EXISTENCE OF SOLUTIONS OF A SPECIAL CLASS
OF FUZZY INTEGRAL EQUATIONS
K. BALACHANDRAN AND K. KANAGARAJAN
Received 30 July 2005; Revised 25 February 2006; Accepted 26 February 2006
We prove an existence theorem for a special class of fuzzy integral equations involving
fuzzy set-valued mappings. The results are obtained by using the contraction mapping
principle.
Copyright © 2006 K. Balachandran and K. Kanagarajan. This is an open access article
distributed under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Chandrasekhar [5] and Crum [6] considered the following integral equation:
H(t) = 1 + H(t)
1
0
t
ψ(s)H(s)ds.
t+s
(1.1)
This equation arises in the study of radiation transfer in a semi-infinite atmosphere. The
first rigorous proof of existence of solutions of (1.1) was given in [6]. By using operators
on a Banach algebra and a fixed point theorem of Darbo for a set contraction map, Legget
[8] proved an existence theorem for an equation of the form
x = x0 + xKx,
(1.2)
where K is a compact operator on the Banach algebra. His abstract theorems are applied
to the integral equation of the form
x(t) = x0 (t) + x(t)
Ω
K(t,s) f s,x(s) ds,
Hindawi Publishing Corporation
Journal of Applied Mathematics and Stochastic Analysis
Volume 2006, Article ID 52620, Pages 1–8
DOI 10.1155/JAMSA/2006/52620
t ∈ Ω, Ω ⊂ Rn .
(1.3)
2
Fuzzy integral equations
Cahlon and Eskin [4] considered the equation
H(t) = 1 + H(t)
1
0
t
ψ(s)H(s)ds +
t+s
1
0
P t,s,H(t),H(s) ds.
(1.4)
This equation is a generalization of (1.3), where P is the perturbation of Chandrasekhar
H-equation.
The problem of existence of solutions of fuzzy integral equations has been studied
by many authors [1, 2, 9–11, 14–16]. Kaleva [7] and Seikkala [13] have discussed the
existence of solutions of fuzzy differential equations. Subrahmanyam and Sudarsanam
[14] studied existence results for fuzzy Volterra integral equation of the form
x(t) = φ(t) +
t
0
g t,s,x(s) ds,
(1.5)
where as Park et al. [11] proved the existence of solutions of fuzzy integral equation of
the form
φ(u) = w0 +
u
u0
F u,s,φ(s) ds
φ u0 = w0 .
(1.6)
Balachandran and Dauer [2] established the local existence of solutions and approximate
solutions of the perturbed fuzzy integral equation. Balachandran and Prakash [3] studied
the existence of solutions of nonlinear fuzzy Volterra integral equations of the form
x(t) = φ(t) + f t,x(t),
t
0
g t,s,x(s) ds .
(1.7)
In this paper we prove the existence of solutions of fuzzy integral equations of the form
x(t) = φ(t) + x(t)
t
0
k(t,s) f s,x(s) ds +
t
0
g t,s,x(s) ds,
(1.8)
where φ : [0,T] → En , k : [0,T] × [0,T] → R, f : [0,T] × En → En , and g : [0,T] × [0,T] ×
En → En are continuous functions. This equation is a generalization of Chandrasekhartype equation in fuzzy setting.
The outlay of the paper is as follows. In Section 2 we give some basic definitions for
our study and in Section 3 we prove the main theorem on the existence of solutions of
fuzzy integral equation (1.8). In Section 4 we state a theorem on the existence of solutions
of a generalization of (1.8).
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. U denotes the closure of U,
where U is contained in Rn . Let I = [0,1] ⊆ R be a compact interval and denote
En = u : Rn −→ [0,1] : u satisfies (i)–(iv) below ,
(2.1)
K. Balachandran and K. Kanagarajan 3
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
g(u,v)(z) = sup min u(x),v(y) .
(2.2)
z=g(x,y)
It is well known that [g(u,v)]α = g([u]α ,[v]α ) for all u,v ∈ En , 0 ≤ α ≤ 1, and continuous
function g. In addition the above equation gives [u + v]α = [u]α + [v]α . The real numbers
can be embedded in En by the rule c → c
(t), where
⎧
⎨1
for t = c,
c
(t) = ⎩
0 elsewhere.
(2.3)
We can also generalize the multiplication by a real number and for any real number c we
get [cu]α = c[u]α , where 0 ≤ α ≤ 1 and u ∈ En .
Let D : En × En → R+ ∪ {0} be defined by D(u,v) = sup0≤α≤1 H([u]α ,[v]α ), where H
is the Hausdorff metric defined in PK (Rn ). Then D is a metric on En . Further, (En ,D)
is a complete metric space [7, 12]. Also D(u + w,v + w) = D(u,v) for every u,v,w ∈ En .
Furthermore, D(λu,λv) = |λ|D(u,v) for every u,v ∈ En and λ ∈ R.
