EXISTENCE OF SOLUTIONS OF A SPECIAL CLASS OF FUZZY INTEGRAL EQUATIONS

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.


Introduction
Chandrasekhar [5] and Crum [6] considered the following integral equation: 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 where K is a compact operator on the Banach algebra.His abstract theorems are applied to the integral equation of the form 2 Fuzzy integral equations Cahlon and Eskin [4] considered the equation 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][10][11][14][15][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 where as Park et al. [11] proved the existence of solutions of fuzzy integral equation of the form 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 In this paper we prove the existence of solutions of fuzzy integral equations of the form where φ : 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).

Preliminaries
Let P k (R n ) denote the family of all nonempty, compact, convex subsets of R n .Addition and scalar multiplication in P k (R n ) are defined as usual.U denotes the closure of U, where U is contained in R n .Let I = [0,1] ⊆ R be a compact interval and denote where (i) u is normal, that is, there exists an If g : R n × R n → R n is a function, then using Zadeh's extension principle we can extend (2. 2) , and continuous function g.In addition the above equation gives The real numbers can be embedded in E n by the rule c → c(t), where We can also generalize the multiplication by a real number and for any real number c we get , where H is the Hausdorff metric defined in P K (R n ).Then D is a metric on E n .Further, (E n ,D) is a complete metric space [7,12].Also D(u It can be proved straight away that D(u where H is the Hausdorff metric on P k (R n ) induced by the norm in R n .)Definition 2.1 [1].Let I be [0,1] and for each t in I, let F(t) be a nonempty subset of R n .Let Ᏺ be the set of all point-valued functions f from I to R n such that f is integrable over I and f (t) ∈ F(t) for all t in I.
(2.4) Definition 2.2 [7].A mapping F : I → E n is strongly measurable if for all α ∈ [0,1] the set-valued map F α : ] α is Lebesgue measurable when P k (R n ) has the topology induced by the Hausdorff metric H.
Definition 2.3 [7].A mapping F : I → E n is said to be integrably bounded if there is an integrable function h such that x ≤ h(t) for every x ∈ F 0 (t).
Definition 2.4 [12].The integral of a fuzzy mapping It has been proved by Puri and Ralescu [12] that a strongly measurable and integrably bounded mapping F : I → E n is integrable (i.e., I F(t)dt ∈ E n ).The concept of a fuzzy integral generalizes the Aumann integral of a set-valued mapping.The following results are proved in [7].Suppose that

Existence theorem
To prove A : Ꮿ → Ꮿ, we have to prove that Aψ is continuous and Clearly the right-hand side of (3.5) is less than as h → 0. So Aψ is continuous.Consider  So Aψ ∈ Ꮿ and A maps Ꮿ into itself.We show that Ꮿ is a closed subset of C([0,T],E n ) a complete metric space with the metric H 1 (see [7]).
Let (ψ n ) be a sequence in Ꮿ converging to ψ in C([0,T],E n ).Consider for sufficiently large n and all positive .So ψ ∈ Ꮿ.This implies that Ꮿ is a closed subset of C([0,T],E n ).Therefore Ꮿ is a complete metric space.We prove that A is a contraction mapping.For ψ 1 ,ψ 2 ∈ Ꮿ, ≤ αH 1 ψ 1 ,ψ 2 where α ∈ (0,1). (3.8) 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).

General equations
As a generalization of (1.8) we consider the following fuzzy integral equation: