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East Asian Math. J.
Vol.32 (2016), No. 5, pp. 685–700
http://dx.doi.org/10.7858/eamj.2016.048
FUZZY GENERAL NONLINEAR ORDERED RANDOM
VARIATIONAL INEQUALITIES IN ORDERED BANACH
SPACES
Salahuddin and Byung-Soo Lee
Abstract. The main object of this work to introduced and studied a new
class of fuzzy general nonlinear ordered random variational inequalities
in ordered Banach spaces. By using the random B-restricted accretive
mapping with measurable mappings α, α0: Ω →(0,1), an existence of
random solutions for this class of fuzzy general nonlinear ordered random
variational inequality (equation) with fuzzy mappings is established, a
random approximation algorithm is suggested for fuzzy mappings, and the
relation between the first value x0(t) and the random solutions of fuzzy
general nonlinear ordered random variational inequality is discussed.
1. Introduction
The variational inclusions, which was introduced and studied by Hassouni
and Moudafi [19] is a useful and important extension of variational inequalities.
In last decades, monotonicity techniques were extended and applied because
of their importance in theory of variational inequality, complementarity prob-
lems and variational inclusions. Recently some systems of variational inequali-
ties, variational inclusions, complementarity problems and equilibrium problems
have been studied by some authors in recent years because of their close rela-
tions to Nash equilibrium problems. Huang and Fang [21] introduced a system
of ordered complementarity problems and established some existence results for
using fixed point theory. Verma [32] introduced and studied the some systems
of variational inequalities and developed some random iterative algorithm for
approximation of random solutions for system of variational inequalities.
On the other hand in 1972, the number of solutions of nonlinear equation has
been introduced and studied by Amann [6] and recent years, the nonlinear
mapping, fixed point theory and application have been extensively studied in
ordered Banach spaces, [16, 17, 18]. Very recently Li [22, 23, 24] has studied
Received January 29, 2016; Accepted June 15, 2016.
2010 Mathematics Subject Classification. 49J40, 47H06.
Key words and phrases. Fuzzy general nonlinear ordered random variational inequality,
Ordered Banach spaces, Random B-restricted accretive mappings, Random algorithm, ran-
dom compression mapping, Fuzzy mappings.
c
2016 The Youngnam Mathematical Society
(pISSN 1226-6973, eISSN 2287-2833)
685
686 SALAHUDDIN AND B. S. LEE
the approximation solution for general nonlinear ordered variational inequalities
and ordered equations in ordered Banach spaces.
Fuzzy sets were founded by Professor L. A. Zadeh in year 1965 [34]. The ad-
dress of fuzzy set theory, since its introduction has been dramatic and breathtak-
ing, several research papers have published in different journals devoted entirely
to theoretical and application aspects of fuzzy sets. In 1989, Chang and Zhu
[9] introduced the concept of variational inequalities in fuzzy mappings in ab-
stract spaces and investigated existence theorem for some kinds of variational
inequalities for fuzzy mappings. Very recently, the problems of random general-
ized fuzzy variational inclusions involving random nonlinear mapping have been
studied by Zhang and Bi [33] in Hilbert spaces. Afterwards, on several kinds
of variational inequalities, variational inclusions and complementarity problems
for fuzzy mappings were considered and studied by many authors see for in-
stance, Ahmad and Salahuddin [1, 2, 3], Ahmad and Bazan [4], Agarwal et al.
[5], Anastassiou et al. [7], Chang and Huang [10], Chang and Salahuddin [11],
Chang et al. [12], Cho et al. [14], Ding and Park [15], Huang [20], Lee et al.
[25, 26, 27], Salahuddin [28], Salahuddin and Ahmad [30], Salahuddin et al. [31]
and Salahuddin and Verma [29], etc.
Inspired and motivated by recent works, in this communication, fuzzy general
nonlinear ordered random variational inequalities and an operator ⊕is intro-
duced and the qualities of an operator ⊕is studied in ordered Banach spaces.
Applying the random B-restricted accretive method of random mapping Awith
measurable operators α, α0, an existence theorem of random solutions for this
class of fuzzy general nonlinear ordered variational inequalities is established,
a random approximation algorithm is suggested and the relation between the
values x0(t) and the random solutions of the fuzzy general nonlinear ordered
random variational inequality is discussed.
2. Preliminaries
Throughout this work, we assume that (Ω,Σ, µ) is a complete σ- finite mea-
surable space and Xis a separable real Banach space endowed with dual space
X∗, the norm k·k and the dual pair h·,·i between Xand X∗. We denote by
B(X) the class of Borel σ- field in X. Let 2Xand C B(X) denote the family of
all nonempty subset of Xand the family of all nonempty bounded closed sets
of X, respectively.
Definition 1. A mapping x: Ω →Xis said to be measurable if for any
B∈B(X),{t∈Ω, x(t)∈B} ∈ Σ.
Definition 2. A mapping f: Ω ×X→Xis called a random operator if for
any x∈X, f (t, x) = x(t) is a measurable. A random operator fis said to be
continuous if for any t∈Ω,the mapping f(t, ·) : X→Xis continuous.
