Báo cáo " Asymptotic equilibrium of the delay differential equation "

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In this paper, we show that if the operator $A(\cdot)$ is strongly continuous on Hilbert space $\mathbb H,$ $A(t)=A^*(t),$ $\sup\limits_{\|h\|\le 1}\int\limits_{T}^{+\infty}\|A(t)h\|dt\le q

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VNU Journal of Science, Mathematics - Physics 23 (2007) 92-98 Asymptotic equilibrium of the delay differential equation Nguyen Minh Man∗, Nguyen Truong Thanh Department of Mathematics, Hanoi University of Mining and Geolory, Dong Ngac, Tu Liem, Hanoi, Vietnam Received 16 April 2007; received in revised form 15 August 2007 Abstract. In this paper, we show that if the operator A(·) is strongly continuous on Hilbert +∞ space H, A(t) = A∗ (t), sup A(t)h dt ≤ q < 1 then the equation h ≤1 T d x(t) = A(t)x(t − r ), ∀t ≥ 0, r is a given positive constant, dt is asymptotic equilibrium. 1. Introduction Consider the delay differential equation d x(t) = A(t)x(t − r ) (t ≥ 0), (1) dt where r is a given positive constant, A(·) ∈ C (R+ , L(H )), H is a Hilbert space. We will show a condition for the asymptotic equilibrium of Eq (1)by extending some results obtained from the equation d x(t) = A(t)x(t) (t ≥ 0), (2) dt Finding conditions for the asymptotic equilibrium of a differential equation is considered by many mathematicians. Some of the mathmaticians are L. Cezari, A. Winter, A. Ju. Levin, Nguyen The Hoan,.etc. In a paper, L. Cezari asserted that If A(t) ∈ L1 (R+ , Rn ) then Eq (2) is asymptotic equilibrium The result was devoloped by A. Winter (1954) (see [2]), and A. Ju. Levin (1967) (see [3]). However, those results were restricted to finite dimentional spaces. Nguyen The Hoan extended them into any Hilbert spaces. From this result, we extend on Eq (1) and obtain a similar result (theorem 3.3) ∗ Corresponding author. Tel.: 84-4-8387564. E-mail: ngmman@yahoo.com 92
N.M. Man, N.T. Thanh / VNU Journal of Science, Mathematics - Physics 23 (2007) 92-98 93 2. Preliminaries The section will be devoted entirely to the notation and concept of asymptotic equilibrium of differential equations. Almost all results of this section are more or less known. However, for the reader’s convenience we will quote them here and even verify several results which seem to be obviuous but not available in the mathematical literature. Thoughout this paper we will use the following notations: H is a given hilbert space. r is a given positive constant. C ([a; b], H ) stands for the space of all continuous functions from the interval [a; b] into H. L(H ) is the set of all continous operators from H into itself. The purpose to introduce Pro.The Hoan’s theorem 1, we consider the following equation: d x(t) = A(t)x(t), ∀t ∈ R+ , (3) dt where A(·) : R+ → L(H), A(t) = A∗ (t) (∀t ∈ R), A(·) is strongly continuous. Definition 2.1. x(·) is called a solution of Eq (3) if there is such t0 ∈ R+ , x0 ∈ H that x(·) is a solution of the following Cauchy problem: d dt x(t) = A(t)x(t) (t ≥ t0 ), x(t0 ) = x0 . Definition 2.2. [Asymptotic equilibrium] The equation (3) is an asymptotic equilibrium equation if all solutions of the equation satisfy: i) if x(·) is an solution of Eq (3) then