Abstract. In this paper, we investigate the existence and uniqueness of fuzzy solution
for a class of general hyperbolic equations with state-dependent delays. We will prove
the well-posedness of problem doesn’t depend on the domain and boundary data as well
as initial data. Our method is based on Banach fixed point theorem in completely new
weighted metric space. Fuzzy solutions in these cases are comprehended in the sense of
Buckley and Feuring. Moreover, some examples are presented to illustrated the results.
In these we use the continuity of the Zadeh’s extension principle combined with Matlab
simulation to show the surface of numerical fuzzy solutions.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0028
Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 14-29
This paper is available online at
EXISTENCE AND UNIQUENESS OF SOLUTIONS
OF SOME GENERAL FUZZY PARTIAL HYPERBOLIC
FUNCTIONAL DIFFERENTIAL EQUATIONS
Nguyen Phuong Dong1, Hoang Thi Phuong Thao2, Vu Tuan3 and Pham Thi Thao3
1Hanoi Pedagogical University Number 2
2Foreign Language Specialized School,
University of Languages and International Studies, VNU, Hanoi
3Faculty of Information Technology,
People’s Police University of Technology and Logistics, Bac Ninh
Abstract. In this paper, we investigate the existence and uniqueness of fuzzy solution
for a class of general hyperbolic equations with state-dependent delays. We will prove
the well-posedness of problem doesn’t depend on the domain and boundary data as well
as initial data. Our method is based on Banach fixed point theorem in completely new
weighted metric space. Fuzzy solutions in these cases are comprehended in the sense of
Buckley and Feuring. Moreover, some examples are presented to illustrated the results.
In these we use the continuity of the Zadeh’s extension principle combined with Matlab
simulation to show the surface of numerical fuzzy solutions.
Keywords: Partial hyperbolic Functional differential equations, fuzzy solution, local
conditions, boundary conditions, contraction operator.
1. Introduction
Fuzzy partial differential equations (FPDEs) were first introduced by Buckley and Feuring
in [1]. They used the concepts Hukuhara-differentiability to obtain a strategy to find BF-solution
or S-solution of some elementary FPDEs by crisp solution and the extension principle. Since then
there appeared some papers concerning different approaches to not only the theoretical field of
FPDEs, but also the applications of FPDEs, fuzzy partial differential inclusions and numerical
methods for these equations. For instance, Allahviranloo et al. [2] used the same strategy of
Buckley and Feuring to find the exact solutions of fuzzy wave-like equations in one and two
dimensions with variable coefficients by a variational iteration method. Bertone et al. [3] applied
a fuzzification process using Zadeh’s extension principle to obtain a fuzzy solution for some type
of heat, wave and Poisson equations.
In this paper, we will investigate the well-posedness of a general class of fuzzy hyperbolic
equations with state-dependent delays, known as fuzzy partial hyperbolic functional differential
Received July 6, 2016. Accepted August 5, 2016.
Contact Nguyen Phuong Dong, e-mail address: phuongdongspt@gmail.com
14
Existence and uniqueness of solutions of some general fuzzy...
equations (FPHFDEs), as they can provide good models for dynamics of real phenomena which
are subjected to one kind of uncertainties [4, 5, 6]. This kind of equations was initialed by Arara
et al. in [7], in which they considered the local and nonlocal initial problem for some classes
of hyperbolic equations. However, this results based on some complicated conditions on data
and domain. Generally, existence theorems need a condition which may restrict the domain to
a small scale. To overcome this restriction, in this paper we introduce new weighted metric space.
For which with a suitable choice of weighted numbers we prove some existence and uniqueness
theorems for whole domain without any constraint on the boundary of domain.
Moreover, in this paper, we introduce one class of FPHFDEs with integral boundary
conditions after that we investigate some results on the existence and uniqueness of fuzzy solutions
for this class of equations. As we know, integral boundary conditions have various applications in
applied fields such as blood flow problems, chemical engineering, thermoplasticity, underground
water flow, population dynamics, and so forth.
