Journal of Inequalities in Pure and
Applied Mathematics
GENERALIZED NONLINEAR MIXED QUASI-VARIATIONAL
INEQUALITIES INVOLVING MAXIMAL η-MONOTONE MAPPINGS
volume 7, issue 3, article 114,
2006.
MAO-MING JIN
Department of Mathematics
Fuling Teachers College
Fuling, Chongqing 408003
People’s Republic of China.
Received 27 July, 2004;
accepted 09 May, 2006.
Communicated by: S.S. Dragomir
EMail: mmj1898@163.com
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Victoria University
ISSN (electronic): 1443-5756
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Abstract
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 quasivariational inequalities.
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
2000 Mathematics Subject Classification: 49J40; 47H10.
Key words: Mixed quasi-variational inequality, Maximal η-monotone mapping, Resolvent operator, Convergence, stability.
This work was supported by the National Natural Science Foundation of China
(69903012) and the Educational Science Foundation of Chongqing, Chongqing of
China (021301).
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Iterative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References
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1.
Introduction
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.
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 Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
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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.
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
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2.
Preliminaries
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 a multivalued mapping
such that for each
T
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
0 ∈ N (x, y) + M (g(u), z)).
(2.1)
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].
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
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(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 mixed quasi-variational
inequality.
S
(III) If M (·, t) = ∂ϕ(·, t), where ϕ : H × H → R {+∞}
S 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)
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 quasivariational-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 the η-subdifferential
of a proper convex lower semicontinuous
S
function ϕ : H → R {+∞}, then problem (2.1) reduces to the following
problem:
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
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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 mapping,
S η(x, y) = x−y and M (·, t) = ∂ϕ for each t ∈
H, where ϕ : H → R T
{+∞} 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
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
(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 setvalued 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.
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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 }∞
n=0 in H.
Suppose that {x ∈ H : T x = x} 6= ∅ and {xn }∞
converges
to
a
fixed
point
n=0
∗
x of T . Let {yn } ⊂ H and let n = ||yn+1 − f (T, yn )||. If lim n = 0 implies
n→∞
that lim yn = x∗ , then the iteration procedure defined by xn+1 = f (T, xn ) is
n→∞
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;
(ii) β-Lipschitz continuous if there exists a constant β > 0 such that
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
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kg(u1 ) − g(u2 )k ≤ βku1 − u2 k,
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for all ui ∈ H, i = 1, 2.
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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
2
hN (x1 , ·) − N (x2 , ·), u1 − u2 i ≥ µku1 − u2 k ,
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
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
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for all u, v ∈ H, x ∈ M u, y ∈ M v;
J. Ineq. Pure and Appl. Math. 7(3) Art. 114, 2006
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(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.
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
Remark 1.
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 2.2 ([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};
−1
2. the inverse mapping (I + λM )
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is single-valued for any λ > 0.
J. Ineq. Pure and Appl. Math. 7(3) Art. 114, 2006
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Based on Lemma 2.2, we can define the resolvent operator for a maximal
η-monotone mapping M as follows:
(2.7)
JρM (z) = (I + ρM )−1 (z)
for all z ∈ H,
where ρ > 0 is a constant and η : H × H → H is a strictly monotone mapping.
Lemma 2.3 ([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.,
kJρM (u) − JρM (v)k ≤
τ
ku − vk
δ
for all u, v ∈ H.
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
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3.
Iterative Algorithms
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 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 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 )) .
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
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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

un+1 = (1


h − λ)un
i


M (·,z )

+λ un − g(un ) + Jρ n (g(un ) − ρN (xn , yn )) ,

(3.2)
xn ∈ Sun , kxn − xn+1 k ≤ (1 + (1 + n)−1 )H(Sun , Sun+1 ),



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:
Algorithm 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 η-monotone mapping satisfying g(u) ∈ Dom(M (·, u)).
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
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For given u0 ∈ H, the perturbed iterative sequence {un } is defined by
(3.3)

un+1 = (1 − αn )un + αn [vn − g(vn )



M (·,v )

+Jρ n (g(vn ) − ρN (Svn , T vn ))] + αn en ,



vn = (1 − βn )un + βn [un − g(un )



M (·,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
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
αn = ∞.
n=0
Let {yn } be any sequence in H and define {n } by

n
h

n = yn+1 − (1 − αn )yn + αn xn − g(xn )



i
o


M (·,xn )

+J
(g(x
)
−
ρN
(Sx
,
T
x
))
+
α
e
,

ρ
n
n
n
n
n

(3.4)
h



x
=
(1
−
β
)y
+
β
yn − g(yn )

n
n
n
n


i


M (·,yn )

+Jρ
(g(yn ) − ρN (Syn , T yn )) + βn fn ,
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for n = 0, 1, 2, . . . .
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4.
Main Results
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 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)
 √
[τ r−δ(1−h)ζβ]2 −(ξ 2 α2 −ζ 2 β 2 )(τ 2 −δ 2 (1−h)2 )

τ r−δ(1−h)ζβ 
,
ρ − τ (ξ2 α2 −ζ 2 β 2 ) <

τ (ξ 2 α2 −ζ 2 β 2 )







 τ r > δ(1 − h)ζβ
p
(ξ 2 α2 − ζ 2 β 2 )(τ 2 − δ 2 (1 − h)2 ), ξα > ζβ,
+





p


−1

h
=
(1
+
δτ
)
1 − 2ν + µ2



+κγ, ρτ ζβ < δ(1 − h), h < 1.
Generalized Nonlinear Mixed
Quasi-Variational Inequalities
Involving Maximal η-Monotone
Mappings
Mao-Ming Jin
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Then the iterative sequences {un }, {xn }, {yn } and {zn } generated by Algorithm
1 converge strongly to u∗ , x∗ , y ∗ and z ∗ , respectively and (u∗ , x∗ , y ∗ , z ∗ ) is a
solution of problem (2.1).
Proof. It follows from (3.2), (4.1) and Lemma 2.3 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
δ
+ λκkzn − zn−1 k
≤ (1 − λ)kun − un−1 k
τ
kun − un−1 − (g(un ) − g(un−1 ))k
+λ 1+
δ
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(4.3)
τ
+ λ kun − un−1 − ρ(N (xn , yn ) − N (xn−1 , yn ))k
δ
τ
+ 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
2
kun − un−1 − ρ(N (xn , yn ) − N (xn−1 , yn ))k
= 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 .
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Further, since T, G are H-Lipschitz continuous and N is Lipschitz continuous
with respect to the second argument, we get
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(4.6)
kN (xn−1 , yn ) − N (xn−1 , yn−1 )k ≤ ζkyn − yn−1 k
≤ ζβ(1 + n−1 )kun − un−1 k,
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(4.7)
kzn − zn−1 k ≤ γ(1 + n−1 )kun − un−1 k.
By (4.3) – (4.7), we obtain
(4.8)
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
= θn kun − un−1 k,
Generalized Nonlinear Mixed
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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
p
t(ρ) = τ δ −1 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,
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(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
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g(u∗ ) = JρM (·,z ) (g(u∗ ) − ρN (x∗ , y ∗ )).
Quit
By Lemma 3.1, (u∗ , x∗ , y ∗ , z ∗ ) is a solution of problem (2.1). This completes
the proof.
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∗
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Remark 3. 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 2.
Theorem 4.2. 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
(z) − JρM (·,y) (z) ≤ κkx − yk,
(4.9)
and
(4.10)
 √
[τ r−δ(1−h)ζβ]2 −(ξ 2 α2 −ζ 2 β 2 )(τ 2 −δ 2 (1−h)2 )

τ r−δ(1−h)ζβ 
,
ρ − τ (ξ2 α2 −ζ 2 β 2 ) <

τ (ξ 2 α2 −ζ 2 β 2 )







 τ r > δ(1 − h)ζβ
p
(ξ 2 α2 − ζ 2 β 2 )(τ 2 − δ 2 (1 − h)2 ), ξα > ζβ,
+





p


−1

h
=
(1
+
δτ
)
1 − 2ν + µ2 + κ,



ρτ ζβ < δ(1 − h), h < 1.
Generalized Nonlinear Mixed
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If lim ken k = 0, lim kfn k = 0, then
n→∞
n→∞
(I) The sequence {un } defined by Algorithm 2 converges strongly to the unique
solution u∗ of problem (2.3).
P
∗
(II) If ∞
n=0 n < ∞, then lim yn = u .
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∗ )).
Generalized Nonlinear Mixed
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Involving Maximal η-Monotone
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From (4.9), (4.11) and Algorithm 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∗ ))
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∗
−JρM (·,u ) (g(u∗ ) − ρN (Su∗ , T u∗ ))
(4.12)
≤ (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
δ
+ αn κkvn − u∗ k
≤ (1 − αn )kun − u∗ k
τ
+ 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.
δ
By the Lipschitz continuity of N, S, T, g and the strong monotonicity of S and
g, we obtain
(4.13)
∗
∗
2
2
∗ 2
kvn − u − (g(vn ) − g(u ))k ≤ (1 − 2ν + µ )kvn − u k ,
(4.14) kvn − u∗ − ρ(N (Svn , T vn ) − N (Su∗ , T vn ))k2
≤ (1 − 2ρr + ρ2 ξ 2 α2 )kvn − u∗ k2 ,
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(4.15)
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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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)
Generalized Nonlinear Mixed
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Involving Maximal η-Monotone
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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,
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∗
where 0 < θ < 1 and so u = u. This completes the proof of Conclusion (I).
J. Ineq. Pure and Appl. Math. 7(3) Art. 114, 2006
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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 .
Generalized Nonlinear Mixed
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Involving Maximal η-Monotone
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As the proof of inequality (4.18), we have
Mao-Ming Jin
(4.20) (1 − αn )yn + αn xn − g(xn )
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+
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 .
(4.21)
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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∗ ≤ kyn+1 − u∗ k + (1 − αn (1 − θ))kyn − u∗ k + αn (1 − θ)dn → 0
as n → ∞. This completes the proof.
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Involving Maximal η-Monotone
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