Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 3, Article 114, 2006
GENERALIZED NONLINEAR MIXED QUASI-VARIATIONAL INEQUALITIES
INVOLVING MAXIMAL η-MONOTONE MAPPINGS
MAO-MING JIN
D EPARTMENT OF M ATHEMATICS
F ULING T EACHERS C OLLEGE
F ULING , C HONGQING 408003
P EOPLE ’ S R EPUBLIC OF C HINA
mmj1898@163.com
Received 27 July, 2004; accepted 09 May, 2006
Communicated by S.S. Dragomir
A BSTRACT. In this paper, we introduce and study a new class of generalized nonlinear mixed
quasi-variational inequalities involving maximal η-monotone mapping. Using the resolvent operator technique for maximal η-monotone mapping, we prove the existence of solution for this
kind of generalized nonlinear multi-valued mixed quasi-variational inequalities without compactness and the convergence of iterative sequences generated by the new algorithm. We also
discuss the convergence and stability of the iterative sequence generated by the perturbed iterative algorithm for solving a class of generalized nonlinear mixed quasi-variational inequalities.
Key words and phrases: Mixed quasi-variational inequality, Maximal η-monotone mapping, Resolvent operator, Convergence, stability.
2000 Mathematics Subject Classification. 49J40; 47H10.
1. I NTRODUCTION
In recent years, variational inequalities have been generalized and extended in many different directions using novel and innovative techniques. These have been used to study wider
classes of unrelated problems arising in optimization and control, economics and finance, transportation and electrical networks, operations research and engineering sciences in a general and
unified framework, see [1] – [15], [18] – [27] and the references therein. An important and useful generalization of variational inequality is called the variational inclusion. It is well known
that one of the most important and interesting problems in the theory of variational inequalities
is the development of an efficient and implementable algorithm for solving variational inequalities. For the past years, many numerical methods have been developed for solving various
classes of variational inequalities, such as the projection method and its variant forms, linear
approximation, descent, and Newton’s methods.
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
This work was supported by the National Natural Science Foundation of China (69903012) and the Educational Science Foundation of
Chongqing, Chongqing of China (021301).
144-04
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Recently, Huang and Fang [10] introduced a new class of maximal η-monotone mappings
and proved the Lipschitz continuity of the resolvent operator for maximal η-monotone mappings in Hilbert spaces. They also introduced and studied a new class of generalized variational
inclusions involving maximal η-monotone mappings and constructed a new algorithm for solving this class of generalized variational inclusions by using the resolvent operator technique for
maximal η-monotone mappings.
The main purpose of this paper is to introduce and study a new class of generalized nonlinear
mixed quasi-variational inequalities involving maximal η-monotone mappings. Using the resolvent operator technique for maximal η-monotone mappings, we prove the existence of a solution
for this kind of generalized nonlinear multivalued mixed quasi-variational inequalities without
compactness and the convergence of iterative sequences generated by the new algorithm. We
also discuss the convergence and stability of the iterative sequence generated by the perturbed
iterative algorithm for solving a class of generalized nonlinear mixed quasi-variational inequalities. The results shown in this paper improve and extend the previously known results in this
area.
2. P RELIMINARIES
Let H be a real Hilbert space endowed with a norm k·k and an inner product h·, ·i, respectively. Let 2H , CB(H), and H(·, ·) denote the family of all the nonempty subsets of H, the
family of all the nonempty closed bounded subsets of H, and the Hausdorff metric on CB(H),
respectively. Let η, N : H × H → H be two single-valued mappings with two variables and
g : H → H be a single-valued mapping. Let S, T, G : H → CB(H) be three multivalued
mappings and M : H × H → 2H be aT
multivalued mapping such that for each t ∈ H, M (·, t)
is maximal η-monotone with Ran(g) Dom M (·, t) 6= ∅. Now we consider the following
problem:
Find u ∈ H, x ∈ Su, y ∈ T u, and z ∈ Gu such that g(u) ∈ Dom(M (·, z)) and
(2.1)
0 ∈ N (x, y) + M (g(u), z)).
Problem (2.1) is called a generalized nonlinear multivalued mixed quasi-variational inequality.
Some special cases of the problem (2.1):
(I) If η(x, y) = x − y for all x, y in H and G is the identity mapping, then problem (2.1)
reduces to finding u ∈ H, x ∈ Su, y ∈ T u such that g(u) ∈ Dom(M (·, u)) and
(2.2)
0 ∈ N (x, y) + M (g(u), u).
Problem (2.2) is called the multivalued quasi-variational inclusion, which was studied
by Noor [18] – [22].
(II) If S, T are single-valued mappings and G is the identity mapping, then problem (2.1) is
equivalent to finding u ∈ H such that g(u) ∈ Dom(M (·, u)) and
(2.3)
0 ∈ N (Su, T u) + M (g(u), u)).
Problem (2.3) is called a generalized nonlinear
Smixed quasi-variational inequality.
(III) If M (·, t) = ∂ϕ(·, t), where ϕS: H × H → R {+∞} is a functional such that for each
fixed t in H, ϕ(·, t) : H → R {+∞} is lower semicontinuous and η-subdifferentiable
on H, and ∂ϕ(·, t) denotes the η-subdifferential of ϕ(·, t), then problem (2.1) reduces
to the following problem:
Find u ∈ H, x ∈ Su and y ∈ T u such that
(2.4)
hN (x, y), η(v, g(u))i ≥ ϕ(g(u), z) − ϕ(v, z)
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M IXED Q UASI -VARIATIONAL I NEQUALITIES
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for all v in H, which which appears to be a new one. Furthermore, if N (x, y) = x − y
for all x, y in H, S, T are single-valued mappings and G is the identity mapping, then
problem (2.4) reduces to the general quasi-variational-like inclusion considered by Ding
and Luo [3].
(IV) If S, T : H → H are single-valued mappings, g is an identity mapping, N (x, y) = x−y
for all x, y in H, and M (·, t) = ∂ϕ for all t in H, where ∂ϕ denotes
S the η-subdifferential
of a proper convex lower semicontinuous function ϕ : H → R {+∞}, then problem
(2.1) reduces to the following problem:
Find u ∈ H such that
(2.5)
hSu − T u, η(v, u)i ≥ ϕ(u) − ϕ(v)
for all v in H, which is called the strongly nonlinear variational-like inclusion problem
considered by Lee et al. [15].
(V) If G is an identity
S mapping, η(x, y) = x − y and M (·, t) = ∂ϕ for each t ∈ H, where
ϕ : HT → R {+∞} is a proper convex lower semicontinuous function on H and
g(H) Dom(∂ϕ(·, t)) 6= ∅ for each t ∈ H and ∂ϕ(·, t) denotes the subdifferential of
function ϕ(·, t), then problem (2.1) reduces to finding u ∈ H, x ∈ Su and y ∈ T u such
that g(u) ∈ Dom(∂ϕ(·, t)) and
(2.6)
hN (x, y), v − g(u)i ≥ ϕ(g(u)) − ϕ(v)
for all v in H. Furthermore, if N (x, y) = x − y for all x, y in H, and g is an identity
mapping, then the problem (2.6) is equivalent to the set-valued nonlinear generalized
variational inclusion considered by Huang [6] and, in turn, includes the variational inclusions studied by Hassouni and Moudafi [5] and Kazmi [14] as special cases.
For a suitable choice of N, η, M, S, T, G, g, and for the space H, one can obtain a number
of known and new classes of variational inclusions, variational inequalities, and corresponding
optimization problems from the general set-valued variational inclusion problem (2.1). Furthermore, these types of variational inclusions enable us to study many important problems arising
in the mathematical, physical, and engineering sciences in a general and unified framework.
Definition 2.1. Let T be a selfmap of H, x0 ∈ H and let xn+1 = f (T, xn ) define an iteration
procedure which yields a sequence of points {xn }∞
6
n=0 in H. Suppose that {x ∈ H : T x = x} =
∗
∞
∅ and {xn }n=0 converges to a fixed point x of T . Let {yn } ⊂ H and let n = ||yn+1 −f (T, yn )||.
If lim n = 0 implies that lim yn = x∗ , then the iteration procedure defined by xn+1 =
n→∞
n→∞
f (T, xn ) is said to be T -stable or stable with respect to T .
Lemma 2.1 ([16]). Let {an }, {bn }, and {cn } be three sequences of nonnegative numbers satisfying the following conditions: there exists n0 such that
an+1 ≤ (1 − tn )an + bn tn + cn ,
P
P
for all n ≥ n0 , where tn ∈ [0, 1], ∞
lim bn = 0 and ∞
n=0 tn = ∞, n→∞
n=0 cn < ∞. Then
lim an = 0.
n→∞
Definition 2.2. A mapping g : H → H is said to be