2. Content
2.1. Preliminaries
We denote the set consisting of all nonempty compact, convex subsets of Rn by KnC . Let
A and B be two nonempty bounded subsets of KnC . Denote by ||.|| a norm in Rn. The distance
between A and B is defined by the Hausdorff metric
dH(A,B) = max
{
sup
a∈A
inf
b∈B
||a− b|| , sup
b∈B
inf
a∈A
||a− b||
}
,
then (KnC , dH) is a complete metric space.
Let En be the space of fuzzy sets on Rn, that are nonempty subsets {(x, u(x)) : x ∈ Rn}
in Rn × [0, 1] of certain functions u: Rn → [0, 1] being normal, fuzzy convex, semi-continuous
and compact support, where [u]0 = {x ∈ Rn : u(x) > 0} is denoted for support of u. The α-cuts
or level sets of u are defined by
[u]α = {x ∈ Rn : u(x) ≥ α} for each 0 < α ≤ 1.
Supremum metric on En is defined by the Hausdorff metric distance between the level sets
of the fuzzy numbers
d∞(u, v) = sup
0<α≤1
Hd ([u]
α , [v]α)
for all u, v ∈ En. It is obviously that (En, d∞) is a complete metric space.
Definition 2.1. A mapping f : J = [x1, x2]× [y1, y2]→ En is called continuous at (s0, t0) ∈ J
if the multi-valued mapping fα (s, t) = [f (s, t)]
α is continuous at (s0, t0) with respect to the
Hausdorff metric dH for all α ∈ [0, 1].
For any positive real numbers r, a, b, we denote Ja = [0, a], Jb = [0, b], J0 = [−r, 0] ×
[−r, 0], Jr = [−r, a]× [−r, b], J¯r = Jr\(0, a] × (0, b] and Jab = [0, a]× [0, b].
The supremum metric d0∞ on C(J0, En) the space of all continuous functions
f : J0 → En is defined by
d0∞(f, g) := max
(ω,θ)∈J0
d∞(f(ω, θ), g(ω, θ))
15
Nguyen Phuong Dong, Vu Tuan and Hoang Thi Phuong Thao
for all f, g ∈ C(J0, En) and (ω, θ) ∈ J0.
In this paper, we also denote by C(Jr, En) the space of all continuous functions f : Jr → En
with the supremum weighted metric Hλ defined by
Hλ(f, g) = sup
(s,t)∈Jr
{
d∞(f (s, t) , g (s, t))e−λ(s+t)
}
.
Since (En, d∞) is a complete metric space, it can be shown that
Proposition 2.1. (C(Jr, E
n),Hλ) is a complete metric space for λ > 0 arbitrary.
Definition 2.2. A mapping f : J ×En → En is called continuous at point (s0, t0, u0) ∈ J ×En
provided that for any fixed α ∈ [0, 1] and arbitrary ǫ > 0, there exists δ(ǫ, α) > 0 such that
dH ([f (s, t, u)]
α , [f (s0, t0, u0)]
α) < ǫ
whenever (s, t, u) ∈ J × En satisfying
max
{
|s− s0| , |t− t0| , dH ([u]α , [u0]α)
}
< δ(ǫ, α).
The notion of integrability considered is the Aumann-integrability defined as
Definition 2.3. We say that f : J → En is strongly measurable if, for all α ∈ (0, 1], the set-valued
mapping fα : J → KnC given by fα(t) = [f(t)]α is Lebesgue-measurable. The integral of f on J
is defined levelsetwise by[∫ x2
x1
∫ y2
y1
f (s, t) dsdt
]α
=
∫ x2
x1
∫ y2
y1
[f (s, t)]αdsdt
= {
∫ x2
x1
∫ y2
y1
v (s, t) dsdt|v : J → Rnis a measurable selection for [f (s, t)]α}
for all α ∈ (0, 1]. We denote it by ∫ x2x1 ∫ y2y1 f (s, t) dsdt. A function f : J → En is integrable on J
if
∫ x2
x1
∫ y2
y1
f (s, t) dsdt ∈ En.
Definition 2.4. Given u, v ∈ En, if there exists w ∈ En such that u = v + w, we call w = u− v
the Hukuhara difference of u and v.
