EXISTENCE OF SOLUTIONS OF GENERAL NONLINEAR FUZZY VOLTERRA-FREDHOLM INTEGRAL EQUATIONS

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


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 f t,s,x(s) ds T 0 g t,s,x(s) ds , ( 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 g t,s,x(s) ds, T 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

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.Let A and B be two nonempty bounded subsets of R n .The distance between A and B is defined by the Hausdorff metric d(A,B) = max{sup a∈A inf b∈B a − b ,sup b∈B inf a∈A a − b }, where • denote the usual Euclidean norm in R n .Then it is clear that (P K (R n ),d) becomes a metric space.
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 g to E n × E n → E n by the equation g(u,v)(z) = sup z=g(x,y) min u(x),v(y) . (2.2) where k ∈ R. The real numbers can be embedded in E n by the rule c → ĉ(t) where , where d is the Hausdorff metric defined in P K (R n ).Then D is a metric in E n and (E n ,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 ∈ E n 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 ∈ E n .
Definition 2.1 [5].A mapping F : I → E n is strongly measurable if for all α ∈ [0,1], the set-valued map F α : It has been proved by Puri and Ralescu [10] that a strongly measurable and integrably bounded mapping F : Now we make the following assumptions.

Main results
Theorem 3.1.If the assumptions (A 1 )-(A 6 ) are satisfied, then there exists a continuous solution x of (1.3).The sequence {x n } defined by (2.14) converges uniformly on J to x, and the following estimates:

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 {x n (t)}, t ∈ J, fulfils the condition Obviously, we see that D(x 0 (t), 0) = 0 ≤ u(t), t ∈ J. Further, if we suppose that inequality (3.3) is true for n ≥ 0, then By (3.3), we have 340 Fuzzy Volterra-Fredholm integral equations Suppose that (3.5) is true for n,r ≥ 0, then 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 x defined in J such that x(t) = x(t) and x(t) ≤ u(t) for t ∈ J. From (3.1) we get D( x(t),x n (t)) ≤ u n (t), t ∈ J, n = 0,1,2,... and it follows that x(t) = x(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.and by (3.8) we conclude that y(t) ≡ 0 for t ∈ J, that is, x(t) = x(t), t ∈ J.This contradiction proves our Theorem 3.2.
Lebesgue measurable when P K (R n ) has the topology induced by the Hausdorff metric d.
To prove the convergence of the sequence {x n } to the solution x of (1.3), we define the sequence {u n } by the relations x n (t),t 0 f 1 t,s,x n (s) ds,..., t 0 f m t,s,x n (s) ds, T 0 g 1 t,s,x n (s) ds,..., T 0 g m t,s,x n (s) ds (2.14)for n = 0,1,2,....
x n+r (t), Because of Lemma 2.6, lim n→∞ u n (t) = 0 in J and we have from (3.5) that x n → x in J.The continuity of x follows from the uniform convergence of the sequence {x n } and the continuity of all functions x n .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).