EXISTENCE OF SOLUTIONS OF GENERAL NONLINEAR
FUZZY VOLTERRA-FREDHOLM INTEGRAL EQUATIONS
K. BALACHANDRAN AND K. KANAGARAJAN
Received 3 April 2004 and in revised form 31 October 2004
We study the problem of existence and uniqueness of solutions of a class of nonlinear
fuzzy Volterra-Fredholm integral equations.
1. Introduction
Fuzzy differential and integral equations have been studied by many authors [1, 2, 5, 6,
7, 14]. Kaleva [5] discussed the properties of differentiable fuzzy set-valued mappings by
means of the concept of H-differentiability introduced by Puri and Ralescu [9]. Seikkala
[11] defined the fuzzy derivative which is a generalization of the Hukuhara derivative [9]
and the fuzzy integral which is the same as that proposed by Dubois and Prade [3, 4].
Balachandran and Dauer [1] established the existence of solutions of perturbed fuzzy
integral equations. Subrahmanyam and Sudarsanam [13] studied fuzzy Volterra integral
equations. Park and Jeong [8] proved the existence and uniqueness of solutions of fuzzy
Volterra-Fredholm integral equations of the form
x(t) = F t,x(t),
t
0
T
f t,s,x(s) ds
0
g t,s,x(s) ds ,
(1.1)
and Balachandran and Prakash [2] studied the same problem for the nonlinear fuzzy
Volterra-Fredholm integral equations of the form
x(t) = f t,x(t) + F t,x(t),
t
0
T
g t,s,x(s) ds,
0
h t,s,x(s) ds .
(1.2)
The purpose of this paper is to prove the existence and uniqueness of solutions of general
nonlinear fuzzy Volterra-Fredholm integral equations of the form
x(t) = F t,x(t),
T
0
t
0
t
f1 t,s,x(s) ds,... ,
g1 t,s,x(s) ds,...,
T
0
0
fm t,s,x(s) ds,
gm t,s,x(s) ds ,
Copyright © 2005 Hindawi Publishing Corporation
Journal of Applied Mathematics and Stochastic Analysis 2005:3 (2005) 333–343
DOI: 10.1155/JAMSA.2005.333
(1.3)
0 ≤ t ≤ T.
334
Fuzzy Volterra-Fredholm integral equations
2. Preliminaries
Let PK (Rn ) denote the family of all nonempty, compact, convex subsets of Rn . Addition
and scalar multiplication in PK (Rn ) are defined as usual. Let A and B be two nonempty
bounded subsets of Rn . The distance between A and B is defined by the Hausdorff metric d(A,B) = max{supa∈A inf b∈B a − b,supb∈B inf a∈A a − b}, where · denote the
usual Euclidean norm in Rn . Then it is clear that (PK (Rn ),d) becomes a metric space.
Let I = [0,1] ⊆ R be a compact interval and denote
En = u : Rn → I : u satisfies (i)–(iv) below ,
(2.1)
where
(i) u is normal, that is, there exists an x0 ∈ Rn such that u(x0 ) = 1,
(ii) u is fuzzy convex,
(iii) u is upper semicontinuous,
(iv) [u]0 = cl{x ∈ Rn : u(x) > 0} is compact.
For 0 < α ≤ 1 denote [u]α = {x ∈ Rn : u(x) ≥ α}. Then from (i)–(iv) it follows that the
α-level set [u]α ∈ PK (Rn ) for all 0 ≤ α ≤ 1.
If g : Rn × Rn → Rn is a function, then using Zadeh’s extension principle we can extend
g to En × En → En by the equation
g̃(u,v)(z) = sup min u(x),v(y) .
(2.2)
z=g(x,y)
It is well known that [g̃(u,v)]α = g([u]α ,[v]α ) for all u,v ∈ En , 0 ≤ α ≤ 1, and continuous
function g. Further, we have [u + v]α = [u]α + [v]α , [ku]α = k[u]α , where k ∈ R. The real
numbers can be embedded in En by the rule c → ĉ(t) where

1
for t = c,
ĉ(t) = 
0 elsewhere.