Definition 2.5. Given mapping f : J → En, we say that f is Hukuhara partial differentiable with
respect to x at (x0, y0) ∈ J if for each h > 0 the Hukuhara-difference f(x0 +∆t, y) − f(x0, y)
and f(x0, y)−f(x0−∆t, y0) exists inEn for every 0 < ∆t < h and if it exists ∂f (x0, y0)
∂x
∈ En
such that
lim
h→0+
d∞
(
f (x0 +∆t, y0)− f (x0, y0)
h
,
∂f (x0, y0)
∂x
)
= 0
and
lim
h→0+
d∞
(
f (x0, y0)− f (x0 −∆t, y0)
h
,
∂f (x0, y0)
∂x
)
= 0.
In this case,
∂f (x0, y0)
∂x
∈ En is called the Hukuhara partial derivative of f at (x0, y0).
The fuzzy partial derivative of f with respect to y and higher order of fuzzy partial derivative
of f at the point (x0, y0) ∈ J are defined similarly.
16
Existence and uniqueness of solutions of some general fuzzy...
2.2. Fuzzy partial hyperbolic functional differential equations with
local conditions
For (x, y) ∈ Jab, we denote the state-dependent delays u(x,y) : J0 → En, by
u(x,y)(s, t) = u(x+ s, y + t), (s, t) ∈ J0.
In this section, we consider the FPHFDEs which is the generalization models of Arara et al. [7]
∂2u(x, y)
∂x∂y
=
∂(p(x, y)u(x, y))
∂y
+
∂(q(x, y)u(x, y))
∂x
+ c(x, y)u(x, y) + f(x, y, u(x,y)) (2.1)
for (x, y) ∈ Jab with the initial condition
u(x, y) = ϕ(x, y), (x, y) ∈ J¯r (2.2)
and the local conditions are
u(x, 0) = η1(x), x ∈ Ja, u(0, y) = η2(y), y ∈ Jb, ϕ(0, 0) = η1(0) = η2(0), (2.3)
where p, q, c ∈ C(Jab,R), η1 ∈ C(Ja, En), η2 ∈ C(Jb, En) are given functions, ϕ ∈ C(J¯r, En)
and f : Jab × C(J0, En)→ En is a L1− Carathéodory function.
Definition 2.6. Mapping f : Jab × C(J0, En) → En is called satisfying Lipschitz condition if
there exists a positive real number K such that
d∞(f(x, y, u), f(x, y, v)) ≤ Kd0∞(u, v) (2.4)
holds for all u, v ∈ C(J0, En), (x, y) ∈ Jab.
Definition 2.7. A function u ∈ C(Jr, En) is called a solution of the problem (2.1)- (2.3) if it
satisfies
u(x, y) =ψ(x, y) +
∫ x
0
p(s, y)u(s, y)ds +
∫ y
0
q(x, t)(x, t)dt +
∫ x
0
∫ y
0
c(s, t)u(s, t)dsdt
+
∫ x
0
∫ y
0
f
(
s, t, u(s,t)
)
dsdt,
where ψ(x, y) is a fuzzy function that satisfies
ψ(x, y) + ϕ(0, 0) +
∫ x
0
p(s, 0)η1(s)ds +
∫ y
0
q(0, t)η2(t)dt = η1(x) + η2(y)
for all (x, y) ∈ Jab and u(x, y) = ϕ(x, y), (x, y) ∈ J¯r .
Theorem 2.1. Suppose that function f : Jab × C(J0, En) → En satisfies Lipschitz condition
(2.4). Then the problem (2.1)- (2.3) has a unique solution in C(Jr, En).
17
Nguyen Phuong Dong, Vu Tuan and Hoang Thi Phuong Thao
Proof. We transform problem (2.1)-(2.3) into a fixed point problem as flowing steps
Step 1. Base on Definition 2.7, we construct an operator N1 : C(Jr, En) → C(Jr, En)
defined by
N1(u(x, y)) =
ϕ(x, y), if (x, y) ∈ J¯r
ψ(x, y) +
∫ x
0 p(s, y)u(s, y)ds +
∫ y
0 q(x, t)u(x, t)dt
+
∫ x
0
∫ y
0 c(s, t)u(s, t)dsdt +
∫ x
0
∫ y
0 f
(
s, t, u(s,t)
)
dsdt, if (x, y) ∈ Jab
where
ψ(x, y) + ϕ(0, 0) +
∫ x
0
p(s, 0)η1(s)ds+
∫ y
0
q(0, t)η2(t)dt = η1(x) + η2(y).
