Advanced
CONVERGENCE OF PARALLEL ITERATIVE ALGORITHMS FOR A SYSTEM OF NONLINEAR VARIATIONAL INEQUALITIES IN BANACH SPACES†
CONVERGENCE OF PARALLEL ITERATIVE ALGORITHMS FOR A SYSTEM OF NONLINEAR VARIATIONAL INEQUALITIES IN BANACH SPACES†
Journal of Applied Mathematics & Informatics. 2016. Jan, 34(1_2): 61-73
Copyright © 2016, Korean Society of Computational and Applied Mathematics
  • Received : June 30, 2015
  • Accepted : October 10, 2015
  • Published : January 30, 2016
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
About the Authors
JAE UG JEONG

Abstract
In this paper, we consider the problems of convergence of parallel iterative algorithms for a system of nonlinear variational inequalities and nonexpansive mappings. Strong convergence theorems are established in the frame work of real Banach spaces. AMS Mathematics Subject Classification: 47J20; 65K10; 65K15; 90C33.
Keywords
1. Introduction
Let ( E , ║ · ║) be a Banach space and C be a nonempty closed convex subset of E . This paper deals with the problems of convergence of iterative algorithms for a system of nonlinear variational inequalities: Find ( x , y ) ∈ C × C such that
PPT Slide
Lager Image
where T 1 , T 2 : C × C E , g 1 , g 2 : C C are nonlinear mappings, J is the normalized duality mapping, j J and ρ 1 , ρ 2 are two positive real numbers.
If T 1 , T 2 : C E are nonlinear mappings and g 1 = g 2 = I ( I denotes the identity mapping), then (1.1) reduces to finding ( x , y ) ∈ C × C such that
PPT Slide
Lager Image
which was considered by Yao et al. [13] .
If E = H is a real Hilbert space and T 1 , T 2 : C H are nonlinear mappings and g 1 = g 2 = g , then (1.1) reduces to finding ( x , y ) ∈ C × C such that
PPT Slide
Lager Image
which was studied by Yang et al. [12] .
If g = I , then (1.3) reduces to finding ( x , y ) ∈ C × C such that
PPT Slide
Lager Image
which was introduced by Ceng et al. [2] .
In particular, if T 1 = T 2 = T , then (1.4) reduces to finding ( x , y ) ∈ C × C such that
PPT Slide
Lager Image
which is defined by Verma [9] .
Further, if x = y , then (1.5) reduces to the following classical variational inequality (VI( T,C )) of finding x C such that
PPT Slide
Lager Image
We can see easily that the variational inequality (1.6) is equivalent to a fixed point problem. An element x C is a solution of the variational inequality (1.6) if and only if x C is a fixed point of the mapping PC ( I − λT ), where PC is the metric projection and λ is a positive real number. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.
Recent development of the variational inequality is to design efficient iterative algorithms to compute approximate solutions for variational inequalities and their generalization. Up to now, many authors have presented implementable and significant numerical methods such as projection method and it’s variant forms, linear approximation, descent method, Newton’s method and the method based on auxiliary principle technique.
However, these sequential iterative methods are only suitable for implementing on the traditional single-processor computer. To satisfy practical requirements of modern multiprocessor systems, efficient iterative methods having parallel characteristics need to be further developed for the system of variational inequalities (see [1 , 4 , 5 , 6 , 12 , 14] ).
Motivated and inspired by the research work going on this field, in this paper, we construct an parallel iterative algorithm for approximating the solution of a new system of variational inequalities involving four different nonlinear mappings. Finally, we prove the strong convergence of the purposed iterative scheme in 2-uniformly smooth Banach spaces.
2. Preliminaries
Let C be a nonempty closed convex subset of a Banach space E with the dual space E . Let ⟨·, ·⟩ denote the dual pair between E and E . Let 2 E denote the family of all the nonempty subsets of E . For q > 1, the generalized duality mapping Jq : E → 2 E is defined by
PPT Slide
Lager Image
In particular, J = J 2 is the normalized duality mapping. It is known that Jq ( x ) = ║ x q−2 J ( x ) for all x E and Jq is single-valued if E is strictly convex or E is uniformly smooth. If E = H is a Hilbert space, J = I , the identity mappings.