FUZZY ORDERED RANDOM VARIATIONAL INEQUALITIES 687
Definition 3. A set valued mapping T: Ω ×X→2Xis said to be measurable
if for any B ∈B(X), T −1(B) = {t∈Ω, T (t)∩B6=∅} ∈ Σ.
Definition 4. A mapping u: Ω →Xis called a measurable selection of a
set valued measurable mapping T: Ω →2Xif uis a measurable and for any
t∈Ω, u(t)∈T(t).
Definition 5. A mapping T: Ω ×X→2Xis called a random set valued
mapping if for any x∈X, T (·, x) is a measurable. A random set valued mapping
T: Ω ×X→CB (X) is said to be H-continuous if for any t∈Ω, T (t, ·) is
continuous in Hausdorff metric.
Definition 6. A fuzzy mapping F: Ω →F(X) is called measurable if for any
α∈(0,1),(F(·))α: Ω →2Xis a measurable set valued mappings.
Definition 7. A fuzzy mapping F: Ω ×X→F(X) is called a random fuzzy
mapping if for any x∈X, F (·, x)→F(X) is a measurable fuzzy mapping.
Let F(X) be a collection of all fuzzy sets over X. A mapping Ffrom X
to F(X) is called a fuzzy mapping on X. If Fis a fuzzy mapping on X, the
F(x) (denote it by Fx,in the sequel) is a fuzzy set on Xand (Fx)(y) is the
membership function of yin Fx. Let N∈F(x), q ∈[0,1],then the set
(N)q={x∈X:N(x)≥q}
is called a q-cut set of N.
Let T: Ω ×X→F(X) be the random fuzzy mapping satisfying the following
condition (C):
(C): There exists a mapping a:X→[0,1] such that (Tt,x)a(x)∈CB(X)∀(t, x)∈
Ω×X. By using the random fuzzy mapping Twe can define random set valued
mapping ˜
Tas follows:
˜
T: Ω ×X→C B(X), x →(Tt,x )a(x),∀(t, x)∈Ω×Xwhere Tt,x =T(t, x(t)).
Let Xbe a real ordered Banach space with a norm k·kand θbe a zero in the
X. Let Pbe a normal cone of Xand ≤be a partial ordered relation defined
by the cone P. Given a mapping a:X→[0,1],random fuzzy mapping T:
Ω×X→F(X),let A, g, f : Ω ×X→Xbe the single valued random nonlinear
ordered comparison mappings and range g(x(t), t)∩domA(·, t) = ∅ ∀ t∈Ω, we
consider the following problem:
Find a measurable mapping x, u : Ω →Xsuch that for all t∈Ω, x(t)∈
X, T(t,x(t))(u(t)) ≥a(x(t)) and g(t, x(t)) ∩dom A(·, t)6=∅for t∈Ω such that
A(g(x(t), t), t) + f(u(t), t)≥θ. (1)
The problem (2.1) is called a fuzzy general nonlinear ordered random variational
inequalities in ordered Banach spaces.
Definition 8. [13] Let Xbe a real Banach space with a norm k · k, θ be a zero
element in the X. A nonempty closed convex subsets Pof Xis said to be a
cone if
688 SALAHUDDIN AND B. S. LEE
(a) for any x∈P, and any λ > 0, λx ∈Pholds;
(b) if x∈Pand −x∈Pthen x=θ.
Definition 9. [22] Let Pbe a cone of X. P is said to be a normal cone if and
only if there exists a constant N > 0 such that for θ≤x≤y, hold kxk ≤ Nkyk
where Nis called normal constant of P.
Lemma 2.1. [13] Let Pbe a cone in X, for arbitrary x, y ∈X, x ≤yif and
only if x−y∈P, then the relation ≤in Xis a partial ordered relation in X
where the Banach space Xwith an ordered relation ≤defined by a normal cone
Pis called an ordered Banach space.
Definition 10. [13] Let Xbe an ordered Banach space and Pbe a cone of X.
The ≤is a partial ordered relation defined by the cone Pfor all x, y ∈Xif hold
x≤y( or y≤x) then xand yis said to be the comparison between each other
(denoted by x∝yfor x≤yand y≤x).
Definition 11. [13] Let Xbe an ordered Banach space and Pbe a cone of
X. The ≤is a partial ordered relation defined by the cone P, for arbitrary
x, y ∈X, lub{x, y}and glb{x, y}express the least upper bound of the set {x, y}
and the greatest lower bound of the set {x, y}on the partial ordered relation ≤
respectively. Suppose lub{x, y}and glb{x, y}exists some binary operator can
be defined as follows:
(i) x∨y=lub{x, y };
(ii) x∧y=glb{x, y };
(iii) x⊕y= (x−y)∨(y−x).
∨,∧and ⊕is called OR, AN D and X OR operations, respectively. For arbitrary
x, y, w ∈Xthen holds the following relations:
(1) if x≤ythen x∨y=y, x ∧y=x;
(2) if xand ycan be compared then θ≤x⊕y;
(3) (x+w)∨(y+w) exists and (x+w)∨(y+w)=(x∨y) + w;
(4) (x∧y) = (x+y)−(x∨y);
(5) if λ≥0 then λ(x∨y) = λx ∨λy;
(6) if λ≤0 then λ(x∧y) = λx ∨λy;
(7) if x6=ythen the converse holds for (5) and (6);
(8) if for any x, y ∈X, either x∨yand x∧yexists, then Xis a lattice;
(9) (x+w)∧(y+w) exists and (x+w)∧(y+w)=(x∧y) + w;
(10) (x∧y) = −(−x∨ −y);
(11) (−x)∧(x)≤θ≤(−x)∨x.
Lemma 2.2. [16] If x∝ythen lub {x, y}and glb {x, y}exist, x−y∝y−x,
and θ≤(x−y)∨(y−x).
Lemma 2.3. [16] If for any natural number n, x ∝ynand yn→y∗(n→ ∞)
then x∝y∗.
FUZZY ORDERED RANDOM VARIATIONAL INEQUALITIES 689
Lemma 2.4. [22] Let Xbe an ordered Banach space and Pbe a cone of X.
The ≤is a partial ordered relation defined by the cone P, if for x, y, z , w ∈X
they can be compared each other, then holds the following relations:
(1) x⊕y=y⊕x;
(2) x⊕x=θ;
(3) θ≤x⊕θ;
(4) let λbe a real then (λx)⊕(λy) =|λ|(x⊕y);
(5) if x, y and wcan be comparative each other then (x⊕y)≤x⊕w+w⊕y;
(6) let (x+y)∨(u+v)exists and if x∝u, v and y∝u, v then;
(x+y)⊕(u+v)≤(x⊕u+y⊕v)∧(x⊕v+y⊕u);
(7) if x, y, z, w can be compared with each other then
(x∧y)⊕(z∧w)≤((x⊕z)∨(y⊕w)) ∧((x⊕w)∨(y⊕z));
(8) αx ⊕βx =|α−β|x+ (α⊕β)xif x∝θ.
Lemma 2.5. [8] Let T: Ω×X→CB(X)be a H-continuous random set valued
mapping. Then for any measurable mapping w: Ω →X, the set valued mapping
T(·, w(·)) : Ω →CB(X)is a measurable.
Lemma 2.6. [8] Let T, S : Ω →CB(X)be the two measurable set valued
mappings, > 0be a constant and υ: Ω →Hbe a measurable selection of S
then there exists a measurable selection w: Ω →Hof Tsuch that for all t∈Ω
kυ(t)−w(t)k ≤ (1 + )H(S(t), T (t)).
Definition 12. Let Xbe a real ordered Banach space and let A, B : Ω×X→X
be the two random mappings.
(i) A(t) is said to be randomly comparison if for any t∈Ω and each
x(t), y(t)∈X, x(t)∝y(t) then A(x(t), t)∝A(y(t), t), x(t)∝A(x(t), t)
and y(t)∝A(y(t), t).
(ii) A(t) and B(t) are said to be randomly comparison with each other if for
each t∈Ω, x(t)∈X, A(x(t), t)∝B(x(t), t) (denoted by A(t)∝B(t)).
Obviously, if A(t) is a randomly comparison, then A(t)∝I(t) (where I(t) is an
random identity mapping on the X).
Definition 13. Let Xbe a real ordered Banach space, Pbe a normal cone
with normal constant Nin X, Ω be a set in X, A : Ω ×X→Xbe a random
mapping. A random mapping A(t) is said to be randomly β(t)-order compres-
sion with respect to a measurable mapping β: Ω →(0,1) if A(t) is a randomly
comparative with respect to the measurable mapping β: Ω →(0,1) such that
for any t∈Ω,
A(x(t), t)⊕A(y(t), t)≤β(t)(x(t)⊕y(t)),
holds.
690 SALAHUDDIN AND B. S. LEE
Definition 14. Let Xbe a real ordered Banach space, Pbe a normal cone
with normal constant Nin the X, Ω be a nonempty open subset of Xin which
the ttakes values, A, B : Ω ×X→Xbe the two random mappings, Ibe an
identity mapping on the X×X.
(i) A mapping A: Ω ×X→Xis said to be randomly restricted accretive
mapping if A(t) is randomly comparative and there exists two measur-
able mappings α, α0: Ω →(0,1) such that for all t∈Ω, x(t), y(t)∈X,
(A(x(t), t) + I(x(t), t)) ⊕(A(y(t), t) + I(y(t), t))
≤α(t)(A(x(t), t)⊕A(y(t), t)) + α
0(t)(x(t)⊕y(t))
holds where Iis an random identity mapping on Ω ×X.
(ii) A mapping A: Ω ×X→Xis said to be randomly B(t)-restricted ac-
cretive mapping, if A(t), B(t)∈X, t ∈Ω and A(t)∧B(t):Ω×X→
A(x(t), t)∧B(x(t), t)∈Xfor all t∈Ω all are randomly comparative
and they are randomly comparison for t∈Ω and there exists two mea-
surable mappings α, α0: Ω →(0,1) such that for any t∈Ω,and an
arbitrary x(t), y(t)∈X, holds
(A(x(t), t)∧B(x(t), t) + I(x(t), t)) ⊕(A(y(t), t)∧B(y(t), t) + I(y(t), t))
≤α(t)((A(x(t), t)∧B(x(t), t)) ⊕(A(y(t), t)∧B(y(t), t)))
+α
0(t)(x(t)⊕y(t)),
where I(x(t), t) = x(t) : X×Ω→Xis a random identity mapping.
