J. Nonlinear Sci. Appl.3 (2010), no. 1, 63–77
The Journal of Nonlinear Science and Applications
http://www.tjnsa.com
GENERALIZED FUZZY RANDOM SET-VALUED MIXED VARIATIONAL
INCLUSIONS INVOLVING RANDOM NONLINEAR (Aω , η ω )-ACCRETIVE
MAPPINGS IN BANACH SPACES
HONG GANG LI∗
Abstract. The main purpose of this paper is to introduce and study a new class of random generalized fuzzy set-valued mixed variational inclusions involving random nonlinear
(Aω , ηω )-accretive mappings in Banach Spaces. By using the random resolvent operator
associated with random nonlinear (Aω , ηω )-accretive mappings, an existence theorem of solutions for this kind of random generalized fuzzy set-valued mixed variational inclusions is
established and a new iterative algorithm with an random error is suggested and discussed.
The results presented in this paper generalize, improve, and unify some recent results in this
field.
1. Introduction
Variational inclusions are an important and generalization of classical variational inequalities which have wide applications to many fields including, for example, mechanics, physics,
optimization and control, nonlinear programming, economics, and engineering sciences and
in face, the problems for random variational inclusions(inequalities) are just so. Motivated
and inspired by the recent research works in these fascinating areas , the random variational inclusion(inequalities, equalities, quasi-variational inclusions, quasi-complementarity)
problems have been introduced and studied by Ahmad and Bazán [1], Chang[5], Chang and
Huang [7], Cho et al. [8], Ganguly and Wadhwa[15], Huang [16], Haung and Cho[17], Huang
et al.[18], Khan et al.[24], Lan[25] Noor and Elsanosi[33]. Very recently, the problems of
Date: Received: 28 Aut. 2009.
∗
Corresponding author.
2000 Mathematics Subject Classification. 49J40, 47H06.
Key words and phrases. Generalized fuzzy random set-valued mixed variational inclusions; random nonlinear (Aω , ηω )-accretive mappings; random resolvent operator; random fuzzy set-valued mapping; convergence;
iterative algorithm with an random error.
63
64
HONG GANG LI
random fuzzy generalized variational inclusions involving Random nonlinear mappings have
been studied by Zhang and Bi[42] in Hilbert Spaces. On the other hand, Monotonicity
techniques were extended and applied in recent years because of their importance in the
theory of variational inequalities, complementarity problems, and variational inclusions. In
2003, Huang and Fang[19] introduced a class of generalized monotone mappings, maximal
η-monotone mappings, and defined an associated resolvent operator. Using resolvent operator methods, which is a very important method to find solutions of variational inequality
and variational inclusion problems, they developed some iterative algorithms to approximate
the solution of a class of general variational inclusions involving maximal η-monotone operators. Huang and Fang, s method extended the resolvent operator method associated with
an η-subdifferential operator due to Ding and Luo[12]. In [13], Fang and Huang introduced
another class of generalized monotone operators, H-monotone operators, and defined an associated resolvent operator. They also established the Lipschitz continuity of the resolvent
operator and studied a class of variational inclusions in Hilbert spaces using the resolvent
operator associated with H-monotone operators. In a paper[14], Fang and Huang et al. further introduced a new class of generalized monotone operators, (H, η)-monotone operators,
which provide a unifying framework for classes of maximal monotone operators, maximal ηmonotone operators, and H-monotone operators. Recently, Lan et al.[27] introduced a new
concept of (A, η)-accretive mappings, which generalizes the existing monotone or accretive
operators, and studied some properties of (A, η)-accretive mappings and defined resolvent
operators associated with (A, η)-accretive mappings. They also studied a class of variational
inclusions using the resolvent operator associated with (A, η)-accretive mappings. For these
reasons, various variational inclusions have been intensively studied in recent years. For
details, we refer the reader to [1-23, 25-42] and the references therein.
Inspired and motivated by recent research works in this field, the main purpose of this
paper is to introduce and study a new class of random generalized fuzzy set-valued mixed variational inclusions involving random nonlinear (A, η)-accretive mappings in Banach Spaces.
By using the resolvent operator associated with random nonlinear (A, η)-accretive mappings,