Let B = { x E : ║ x ║ = 1}. A Banach space E is said to be smooth if the limit
PPT Slide
Lager Image
exists for all x, y B . The modulus of smoothness of E is the function ρE : [0,∞) → [0,∞) defined by
PPT Slide
Lager Image
A Banach space E is called uniformly smooth if
PPT Slide
Lager Image
E is called q -uniformly smooth if there exists a constant c > 0 such that
PPT Slide
Lager Image
If E is q -uniformly smooth, then q ≤ 2 and E is uniformly smooth.
Definition 2.1. Let T : C × C E be a mapping. T is said to be
(i) ( δ, ξ )-relaxed cocoercive with respect to the first argument if there exist j ( x − y ) ∈ J ( x − y ) and constants δ, ξ > 0 such that
PPT Slide
Lager Image
for all x, y C ;
(ii) μ -Lipschitz continuous with respect to the first argument if there exists a constant μ > 0 such that
PPT Slide
Lager Image
for all x, y C ;
(iii) γ -Lipschitz continuous with respect to the second argument if there exists a constant γ > 0 such that
PPT Slide
Lager Image
for all x, y C .
Definition 2.2. Let g : C C be a mapping. g is said to be
(i) ζ -strongly accretive if there exists a constant ζ > 0 such that
PPT Slide
Lager Image
for all x, y C .
(ii) η -Lipschitz continuous if there exists a constant η > 0 such that
PPT Slide
Lager Image
for all x, y C .
Let D be a subset of C and Q be a mapping of C into D . Then Q is said to be sunny if
PPT Slide
Lager Image
whenever Q ( x ) + t ( x Q ( x )) ∈ C for x C and t ≥ 0. A mapping Q of C into itself is called a retraction if Q 2 = Q . If a mapping Q of C into itself is a retraction, then Q ( z ) = z for all z R ( Q ), where R ( Q ) is the range of Q . A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D .
In order to prove our main results, we also need the following lemmas.
Lemma 2.3 ( [11] ). Let E be a real 2-uniformly smooth Banach space. Then
PPT Slide
Lager Image
where K is the 2-uniformly smooth constant of E .
Lemma 2.4 ( [7] ). Let C be a nonempty closed convex subset of a smooth Banach space E and let QC be a retraction from E onto C. Then the following are equivalent:
(i) QC is both sunny and nonexpansive;
(ii) x − QC ( x ), j ( y − QC ( x ))⟩ ≤ 0 for all x E and y C .
Lemma 2.5 ( [10] ). Suppose { δn } is a nonnegative sequence satisfying the fol- lowing inequality:
PPT Slide
Lager Image
with λn ∈ [0, 1],
PPT Slide
Lager Image
and σn = 0( λn ). Then lim n→∞ δn = 0.
Lemma 2.6 ( [3] ). Let { cn } and { kn } be two real sequences of nonnegative num-bers that satisfy the following conditions:
(i) 0 ≤ kn ≤ 1 for n = 1, 2, · · · and lim sup n kn < 1 ;
(ii) c n+1 kncn for n = 1, 2, · · · .
Then cn converges to 0 as n → ∞.
3. Iterative algorithms
In this section, we suggest a parallel iterative algorithm for solving the system of nonlinear variational inequality (1.1). First of all, we establish the equivalence between the system of variational inequalities and fixed point problems.
Lemma 3.1. Let C be a nonempty closed convex subset of a smooth Banach space E. Let QC : E C be a sunny nonexpansive retraction , Ti : C × C E and gi : C C be mappings for i = 1, 2. Then ( x , y ) with x , y C is a solution of problem (1.1) if and only if
PPT Slide
Lager Image
Proof . Applying Lemma 2.4, we have that
PPT Slide
Lager Image
That is,
PPT Slide
Lager Image
This completes the proof. □
This fixed point formulation allow us to suggest the following parallel iterative algorithms.
Algorithm 3.1. For any given x 0 , y 0 C , computer the sequences { xn } and { yn } defined by
PPT Slide
Lager Image
where ρ 1 , ρ 2 are positive real numbers.
