有關無窮維度差分方程系統全解之研究

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行政院國家科學委員會專題研究計畫成果報告

有關無窮維度差分方程系統全解之研究

研究成果報告(精簡版)

計畫類別：個別型計畫編號： NSC 97-2115-M-004-001- 執行期間： 97 年 08 月 01 日至 98 年 07 月 31 日執行單位：國立政治大學應用數學學系計畫主持人：符聖珍計畫參與人員：碩士班研究生-兼任助理人員：林明黎處理方式：本計畫可公開查詢

中華民國 98 年 08 月 06 日

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Oscillation and nonoscillation criteria for

linear dynamic systems on time scales ?

Sheng-Chen Fu and Ming-Li Lin

Deparment of Mathematical Sciences, National Chengchi University, 64, S-2 Zhi-nan Road, Taipei 116, Taiwan

Abstract

In this paper we establish oscillation and nonoscillation criteria for the linear dynamic system

u∆= pv, v∆= −quσ.

Here we assume that p and q are nonnegative, rd-continuous functions on T, where T is a time scale such that supT = ∞. Indeed, we extend some known oscillation theories on differential systems and difference systems to the so-called dynamic systems.

Key words: Oscillation; Linear dynamic systems; Time scale

1 Introduction

Let T be a time scale, i.e., a nonempty closed subset of R, which is un-bounded above. This paper is concerned with the linear dynamic system

u∆= pv, v∆ = −quσ, (1) where p and q are nonnegative, rd-continuous functions on T.

The global existence and uniqueness of solutions of (1) can be easily ver-ified by applying Theorem 5.8 of [12]. We say that a solution (u, v) of (1) is nonoscillatory if both u and v are either eventually positive or eventually

? This work is partially supported by National Science Council of the Republic of China under the contract 97-2115-M-004-001.

Corresponding author: Sheng-Chen Fu.

Email addresses: [email protected] (Sheng-Chen Fu); [email protected] (Ming-Li (Ming-Lin).

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negative. Otherwise, it is oscillatory. System (1) is called oscillatory if all its solutions are oscillatiory. Otherwise, it is nonoscillatory.

Oscillation for system (1) has received a lot of attention by many re-searchers. When T = R, system (1) is equivalent to the linear differential system

u0 = pv, v0 = −qu. (2) The oscillatory property of system (2) has been extensively studied, see for example [1], [2], [3], [4], [10] and the references cited therein. When T = Z, system (1) is equivalent to the linear difference system

∆xn= pnyn, ∆yn−1 = −qnxn. (3)

For papers dealing with oscillatory property of system (3), the reader is re-ferred to [5], [6], [7], [8] and the references cited therein. When p(t) 6= 0 for all t ∈ T, system (1) can be reduced to a single dynamic equation

(1 pu

∆

)∆+ quσ = 0,

which has been studied by many authors (see, for example, [9], [12], and the references cited therein).

Since there are few works about oscillation of dynamic systems on time scales (see [15]), motivated by [4] and [8], in the present paper we investigate oscillatory property for system (1).

The remaider of this paper is organized as follows. Section 2 contains some basic definitions and the necessary results about time scales. In Section 3, we present our main results, which include some oscillation and nonoscillation criteria for system (1). In Section 4, we provide some useful lemmas. In Section 5, we prove the main results. Finally, in Section 6, we give several examples to illustrate the applicability of the obtained results.

2 Preliminary

For completeness, we state some fundamental definitions and results con-cerning time scales that we will use in the sequel. More details can be found in [11], [12], and [13] . In this section, we assume that T is an arbitrary time scale. Definition 2.1 The mappings σ, ρ : T → T defined by

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are called the forward and backward jump operators respectively. In this defi-nition, we put inf ∅ = supT and sup∅ = inf T. Graininess function µ : T → [0, ∞) is defined by µ(t) := σ(t) − t.

Definition 2.2 A point t in T is said to be right-dense if t < supT and σ(t) = t, and left-dense if t > inf T and ρ(t) = t. Let ¯T = T∪{supT}∪{inf T}. We call ∞ left-dense if ∞ ∈ ¯_{T, and −∞ right-dense if −∞ ∈}_T.¯

Note that for any left-dense t0 ∈ T and any ε > 0, Lε(t0) = {t ∈ T|0 < t0−t <

ε} is nonempty. If ∞ ∈ ¯_{T, L}ε(∞) = {t ∈ T|t > 1_ε} is nonempty.

Definition 2.3 If a function f maps T into R, we define fσ by fσ(t) = f (σ(t)) which maps T into R.

Definition 2.4 Let TK=     

T\(ρ(sup T), sup T], if sup T < ∞,

T, otherwise .

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A function f : T → R is called (delta) differentiable at t ∈ TK if lim

s→t

fσ_{(t) − f (s)}

σ(t) − s

exists, saying f∆_{(t), where s ∈ (t − δ, t + δ) ∩ T for some δ > 0. In this case,} we call f∆_{(t) the (delta) derivative of f at t.}

Definition 2.5 A function on T is called rd-continuous if it is continuous at all right-dense points in T and its left-sided limits exist at all left-dense points in T. The set of all rd-continuous functions on T is denoted by Crd(T).