(2.3)
Let D : En × En → R+ be defined by D(u,v) = sup0≤α≤1 d([u]α ,[v]α ), where d is the
Hausdorff metric defined in PK (Rn ). Then D is a metric in En and (En ,D) is a complete
metric space [5, 10]. Further D(u + w,v + w) = D(u,v) and D(λu,λv) = |λ|D(u,v) for
every u,v,w ∈ En and λ ∈ R.
It can be proved that D(u + v,w + z) ≤ D(u,w) + D(v,z) for u, v, w, and z ∈ En .
Definition 2.1 [5]. A mapping F : I → En is strongly measurable if for all α ∈ [0,1], the
set-valued map Fα : I → PK (Rn ) defined by Fα (t) = [F(t)]α is Lebesgue measurable when
PK (Rn ) has the topology induced by the Hausdorff metric d.
Definition 2.2 [5]. A mapping F : I → En is said to be integrably bounded if there is an
integrable function h(t) such that x(t) ≤ h(t) for every x ∈ F0 (t).
K. Balachandran and K. Kanagarajan 335
Definition 2.3 [10]. The integral of a fuzzy mapping F : I → En is defined level-wise by
I
α
=
F(t)dt
I
Fα (t)dt
=
I
(2.4)
f (t)dt | f : I → Rn is a measurable selection for Fα
for all α ∈ [0,1].
It has been proved by Puri and Ralescu [10] that
a strongly measurable and integrably
bounded mapping F : I → En is integrable (i.e., I F(t)dt ∈ En ).
Theorem 2.4. If F : I → En is continuous, then it is integrable.
n
and λ ∈ R. Then
Theorem
E be integrable
2.5. Let F,G : I →
=
F(t)dt
+
G(t)dt,
(i) I (F(t) + G(t))dt
I
I
(ii) I λF(t)dt = λ I F(t)dt,
(iii) D(F,G)
is integrable,
(iv) D( I F(t)dt, I G(t)dt) ≤ I D(F(t),G(t))dt.
Now we make the following assumptions.
(A1 ) Let J = [0,T], ∆ = {(t,s) : 0 ≤ s ≤ t ≤ T }. If fi ,gi ∈ C(∆ × En ,En ), i = 1,2,...,m,
F ∈ C(J × E(2m+1)n ,En ) and if x ∈ C(J,En ) and
t
z(t) = F t,x(t),
T
0
t
f1 t,s,x(s) ds,... ,
T
g1 t,s,x(s) ds,... ,
0
0
0
fm t,s,x(s) ds,
(2.5)
gm t,s,x(s) ds ,
then z ∈ C(J,En ).
(A2 ) There exist functions ωi (t,s, p), ω̂i (t,s, p) such that ωi , ω̂i ∈ C(∆ × R+ ,R+ ), R+ =
[0, ∞), which are nondecreasing in p and fulfil the conditions
≤ ωi t,s,D x(s),x(s) ,
D gi t,s,x(s) ,gi t,s,x(s) ≤ ω̂i t,s,D x(s),x(s)
for x,x ∈ C J,En , i = 1,2,...,m.