(i) α-strongly monotone if there exists a constant α > 0 such that
hg(u1 ) − g(u2 ), u1 − u2 i ≥ αku1 − u2 k2 ,
for all ui ∈ H, i = 1, 2;
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M AO -M ING J IN
(ii) β-Lipschitz continuous if there exists a constant β > 0 such that
kg(u1 ) − g(u2 )k ≤ βku1 − u2 k,
for all ui ∈ H, i = 1, 2.
Definition 2.3. A multivalued mapping S : H → CB(H) is said to be
(i) H-Lipschitz continuous if there exists a constant γ > 0 such that
H(Su1 , Su2 ) ≤ γku1 − u2 k,
for all ui ∈ H, i = 1, 2;
(ii) strongly monotone with respect to the first argument of N (·, ·) : H × H → H, if there
exists a constant µ > 0 such that
hN (x1 , ·) − N (x2 , ·), u1 − u2 i ≥ µku1 − u2 k2 ,
for all xi ∈ Sui , ui ∈ H, i = 1, 2.
Definition 2.4. A mapping N (·, ·) : H ×H → H is said to be Lipschitz continuous with respect
to the first argument if there exists a constant ν > 0 such that
kN (u1 , ·) − N (u2 , ·)k ≤ νku1 − u2 k,
for all ui ∈ H, i = 1, 2.
In a similar way, we can define Lipschitz continuity of N (·, ·) with respect to the second
argument.
Definition 2.5. Let η : H × H → H be a single-valued mapping. A multivalued mapping
M : H → 2H is said to be
(i) η-monotone if
hx − y, η(u, v)i ≥ 0
for all u, v ∈ H, x ∈ M u, y ∈ M v;
(ii) strictly η-monotone if
hx − y, η(u, v)i ≥ 0
for all u, v ∈ H, x ∈ M u, y ∈ M v
and equality holds if and only if u = v;
(iii) strongly η-monotone if there exists a constant r > 0 such that
hx − y, η(u, v)i ≥ rku − vk2
for all u, v ∈ H, x ∈ M u, y ∈ M v;
(iv) maximal η-monotone if M is η-monotone and (I + λM )(H) = H for any λ > 0.
Remark 2.2.
(1) If η(u, v) = u − v for all u, v in H, then (i)-(iv) of Definition 2.5 reduce to the classical definitions of monotonicity, strict monotonicity, strong monotonicity, and maximal
monotonicity, respectively.
(2) Huang and Fang gave one example of maximal η-monotone mapping in [10].
Lemma 2.3 ([10]). Let η : H × H → H be strictly monotone and M : H → 2H be a maximal
η-monotone mapping. Then the following conclusions hold:
(1) hx−y, η(u, v)i ≥ 0 for all (v, y) ∈ Graph(M ) implies that (u, x) ∈ Graph(M ), where
Graph(M ) = {(u, x) ∈ H × H : x ∈ M u};
(2) the inverse mapping (I + λM )−1 is single-valued for any λ > 0.
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M IXED Q UASI -VARIATIONAL I NEQUALITIES
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Based on Lemma 2.3, we can define the resolvent operator for a maximal η-monotone mapping M as follows:
JρM (z) = (I + ρM )−1 (z)
(2.7)
for all z ∈ H,
where ρ > 0 is a constant and η : H × H → H is a strictly monotone mapping.
Lemma 2.4 ([10]). Let η : H × H → H be strongly monotone and Lipschtiz continuous with
constants δ > 0 and τ > 0, respectively. Let M : H → 2H be a maximal η-monotone mapping.
Then the resolvent operator JρM for M is Lipschitz continuous with constant τ /δ, i.e.,
τ
for all u, v ∈ H.
kJρM (u) − JρM (v)k ≤ ku − vk
δ
3. I TERATIVE A LGORITHMS
We first transfer problem (2.1) into a fixed point problem.
Lemma 3.1. For given u ∈ H, x ∈ Su, y ∈ T u, and z ∈ Gu, (u, x, y, z) is a solution of the
problem (2.1) if and only if
g(u) = JρM (·,z) (g(u) − ρN (x, y)),
(3.1)
M (·,z)
where Jρ
= (I + ρM (·, z))−1 and ρ > 0 is a constant.
M (·,u)
Proof. This directly follows from the definition of Jρ
.
Remark 3.2. Equality (3.1) can be written as
u = (1 − λ)u + λ[u − g(u) + JρM (·,z) (g(u) − ρN (x, y))],
where 0 < λ ≤ 1 is a parameter and ρ > 0 is a constant. This fixed point formulation enables
us to suggest the following iterative algorithm for problem (2.1) as follows:
Algorithm 3.1. Let η, N : H × H → H, g : H → H be single-valued mappings and S, T, G :
H → CB(H) be multivalued mappings. Let M : H × H → 2H be such that for each fixed
t ∈ H, M (·, t) : H → 2H is a maximal η-monotone mapping satisfying g(u) ∈ Dom(M (·, z)).
For given λ ∈ [0, 1], u0 ∈ H, x0 ∈ Su0 , y0 ∈ T u0 and z0 ∈ Gu0 , let
u1 = (1 − λ)u0 + λ u0 − g(u0 ) + JρM (·,z0 ) (g(u0 ) − ρN (x0 , y0 )) .
By Nadler [17], there exist x1 ∈ Su1 , y1 ∈ T u1 and z1 ∈ Gu1 such that
kx0 − x1 k ≤ (1 + 1)H(Su0 , Su1 ),
ky0 − y1 k ≤ (1 + 1)H(T u0 , T u1 ),
kz0 − z1 k ≤ (1 + 1)H(Gu0 , Gu1 ).
Let
u2 = (1 − λ)u1 + λ u1 − g(u1 ) + JρM (·,z1 ) (g(u1 ) − ρN (x1 , y1 )) .
By induction, we can obtain sequences {un }, {xn }, {yn } and {zn } satisfying

