Available online at http://scik.org Adv. Inequal. Appl. 2014, 2014:4 ISSN: 2050-7461
A GENERALIZED SYSTEM OF VARIATIONAL INEQUALITIES IN A BANACH SPACE
H. ZHANG
Department of Mathematics and statistics, York University, Toronto, M3J 1P3, Canada
Copyright c2014 H. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract. In this paper, we investigate a generalized system for nonlinear variational inequalities based on a two-step algorithm. Convergence theorems of solutions are established in the framework of Banach spaces. Keywords: variational inequality; Banach space; nonexpansive mapping; solution.
2000 AMS Subject Classification:47H05, 47H09, 47H10
1. Introduction-Preliminaries
Systems of variational inequalities have significant applications in various fields of mathe-matics, physics, economics, and engineering sciences. The solvability of systems of variational inequalities based on iterative methods has been extensively investigated; see [1-17] and the references therein.
LetBbe a Banach space and letB∗be the dual space ofB. Leth·,·idenote the duality pairing ofB∗andB. We consider the following the variational inequality: Findx∈Ksuch that
hT x,y−xi ≥0, ∀y∈K, (1.1) E-mail address: [email protected]
Received August 25, 2013
whereKis a nonempty, closed and convex subset ofBandT :K→B∗. A pointx0∈Kis called a solution of the variational inequality (1.1) if, for everyy∈K,hT x0,y−x0i ≥0. The variation-al inequvariation-ality (1.1) has been intensively considered due to its various applications in operations research, economic equilibrium and engineering design. WhenT has some monotonicity, many iterative methods for solving the variational inequality (1.1) have been developed. Recently, applying the generalized projection operator in uniformly convex and uniformly smooth Ba-nach spaces, Li [1] established the convergence of a Mann type iterative scheme for variational inequalities in compact subsets of Banach spaces. Fan [2] also studied the same problem in noncompact subsets of Banach spaces and extended Li’s results to some extent.
In this paper, we consider, based on the generalized projection methods, the approximate solvability of a system of nonlinear variational inequalities in the framework of Banach spaces. Our results obtained in this paper main generalize the results of Li [1], Fan [2], and Verma [3].
LetX,Y be Banach spaces, T :D(T)⊂X →Y, the operatorT is said to be compact if it is continuous and maps the bounded subsets ofD(T)onto the relatively compact subsets ofY; the operatorT is said to be weak to norm continuous if it is continuous from the weak topology of
X to the strong topology ofY . We denote byJ:B→2B∗ the normalized duality mapping from
Bto 2B∗, defined by
J(x):={y∈B∗:hy,xi=kyk2=kxk2}, ∀x∈B.
It is well known thatJ is single-valued and norm to weak∗ continuous if Bis smooth. In [4], Alber introduced the functionalV :B∗×B→Rdefined by
V(ϕ,x) =kϕk2−2hϕ,xi+kxk2,
whereϕ ∈B∗andx∈B. It is easy to see that
(kϕk − kxk)2≤V(ϕ,x)≤(kϕk+kxk)2.
Thus the functionalV :B∗×B→R+ is nonnegative.
point of the functionalV(ϕ,x), i.e., a solution to the minimization problem
V(ϕ,πKϕ) = inf
y∈KV(ϕ,y). (1.2)
It was proved that the generalized projection operator πK :B∗→K is continuous, if B is a
reflexive, strictly convex and smooth Banach space. In what follows, we let Bbe a uniformly convex and uniformly smooth Banach space, unless otherwise specified. Then B is reflexive and strictly convex. Thus the generalized projection operatorπK :B∗→K is continuous. The
functionalV2:B×B→Ris defined by
V2(x,y) =V(Jx,y),∀x,y∈B.
The following properties of the operatorsπK,V are useful for our paper. (See, for example,
[1,2,11].)
(1)V :B∗×B→Ris continuous. (2)V(ϕ,x) =0 if and only ifϕ=J(x).
(3)V(JπKϕ,x)≤V(ϕ,x)for allϕ∈B∗andx∈B.