D fi t,s,x(s) , fi t,s,x(s)
(2.6)
(A3 ) There exists a function H(t, p1 , p2 ,..., p2m+1 ) defined for t ∈ J and 0 ≤ p1 ≤ p2 ≤
· · · ≤ p2m+1 < ∞ such that
(i) if u ∈ C(J,J) and
v(t) = H t,u(t),
T
0
then v ∈ C(J,J);
t
0
t
ω1 t,s,u(s) ds,... ,
ω̂1 t,s,u(s) ds,... ,
T
0
0
ωm t,s,u(s) ds,
ω̂m t,s,u(s) ds ,
(2.7)
336
Fuzzy Volterra-Fredholm integral equations
(ii) if u,u ∈ C(J,J) and u(t) ≤ u(t) for t ∈ J, then
t
H t,u(t),
T
0
0
t
ω1 t,s,u(s) ds,... ,
ω̂1 t,s,u(s) ds,... ,
≤ H t,u(t),
T
t
0
0
T
t
ω1 t,s,u(s) ds,... ,
ω̂m t,s,u(s) ds
0
T
ω̂1 t,s,u(s) ds,...,
0
ωm t,s,u(s) ds,
0
ωm t,s,u(s) ds,
for t ∈ J;
ω̂m t,s,u(s) ds
0
(2.8)
(iii) if un ∈ C(J,J), un+1 (t) ≤ un (t), t ∈ J, n = 0,1,2,..., and limn→∞ un (t) = u(t),
then
t
lim H t,un (t),
n→∞
T
0
0
T
ω̂1 t,s,un (s) ds,... ,
= H t,u(t),
T
0
t
ω1 t,s,un (s) ds,... ,
t
0
0
0
t
T
ω̂1 t,s,u(s) ds,... ,
ω̂m t,s,un (s) ds
ω1 t,s,u(s) ds,... ,
ωm t,s,un (s) ds,
0
0
(2.9)
ωm t,s,u(s) ds,
ω̂m t,s,u(s) ds .
(A4 )
D F t,x1 (t),x2 (t),... ,x2m+1 (t) ,F t,x1 (t),x2 (t),...,x2m+1 (t)
≤ H t,D x1 (t),x1 (t) ,D x2 (t),x2 (t) ,...,D x2m+1 (t),x2m+1 (t)
(2.10)
holds for xi ,xi ∈ C(J,En ), t ∈ J, i = 1,2,...,(2m + 1).
(A5 ) There exists a nonnegative continuous function u : J → R+ being the solution of
the inequality,
t
H t,u(t),
T
0
0
w1 t,s,u(s) ds,... ,
T
ŵ1 t,s,u(s) ds,... ,
where
q(t) = sup D F t, 0̂,
t ∈J
t
0
0
t
0
(2.11)
ŵm t,s,u(s) ds + q(t) ≤ u(t),
t
f1 (t,s, 0̂)ds,... ,
T
0
wm t,s,u(s) ds,
T
g1 (t,s, 0̂)ds,... ,
0
0
fm (t,s, 0̂)ds,
gm (t,s, 0̂)ds , 0̂ .
(2.12)
K. Balachandran and K. Kanagarajan 337
(A6 ) In the class of functions satisfying the condition 0 ≤ u(t) ≤ u(t), t ∈ J, the function u(t) ≡ 0, t ∈ J, is the only solution of the equation
u(t) = H t,u(t),
T
0
t
0
t
w1 t,s,u(s) ds,... ,
T
ŵ1 t,s,u(s) ds,... ,
0
wm t,s,u(s) ds,
0
(2.13)
ŵm t,s,u(s) ds .
In order to prove the existence of a solution of (1.3), we define the sequence
xn+1 (t) = F t,xn (t),
T
0
x0 (t) ≡ 0̂,
t
t
f1 t,s,xn (s) ds,... ,
0
T
g1 t,s,xn (s) ds,... ,
0
0
fm t,s,xn (s) ds,
(2.14)
gm t,s,xn (s) ds
for n = 0,1,2,....
To prove the convergence of the sequence {xn } to the solution x of (1.3), we define the
sequence {un } by the relations
un+1 (t) = H t,un (t),
T
0
u0 (t) = u(t),
t
0
t
w1 t,s,un (s) ds,... ,
T
ŵ1 t,s,un (s) ds,... ,
0
0
wm t,s,un (s) ds,
(2.15)
ŵm t,s,un (s) ds
for n = 0,1,2,..., where the function u(t) is from the assumptions (A5 ) and (A6 ).