Definition 2.6 A function F : T → R is called an antiderivative of f : T → R if F∆(t) = f (t), and we define

Z s

r f (t)∆t = F (s) − F (r) for all r, s ∈ T.

Note that every rd-continuous function has an antiderivative.

Lemma 2.7 Assume that f, g : T → R are differentiable at t ∈ TK and let α ∈ R be a constant. Then the following statements are valid:

(1) fσ_{(t) = f (t) + µ(t)f}∆_(t).

(2) f + g is differentiable at t and (f + g)∆(t) = f∆(t) + g∆(t). (3) αf is differentiable at t and

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(4) f g is differentiable at t and

(f g)∆(t) = f∆(t)g(t) + fσ(t)g∆(t) = g∆(t)f (t) + gσ(t)f∆(t). (5) If f (t)fσ_{(t) 6= 0, then 1/f is differentiable at t and}

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∆_{(t) = −} f∆(t)

f (t)fσ_(t).

(6) If f (t)fσ(t) 6= 0, then g/f is differentiable at t and (g f) ∆ (t) = f (t)g ∆_{(t) − f}∆_(t)g(t) f (t)fσ_(t) .

Lemma 2.8 (1) If f ∈ Crd(T) and t ∈ TK, then

Rσ(t)

t f (τ )∆τ = µ(t)f (t).

(2) If f∆≥ 0, then f is nondecreasing.

Lemma 2.9 If a, b, c ∈ T, α ∈ R, and f , g ∈ Crd, then:

(1) Rb a[f (t) + g(t)]∆t = Rb af (t)∆t + Rb ag(t)∆t. (2) Rb a(αf )(t)∆t = α Rb af (t)∆t. (3) Rb af (t)g∆(t)∆t = (f g)(b) − (f g)(a) − Rb a f∆(t)gσ(t)∆t. (Integration by Parts)

Lemma 2.10 (Chain Rule) Assume that g : R → R is continuously differen-tiable and f : T → R is delta differendifferen-tiable. Then g◦f : T → R is differendifferen-tiable and (g ◦ f )∆(t) = Z 1 0 g0f (t) + hµ(t)f∆(t)dh f∆(t).

Lemma 2.11 (L’Hˆopital’s Rule) Assume that f and g are differentiable on T. If lim_t→t

0−

g(t) = ∞ for some left-dense t0 ∈ ¯T, and there exists ε > 0 such

that g(t) > 0 , g∆_{(t) > 0 for all t ∈ L}

ε(t0), then lim t→t₀₋ f∆_(t) g∆_(t) = r ∈ ¯R implies lim t→t₀₋ f (t) g(t) = r. 3 Main results

Let t0 be an arbitrary point in T.

Theorem 3.1 If Z ∞ t0 p(s)∆s = ∞, Z ∞ t0 q(s)∆s = ∞, (5)

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then system (1) is oscillatory. Theorem 3.2 If Z ∞ t0 p(s)∆s < ∞, Z ∞ t0 q(s)∆s < ∞, (6) then system (1) is nonoscillatory.

In the sequel, we are going to focus on the case

Z ∞ t0 p(s)∆s = ∞, Z ∞ t0 q(s)∆s < ∞. (7)

For convenience, we put

f (t) =

Z t

t0

p(s)∆s. (8)

Theorem 3.3 Suppose that lim

t→∞

µ(t)p(t)

f (t) = 0 (9)

and (7) hold. Suppose also that there exists λ ∈ (0, 1) such that

Z ∞

t0

fλ(s)q(s)∆s = ∞. (10) Then system (1) is oscillatory.

According to Theorem 3.3, we can furthermore restrict to the case:

Z ∞

t0

fλ(s)q(s)∆s < ∞ f or all λ ∈ [0, 1). (11) For convenience, we define

g(t, λ) =      f1−λ_(t)R∞ t fλ(s)q(s)∆s, if λ < 1, f1−λ(t)Rt t0f λ_(s)q(s)∆s, _{if λ > 1,} (12) and we set g∗(λ) = lim inf t→∞ g(t, λ), g∗(λ) = lim sup t→∞ g(t, λ). Theorem 3.4 Let (7), (9) and (11) hold. If

g∗(0) >

1

4 or g∗(2) > 1

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then system (1) is oscillatory.

Theorem 3.5 Let g∗(0) ≤ 1₄, g∗(2) ≤ 1₄, (7), (9) and (11) hold. Suppose that

there exists λ ∈ [0, 1) such that g∗(λ) > λ 2 4(1 − λ) + 1 2 1 +q1 − 4g∗(2) . (14)

Then system (1) is oscillatory.

Corollary 3.6 Let g∗(0) ≤ 1₄, g∗(2) ≤ 1₄ and (7), (9) and (11) hold. If

g∗(0) > 1 2 1 +q1 − 4g∗(2) , then system (1) is oscillatory.