h
i
M (·,zn )


u
=
(1
−
λ)u
+
λ
u
−
g(u
)
+
J
(g(u
)
−
ρN
(x
,
y
))
,
ρ
n+1
n
n
n
n
n
n


−1
xn ∈ Sun , kxn − xn+1 k ≤ (1 + (1 + n) )H(Sun , Sun+1 ),
(3.2)

yn ∈ T un , kyn − yn+1 k ≤ (1 + (1 + n)−1 )H(T un , T un+1 ),



zn ∈ gun , kzn − zn+1 k ≤ (1 + (1 + n)−1 )H(Gun , Gun+1 ),
for n = 1, 2, 3, . . . , where 0 < λ ≤ 1 and ρ > 0 are both constants.
Now we construct a new pertured iterative algorithm for solving the generalized nonlinear
mixed quasi-variational inequality (2.3) as follows:
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Algorithm 3.2. Let η, N : H × H → H and S, T : H → H be single-valued mappings. Let
M : H × H → 2H be such that for each fixed t ∈ H, M (·, t) : H → 2H is a maximal ηmonotone mapping satisfying g(u) ∈ Dom(M (·, u)). For given u0 ∈ H, the perturbed iterative
sequence {un } is defined by

M (·,v )
 un+1 = (1 − αn )un + αn [vn − g(vn ) + Jρ n (g(vn ) − ρN (Svn , T vn ))] + αn en ,
(3.3)

M (·,un )
vn = (1 − βn )un + βn [un − g(un ) + Jρ
(g(un ) − ρN (Sun , T un ))] + βn fn ,
for n = 0, 1, 2, . . . , where {en } and {fn } are two sequences of the elements of H introduced
to take into account possible inexact computation and the sequences {αn }, {βn } satisfy the
following conditions:
0 ≤ αn , βn ≤ 1(n ≥ 0),
and
∞
X
αn = ∞.
n=0
Let {yn } be any sequence in H and define {n } by

h
n

n = yn+1 − (1 − αn )yn + αn xn − g(xn )



i
o


M (·,xn )
+Jρ
(g(xn ) − ρN (Sxn , T xn )) + αn en ,
(3.4)



