is Lipschitz continuous, that is,
where
In the study of characteristic inequalities in uniformly smooth Banach spaces, Xu [18] proved the following result.
Lemma 2.10 (see [18]).
Let be a real uniformly smooth Banach space. Then is uniformly smooth if and only if there exists a constant such that for all ,
3. The Existence of Solutions
Now, we are studing the existence for solutions of problem (2.5).
Lemma 3.1.
Let be a Banach space. Let be a Lipschtiz continuous mapping, be an strongly accretive mapping, and an accretive mapping. Then the following statements are mutually equivalent.
(i)An element is a solution of problem (2.5).

(ii)
For a and any , there exists such that
(3.1)
where is a constant.
Proof.
This directly follows from the definition of .
Theorem 3.2.
Let be a uniformly smooth Banach space. Let ; be singlevalued mappings, and be a Lipschtiz continuous mapping, a strongly accretive and Lipschitz continuous mapping, be a strongly accretive and Lipschitz continuous mapping, and a Lipschitz continuous mapping, respectively. Let be Lipschitz continuous, and relaxed cocoercive with respect to in the first argument. Let be a setvalued accretive mapping. If the following condition holds:
where is the same as in Lemma 2.10, and , then the problem (2.5) has a solution .
Proof.
Define a mapping as follows:
For elements , if we let and
then by (3.1), (3.3), and Lemma 2.10, we have
Since is relaxed cocoercive with respect to in the first argument and is a Lipschitz continuous mapping so we obtain
By strongly accretivity of , we have
Combining(3.5), (3.6), (3.7), and (3.8), we can get
where
It follows from (3.2) and (3.9) that has a fixed point in , that is, there exists a point such that , and
where . This completes the proof.
4. IshikawaHybrid Proximal Point Algorithm
Based on Lemma 3.1, we develop an Ishikawahybrid proximal point algorithm for finding an iterative sequence solving problem (2.5) as follows.
Algorithm 4.1.
Let be a solution of problem ( 2.5 ). Let , , , and , be five nonnegative sequences such that
Step 1.
For an arbitrarily initial point , we choose suitable , letting
Step 2.
The sequences and are generated by an iterative procedure
where
Remark 4.2.
For a suitable choice of the mappings , space , and nonnegative sequences , , Algorithm 4.1 can be degenerated to a number of algorithms involving many known algorithms which are due to classes of variational inequalities and variational inclusions [12â€“14].
Theorem 4.3.
Let be the same as in Theorem 3.2, then condition (3.2) holds. Let , , , and be the same as in Algorithm 4.1. Then the sequence generated by hybrid proximal point Algorithm 4.1 converges linearly to a solution of problem (2.5) as
where is the same as in Lemma 2.10, , and the convergence rate is
Proof.
Suppose that the sequence is the the sequence generated by the Ishikawahybrid proximal point Algorithm 4.1, and that is a solution of problem (2.5). From Lemma 3.1 and condition , we can get
where .
For all , and , setting
we find the estimation
By the conditions and Lemma 2.10, we have
It follows from (4.8)â€“(4.10) that
where
Since and (4.3), and
Next, we calculate
This implies that
letting
For all , set
For the same reason,
where
Furthermore,
Combining (4.16)(4.22), then we have
By (4.4) and the condition , we can see that
and the convergence rate is .By (4.4), if , then it follows that and . Therefor, the sequence generated hybrid proximal point Algorithm 4.1 converges linearly to a solution of problem (2.5) with convergence rate . This completes the proof.
Remark 4.4.
For a suitable choice of the mappings , we can obtain several known results [12â€“14, 17] as special cases of Theorem 3.2 and Theorem 4.3.
References
Di Bella B: An existence theorem for a class of inclusions. Applied Mathematics Letters 2000,13(3):15â€“19. 10.1016/S08939659(99)001792
Huang NJ, Tang YY, Liu YP: Some new existence theorems for nonlinear inclusion with an application. Nonlinear Functional Analysis and Applications 2001,6(3):341â€“350.
Jeong JU: Generalized setvalued variational inclusions and resolvent equations in Banach spaces. Computers & Mathematics with Applications 2004,47(8â€“9):1241â€“1247. 10.1016/S08981221(04)901186
Ding XP, Luo CL: Perturbed proximal point algorithms for general quasivariationallike inclusions. Journal of Computational and Applied Mathematics 2000,113(1â€“2):153â€“165. 10.1016/S03770427(99)002502
Verma RU: Approximationsolvability of a class of Amonotone variational inclusion problems. Journal KSIAM 2004,8(1):55â€“66.
Huang NJ: Nonlinear implicit quasivariational inclusions involving generalized m accretive mappings. Archives of Inequalities and Applications 2004,2(4):413â€“425.
Fang YP, Huang NJ: H accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces. Applied Mathematics Letters 2004,17(6):647â€“653. 10.1016/S08939659(04)900997
Lan HY, Cho YJ, Verma RU: Nonlinear relaxed cocoercive variational inclusions involving accretive mappings in Banach spaces. Computers & Mathematics with Applications 2006,51(9â€“10):1529â€“1538. 10.1016/j.camwa.2005.11.036
Fang YP, Huang NJ, Thompson HB: A new system of variational inclusions with monotone operators in Hilbert spaces. Computers & Mathematics with Applications. An International Journal 2005,49(2â€“3):365â€“374. 10.1016/j.camwa.2004.04.037
Zhang QB, Zhanging XP, Cheng CZ: Resolvent operator technique for solving generalized implicit variationallike inclusions in Banach space. Journal of Mathematical Analysis and Applications 2007, (20):216â€“221.
Eckstein J, Bertsekas DP: On the DouglasRachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming 1992,55(3):293â€“318. 10.1007/BF01581204
Verma RU: A hybrid proximal point algorithm based on the maximal monotonicity framework. Applied Mathematics Letters 2008,21(2):142â€“147. 10.1016/j.aml.2007.02.017
Li HG: Perturbed Ishikawa iterative algorithm and stability for nonlinear mixed quasivariational inclusions involving accretive mappings. Advances in Nonlinear Variational Inequalities 2008,11(1):41â€“50.
Huang NJ, Fang YP: Generalized m accretive mappings in Banach spaces. Journal of Sichuan University 2001,38(4):591â€“592.
Weng XL: Fixed point iteration for local strictly pseudocontractive mapping. Proceedings of the American Mathematical Society 1991,113(3):727â€“731. 10.1090/S00029939199110863458
Shim SH, Kang SM, Huang NJ, Cho YJ: Perturbed iterative algorithms with errors for completely generalized strongly nonlinear implicit quasivariational inclusions. Journal of Inequalities and Applications 2000,5(4):381â€“395. 10.1155/S1025583400000205
Agarwal RP, Cho YJ, Huang NJ: Sensitivity analysis for strongly nonlinear quasivariational inclusions. Applied Mathematics Letters 2000,13(6):19â€“24. 10.1016/S08939659(00)000483
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis. Theory, Methods & Applications 1991,16(12):1127â€“1138. 10.1016/0362546X(91)90200K
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Li, H., Xu, A. & Jin, M. An IshikawaHybrid Proximal Point Algorithm for Nonlinear SetValued Inclusions Problem Based on Accretive Framework. Fixed Point Theory Appl 2010, 501293 (2010). https://doi.org/10.1155/2010/501293
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DOI: https://doi.org/10.1155/2010/501293