an existence theorem of solutions for this kind of fuzzy set-variational inclusions is established and a new iterative algorithm is suggested and discussed. The results presented in
this paper generalize, improve, and unify some recent results in this field.
2. Generalized Fuzzy Random Set-Valued
Variational Inclusions
Throughout this paper, we suppose that (Ω, <, µ) is a complete σ-finite measure space
and X is a separable real Banach Space endowed with dual space X ∗ , the norm k · k and the
dual pair h·, ·i, between X and X ∗ . We denote by =(X) the class of Borel σ-fields in X. Let
2X , and CB(X) denote the family of all the nonempty subset of X, and the family of all the
nonempty bounded closed sets of X, respectively. The generalized dual mapping Jq :X → 2X
is defined by
Jq (x) = {f ∗ ∈ X ∗ : hx, f ∗ i = kxkq , kf ∗ k = kxkq−1 }, ∀x ∈ X,
where q > 1 is a constant. In particular, J2 is the usual normalized duality mapping. It is
known that, Jq (x) = kxkq−2 J2 (x) f or all x =
6 0, Jq is single-valued if X ∗ is strictly convex,
RANDOM GENERALIZED FUZZY SET-VALUED MIXED VARIATIONAL INCLUSIONS
65
and if X = H, the Hilbert space, then J2 becomes the identity mapping on H. The modulus
of smoothness of X is the function πX : [0, ∞) → [0, ∞) defined by
1
πX (t) = sup{ (kx + yk + kx − yk) − 1 : kxk ≤ 1, kyk ≤ t}.
2
A Banach space X is called uniformly smooth if
πX (t)
= 0.
t→0
t
X is called q-uniformly smooth if there exists a constant c > 0 such that
lim
πX (t) ≤ ctq , (q > 1).
Remark 2.1. That Jq is single-valued if X is uniformly smooth, and Hilbert space and
Lp (2 ≤ p < ∞) space are 2-uniformly smooth Banach space.In what follows we shall denote
the single-valued generalized duality mapping by Jq .
Definition 2.2. A mapping x : Ω → X is said to be measurable if, for any B ∈ =(X),
{ω ∈ Ω : x(ω) ∈ B ∈ <}.
Definition 2.3. A mapping f : Ω × X → X is called a random mapping if, for any x ∈
X, f(ω, x) = y(ω) is measurable. A random mapping f is said to be continuous (resp., linear,
bounded) if for any ω ∈ Ω, the mapping f(ω, ·) : X → X is continuous (resp., linear,
bounded).
Similarly, we can define a random mapping h : Ω × X × X → X. We shall write x = x(ω),
y = y(ω), fω = f(ω, x(ω)) and hω (x, y) = b(ω, x(ω), y(ω)) for all ω ∈ Ω and x(ω), y(ω) ∈ X.
It is well-known that a measurable mapping is necessarily a random mapping.
Definition 2.4. A set-valued mapping
Q : Ω → 2X is said to be measurable if, for any
T
B ∈ =(X), Q−1 (B) = {ω ∈ Ω : Q(ω) B 6= ∅} ∈ <.
Definition 2.5. A mapping u : Ω → X is called a measurable selection of a set-valued
measurable mapping Q : Ω → 2X if, for any ω ∈ Ω, u(ω) is measurable and u(ω) ∈ Q(ω).
Definition 2.6. A set-valued mapping Q : Ω × X → 2X is called a random set-valued
mapping if, for any x ∈ X, Q(·, x) is measurable(denoted by Qω,x , or Q).
Let F(X) be a collection of all fuzzy sets over X. A mapping Fb : X → FF (X) is called
a fuzzy mapping. For each x ∈ X, Fb(x)(denote it by Fbx , in the sequel) is a fuzzy set on X
and Fbx (y) is the membership function of y in Fbx .
b ∈ F(X), q ∈ [0, 1]. Then the set
Let B
b q = {x ∈ X : B(x)
b
(B)
≥ q}
b
is called a q-cut set of B.
b : Ω → F(X) is called a measurable if, for any a ∈
Definition 2.7. A fuzzy mapping Q
b a : Ω → 2X is a measurable set-valued mapping.
(0, 1], (Q(·))
66
HONG GANG LI
b : Ω × X → F(X) is a random fuzzy mapping if, for any
Definition 2.8. A fuzzy mapping Q
b x) : Ω × X → F(X) is a measurable fuzzy mapping(denoted by Q
b ω,x , short down
x ∈ X, Q(·,
b
Q).
bω , T
b ω, G
b ω and P
b ω : Ω × X → F(X) be four random fuzzy mappings satisfying the
Let S
condition (∗):
(*) there exists four functions a, b, c, d : X → (0, 1] such that for all (ω, x) ∈ Ω × X, we
b ω )a(x) , (T
b ω )b(x) , (G
b ω )c(x) , (P
b ω )d(x) ∈ CB(X), where CB(X) denotes the family of all
have (S
nonempty bounded closed subsets of X.
bω , T
b ω, G
b ω and P
b ω , we can define four random
By using the random fuzzy mappings S
set-valued mappings Sω , Tω , Gω , Pω : X → CB(X) by
b ω )a(x) ,
Sω = ( S
b ω )b(x) ,
Tω = ( T
b ω )c(x) ,
Gω = ( G
b ω )d(x) ,
Pω = (P
for each x ∈ X. In the sequel, Sω , Tω , Gω and Pω are called the set-valued mappings induced
bω , T
b ω, G
b ω and P
b ω , respectively.
by the fuzzy mappings S
Let Aω : Ω×X → X, η ω : Ω×X×X → X and Fω : Ω×X×X → X be single-valued random
bω , T
b ω, G
b ω and P
b ω be four random fuzzy set-valued mappings. Let a, b, c, d :
mappings, and S
X → (0, 1] be four functions, and Mω : Ω × X × X → 2X be a random set-valued mapping
X
such that for