(4) The operatorπK isJfixed in each pointx∈K, i.e.,πK(Jx) =x.
(5) If the Banach spaceBis uniformly smooth, then for allϕ1,ϕ2∈B∗, we have
kπKϕ1−πKϕ2k ≤2R1g−B1(kϕ1−ϕ2k/R1),
whereR1= (kπKϕ1k2+kπKϕ2k2)1/2, andg−B1is the inverse function togB that is defined by
the modulus of smoothness for an uniformly smooth Banach space.
(6) If B is smooth, then operator πK :B∗→K is single valued and for any given ϕ ∈B∗,
hϕ−Jϕ,e ϕe−xi ≥0, ∀x∈K⇔ϕe=πKϕ. In 1996, Alber proved the following theorem.
Theorem A[6]Let B be a reflexive, strictly convex and smooth Banach space with dual B∗. Let T be an arbitrary operator from Banach space B to B∗, α an arbitrary fixed positive number.
Then the point x∈K⊂B is a solution of variational inequality (1.1) if and only if x is a solution
of the operator equation in B
In this paper, we investigate the following system of nonlinear variational inequality (SNVI) problem:
Findx∗,y∗∈K such that
hsT1y∗+Jx∗−Jy∗,x−x∗i ≥0, ∀x∈K,s>0, (1.3) htT2x∗+Jy∗−Jx∗,x−x∗i ≥0, ∀x∈K,t>0, (1.4) whereTi:K→B∗are nonlinear mappings for i=1,2 andJ:B→B∗ is a normalized duality
mapping.
One can easily see the SNVI problems (1.3) and (1.4) are equivalent to the following projec-tion formulas:
x∗=πK[Jy∗−sT1y∗], s>0, (1.5)
y∗=πK[Jx∗−tT2x∗], t >0, (1.6)
Next, we consider some special classes of the SNVI problems (1.3) and (1.4) as follows: (I) Ift=0 then the SNVI problems (1.3) and (1.4) collapse to the following NVI: Findx∗∈Ksuch that
hsT x∗,x−x∗i ≥0, ∀x∈K,s>0, (1.7) which considered by Fan [9].
(II) IfT1=T2=T,then the SNVI problems (1.3) and (1.4) reduce to the following nonlinear system of variational inequality problem (SNVI):
Findx∗,y∗∈K such that
hsTy∗+Jx∗−Jy∗,x−x∗i ≥0, ∀x∈K,s>0, (1.8) htT x∗+Jy∗−Jx∗,x−x∗i ≥0, ∀x∈K,t>0, (1.9) whereT :K→B∗be a nonlinear mapping.
(III) IfKis a closed convex cone ofB, then the SNVI problems (1.3) and (1.4) are equivalent to the following system (SNC) of nonlinear complementarity problems:
Findx∗,y∗∈K such that
hsT1y∗+Jx∗−Jy∗,x∗i=0, s>0, (1.10) htT2x∗+Jy∗−Jx∗,x∗i=0, t>0, (1.11) whereK∗is the polar cone toCdefined by
K∗={f ∈B:hf,xi ≥0, ∀x∈K}.
(IV) IfB=H, a Hilbert space then the SNVI problems (1.3) and (1.4) reduce to the following nonlinear system of variational inequality problem (SNVI):
Findx∗,y∗∈K such that
hsT1y∗+x∗−y∗,x−x∗i ≥0, ∀x∈K,s>0, (1.12)
htT2x∗+y∗−x∗,x−x∗i ≥0, ∀x∈K,t >0, (1.13)
whereTi:K→B∗are nonlinear mappings fori=1,2.
(V) IfB=H andT1=T2, a Hilbert space then the SNVI problems (1.3) and (1.4) reduce to the following nonlinear system of variational inequality problem (SNVI):
Findx∗,y∗∈K such that
hsTy∗+x∗−y∗,x−x∗i ≥0, ∀x∈K,s>0, (1.14)
htT x∗+y∗−x∗,x−x∗i ≥0, ∀x∈K,t >0, (1.15) whereT :K→B∗is a nonlinear mapping.