Theorem 3.7 Let g∗(0) ≤ 1₄, g∗(2) ≤ 1₄, and (7), (9) and (11) hold. Assume

that there exists λ ∈ [0, 1) such that g∗(0) > λ(2 − λ) 4 (15) and g∗(λ) > g∗(0) 1 − λ+ 1 2 q 1 − 4g∗(0) + q 1 − 4g∗(2) . (16)

Then system (1) is oscillatory.

Corollary 3.8 Let 0 < g∗(0) ≤ 1₄, g∗(2) ≤ 1₄, and (7), (9) and (11) hold. If

g∗(0) > g∗(0) + 1 2 q 1 − 4g∗(0) + q 1 − 4g∗(2) , (17)

then system (1) is oscillatory.

Remark (i) We consider the more general first-order linear dynamic system u∆= au + bv, v∆= cu + dv, (18) where a, b, c, d ∈ Crd and b ≥ 0, c ≤ 0. Suppose that 1 + µa > 0 and

1 + µ(a + d) + µ2_{(ad − bc) > 0. Then one can easily verify that system (18) is}

equivalent to system (1) with p(t) = b(t)

[eα(t, t0)(1 + µa)]

≥ 0, q(t) = −c(t)eα(σ(t), t0) 1 + µa ≥ 0,

where α(t) = [a − d + µ(a2− ad + bc)]/[1 + µ(a + d) + µ2_{(ad − bc)]. Hence the}

oscillation and nonoscillation criteria for system (18) can be obtained from Theorem 3.1-3.5, 3.7 and Corollary 3.6, 3.8.

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(ii) We consider the case Z ∞ t0 p(s)∆s < ∞, Z ∞ t0 q(s)∆s = ∞. (19) Notice that if (u, v) is a solution of (1) then (˜u, ˜v) = (v, −u) is a solution of

˜

u∆= −µpq ˜u + q˜v, ˜v∆= −p˜u. (20) From Remark (i), we see that if µ2_{pq < 1 then system (20) is equivalent to}

˜ u∆= ˜p˜v, v˜∆ = −˜q ˜uσ, with ˜ p = q [eα(t, t0)(1 − µ2pq)] ≥ 0, q =˜ peα(σ(t), t0) 1 − µ2_pq ≥ 0.

Hence if (19) holds and µ2_{pq < 1 then the oscillation criteria for system (1)}

can be obtained from Theorem 3.3-3.5, 3.7 and Corollary 3.6, 3.8 by replacing p(t) and q(t) by ˜p(t) and ˜q(t) respectively.

4 Some auxiliary lemmas

In this section we establish some lemmas which will be needed to prove our main results. Hereafter [t0, ∞)T represents an interval on T, that is, [t0, ∞)T =

[t0, ∞) ∩ T. For convenience, we let

w(t) = v(t)

u(t), r = lim inft→∞ f (t)w(t), and R = lim sup_t→∞ f (t)w(t). (21)

Lemma 4.1 Let (7) hold. If (u, v) is a nonoscillatory solution of (1), then uv is eventually positive.

Proof. Without loss of generality, we may assume that

u(t) > 0 f or all t ∈ [t0, ∞)T. (22)

From (1) we have v∆_{≤ 0 for all t ∈ [t}

0, ∞)T. Hence v(t) > 0 for all t ∈ [t0, ∞)T.

Otherwise, there exists t1 ∈ [t0, ∞)T such that v(t) < 0 for all t ∈ [t1, ∞)T. In

this case, u∆(t) ≤ 0. Then Rt

t1u ∆_{(s)∆s =}Rt t1p(s)v(s)∆s implies u(t) = u(t1) + Z t t1 p(s)v(s)∆s ≤ u(t1) + v(t1) Z t t1 p(s)∆s. Letting t → ∞ and using (7), we get lim

t→∞u(t) = −∞, and this contradicts

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Lemma 4.2 Let (7) hold. If (u, v) is a nonoscillatory solution of (1), then we have eventually

0 ≤ f wσ ≤ f w ≤ 1.

Proof. By applying Lemma 4.1 and (1), without loss of generality, we may assume that u(t) > 0, v(t) > 0, u∆(t) ≥ 0, and v∆(t) ≤ 0 for all t ∈ [t0, ∞)T.

Then we have w(t) > 0 and

w∆(t) ≤ −q(t) − p(t)w(t)wσ(t), (23) for all t ∈ [t0, ∞)T. Hence we obtain

w∆(t) ≤ 0, t ∈ [t0, ∞)T. (24)

It follows from (23) and (24) that

w∆(t) ≤ −q(t) − p(t)(wσ(t))2, t ∈ [t0, ∞)T. (25) Rewrite (25) as q(t) ≤ −w∆(t) − p(t)(wσ(t))2, t ∈ [t0, ∞)T. (26) By (23) we get on [t0, ∞)T, −w∆_(t) w(t)wσ_(t) ≥ q(t) w(t)wσ_(t) + p(t) ≥ p(t), and therefore Z t t0 −w∆_(s) w(s)wσ_(s)∆s ≥ Z t t0 p(s)∆s = f (t). (27) Notice that Z t t0 −w∆_(s) w(s)wσ_(s)∆s = Z t t0 1 w(s) !∆ ∆s = 1 w(t) − 1 w(t0) ≤ 1 w(t). (28) Combining (27) and (28), we deduce that

w(t)f (t) ≤ 1, t ∈ [t0, ∞)T. (29)

By (24) and (29), we have

0 ≤ f (t)wσ(t) ≤ f (t)w(t) ≤ 1, t ∈ [t0, ∞)T. (30)

2

Lemma 4.3 Assume that (7) and (9) hold and let λ ∈ [0, 1). Then for any ε > 0 there exists t1 ∈ [t0, ∞)T such that

Z ∞ t h (fλ)∆(s)i2 p(s)fλ_(s) ∆s ≤ λ2 1 − λ(1 + ε) 2−λ_fλ−1_(t) ₍₃₁₎

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and Z ∞ t p(s)fλ−2(s)∆s ≤ (1 + ε) 2−λ 1 − λ f λ−1_(t), ₍₃₂₎ for all t ∈ [t1, ∞)T.

Proof. For s ∈ [t0, ∞)T, by Lemma 2.10, we have

(fλ)∆(s) ≤ λfλ−1(s)p(s) (33) and

(fλ−1)∆(s) ≤ (λ − 1)fσ(s)λ−2p(s). (34) From (33) and (34), we get

h (fλ)∆(s)i2 p(s)fλ_(s) ≤ λ 2_p(s)fλ−2_{(s) ≤} −λ2 1 − λ f (s) fσ_(s) !λ−2 (fλ−1)∆(s). (35) Let ε > 0 be given. From (9), there exists t1 ∈ [t0, ∞)T such that

u(s)p(s)

f (s) < ε, (36)

for all s ∈ [t1, ∞)T. Then for any s ∈ [t1, ∞)T we have

fσ(s)

f (s) = 1 +

µ(s)p(s)

f (s) < 1 + ε. (37) Integrating (35) from t to ∞ and by (7) and (37), we have

Z ∞ t h (fλ₎∆_(s)i2 p(s)fλ_(s) ∆s ≤ −λ2 1 − λ(1 + ε) 2−λZ ∞ t (fλ−1)∆(s)∆s = λ 2 1 − λ(1 + ε) 2−λ_fλ−1_(t),

for all t ∈ [t1, ∞)T. Therefore we get (31).

With the similar process, we can also get (32). Indeed, it follows from (7), (34) and (37) that R∞ t p(s)fλ−2(s)∆s = Z ∞ t p(s)fσ(s)λ−2 f (s) fσ_(s) !λ−2 ∆s ≤ (1 + ε)2−λZ ∞ t p(s)fσ(s)λ−2∆s ≤ (1 + ε)2−λZ ∞ t fλ−1_(s) λ − 1 !∆ ∆s = (1 + ε) 2−λ 1 − λ f λ−1_(t),

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for all t ∈ [t1, ∞)T. Therefore (32) holds. 2

Lemma 4.4 Assume that (7), (9) and (11) hold. If system (1) is nonoscilla-tory, then

r2− r + g∗(0) ≤ 0, (38)

R2− R + g∗(2) ≤ 0. (39)

Proof. Let (u,v) be a nonoscillatory solution of (1). By applying Lemma 4.1 and (1), without loss of generality, we may assume that u(t) > 0, v(t) > 0, u∆(t) ≥ 0, and v∆(t) ≤ 0 for all t ∈ [t0, ∞)T. Then by (7) and (30) we obtain

that 0 ≤ r ≤ R ≤ 1 and limt→∞w(t) = 0.

First, we derive (38). If g∗(0) = 0, then (38) is clear. Now we suppose that

g∗(0) 6= 0. Integrating (25) from t to ∞ and then multiplying by f (t), we have

f (t) Z ∞ t w∆(s)∆s ≤ −f (t) Z ∞ t q(s)∆s − f (t) Z ∞ t p(s)(wσ(s))2∆s, and hence f (t)w(t) ≥ f (t) Z ∞ t q(s)∆s + f (t) Z ∞ t p(s)(wσ(s))2∆s, (40) for all t ∈ [t0, ∞)T. We divide into two cases: 0 < r ≤ 1 or r = 0. For the case

0 < r ≤ 1, we let 0 < ε1 < min{r, g∗(0)}. From (9) and the definitions of r

and g∗(0), there exists t1 ∈ [t0, ∞)T such that

f fσ > 1 − ε1, f (t)w(t) > r − ε1, and f (t) Z ∞ t q(s)∆s > g∗(0) − ε1, (41)

for all t ∈ [t1, ∞)T. Since

Z ∞ t f∆(s) f (s)fσ_(s)∆s = 1 f (t), (42) we obtain on [t1, ∞)T, f (t) Z ∞ t p(s)(wσ(s))2∆s = f (t) Z ∞ t f∆_(s) f (s)fσ_(s) f (s) fσ_(s)(f σ_(s)wσ_(s))2_∆s ≥ (1 − ε1)(r − ε1)2f (t) Z ∞ t f∆_(s) f (s)fσ_(s)∆s = (1 − ε1)(r − ε1)2. (43)

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By (40), (41) and (43), we get

f (t)w(t) ≥ (g∗(0) − ε1) + (1 − ε1)(r − ε1)2, t ∈ [t1, ∞)T. (44)