Also, we propose a relaxed parallel algorithm which can be applied to the approximation of solution of the problem (1.1) and common fixed point of two mappings.
Algorithm 3.2. For any given x 0 , y 0 C , compute the sequences { xn } and { yn } defined by
PPT Slide
Lager Image
where S 1 , S 2 : C C are nonexpansive mappings, { αn }, { βn } are sequences in [0,1], κ ∈ (0, 1) and ρ 1 , ρ 2 are positive real numbers.
If T 1 , T 2 : C E are nonlinear mappings and g 1 = g 2 = I , then the algorithm 3.1 reduces to the following parallel iterative method for solving problem (1.2).
Algorithm 3.3. For any given x 0 , y 0 C , compute the sequences { xn } and { yn } defined by
PPT Slide
Lager Image
where ρ 1 , ρ 2 are positive real numbers.
If E = H is a Hilbert space, T 1 , T 2 : C H are nonlinear mappings and g 1 = g 2 = g , Algorithm 3.1 reduces to the following parallel iterative method for solving problem (1.3).
Algorithm 3.4. For any given x 0 , y 0 C , compute the sequences { xn } and { yn } defined by
PPT Slide
Lager Image
where ρ 1 , ρ 2 are positive real numbers.
4. Main results
We now state and prove the main results of this paper.
Theorem 4.1. Let E be a 2 -uniformly smooth Banach space with the 2- uniformly smooth constant K, C be a nonempty closed convex subset of E and QC be a sunny nonexpansive retraction from E onto C. Let Ti : C × C E be a non-linear mapping such that ( δi, ξi ) -relaxed cocoercive, μi-Lipschitz continuous with respect to the first argument and γi-Lipschitz continuous with respect to the sec-ond argument for i = 1, 2. Let gi : C C be a ηi-Lipschitz continuous and ζi-strongly accretive mapping for i = 1, 2. Assume that the following assump-tions hold:
PPT Slide
Lager Image
PPT Slide
Lager Image
where τ 1 = m 1 + m 2 + ρ 2 γ 2 , τ 2 = m 1 + m 2 + ρ 1 γ 1 ,
PPT Slide
Lager Image
Then there exist x , y E, which solves the problem (1.1). Moreover, the parallel iterative sequences { xn } and { yn } generated by the Algorithm 3.1 con-verge to x and y , respectively .
Proof . To proof the result, we first need to evaluate ║ x n+1 xn ║ for all n ≥ 0. From Algorithm 3.1 and the nonexpansive property of the sunny nonexpansive retraction QC , we can get
PPT Slide
Lager Image
Using the strongly accretivity and Lipschitz continuity of g 1 and Lemma 2.3, we find that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
which imply that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where
PPT Slide
Lager Image
. Since T 1 is ( δ 1 , ξ 1 )-relaxed cocoercive and μ 1 -Lipschitz continuous with respect to the first argument, we have
PPT Slide
Lager Image
Also,using the Lipschitz continuity of T 1 with respect to second argument,
PPT Slide
Lager Image
Combining (4.3)-(4.7), we have
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Similarly, since g 2 is η 2 -Lipschitz continuous and ζ 2 -strongly accretive, T 2 is ( δ 2 , ξ 2 )-relaxed cocoercive, μ 2 -Lipschitz continuous with respect to the first argument and γ 2 -Lipschitz continuous with respect to the second argument, we obtain
PPT Slide
Lager Image
where
PPT Slide
Lager Image
. It follows from (4.8) and (4.9) that
PPT Slide
Lager Image
where k = max{ m 1 + m 2 + θ 2 + ρ 1 γ 1 , m 1 + m 2 + θ 1 + ρ 2 γ 2 }. From (4.1) and (4.2), we know that 0 ≤ k < 1. Let cn = ║ xn x n−1 ║ + ║ yn y n−1 ║. Then (4.10) can be rewritten as
PPT Slide
Lager Image
It follows from Lemma 2.6 that { xn } and { yn } are both Cauchy sequences in E . There exist x , y E such that xn x and yn y as n → ∞. By continuity, we know that x , y satisfy
PPT Slide
Lager Image
It follows from Lemma 3.1 that ( x , y ) is a solution of problem (1.1). This completes the proof. □
If T 1 , T 2 : C E are nonlinear mappings and g 1 = g 2 = I , the the following corollary follows immediately from Theorem 4.1.