Lemma 2.7. [22] Let Xbe an ordered Banach space, Pbe a normal cone with
normal constant Nin X, A :X→Xbe a comparative then for any x, y ∈X
(1) kθ⊕θk=kθk= 0,
(2) kx∨yk≤kxk∨kyk≤kxk+kyk,
(3) kx⊕yk≤kx−yk ≤ Nkx⊕yk,
(4) if x∝y, then kx⊕yk=kx−yk,
(5) limx→x0kA(x)−A(x0)k= 0,if and only if
lim
x→x0
A(x)⊕A(x0) = θ.
3. Main Results
In this section, we will show the convergence of the approximation of random
sequences for finding random solutions of the problem (2.1) and discussed the
relation between the initial random values x0(t) and the random solution of the
problem (2.1).
Theorem 3.1. Assume that (Ω,Σ, µ)is a complete σ-finite measurable space
and Xis a separable real Banach space, Pa normal cone with normal constant
N, in X, ≤is an ordered relation defined by the cone P, Ωis a nonempty
open subset of X in which the t∈Ω, let T: Ω ×X→F(X)be the random
fuzzy mapping satisfying condition (C)and ˜
T: Ω ×X→C B(X)be the random
FUZZY ORDERED RANDOM VARIATIONAL INEQUALITIES 691
continuous set valued mapping induced by Trespectively. Let a mapping ˜
Tbe the
random H-continuous ordered compression mapping with the measure η: Ω →
(0,1).Let A, g, f, ˜
T , B and (A+f):Ω×X→Xbe some random comparison
mappings to each others and A(t), B(t)be the random comparison mapping. Let
A(t)be the random β(t)-ordered compression measurable mapping with measure
β: Ω →(0,1). Let fbe a random σ(t)-ordered compression mapping with
measure σ: Ω →(0,1) and gbe a random γ(t)-ordered compression mapping
with measure γ: Ω →(0,1). If A+fis a random B(t)-restricted accretive
mapping for two random measurable mappings α, α0: Ω →(0,1) and ρ: Ω →
(0,1) is a any measure
ρ(t)[β(t)γ(t) + σ(t)η(t)] <1−α0(t)
α(t)(2)
holds. Then the fuzzy general nonlinear ordered random variational inequality
problem
[A(g(x(t), t), t) + f(u(t), t)] ≥θ, ∀x(t)∈X, u(t)∈˜
T(x(t), t), t ∈Ω,(3)
there exists a random solutions x∗(t)∈X, u∗(t)∈˜
T(x∗(t), t),for t∈Ωand for
any x0(t)∈X
kx∗(t)−x0(t)k≤{1 + N(α(t)ρ(t)(β(t)γ(t) + σ(t)η(t)) + α0(t))
1−(α(t)ρ(t)(β(t)γ(t) + σ(t)η(t)) + α0(t))}×
kA(g(x0(t), t), t) + f(u0(t), t)−x0(t)k.
Moreover, xn(t)→x∗(t), un(t)→u∗(t)where {xn(t)}and {un(t)}are the
random sequences obtained by random iterative algorithm.
Proof. Let Xbe a real ordered Banach space, Pbe a normal cone with normal
constant Nin the X, ≤be an ordered relation defined by the cone P, Ω be
a nonempty open subset of Xin which the ttakes values. For any t∈Ω and
x1(t), x2(t)∈Xand ρ: Ω →(0,1) is a measurable mapping. Let x0(t)∝x1(t).
Then for x0(t)∈Xand
x1(t) = ρ(t)[A(g(x0(t), t), t) + f(u0(t), t)] ∧B(x0(t), t) + I(x0(t), t),
for u0(t)∈˜
T(x0(t), t).
Since A(t) and B(t) be the randomly ordered comparison to each other so
that x0(t)∝x1(t).Further we can have a random iterative algorithm for fuzzy
general nonlinear ordered random variational inequalities (1), i.e.