Lemma 2.6. If the conditions (A3 ), (A5 ), and (A6 ) are satisfied, then
0 ≤ un+1 (t) ≤ un (t) ≤ u(t),
lim un (t) = 0,
n→∞
t ∈ J, n = 0,1,2,...,
t ∈ J,
and the convergence is uniform in each bounded set.
(2.16)
338
Fuzzy Volterra-Fredholm integral equations
Proof. From (2.11) and (2.15) we have
u1 (t) = H t,u0 (t),
T
0
0
0
T
ŵ1 t,s,u0 (s) ds,... ,
t
0
t
w1 t,s,u0 (s) ds,... ,
≤ H t,u(t),
T
t
T
ŵ1 t,s,u(s) ds,... ,
0
t
ŵm t,s,u0 (s) ds
0
w1 t,s,u(s) ds,... ,
0
wm t,s,u0 (s) ds,
0
wm t,s,u(s) ds,
(2.17)
ŵm t,s,u(s) ds + q(t)
≤ u(t) = u0 (t)
for t ∈ J. Further, we obtain (2.16) by induction. But (2.16) implies the convergence of
the sequence {un (t)} to some nonnegative function φ(t) for t ∈ J. By Lebesgue’s theorem
and the continuity of H, it follows that the function φ(t) satisfies (2.13). Now from assumptions (A5 ) and (A6 ), we have φ(t) ≡ 0, t ∈ J. Hence by the Dini theorem [12], the
sequence {un } converges uniformly in J.
3. Main results
Theorem 3.1. If the assumptions (A1 )–(A6 ) are satisfied, then there exists a continuous
solution x of (1.3). The sequence {xn } defined by (2.14) converges uniformly on J to x, and
the following estimates:
D x(t),xn (t) ≤ un (t),
t ∈ J, n = 0,1,2,...,
t∈J
D x(t), 0̂ ≤ u(t),
(3.1)
(3.2)
hold. The solution x of (1.3) is unique in the class of functions satisfying the condition (3.2).
Proof. We first prove that the sequence {xn (t)}, t ∈ J, fulfils the condition
t ∈ J, n = 0,1,2,....
D xn (t), 0̂ ≤ u(t),
(3.3)
Obviously, we see that D(x0 (t), 0̂) = 0 ≤ u(t), t ∈ J. Further, if we suppose that inequality
(3.3) is true for n ≥ 0, then
D xn+1 (t), 0̂
≤ D xn+1 (t),x1 (t) + D x1 (t), 0̂
t
t
≤ D F t,xn (t), f1 t,s,xn (s) ds,... , fm t,s,xn (s) ds,
0
T
0
0
g1 t,s,xn (s) ds,...,
T
0
gm t,s,xn (s) ds ,
K. Balachandran and K. Kanagarajan 339
t
F t, 0̂,
f1 (t,s, 0̂)ds,... ,
0
t
0
f1 (t,s, 0̂)ds,... ,
≤ H t,D xn (t), 0̂ ,D
D
D
0
T
0
0
gm (t,s, 0̂)ds
0
+ q(t)
T
0
D g1 (t,s,xn (s) ,g1 (t,s, 0̂) ds,... ,
t
0
T
t
0
T
T
0
t
w1 t,s,u(s) ds,...,
t
w1 t,s,D xn (s), 0̂ ds,... ,
ŵ1 t,s,D xn (s), 0̂ ds,... ,
≤ H t,u(t),
≤ u(t)
g1 (t,s, 0̂)ds ,...,
D gm t,s,xn (s) ,gm (t,s, 0̂) ds + q(t)
T
0
D f1 t,s,xn (s) , f1 (t,s, 0̂) ds,... ,
≤ H t,D xn (t), 0̂ ,
t
gm (t,s, 0̂)ds , 0̂
0
f1 (t,s, 0̂)ds ,... ,
0
T
g1 (t,s, 0̂)ds,...,
0
gm t,s,xn (s) ds,
0
0
0
fm (t,s, 0̂)ds ,
D fm t,s,xn (s) , fm (t,s, 0̂) ds,
T
T
T
≤ H t,D xn (t), 0̂ ,
t
t
g1 t,s,xn (s) ds,
0
0
fm t,s,xn (s) ds,
0
gm (t,s, 0̂)ds
T
t
f1 t,s,xn (s) ds,
0
t
fm (t,s, 0̂)ds,
t
0
T
0
T
g1 (t,s, 0̂)ds,...,
0
t
D
fm (t,s, 0̂)ds,
0
t
+ D F t, 0̂,
T
ŵ1 t,s,u(s) ds,... ,
0
0
0
wm t,s,D xn (s), 0̂ ds,
ŵm t,s,D xn (s), 0̂ ds + q(t)
wm t,s,u(s) ds,
ŵm t,s,u(s) ds + q(t)
for t ∈ J.