T each ω ∈ Ω,and u ∈ X Mω (u, ·) : X → 2 is (Aω , η ω )-accretive mapping and
range(Pω ) dom(Mω (u, ·)) 6= ∅. We introduce and study the following problem for a new
class of random generalized fuzzy set-valued mixed variational inclusions.
For a given element g : Ω → X and any real-valued random variable k(ω) > 0, finding
measurable mappings x = x(ω), u = u(ω), v = v(ω), z = z(ω), y = y(ω) : Ω → X such that
b ω (u(ω)) ≥ a(x), T
b ω (v(ω)) ≥ b(x), G
b ω (z(ω)) ≥ c(x), P
b ω (y(ω)) ≥ d(x)
S
and
g(ω) ∈ Fω (u, v) + k(ω)Mω (z, y),
(2.1)
which is called generalized fuzzy random set-valued mixed variational inclusions involving
random nonlinear (Aω , η ω )-accretive mappings in Banach Spaces.
If S, T, G, P : Ω × X → CB(X) are four random set-valued mappings, we can define four
bω , T
b ω, G
b ω, P
b ω : X → F(X) by
random fuzzy mappings S
x 7→ χSω , x 7→ χTω , x 7→ χGω , x 7→ χPω
bω , T
b ω, G
b ω, P
b ω , respectively.
where χSω , χTω , χGω and χPω are the characteristic functions of S
Taking a(x) = b(x) = c(x) = 1 for all x ∈ X, then problem (2.1) equivalent to the following
problem:
For a given element g : Ω → X and any real-valued random variable k(ω) > 0, finding
x = x(ω) ∈ X, and measurable mappings u = u(ω) ∈ Sω , v = v(ω) ∈ Tω , z = z(ω) ∈ Gω ,
y(ω) ∈ Gω such that
g(ω) ∈ Fω (u, v) + k(ω)Mω (z, y),
(2.2)
which is called generalized random set-valued mixed variational inclusions involving random
nonlinear (Aω , η ω )-accretive mappings in Banach Spaces.
RANDOM GENERALIZED FUZZY SET-VALUED MIXED VARIATIONAL INCLUSIONS
67
bω , T
b ω, G
b ω, P
b ω , and Sω , Tω , Gω , Pω and the space
For a suitable choice of Aω , η ω , Fω , Mω , S
X, a number of known classes of variational inclusions and variational inequalities can be
obtained as special cases of the general set-valued mixed quasi-variational inclusions (2.2).
Remark 2.9. (i)Some special case of the problem (2.1):
If k(ω) = 1, X = X ∗ = E is a Hilbert space and Mω (z, ·) = ∂φ(·)(∀z ∈ E) is S
the subdifferential of a lower semi-continuous and η-subdifferentiable function φ : E → R {+∞}.
b ω )a(x) = I is
Let for any v ∈ E, η ω (v, gω ) = v − gω (u, x),and Fω (u, v) = fω (x) − pω (v) and (S
a identical mapping in Hilbert Space E, and taking gω = v ∈ H(∀ω ∈ Ω),then problem(2.1)
becomes the following problem:
Find measurable mappings u, x, y : Ω → H, such that for each ω ∈ Ω, v ∈ H, hold
x(ω) ∈ Gω (u)a(u) , y(ω) ∈ Pω (u)b(u) ,
and
hfω (x) − pω (y), v − gω (u, x)i ≥ φ(gω (u, x)) − φ(v),
(2.3)
for all ω ∈ Ω, and each measurable mappings u(ω), v(ω) ∈ H, which is random generalized
nonlinear variational inclusions for random fuzzy mappings in Hilbert space. The form of the
problem (2.3) was studied by Zhang and Bi[42]. And a number of known classes of variational
inclusions and variational inequalities in [Chang[5], Cho and Lan[9], Wadhwa[15], Huang
[16], Haung and Cho[17], Noor and Elsanosi[33], Li[31]] can be obtained as special cases of
the problem (2.3) when the fuzzy random set-valued mappings all are taken as general random
set-valued mappings in the problem.
(ii)Some special case of the problem (2.2):
If Fω (u, v) = fω (v(ω)) + u(ω), Mω (·, w) ≡ Mω (w(ω)), and Sω = Gω = Iω is a identical
mapping, and Pω = pω (x) is a single-valued mapping in Banach Space, then the problem
(2.2) becomes the following problem:
For a given element g : Ω → X and any real-valued random variable k(ω) > 0, finding
measurable mapping x : Ω → X such that
g(ω) ∈ fω (x) + u + k(ω)Mω (x),
(2.4)
(∀ω ∈ Ω, u = u(ω) ∈ S(ω, x), the problem (2.4) was studied by Cho and Lan[9] which is
called generalized nonlinear random (A, η) equations with random relaxed cocoercive mappings in Banach Spaces. A number of known classes of random variational inclusions and
variational inequalities, random quasi-variational inclusions, random variational-like inclusions, random complementarity and random quasi-complementarity problems were studied
previously by many authors(see, Chang[5]–Haung and Cho[17], Noor and Elsanosi[33], [9],
[42]), and for examples, [[16], [25], [3], [9], [11], [31]] can be obtained as special cases of the
problem (2.4).
(iii) If in the problem (2.4), Pω = pω (x), Mω (·, ·) ≡ ∂φ(ω, ·) : Ω × H → H is subdifferentiable, and φ(ω, ·) is the indicator function of a nonempty closed convex set K in H defined
in the form:
½
0
if y ∈ K,
φ(y) =
∞
otherwise,
68
HONG GANG LI
then the problem (2.4) becomes the problem of finding measurable mappings x, u : Ω → X
such that u ∈ Tω (x) and
hfω (x) + u(ω) − gω , y − pω (x)i ≥ 0
(∀ω ∈ Ω, y ∈ K),
(2.5)
Furthermore, these types of variational inclusions can enable us to study many important
nonlinear random problems arising in mechanics, physics, random optimization and random
control, nonlinear random programming, random economics, regional, structural, and applied sciences with respect random things and produce in a general and unified framework.
Let us recall the following results and concepts.
3. Preliminaries
Definition 3.1. Let X be a q-uniformly smooth Banach Space, η ω : Ω × X × X → X and
Aω , Hω : Ω×X → X be random single-valued mappings. Then a random set-valued mapping
Mω : Ω × X → 2X is said to be:
(i) Ĥ-continuous if, for any ω ∈ Ω, Mω (·) is continuous in Ĥ(·, ·)-continuous i.e.: there
exists a real-valued random variable αω > 0 such that
Ĥ(Mω (x1 (ω)), Mω (x2 (ω))) ≤ αω kx1 (ω) − x2 (ω)k ∀x1 (ω), x2 (ω) ∈ X, ω ∈ Ω
where Ĥ(·, ·) is the Hausdorff metric on CB(X) defined as follows: for any A, B ∈ CB(X),
Ĥ(A, B) = max{sup inf d(x, y), sup inf d(x, y)};
x∈A y∈B
y∈B x∈A
(ii) η ω -accretive if, for any ω ∈ Ω,
hu1 (ω) − u2 (ω), Jq η ω (x1 (ω), x2 (ω))i ≥ 0,
for all x1 (ω), x2 (ω) ∈ X, u1 (ω) ∈ Mω (x1 (ω)), u2 (ω) ∈ Mω (x2 (ω));
(iii) strictly η ω -accretive if, for any ω ∈ Ω,
hu1 (ω) − u2 (ω), Jq η ω (x1 (ω), x2 (ω))i ≥ 0,
for all x1 (ω), x2 (ω) ∈ X, u1 (ω) ∈ Mω (x1 (ω)), u2 (ω) ∈ Mω (x2 (ω)); and the equality holds if
and only if x1 (ω) = x2 (ω) for all ω ∈ Ω;
(iv) rω −strongly η-accretive if there exists a real-valued random variable r(ω) > 0 such
that
hu1 (ω) − u2 (ω), Jq η ω (x1 (ω), x2 (ω))i ≥ r(ω)kx1 (ω) − x2 (ω)kq
, for all x1 (ω), x2 (ω) ∈ X, u1 (ω) ∈ Mω (x1 (ω)), u2 (ω) ∈ Mω (x2 (ω));
(v) γω -relaxed η ω -accretive if there exists a real-valued random variable γ(ω) > 0 such that,
for any ω ∈ Ω,
hu1 (ω) − u2 (ω), Jq η ω (x1 (ω), x2 (ω))i ≥ −γ(ω)kx1 (ω) − x2 (ω)kq
, for all x1 (ω), x2 (ω) ∈ X, u1 (ω) ∈ Mω (x1 (ω)), u2 (ω) ∈ Mω (x2 (ω));
(vi) Hω -accretive, if the Mω is accretive and (Hω (·) + ρ(ω)Mω (·))(X) = X for all ω ∈ Ω
and ρ(ω) > 0;
(vii) (Hω , η ω )-accretive, if the Mω is η ω -accretive and (Hω (·) + ρ(ω)Mω (·))(X) = X for
all ω ∈ Ω and ρ(ω) > 0 ;
RANDOM GENERALIZED FUZZY SET-VALUED MIXED VARIATIONAL INCLUSIONS
69
(viii) (Aω , η ω )-accretive, if the Mω is γ-relaxed η ω -accretive and (Aω (·)+ρ(ω)Mω (·))(X) =
X for all ω ∈ Ω and ρ(ω) > 0.
In a similar way, we can define the strictly η ω -accretivity and strongly η ω -accretivity of
the single-valued mapping Aω .
Definition 3.2. Let X be a q-uniformly smooth Banach Space, Aω : Ω × X → X and
Fω : Ω × X × X → X be random single-valued mappings, and Sω , Pω : Ω × X → 2X be two
random set-valued mappings.
(i) A random set-valued mapping Sω is said to be mω -relaxed accretive in the second
argument, if there exist a real-valued random variable m(ω) such that
hu1 (ω) − u2 (ω), jq (x1 (ω) − x2 (ω))i ≤ −m(ω)kx1 (ω) − x2 (ω)kq ,
∀x1 (ω), x2 (ω) ∈ X, u1 (ω) ∈ Sω (x1 (ω)), u2 (ω) ∈ Sω (x2 (ω)), ∀ω ∈ Ω.
(ii) A random set-valued mapping Sω is said to be sω -cocoercive in the second argument,
if there exist a real-valued random variable s(ω) > 0 such that
hu1 (ω) − u2 (ω), jq (x1 (ω) − x2 (ω))i ≤ s(ω)ku1 (ω) − u2 (ω)kq ,
∀x1 (ω), x2 (ω) ∈ X, u1 (ω) ∈ Sω (x1 (ω)), u2 (ω) ∈ Sω (x2 (ω)), ∀ω ∈ Ω.
(iii) A random set-valued mapping Sω is said to be tω -relaxed cocoercive with respct to Aω
in the second argument, if there exist a real-valued random variable t(ω) > 0 such that
hu1 (ω) − u2 (ω), jq (Aω (x1 (ω)) − Aω (x2 (ω)))i ≤ −t(ω)ku1 (ω) − u2 (ω)kq ,
∀x1 (ω), x2 (ω) ∈ X, u1 (ω) ∈ Sω (x1 (ω)), u2 (ω) ∈ Sω (x2 (ω)), ∀ω ∈ Ω.
(iv) A random set-valued mapping Sω is said to be (ςω , κω )-relaxed cocoercive in the second
argument, if there exist two positive real-valued random variables ς(ω) and κ(ω) such that
hu1 (ω)−u2 (ω), jq (Aω (x1 (ω))−Aω (x2 (ω)))i ≤ −ς(ω)ku1 (ω)−u2 (ω)kq +κ(ω)kx1 (ω)−x2 (ω)kq ,
∀x1 (ω), x2 (ω) ∈ X, u1 (ω) ∈ Sω (x1 (ω)), u2 (ω) ∈ Sω (x2 (ω)), ∀ω ∈ Ω.
(v) A random single-valued mapping Fω is said to be (µω , νω )-Lipschitz continuous, if there
exist two random variables µω , νω : Ω → (0, +∞) such that
kFω (x1 (ω), y1 (ω)) − Fω (x2 (ω), y2 (ω))k ≤ µω kx1 (ω) − x2 (ω)k + νω ky1 (ω) − y2 (ω)k,
∀xi (ω), yi (ω) ∈ X, i = 1, 2;
(vi) A random single-valued mapping Fω is said to be (ϕω , ψω )-Sω -relaxed cocoercive with
respect to Aω Pω in the second argument of F(·, ·, ·), if there exist two random variables
ϕ, ψ : Ω → (0, +∞) such that
hFω (u1 , ·) − Fω (u2 , ·), Jq (Aω (y1 ) − Aω (y2 ))i ≥ −ϕω kFω (u1 , ·) − Fω (u2 , ·)kq + ψω kx1 − x2 kq ,
∀xi (ω) ∈ X, ui (ω) ∈ Sω (xi (ω)), yi (ω) ∈ Pω (xi (ω))(i = 1, 2), ω ∈ Ω;
Definition 3.3. The random mapping η ω : Ω × X × X → X is said to be τω -Lipschitz
continuous if there exists a real-valued random variable τ (ω) > 0 such that
kη ω (x(ω), y(ω))k ≤ τ ω kx(ω) − y(ω)k,
∀x(ω), y(ω) ∈ X,
and
∀ω ∈ Ω.
70
HONG GANG LI
Definition 3.4. Let X be a Banach Space , η ω : Ω × X × X → X be a random single-valued
mapping, Aω : Ω × X → X be a strictly η ω -accretive random single-valued mapping and
Mω : Ω × X → 2X be a (Aω , η ω )-accretive random mapping, and ρω : Ω → (0, +∞) be a
A ,η
random variable. The random resolvent operator Rρωω,Mωω : Ω × X → X is defined by
A ,η
Rρωω,Mωω (y) = (Aω + ρω Mω )−1 (y),
for all ω ∈ Ω, y = y(ω) ∈ X, and {ω ∈ Ω : 0 < ρω ∈ B} ∈ <.
Lemma 3.5. ([10])Let X be a Banach Space, Mω : Ω × X → CB(X) be a Ĥ-continuous
random set-valued mapping. then for any measurable mapping x : Ω → X, the random
set-valued mapping Mω (x(ω)) : Ω → CB(X) is measurable.
Lemma 3.6. ([10]) Let Mω , Qω : Ω × X → CB(X) be two measurable set-valued mappings,
ε > 0 be a constant and x : Ω → X be a measurable selection of Mω . Then there exists a
measurable selection y : Ω → X of Qω such that for any ω ∈ Ω,
kx(ω) − y(ω)k ≤ (1 + ε)Ĥ(Mω (·), Qω (·)).
Lemma 3.7. ([27])Let X be a q-uniformly smooth and real separable Banach Space, η ω :
Ω × X × X → X be τω -Lipschtiz continuous mapping, Aω : Ω × X → X be an rω -strongly
η-accretive mapping, and Mω (·, y) : Ω × X × X → 2X (∀y ∈ X) be an (Aω , η ω )-accretive
A ,η
mapping. Then the generalized random resolvent operator Rρωω,Mωω : X → X is τωq−1 /(rω −
mω ρω )-Lipschitz continuous, that is,
τωq−1
kx − yk
for all x, y ∈ X, ω ∈ Ω.
rω − mω ρω
where ρω , rω , mω : Ω → (0, +∞) are real-valued measurable, and 0 < ρω < mrωω .
A ,η
A ,η
kRρωω,Mωω (x) − Rρωω,Mωω (y)k ≤
Lemma 3.8. [38]Let X be a real uniformly smooth Banach space. Then X is q-uniformly
smooth if and only if there exists a constant cq > 0 such that for all x, y ∈ X,
kx + ykq ≤ kxkq + qhy, Jq (x)i + cq kykq
4. Iterative algorithm with an random error of solutions
We first transfer the problem (2.1) into a fixed point problem.
Lemma 4.1. Measurable x, u, v, z, y : Ω → X is a system solution of random generalized
set-valued mixed variational inclusions the problem (2.1) if and only if for each ω ∈ Ω, holds
the following relation
A ,η
y(ω) = Rρωωkω ω,Mω [Aω (y) + ρω gω − ρω Fω (u, v)],
(4.1)
where u ∈ Sω , v ∈ Tω , z ∈ Gω , y ∈ Pω , ρω , kω : Ω → (0, +∞) are two real-valued random
A ,η
variables, and Rρωωkω ω,Mω = (Aω + ρω kω Mω )−1 is a resolvent operator in Banach Space X.
A ,η
ω
Proof. The proof directly follows from the definition of Rρωωkω ,M
and so it is omitted.
ω
Based on Lemma 4.1 and Nadler [32], we can develop a new iterative algorithm for solving
the random generalized fuzzy set-valued mixed variational inclusions (2.1) with random
nonlinear (Aω , η ω )-accretive mappings as follows:
RANDOM GENERALIZED FUZZY SET-VALUED MIXED VARIATIONAL INCLUSIONS
71
bω , T
b ω, G
b ω, P
b ω : Ω × X → F(X) be random fuzzy mappings satisfying
Algorithm 4.2. Let S
condition (∗) and Sω , Tω , Gω , Pω : Ω × X → CB(X) be the set-valued mappings induced by
bω , T
b ω, G
b ω, P
b ω , respectively. Let Aω : Ω×X → X, η ω , Fω : Ω×X×X →
the fuzzy mappings S
X be single-valued mappings and let Mω : Ω × X × X → 2X be a random set-valued mapping
such that for each fixed ω ∈ Ω, and for any a measurable mapping z : Ω ×
TX → X, Mω (z, ·) :
Ω × X × X → 2X be a (Aω , η ω )-accretive mapping and range(Pω ) domMω (·, t) 6= ∅.
For any given x0 : Ω → X, the set-mappings Sω (x0 ), Tω (x0 ), Gω (x0 ), Pω (x0 ) : Ω × X →
CB(X) all are measurable by lemma (3.6). We know that, for any x0 ∈ X, the set-mappings
Sω (x0 ), Tω (x0 ), Gω (x0 ), Pω (x0 ) are measurable and there exists measurable selections u0 ∈
Sω (x0 ), v0 ∈ Tω (x0 ), z0 ∈ Gω (x0 (·)), y0 ∈ P(·, x0 )(see [22]). Set
A ,η
x1 (ω) = (1 − $)x0 + $[x0 − y0 + Rρωωkω ω,Mω (Aω (y0 ) + ρω gω − ρω Fω (u0 , v0 ))] + e0 ,
where kω , ρω , Aω , Mω , Fω are the same as in Lemma (4.1), 1 > $ > 0 is a constant, and
e0 = e0 (ω) : Ω → X is a measurable function which is an random error to take into account a
possible inexact computation of the proximal point. Then, it is easy to know that x1 : Ω → X
is a measurable mapping. Since u0 ∈ Sω (x0 ), v0 ∈ Tω (x0 ), z0 ∈ Gω (x0 ), y0 ∈ Pω (x0 ), by
Lemma (3.7) ,there exists measurable selections u1 ∈ Sω (x1 ), v1 ∈ Tω (x1 ), z1 ∈ Gω (x1 ), y1 ∈
Pω (x1 such that , for all ω ∈ Ω,
1
ku0 − u1 k ≤ (1 + )Ĥ(Sω (x0 ), Sω (x1 )),
1
1
kv0 − v1 k ≤ (1 + )Ĥ(Tω (x0 ), Tω (x1 )),
1
1
kz0 − z1 k ≤ (1 + )Ĥ(Gω (x0 ), Gω (x1 )),
1
1
ky0 − y1 k ≤ (1 + )Ĥ(Pω (x0 ), Pω (x1 )),
1
By induction, we can define a measurable sequences xn , un , vn , zn , and yn : Ω → X inductively
satisfying