h
i


 xn = (1 − βn )yn + βn yn − g(yn ) + JρM (·,yn ) (g(yn ) − ρN (Syn , T yn )) + βn fn ,
for n = 0, 1, 2, . . . .
4. M AIN R ESULTS
In this section, we first prove the existence of a solution of problem (2.1) and the convergence
of an iterative sequence generated by Algorithm 3.1.
Theorem 4.1. Let η : H × H → H be strongly monotone and Lipschitz continuous with
constants δ and τ , respectively. Let S, T, G : H → CB(H) be H-Lipschitz continuous with
constants α, β, γ, respectively, g : H → H be µ-Lipschitz continuous and ν-strongly monotone.
Let N : H ×H → H be Lipschitz continuous with respect to the first and second arguments with
constants ξ and ζ, respectively, and S : H → CB(H) be strongly monotone with respect to the
first argument of N (·, ·) with constant r. Let M : H × H → 2H be a multivalued mapping such
that for each fixed t ∈ H, M (·, t) is maximal η-monotone. Suppose that there exist constants
ρ > 0 and κ > 0 such that for each x, y, z ∈ H,
M (·,x)
Jρ
(z) − JρM (·,y) (z) ≤ κkx − yk,
(4.1)
and
(4.2)
 

ρ −




τ r−δ(1−h)ζβ τ (ξ 2 α2 −ζ 2 β 2 ) √
<
[τ r−δ(1−h)ζβ]2 −(ξ 2 α2 −ζ 2 β 2 )(τ 2 −δ 2 (1−h)2 )
,
τ (ξ 2 α2 −ζ 2 β 2 )
p
τ r > δ(1 − h)ζβ + (ξ 2 α2 − ζ 2 β 2 )(τ 2 − δ 2 (1 − h)2 ),





