ITERATIVE FUNCTIONAL-DIFFERENTIAL SYSTEM WITH RETARDED ARGUMENT ∗

. Existence, uniqueness and data dependence results of solution to the Cauchy problem for iterative functional-diﬀerential system with delays are obtained using weakly Picard operator theory. MSC 2000. 34L05, 47H10.

The literature in differential equations with modified arguments, especially of retarded type, is now very extensive.We refer the reader to the following monographs: J. Hale [2], Y. Kuang [4], V. Mureşan [3], I. A. Rus [7] and to our papers [5], [6].The case of iterative system with retarded arguments has been studied by many authors: I. A. Rus and E. Egri [10], J. G. Si, W. R. Li and S. S. Cheng [11], S. Stanek [12].So our paper complement in this respect the existing literature.
Let us mention that the results from this paper are obtained as a concequence of those from [10] where is considered the case of boundary value problems.

WEAKLY PICARD OPERATORS
In this paper we need some notions and results from the weakly Picard operator theory (for more details see I. A. Rus [9], [8], M. Serban [13]).
Let (X, d) be a metric space and A : X → X an operator.We shall use the following notations: Definition 2.1.Let (X, d) be a metric space.An operator A : X → X is a Picard operator (PO) if there exists x * ∈ X such that: Remark 2.2.Accordingly to the definition, the contraction principle insures that, if A : X → X is a α -contraction on the complet metric space X, then it is a Picard operator.
Theorem 2.3.(Data dependence theorem).Let (X, d) be a complete metric space and A, B : X → X two operators.We suppose that Definition 2.4.Let (X, d) be a metric space.An operator A : X → X is a weakly Picard operator (WPO) if the sequence (A n (x)) n∈N converges for all x ∈ X, and its limit ( which may depend on x ) is a fixed point of A.
Theorem 2.5.Let (X, d) be a metric space and A : X → X an operator.The operator A is weakly Picard operator if and only if there exists a partition of X, where Λ is the indices set of partition, such that: Definition 2.6.If A is weakly Picard operator then we consider the operator A ∞ defined by Definition 2.7.Let A be a weakly Picard operator and c > 0. The operator A is c-weakly Picard operator if Example 2.8.Let (X, d) be a complete metric space and A : X → X a continuous operator.We suppose that there exists α ∈ [0, 1) such that Then A is c-weakly Picard operator with c = 1 1−α .
Theorem 2.9.Let (X, d) be a metric space and A i : X → X, i = 1, 2. Suppose that (i) the operator A i is c i -weakly Picard operator, i = 1, 2; (ii) there exists η > 0 such that Theorem 2.10.(Fibre contraction principle).Let (X, d) and (Y, ρ) be two metric spaces and Then the operator A is Picard operator.

CAUCHY PROBLEM
In what follows we consider the fixed point equations (3a) and (3b).Let given by the relation where A f 1 (x 1 , x 2 )(t) := the right hand side of (3a) and A f 2 (x 1 , x 2 )(t) := the right hand side of (3b). Let ) is a complete metric space with respect to the metric We have Theorem 3.1.We suppose that (i) the conditions (H 1 )-(H 4 ) are satisfied;

b]) a unique solution. Moreover the operator
From (iiia) we have m f i and Therefor if condition (iii) holds, we have satisfied the invariance property for the operator Similarly, for t 1 , t 2 ∈ [t 0 − τ 2 , t 0 ] : that follows from (ii), too.
On the other hand, if t 1 , t 2 ∈ [t 0 , b], we have So we can affirm that ∀t 1 , t 2 ∈ [t 0 , b], t 1 ≤ t 2 , and doe to (iii), A f is L-Lipshitz.Thus, according to the above, we have In the same way Then we have the following relation In what follows, consider the following operator given by the relation where B f 1 (x 1 , x 2 ) := the right hand side of (4a) and B f 2 (x 1 , x 2 ) := the right hand side of (4b).
Theorem 3.2.In the conditions of Theorem 3.1, the operator Proof.The operator B f is a continuous operator but it is not a contraction operator.Let take the following notation: Then we can write (5) We have that X ϕ 1 ×X ϕ 2 ∈ I(B f ) and B f | Xϕ 1 ×Xϕ 2 is a Picard operator because is the operator which appears in the proof of Theorem 3.1.By applying Theorem 2.5, we obtain that B f is WPO.
) be an increasing solution of the system (1.1) and (y 1 , y 2 ) an increasing solution for the system of inequalities Then Proof.In the terms of the operator B f , we have However, from the condition (b), we have that the operator B ∞ f is increasing, Here, for ( x 1 , x 2 ) we used the notation  ≤) be an ordered metric space and A, B, C : X → X such that: In this case we can establish the theorem.
We suppose that 2 ) be an increasing solution of the systems Proof.The operators B j f , j = 1, 2, 3 are WPO.Taking into consideration the condition (a) the operator B 2 f is increasing.From (b) we have that 2 ), j = 1, 2, 3. Now, using the Abstract comparison lemma, the proof is complete.

Proof. Consider the operators
From Theorem 3.1 these operators are contractions. Then

Now the proof follows from Theorem 2.3, with A
From the Theorem above we have: be the solution set of the system (1.1) corresponding to f 1 i and f 2 i , i = 1, 2. Suppose that there exists η i > 0, i = 1, 2 such that where ) and H • C denotes the Pompeiu-Housdorff functional with respect to Proof.We will look for those c 1 and c 2 for which in condition of Theorem 3.1 the operators . So from Theorem 2.5 and Theorem 3.1 we have 6) we obtain that Applying Theorem 2.9 we have that where ) and H • C denotes the Pompeiu-Housdorff functional with respect to