A(g(x(t), t), t) + f(u(t), t)≥0, t ∈Ω, x(t)∈X, u(t)∈˜
T(x(t), t)
in ordered Banach space X:
xn+1(t) = ρ(t)[A(g(xn(t), t), t) + f(un(t), t)] + I(xn(t), t),
for a measurable mapping ρ: Ω →(0,1) where n= 0,1,2,· · · .It follows
from the condition A, g, f, B, ˜
Tand (A+f) : Ω ×X→Xof the random
comparison mappings and t∈Ω, x0(t)∝x1(t) that is xn(t)∝xn+1(t).By
692 SALAHUDDIN AND B. S. LEE
using the random B(t) restricted accretive mapping of A+fand the randomly
β(t)- ordered compression of A(t),the random γ(t)-ordered compression of a
random mapping g: Ω ×X→Xand ˜
Tbe the random H-continuous ordered
compression mapping with the measurable mapping η: Ω →(0,1) and Lemma
2.4(6), we have
θ≤xn+1(t)⊕xn(t)
≤[ρ(t)(A(g(xn(t), t), t) + f(un(t), t)) + I(xn(t), t)]
⊕[ρ(t)(A(g(xn−1(t), t), t) + f(un−1(t), t)) + I(xn−1(t), t)]
≤α(t)[ρ(t)(A(g(xn(t), t), t) + f(un(t), t)) ⊕(ρ(t)(A(g(xn−1(t), t), t)
+f(un−1(t), t)))] + α
0(t)(xn(t)⊕xn−1(t))
≤α(t)ρ(t)[(A(g(xn(t), t), t)⊕A(g(xn−1(t), t), t))
+ (f(un(t), t)⊕f(un−1(t), t))] + α
0(t)(xn(t)⊕xn−1(t))
≤α(t)ρ(t)[β(t)(g(xn(t), t)⊕g(xn−1(t), t)) + σ(t)(un(t)⊕un−1(t))]
+α
0(t)(xn(t)⊕xn−1(t))
≤α(t)ρ(t)[β(t)γ(t)(xn(t)⊕xn−1(t))
+σ(t)H(˜
T(xn(t), t),˜
T(xn−1(t), t))] + α
0(t)(xn(t)⊕xn−1(t))
≤α(t)ρ(t)[β(t)γ(t)(xn(t)⊕xn−1(t)) + σ(t)η(t)(1 + 1
n)(xn(t)⊕xn−1(t))]
+α
0(t)(xn(t)⊕xn−1(t))
≤(α(t)ρ(t)(β(t)γ(t) + σ(t)η(t)(1 + 1
n)) + α
0(t))(xn(t)⊕xn−1(t))
≤(α(t)ρ(t)(β(t)γ(t) + σ(t)η(t)(1 + 1
n)) + α
0(t))nN(x1(t)⊕x0(t)).
Since
un(t)⊕un−1(t) = H(˜
T(xn(t), t),˜
T(xn−1(t), t)) ≤(1 + 1
n)η(t)(xn(t)⊕xn−1(t)).
By Lemma 2.2 and Definition 9, we obtain
kxn(t)−xn−1(t)k≤4nNkx1(t)−x0(t)k(4)
where 4n=α(t)ρ(t)(β(t)γ(t) + σ(t)η(t)(1 + 1
n)) + α0(t).
Let 4=α(t)ρ(t)(β(t)γ(t) + σ(t)η(t)) + α0(t) with
ρ(t)[β(t)γ(t) + σ(t)η(t)] <1−α0(t)
α(t).
Hence for any m > n > 0 we have
kxm(t)−xn(t)k ≤
m−1
X
i=n
kxi+1(t)−xi(t)k ≤ Nkx1(t)−x0(t)k
m−1
X
i=n
4i.
FUZZY ORDERED RANDOM VARIATIONAL INEQUALITIES 693
It follows from the condition (2) that 0 ≤4≤1 and
kxm(t)−xn(t)k → 0,as n→ ∞
and so {xn(t)}is a random Cauchy sequence in complete space X. Let xn(t)→
x∗(t) as n→ ∞(x∗(t)∈X, t ∈Ω).From the condition that A, g, ˜
Tare randomly
continuous and ρ: Ω →(0,1) is a measurable space, we can have
x∗(t) = lim
n→∞
xn+1(t)
= lim
n→∞[ρ(t)(A(g(xn(t), t), t) + f(un(t), t))] + I(xn(t), t)
= lim
n→∞
ρ(t)A(g(xn(t), t), t) + ρ(t) lim
n→∞
f(un(t), t) + lim
n→∞
I(xn(t), t)
=ρ(t)A(g( lim
n→∞
xn(t), t), t) + ρ(t)f( lim
n→∞
un(t), t) + I( lim
n→∞
xn(t), t)
=ρ(t)A(g( lim
n→∞
xn(t), t), t) + ρ(t)f(˜
T( lim
n→∞
xn(t), t)) + I( lim
n→∞
xn(t), t)
=ρ(t)A(g(x∗(t), t), t) + ρ(t)f(u∗(t), t) + x∗(t).
Hence x∗(t) is a solution of equation (1). By random H-continuous order com-
pression of ˜
T, we have
un(t)⊕un−1(t)≤ H(˜
T(xn(t), t),˜
T(xn−1(t), t))
≤(1 + 1
n)η(t)(xn(t)⊕xn−1(t)).
It follows that {un(t)}is also a random Cauchy sequence in Xand completeness
of X, un(t)→u∗(t).Note that un(t)∈˜
T(xn(t), t), we have
ku∗(t)−˜
T(x∗(t), t)k ≤ ku∗(t)−un(t)k+H(˜
T(xn(t), t),˜
T(x∗(t), t))
≤ ku∗(t)−un(t)k+ (1 + 1
n)η(t)kxn(t)−x∗(t)k
→0 as n→ ∞.