2. Algorithms
Algorithm 2.1. For anyx0,y0∈K, compute the sequences{xn},{yn}and{zn}by the iterative
process:
yn=πK[Jxn−tT2xn],
xn+1= (1−αn)xn+αnπK[Jyn−sT1yn],
(2.1)
where{αn}is a sequence in[0,1]for alln≥0.
Algorithm 2.2. For anyx0∈K, compute the sequences{xn}by the iterative process:
xn+1= (1−αn)xn+αnπK[Jxn−sT xn], (2.2)
where{αn}is a sequence in[0,1]for alln≥0.
(II) IfT1=T2=T in Algorithm 2.1, then we have the following:
Algorithm 2.3. For any x0,y0 ∈K, compute the sequences {xn} and {yn} by the iterative process:
yn=πK[Jxn−tT xn],
xn+1= (1−αn)xn+αnπK[Jyn−sTyn],
(2.3)
where{αn}is a sequence in[0,1]for alln≥0.
(III) IfJ=I, the identity mapping in Algorithm 2.1, then we have the following:
Algorithm 2.4. For any x0,y0 ∈K, compute the sequences {xn} and {yn} by the iterative process:
yn=PK[xn−tT2xn],
xn+1= (1−αn)xn+αnPK[yn−sT2yn],
(2.4)
wherePkis a metric projection from a Hilbert space to its closed convex subsetKand{αn}is a
sequence in[0,1]for alln≥0.
(IV) IfJ=IandT1=T2, the identity mapping in Algorithm 2.1, then we have the following:
Algorithm 2.5. For any x0,y0 ∈K, compute the sequences {xn} and {yn} by the iterative process:
yn=PK[xn−tT xn],
xn+1= (1−αn)xn+αnPK[yn−sTyn],
(2.5)
wherePkis a metric projection from a Hilbert space to its closed convex subsetKand{αn}is a
sequence in[0,1]for alln≥0.
In order to prove our main results, we need the following lemmas and definitions.
x1,x2and y∈Br(0):={x∈B:kxk ≤r}and for anyα ∈[0,1],the following inequality holds:
V2(αx1+ (1−α)x2,y)≤αV2(x1,y) + (1−α)V2(x2,y)−α(1−α)g(kx1−x2k).
Lemma 2.4 [8]. Let B be a real Banach space and J : B→B∗ be the normalized duality mapping, then for any x,y∈B the following holds:
kx+yk2≤ kxk2+2hy,J−1(x+y)i, ∀x,y∈B∗.
3. Convergence Theorems
Theorem 3.1. Let B be an uniformly convex and uniformly smooth Banach space and let K be a closed and convex subset of B. Let T1 and T2:K→B∗be mappings on K such that J−sT1
and J−tT2are compact and
hT1x,J∗[Jx−sT1x]i ≥0and hT2x,J∗[Jx−tT2x]i ≥0, (3.1)
for all x∈K, where J∗=J−1is the normalized duality mapping on B∗. Suppose that(x∗,y∗)∈
K×K is a solution to the SNVI problem (1.3)-(1.4). Assume that {xn} and {yn} are the
se-quences generated by Algorithm 2.1. If {αn} ⊂[0,1] satisfies condition0<a≤αn≤b<1. Then the sequences{xn}and{yn}converge strongly to x∗and y∗, respectively.
Proof.In view of Lemma 2.3, we have
V2(xn+1,u)≤(1−αn)V2(xn,u) +αnV2(πK[Jyn−sT1yn],u)
−αn(1−αn)g(kπK[Jyn−sT1yn]−xnk),∀u∈B
By the definition ofV andV2, we have
V2(πK[Jyn−sT1yn],u)
≤V(Jyn−sT1yn,u)
=kJyn−sT1ynk2+kuk2−2hJyn−sT1yn,ui
≤ kJynk2−2shT1yn,J∗(Jyn−sT1yn)i+kuk2−2hJyn−sT1yn,ui
=V(Jyn,u)−2shT1yn,J∗(Jyn−sT1yn)−ui
=V2(yn,u)−2shT1yn,J∗(Jyn−sT1yn)−ui.