(3.4)
Now we obtain (3.3) by induction. Next, we prove that
D xn+r (t),xn (t) ≤ un (t),
t ∈ J, n = 0,1,2,..., r = 0,1,2,....
(3.5)
By (3.3), we have
D xr (t),x0 (t) = D xr (t), 0̂ ≤ u(t) = u0 (t),
t ∈ J, r = 0,1,2,....
(3.6)
340
Fuzzy Volterra-Fredholm integral equations
Suppose that (3.5) is true for n,r ≥ 0, then
D xn+r+1 (t),xn+1 (t)
= D F t,xn+r (t),
T
t
0
T
g1 t,s,xn+r (s) ds,...,
0
t
F t,xn (t),
T
0
0
t
T
g1 t,s,xn (s) ds,...,
gm t,s,xn+r (s) ds ,
fm t,s,xn+r (s) ds,
0
f1 t,s,xn (s) ds,...,
0
t
f1 t,s,xn+r (s) ds,... ,
fm t,s,xn (s) ds,
0
gm t,s,xn (s) ds
0
t
t
f1 t,s,xn+r (s) ds, f1 t,s,xn (s) ds ,...,
≤ H t,D xn+r (t),xn (t) ,D
0
t
D
D
T
0
g1 t,s,xn+r (s) ds,
0
D
T
fm t,s,xn+r (s) ds,
0
0
t
T
0
gm t,s,xn+r (s) ds,
0
g1 t,s,xn (s) ds ,...,
T
fm t,s,xn (s) ds ,
0
gm t,s,xn (s) ds
t ≤ H t,D xn+r (t),xn (t) , D f1 t,s,xn+r (s) , f1 t,s,xn (s) ds,... ,
0
t
0
D fm t,s,xn+r (s) , fm t,s,xn (s) ds,
T
D g1 t,s,xn+r (s) ,g1 t,s,xn (s) ds,... ,
0
T
0
≤ H t,D xn+r (t),xn (t) ,
t
t
0
w1 t,s,D xn+r (s),xn (s) ds,... ,
T
wm t,s,D xn+r (s),xn (s) ds,
0
T
0
D gm t,s,xn+r (s) ,gm t,s,xn (s) ds
0
ŵm t,s,D xn+r (s),xn (s) ds
ŵ1 t,s,D xn+r (s),xn (s) ds,... ,
K. Balachandran and K. Kanagarajan 341
≤ H t,un (t),
T
0
t
0
w1 t,s,un (s) ds,... ,
T
ŵ1 t,s,un (s) ds,... ,
≤ un+1 (t)
t
0
0
wm t,s,un (s) ds,
ŵm t,s,un (s) ds
for t ∈ J.
(3.7)
Now we obtain (3.5) by induction.
Because of Lemma 2.6, limn→∞ un (t) = 0 in J and we have from (3.5) that xn → x in J.
The continuity of x follows from the uniform convergence of the sequence {xn } and the
continuity of all functions xn . If r → ∞, then (3.5) gives estimation (3.1). Estimation (3.2)
implies (3.3). It is obvious that x is a solution of (1.3).