p

h = (1 + δτ −1 ) 1 − 2ν + µ2 + κγ, ρτ ζβ < δ(1 − h),
ξα > ζβ,
h < 1.
Then the iterative sequences {un }, {xn }, {yn } and {zn } generated by Algorithm 3.1 converge
strongly to u∗ , x∗ , y ∗ and z ∗ , respectively and (u∗ , x∗ , y ∗ , z ∗ ) is a solution of problem (2.1).
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M IXED Q UASI -VARIATIONAL I NEQUALITIES
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Proof. It follows from (3.2), (4.1) and Lemma 2.4 that
kun+1 − un k
= (1 − λ)(un − un−1 ) + λ [un − un−1 − (g(un ) − g(un−1 ))
+JρM (·,zn ) (g(un ) − ρN (xn , yn )) − JρM (·,zn−1 ) (g(un−1 ) − ρN (xn−1 , yn−1 )) ≤ (1 − λ)kun − un−1 k + λkun − un−1 − (g(un+1 ) − g(un ))k
+ λ JρM (·,zn ) (g(un ) − ρN (xn , yn )) − JρM (·,zn ) (g(un−1 ) − ρN (xn−1 , yn−1 ))
+ λ JρM (·,zn ) (g(un−1 ) − ρN (xn−1 , yn−1 ))
− JρM (·,zn−1 ) (g(un−1 ) − ρN (xn−1 , yn−1 ))
≤ (1 − λ)kun − un−1 k + λkun − un−1 − (g(un ) − g(un−1 ))k
τ
+ λ kg(un ) − g(un−1 ) − ρ(N (xn , yn ) − N (xn−1 , yn−1 ))k + λκkzn − zn−1 k
δ
τ
≤ (1 − λ)kun − un−1 k + λ 1 +
kun − un−1 − (g(un ) − g(un−1 ))k
δ
τ
+ λ kun − un−1 − ρ(N (xn , yn ) − N (xn−1 , yn ))k
δ
τ
(4.3)
+ λρ kN (xn−1 , yn ) − N (xn−1 , yn−1 )k + λκkzn − zn−1 k.
δ
Since g is strongly monotone and Lipschitz continuous, we obtain
kun − un−1 − (g(un ) − g(un−1 ))k2
= kun − un−1 k2 − 2hun − un−1 , g(un ) − g(un−1 )i + kg(un ) − g(un−1 )k2
(4.4)
≤ (1 − 2ν + µ2 )kun − un−1 k2 .
Since S is H-Lipschitz continuous and strongly monotone with respect to the first argument of
N (·, ·) and N is Lipschitz continuous with respect to the first argument, we have
kun − un−1 − ρ(N (xn , yn ) − N (xn−1 , yn ))k2
= kun − un−1 k2 − 2ρhun − un−1 , N (xn , yn ) − N (xn−1 , yn )i
+ ρ2 kN (xn , yn ) − N (xn−1 , yn )k2
(4.5)
≤ (1 − 2ρr + ρ2 ξ 2 (1 + n−1 )2 α2 )kun − un−1 k2 .
Further, since T, G are H-Lipschitz continuous and N is Lipschitz continuous with respect to
the second argument, we get
(4.6)
kN (xn−1 , yn ) − N (xn−1 , yn−1 )k ≤ ζkyn − yn−1 k ≤ ζβ(1 + n−1 )kun − un−1 k,
(4.7)
kzn − zn−1 k ≤ γ(1 + n−1 )kun − un−1 k.
By (4.3) – (4.7), we obtain
p
kun − un−1 k ≤ (1 − λ + λ(1 + τ δ −1 ) 1 − 2ν + µ2
p
+ λτ δ −1 1 − 2ρr + ρ2 ξ 2 (1 + n−1 )2 α2
+ λρτ δ −1 ζβ(1 + n−1 ) + λκγ(1 + n−1 )
= (1 − λ + λhn + λtn (ρ))kun − un−1 k
(4.8)
= θn kun − un−1 k,
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where
p
hn = (1 + τ δ −1 ) 1 − 2ν + µ2 + κγ(1 + n−1 ),
p
tn (ρ) = τ δ −1 1 − 2ρr + ρ2 ξ 2 (1 + n−1 )2 α2 + ρτ δ −1 ζβ(1 + n−1 ) and