(3.3)
In the same way, we obtain that
V2(yn,u) =V(JπK[Jxn−tT2xn],u)
≤V(Jxn−tT2xn,u)
=kJxn−tT2xnk2+kuk2−2hJxn−tT2xn,ui
≤ kJxnk2−2thT2xn,J∗(Jxn−tT2xn)i+kuk2−2hJxn−tT2xn,ui
=V(Jxn,u)−2thT2xn,J∗(Jxn−tT2xn)−ui =V2(xn,u)−2thT2xn,J∗(Jxn−tT2xn)−ui.
(3.4)
Combining (3.3) and (3.4) yields that
V2(πK[Jyn−sT1yn],u)
≤V2(xn,u)−2thT2xn,J∗(Jxn−tT2xn)−ui −2shT1yn,J∗(Jyn−sT1yn)−ui.
(3.5)
Substitute (3.5) into (3.2) yields that
V2(xn+1,u)≤V2(xn,u)−2αnthT2xn,J∗(Jxn−tT2xn)−ui
−2αnshT1yn,J∗(Jyn−sT1yn)−ui
−αn(1−αn)g(kπK[Jyn−sT1yn]−xnk).
(3.6)
Sinceu∈Bis arbitrary, we, without loss any generality, chooseu=θ, the zero element inB. It
follows from (3.6) and condition (3.1) that
Noticing that 0<a≤αn≤b<1 and taking the sum fori=1,2, . . . ,n,we arrive at a(1−b)
n
∑
i=1
g(kπK[Jyi−sT1yi]−xik)≤V2(x0,θ)−V2(xn+1,θ).
It follows fromV2:B×B→R+is nonnegative andV2(x0,θ)<∞that∑∞i=1g(kπK[Jyi−sT1yi]− xik)<∞.Therefore, we have limn→∞g(kπK[Jyn−sT1yn]−xnk) =0.Notice the properties ofg
implies that
lim
n→∞
kπK[Jyn−sT1yn]−xnk=0. (3.8)
Noticing thatV2(xn+1,θ)≤V2(xn,θ),we havekxn+1k ≤ kxnk, which implies that{xn}is
bound-ed, so are{yn}and{zn}. Since the sequence{yn}is bounded andJ−sT1is compact onK, then the sequence{Jyn−sT1yn}must have a subsequence{Jyni−sT1yni}, which converges to a point
f1∈B∗. By the continuity of the projection operatorπk, we have
lim
i→∞
πK[Jyni−sT1yni] =πK(f1). (3.9)
Letx∗=:πK(f1). On the other hand, we have
kxni−x
∗k ≤ kx
ni−πK[Jyni−sT1yni]k+kπK[Jyni−sT1yni]−x
∗k, (3.10)
which combines with (3.8) and (3.9) yields that limi→∞xni =x
∗. In virtue of arbitrary
subse-quence of{xn}having above property, we have
lim
n→∞
xn=x∗. (3.11)
It follows from (3.8), (3.11) and the continuity properties of the operatorsπK andJ−sT1that
x∗=πK[Jy∗−sT1y∗]. (3.12)
Since the sequence {xn} is bounded and J−tT2 is compact on K, then the sequence {Jxn− tT2xn} must have a subsequence {Jxni−tT2xni}, which converges to a point f2 ∈B∗. By the continuity of the projection operatorπk, we have
lim
i→∞
πK[Jxni−tT2xni] =πK(f2). (3.13)
Lety∗=:πK(f2). On the other hand, we havekyni−y
∗k ≤ ky
ni−πK[Jxni−tT2xni]k+kπK[Jxni−
tT2xni]−y∗k,which combines with (2.1) and (3.13) yields that limi→∞yni =y
∗.Since{y
arbitrary, we have limn→∞yn=y∗. It follows from (2.1) and the continuity properties of the
operatorsπK andJ−tT2thaty∗=πK[Jx∗−tT2x∗].This completes the proof.
Conflict of Interests
The author declares that there is no conflict of interests.
Acknowledgements
The author is grateful to the anonymous reviewer’s suggestions which improved the contents of the article.
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