It can be seen that the local conditions (2.3) imply the continuity of operator N1.
Step 2.We will prove that N1 is a contraction operator.
Indeed, with u, v ∈ C(Jr, En) and (x, y) ∈ Jab we have
d∞(N1(u(x, y)), N1(v(x, y))) ≤ d∞(
∫ x
0
p(s, y)u(s, y)ds,
∫ x
0
p(s, y)v(s, y)ds)
+ d∞(
∫ y
0
q(x, t)u(x, t)dt,
∫ y
0
q(x, t)v(x, t)dt)
+ d∞(
∫ x
0
∫ y
0
c(s, t)u(s, t)dsdt,
∫ x
0
∫ y
0
c(s, t)v(s, t)dsdt)
+ d∞(
∫ x
0
∫ y
0
f(s, t, u(s,t))dsdt,
∫ x
0
∫ y
0
f(s, t, v(s,t))dsdt). (2.5)
For simplicity, we denote p = sup
(s,t)∈Jab
|p(s, t)|, q = sup
(s,t)∈Jab
|q(s, t)|, c = sup
(s,t)∈Jab
|c(s, t)|, and
obtain following assessment
d∞(
∫ x
0
p(s, y)u(s, y)ds,
∫ x
0
p(s, y)v(s, y)ds)
≤ sup
(s,t)∈Ja×Jb
|p(s, t)|d∞(
∫ x
0
u(s, y)ds,
∫ x
0
v(s, y)ds)
≤ p
∫ x
0
d∞(u(s, y), v(s, y))ds.
Then we receive
e−λ(x+y)d∞(
∫ x
0
p(s, y)u(s, y)ds,
∫ x
0
p(s, y)v(s, y)ds)
≤ pe−λ(x+y)
∫ x
0
d∞(u(s, y), v(s, y))e−λ(s+y)eλ(s+y)ds
≤ p
λ
Hλ(u, v). (2.6)
By doing the same arguments, we have
e−λ(x+y)d∞(
∫ y
0
q(x, t)u(x, t)dt,
∫ y
0
q(x, t)v(x, t)dt) ≤ q
λ
Hλ(u, v), (2.7)
18
Existence and uniqueness of solutions of some general fuzzy...
and
e−λ(x+y)d∞(
∫ x
0
∫ y
0
c(s, t)u(s, t)dsdt,
∫ x
0
∫ y
0
c(s, t)v(s, t)dsdt)
≤ ce−λ(x+y)
∫ x
0
∫ y
0
d∞(u(s, t), v(s, t))dsdt
≤ c
λ2
Hλ(u, v). (2.8)
Since f satisfies Lipschitz condition that leads to
e−λ(x+y)d∞(
∫ x
0
∫ y
0
f(s, t, u(s,t))dsdt,
∫ x
0
∫ y
0
f(s, t, v(s,t))dsdt)
≤ Ke−λ(x+y)
∫ x
0
∫ y
0
d0∞(u(s,t), v(s,t))dsdt
≤ K
λ2
Hλ(u, v). (2.9)
Otherwise, when (x, y) ∈ J¯r we have
Hλ(N1(u)(x, y), N1(v)(x, y)) = 0. (2.10)
Hence, from (2.5)- (2.10) we can see that
e−λ(x+y)d∞(N1(u(x, y)), N1(v(x, y))) ≤ [p+ q
λ
+
c+K
λ2
]Hλ(u, v)
holds for all (x, y) ∈ Jr.
That leads to
Hλ(N1(u(x, y)), N1(v(x, y))) ≤ [p+ q
λ
+
c+K
λ2
]Hλ(u, v), (2.11)
for all u, v ∈ C(Jr, En).