A ,η
xn+1 = (1 − $)xn + $[xn − yn + Rρωωkω ω,Mω (Aω (yn ) + ρω gω − ρω Fω (un , vn ))] + en ,




1

 un ∈ Sω (xn ), kun − un+1 k ≤ (1 + n+1 )Ĥ(Sω (xn ), Sω (xn+1 )),
1
(4.2)
vn ∈ Tω (xn ), kvn − vn+1 k ≤ (1 + n+1
)Ĥ(Tω (xn ), Tω (xn+1 )),


1

zn ∈ Gω (xn ), kzn − zn+1 k ≤ (1 + n+1 )Ĥ(Gω (xn ), Gω (xn+1 )),



1
)Ĥ(Pω (xn ), Pω (xn+1 )),
yn ∈ Pω (xn ) kyn − yn+1 k ≤ (1 + n+1
where n = 0, 1, 2, · · ·, 0 < $ < 1 is a constant, en = en (ω) : Ω → X(n ≥ 0) is an random
error to take into account a possible inexact computation of the proximal point.
From Algorithm 4.2, we can get algorithm for solving problems (2.2) as follows:
Algorithm 4.3. For any given for any x0 (·) ∈ X, the set-valued mappings Sω (x0 ), Tω (x0 ),
Gω (x0 ), Pω (x0 ) are measurable and there exists measurable selections u0 ∈ Sω (x0 ), v0 ∈
Tω (x0 ), z0 ∈ Qω (x0 ), y0 ∈ Pω (x0 ), we can get the measurable iterative sequences {xn }, {un },
72
HONG GANG LI
{vn }, {zn } and {yn } : Ω → X as follows:

A ,η
xn+1 = (1 − $)xn + $[xn − yn + Rρωωkω ω,Mω (Aω (yn ) + ρω gω − ρω Fω (un , vn ))] + en ,




1

 un ∈ Sω (xn ), kun − un+1 k ≤ (1 + n+1 )Ĥ(Sω (xn ), Sω (xn+1 )),
1
(4.3)
vn ∈ Tω (xn ), kvn − vn+1 k ≤ (1 + n+1
)Ĥ(Tω (xn ), Tω (xn+1 )),


1

zn ∈ Gω (xn ), kzn − zn+1 k ≤ (1 + n+1 )Ĥ(Gω (xn ), Sω (xn+1 )),



1
yn ∈ Pω (xn ) kyn − yn+1 k ≤ (1 + n+1
)Ĥ(Pω (xn ), Pω (xn+1 )),
where n = 0, 1, 2, · · ·, 0 < $ < 1 is a constant, en (ω) : Ω → X(n ≥ 0) is an random error to
take into account a possible inexact computation of the proximal point.
Remark 4.4. If we choose suitable η, A, F, S, T, G, P and M , then Algorithm 4.3 can be
degenerated to a number of algorithms involving many known algorithms which due to classes
of variational inequalities, and variational inclusions (see, for examples, [4], [11], [16], [25],
[31], [35]).
Now we prove the existence of solutions of problem (2.1) and the convergence of iterative
sequences generated by Algorithm 4.2.
5. Existence and convergence
In this section, we will prove the existence of solution for problem (2.1) and the convergence
of the iterative sequences generated by Algorithm 3.2.
Theorem 5.1. Let X be a q-uniformly smooth and real separable Banach Space, η ω :
Ω × X × X → X be τ ω -Lipschtiz continuous random mapping, Aω : Ω × X → X be
bω , T
b ω, G
b ω,
rω -strongly η ω (·, ·)-accretive random mapping and αω -Lipschitz continuous. Let S
b
Gω : Ω × X → F(X) be fuzzy random mappings satisfying condition (∗) and Sω , Tω , Gω ,
Pω : Ω×X → CB(X) be random set-valued mappings induced by the fuzzy random mappings
bω , T
b ω, G
b ω , and P
b ω , respectively. suppose that Sω , Tω , Gω , Pω be Ĥ-Lipschitz continuous
S
with random variables γω , ξω , ζω , χω , respectively. Let Pω be (ςω , κω )-relaxed cocoercive in the
second argument of Pω (·). Let Fω : Ω × X × X → X be Lipschitz continuous random mapping with random variables (µω , νω ), and Fω be (ϕω , ψω )-Sω -relaxed cocoercive with respect
to Aω Pω in the second argument of Fω (·, ·), and let gω : Ω → X be a real random variable.
Suppose Mω : Ω × X × X → 2X such that for each measurable
y ∈ X, Mω (·, y) : X → 2X be
T
(Aω , η ω )-accretive random mapping and range(Pω ) domMω (·, y) 6= ∅. If for any x, y, z ∈
X there exists a random real-valued variable δω > 0 such that:
A ,η
A ,η
kRρωωkω ω,Mω (·,x) (z) − Rρωωkω ω,Mω (·,y) (z)k ≤ δω kx − yk,
and

1

L = δω χω + [1 + χqω (cq + qςω ) − qκω ] q < 1,




1
q q q
q q q
q q
)
γ
+
qρ
ϕ
µ
γ
µ
+
c
ρ
χ
ρ
ν
ξ
+
(α
< (1 − L)(rω − kω ρω mω )τω1−q ,
ω
ω
q
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω





rω > kω ρω mω ;
(5.1)
(5.2)
RANDOM GENERALIZED FUZZY SET-VALUED MIXED VARIATIONAL INCLUSIONS
73
And
lim ken (ω)k = 0,
n→∞
∞
X
ken (ω) − en−1 (ω)k < ∞,
∀ω ∈ Ω.
(5.3)
n=1
Then the random variable iterative sequences {xn }, {un }, {vn }, {zn } and {yn } : Ω → X
generated by Algorithm 4.2 converge strongly to random variables x∗ , u∗ , v ∗ , z ∗ , and y ∗ :
Ω → X, respectively, and (x∗ , u∗ , v ∗ , z ∗ , y ∗ ) is a solution of problem (2.1).
Proof.. From Algorithm 4.2, Lemma 3.7 and (5.1), for any ω ∈ Ω, and 0 < $ < 1, we
have
kxn+1 − xn k
≤ (1 − $)kxn − xn−1 k + ken − en−1 k + $kxn − xn−1 − (yn − yn−1 )k
A ,η
ω
+$kRρωωkω ,M
(Aω (yn ) − ρω Fω (un , vn ))
ω (zn ,·)
A ,η
ω
−Rρωωkω ,M
(Aω (yn−1 ) − ρω Fω (un−1 , vn−1 ))k
ω (zn ,·)
A ,η
ω
+$kRρωωkω ,M
(Aω (yn−1 ) − ρω Fω (un−1 , vn−1 ))
ω (zn ,·)
A ,η
ω
−Rρωωkω ,M
(Aω (yn−1 ) − ρω Fω (un−1 , vn−1 ))k
ω (zn−1 ,·)
≤ (1 − $)kxn − xn−1 k + ken − en−1 k + ${kxn − xn−1 − (yn − yn−1 )k
τωq−1
+
[kAω (yn ) − Aω (yn−1 ) − ρω (Fω (un , vn−1 ) − Fω (un−1 , vn−1 ))k
rω − kω ρω mω
+ρ(ω)kFω (un , vn ) − Fω (un , vn−1 )k] + δω kyn − yn−1 k}.
(5.4)
Since Pω is the Ĥ-Lipschitz continuous with χω and is (ςω , κω )-relaxed cocoercive in the
second argument of Pω (·), and by Lemma (3.8)[38] and Algorithm 4.2, we obtain
kxn − xn−1 − (yn − yn−1 )kq = kxn − xn−1 kq
−qhyn − yn−1 , jq (xn − xn−1 )i + cq kyn − yn−1 kq
≤ kxn − xn−1 kq + cq (1 + n−1 )q Ĥ q (Pω (xn ), Pω (xn−1 ))
−q(−ςω kyn − yn−1 kq + κω kxn − xn−1 kq )
≤ [1 + (1 + n−1 )q χqω (cq + qςω ) − qκω ]kxn − xn−1 kq .
(5.5)
Since Fω is Ĥ-Lipschitz continuous with (µω , νω ), and is (ϕω , ψω )-Sω -relaxed cocoercive with
respect to Aω Pω in the second argument of Fω (·, ·) and Sω is Ĥ-Lipschitz continuous with
74
HONG GANG LI
γω , we have
kAω (yn ) − Aω (yn−1 ) − ρω (Fω (un , vn−1 ) − Fω (un−1 , vn−1 ))kq
≤ kAω (yn − Aω (yn− )kq + cq ρqω kFω (un , vn−1 ) − Fω (un−1 , vn−1 )kq
−qhρω (Fω (un , vn−1 ) − Fω (un−1 , vn−1 )), Jq (Aω (yn ) − Aω (yn−1 ))i
≤ αωq kyn − yn−1 kq + cq ρqω µqω kun − un−1 kq
−qρω (−ϕω kFω (un , vn−1 ) − Fω (un−1 , vn−1 )kq + ψω kxn − xn−1 kq )
≤ αωq kyn − yn−1 kq + cq (1 + n−1 )q ρqω µq (ω)Ĥ q (Sω (xn ), Sω (xn−1 ))
+qρω ϕω γωq kun − un−1 kq − qρω ψω kxn − xn−1 kq
≤ [αωq (1 + n−1 )q χqω + cq (1 + n−1 )q ρqω µqω γωq
+qρω ϕω µqω (1 + n−1 )q γωq ]kxn − xn−1 kq .
(5.6)
Further, By assumptions
kFω (un , vn ) − Fω (un , vn−1 )k ≤ νω ξω (1 + n−1 )kxn − xn−1 k,
kyn − yn−1 k ≤ (1 + n−1 )χω kxn − xn−1 k.
(5.7)
(5.8)
From (5.4)∼(5.8), It follows that
kxn+1 − xn k ≤ (1 − $ + $hn )kxn − xn−1 k + ken − en−1 k
= θn kxn − xn−1 k + ken − en−1 k
(5.9)
where
θn = 1 − $ + $hn ,
1
hn = δω χω (1 + n−1 ) + [1 + (1 + n−1 )q χqω (cq + qςω ) − qκω ] q
τωq−1
+
[ρω νω ξω (1 + n−1 ) + (αωq (1 + n−1 )q χqω
rω − kω ρω mω
1
+cq (1 + n−1 )q ρqω µqω γωq + qρω ϕω µqω (1 + n−1 )q γωq ) q ].
Letting
θ = 1 − λ + λh
1
h = δω χω + [1 + χqω (cq + qςω ) − qκω ] q
1
τωq−1
+
[ρω νω ξω + (αωq χqω + cq ρqω µqω γωq + qρω ϕω µqω γωq ) q ].
rω − kω ρω mω
we have that hn → h and θn → θ as n → ∞. It follows from condition (5.2) and 0 < $ < 1