θn = 1 − λ + λhn + λtn (ρ).
Letting θ = 1 − λ + λh + λt(ρ), where
p
h = (1 + τ δ −1 ) 1 − 2ν + µ2 + κγ and
t(ρ) = τ δ −1
p
1 − 2ρr + ρ2 ξ 2 α2 + ρτ δ −1 ζβ,
we have that hn → h, tn (ρ) → t(ρ) and θn → θ as n → ∞. It follows from condition (4.2) that
θ < 1. Hence θn < 1 for n sufficiently large. Therefore, (4.8) implies that {un } is a Cauchy
sequence in H and so we can assume that un → u∗ ∈ H as n → ∞. By the Lipschitz continuity
of S, T and G we obtain
kxn − xn−1 k ≤ (1 + (1 + n)−1 )H(Sun , Sun−1 ) ≤ α(1 + (1 + n)−1 )kun − un−1 k,
kyn − yn−1 k ≤ (1 + (1 + n)−1 )H(T un , T un−1 ) ≤ β(1 + (1 + n)−1 )kun − un−1 k,
kzn − zn−1 k ≤ (1 + (1 + n)−1 )H(Gun , Gun−1 ) ≤ γ(1 + (1 + n)−1 )kun − un−1 k.
It follows that {xn }, {yn } and {zn } are also Cauchy sequences in H. We can assume that
xn → x∗ , yn → y ∗ and zn → z ∗ , respectively. Note that for xn ∈ Sun , we have
d(x∗ , Su∗ ) ≤ kx∗ − xn k + d(xn , Su∗ )
≤ kx∗ − xn k + H(Sun , Su∗ )
≤ kx∗ − xn k + αkun − u∗ k → 0,
as n → ∞. Hence we must have x∗ ∈ Su∗ . Similarly, we can show that y ∗ ∈ T u∗ and
z ∗ ∈ Gu∗ . From
un+1 = (1 − λ)un + λ un − g(un ) + JρM (·,zn ) (g(un ) − ρN (xn , yn )) ,
it follows that
∗
g(u∗ ) = JρM (·,z ) (g(u∗ ) − ρN (x∗ , y ∗ )).
By Lemma 3.1, (u∗ , x∗ , y ∗ , z ∗ ) is a solution of problem (2.1). This completes the proof.
Remark 4.2. For an appropriate and suitable choice of the mappings η, N, S, T, G, g, M and
the space H, we can obtain several known results in [1], [3], [5] – [8], [14], [18] – [22], [24] –
[26] as special cases of Theorem 4.1.
Now we prove the convergence and stability of the iterative sequence generated by the Algorithm 3.2.
Theorem 4.3. Let η : H × H → H be strongly monotone and Lipschitz continuous with
constants δ and τ , respectively. Let S, T : H → H be Lipschitz continuous with constants
α, β, respectively, g : H → H be µ-Lipschitz continuous and ν-strongly monotone. Let N :
H × H → H be Lipschitz continuous with respect to the first and second arguments with
constants ξ and ζ, respectively, and S : H → H be strongly monotone with respect to the first
argument of N (·, ·) with constant r. Let M : H × H → 2H be a multivalued mapping such that
for each fixed t ∈ H, M (·, t) is maximal η-monotone. Suppose that there exist constants ρ > 0
and κ > 0 such that for each x, y, z ∈ H,
M (·,x)
Jρ
(4.9)
(z) − JρM (·,y) (z) ≤ κkx − yk,
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M IXED Q UASI -VARIATIONAL I NEQUALITIES
and
(4.10)
 