x(t) tends to the finite limit, as t −→ +∞. ii) For a given c belonging to H, there is such a solution x(·) of Eq (3) that lim x(t) = c. t→+∞ +∞ Theorem 2.3 If A(·) satisfies sup A(t)h dt < q < 1, where T, q are given, then Eq (3) is h ≤1 T asymptotic equilibrium. To prove this theorem, we must solve the following lemma. Note: We usually assume that A(·) satifies all the conditions of theorem (2.3). Lemma 2.4. If x(·) is a solution of Eq (3) then x(·) satisfies the condition (i) of definition (2.2). Proof. Firstly, we see, for t ≥ T, h ∈ H : h ≤ 1, t < x(t), h > =< x(T ), h > + < A(τ )x(τ ), h > dτ, T t =< x(T ), h > + < x(τ ), A(τ )h > dτ. T Hence, t x(t) = sup < x(t), h > ≤ x(T ) + A(t)h x(τ ) dτ h ≤1 T
N.M. Man, N.T. Thanh / VNU Journal of Science, Mathematics - Physics 23 (2007) 92-98 94 By the Gronwall-Bellman inequality, we have t A(τ )h dτ ≤ x(T ) eq < +∞. x(t) ≤ x(T ) eT Let M := sup x(t) . From t∈ R x(t) − x(s) = sup < x(t), h > − < x(s), h > h ≤1 t = sup < x(τ ), A(τ )h > dτ h ≤1 s t ≤ M sup A(τ )h dτ → 0, h ≤1 s as t ≥ s → +∞. This lemma is proved. The proof of theorem [2.3] Let a fixed h0 ∈ H. Consider the function: +∞ ξ1 (t, h) =< h0 , h > − < A(τ )h0 , h > dτ, t ≥ T, h ∈ H. t It is easy to show that i) ξ1 (t, h) ≤ (1 + q ) h0 , h ≤ 1. ii) ξ1 (t, h) ≤ ( h0 + q ) h h0 , ∀h ∈ H. Hence, ξ1 (t, ·) ∈ H ∗ = L(H, R). By theorem Riezs, there is a x1 (t) ∈ H : ξ1 (t, h) =< x1 (t), h > and x1 (t) ≤ (1 + q ) h0 . Let x0 (·) ≡ h0 . Obviously, d x1 (t) = A(t)x0 (t), ∀t ≥ T. dt By the same way, we establish two sequences {ξn (·, ·)},{xn(·)}: +∞  ξn (t, h) =< h0 , h > − < A(τ )xn−1 (τ ), h > dτ (t ≥ T, n ∈ N),    t   ξn (t, h) =< xn (t), h >, (4)  xn (t) ≤ (1 + q + · · · + q n ) h0 ≤ 1 h0 ,  1−q    d xn (t) = A(t)xn−1 (t) (t ≥ T ). dt Moreover, xn+1 (t) − xn (t) = sup < xn+1 (t) − xn (t), h > h ≤1 +∞ ≤ sup xn (τ ) − xn−1 (τ ) A(τ )h dτ h ≤1 T q n+1 h0 (∀n ∈ N). ≤
N.M. Man, N.T. Thanh / VNU Journal of Science, Mathematics - Physics 23 (2007) 92-98 95 Consequently, {xn (·)} converges uniformly on [T, +∞). Let the limit of {xn (·)} be x(·) belonging to C ([T, +∞), H). For a given T1 > T, By the strongly continuous property of A(·), sup A(t)h = t∈[T ,T1] Mh < +∞. It follows from the uniformly bounded priciple, there is such a positive M1 that sup A(t) = M1 < +∞. t∈[T ,T1] From d d xn+1 (t) − xn (t) = sup < A(t)xn (t) − A(t)xn−1 (t), h > dt dt h ≤1 = sup < xn (t) − xn−1 (t), A(t)h > h ≤1 ≤M1 xn (t) − xn−1 (t) ≤ M1 q n h0 , ∀t ∈ [T, T1], d d the sequence { dt xn (·)} converges uniformly to dt x(·) on (T, T1). On the other hand, +∞ < xn (t), h >=< h0 , h > − < xn−1 (t), A(t)h > dt (t ∈ [T, T1]). t Letting n → +∞, we have +∞ < x(t), h >=< h0 , h > − < x(t), A(t)h > dt (t ∈ [T, T1]). t d dt x(t), h >=< A(t)x(t), h >, ∀h ∈ H, t ∈ (T, T1). By a any given T1 > T, It leads to < d x(t) = A(t)x(t), ∀t > T. dt Observe that +∞ x(t) − h0 = sup < x(t) − h0 , h > = sup < x(τ ), A(τ )h > dτ h ≤1 h ≤1 t +∞ 1 ≤ sup A(τ )h dτ h0 → 0, 1−q h ≤1 t as t → +∞. This is proved the theorem. 3. The main result In the section, we will extend Pro.The Hoan’s result to the following delay differential equation: d x(t) = A(t)x(t − r ) (t ∈ R+ ), (5) dt where r is a given positive constant, A(·) satisfies all the conditions in the section 2.