If we choose weighted number λ > 0 satisfying
p+ q
λ
+
c+K
λ2
p+ q +
√
(p+ q)2 + 4c+ 4K
2
(2.12)
then N1 is a contraction operator.
Step 3. By Banach’s fixed point theorem, N1 has a unique fixed point u with N1(u) = u,
i.e., u is a solution of Problem (2.1)-(2.3). From Definition 2.7, we can see that the fixed point u
is a unique solution of the problem (2.1)- (2.3).
Example 2.1.We consider the following FPHFDEs
Uxy − Uy + (x exp(−x− y)U)x + U(x, y)
= exp(−x− y + 2
3
)U(x,y)(−
1
3
,−1
3
) + C(x+ 2) exp(x+ y) + Cx+
1
3
C, (2.13)
19
Nguyen Phuong Dong, Vu Tuan and Hoang Thi Phuong Thao
for all (x, y) ∈ J12 = [0, 1] × [0, 2], with the local conditions
U(x, 0) = C(x+ 1) exp(x), U(0, y) = C exp(y), U(0, 0) = C (2.14)
for x ∈ [0, 1], y ∈ [0, 2], and C is a fuzzy number in universal interval I = [0,M ],M > 0. And
U(x, y) = C(x exp(y) + xy + 1) exp(x+ y) (2.15)
for (x, y) ∈ J¯ 1
3
:= [−1
3
, 1] × [−1
3
, 2]\(0, 1] × (0, 2].
From (2.13), we have
F (x, y, U(x,y)) = exp(−x− y +
2
3
)U(x− 1
3
, y − 1
3
) + C(x+ 2) exp(x+ y) + Cx+
1
3
C,
satisfies Lipschitz condition with K = exp(
2
3
). Due to p = 1, q =
1
exp(1)
, c = 1,K = exp(
2
3
)
we can choose
λ >
1 + exp(1) +
√
1 + 2 exp(1)− 4 exp(83)− 3 exp(2)
2 exp(1)
,
It implies that the conditions of Theorem 3.1 are hold. Therefore, there exists a unique fuzzy
solution U of this problem. The deterministic solution of the crisp hyperbolic functional PDEs
corresponding to (2.13)-(2.15) is
u(x, y) = g(x, y, c) = c(x+ 1) exp(x+ y).
We now fuzzify this crisp solution to find fuzzy solution of FPDEs (2.13)-(2.15). To this end, let us
denote h(x, y, c) = c(x+2) exp(x+ y)+ cx+
1
3
c. We apply the fuzzification in c, and supposed
that the parametric form of corresponding fuzzy number C is
[C]α = [c1(α), c2(α)], α ∈ [0, 1].
By using the extension principle, we compute H from h and Y from g. We will show that
Y is the fuzzy solution of this problem. Indeed, since all the partials of h and g with respect to k
are positive. We see that
[H(x, y, C)]α =[H1(x, y, α),H2(x, y, α)]
=[c1(α)(x+ 2) exp(x+ y) + c1(α)x+
1
3
c1(α),
c2(α)(x + 2) exp(x+ y) + c2(α)x +
1
3
c2(α)]
satisfied conditions h1-h3. Therefore [H(x, y, C)]α is the α−cuts of fuzzy number C(x +
2) exp(x+ y) + Cx+
1
3
C.
Similarly
[Y (x, y, C)]α = [Y1(x, y, α), Y2(x, y, α)]
= [c1(α)(x + 1) exp(x+ y), c2(α)(x+ 1) exp(x+ y)]
20
Existence and uniqueness of solutions of some general fuzzy...