Corollary 4.2. Let E be a 2 -uniformly smooth Banach space with the 2- uniformly smooth constant K, C be a nonempty closed convex subset of E and QC be a sunny nonexpansive retraction from E onto C. Let Ti : C E be a ( δi, ξi ) -relaxed cocoercive and μi-Lipschitz continuous mapping for i = 1, 2. Assume that the following assumptions hold:
PPT Slide
Lager Image
Then there exist x , y E, which solves the problem (1.2). Moreover, the par-allel iterative sequences { xn } and { yn } generated by the Algorithm 3.3 converge to x and y , respectively .
Remark 4.1. (i) We note that Hilbert spaces and Lp ( p ≥ 2) spaces are 2-uniformly smooth.
(ii) If E = H is a Hilbert space, then a sunny nonexpansive retraction QC is coincident with the metric projection PC from H onto C .
(iii) It is well known that the 2-uniformly smooth constant
PPT Slide
Lager Image
in Hilbert spaces.
We can obtain the following result immediately.
Corollary 4.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Ti : C H be a ( δi, ξi ) -relaxed cocoercive and μi-Lipschitz continuous mapping for i = 1, 2. Let g : C C be a η-Lipschitz continuous and ζ-strongly monotone mapping. Assume that the following assumptions hold:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Then there exist x , y H, which solve the problem (1.3). Moreover, the par-allel iterative sequences { xn } and { yn } generated by the Algorithm 3.4 converge to x and y , respectively .
Let Fix ( Si ) denote the set of fixed points of the mapping Si , i.e., Fix ( Si ) = { x C : Six = x } and Ω the set of solutions of the problem (1.1).
Theorem 4.4. Let E be a 2- uniformly smooth Banach space with the 2- uniformly smooth constant K, C be a nonempty closed convex subset of E and QC be a sunny nonexpansive retraction from E onto C. Let Ti : C × C E be a non-linear mapping such that ( δi, ξi ) -relaxed cocoercive, μi-Lipschitz continuous with respect to the first argument and γi-Lipschitz continuous with respect to the sec-ond argument for i = 1, 2. Let gi : C C be a ηi-Lipschitz continuous and ζi-strongly accretive mapping for i = 1, 2. Let Si : C C be a nonexpansive mapping with a fixed point for i = 1, 2. Let { αn }, { βn } be sequences in [0, 1]. Assume that the following assumptions hold:
(C1) 0 < Θ 1,n = αn (1 − κ − (1 − κ )( m 1 + ρ 1 γ 1 )) − βn (1 − κ )( m 2 + θ 2 ) < 1,
(C2) 0 < Θ 2,n = βn (1 − κ − (1 − κ )( m 2 + ρ 2 γ 2 )) − αn (1 − κ )( m 1 + θ 1 ) < 1,
(C3)
PPT Slide
Lager Image
, where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
If Ω ∩ Fix ( S 1 ) ∩ Fix ( S 2 ) ≠ ϕ, then the sequences { xn } and { yn } generated by the Algorithm 3.2 converge to x and y , respectively, where ( x , y ) ∈ Ω and x , y Fix ( S 1 ) ∩ Fix ( S 2 ).