Taking lim inf

t→∞ on (44) and letting ε1 → 0, we have r ≥ g∗(0) + r

2_{. Hence}

(38) follows. For the case r = 0, it suffices to show that g∗(0) ≤ 0. Let

0 < ε1 < g∗(0). As above, there exists t1 ∈ [t0, ∞)T such that (41) hold on

[t1, ∞)T . Then by (40) and (41), we get

f (t)w(t) ≥ g∗(0) − ε1, t ∈ [t1, ∞)T. (45)

Taking lim inf

t→∞ on (45) and letting ε1 → 0, we get g∗(0) ≤ 0. Now we derive

(39). If g∗(2) = 0, then (39) is clear. Now we suppose that g∗(2) 6= 0. By

Lemma 2.7 (1) and Lemma 2.10, we have on [t0, ∞)T,

(f2(s))∆≤ 2fσ_{(s)p(s) = 2 (f (s) + µ(s)p(s)) p(s).} ₍₄₆₎

Hence, multiplying (26) by f2(t) and integrating from t0 to t and using (46),

we get on [t0, ∞)T, Rt t0f 2_{(s)q(s)∆s ≤ −}Rt t0f 2_(s)w∆_{(s)∆s −}Rt t0f 2_(s)p(s)(wσ_(s))2_∆s = −f2_{(t)w(t) + f}2_(t 0)w(t0) +Rtt0(f 2_(s))∆_wσ_(s)∆s −Rt t0f 2_(s)p(s)(wσ_(s))2_∆s ≤ −f2_{(t)w(t) + f}2_(t 0)w(t0) +Rtt02µ(s)(p(s)) 2_wσ_(s)∆s +Rt t02f (s)p(s)w σ_{(s)∆s −}Rt t0f 2_(s)p(s)(wσ_(s))2_∆s ≤ −f2_{(t)w(t) + f}2_(t 0)w(t0) +Rtt02µ(s)(p(s)) 2_wσ_(s)∆s +Rt t0f (s)p(s)w σ_{(s)[2 − f (s)w}σ_(s)]∆s.

Dividing this inequality by f (t) and rearranging, we get on [t0, ∞)T,

f (t)w(t) ≤ − 1 f (t) Rt t0f 2_(s)q(s)∆s + 1 f (t) h f2(t0)w(t0) +Rtt02µ(s)(p(s)) 2_wσ_(s)∆si + 1 f (t) Rt t0f (s)p(s)w σ_{(s) [2 − f (s)w}σ_{(s)] ∆s.} (47)

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We consider the second part of the right hand side of (47). Applying Lemma 2.11 and using (9) and (30), we have

lim t→∞ f2(t0)w(t0) + Rt t02µ(s)(p(s)) 2_wσ_(s)∆s f (t) = lim t→∞2µ(t)p(t)w σ_(t) ≤ lim t→∞ 2µ(t)p(t) f (t) = 0. (48)

Now we divide into two cases: 0 ≤ R < 1 or R = 1. For the case 0 ≤ R < 1, we let 0 < ε2 < min{1 − R, g∗(2)}. From (9) and the definitions of R and

g∗(2), there exists t2 ∈ [t0, ∞)T such that

0 < f (t)w(t) < R + ε2 and 1 f (t) Z t t0 f2(s)q(s)∆s > g∗(2) − ε2, (49)

for all t ∈ [t2, ∞)T. Since 0 < f (s)w

σ_{(s) ≤ f (s)w(s) < R + ε} 2 < 1 on [t2, ∞)T, we obtain that on [t2, ∞)T, 1 f (t) Z t t0 f (s)p(s)wσ(s)[2 − f (s)wσ(s)]∆s < (R + ε2)(2 − R − ε2). (50) By (47), (49) and (50), we get on [t2, ∞)T, f (t)w(t) ≤ (−g∗(2) + ε2) + 1 f (t)[f 2_(t 0)w(t0) +Rtt02µ(s)(p(s)) 2_wσ_(s)∆s] +(R + ε2)(2 − R − ε2). (51)

Taking lim sup

t→∞

on (51) and letting ε2 → 0 , we have R ≤ −g∗(2) + R(2 − R).

Hence we get (39). For the case R = 1, we let 0 < ε2 < g∗(2). As above, there

exists t2 ∈ [t0, ∞)T such that (49) holds. Since 0 < f (s)wσ(s) ≤ 1 = R, we

obtain that 1 f (t) Z t t0 f (s)p(s)wσ(s)[2 − f (s)wσ(s)]∆s ≤ R(2 − R). (52) Then, using (47), (49) and (52), we get on [t2, ∞)T,

f (t)w(t) ≤ (−g∗(2) + ε2) +( 1 f (t)[f 2_(t 0)w(t0) +Rtt02µ(s)(p(s)) 2_wσ_(s)∆s]) +R(2 − R). (53)

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Taking lim sup

t→∞

on (53) and letting ε2 → 0, we have R ≤ −g∗(2) + R(2 − R).

Hence we can also easily get (39). Therefore the lemma follows. 2

5 Proof of the main results

Proof of Theorem 3.1. For contradiction, we assume that (1) has a nonoscil-latory solution (u, v). By applying Lemma 4.1 and (1), without loss of gener-ality, we may assume that u(t) > 0, v(t) > 0, u∆(t) ≥ 0, and v∆(t) ≤ 0 for all t ∈ [t0, ∞)T. Then

u(t) ≥ u(t0) > 0 f or all t ∈ [t0, ∞)T. (54)

Integrating the second equation of (1) from t0 to ∞ and using (54), we obtain

Z ∞ t0 v∆(s)∆s = − Z ∞ t0 q(s)uσ(s)∆s ≤ −u(t0) Z ∞ t0 q(s)∆s.

It follows that u(t0)

Z ∞

t0

q(s)∆s ≤ v(t0)−v(∞) ≤ v(t0) < ∞, which contradicts

(5). 2

Proof of Theorem 3.2. Let c, d > 0 be given. From (6), there exists T ∈ [t0, ∞)T

such that Z ∞ T p(s)(d + c Z ∞ s q(τ )∆τ )∆s < a,

where a = c/2. Let U be the Banach space of all real-valued rd-continuous functions on [T, ∞)_T endowed with the norm kuk = sup

t∈[T ,∞)_T

| u(t) |, i.e,

U = {u : [T, ∞)_T _{→ R | kuk = sup}

t∈[T,∞)_T

| u(t) |< ∞},

and with the usual pointwise ordering ≤. First, we define a subset Ω of U as : Ω = {u ∈ U | a ≤ u(t) ≤ 2a, ∀t ∈ [T, ∞)_T}.

Then inf B ∈ Ω and sup B ∈ Ω for any nonempty subset B of Ω. Second, we define an operation F : Ω → U as : (F u)(t) = a + Z t T p(s)(d + Z ∞ s q(τ )uσ(τ )∆τ )∆s, t ∈ [T, ∞)_T.

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Then F is nondecreasing. Also, if u ∈ Ω, by the definition of F , it is clear that (F u)(t) ≥ a and (F u)(t) ≤ a +Rt T p(s)(d + R∞ s q(τ )(2a)∆τ )∆s = a +Rt T p(s)(d + c R∞ s q(τ )∆τ )∆s ≤ 2a,

for all t ∈ [T, ∞)_T. Hence F maps into itself. By Knaster’s fixed-point theorem [14], there exists an u ∈ Ω such that u = F u. Set

v(t) = d +

Z ∞

t

q(k)uσ(τ )∆τ, t ∈ [T, ∞)_T.

It is easy to see that v∆_{(t) = −q(t)u}σ_{(t), lim}

t→∞v(t) = d > 0 and u

∆_{(t) =}

p(t)v(t). Hence (u, v) is a nonoscillatory solution of (1). Therefore we conclude that system (1) is nonoscillatory. 2

Proof of Theorem 3.3. For contradiction, we assume that (1) has a nonoscil-latory solution (u, v). By applying Lemma 4.1 and (1), without loss of gener-ality, we may assume that u(t) > 0, v(t) > 0, u∆_{(t) ≥ 0, and v}∆_{(t) ≤ 0 for all}

t ∈ [t0, ∞)T. Then by (30), we obtain that

0 < f (t)wσ(t) ≤ 1, t ∈ [t0, ∞)T,

and hence there exists a positive constant c such that | f (t)wσ_{(t)(λ − f (t)w}σ_{(t)) |< c,} _{t ∈ [t}

0, ∞)T. (55)

Multiplying (26) by fλ_{(t) and then integrating from t}

0 to t, we have Rt t0f λ_(s)q(s)∆s ≤ −Rt t0f λ_(s)w∆_{(s)∆s −}Rt t0f λ_(s)p(s)(wσ_(s))2_∆s (56)

Applying integration by part and Lemma 2.10, we get

Rt t0f λ_(s)w∆_(s)∆s = (fλw)(t) − (fλw)(t0) −Rtt0(f λ_(s))∆_wσ_(s)∆s ≥ (fλ_{w)(t) − (f}λ_w)(t 0) − Rt t0λf λ−1_(s)p(s)wσ_(s)∆s. (57)

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From (55), (56), (57) and Lemma 4.3, there exists t1 ∈ [t0, ∞)T such that Rt t0f λ_(s)q(s)∆s ≤ fλ_(t 0)w(t0) + Rt t0p(s)f λ−2_{(s)[f (s)w}σ_{(s)(λ − f (s)w}σ_(s))]∆s ≤ fλ_(t 0)w(t0) + cRtt0p(s)f λ−2_(s)∆s ≤ fλ_(t 0)w(t0) + c Rt1 t0 p(s)f λ−2_{(s) +}c(1 + ε) 2−λ 1 − λ f λ−1_(t 0), (58)

for all t ∈ [t1, ∞)T. Therefore we have

R∞

t0 f

λ_{(s)q(s)∆s < ∞, which contradicts}

(10). 2

Proof of Theorem 3.4. For contradiction, we assume that (1) is nonoscillatory. Then Lemma 4.4 implies that g∗(0) ≤ r − r2 ≤ 1₄ and g∗(2) ≤ R − R2 ≤ 1₄,

which contradicts (13). 2

Proof of Theorem 3.5. For contradiction, we assume that (1) has a nonoscil-latory solution (u, v). By applying Lemma 4.1 and (1), without loss of gener-ality, we may assume that u(t) > 0, v(t) > 0, u∆_{(t) ≥ 0, and v}∆_{(t) ≤ 0 for all}

t ∈ [t0, ∞)T. Then Lemma 4.4 implies

R ≤ 1 2 1 +q1 − 4g∗(2) . (59)

Multiplying (26) by fλ(t) and then integrating from t to ∞, we have

Z ∞ t fλ(s)q(s)∆s ≤ − Z ∞ t fλ(s)w∆(s)∆s − Z ∞ t fλ(s)p(s)(wσ(s))2∆s = −fλ_{(∞)w(∞) + f}λ_{(t)w(t) +}Z ∞ t (fλ(s))∆wσ(s)∆s − Z ∞ t fλ(s)p(s)(wσ(s))2∆s = fλ_{(t)w(t) +}1 4 Z ∞ t h (fλ)∆(s)i2 p(s)fλ_(s) ∆s − Z ∞ t p1/2(s)fλ/2(s)wσ(s) −1 2(f −λ/2 (s)p−1/2(s)(fλ(s))∆ 2 ∆s ≤ fλ_{(t)w(t) +}1 4 Z ∞ t h (fλ₎∆_(s)i2 p(s)fλ_(s) ∆s.

Then multiplying this inequality by f1−λ_{(t) and applying Lemma 4.3, we have}

on [t1, ∞)T, f1−λ(t) Z ∞ t fλ(s)q(s)∆s ≤ f (t)w(t) + λ 2 4(1 − λ)(1 + ε) 2−λ_, ₍₆₀₎

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where t1 ∈ [t0, ∞)T is given in Lemma 4.3. Taking lim sup t→∞

on (60), letting ε → 0, and using (59), we obtain

g∗(λ) ≤ R + λ 2 4(1 − λ) ≤ 1 2 1 +q1 − 4g∗(2) + λ 2 4(1 − λ), which contradicts (14). 2

Proof of Corollary 3.6. By taking λ = 0 in Theorem 3.5, the proof is done. 2 Proof of Theorem 3.7. For contradiction, we assume that (1) has a nonoscil-latory solution (u, v). By applying Lemma 4.1 and (1), without loss of gener-ality, we may assume that u(t) > 0, v(t) > 0, u∆(t) ≥ 0, and v∆(t) ≤ 0 for all t ∈ [t0, ∞)T. Then Lemma 4.4 implies

r ≥ 1 2 1 −q1 − 4g∗(0) , m, R ≤ 1 2 1 +q1 − 4g∗(2) , M.

By (15), we have 1 − 4g∗(0) < 1 − λ(2 − λ) = (1 − λ)2, and hence

m = 1 2 1 −q1 − 4g∗(0) > 1 2(1 − (1 − λ)) = λ 2.

Take ε > 0 such that (1 − ε)(m − ε) ≥ λ/2. Then there exists t1 ∈ [t0, ∞)T

such that

m − ε < f (t)w(t) < M + ε (61) and

f

fσ > 1 − ε (62)

for all t ∈ [t1, ∞)T. Multiplying (26) by fλ(s) and then integrating from t to

∞, we have on [t1, ∞)T, Z ∞ t fλ(s)q(s)∆s ≤ − Z ∞ t fλ(s)w∆(s)∆s − Z ∞ t p(s)fλ(s)(wσ(s))2∆s ≤ fλ_{(t)w(t) +}Z ∞ t (fλ(s))∆wσ(s)∆s − Z ∞ t p(s)fλ(s)(wσ(s))2∆s ≤ fλ_{(t)w(t) +}Z ∞ t λfλ−1(s)p(s)wσ(s)∆s − Z ∞ t p(s)fλ(s)(wσ(s))2∆s.

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Multiplying this inequality by f1−λ_{(t) and using (61), (62) and Lemma 4.3,} we have f1−λ_(t)Z ∞ t fλ(s)q(s)∆s ≤ f (t)w(t) + f1−λ_(t)Z ∞ t p(s)fλ−2(s)[f (s)wσ(s)(λ − f (s)wσ(s))]∆s < (M + ε) + (1 − ε)(m − ε)[λ − (1 − ε)(m − ε)]f1−λ(t) Z ∞ t p(s)fλ−2(s)∆s < (M + ε) + (1 − ε)(m − ε)[λ − (1 − ε)(m − ε)](1 + ε) 2−λ 1 − λ , for all t ∈ [t1, ∞)T. Taking lim sup

t→∞

and letting ε → 0, we have

g∗(λ) ≤ M + m(λ − m) 1 − λ = g∗(0) 1 − λ + 1 2 q 1 − 4g∗(0) + q 1 − 4g∗(2) , which contradicts (16). 2

Proof of Corollary 3.8. By taking λ = 0 in Theorem 3.7, the proof is done. 2

6 Some examples

Example 1 Consider the system u∆(t) = 1 t ln tv(t), v ∆_{(t) = −(t + σ(t))u}σ_(t), where T = aN0_{, N} 0 = {0} ∪ N, and a > 1. Let p(t) = 1 t ln t and q(t) = t + σ(t). Since Z ∞ a p(s)∆s = ∞ X i=1 a − 1 i ln a = a − 1 ln a ∞ X i=1 1 i = ∞ and Z ∞ a q(s)∆s = ∞ X i=1 a2i(a2− 1) = ∞, the system is oscillatory by Theorem 3.1.

Example 2 Consider the system u∆(t) = 1

t2v(t), v

∆_{(t) = −} 1

tσ(t)u

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where T = aN0_{, N} 0 = {0} ∪ N, and a > 1. Let p(t) = 1 t2 and q(t) = 1 tσ(t). Since Z ∞ 1 p(s)∆s = ∞ X i=0 a − 1 ai = a < ∞ and Z ∞ 1 q(s)∆s = ∞ X i=0 a − 1 ai+1 = 1 < ∞,

the system is nonoscillatory by Theorem 3.2. Example 3 Consider the system

u∆(t) = tv(t), v∆(t) = −1 t2u

σ_(t),

where T = aN = {an | n ∈ N} and a is a positive constant. Let p(t) = t and q(t) = 1 t2. Since Z ∞ a p(s)∆s = a2 ∞ X i=1 i = ∞, f (t) = Z t a p(s)∆s = a2 n X i=1 i = a 2_{n(n + 1)} 2 , ∀ t = an ∈ T. lim t→∞ µ(t)p(t) f (t) = limn→∞ 2 n + 1 = 0, Z ∞ a q(s)∆s = ∞ X i=1 a (ai)2 = 1 a ∞ X i=1 1 i2 < ∞, and Z ∞ a fλ(s)q(s)∆s = a 2λ−1 2λ ∞ X i=1 [i(i + 1)]λ i2 > a2λ−2 2λ ∞ X i=1 i2λ−2= ∞ if 1 2 ≤ λ < 1, the system is oscillatory by Theorem 3.3.

Example 4 Consider the system

u∆(t) = v(t), v∆(t) = −1 t2u

σ_(t),

where T = aN = {an | n ∈ N} and a is a positive constant. Let p(t) = 1 and q(t) = 1 t2. We compute Z ∞ a p(s)∆s = Z ∞ a 1∆s = ∞,

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f (t) = Z t a 1∆s = n X i=1 a = an, ∀ t = an ∈ T, and lim t→∞ µ(t)p(t) f (t) = limn→∞ 1 n = 0. From Example 3, we know that

Z ∞ a q(s)∆s < ∞. Since Z ∞ a fλ(s)q(s)∆s = aλ−1 ∞ X i=1 1 i2−λ < ∞ f or all λ ∈ [0, 1) and g∗(0) = lim inf n→∞ n ∞ X i=n 1 i2 > lim infn→∞ n Z ∞ n 1 x2dx = 1 > 1 4, the system is oscillatory by Theorem 3.4.

Example 5 Consider the system

u∆(t) = p(t)v(t), v∆(t) = −q(t)uσ(t), where T = 2N = {2n | n ∈ N}, p(t) = 1 +1 t and q(t) =      1 28 −k_{, t = 8}k 0, t 6= 8k, k = 1, 2, .... First, we compute Z ∞ 4 p(s)∆s = ∞ X i=2 2 1 + 1 2i = ∞, f (t) = Z t 4 p(s)∆s = n X i=2 2 1 + 1 2i = 2n − 2 + n X i=2 1 i, ∀t = 2n ∈ T. lim t→∞ µ(t)p(t) f (t) = limn→∞ 2 1 + 1 2n 2n − 2 + n X i=2 1 i = 0, and Z ∞ 4 q(s)∆s = ∞ X k=1 2q(8k) = ∞ X k=1 8−k = 1 7 < ∞.

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Since λ < 1, we have Z ∞ 4 fλ(s)q(s)∆s = ∞ X k=1 2fλ(8k)q(8k) = ∞ X k=1 " _k X l=1 2 1 + 1 8l #λ 8−k < ∞ X k=1 2k + 2 7 1 − 1 8k 8−k < ∞. For t = 8m+ 2, m = 1, 2, ..., g(t, 0) =  8 m + 8m_/2+1 X i=2 1 i   ∞ X k=m+1 8−k → 1 7, as m → ∞. For t = 8m_{, m = 1, 2, ...,} g(t, 0) =  8m− 2 + 8m_/2 X i=2 1 i   ∞ X k=m 8−k → 8 7, as m → ∞. For t = 8m_{− 2, m = 1, 2, ..., we have} g(t, 2) =  8m− 4 + 8m_/2−1 X i=2 1 i   −1 m−1 X k=1  8 k_{− 2 +} 8k_/2 X l=2 1 l   2 8−k → 1 7, as m → ∞. Therefore we get g∗(0) ≤ 1 7 ≤ 1 4, g∗(2) ≤ 1 7 ≤ 1 4 and g ∗_{(0) ≥} 8 7. Hence the system is oscillatory by Corollary 3.6 and 3.8.

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