LIMITS OF SOLUTIONS OF A PERTURBED LINEAR DIFFERENTIAL EQUATION

Using interesting techniques, an existence result for the problem  + 2f (t) _ x + x + g (t; x) = 0; lim t!+1 x (t) = lim t!+1 _ x (t) = 0; is given in [2]. This note treates the same problem via Schauder-Tychono and Banach theorems.

( To build the operator H one changes firstly the equation (1.1) to a system ż = A (t) z + F (t, z) , which is a perturbed system for ż = A (t) z.
(1.5) (here A is a quadratic matrix 2 × 2, z = x y and F is a function with values in IR 2 ; the expressions of A and F will be given in the next section) It is known that if (1.5) is uniformly asymptotically stable, then for "small" perturbations F (1.4) is uniformly asymptotically stable; in the case considered by T. A. Burton and T. Furumochi, the system (1.5) is only asymptotically stable and it is not uniformly asymptotically stable, which praises again the importance of the obtained result.
Let Z = Z (t) be the fundamental matrix of the system (1.5) which is principal in 0; as is well known, if w : IR + → IR 2 is a continuous function, then the solutions of the system (1.4) equipped with the condition coincide with the fixed points of the operator H defined by The problem of the existence of solutions for the system (1.4) for which z (+∞) does exist (in particular, z (+∞) = 0) has been treated in the general case in [2], if the system (1.5) is uniformly asymptotically stable.In this case, the operator H is considered in C l , the subspace of C, where By using this space one gets an advantage since an efficient compactness criterion holds (see [1]), which in C it doesn't happen; from this criterion it can be easily proved that H is a compact operator in C l .In the case when (1.5) is only asymptotically stable the operator H is not compact in C l .To obviate this difficulty, T.A. Burton and T. Furumochi use the another (equivalent) variant of Schauder's theorem, by identifying a compact convex nonempty subset S of C i.e.
where S is equi-continuous and q : IR + → IR is a positive continuous function with lim t→+∞ q (t) = 0.By showing that HS ⊂ S and H is continuous, one can apply Schauder's theorem.
In the present paper we show firstly that for initial data small enough the equation (1.1) admits solutions defined on IR + ; next we prove that each such a solution fulfills (1.2) .To this aim we use, as in [2] an equivalent equation of type (1.6) .The admitted hypotheses will be analogous with the ones from [2].
Unlike T.A. Burton and T. Furumochi, we shall apply Schauder-Tychonoff' fixed point theorem in the Fréchet space endowed with a family of seminorms as chosen as to determine the convergence on compact subsets of IR + with the usual topology.
Furthermore we shall indicate the possibility to apply Banach's fixed point theorem in C c .

Admit the following hypotheses:
(2.1) (iii) there exists M > 0 and α > 1 such that The main result is contained in the following theorem.Theorem 2.1 Under hypotheses (i), (ii) and (iii) there exists an a > 0 such that every solution x of the equation (1.1) with |x (0)| < a is defined on IR + and satisfies the condition (1.2) .
The proof will be made in more steps and it needs some preliminary notations.
Firstly, as in [2] one changes the equation (1.1) to a system and Attach to system (2.3) the initial condition Denote by Z the fundamental matrix of the system which is principal in 0. An easy computation shows that Consider as fundamental the space C c is a Fréchet space (i.e. a complete metrizable real linear space) with respect to the family of seminorms (2.7) We mention that the topology defined by the family of seminorms (2.7) is the topology of the convergence on compact subsets of IR + ; in addition, a family A ⊂ C c is relatively compact if and only if it is equi-continuous and uniformly bounded on compacts subsets of IR + (Ascoli-Arzela' theorem).
Define in C c the operator It is obvious that the set of solutions for the problem (2.3) , (2.4) coincides with the set of fixed points of H.
where ρ > 0 is a fixed number; obviously, B ρ is a closed bounded convex nonempty subset of C c .Lemma 2.1 There exists a number h > 0 such that for every ρ ∈ (0, h) there exists a number a > 0 with the property for every z 0 with |z 0 | ∈ (0, a) , (2.9) Proof.Let w ∈ B ρ and z = Hw; therefore z is given by (2.8) .
We have the following estimates: which are proved in [2].By substituting in (2.11) and (2.12) the inequality |w (s Indeed, one has successively (2.14) Obviously, a > 0; in addition it results that which ends the proof. 2 Lemma 2.2 If z is a solution of the problem (2.3) , (2.4) defined on IR + , then for |z 0 | small enough, z (+∞) = 0.
Evaluating r (t) one finds By Ciaplyghin's lemma we get r (t) ≤ q (t) , t ∈ IR + , where q = q (t) is the unique solution of the (Bernoulli) problem We have an equivalent form of q (t) , , by hypotheses (i), (ii) and (iii) we find lim t→+∞ q (t) = 0. Hence, The proof is now complete.2 To prove Theorem 2.1 it remains to show that for every z 0 with |z 0 | small enough, the problem (2.3) , (2.4) admits solutions on IR + .To this aim we use the following theorem.
Theorem (SCHAUDER-TYCHONOFF) Let E be a Fréchet space, S ⊂ E be a closed bounded convex nonempty subset of E and H : S → S be a continuous operator.If HS is relatively compact in E, then H admits fixed points.
Setting E = C c , H given by (2.8) , and S = B ρ it remains to prove the continuity of H and the relatively compactness of HS. Let where So, having the family of derivatives uniformly bounded, HB ρ is equicontinuous on the compact subsets of IR + .The proof is now complete.2

. The case when H is contractant
In [2] the mentioned result is obtained by admitting in addition a supplementary hypothesis i.e. (iv) for every δ > 0 there exists L (δ) > 0 such that and L (δ) is a continuous and increasing function.
If we admit a Lipschitz condition, then one can prove that the problem (2.3) , (2.4) admits an unique solution which is convergent to zero for every z 0 with |z 0 | < a. Obviously, we can separately prove the uniqueness, but we prefer to obtain simultaneously the existence and the uniqueness, by admitting a weaker condition than (iv).
To this aim, we shall use the Banach's fixed point theorem which in Fréchet spaces has the following statement.
Theorem Then H admits an unique fixed point.
To apply the stated Banach's theorem we shall consider in the space C c another family of seminorms, topologically equivalent with (2.9) , i.e.
where λ n > 0 are arbitrary numbers.Theorem 3.1 Suppose that the hypotheses (i), (ii) and (iii) are fulfilled and in addition for every n ∈ IN * there exists L n ∈ [0, 1) such that for every Then the problem (2.3) , (2.4) admits an unique solution which is convergent to zero.
Proof.Consider the same operator H with HB ρ ⊂ B ρ .Let z i = (x i , y i ) ∈ B ρ , i ∈ 1, 2; since
BANACH) Let E be a Fréchet space endowed with a family of seminorms |•| n and let S ⊂ E be a closed subset of E. Let H : S → S be an operator fulfilling the following condition: for every n ∈ IN * there exists a positive number L n ∈ [0, 1) such that for every x, y ∈ S, |Hx − Hy| n ≤ L n |x − y| n .
z) is uniformly continuous for t ∈ [0, n] and |z| ≤ ρ, it follows that the sequence F (t, w m (t)) converges uniformly on [0, n] to F (t, w (t)) , which finally proves the continuity of H.Let us show that HB ρ is relatively compact; from HB ρ ⊂ B ρ it results that HB ρ is unifomly bounded in C c .Let w ∈ B ρ be arbitrary; since z = Hw ∈ B ρ and z = A (t) z + B (t) w + F (t, w)