SOME FUNCTIONAL DIFFERENTIAL EQUATIONS WITH BOTH RETARDED AND ADVANCED ARGUMENTS

. In this paper we shall study a functional diﬀerential equations of mixed type. This equation is a generalization of some equations from medicine. Related to this equation we study the existence of the solution by contraction’s principle and Schauder’s ﬁxed point theorem.


INTRODUCTION
Functional differential equations with advanced and retarded arguments (called here as functional differential equations of mixed type -MFDE) have had a slower contributions in mathematical researches, that is if we compare them with functional differential equations with delay.This is due the fact that in our type of equations we have in the same time both advanced and retarded argument, each of them having different behavior.These equations have the important feature that both the history and the future status of the system affect their change rate at the present time.
Here we have a general form of MFDE problem (1) x (t) = f (t, x(t), x(t − h), x(t + h)) + λ, t ∈ (a, b), (2) where The first part of the results from this paper appear in I.A. Rus-V.Dârzu-Ilea [6] and contains the existence and the uniqueness of the solution of the problem (1)-( 2)-(3) studied in different spaces.
This type of equations came from different fields of applications.For example, A. Rustichini [7], [8] investigated a specific mixed functional differential equation arising in a special way from a competitive economy; Schulman [9] gave a physical justification to MFDE.Some other fields of interest for this type of equations would be: population genetic (D.G.Aronson and H.F. Weinberger [1]), population growth, mathematical biology.Some other names in the study of MFDE are: J. Mallet-Paret [2], [3], [4], J. Wu and X. Zou [10], R. Precup [5].
The biologic signification of a MFDE can be given as follows.
The equation ( 1) is a model for a certain disease.This depends on the physical state of the subject -the delay argument; the treatment that should be given to the patient -the advanced argument; the parameter is an outside factor that can influence the physical state of the subject; condition (2)represents the statistics observations obtained before from other subjects; condition (3) -represents also, from a statistical point of view, the expectations of the evolution of the disease.
From the study of population growth we can explain the above model as follows: ϕ-the state of some population in an chosen environment, ζ-the state that should have the population, λ-a control parameter, x (t) is the speed of grow of the population with the low x = f + λ.
If some part of the population is sacrificed then λ < 0, and if the population is extending numerically then λ > 0.

HOW TO OBTAIN A FREDHOLM-VOLTERRA INTEGRAL EQUATION
Let (x, λ) a solution for (1)-( 2)- (3).It follows that: From the continuity in t = b we have Thus the problem (1)-( 2)-( 3) is equivalent with x = A(x) and λ = the right hand side of (5), By the contraction principle we obtain the following existence theorem.

MAIN RESULT
We generalize the above problem considering the next Fredholm-Volterra integral equation where given by the formula (F 1) . Our purpose here is to study the existence of the solution of the equation ( 8) with contraction's principle and Schauder's types theorems.
The problem (8)-( 9) is equivalent with x = T (x), where T : By the contraction principle we obtain the following existence theorem.
Theorem 2. If we have the following conditions: (i) there exist L f > 0 such as Then the problem (8)-( 9) has a unique solution, moreover the solution x * can be obtain by the method of successive approximation beginning from any element from the space Proof.Let the operator (11) follows T is a contraction.We can apply now the principle of contraction and follows the conclusion from the theorem.This is a classical problem of Krasnoselskii type, but by applying this type of theorem in space C[a − h, b + h] we obtain to many conditions on the date K, thus we apply Schauder type theorem in order to obtain optimality of the conditions on equation's data.
Let the Banach space C[a − h, b + h] with the Chebyshev norm, • .
Theorem 3. If we have the following conditions given by (F 1); (ii) there exists M ∈ R + such that

Proof. Let the operator
From (i) the operator T is well defined and complete continuous.
In what follows we use the conditions (ii) in order to prove the invariance on sphere Now we can say that for R greater than 2M (b − a) the operator T satisfy the invariance condition.Thus by applying Schauder's theorem it follows that there exist at least one solution x * and for this solution we have established that Now we consider the following equation with the initial conditions (15) where The problem ( 14)-( 15) is equivalent with x = T (x), where T : The existence of the solution of the equation ( 14) with contraction's principle is trivial.
Theorem 4. If we have the following conditions: (i) there exist L i > 0 such as given by (F 1); (ii) there exist real numbers α, β, γ, δ such as From (i) the operator T is well defined and complete continuous.
In what follows we use the conditions (ii), (iii) in order to prove the invariance on sphere the operator T satisfy the invariance condition.Thus by applying the Schauder theorem it follows that there exist at least one solution x * and for this solution we have established that x * ≤ ϕ(b) + R. T is well defined and complete continuous.
In what follows we prove the invariance on sphere T (x)(t) − t ≤ R, with x ≤ g +R and R > 0 (x ∈ B(g; R) ⇒ x(t) ∈ J, where J = [−j, j], with j = g +R),

then the equation ( 8 )
has at least one solution x * ∈ C([a, b]) with the propriety that x * ≤ R, where R is a number greater than 2M (b − a).

T
(x)(t) − t ≤ ≤ b−t b−a t a [|x(s)| + |x(s − h)|]ds + (b−t)(t−a) b−a b a |x(s + h)|ds ≤ (b − a)(1 + b − a) x ≤ (b − a)(1 + b − a)(b + R) < R. Now we can say that if (b−a)(1+b−a) < 1, for R greater than b(b−a)(1+b−a) 1−(b−a)(1+b−a)the operator T satisfy the invariance condition.Thus by applying the Schauder theorem it follows that there exist at least one solution x * and for this solution we have established that x * ≤ ϕ(b) + R.