is the α−cuts of fuzzy number C(x+ 1) exp(x+ y). Define differential operator
ϕ(Dx,Dy)U(x, y) =Uxy − Uy + (x exp(−x− y)U)x + U(x, y)
− exp(−x− y + 2
3
)U(x− 1
3
, y − 1
3
)
and
S(x, y, α) = [ϕ(Dx,Dy)Y1(x, y, α), ϕ(Dx,Dy)Y2(x, y, α)]
We first check to see if Y (x, y) is differentiable. We compute
[ϕ(Dx,Dy)Y1(x, y, α), ϕ(Dx,Dy)Y2(x, y, α)]
which equals
[c1(α)(x + 2) exp(x+ y) + c1(α)x +
1
3
c1(α), c2(α)(x + 2) exp(x+ y) + c2(α)x+
1
3
c2(α)]
which are the α-cuts of fuzzy number C(x + 2) exp(x + y) + Cx +
1
3
C. Hence,Y (x, y) is
differentiable. Because all partials of h and g with respect to c are positive. Therefore, Y (x, y)
is a the fuzzy solution of (2.13)-(2.15) in the sense of Buckley and Feuring (see in [2, 1]), which
satisfies the boundary conditions. This solution may be written as
U(x, y) = C(x+ 1) exp(x+ y).
In order to visualize the surface of fuzzy solution, we represent some α−cut of the solutions
by consider fuzzy number C with membership function is Laplace function [8]
C(t) = 2
(
1 + exp
(π|t− c|
σ
√
6
))−1
, t > 0.
We can transform this fuzzy number into parametric form
[C]α = [c− σ
√
6
π
ln
( 2
α
− 1), c+ σ
√
6
π
ln
( 2
α
− 1)], α ∈ [0, 1].
The continuity of extension principle states that the α-cuts of
U(x, y) = C(x+ 1) exp(x+ y)
(for any fixed pair (x, y)) are
[U(x, y)]α = (x+ 1) exp(x+ y)
[
c− σ
√
6
π
ln
( 2
α
− 1), c+ σ
√
6
π
ln
( 2
α
− 1)].
Obviously, the deterministic solution is the preferred solution [U(x, y)]1, which means that it has
membership degree 1. So the membership function of U(x, y) is
U(x, y)(t) = 2
(
1 + exp
(π| t(x+1) exp(x+y) − c|
σ
√
6
))−1
.
Figure 1 shows the surface of fuzzy solutions U(x, y)(t) in the whole domain, the dark area
corresponds to surface of solution in phase space J0, the light area is the surface of sulution in
J12.
21
Nguyen Phuong Dong, Vu Tuan and Hoang Thi Phuong Thao
Figure 1. Surface of fuzzy solutions U(x, y)(t) by fuzzifying Laplacian fuzzy number C
2.3. Fuzzy solutions of fuzzy partial hyperbolic functional differential
equations with boundary conditions
In this section, we consider the general FPHFDEs in the form
∂2u(x, y)
∂x∂y
=
∂(p(x, y)u(x, y))
∂y
+
∂(q(x, y)u(x, y))
∂x
+ c(x, y)u(x, y) + f(x, y, u(x,y)) (2.16)
for (x, y) ∈ Jab, with the following integral boundary conditions
u(x, 0) +
∫ y
0
k1(x)u(x, t)dt = g1(x, y), (x, y) ∈ Jab (2.17)
u(0, y) +
∫ x
0
k2(y)u(s, y)ds = g2(x, y), (x, y) ∈ Jab (2.18)
and
u(x, y) = ϕ(x, y), (x, y) ∈ J¯r (2.19)
where p, q, c ∈ C(Jab,R), g1, g2 ∈ C(Jab, En), k1 ∈ C(Ja, En), k2 ∈ C(Jb, En) are given
functions, , ϕ ∈ C(J¯r, En) and f : Jab × C(J0, En)→ En is a L1− Carathéodory function and
satisfies the condition L.
Definition 2.8. A function u ∈ C(Jr, En) is called a solution of the problem (2.16) - (2.19) if it
satisfies integral equation
u(x, y) = φ(x, y)− k1(0)
∫ y
0
∫ x
0
k2(t)u(s, t)dsdt−
∫ y
0
k1(x)u(x, t)dt
−
∫ x
0
k2(y)u(s, y)ds +
∫ x
0
p(s, y)u(s, y)ds −
∫ x
0
p(s, 0)u(s, 0)ds +
∫ y
0
q(x, t)u(x, t)dt
−
∫ y
0
q(0, t)u(0, t)dt +
∫ x
0
∫ y
0
c(s, t)u(s, t)dsdt +
∫ x
0
∫ y
0
f(s, t, u(s,t))dsdt
22
Existence and uniqueness of solutions of some general fuzzy...
where φ(x, y) is a fuzzy function that satisfies
φ(x, y) + g1(0, y) = g1(x, y) + g2(x, y) + k1(0)
∫ y
0
g2(x, t)dt,
for all (x, y) ∈ Jab and u(x, y) = ϕ(x, y), (x, y) ∈ J¯r .