that 0 < θ < 1 and hence there exists N0 > 0 and θ∗ ∈ (θ, 1) such that θn < θ∗ for all
n ≥ N0 . Therefore, by (5.9), we have
kxn+1 − xn k ≤ θ∗ kxn − xn−1 k + ken − en−1 k, ∀n ≥ N0 .
Without loss of generality we assume
kxn+1 − xn k ≤ θ∗ kxn − xn−1 k + ken − en−1 k, ∀n ≥ 1,
RANDOM GENERALIZED FUZZY SET-VALUED MIXED VARIATIONAL INCLUSIONS
75
Hence, for any m > n > 0, we have
kxm − xn )k ≤
m−1
X
kxi+1 − xi k ≤
i=n
m−1
X
i=n
θ∗i kx1 − x0 k +
m−1
i
XX
θ∗i−j kej − ej−1 k.
i=n j=1
It follows from condition (4.3) that kxm − xn k → 0, as n → ∞, and so {xn } is a Cauchy
sequence in X. Let xn → x∗ as n → ∞. By the Lipschitz continuity of Sω (·), Tω (·), Gω (·)
and Pω (·), we obtain
kun+1 − un k ≤ (1 + n−1 )Ĥ(Sω (xn+1 ), Sω (xn )) ≤ γω (1 + n−1 )kxn+1 − xn k,
kvn+1 − vn k ≤ (1 + n−1 )Ĥ(Tω (xn+1 ), Tω (xn )) ≤ ξω (1 + n−1 )kxn+1 − xn k,
kzn+1 − zn k ≤ (1 + n−1 )Ĥ(Gω (xn+1 ), Gω (xn )) ≤ ζω (1 + n−1 )kxn+1 − xn k,
kyn+1 − yn k ≤ (1 + n−1 )Ĥ(Pω (xn+1 ), Pω (xn )) ≤ χω (1 + n−1 )kxn+1 − xn k.
It follows that {un }, {vn }, {zn }, and {yn } are also Cauchy sequences in X. We can assume
that un → u∗ , vn → v ∗ , zn → z ∗ , and yn → y ∗ respectively. Note that un ∈ Sω (xn ) , we have
d(u∗ , Sω (x∗ )) ≤ ku∗ − un k + d(un , Sω (x∗ ))
≤ ku∗ − un k + Ĥ(Sω (xn ), Sω (x∗ ))
≤ ku∗ − un k + γω kxn − x∗ k → 0(n → ∞).
Hence d(u∗ , Sω (x∗ )) = 0 and therefore u∗ ∈ Sω (x∗ ). Similarly, we can prove that v ∗ ∈ Tω (x∗ ),
z ∗ ∈ Gω (x∗ ), and y ∗ ∈ Pω (x∗ )).
By the Lipschitz continuity of Sω (·), Tω (·), Gω (·) and Pω (·), and Lemma 4.1, condition
(5.1) and lim ken (ω)k = 0, we have
n→∞
x∗ (ω) = (1 − $)x∗ (ω) + $[x∗ (ω) − y ∗ (ω)
A (y ∗ (ω)),η
+Rρωωkω ,Mω (z∗ ω(ω),·) (Aω (y ∗ (ω)) + ρω gω − ρω Fω (u∗ (ω), v ∗ (ω)))].
By Lemma 3.1, we know that (x∗ , u∗ , v ∗ , z ∗ , y ∗ ) is a solution of problem (2.1). This completes
the proof. From Theorem 5.1, we have the following theorem.
Theorem 5.2. Let Aω , gω , η ω , Fω , Mω , X be the same as in Theorem 5.1, and Sω , Tω , Gω , Pω :
Ω × X → CB(X) be D-Lipschitz continuous with random variables γω , ξω , ζω , χω , respectively, and let Pω be (ςω , κω )-relaxed cocoercive in the second argument of Pω (·). Let Fω :
Ω × X × X → X be Lipschitz continuous with random variables (µω , νω ), and Fω be (ϕω , ψω )Sω -relaxed cocoercive with respect to Aω Pω in the second argument of Fω (·, ·). If conditions
(5.1)∼(5.3) of Theorem 5.1 hold, then the random variable iterative sequences {xn }, {un }, {vn },
{zn } and {yn } : Ω → X generated by Algorithm 4.3 converge strongly to random variables
x∗ , u∗ , v ∗ , z ∗ and y ∗ : Ω → X, respectively, and (x∗ , u∗ , v ∗ , z ∗ , y ∗ ) is a solution of the problem
(2.2).
Remark 5.3. For a suitable choice of the mappings Aω , gω , η ω , Fω , Mω , Sω , Tω , Gω , Pω and
Xω , we can obtain several known results [[4], [11], [16], [25], [31], [35] et al.] as special cases
of Theorem 5.2.
Acknowledgments The authors acknowledgment support of the Educational Science
Foundation of Chongqing, Chongqing(KJ091315).
76
HONG GANG LI
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∗
Institute of Applied Mathematics Research Chongqing University of Posts and TeleCommunications Chongqing 400065, China
E-mail address: lihg12@126.com