ρ −




τ r−δ(1−h)ζβ τ (ξ 2 α2 −ζ 2 β 2 ) √
<
9
[τ r−δ(1−h)ζβ]2 −(ξ 2 α2 −ζ 2 β 2 )(τ 2 −δ 2 (1−h)2 )
,
τ (ξ 2 α2 −ζ 2 β 2 )
p
τ r > δ(1 − h)ζβ + (ξ 2 α2 − ζ 2 β 2 )(τ 2 − δ 2 (1 − h)2 ), ξα > ζβ,





p

h = (1 + δτ −1 ) 1 − 2ν + µ2 + κ, ρτ ζβ < δ(1 − h), h < 1.
If lim ken k = 0, lim kfn k = 0, then
n→∞
n→∞
(I) The sequence {un } defined by Algorithm 3.2 converges strongly to the unique solution
u∗ P
of problem (2.3).
(II) If ∞
lim yn = u∗ .
n=0 n < ∞, then n→∞
(III) If lim yn = u∗ , then lim n = 0.
n→∞
n→∞
Proof. (I) It follows from Theorem 4.1 that there exists u∗ ∈ H which is a solution of problem
(2.3) and so
(4.11)
∗
g(u∗ ) = JρM (·,u ) (g(u∗ ) − ρN (Su∗ , T u∗ )).
From (4.9), (4.11) and Algorithm 3.2, it follows that
kun+1 − u∗ k
h
∗
= (1 − αn )(un − u ) − αn vn − u∗ − (g(vn ) − g(u∗ ))
+ JρM (·,vn ) (g(vn ) − ρN (Svn , T vn ))
i
∗
− JρM (·,u ) (g(u∗ ) − ρN (Su∗ , T u∗ )) + αn en ≤ (1 − αn )kun − u∗ k + αn kvn − u∗ − (g(vn ) − g(u∗ ))k + αn ken k
+ αn JρM (·,vn ) (g(vn ) − ρN (Svn , T vn )) − JρM (·,vn ) (g(u∗ ) − ρN (Su∗ , T u∗ ))
∗
+ αn JρM (·,vn ) (g(u∗ ) − ρN (Su∗ , T u∗ )) − JρM (·,u ) (g(u∗ ) − ρN (Su∗ , T u∗ ))
≤ (1 − αn )kun − u∗ k + αn kvn − u∗ − (g(vn ) − g(u∗ ))k + αn ken k
τ
+ αn kg(vn ) − g(u∗ ) − ρ(N (Svn , T vn ) − N (Su∗ , T u∗ ))k + αn κkvn − u∗ k
δ
τ
≤ (1 − αn )kun − u∗ k + αn 1 +
kun − u∗ − (g(vn ) − g(u∗ ))k + αn ken k
δ
τ
+ αn kvn − u∗ − ρ(N (Svn , T vn ) − N (Su∗ , T vn ))k
δ
τ
+ αn ρ kN (Su∗ , T vn ) − N (Su∗ , T u∗ )k + αn κkvn − u∗ k.
(4.12)
δ
By the Lipschitz continuity of N, S, T, g and the strong monotonicity of S and g, we obtain
(4.13)
(4.14)
(4.15)
kvn − u∗ − (g(vn ) − g(u∗ ))k2 ≤ (1 − 2ν + µ2 )kvn − u∗ k2 ,
kvn − u∗ − ρ(N (Svn , T vn ) − N (Su∗ , T vn ))k2 ≤ (1 − 2ρr + ρ2 ξ 2 α2 )kvn − u∗ k2 ,
kN (Su∗ , T vn ) − N (Su∗ , T u∗ ))k ≤ ζβkvn − u∗ k.
It follows from (4.12) – (4.15) that
(4.16)
kun+1 − u∗ k ≤ (1 − αn )kun − u∗ k + θαn kvn − u∗ k + αn ken k,
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10
M AO -M ING J IN
where
p
p
θ = κ + (1 + τ δ −1 ) 1 − 2ν + µ2 + τ δ −1 1 − 2ρr + ρ2 ξ 2 α2 + ρτ δ −1 ζβ.
Similarly, we have
(4.17)
kvn − u∗ k ≤ (1 − βn )kun − u∗ k + θβn kun − u∗ k + βn kfn k.
From (4.16) and (4.17), we have
kun+1 − u∗ k ≤ [1 − αn (1 − θ)(1 + θβn )]kun − u∗ k + αn βn θkfn k + αn ken k
Condition (4.10) implies that 0 < θ < 1, and so
(4.18)
kun+1 − u∗ k ≤ [1 − αn (1 − θ)]kun − u∗ k + αn (1 − θ)dn ,
where dn = (βn θkfn k + ken k)(1 − θ)−1 → 0, as n → ∞. It follows from (4.18) and Lemma
2.1 that un → u∗ as n → ∞.
Now we prove that u∗ is a unique solution of problem (2.3). In fact, if u is also a solution of
problem (2.3), then
g(u) = JρM (·,u) (g(u) − ρN (Su, T u)),
and, as the proof of (4.16), we have
ku∗ − uk ≤ θku∗ − uk,
where 0 < θ < 1 and so u∗ = u. This completes the proof of Conclusion (I).
Next we prove Conclusion (II). Using (3.4) we obtain
kyn+1 − u∗ k
≤ yn+1 − (1 − αn )yn + αn xn − g(xn )
(4.19)
+ JρM (·,xn ) (g(xn ) − ρN (Sxn , T xn )) + αn en + (1 − αn )yn + αn xn − g(xn )
+ JρM (·,xn ) (g(xn ) − ρN (Sxn , T xn )) + αn en − u∗ = (1 − αn )yn + αn xn − g(xn )
+ JρM (·,xn ) (g(xn ) − ρN (Sxn , T xn )) + αn en − u∗ + n .
As the proof of inequality (4.18), we have
(4.20) (1 − αn )yn + αn xn − g(xn ) + JρM (·,xn ) (g(xn ) − ρN (Sxn , T xn )) + αn en − u∗ ≤ (1 − αn (1 − θ))kyn − u∗ k + αn (1 − θ)dn .
It follows from (4.19) and (4.20) that
kyn+1 − u∗ k ≤ (1 − αn (1 − θ))kyn − u∗ k + αn (1 − θ)dn + n .
P
P∞
Since ∞
n=0 n < ∞, dn → 0, and
n=0 αn < ∞. It follows that (4.21) and Lemma 2.1 that
∗
lim yn = u .
n→∞
Now we prove Conclusion (III). Suppose that lim yn = u∗ . Then we have
n→∞
n = yn+1 − (1 − αn )yn + αn xn
− g(xn ) + JρM (·,xn ) (g(xn ) − ρN (Sxn , T xn )) + αn en ≤ kyn+1 − u∗ k + (1 − αn )yn + αn xn
− g(xn ) + JρM (·,xn ) (g(xn ) − ρN (Sxn , T xn )) + αn en − u∗ (4.21)
≤ kyn+1 − u∗ k + (1 − αn (1 − θ))kyn − u∗ k + αn (1 − θ)dn → 0
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M IXED Q UASI -VARIATIONAL I NEQUALITIES
11
as n → ∞. This completes the proof.
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