It can be proved straight away that D(u + v,w + z) ≤ D(u,w) + D(v,z) for u, v, w,
and z ∈ En . (The proof is based on the fact H(A1 + A2 ,B1 + B2 ) ≤ H(A1 ,B1 ) + H(A2 ,B2 ),
where H is the Hausdorff metric on Pk (Rn ) induced by the norm in Rn .)
Definition 2.1 [1]. Let I be [0,1] and for each t in I, let F(t) be a nonempty subset of Rn .
Let Ᏺ be the set of all point-valued functions f from I to Rn such that f is integrable over
I and f (t) ∈ F(t) for all t in I. Then
I
F(t)dt =
I
f (t)dt : f ∈ Ᏺ .
(2.4)
Definition 2.2 [7]. 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 H.
Definition 2.3 [7]. A mapping F : I → En is said to be integrably bounded if there is an
integrable function h such that x ≤ h(t) for every x ∈ F0 (t).
4
Fuzzy integral equations
Definition 2.4 [12]. The integral of a fuzzy mapping F : [0,1] → En is defined levelwise by
α
[0,1]
=
F(t)dt
[0,1]
Fα (t)dt
n
=
[0,1]
f (t)dt : f : [0,1] −→ R is a measurable selection for Fα
(2.5)
for all α ∈ [0,1].
It has been proved by Puri and Ralescu [12] that
a strongly measurable and integrably
bounded mapping F : I → En is integrable (i.e., I F(t)dt ∈ En ). The concept of a fuzzy
integral generalizes the Aumann integral of a set-valued mapping. The following results
are proved in [7].
Theorem 2.5. If F : I → En is continuous, then it is integrable.
n
and λ ∈ R. Then
Theorem
E be integrable
2.6. Let F,G : I →
(i) I (F(t) + G(t))dt = I F(t)dt + I G(t)dt,
(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.
3. Existence theorem
Theorem 3.1. Let a and b be positive numbers such that
b = max D ψ(t)
0≤t ≤a
t
0
t
k(t,s) f s,ψ(s) ds +
0
g t,s,ψ(s) ds, 0 .
(3.1)
Suppose that
(i) φ : [0,a] → En is continuous,
(ii) f : [0,T] × En → En and k : [0,T] × [0,T] → R are continuous and there exists a
constant L > 0 such that
t
D x(t)
0
k(t,s) f s,x(s) ds, y(t)
t
0
k(t,s) f s, y(s) ds ≤ LD(x, y)
(3.2)
for x, y ∈ En ,
(iii) g : U → En is continuous, where U = {(t,s,x) : 0 ≤ s ≤ t ≤ a, x ∈ En and D(x,φ(t)) ≤
b} and satisfies Lipschitz condition with respect to x on U, that is, there exists a constant M > 0 such that
D g(t,s,x),g(t,s, y) ≤ MD(x, y) if (t,s,x),(t,s, y) ∈ U.
(3.3)
If c = (α − L)/M for some fixed α ∈ (0,1), then there is a unique solution of (1.7) on [0,T],
where T = min{a,b,c}.
K. Balachandran and K. Kanagarajan 5
Proof. Let Ꮿ be the space of continuous functions from [0, T] into (En ,D) with H1 (ψ,φ) ≤
b, that is, Ꮿ = {ψ : ψ : [0,T] → En is continuous and H1 (ψ,φ) ≤ b}, where H1 (ψ,φ) =
sup0≤t≤T D(ψ(t),φ(t)). Define an operator A : Ꮿ → Ꮿ by
Aψ(t) = φ(t) + ψ(t)
t
0
t
k(t,s) f s,ψ(s) ds +
g t,s,ψ(s) ds.
0
(3.4)
To prove A : Ꮿ → Ꮿ, we have to prove that Aψ is continuous and H1 (Aψ,φ) ≤ b whenever
ψ ∈ Ꮿ. Consider
D Aψ(t + h),Aψ(t)
t+h
t+h
k(t + h,s) f s,ψ(s) ds +
g t + h,s,ψ(s) ds,φ(t)
= D φ(t + h) + ψ(t + h)
0
t
+ ψ(t)
0
t+h
+ D ψ(t + h)
+D
≤
3
0
+
0
t
g t + h,s,ψ(s) ds,
0
t+h
0
0
0
t
0
g t,s,ψ(s) ds
k(t,s) f s,ψ(s) ds
t
k(t + h,s) f s,ψ(s) ds,ψ(t)
g t,s,ψ(s) ds ≤ D φ(t + h),φ(t)
k(t + h,s) f s,ψ(s) ds,ψ(t)
0
+ D ψ(t + h)
t
k(t,s) f s,ψ(s) ds +
t+h
t
D g t + h,s,ψ(s) ds,g t,s,ψ(s) ds +
t+h
t
0
k(t,s) f s,ψ(s) ds
D g t + h,s,ψ(s) ds, 0
ds.