N.M. Man, N.T. Thanh / VNU Journal of Science, Mathematics - Physics 23 (2007) 92-98 96 Definition 3.1. x(·) is called a solution of Eq (5) if there is such t0 ∈ R+ , ϕ ∈ C ([t0 − r, t0], H) that x(·) is a solution of the following Cauchy problem: d dt x(t) = A(t)x(t − r ) (t ≥ t0 ), xt0 = ϕ. Lemma 3.2. If x(·) is a solution of Eq (5), then x(·) satifies the condition (i) of definition [2.2]. Proof. From t < x(t), h >=< x(s), h > + < A(τ )x(τ − r ), h > dτ (t ≥ s ≥ T ), s we have t x(t) = sup < x(t), h > ≤ x(s) + sup A(τ )h x(τ − r ) dτ, t ≥ s > T + t0 . h ≤1 h ≤1 s Hence, x(s) |x(t) | ≤ x(s) + q |x(t) | or |x(t) | ≤ t ≥ s > T + t0 , , 1−q where |x(t) | = sup x(ξ ) . Let M := sup x(t) . t≥t0 −r t0 −r ≤ξ ≤t On the other hand, t x(t) − x(s) = sup < x(t) − x(s), h > = sup < x(τ − r ), A(τ ) > dτ h ≤1 h ≤1 s t ≤ M sup A(τ )h dτ → 0, h ≤1 s as t ≥ s → +∞. The lemma is proved. Theorem 3.3. The Eq (5) is asymptotic equilibrium. Proof. Let a ﬁxed h0 ∈ H. We consider the following function: +∞ ξ1 (t, h) =< h0 , h > − < A(τ )h0 , h > dτ (t ≥ T ). t Let x0 (t) ≡ 0. Use exactly the argument of the proof of theorem 2.3, we establish the functions x1 (·),x1(·) which satisfy : For t ≥ T, i) ξ1 (t, h) =< x1 (t), h >, d ii) dt x1 (t) = A(t)x0 (t), iii) x1 (t) = x1 (t), iv) x1 (t) ≤ (1 + q ) h0 . By the same way, we have three sequences {ξn (t, h)}, {xn(t)}, {xn(t)} which satisfy :
N.M. Man, N.T. Thanh / VNU Journal of Science, Mathematics - Physics 23 (2007) 92-98 97 +∞ i) ξn (t, h) =< xn (t), h >=< h0 , h > − < A(τ )xn−1 (τ − r ), h > dτ (t ≥ T + r ). t d ii) dt xn (t) = A(t)xn−1 (t − r ), ∀t > T + r. iii) xn (t), t ≥ T + r, xn (t) = xn (T + r ), T + r ≥ t ≥ T. 1 iv) xn (t) ≤ (1 + q + · · · + q n ) h0 ≤ t ≥ T. h0 , 1−q We see that xn+1 (t) − xn (t) = sup < xn+1 (t) − xn (t), h > h ≤1 +∞ ≤ sup xn (τ − r ) − xn−1 (τ − r ) A(τ )h dτ h ≤1 T q n+1 h0 ≤ (∀n ∈ N, t ≥ T + r ). Moreover, xn+1 (t) − xn (t) ≤ q n+1 h0 , ∀t ≥ T, n ∈ N. Consequently, {xn (·)} converges uniformly on [T, +∞). Let the limit of {xn (·)} be x(·) belonging to C ([T, +∞), H). From +∞ < xn+1 (t), h >=< h0 , h > − < xn (τ − r ), A(τ )h > dτ (t ≥ T + r ), t letting n → +∞, we have +∞ < x(t), h >=< h0 , h > − < x(τ − r ), A(τ )h > dτ (t ≥ T + r ). t d On the other hand, for a given any T1 > T + r, the sequence { dt xn (t)} converges uniformly on (T + r, T1](see proof of theorem 2.3). This leads to d d x(t) = lim xn (t) ∀t ≥ T + r. dt n→+∞ dt Hence, d x(t) = A(t)x(t − r ) ∀t > T + r. dt 1 By x(t)| ≤ h0 , ∀t ≥ T, 1−q +∞ x(t) − h0 = sup < x(t) − h0 , h > = sup < x(τ − r ), A(τ )h > dτ h ≤1 h ≤1 t +∞ 1 ≤ sup A(τ )h dτ h0 → 0, 1−q h ≤1 t as t → +∞. This is proved the theorem.
N.M. Man, N.T. Thanh / VNU Journal of Science, Mathematics - Physics 23 (2007) 92-98 98 References [1] L.Cezari, Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations, ”Mir” Moscow, 1964. [2] A.Winter, On a Theorem of Bocher in The Theory of Ordinary Linear Differential Equations, Amer J.Math. 76 (1954) 183. [3] A.Ju.Levin, Limiting Transition for The Nonlinear Systems, Dokl. Akad. Nauk SSSR 176 (1967) 774 (in Russian). [4] Nguyen The Hoan, Asymptotic Bahaviour of Solutions of Non Linear Systems of Differential Equations, Differentilnye Uravnenye 12 (1981) 624 (in Russian).