Proof . Letting ( x , y ) ∈ Ω, we obtain from Lemma 3.1 that
PPT Slide
Lager Image
Since x , y Fix ( S 1 ) ∩ Fix ( S 2 ), we have
PPT Slide
Lager Image
Putting e 1,n = κ S 1 ( xn ) + (1 − κ )( xn g 1 ( xn ) + QC [ g 1 ( yn ) − ρ 1 T 1 ( yn , xn )]) for each n = 0, 1, 2, · · · , we arrive at
PPT Slide
Lager Image
Using the arguments as in the proof of Theorem 4.1, we obtain
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where
PPT Slide
Lager Image
From (4.11), we have
PPT Slide
Lager Image
It follows that
PPT Slide
Lager Image
Similarly, we obtain
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Now (4.12) and (4.13) imply
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Define the norm ║·║ on E × E by
PPT Slide
Lager Image
Then ( E × E , ║·║ ) is a Banach space. Hence, (4,14) implies that
PPT Slide
Lager Image
From the conditions (C1)-(C3) and Lemma 2.5 to (4.15), we obtain that
PPT Slide
Lager Image
Therefore, the sequences { xn } and { yn } converge to x and y , respectively. This completes the proof. □
Remark 4.2. Theorem 4.1 and 4.4 extend the solvability of the systems of variational inequalities (1.2)-(1.6) to the more general system of variational inequalities (1.1). The underlying mapping Ti : C × C E ( i = 1, 2) in our paper needs to be relaxed ( δi, ξi )-relaxed cocoercive while the underlying operators A,B in [13] needs to inverse strongly accretive. Hence, Theorem 4.1 and 4.4 extend and improve the main results of [9 , 12 , 13] .
BIO
Jae Ug Jeong received M.Sc. from Busan National University and Ph.D at Gyeongsang National University. Since 1982 he he has been at Dongeui University. His research interests include fixed point theory and variational inequality problems.
Department of Mathematics, Dongeui University, Busan 614-714, South Korea.
e-mail: jujeong@deu.ac.kr
References
Bertsekas D. , Tsitsiklis J. 1989 Parallel and Distributed Computation, Numerical Methods Prentice-Hall Englewood Cliffs, NJ
Ceng L.C. , Wang C. , Yao J.C. (2008) Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities Math. Methods Oper. Res. 67 375 - 390    DOI : 10.1007/s00186-007-0207-4
Fang Y.P. , Huang N.J. , Thompson H.B. (2005) A new system of variational inclusions with (H, η)-monotone operators in Hilbert spaces Comput. Math. Appl. 49 365 - 374    DOI : 10.1016/j.camwa.2004.04.037
Hoffmann K.H. , Zou J. (1992) Parallel algorithms of Schwarz variant for variational inequalities Numer. Funct. Anal. Optim. 13 449 - 462    DOI : 10.1080/01630569208816491
Hoffmann K.H. , Zou J. (1996) Parallel solution of variational inequality problems with nonlinear source terms IMA J. Numer. Anal. 16 31 - 45    DOI : 10.1093/imanum/16.1.31
Lions J.L. (1999) Parallel algorithms for the solution of variational inequalities Interfaces Free Bound. 1 13 - 16
Reich S. (1979) Weak convergence theorems for nonexpansive mappings in Banach spaces J. Math. Anal. Appl. 67 274 - 276    DOI : 10.1016/0022-247X(79)90024-6
Thakur B.S. , Khan M.S. , Kang S.M. (2013) Existence and approximation of solutions for system of generalized mixed variational inequalities, Fixed Point Theory Appl. 2013 (108)
Verma R.U. (1999) On a new system of nonlinear variational inequalities and associated iterative algorithms Math. Sci. Res. 3 65 - 68
Weng X.L. (1991) Fixed point iteration for local strictly pseudocontractive mapping Proc. Amer. Math. Soc. 113 727 - 731    DOI : 10.1090/S0002-9939-1991-1086345-8
Xu H.K. (1991) Inequalities in Banach spaces with applications Nonlinear Anal. 16 1127 - 1138    DOI : 10.1016/0362-546X(91)90200-K
Yang H. , Zhou L. , Li Q. (2010) A parallel projection method for a system of nonlinear variationalinequalities Appl. Math. Comput. 217 1971 - 1975    DOI : 10.1016/j.amc.2010.06.053
Yao Y. , Liou Y.C. , Kang S.M. , Yu Y. (2011) Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces Nonlinear Anal. 74 6024 - 6034    DOI : 10.1016/j.na.2011.05.079
Jeong J.U. (2012) Iterative algorithm for a new system of generalized set-valued quasi-variational-like inclusions with (A, η)-accretive mappings in Banach spaces J. Appl. Math. & Informatics 30 935 - 950