(3.5)
Clearly the right-hand side of (3.5) is less than as h → 0. So Aψ is continuous. Consider
t
H1 (Aψ,φ) = sup D Aψ(t),φ(t)
0≤t ≤T
= sup D φ(t) + ψ(t)
0≤t ≤T
= sup D ψ(t)
0≤t ≤T
t
0
0
t
k(t,s) f s,ψ(s) ds +
k(t,s) f s,ψ(s) ds +
t
0
0
g t,s,ψ(s) ds,φ(t)
g t,s,ψ(s) ds, 0
(3.6)
≤ b.
So Aψ ∈ Ꮿ and A maps Ꮿ into itself. We show that Ꮿ is a closed subset of C([0,T],En ) a
complete metric space with the metric H1 (see [7]).
6
Fuzzy integral equations
Let (ψn ) be a sequence in Ꮿ converging to ψ in C([0,T],En ). Consider
H1 (ψ,φ) = sup D ψ(t),φ(t)
0≤t ≤T
= sup D ψn (t),ψ(t) + D ψn (t),φ(t)
0≤t ≤T
≤ H1 ψn ,ψ + H1 ψn ,φ
(3.7)
≤ +b
for sufficiently large n and all positive . So ψ ∈ Ꮿ. This implies that Ꮿ is a closed subset
of C([0,T],En ). Therefore Ꮿ is a complete metric space. We prove that A is a contraction
mapping. For ψ1 ,ψ2 ∈ Ꮿ,
H1 Aψ1 ,Aψ2
= sup D Aψ1 (t),Aψ2 (t)
0≤t ≤T
t
t
= sup D φ(t) + ψ1 (t) k(t,s) f s,ψ1 (s) ds + g t,s,ψ1 (s) ds,φ(t)
0
0≤t ≤T
+ ψ2 (t)
0≤t ≤T
D ψ1 (t)
t
+
0
0
0
t
k(t,s) f s,ψ2 (s) ds +
t
≤ sup
0
t
0
t
g t,s,ψ2 (s) ds
k(t,s) f s,ψ1 (s) ds,ψ2 (t)
0
k(t,s) f s,ψ2 (s) ds
(3.8)
D g t,s,ψ1 (s) ,g t,s,ψ2 (s) ds
≤ (L + MT)H1 ψ1 ,ψ2
≤ (L + Mc)H1 ψ1 ,ψ2
where α ∈ (0,1).
≤ αH1 ψ1 ,ψ2
So A : Ꮿ → Ꮿ is a contraction map. Since Ꮿ is a complete metric space and A is a contracting self-map on Ꮿ, it has a unique fixed point x ∈ Ꮿ. This fixed point is the required
unique solution to (1.8).
4. General equations
As a generalization of (1.8) we consider the following fuzzy integral equation:
x(t) = φ(t) + h t,x(t)
t
0
k(t,s) f s,x(s) ds +
t
0
g t,s,x(s) ds,
(4.1)
where h : [0,T] × En → En is continuous and all other conditions are as before. Now we
state without proof an existence theorem for (4.1).
K. Balachandran and K. Kanagarajan 7
Theorem 4.1. Let a∗ and b∗ be positive numbers such that
b∗ = max∗ D h t,ψ(t)
t
0≤t ≤a
0
t
k(t,s) f s,ψ(s) ds +
0
g t,s,ψ(s) ds, 0
.
(4.2)
Suppose that
(i) φ : [0,a∗ ] → En is continuous,
(ii) f ,h : [0,T] × En → En and k : [0,T] × [0,T] → R are continuous and there exists a
constant L∗ > 0 such that
D h t,x(t)
t
0
k(t,s) f s,x(s) ds,h t, y(t)
≤ L∗ D(x, y)
t
0
k(t,s) f s, y(s) ds
(4.3)
for x, y ∈ En ,
(iii) g : U → En is continuous where U = {(t,s,x) : 0 ≤ s ≤ t ≤ a∗ , x ∈ En and D(x,
φ(t)) ≤ b∗ } and satisfies Lipschitz condition with respect to x on U, that is, there
exists a constant M ∗ > 0 such that
D g(t,s,x),g(t,s, y) ≤ M ∗ D(x, y) if (t,s,x),(t,s, y) ∈ U.
(4.4)
If c∗ = (α − L∗ )/M for some fixed α ∈ (0,1), then there is a unique solution of (4.1) on
[0,T], where T = min{a∗ ,b∗ ,c∗ }.
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Fuzzy integral equations
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K. Balachandran: Department of Mathematics, Bharathiar University,
Coimbatore 641046, India
E-mail address: :balachandran k@lycos.com
K. Kanagarajan: Department of Mathematics, Karpagam College of Engineering,
Coimbatore 641032, India
E-mail address: kanagarajank@rediffmail.com