Hence ku∗(t)−un(t)k= 0 and therefore u∗(t)∈˜
T(x∗(t), t) is also a random
solution of (1). We know that (x∗(t), u∗(t)) is a random solution set of equation
(1). It follows that
A(g(xn(t), t), t) + f(un(t), t)∝x∗(t), n = 0,1,2,· · · , t ∈Ω
from Lemma 2.3 and (4)
kx∗(t)−x0(t)k= lim
n→∞ kxn(t)−x0(t)k
≤lim
n→∞
n
X
i=1
kxi+1(t)−xi(t)k
≤lim
n→∞
N
n
X
i=2
4n−1kx1(t)−x0(t)k+kx1(t)−x0(t)k
≤1 + N(α(t)ρ(t)(β(t)γ(t) + σ(t)η(t)) + α0(t))
1−(α(t)ρ(t)(β(t)γ(t) + σ(t)η(t)) + α0(t))×
694 SALAHUDDIN AND B. S. LEE
kA(g(x0(t), t), t) + f(u0(t), t)−x0(t)k
holds. This complete the proof.
Lemma 3.2. Assume that (Ω,Σ, µ)is a complete σ-finite measurable space and
Xis a separable real Banach space, Pa normal cone with normal constant N
in the space X, ≤is a partial ordered relation defined by the cone P, Ωis a
nonempty open subset of Xin which the t∈Ω. Let T: Ω ×X→F(X)be
the random fuzzy mapping satisfy the conditions (C)and ˜
T: Ω ×X→C B(X)
be the random continuous set valued mapping induced by Trespectively. Let
a mapping ˜
Tbe the random H-continuous ordered compression mapping with
measure η: Ω →(0,1).Let f, A, g, B , ˜
T , A +fand (A+f)∧B: Ω ×X→X
be randomly comparison mapping respectively and two of them can be compared
each other. If a equation
(A(g(x(t), t), t)+f(u(t), t))∧B(x(t), t) = θ, θ ∈X, t ∈Ω, u(t)∈˜
T(x(t), t) (5)
has a random solution sets (x∗(t), u∗(t)).Then (x∗(t), u∗(t)) is a random solu-
tion sets of fuzzy general nonlinear ordered random variational inequalities in
ordered Banach spaces.
Proof. This directly follows from the definition of the ∧and the condition that
A, g, B, f, ˜
T , A +fand (A+f)∧B: Ω ×X→Xbe randomly comparison
respectively and any two of them can compared each other.
From Theorem 3.1 and Lemma 3.2, we have the following Theorem.
Theorem 3.3. Assume that (Ω,Σ, µ)is a complete σ-finite measurable space
and Xis a separable real Banach space, Pa normal cone with normal constant
Nin X, ≤is an ordered relation defined by the cone P, Ωis a nonempty open
subset of Xin which the t∈X. Let T: Ω ×X→F(X)be the random fuzzy
mapping satisfying condition (C)and ˜
T: Ω ×X→CB(X)be the continuous
random set valued mapping induced by T, respectively. Let mapping ˜
Tbe the
randomly H-continuous ordered comparison mapping with the measure η: Ω →
(0,1). Let A, g, f, A +f, ˜
T , B and (A+f)∧B: Ω ×X→Xbe the some
comparison random mappings to each other and A(t), B(t)be the random βi(t)-
ordered compression measurable mapping with measure βi(t):Ω→(0,1) for
i= 1,2.Let fbe a random σ-ordered compression mapping with measure σ:
Ω→(0,1) and gbe a randomly γ(t)-ordered compression mapping with measure
γ: Ω →(0,1).If A+fis a randomly B(t)- restricted accretive mapping with
respect to t∈Ω, for two measurable mappings α, α0: Ω →(0,1) and ρ: Ω →
(0,1) is a any measure
ρ(t)[(β1(t)γ(t) + σ(t)η(t)) ∨β2(t)] <1−α0(t)
α(t)(6)
holds. Then the fuzzy general nonlinear ordered random variational inequalities
[A(g(x(t), t), t)+f(u(t), t)]∧B(x(t), t) = θfor x(t)∈X, u(t)∈˜
T(x(t), t), t ∈Ω.
(7)
FUZZY ORDERED RANDOM VARIATIONAL INEQUALITIES 695
There exists x∗(t)∈X, u∗(t)∈˜
T(x∗(t), t)for t∈Ω, which is a random solution
sets of problem (1) and for any x0(t)∈X,
kx∗(t)−x0(t)k ≤ 1 + N(α(t)ρ(t)((β1(t)γ(t) + σ(t)η(t)) ∨β2(t)) + α0(t))
1−(α(t)ρ(t)((β1(t)γ(t) + σ(t)η(t)) ∨β2(t)) + α0(t))×
NkA(g(x0(t), t), t) + f(u0(t), t)k∧kB(x0(t), t)k.
Moreover xn(t)→x∗(t), un(t)→u∗(t)where {xn(t)}and {un(t)}are the ran-
dom sequences obtained by random iterative algorithm for fuzzy mapping.
Proof. Let Xbe a real ordered Banach space, Pbe a normal cone with normal
constant Nin the X, ≤be an ordered relation defined by the cone P, Ω be
a nonempty open subset of Xin which the ttakes values. For any t∈Ω and
x1(t), x2(t)∈Xand ρ: Ω →(0,1) is a measurable mapping. Let x0(t)∝x1(t).