Theorem 2.2. The problem (2.16) - (2.19) has a unique solution in C(Jr, En) if the function
f : Jab × C(J0, En)→ En satisfies the condition L.
Proof. By doing the same arguments in Theorem 2.1 we transform problem (2.16) - (2.19) into
fixed point problem.
Step 1. Consider operator N2 : C(Jr, En)→ C(Jr, En) defined by
N2(u(x, y)) =
ϕ(x, y) if (x, y) ∈ J¯r
φ(x, y) − k1(0)
y∫
0
x∫
0
k2(t)u(s, t)dsdt
−
y∫
0
k1(x)u(x, t)dt −
x∫
0
k2(y)u(s, y)ds
+
x∫
0
p(s, y)u(s, y)ds −
x∫
0
p(s, 0)u(s, 0)ds
+
y∫
0
q(x, t)u(x, t)dt −
y∫
0
q(0, t)u(0, t)dt
+
x∫
0
y∫
0
c(s, t)u(s, t)dsdt+
x∫
0
y∫
0
f
(
s, t, u(s,t)
)
dsdt if (x, y) ∈ Jab
where
φ(x, y) + g1(0, y) = g1(x, y) + g2(x, y) + k1(0)
∫ y
0
g2(x, t)dt,
Step 2. We will prove that N2 is a contraction operator. In fact, for u, v ∈ C(Jr, En) and
23
Nguyen Phuong Dong, Vu Tuan and Hoang Thi Phuong Thao
when (x, y) ∈ Jab we have
d∞(N2(u(x, y)), N2(v(x, y)) ≤ d∞(
∫ y
0
k1(x)u(x, t)dt,
∫ y
0
k1(x)v(x, t)dt)
+ d∞(
∫ x
0
k2(y)u(s, y)ds,
∫ x
0
k2(y)v(s, y)ds)
+ d∞(k1(0)
∫ y
0
∫ x
0
k2(t)u(s, t)dsdt, k1(0)
∫ y
0
∫ x
0
k2(t)v(s, t)dsdt)
+ d∞(
∫ x
0
p(s, y)u(s, y)ds,
∫ x
0
p(s, y)v(s, y)ds)
+ d∞(
∫ x
0
p(s, 0)u(s, 0)ds,
∫ x
0
p(s, 0)v(s, 0)ds)
+ d∞(
∫ y
0
q(x, t)u(x, t)dt,
∫ y
0
q(x, t)v(x, t)dt)
+ d∞(
∫ x
0
q(0, t)u(0, t)dt,
∫ y
0
q(0, t)v(0, t)dt)
+ d∞(
∫ x
0
∫ y
0
c(s, t)u(s, t)dsdt,
∫ x
0
∫ y
0
c(s, t)v(s, t)dsdt)
+ d∞(
∫ x
0
∫ y
0
f(s, t, u(s,t))dsdt,
∫ x
0
∫ y
0
f(s, t, v(s,t))dsdt) (2.20)
Let p = sup
(s,t)∈Jab
|p(s, t)|, q = sup
(s,t)∈Jab
|q(s, t)|, c = sup
(s,t)∈Jab
|c(s, t)|, k1 = sup
x∈Ja
|k1(x)| and
k2 = sup
y∈Jb
|k2(y)|.
First of all, we estimate the first term in the right side of (2.20) after multiplying with e−λ(x+y)
e−λ(x+y)d∞(
∫ y
0
k1(x)u(x, t)dt,
∫ y
0
k1(x)v(x, t)dt)
≤ k1e−λ(x+y)
∫ y
0
d∞(u(x, t), v(x, t))dt
≤ k1
λ
Hλ(u, v)e
−λ(x+y)(eλ(x+y) − eλx)
≤ k1
λ
Hλ(u, v) (2.21)
Similarly, we obtai