Study on the Oscillation of a Class of Nonlinear Delay Functional Differential Equations

The research is financed by Hunan Provincial Natural Science Fund (No. 060D74). (Sponsoring information) Abstract In this paper, a class of nonlinear delay functional differential equations with variable coefficients is linearized, and through analogizing the oscillation theory of linear functional differential equation, we obtain many oscillation criteria of this class of equation by using the Schauder fixed point theorem.


Introduction
There are many researchers about the oscillation of the linear delay functional differential equation with constant coefficients and the linear delay functional differential equation with variable coefficients, and a series of conclusions has been acquired.However, the literatures about the nonlinear delay functional differential equation with variable coefficients are very few.In the following study, we suppose the functional differential equation accords with the whole existence of solution, and we will use the Schauder fixed point theorem when proving the existence of positive solution.
Consider the nonlinear delay functional differential equation with variable coefficients and the linear delay functional differential equation with constant coefficients Replace the variable coefficients in the equation (1) by the constant q i , we can obtain the equation Gyori's article (Gyori, 1991) studied the oscillation of equation ( 3) and proved that if the following conditions (H 1 ) lim u, and when u ∈ (−σ, 0], f (u) u comes into existence, so the sufficient and necessary condition of the oscillation of differential equation ( 3) is the equation ( 2) is oscillatory.

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In the article, we will discuss the oscillation of the equation (1) which is more common than the equation (3), and the result will extend the conclusion in Gyori's article.To prove the main result, we first introduce the following lemma.
Make integral to the above inequation from s to s + τ 2 , we can obtain Change s in (5) by t, and from (6), we can obtain , and when t is enough big, the following inequation comes into existence.
Lemma 1.3 (Gyori, 1991): The sufficient and necessary condition of the oscillation of the differential equation ( 2) is the characteristic equation λ + n i=1 q i e −τ i λ = 0 has no real root.
Lemma 1.4 (Zhang, 1987) (Schauder fixed point theory): Suppose M is the closed convex subset in the Banach space X, T : M → M is continuous, and is the relative compact subset of X, so T must have a fixed point x ∈ M to make T x = x.

Main results and proofs
For the need of following proofs, we give following conditions after (H 1 ), (H 2 ) and (H 3 ).
(H 4 ) lim Prove: Suppose x(t) is the non-oscillatory solution and the finally positive solution of the equation ( 1), and for the situation of finally negative solution, we can prove it analogously.From the equation (1), we can obtain So x(t) is finally monotonically decreasing function, and suppose lim t→∞ x(t) = l, so l = 0, or else, l > 0, from the equation (1), we can obtain The above equation indicates lim t→∞ x(t) = −∞, that is contrary with the condition that x(t) is the finally positive solution.So the theorem is proved.
Theorem 2.2: Under the condition of (H 6 ), if the equation ( 2) is oscillatory, so one j 0 exists at least and makes q j 0 > 0 and τ j 0 > 0.
Prove: Because the equation ( 2) is oscillatory, from Lemma 1.3 (Gyori, 1991), we know the characteristic equation has no real root.And because F(∞) > 0, F(0) = n i=1 q i > 0, so one j 0 exists at least to make q j 0 > 0 and q i < 0 is one negative real root of the characteristic equation q i e −τ i λ = 0, but that is impossible.The theorem is proved.
Prove: Because x(t) is the finally positive solution, according to the conditions of (H 1 ), (H 4 ) and Theorem 2.1, we can obtain So, to any appointed positive number ε ∈ (0, 1), enough big T 0 t 0 exists, and when t T 0 , the following inequation exists.
Prove: Otherwise, the equation( 1) has the non-oscillatory solution x(t).Suppose x(t) is the finally positive solution, we can analogously prove the situation of finally negative solution.From the theorem 2.3, the set Λ ≡ {λ 0 : x (t) + λx(t − τ j 0 ) 0, t T 0 } is nonempty and bounded.

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Because the equation ( 2) is oscillatory, from Lemma 1.3, we can obtain the characteristic equation has not real root.Suppose K = min λ∈R F(λ), so the inequation exists.
Prove: Otherwise, the equation ( 2) is non-oscillatory.From Lemma 1.3, we know the characteristic equation q i e −τ i λ = 0 has real root u, and u < 0. If τ = max 1 i n {τ i }, X is the Banach space which is composed by the collectivity of bounded continuous function with supremum norm in [t 0 − τ, ∞], M in X is the set composed by the function x(t) which could fulfill following characters.
Next, we will use Lemma 1.4 (Schauder fixed point theorem) to prove that the fixed point exists in T on M. Obviously, (T x)(t)is the continuously monotonically decreasing function, and (T x)(t) x 0 .
When t t 0 , we can obtain the following inequations.

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March, 2009 From ( 18) and ( 19), we can obtain (T x)(t) ∈ M, and the set M is the closed convex nonempty set.Next, we prove the M is relatively compact subset of X, and we only need to prove (T x)(t) is equicontinuous, i.e. d(T x)(t) dt is uniformly bounded.In fact, dt is uniformly bounded.
From above proofs, we can see that the mapping (T x)(t) from M to M fulfills the condition of Schauder fixed point theorem, so the fixed point x(t) exists and (T x)(t) = x(t), and x(t) > 0 fulfills the equation (1), i.e. the equation (1) has finally positive solution, which is contrary with the condition that the equation ( 1) is oscillatory.The theorem is proved.From Theorem 2.4 and Theorem 2.5, we can obtain following deductions.
Example: We know the nonlinear functional differential equation e − 3π 4 , f (u) = arctan u, so the equation ( 20) is oscillatory.