Then for x0(t)∈Xand u0(t)∈˜
T(x0(t), t),
x1(t) = ρ(t)[A(g(x0(t), t), t) + f(u0(t), t)] ∧B(x0(t), t) + I(x0(t), t).
Since A(t) and B(t) are the randomly comparison to each other so that x0(t)∝
x1(t).Further we can have a random iterative algorithm for fuzzy general non-
linear ordered random variational inequalities (7) in ordered Banach space X.
xn+1(t) = ρ(t)[A(g(xn(t), t), t) + f(un(t), t)] ∧B(xn(t), t) + I(xn(t), t),
for a measurable mapping ρ: Ω →(0,1) where n= 0,1,2,· · · .It follows from
the condition A, g, f, A +f , B, ˜
Tand (A+f)∧B: Ω ×X→Xof the random
comparison mappings and t∈Ω, x0(t)∝x1(t) that is xn(t)∝xn+1(t).By using
the random B(t)-restricted accretive mapping and the randomly β(t)-ordered
compression of A(t),the random γ(t)-ordered compression of a random mapping
g: Ω ×X→Xand ˜
Tbe the random H-continuous ordered compression
mapping with the measurable mapping η: Ω →(0,1) and Lemma 2.4(7), we
696 SALAHUDDIN AND B. S. LEE
have
θ≤xn+1(t)⊕xn(t)
≤[ρ(t)(A(g(xn(t), t), t) + f(un(t), t)) ∧B(xn(t), t) + I(xn(t), t)]
⊕[ρ(t)(A(g(xn−1(t), t), t) + f(un−1(t), t)) ∧B(xn−1(t), t) + I(xn−1(t), t)]
≤α(t)[(ρ(t)(A(g(xn(t), t), t) + f(un(t), t)) ∧B(xn(t), t))
⊕(ρ(t)(A(g(xn−1(t), t), t) + f(un−1(t), t)) ∧B(xn−1(t), t))]
+α
0(t)(xn(t)⊕xn−1(t))
≤α(t)ρ(t)[(A(g(xn(t), t), t) + f(un(t), t)) ⊕(A(g(xn−1(t), t), t)
+f(un−1(t), t)) ∨(B(xn(t), t)⊕B(xn−1(t), t))] + α
0(t)(xn(t)⊕xn−1(t))
≤ρ(t)α(t)[((A(g(xn(t), t), t)⊕A(g(xn−1(t), t), t))
+ (f(un(t), t)⊕f(un−1(t), t))) ∨(B(xn(t), t)⊕B(xn−1(t), t))]
+α
0(t)(xn(t)⊕xn−1(t))
≤ρ(t)α(t)[β1(t)(g(xn(t), t)⊕g(xn−1(t), t)) + σ(t)(un(t)⊕un−1(t))
∨β2(t)(xn(t)⊕xn−1(t))] + α
0(t)(xn(t)⊕xn−1(t))
≤ρ(t)α(t)[β1(t)γ(t)(xn(t)⊕xn−1(t)) + σ(t)H(˜
T(xn(t), t),˜
T(xn−1(t), t))
∨β2(t)(xn(t)⊕xn−1(t))] + α
0(t)(xn(t)⊕xn−1(t))
≤ρ(t)α(t)[β1(t)γ(t)(xn(t)⊕xn−1(t)) + σ(t)η(t)(1 + 1
n)(xn(t)⊕xn−1(t))
∨β2(t)(xn(t)⊕xn−1(t))] + α
0(t)(xn(t)⊕xn−1(t))
≤(ρ(t)α(t)[(β1(t)γ(t) + σ(t)η(t)(1 + 1
n)) ∨β2(t)] + α
0(t))(xn(t)⊕xn−1(t)).
Since ˜
Tis randomly H-continuous ordered compression mapping with measur-
able mapping η: Ω →(0,1), we have
(un(t)⊕un−1(t)) ≤ H(˜
T(xn(t), t),˜
T(xn−1(t), t))
≤η(t)(1 + 1
n)(xn(t)⊕xn−1(t)).
Now continuing these process, we have
0≤xn(t)⊕xn−1(t)
≤(ρ(t)α(t)[(β1(t)γ(t) + σ(t)η(t)(1 + 1
n)) ∨β2(t)] + α
0(t))(xn(t)⊕xn−1(t))
≤(ρ(t)α(t)[(β1(t)γ(t) + σ(t)η(t)(1 + 1
n)) ∨β2(t)] + α
0(t))nN(x1(t)⊕x0(t)).
By Lemma 2.2 and Definition 9, we obtain
kxn(t)−xn−1(t)k≤4nNkx1(t)−x0(t)k(8)
FUZZY ORDERED RANDOM VARIATIONAL INEQUALITIES 697
where 4n= (α(t)ρ(t)[(β1(t)γ(t) + σ(t)η(t)(1 + 1
n)) ∨β2(t)] + α0(t)).
Let 4= (α(t)ρ(t)[(β1(t)γ(t) + σ(t)η(t)) ∨β2(t)] + α0(t)) with
ρ(t)[(β1(t)γ(t) + σ(t)η(t)) ∨β2(t)] <1−α0(t)
α(t).