To prove that the solution x is a unique solution of (1.3) in the class of functions
satisfying the condition (3.2), we suppose that there exists another solution x̂ defined in J
such that x(t) = x̂(t) and x̂(t) ≤ u(t) for t ∈ J. From (3.1) we get D(x̂(t),xn (t)) ≤ un (t),
t ∈ J, n = 0,1,2,... and it follows that x(t) = x̂(t) for t ∈ J. This contradiction proves the
uniqueness of x in the class of functions satisfying the relation (3.2). This completes the
proof of the theorem.
Theorem 3.2. If the assumptions (A1 )–(A4 ) are satisfied and the function y(t) ≡ 0, t ∈ J,
is the only nonnegative continuous solution of the inequality
y(t) ≤ H t, y(t),
T
0
t
0
t
w1 t,s, y(s) ds,...,
T
ŵ1 t,s, y(s) ds,...,
0
0
wm t,s, y(s) ds,
ŵm t,s, y(s) ds ,
(3.8)
t ∈ J,
then (1.3) has at most one solution in J.
Proof. We suppose that there exist two solutions x and x̂ of (1.3) such that x(t) = x̂(t),
t ∈ J. Put y(t) = D(x(t), x̂(t)), t ∈ J, then
y(t) = D x(t), x̂(t)
= D F t,x(t),
T
0
t
0
0
T
t
F t, x̂(t),
0
t
g1 t,s,x(s) ds,... ,
T
f1 t,s,x(s) ds,... ,
0
0
g1 t,s, x̂(s) ds,... ,
0
gm t,s,x(s) ds ,
f1 t,s, x̂(s) ds,... ,
T
t
fm t,s,x(s) ds,
0
fm t,s, x̂(s) ds,
gm t,s, x̂(s) ds
342
Fuzzy Volterra-Fredholm integral equations
≤ H t,D x(t), x̂(t) ,D
t
D
D
D
T
0
T
0
t
0
g1 t,s, x̂(s) ds ,...,
T
gm t,s,x(s) ds,
0
f1 t,s, x̂(s) ds ,...,
fm t,s, x̂(s) ds ,
0
g1 t,s,x(s) ds,
0
≤ H t,D x(t), x̂(t) ,
t
t
t
f1 t,s,x(s) ds,
0
T
0
fm t,s,x(s) ds,
0
t
0
gm t,s, x̂(s) ds
D f1 t,s,x(s) , f1 t,s, x̂(s) ds,...,
D fm t,s,x(s) , fm t,s, x̂(s) ds,
T
D g1 t,s,x(s) ,g1 t,s, x̂(s) ds,... ,
0
T
D gm t,s,x(s) ,gm t,s, x̂(s) ds
0
t ≤ H t,D x(t), x̂(t) , w1 t,s,D x(s), x̂(s) ds,... ,
0
t
0
T
0
T
wm t,s,D x(s), x̂(s) ds,
0
ŵ1 t,s,D x(s), x̂(s) ds,... ,
ŵm t,s,D x(s), x̂(s) ds
≤ H t, y(t),
T
0
t
0
t
w1 t,s, y(s) ds,... ,
ŵ1 t,s, y(s) ds,... ,
T
0
0
wm t,s, y(s) ds,
ŵm t,s, y(s) ds
(3.9)
and by (3.8) we conclude that y(t) ≡ 0 for t ∈ J, that is, x(t) = x̂(t), t ∈ J. This contradic
tion proves our Theorem 3.2.
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K. Balachandran and K. Kanagarajan 343
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K. Balachandran: Department of Mathematics, Bharathiar University, Coimbatore 641 046, India
E-mail address: balachandran k@lycos.com
K. Kanagarajan: Department of Mathematics, Karpagam College of Engineering, Coimbatore
641 032, India
E-mail address: kanagarajank@lycos.com