Let
- 格式：pdf
- 大小：184.37 KB
- 文档页数：13
= 1. ∆ x(t) = −ξ (t), t > T , so
Ψ` ξ (t) = o(1) as t → ∞.
If ξ ∈ S [T, ∞), then by (1.1) Ψξ solves the equation that (1.2) holds for ℓ = 1. Let ε > 0 be given arbitrarily. Since, by hypothesis, (1.1) converges uniformly on [T − σ, T ), there exists P ∈ N such that (1.4)
57
The conditions we always assume for f and g are as follows: (B) (a) (b) is continuous, and tlim !1 g (t) = ∞; f : [t0 , ∞) × R → R is continuous and satis es
Let ℓ > 1 be an integer. If ξ ∈ S ` [T, ∞), then Ψ` ξ is a solution of the difference equation
Lemma 1.
(1.2)
and satisfies
∆` x(t) = (−1)` ξ (t),
t > T,
(1.3)
Y. Kitamura, Nagasaki, T. Kusano,* Hiroshima, B. S. Lalli, Saskatoon
(Received September 8, 1993)
Summary. For a certain class of functional di erential equations with perturbations conditions are given such that there exist solutions which converge to solutions of the equations without perturbation. Keywords : neutral functional di erential equations, unperturbed equations MSC 1991 : 34K15
case m = 1. For related results regarding neutral equations of the form (A) with ∆ replaced by ∆ ; , where
∆ ; x(t) = x(t) − λx(t − σ), λ = 1,
we refer to Jaro and Kusano 1], Kitamura and Kusano 3], Naito 4, 5] and Ruan 7].
(A)
= 0,
t > t0 ,
where m > 1, n > 1, t0 > 0, and ∆m is the m-th iterate of ∆ , i.e.
∆m xபைடு நூலகம்t) =
m X i=0
(−1)i
m x(t − iσ ). i
* This work was done while visiting the University of Saskatchewan as a visiting Professor of Mathematics.
0. Introduction
Let
∆ x(t) = x(t) − x(t − σ), σ>0
being a constant, and consider the neutral functional di erential equation
? ? dn ∆m x(t) + f t, x g(t) n dt
xk (t) = ctk + o(1) as t → ∞, k = m, m + 1, . . . , m + n − 1. xj (t) = ω (t)tj + o(1)
as t → ∞,
j = 0, 1, . . . , m − 1,
It is clear that the solutions of the type (II) are nonoscillatory whereas the solutions of the type (I) are oscillatory or nonoscillatory according to whether the periodic functions ω (t) involved are oscillatory or nonoscillatory. The construction of solutions of types (I) and (II) of (A) is presented in Section 2. Our main tool is the Schauder-Tychono xed point theorem applied to nonlinear dn operators formed by suitably chosen \inverses" of the di erential operator d tn and m the iterated di erence operator ∆ . Preliminary results needed in Section 2 are collected in Section 1. It seems that very little is known about the existence of solutions, oscillatory or nonoscillatory, of neutral equations whose leading parts contain the di erence operator ∆ and/or its iterate. To the best of the authors' knowledge Jaro and Kusano 2] and Naito 6] are the only references dealing with this subject for the 58
g : [t0 , ∞) → [0, ∞) |f (t, x)| 6 F (t, |x|), (t, x) ∈ [t0 , ∞) × R ,
for some continuous function F (t, u) on [t0 , ∞) × [0, ∞) which is nondecreasing in u for each xed t > t0 . By a solution of (A) we mean a continuous function x : [Tx − mσ, ∞) → R , Tx > t0 , such that ∆m x(t) is n-times continuously di erentiable and satis es the equation for t > Tx . The solutions vanishing in a neighborhood of in nity will be excluded from our consideration. A solution is said to be oscillatory if it has an in nite sequence of zeros clustering at t = ∞; otherwise a solution is said to be nonoscillatory. The objective of this paper is to develop an existence theory enabling us to construct various types of oscillatory and nonoscillatory solutions of neutral equations of the formn(A). We make use of the observation that the associated unperturbed d m equation d tn ∆ x(t) = 0 has the solutions (0.1) (0.2)
S ` [T, ∞) = ξ ∈ S `?1 [T, ∞) : Ψ`?1 ξ ∈ S T − (ℓ − 1)σ, ∞ , ℓ = 1, 2, . . . ,
where it is understood that Ψ0 = id (identity mapping) and S 0 [T, ∞) = C [T, ∞).
120 (1995)
MATHEMATICA BOHEMICA
No. 1, 57{69
EXISTENCE OF OSCILLATORY AND NONOSCILLATORY SOLUTIONS FOR A CLASS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS
1 X
i=p+1
ξ (t + iσ ) < ε
Proof. Let ℓ
for all t ∈ [T − σ, T ) and p > P. 59
Let t > t1 = T + P σ and choose p ∈ N such that t − pσ ∈ [T − σ, T ). Then
p> t−T σ
ω (t)tj , ctk , k = m, m + 1, . . . , m + n − 1, j = 0, 1, . . . , m − 1,
where ω (t) is an arbitrary σ-periodic function and c is an arbitrary constant, and intend to establish the existence of solutions xj (t), xk (t) of (A) which are asymptotic to the functions (0.1) and (0.2) in the sense that (I) (II)
Ψ` ξ (t) = o(1) as t → ∞.
If ξ ∈ S [T, ∞), then by (1.1) Ψξ solves the equation that (1.2) holds for ℓ = 1. Let ε > 0 be given arbitrarily. Since, by hypothesis, (1.1) converges uniformly on [T − σ, T ), there exists P ∈ N such that (1.4)
57
The conditions we always assume for f and g are as follows: (B) (a) (b) is continuous, and tlim !1 g (t) = ∞; f : [t0 , ∞) × R → R is continuous and satis es
Let ℓ > 1 be an integer. If ξ ∈ S ` [T, ∞), then Ψ` ξ is a solution of the difference equation
Lemma 1.
(1.2)
and satisfies
∆` x(t) = (−1)` ξ (t),
t > T,
(1.3)
Y. Kitamura, Nagasaki, T. Kusano,* Hiroshima, B. S. Lalli, Saskatoon
(Received September 8, 1993)
Summary. For a certain class of functional di erential equations with perturbations conditions are given such that there exist solutions which converge to solutions of the equations without perturbation. Keywords : neutral functional di erential equations, unperturbed equations MSC 1991 : 34K15
case m = 1. For related results regarding neutral equations of the form (A) with ∆ replaced by ∆ ; , where
∆ ; x(t) = x(t) − λx(t − σ), λ = 1,
we refer to Jaro and Kusano 1], Kitamura and Kusano 3], Naito 4, 5] and Ruan 7].
(A)
= 0,
t > t0 ,
where m > 1, n > 1, t0 > 0, and ∆m is the m-th iterate of ∆ , i.e.
∆m xபைடு நூலகம்t) =
m X i=0
(−1)i
m x(t − iσ ). i
* This work was done while visiting the University of Saskatchewan as a visiting Professor of Mathematics.
0. Introduction
Let
∆ x(t) = x(t) − x(t − σ), σ>0
being a constant, and consider the neutral functional di erential equation
? ? dn ∆m x(t) + f t, x g(t) n dt
xk (t) = ctk + o(1) as t → ∞, k = m, m + 1, . . . , m + n − 1. xj (t) = ω (t)tj + o(1)
as t → ∞,
j = 0, 1, . . . , m − 1,
It is clear that the solutions of the type (II) are nonoscillatory whereas the solutions of the type (I) are oscillatory or nonoscillatory according to whether the periodic functions ω (t) involved are oscillatory or nonoscillatory. The construction of solutions of types (I) and (II) of (A) is presented in Section 2. Our main tool is the Schauder-Tychono xed point theorem applied to nonlinear dn operators formed by suitably chosen \inverses" of the di erential operator d tn and m the iterated di erence operator ∆ . Preliminary results needed in Section 2 are collected in Section 1. It seems that very little is known about the existence of solutions, oscillatory or nonoscillatory, of neutral equations whose leading parts contain the di erence operator ∆ and/or its iterate. To the best of the authors' knowledge Jaro and Kusano 2] and Naito 6] are the only references dealing with this subject for the 58
g : [t0 , ∞) → [0, ∞) |f (t, x)| 6 F (t, |x|), (t, x) ∈ [t0 , ∞) × R ,
for some continuous function F (t, u) on [t0 , ∞) × [0, ∞) which is nondecreasing in u for each xed t > t0 . By a solution of (A) we mean a continuous function x : [Tx − mσ, ∞) → R , Tx > t0 , such that ∆m x(t) is n-times continuously di erentiable and satis es the equation for t > Tx . The solutions vanishing in a neighborhood of in nity will be excluded from our consideration. A solution is said to be oscillatory if it has an in nite sequence of zeros clustering at t = ∞; otherwise a solution is said to be nonoscillatory. The objective of this paper is to develop an existence theory enabling us to construct various types of oscillatory and nonoscillatory solutions of neutral equations of the formn(A). We make use of the observation that the associated unperturbed d m equation d tn ∆ x(t) = 0 has the solutions (0.1) (0.2)
S ` [T, ∞) = ξ ∈ S `?1 [T, ∞) : Ψ`?1 ξ ∈ S T − (ℓ − 1)σ, ∞ , ℓ = 1, 2, . . . ,
where it is understood that Ψ0 = id (identity mapping) and S 0 [T, ∞) = C [T, ∞).
120 (1995)
MATHEMATICA BOHEMICA
No. 1, 57{69
EXISTENCE OF OSCILLATORY AND NONOSCILLATORY SOLUTIONS FOR A CLASS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS
1 X
i=p+1
ξ (t + iσ ) < ε
Proof. Let ℓ
for all t ∈ [T − σ, T ) and p > P. 59
Let t > t1 = T + P σ and choose p ∈ N such that t − pσ ∈ [T − σ, T ). Then
p> t−T σ
ω (t)tj , ctk , k = m, m + 1, . . . , m + n − 1, j = 0, 1, . . . , m − 1,
where ω (t) is an arbitrary σ-periodic function and c is an arbitrary constant, and intend to establish the existence of solutions xj (t), xk (t) of (A) which are asymptotic to the functions (0.1) and (0.2) in the sense that (I) (II)