From the assumption (8) and Lemma 2.2. Hence for any m > n > 0 we have
kxm(t)−xn(t)k ≤ N
m−1
X
i=n
kxi+1(t)−xi(t)k ≤ Nkx1(t)−x0(t)k
m−1
X
i=n
4i.
It follows from the condition (6) that 0 ≤4≤1 and
kxm(t)−xn(t)k → 0,as n→ ∞ for t∈Ω.
So {xn(t)}is a random Cauchy sequence in complete space X. Let xn(t)→x∗(t)
as n→ ∞(x∗(t)∈X, t ∈Ω).From the condition that A, g, ˜
T , f , A +fand
(A+f)∧B: Ω ×X→Xare randomly continuous and ρ: Ω →(0,1), we can
have
x∗(t) = lim
n→∞
xn+1(t)
= lim
n→∞(ρ(t)(A(g(xn(t), t), t) + f(un(t), t)) ∧B(xn(t), t) + I(xn(t), t))
= (ρ(t) lim
n→∞
A(g(xn(t), t), t) + ρ(t) lim
n→∞
f(un(t), t)) ∧lim
n→∞
B(xn(t), t)
+ lim
n→∞
I(xn(t), t)
= (ρ(t)A(g( lim
n→∞
xn(t), t), t) + ρ(t)f( lim
n→∞
un(t), t)) ∧B( lim
n→∞
xn(t), t)
+I( lim
n→∞
xn(t), t)
= (ρ(t)A(g( lim
n→∞
xn(t), t), t) + ρ(t)f(˜
T( lim
n→∞
xn(t), t))) ∧B( lim
n→∞
xn(t), t)
+I( lim
n→∞
xn(t), t)
= (ρ(t)A(g(x∗(t), t), t) + ρ(t)f(u∗(t), t)) ∧B(x∗(t), t) + x∗(t),
where
lim
n→∞
un(t) = ˜
T( lim
n→∞
xn(t), t) = ˜
T(x∗(t), t) = u∗(t).
Hence x∗(t) is a solution of equation (1), i.e.,
(A(g(x(t), t), t) + f(u(t), t)) ∧B(x(t), t) = 0.
By random H-continuous order compression of ˜
Twe have
un+1(t)⊕un(t)≤ H(˜
T(xn+1(t), t),˜
T(xn(t), t))
≤(1 + 1
n)η(t)(xn+1(t)⊕xn(t)).
It follows that {un(t)}is also a random Cauchy sequence in Xand completeness
of X, un(t)→u∗(t) as n→ ∞.
698 SALAHUDDIN AND B. S. LEE
Note that un(t)∈˜
T(xn(t), t), we have
ku∗(t)−˜
T(x∗(t), t)k≤ku∗(t)−un(t)k+H(˜
T(xn(t), t),˜
T(x∗(t), t))
≤ ku∗(t)−un(t)k+ (1 + 1
n)η(t)kxn(t)−x∗(t)k
→0 as n→ ∞.
Hence
ku∗(t)−˜
T(x∗(t), t)k= 0
and therefore u∗(t)∈˜
T(x∗(t), t).
We know that (x∗(t), u∗(t)) is a random solution sets of equation (7). It follows
that
(A(g(xn(t), t), t) + f(un(t), t)) ∧B(xn(t), t)∝x∗(t), n = 0,1,2,· · · , t ∈Ω,
from Lemma 2.3 and (8)
kx∗(t)−x0(t)k= lim
n→∞ kxn(t)−x0(t)k ≤ lim
n→∞
n
X
i=1
kxi+1(t)−xi(t)k
≤lim
n→∞
N
n
X
i=2
4n−1kx1(t)−x0(t)k+kx1(t)−x0(t)k
≤1 + N(α(t)ρ(t)(β1(t)γ(t) + σ(t)η(t)) ∨β2(t) + α0(t))
1−(α(t)ρ(t)(β1(t)γ(t) + σ(t)η(t)) ∨β2(t) + α0(t))k(A(g(x0(t), t), t)
+f(u0(t), t)) ∧B(x0(t), t)k
≤1 + N(α(t)ρ(t)(β1(t)γ(t) + σ(t)η(t)) ∨β2(t) + α0(t))
1−(α(t)ρ(t)(β1(t)γ(t) + σ(t)η(t)) ∨β2(t) + α0(t))N{kA(g(x0(t), t), t)
+f(u0(t), t)k} ∧ kB(x0(t), t)k
≤1+ N(α(t)ρ(t)(β1(t)γ(t) + σ(t)η(t)) ∨β2(t) + α0(t))
1−(α(t)ρ(t)(β1(t)γ(t) + σ(t)η(t)) ∨β2(t) + α0(t))N{kA(g(x0(t), t), t)k
+kf(u0(t), t)k} ∧ kB(x0(t), t)k
holds. This complete the proof.
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Salahuddin
Department Of Mathematics
Jazan University, Jazan,
Kingdom of Saudi Arabia
E-mail address:salahuddin12@mailcity.com
Byung-Soo Lee
Department of Mathematics
Kyungsung University, Busan
608-736, Korea
E-mail address:bslee@ks.ac.kr
