Bounded and Periodic Solutions of Nonlinear Integro-differential Equations with Infinite Delay

By using the concept of integrable dichotomy, the fixed point theory, functional analysis methods and some new technique of analysis, we obtain new criteria for the existence and uniqueness of bounded and periodic solutions of general and periodic systems of nonlinear integro-differential equations with infinite delay.

In particular, the existence of bounded and periodic solutions of several families of quasilinear systems has been advantageously studied with the help of the Green matrix G(t, s) of system (1) and concluding that for any bounded function f is a bounded solution of the non-homogeneous linear system Nevertheless, notice that similar results can be obtained by the more general condition: that is, if system (1) has an integrable dichotomy.For example, condition (4) holds for any integrable (h, k)-dichotomy without need of being exponential [7,8,14].Under the assumptions that A and f are periodic and (1) has an integrable dichotomy, we will study the periodicity of the solution x given by (2).The existence of periodic solutions of functional differential equations has been discussed extensively in theory and in practice (for example, see [1][2][3][7][8][9][10][14][15][16][17][18][19] and the references cited therein), but there are few papers considering integrable dichotomies.
2 Bounded and periodic solutions of nonhomogeneous systems.
Let C n and R n denote the sets of complex and real vectors, and |x| any convenient norm for x ∈ C n , also let Now, we recall some of the definitions (see [1,14,15,[24][25][26]), concerning integrable dichotomy and the notion of (h, k)−dichotomy for linear nonautonomous ordinary differential equations .A solution-matrix Φ(t) of system (1) is said to be a fundamental-matrix, if Φ(0) = I.For a projection matrix P, we define G = G P a Green matrix as: Definition 1 .System (1) is said to have an integrable dichotomy , if there exist a projection P and µ ∈ R + such that its Green matrix G satisfies: Definition 2 Let h, k : R → R + be two positive continuous functions.
The linear system (1) is said to possess an (h, k)−dichotomy, if there are a projection matrix P and a positive constant K such that for all t, s ∈ R the following inequality holds: where Definition 3 We say that system (1) has an integrable (h, k)-dichotomy if system (1) has an (h, k)-dichotomy for which there exists Let us state our main hypothesis on the linear system (1).(I) System (1) has an integrable dichotomy .
If the system (1) has an (h, k)-dichotomy integrable then necessarily the dichotomy satisfies condition (D).
Corollary 1.For every integrable (h, k)-dichotomy, there exist constants α, M > 0 such that h(t) ≤ Me −αt for all t ≥ 0; k(t) −1 ≤ Me αt for all t ≤ 0. Proposition 1.If system (1) has an integrable dichotomy, then x(t) = 0 is the unique bounded solution of system (1).EJQTDE, 2009 No. 46, p. 5 Proof.Define B 0 ⊂ C n to be the set of initial conditions ξ ∈ C n pertaining to bounded solutions of Eq (1).Take any vector ξ ∈ C n and assume first that (I − P )ξ = 0. Define φ(t) −1 = |Φ(t)(I − P )ξ|, we may write So using the integrability of the dichotomy, we have uniformly in t.
Thus B 0 = {0} and the only bounded solution of system (1) is x(t) = 0. Proposition 2. If system (1) satisfies condition (I) then the system (3) has exactly one bounded solution x, which can be represented by (2).
Proof.It is not difficult to check that x(t), given by ( 2), is a bounded solution of (3).If there exists another bounded solution z(t), then x(t) − z(t) is a bounded solution of the homogeneous linear system (1).By Proposition 1, x(t) ≡ z(t).The uniqueness of the bounded solution of (3) is proved.
From now on, the boundedness of Φ(t)P Φ −1 (t) is fundamental.
Proposition 3. If the linear system (1) satisfies hypothesis (D), then the projector P is unique, i.e., P is decided uniquely by the integrable dichotomy.
Finally, we obtain two important consequences to the non homogeneous linear system (3).
Proposition 5. Let all conditions in Proposition 4 hold and f (t) is a T -periodic function.Then system (3) has exactly one T −periodic solution, which can be represented as (2).Proof.By Proposition 4, it is not difficult to check that x(t), given by ( 2), is a T − periodic solution and so it is a bounded solution.Then, by Proposition 2, the result follows.

Existence of bounded and periodic solutions
In this section, we will prove some results about the existence and uniqueness of bounded and periodic solutions of system (5).Let the corresponding space of the initial conditions ϕ: where F 1 involves bounded delay and F 2 unbounded delay.Typically system (15) has the form: and its linear system (1) has an integrable dichotomy , where

Now we introduce the following conditions:
Dichotomy conditions: • (I) The linear system (1) possesses an integrable dichotomy with projection P and constant µ.
• (C 1 ) The function g: For any > 0 there exist δ > 0 and γ : Now we are ready to state our main results.
Remark 7 As a special case, when λ(t, s) = λ 1 (t−s), the smallness condition in (L 2 ) can be reduced to We will prove only Theorems 3 and 4 about periodic solutions because, with the obvious differences, the proofs of Theorems 1 and 2 are respectively similar.In Theorems 1 and 2, we will use Proposition 2, while in Theorems 3 and 4, we will use Proposition 5.In all of them, we need the same operator defined on the Banach space B = {u : R → C n |u is continuous and bounded} provided with the supremum-norm.Proof of Theorem 3. Consider the Banach space P = { u : R → C n |u(t) is continuous T − periodic function} provided with the norm u = sup {|u(t)| : 0 ≤ t ≤ T } .For any u ∈ P, consider the integro-differential correspondence: where F is the functional: By the conditions (D) and (P) and Proposition 5, F (t, u) is T-periodic in t and system (18) has exactly one T −periodic solution which can be written as So, the operator Γ : P → P given by is well defined and any fixed point of Γ is a T-periodic solution of system (18).By (L 1 ) and (L 2 ), we shall prove that Γ is a contraction mapping in P .In fact, for any u 1 , u 2 ∈ P, it follows from ( 19), (20) and the conditions in Theorem 3 that It follows from µ(2L 1 + L 2 ) < 1, that Γ is a contraction mapping.Therefore Γ has exactly one fixed point u in P. It is easy to check that u is the unique T -periodic solution of (18).
Proof of Theorem 4. Take the Banach space P and the operator Γ defined in the proof of Theorem 3. Now by using Schauder's fixed point theorem, we shall prove that Γ has at least one fixed point under the assumption of Theorem 4.
Proof.If not, for any n ∈ N, there exists u n ∈ B n such that Γu n > n.For any sufficiently small , it follows from (E 1 ) and (E 2 ) that there exists sufficiently large N ∈ N such that if n > N then As µ(2c 1 +c 2 ) < 1, taking sufficiently small , we have µ(2c 1 +c 2 )+µ < 1.Therefore, it follows that lim Therefore, {Γu(t)/u ∈ B N } is equicontinuous.It follows from Ascoli-Arzela theorem that ΓB N is a relatively compact subset of B. Proof.Since g(t, x, y) is uniformly continuous on [0, T ] × C N and g(t + T, x, y) = g(t, x, y), g(t, x, y) is uniformly continuous on R × C N .Therefore, has at least one T-periodic solution.EJQTDE, 2009 No. 46, p. 10

Lemma 3 :
Γ is continuous on B N .