Electronic Journal of Qualitative Theory of Differential Equations

In the present Note an existence result of asymptotically stable solutions for the integral equation x (t) = q (t) + Z t 0 K (t; s; x (s)) ds + Z 1 0 G (t; s; x (s)) ds is presented.


Introduction
In this Note we will present an existence result of asymptotically stable solutions to the equation x (t) = q (t) + t 0 K (t, s, x (s)) ds + ∞ 0 G (t, s, x (s)) ds, (1.1) under hypotheses which will be given in Section 2. We call the integral equation (1.1) to be of mixed type, since within its form an operator of Volterra type and an operator of Uryson type appear.The notion of asymptotically stable solution to the functional equation has been recently introduced in [6] and reconsidered in a more general framework in [7].Let F : BC → BC be an operator, where BC := BC IR + , IR d = {x : IR + → IR d , x bounded and continuous}, IR + := [0, ∞), d ≥ 1.Let x ∈ BC be a solution to Eq. (1.2).Definition 1.1 The function x is said to be an asymptotically stable solution of (1.1) if for any ε > 0 there exists T = T (ε) > 0 such that for every t ≥ T and for every other solution y of (1.1) , then where Remark that in [6] the case d = 1 is considered, unlike [7] wherein the general case is treated.In our papers [2]- [4] we studied the existence of asymptotically stable solutions for certain particular cases of Eq. (1.2) , in which integral operators appear.Eq. (1.1) considered in the present Note is more general than those of [2]- [4].
Notice that Definition 1.1 may be stated on other spaces of functions defined on IR + , not necessarily bounded.Since the method used in all the works cited above consists in the application of Schauder's fixed point Theorem, it is enough to suppose Definition 1.1 fulfilled only on the set on which the fixed point theorem is applied.
Each of these two families determine on C c a structure of Fréchet space (i.e. a linear, metrisable, and complete space), its topology being the one of the uniform convergence on compact subsets of IR + , for every sequence λ n .We also mention that a family A ⊂ C c is relatively compact if and only if for each n ≥ 1, the restrictions to [0, n] of all functions from A form an equicontinuous and uniformly bounded set.

Main result
In this section we will admit the following hypotheses: (k) there exist continuous functions α, β : for all (t, s) ∈ ∆ and all x, y ∈ IR d ; (g) there exist continuous functions a, b : for all (t, s) ∈ ∆ and all x ∈ IR d .
Lemma 3.1 Let z : IR + → IR + be a continuous function, satisfying the condition where γ : IR + → IR + is continuous function.Then, there exists a continuous function h : and, since (3.2) , we obtain Proposition 3.1 (Banach) Every contraction admits a unique fixed point.
The proof is classical and follows the proof of the known Banach's Contraction Principle.We remark that the result still holds if (3.5) is fulfilled only on a closed set M , for which H (M ) ⊂ M. Finally, notice that Proposition 3.1 is a particular case of a more general result due to Cain & Nashed (see [8]).(i) A is contraction; (ii) B is compact operator; (iii) the set y = λA y λ + λBy, y ∈ C c , λ ∈ (0, 1) is bounded.Then there exists x ∈ S, such that x = Ax + Bx.
The result contained in Proposition 3.2 has been obtained in the case of a normed space by Burton & Kirk (see [5]) and it represents the generalization of a known theorem of Krasnoselskii.The result of Burton & Kirk has been extended in [1] in the case of a Fréchet space.Lemma 3.2 Admit that hypothesis (k) is fulfilled.Then the equation admits a unique solution in C c .
Proof.We define the operator H : Let n ≥ 1 be fixed.Obviously, for t ∈ [0, n] , where where h (t) is given by (3.4) , with γ (t) = a (t) ∞ 0 b (s) ds.If, in addition, lim t→∞ h (t) = 0, then every solution x ∈ U to (1.1) is asymptotically stable and moreover, for every solution x ∈ U to (1.1) we have (i) As in the proof of Lemma 3.2 it follows that A is contraction.
(ii) We prove that B is compact operator.First, since hypothesis (g), the convergence of the integral ∞ 0 G (t, s, y (s) + ξ (s)) ds is uniform with respect to t on each compact subset of IR + , and so (By) (t) is a continuous function of t.
Let us consider Let us fix n ≥ 1.From the convergence of {y m } m and the continuity of ξ, there is r ≥ 0 such that |y m + ξ| n ≤ r, |y + ξ| n ≤ r, ∀m.

Proposition 3 . 2 (
[9]) Let A, B : C c → C c be two operators fulfilling the following hypotheses: