Salhin et al. SpringerPlus 2014, 3:300 http://www.springerplus.com/content/3/1/300
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Oscillation theorems for second order nonlinear forced differential equations Ambarka A Salhin*, Ummul Khair Salma Din, Rokiah Rozita Ahmad and Mohd Salmi Md Noorani
Abstract In this paper, a class of second order forced nonlinear differential equation is considered and several new oscillation theorems are obtained. Our results generalize and improve those known ones in the literature. Keywords: Oscillation; Forced nonlinear differential equations of second order
Introduction We consider the oscillation behavior of solutions of second order forced nonlinear differential equation α−1 ′ r ðt Þψ ðxðt ÞÞf x′ ðt Þ f x′ ðt Þ þ qðt Þg ðxðt ÞÞð1:1Þ (1.1) ¼ H t; xðt Þ; x′ ðt Þ ; t∈½t 0 ; ∞Þ; and ′ r ðt Þψ ðxðt ÞÞf x′ ðt Þ þ qðt Þg ðxðt ÞÞ ¼ H t; xðt Þ; x′ ðt Þ ; t∈½t 0 ; ∞Þ;
ð1:2Þ
where r, q ∈ C([t0, ∞), ℝ), and f, ψ, g ∈ C(ℝ, ℝ) and H is a continuous function on [t0, ∞) × ℝ2, α is a positive real number. Throughout the paper, it is assumed that the following conditions are satisfied: (A1) r(t) > 0, t ≥ 0; (A2) xg(x) > 0, g ∈ C1(ℝ) for x ≠ 0; Þ (A3) H ðgt;x;y ðxÞ ≤ pðt Þ∀t∈½t 0 ; ∞Þ; x; y∈ℝ and x≠0: We restrict our attention only to the solutions of the differential equations (1.1) and (1.2) that exist on some ray [t0, ∞), where t0 ≥ t, to may depend on the particular solutions. Such a solution is said to be oscillatory if it has arbitrarily large zeros, and otherwise, it is said to be nonoscillatory. Equations (1.1) and (1.2) are called oscillatory if all its solutions are oscillatory. The problem of finding oscillation criteria for second order nonlinear ordinary differential equations, which
involve the average of integral of the alternating coefficient, has received the attention of many authors because in the fact there are many physical systems are modeled by second order nonlinear ordinary differential equations; for example, the so called Emden – Fowler equation arises in the study of gas dynamics and fluid mechanics. This equation appears also in the study of relativistic mechanics, nuclear physics and in the study of chemically reacting systems. The oscillatory theory as a part of the qualitative theory of differential equations has been developed rapidly in the last decades, and there has been a great deal of work on the oscillatory behavior of differential equations; see e.g. (Agarwal et al. 2010; Beqiri and Koci 2012; Bihari 1963; Elabbasy and Elsharabasy 1997; Elabbasy and Elhaddad 2007; Grace et al. 1984, 1988; Grace and Lalli 1987, 1989, 1990; Grace 1989, 1990, 1992; Greaf and Spikes 1986; Graef et al. 1978; Lee and Yeh 2007; Kamenev 1978; Kartsatos 1968; Li and Agarwal 2000; Meng 1996; Nagabuchi and Yamamoto 1988; Ohriska and Zulova 2004; Ouyang et al. 2009; Philos 1983, 1984, 1985; Remili 2010; Salhin 2014; Tiryaki and Basci 2008; Tiryaki 2009; Temtek and Tiryaki 2013; Yan 1986; Yibing et al. 2013a, b; Zhang and Wang 2010). Remili (2010), studied the equation ′ r ðt Þx′ ðt Þ þ Qðt; xÞ ¼ H t; x′ ðt Þ; x ðt Þ ;
ð1:3Þ
and derived some oscillation criteria for the equation (1.3), where new results with additional suitable weighted
* Correspondence: [email protected] School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia © 2014 Salhin et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
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function are investigated. Zhang and Wang (2010), studied the following equation
positive on [T, ∞), T ≥ t0. A similar argument holds for the case when x(t) is negative. Let
′ r ðt Þψ ðxðt ÞÞx′ ðt Þ þ Qðt; xÞ ¼ H t; x′ ðt Þ; xðt Þ :
α−1 ρðt Þr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ f ðx′ ðt ÞÞ wðt Þ ¼ ; t ≥ T 0: g ðxðt ÞÞ ð2:5Þ
ð1:4Þ Temtek and Tiryaki (2013) obtained several new oscillation results for the equation
α−1 r ðt Þψ ðxðt ÞÞx′ ðt Þ x′ ðt ÞÞ′ þ Qðt; xÞ ¼ H t; x′ ðt Þ; xðt Þ ;
Then differentiating (2.5), (1.1) and take in account assumptions (A1) - (A3), (2.2) we have ρ′ ðt Þ jwðt Þj ρðt Þ α−1 2 1 ρðt Þr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ f ðx′ ðt ÞÞg ′ ðxÞ : − g 2 ð xÞ k2
ð1:5Þ and its special cases by using generalized Riccati transformation and well known techniques. In this paper, we continue in this direction the study of oscillatory properties of equations (1.1) and (1.2). The purpose of this paper is to improve and extend the above mentioned results. Our results are more general than the previous results. The relevance of our results becomes clear due to some carefully selected examples.
Main results In this section we prove our main results. Theorem 2.1. Suppose that, conditions (A1) – (A3) hold, and g ′ ð xÞ 1 α−1 α ψ ð xÞ j g ð xÞ j
≥k > 0
f ðyÞ 0 < k1 ≤ ≤ k2 y
for all x ∈ ℝ
w′ ðt Þ ≤ −ρðt Þ ½qðt Þ−pðt Þ þ
ð2:6Þ In view of (2.1) we conclude that w′ ðt Þ ≤ −ρðt Þ ½qðt Þ−pðt Þ þ
lim
t→∞
T0
1=
α
DX−EX αþ1 ≤
ð2:1Þ
for all
′
y ¼ x ðt Þ ≠ 0:
ð2:2Þ
α ds
¼ ∞;
ð2:3Þ
Dαþ1 E −α ; D ≥ 0; E > 0; X ≥ 0:
(2.3)
ð2:8Þ
Integrating (2.8) from T to t, we get Zt
"
ρðsÞ qðsÞ−pðsÞ−λr ðsÞ
αþ1 # ds; t ≥ T ≥ T 0 :
ρ′ ðsÞ ρðsÞ
T
Z ðsÞds ¼ ∞;
ð2:9Þ
ð2:4Þ
T0
′ αþ1 where Z ðsÞ ¼ ρðsÞ ðqðsÞ−pðsÞÞ−λr ðsÞ ρρððssÞÞ and λ¼
ðα þ 1Þαþ1
αþ1
Zt t→∞
αα
r ðt Þðρ′ ðt ÞÞ w′ ðt Þ ≤ −ρðt Þ ½qðt Þ−pðt Þ þ λ ðρðt ÞÞα ′ αþ1 : ¼ −ρðt Þ qðt Þ−pðt Þ−λr ðt Þ ρρððttÞÞ
wðt Þ ≤ wðT Þ−
lim sup
(2.7)
Now, by applying this inequality we have
1 ðρðsÞr ðs ÞÞ
ð2:7Þ
By using the extremum of one variable function it can be proved that
Let ρ be a positive continuously differentiable function over [T, ∞) such that ρ′(t) ≥ 0 over [T0, ∞); Zt
αþ1
ρ′ ðt Þ k jwðt Þj α jwðt Þj− ρðt Þ k 2 ðρðt Þr ðt ÞÞ1 =α :
α ðkk 1 Þαþ1 ; αþ1
Taking the limit for both sides of (2.9) and using (2.4), we find w(t) → − ∞. Hence, there exists T1 ≥ T such that f(x′(t)) < 0 ⇒ x′(t) < 0, ∀t ≥ T1. Z∞
and there exists T2 ≥ T1such that Then all solutions of equation (1.1) are oscillatory. Proof. Let x(t) be a non-oscillatory solution on [T, ∞), T ≥ T0 of the equation (1.1). We assume that x(t) is
ρðsÞ½qðsÞ−pðsÞds ¼ ∞;
Condition (2.4) also implies that
ZT 2
Zt
ρðsÞ½qðsÞ−pðsÞ ds ¼ 0 and T1
T
ρðsÞ½qðsÞ−pðsÞ ds ≥ 0; ∀t ≥ T 2 : T2
Salhin et al. SpringerPlus 2014, 3:300 http://www.springerplus.com/content/3/1/300
Page 3 of 7
"
Multiplying Eq. (1.1) by ρ(t) and integrating by parts on [T2, t], we have α ′ ρðt Þ r ðt Þψ ðxðt ÞÞf x′ ðt Þ ≤ −ρðt Þg ðxðt ÞÞ½qðt Þ−pðt Þ:
¼
Now, integrating by parts, we get α −ρðt Þ r ðt Þψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ þ C T 2 ≤ Zt −
α ρ′ ðsÞr ðsÞψ ðxðsÞÞ −f x′ ðsÞ ds
Zt
sint xð t Þ þ tþ t
2x8 sint cosðx′ ðt Þ þ 1Þ π ; t≥ ; ðx7 þ 1Þt 3 2
Z∞
ρðsÞg ðxðsÞÞ½qðsÞ−pðsÞds;
ρðsÞ½qðsÞ−pðsÞ ds < ∞;
T2
ð2:10Þ
T0
where
2
α
CT 2
#′
Evidently, if we take pðt Þ ¼ t23 ; ρðt Þ ¼ t and α ¼ 2: Then all conditions of Theorem 2.1 are satisfied, hence, all the solutions are oscillatory. Theorem 2.3. If (A1) – (A3), conditions (2.1) – (2.3) hold, and
T2
−
1 x′ ðt Þ ð13x′ ðt Þ þ t ðx′ ðt ÞÞ2 þ 1
ρðT 2 Þr ðT 2 Þψ ðxðT 2 ÞÞð−f ðx′ ðT 2 ÞÞÞ ¼ > 0: g ðxðT 2 ÞÞ
limt→∞ inf 4
Zt
3 Z ðsÞds5 ≥ 0 for all large
T;
T
α ρðt Þ r ðt Þψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ ≥ C T 2 þ g ðxðt ÞÞ Zt −
x′ ðsÞg ′ ðxðsÞÞ
T2
Zt
þ
Zt
ð2:11Þ
ρðsÞ½qðsÞ − pðsÞds
Zt
T2
Zs
lim
ρðuÞ½qðuÞ−pðuÞduds
t→∞
0 @
T0
T2
ρ′ ðsÞr ðsÞψ ðxðsÞÞ −f x′ ðsÞ
α
ds ≤ C T 2 ;
1 ρðsÞr ðsÞ
Z∞
11 =α ds¼∞; Z ðuÞ d A
ð2:12Þ
s
and
T2
∀t ≥ T 1 :
Therefore, α ρðt Þ r ðt Þψ ðxðt ÞÞ −f x′ ðt Þ ≥ CT 2 From (2.1) and (2.2), we find CT 2 ; r ðt Þρðt Þ 1
1 CT 2 α; ′ ðψ ðxðt ÞÞÞα ð−f ðx ðt ÞÞÞ ≥ r ðsÞρðsÞ 1
1 Zt Zt −C T α ds; 2 k 2 ðψ ðxðsÞÞÞα x′ ðsÞds ≤ r ðsÞρðsÞ α
ψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ ≥
T2
ZxðtÞ xð T 2 Þ
T2
1 k 2 ðψ ðyÞÞα dy ≤
1 −C T 2 α ds: r ðsÞρðsÞ
Zt T2
From (2.3) and 0 < x(t) ≤ x(T2), this implies that ZxðtÞ 1 k 2 ðψ ðyÞÞα dy is lower bounded, but the right side xð T 2 Þ
of it tends to mines infinity. Then, this is a contradiction. Example 2.2. Consider the following differential equation
Z∞ ε
1 ψ ðyÞ α dy < ∞ g ðyÞ
for every
ε > 0:
ð2:13Þ
Thus all solutions of Eq. (1.1) are oscillatory. Proof. Let x(t) be a non-oscillatory solution on [T, ∞), T ≥ T0 of Eq. (1.1). Let us assume that x(t) is positive on [T, ∞) and consider the following three cases for the behavior of x′(t). Case 1: x′(t) > 0 for T1 ≥ T for some t ≥ T1; then from (2.10), we obtain Zt T1
α−1 r ðT 1 ÞρðT 1 Þψ ðxðT 1 ÞÞf ðx′ ðT 1 ÞÞ f ðx′ ðT 1 ÞÞ Z ðsÞds ≤ g ðxðT 1 ÞÞ α
−
ρðt Þr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ : g ðxðt ÞÞ
From (2.1) and (2.2), we obtain 1 r ðt Þρðt Þ
Zt
α
Z ðsÞds ≤ T1
Hence, for all t ≥ T1
ψ ðxðt ÞÞf ðx′ ðt ÞÞ g ðxðt ÞÞ
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0 B @ Zt T1
1 r ðt Þρðt Þ 0
Z∞
Z∞
≤ k1 xðT 1 Þ
" !#′
5 ′ x4 ð t Þ ð x ð t Þ Þ 1 7x′ ðt Þ þ þ 3 x3 ðt Þ t 4 4 ′ x ðt Þ þ 1 t ðx ðt ÞÞ þ 1 x3 cosx sin2x′ ðt Þ ¼ ; t > 1; t4
11=α C Z ðsÞdsA
T1
@ 1 r ðsÞρðsÞ
Page 4 of 7
Z∞
≤
ψ ðxðt ÞÞ
s
ψ ðy Þ g ðy Þ
f ðx ðt ÞÞ
Zt ds ≤ k 1
ψ ðxðsÞÞ1=α x′ ðsÞ g ðxðsÞÞ1=α
T1
1=α
′
g ðxðt ÞÞ1=α
11=α Z ðuÞduA
1=α
ds;
dy:
Evidently, if we take pðt Þ ¼ t14 ; ρðt Þ ¼ t and α ¼ 1: Then the equation given in Example 2.2 is oscillatory by Theorem 2.2. Z∞ Remark 2.1. Condition (2.10) implies that Z ðsÞ ≥ 0
Using (2.13), we obtain Zt T1
0 @ 1 r ðsÞρðsÞ
Z∞
11 =α ds < ∞;
Z ðsÞds ¼
and lim inf t→∞
Z ðuÞduA
T
Z∞
s
T
Z∞
Z∞
Z ðsÞds; hence (2.11) takes T
Z ðsÞ ≥ 0; for all large T.
the form of T
which contradicts to the condition (2.13). Case 2: If x′(t) is oscillatory, then there exists a sequence {αn} → ∞ on [T, ∞) such that x′(αn) < 0. Let us assume that N is sufficiently large so that Z∞ Z ðsÞds ≥ 0:
Remark 2.2. when α = 1, ψ(x(t)) = 1 and f(x′(t)) = x′(t), Theorem 2.1 and 2.2 reduce to Theorem 1 and 2 Remili (2010) and Theorems 2.1 and 2.3 are obtained by analogy with Theorems 2.1 and 2.2 from (Temtek and Tiryaki 2013). Theorem 2.5. Assume that f ðyÞ ≥ by for all y ∈ ℝ and for some constant b > 0;
αN
ð2:14Þ
Then, from (2.1), (2.2) and (2.7), we have Zt −C αN − αN
Zt
C αN þ αN
Zε
ρðt Þr ðt Þψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ Z ðsÞds ≥ − g ðxðt ÞÞ ρðt Þr ðt Þψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ Z ðsÞds ≤ g ð xð t Þ Þ
α
α
0< 0
ψ ðu Þ du < ∞ for all ε > 0: g ðu Þ
Furthermore, assume that there exist a constant A such that lim supRðt Þ ¼ A < ∞; Zt ds where Rðt Þ ¼ ; and r ðsÞ t→∞
Thus α
≤ lim inf t→∞
ρðt Þr ðt Þψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ ≥ C αN g ðxðt ÞÞ Zt
þ lim inf
Zt lim sup
t→∞
t0
Z ðsÞds > 0;
t→∞
αN
which contradicts to the assume that x′(t) oscillates. Case 3: Let x′(t) < 0 for t ≥ T1. Condition (2.11) implies that for any t0 ≥ T1 such that Z∞ ρðsÞ½qðsÞ−pðsÞds ≥ 0 for all t ≥ T1. t
The remaining part of the proof is similar to that of Theorem 2.1 then will be omitted. Example 2.4. Let us consider the following equation
ð2:15Þ
ð2:16Þ
t0
Zs 1 ½qðuÞ−pðuÞduds ¼ ∞: r ðsÞ
ð2:17Þ
t0
Then the differentia Eq. (1.2) is oscillatory. Proof. Without loss of generality, let assume that there exists a solution x(t) of (1.2) such that x(t) > 0 on [T, ∞) for some T ≥ t0. A similar argument holds also for the case when x(t) < 0. Let w(t) be defined by the Riccati Transformation wðt Þ ¼
r ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ ; t ≥ T: g ðxðt ÞÞ
Derivation this equality we have
Salhin et al. SpringerPlus 2014, 3:300 http://www.springerplus.com/content/3/1/300
ðr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞÞ w ðt Þ ¼ g ðxðt ÞÞ ′
−
Page 5 of 7
function ρ : [t0, ∞) → (0, ∞) such that ρ′(t) ≥ 0 for all t ≥ t0, and 0 s 1 Zt Z 1 @ ρðuÞ½qðuÞ − pðuÞduAds ¼ ∞: lim sup t→∞ ρðsÞr ðsÞ
′
r ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞg ′ ðxðt ÞÞx′ ðt Þ : g 2 ðxðt ÞÞ
t0
t0
ð2:18Þ
This, and (1.2) imply w′ ðt Þ ≤ pðt Þ − qðt Þ
t ≥ T:
Integrating this inequality from T to t( ≥ T), we obtain Zt wðt Þ ≤ wðT Þ−
½qðsÞ − pðsÞds T
Then the differential equation (1.2) is oscillatory. Proof. Without loss of generality, let assume that there exists a solution x(t) of (1.2) such that x(t) > 0 on [T, ∞) for some T ≥ t0. Let w(t) be defined by the Riccati Transformation wðt Þ ¼ ρðt Þ
By condition (2.14), we get
r ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ ; t ≥ T: g ðxðt ÞÞ
Derivation this equality we have ′
b
′
r ðt Þψ ðxðt ÞÞx ðt Þ r ðt Þψ ðxðt ÞÞf ðx ðt ÞÞ ≤ g ð xð t Þ Þ g ðxðt ÞÞ Zt ≤ wðT Þ− ½qðsÞ − pðsÞds; b > 0
w ′ ðt Þ ¼
′
ρðt Þðr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞÞ ρ′ ðt Þr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ þ g ðxðt ÞÞ g ðxðt ÞÞ −
T
ρðt Þr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞg ′ ðxðt ÞÞx′ ðt Þ : g 2 ðxðt ÞÞ
This, and (1.2) imply Integrating the above inequality multiplied by from T to t( ≥ T), we have Zt b T
ψ ðxðsÞÞx′ ðsÞ ds ≤ g ðxðsÞÞ
Zt T
1 r ðt Þ
ψ ðxðsÞÞf ðx′ ðsÞÞ ds g ðxðsÞÞ Zt
≤ wðT ÞRðt Þ− T
1 r ðsÞ
Zs ½qðuÞ−pðuÞduds: T
From condition (2.16) and (2.17), we get that Zt
w′ ðt Þ ≤ − ρðt Þ½qðt Þ−pðt Þ þ
Hence for all t ≥ T, we obtain Zt
Zt ρðsÞ½qðsÞ − pðsÞ ds ≤ − T
Zt −
Now, if x(t) ≥ x(T) for large t then θ(t) ≥ 0, which is a contradiction. Hence for large t, x(t) ≤ x(T), so
Zt
θ ðt Þ ¼ − xð t Þ
ψ ðuÞ du > − g ðu Þ
ZxðT Þ 0
ψ ðu Þ du > −∞; g ðuÞ
d ρðsÞ ds
wðsÞ ds: ρðsÞ
T0
−
d ρðsÞ ds
wðt Þ wðT 0 Þ þ ρðt Þ ρðt Þ ρðT 0 Þ
wðsÞ wðT 0 Þ ds ¼ −wðt Þ þ Bρðt Þ ; B ¼ : ρðsÞ ρðT 0 Þ ð2:20Þ
By (2.19) and (2.20) we get Zt ρðsÞ½qðsÞ−pðsÞ ds ¼ −wðt Þ þ Bρðt Þ:
which is again a contradiction. This completes proof the Theorem 2.3. Theorems 2.6. Suppose that conditions (2.14), (2.15) and (2.16) hold. Furthermore, suppose that, there exist a
ð2:19Þ
Zt wðsÞ d wðsÞ ds ¼ −ρðt Þ ds ρðsÞ ds ρðsÞ ¼ −ρðt Þ
T
ZxðT Þ
d ds
By the Bonnet’s Theorem that for each t ≥ T, there exist a T0 ∈ [T, t] such that
T
T
ρðsÞ T
ψ ðxðsÞÞx′ ðsÞ ds→−∞ as t→∞: g ðxðsÞÞ
θ ðt Þ ¼
ρ′ ðt Þ wðt Þ: ρðt Þ
ð2:21Þ
T
Integrating the above inequality multiplied by from T to t( ≥ T), we obtain
1 r ðt Þρðt Þ
Salhin et al. SpringerPlus 2014, 3:300 http://www.springerplus.com/content/3/1/300
Zt b T
ψ ðxðsÞÞx′ ðsÞ ds ≤ g ðxðsÞÞ
Page 6 of 7
Zt
ψ ðxðsÞÞf ðx′ ðsÞÞ ds g ðxðsÞÞ T 0 1 Zs Zt 1 ρðuÞ½qðuÞ−pðuÞduAds: ≤ BRðt Þ − @ r ðsÞρðsÞ T
T
Zt t0
0
Zs
1 ρðuÞ½qðuÞ − pðuÞduAds r ðsÞρðsÞ 0 ts0 1 Zt Z 2u 1 e þ sinu − sinu duAds ¼ ∞; ¼ @ s e @
T
From (2.16) and (2.18), we have Zt θ ðt Þ ¼ T
Competing interests The authors declare that they have no competing interests.
Now, if x(t) ≥ x(T) for large t, then θ(t) ≥ 0, which is a contradiction. Hence for large t, x(t) ≤ x(T), so
θ ðt Þ ¼ − xð t Þ
ψ ðuÞ du > − g ðu Þ
ZxðT Þ 0
x4 ð t Þ x′ ð t Þ 4 x ðt Þ þ 1
et ¼
x7 ðt Þ sint
ψ ðu Þ du > −∞; g ðuÞ
Received: 4 April 2014 Accepted: 16 May 2014 Published: 18 June 2014
′ þ ðe2t þ sint Þx3 ðt Þ 2
ð x′ ð t Þ Þ
ð1 þ x4 ðt ÞÞ2 ðx′ ðt ÞÞ2 þ 1
;
t ≥ 0:
Here, r ðt Þ ¼ et ; qðt Þ ¼ e2t þ sint; ψ ðxðt ÞÞ ¼
x4 ð t Þ ; þ1
x4 ð t Þ
2 x7 ðt Þ sint ð x′ ð t Þ Þ ; g ðxÞ ¼ x3 ; H t; xðt Þ; x′ ðt Þ ¼ ð1 þ x4 ðt ÞÞ2 ðx′ ðt ÞÞ2 þ 1 2
H ðt; xðt Þ; x′ ðt ÞÞ x7 ðt Þ sint ð x′ ð t Þ Þ ¼ g ðxÞ ð1 þ x4 ðt ÞÞ2 ðx′ ðt ÞÞ2 þ 1
1 ≤ sint ¼ pðt Þ: x3
So, can note that Zt limt→∞ supRðt Þ ¼ limt→∞ sup Zε 0
ds < ∞; es
t0
u ðu2 Þ2
þ1
du ¼
1 tan−1 ε2 < ∞: 2
Let us take ρ(t) = 1 we have
Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgement This research has been completed with the support of these grants: DIP2012-31, FRGS/2/2013/SG04/UKM/02/3 and FRGS/1/2012/SG04/UKM/01/1.
which is again a contradiction. This completes proof the Theorem 2.6. Example 2.7. Consider the differential equation
T
then, Theorem 2.4 ensures that every solution of the equation given oscillates.
ψ ðxðsÞÞx′ ðsÞ ds→−∞ as t→∞: g ðxðsÞÞ
ZxðT Þ
1
References Agarwal RP, Avramescu C, Mustafa OG (2010) On the oscillation theory of a secondorder strictly sublinear differential equation. Can Math Bull 53(2):193–203 Beqiri XH, Koci E (2012) Oscillation criteria for second order nonlinear differential equations. British Journal of Science 6(2):73–80 Bihari I (1963) An oscillation theorem concerning the half linear differential equation of the Second order. Magyar Tud Akad Mat Kutato Int Kozl 8:275–280 Elabbasy EM, Elhaddad WW (2007) Oscillation of second order nonlinear differential equations with damping term. Electron J Qual Theor Differ Equat 25:1–19 Elabbasy EM, Elsharabasy MA (1997) Oscillation properties for second order nonlinear differential equations. Kyungpook Math J 37:211–220 Grace SR (1989) Oscillation theorems for second order nonlinear differential equations with damping. Math Nachr 141:117–127 Grace SR (1990) Oscillation criteria for second order differential equations with damping. J Austral Math Soc (Series A) 49:43–54 Grace SR (1992) Oscillation theorems for nonlinear differential equations of second order. Math Anal And Appl 171:220–241 Grace SR, Lalli BS (1987) On the second order nonlinear oscillations. Bull Inst Math Acad Sinica 15(no. 3):297–309 Grace SR, Lalli BS (1989) Oscillation theorems for second order nonlinear differential equations with a damping term, Comment. Math Univ Carolinae 30(4):691–697 Grace SR, Lalli BS (1990) Integral averaging technique for the oscillation of second order nonlinear differential equations. J Math Anal Appl 149:277–311 Grace SR, Lalli BS, Yeh CC (1984) Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term. SIAM J Math Anal 15:1082–1093 Grace SR, Lalli BS, Yeh CC (1988) Addendum: Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term. SIAM J Math Anal 19(5):1252–1253 Graef JR, Rankin SM, Spikes PW (1978) Oscillation theorems for perturbed non-linear differential equation. J Math Anal Appl 65:375–390 Greaf JR, Spikes PW (1986) On the oscillatory behaviour of solutions of second order nonlinear differential equation. Czech Math J 36:275–284 Kamenev IV (1978) Integral criterion for oscillation of linear differential equations of second order. Math Zametki 23:249–251 Kartsatos AG (1968) On oscillation of nonlinear equations of second order. J Math Anal Appl 24:665–668 Lee CF, Yeh CC (2007) An Oscillation theorems. Appl Math Lett 20:238–240 Li WT, Agarwal RP (2000) Interval oscillation criteria for second order nonlinear equations with damping. Computers Math Applic 40:217–230 Meng FW (1996) An oscillation theorem for second order superlinear differential equations. Ind J Pure Appl Math 27:651–658
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Nagabuchi Y, Yamamoto M (1988) Some oscillation criteria for second order nonlinear ordinary differential equations with damping. Proc Japan Acad 64:282–285 Ohriska J, Zulova A (2004) Oscillation criteria for second order nonlinear differential equation. IM Preprint Series A 10:1–11 Ouyang Z, Zhong J, Zou S (2009) Oscillation criteria for a class of second-order nonlinear differential equations with damping term. Abstr Appl Anal 2009:1–12 Philos CHG (1983) Oscillation of sublinear differential equations of second order. Nonlinear Anal 7(10):1071–1080 Philos CHG (1984) On second order sublinear oscillation. Aequations Math 27:242–254 Philos CHG (1985) Integral averages and second order superlinear oscillation. Math Nachr 120:127–138 Remili M (2010) Oscillation criteria for second order nonlinear perturbed differential equations. Electron J Qual Theor Differ Equat 25:1–11 Salhin AA (2014) Oscillation criteria of second order nonlinear differential equations with variable Coefficients. Discret Dyn Nat Soc 2014:1–9 Temtek P, Tiryaki A (2013) Oscillation criteria for a certain second-order nonlinear perturbed differential equations. Journal of Inequalities and Applications 524:1–12 Tiryaki A (2009) Oscillation criteria for a certain second-order nonlinear differential equations with deviating arguments. Electron J Qual Theor Differ Equat 61:1–11 Tiryaki A, Basci Y (2008) Oscillation theorems for certain even-order nonlinear damped differential equations. Rocky Mt J Math 38(3):1011–1035 Yan J (1986) Oscillation theorems for second order linear differential equations with damping. Proc Amer Math Soc 98(2):276–282 Yibing S, Zhenlai H, Shurong S, Chao Z (2013a) Interval Oscillation Criteria for Second-Order Nonlinear Forced Dynamic Equations with Damping on Time Scales. Abstr Appl Anal 2013:1–11 Yibing S, Zhenlai H, Shurong S, Chao Z (2013b) Fite-Wintner-Leighton-Type Oscillation Criteria for Second-Order Differential Equations with Nonlinear Damping. Abstr Appl Anal 1–10:2012 Zhang Q, Wang L (2010) Oscillatory behavior of solutions for a class of second-order nonlinear differential equation with perturbation. Acta Appl Math 110:885–893 doi:10.1186/2193-1801-3-300 Cite this article as: Salhin et al.: Oscillation theorems for second order nonlinear forced differential equations. SpringerPlus 2014 3:300.
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Oscillation theorems for second order nonlinear forced differential equations Ambarka A Salhin*, Ummul Khair Salma Din, Rokiah Rozita Ahmad and Mohd Salmi Md Noorani
Abstract In this paper, a class of second order forced nonlinear differential equation is considered and several new oscillation theorems are obtained. Our results generalize and improve those known ones in the literature. Keywords: Oscillation; Forced nonlinear differential equations of second order
Introduction We consider the oscillation behavior of solutions of second order forced nonlinear differential equation α−1 ′ r ðt Þψ ðxðt ÞÞf x′ ðt Þ f x′ ðt Þ þ qðt Þg ðxðt ÞÞð1:1Þ (1.1) ¼ H t; xðt Þ; x′ ðt Þ ; t∈½t 0 ; ∞Þ; and ′ r ðt Þψ ðxðt ÞÞf x′ ðt Þ þ qðt Þg ðxðt ÞÞ ¼ H t; xðt Þ; x′ ðt Þ ; t∈½t 0 ; ∞Þ;
ð1:2Þ
where r, q ∈ C([t0, ∞), ℝ), and f, ψ, g ∈ C(ℝ, ℝ) and H is a continuous function on [t0, ∞) × ℝ2, α is a positive real number. Throughout the paper, it is assumed that the following conditions are satisfied: (A1) r(t) > 0, t ≥ 0; (A2) xg(x) > 0, g ∈ C1(ℝ) for x ≠ 0; Þ (A3) H ðgt;x;y ðxÞ ≤ pðt Þ∀t∈½t 0 ; ∞Þ; x; y∈ℝ and x≠0: We restrict our attention only to the solutions of the differential equations (1.1) and (1.2) that exist on some ray [t0, ∞), where t0 ≥ t, to may depend on the particular solutions. Such a solution is said to be oscillatory if it has arbitrarily large zeros, and otherwise, it is said to be nonoscillatory. Equations (1.1) and (1.2) are called oscillatory if all its solutions are oscillatory. The problem of finding oscillation criteria for second order nonlinear ordinary differential equations, which
involve the average of integral of the alternating coefficient, has received the attention of many authors because in the fact there are many physical systems are modeled by second order nonlinear ordinary differential equations; for example, the so called Emden – Fowler equation arises in the study of gas dynamics and fluid mechanics. This equation appears also in the study of relativistic mechanics, nuclear physics and in the study of chemically reacting systems. The oscillatory theory as a part of the qualitative theory of differential equations has been developed rapidly in the last decades, and there has been a great deal of work on the oscillatory behavior of differential equations; see e.g. (Agarwal et al. 2010; Beqiri and Koci 2012; Bihari 1963; Elabbasy and Elsharabasy 1997; Elabbasy and Elhaddad 2007; Grace et al. 1984, 1988; Grace and Lalli 1987, 1989, 1990; Grace 1989, 1990, 1992; Greaf and Spikes 1986; Graef et al. 1978; Lee and Yeh 2007; Kamenev 1978; Kartsatos 1968; Li and Agarwal 2000; Meng 1996; Nagabuchi and Yamamoto 1988; Ohriska and Zulova 2004; Ouyang et al. 2009; Philos 1983, 1984, 1985; Remili 2010; Salhin 2014; Tiryaki and Basci 2008; Tiryaki 2009; Temtek and Tiryaki 2013; Yan 1986; Yibing et al. 2013a, b; Zhang and Wang 2010). Remili (2010), studied the equation ′ r ðt Þx′ ðt Þ þ Qðt; xÞ ¼ H t; x′ ðt Þ; x ðt Þ ;
ð1:3Þ
and derived some oscillation criteria for the equation (1.3), where new results with additional suitable weighted
* Correspondence: [email protected] School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia © 2014 Salhin et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Salhin et al. SpringerPlus 2014, 3:300 http://www.springerplus.com/content/3/1/300
Page 2 of 7
function are investigated. Zhang and Wang (2010), studied the following equation
positive on [T, ∞), T ≥ t0. A similar argument holds for the case when x(t) is negative. Let
′ r ðt Þψ ðxðt ÞÞx′ ðt Þ þ Qðt; xÞ ¼ H t; x′ ðt Þ; xðt Þ :
α−1 ρðt Þr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ f ðx′ ðt ÞÞ wðt Þ ¼ ; t ≥ T 0: g ðxðt ÞÞ ð2:5Þ
ð1:4Þ Temtek and Tiryaki (2013) obtained several new oscillation results for the equation
α−1 r ðt Þψ ðxðt ÞÞx′ ðt Þ x′ ðt ÞÞ′ þ Qðt; xÞ ¼ H t; x′ ðt Þ; xðt Þ ;
Then differentiating (2.5), (1.1) and take in account assumptions (A1) - (A3), (2.2) we have ρ′ ðt Þ jwðt Þj ρðt Þ α−1 2 1 ρðt Þr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ f ðx′ ðt ÞÞg ′ ðxÞ : − g 2 ð xÞ k2
ð1:5Þ and its special cases by using generalized Riccati transformation and well known techniques. In this paper, we continue in this direction the study of oscillatory properties of equations (1.1) and (1.2). The purpose of this paper is to improve and extend the above mentioned results. Our results are more general than the previous results. The relevance of our results becomes clear due to some carefully selected examples.
Main results In this section we prove our main results. Theorem 2.1. Suppose that, conditions (A1) – (A3) hold, and g ′ ð xÞ 1 α−1 α ψ ð xÞ j g ð xÞ j
≥k > 0
f ðyÞ 0 < k1 ≤ ≤ k2 y
for all x ∈ ℝ
w′ ðt Þ ≤ −ρðt Þ ½qðt Þ−pðt Þ þ
ð2:6Þ In view of (2.1) we conclude that w′ ðt Þ ≤ −ρðt Þ ½qðt Þ−pðt Þ þ
lim
t→∞
T0
1=
α
DX−EX αþ1 ≤
ð2:1Þ
for all
′
y ¼ x ðt Þ ≠ 0:
ð2:2Þ
α ds
¼ ∞;
ð2:3Þ
Dαþ1 E −α ; D ≥ 0; E > 0; X ≥ 0:
(2.3)
ð2:8Þ
Integrating (2.8) from T to t, we get Zt
"
ρðsÞ qðsÞ−pðsÞ−λr ðsÞ
αþ1 # ds; t ≥ T ≥ T 0 :
ρ′ ðsÞ ρðsÞ
T
Z ðsÞds ¼ ∞;
ð2:9Þ
ð2:4Þ
T0
′ αþ1 where Z ðsÞ ¼ ρðsÞ ðqðsÞ−pðsÞÞ−λr ðsÞ ρρððssÞÞ and λ¼
ðα þ 1Þαþ1
αþ1
Zt t→∞
αα
r ðt Þðρ′ ðt ÞÞ w′ ðt Þ ≤ −ρðt Þ ½qðt Þ−pðt Þ þ λ ðρðt ÞÞα ′ αþ1 : ¼ −ρðt Þ qðt Þ−pðt Þ−λr ðt Þ ρρððttÞÞ
wðt Þ ≤ wðT Þ−
lim sup
(2.7)
Now, by applying this inequality we have
1 ðρðsÞr ðs ÞÞ
ð2:7Þ
By using the extremum of one variable function it can be proved that
Let ρ be a positive continuously differentiable function over [T, ∞) such that ρ′(t) ≥ 0 over [T0, ∞); Zt
αþ1
ρ′ ðt Þ k jwðt Þj α jwðt Þj− ρðt Þ k 2 ðρðt Þr ðt ÞÞ1 =α :
α ðkk 1 Þαþ1 ; αþ1
Taking the limit for both sides of (2.9) and using (2.4), we find w(t) → − ∞. Hence, there exists T1 ≥ T such that f(x′(t)) < 0 ⇒ x′(t) < 0, ∀t ≥ T1. Z∞
and there exists T2 ≥ T1such that Then all solutions of equation (1.1) are oscillatory. Proof. Let x(t) be a non-oscillatory solution on [T, ∞), T ≥ T0 of the equation (1.1). We assume that x(t) is
ρðsÞ½qðsÞ−pðsÞds ¼ ∞;
Condition (2.4) also implies that
ZT 2
Zt
ρðsÞ½qðsÞ−pðsÞ ds ¼ 0 and T1
T
ρðsÞ½qðsÞ−pðsÞ ds ≥ 0; ∀t ≥ T 2 : T2
Salhin et al. SpringerPlus 2014, 3:300 http://www.springerplus.com/content/3/1/300
Page 3 of 7
"
Multiplying Eq. (1.1) by ρ(t) and integrating by parts on [T2, t], we have α ′ ρðt Þ r ðt Þψ ðxðt ÞÞf x′ ðt Þ ≤ −ρðt Þg ðxðt ÞÞ½qðt Þ−pðt Þ:
¼
Now, integrating by parts, we get α −ρðt Þ r ðt Þψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ þ C T 2 ≤ Zt −
α ρ′ ðsÞr ðsÞψ ðxðsÞÞ −f x′ ðsÞ ds
Zt
sint xð t Þ þ tþ t
2x8 sint cosðx′ ðt Þ þ 1Þ π ; t≥ ; ðx7 þ 1Þt 3 2
Z∞
ρðsÞg ðxðsÞÞ½qðsÞ−pðsÞds;
ρðsÞ½qðsÞ−pðsÞ ds < ∞;
T2
ð2:10Þ
T0
where
2
α
CT 2
#′
Evidently, if we take pðt Þ ¼ t23 ; ρðt Þ ¼ t and α ¼ 2: Then all conditions of Theorem 2.1 are satisfied, hence, all the solutions are oscillatory. Theorem 2.3. If (A1) – (A3), conditions (2.1) – (2.3) hold, and
T2
−
1 x′ ðt Þ ð13x′ ðt Þ þ t ðx′ ðt ÞÞ2 þ 1
ρðT 2 Þr ðT 2 Þψ ðxðT 2 ÞÞð−f ðx′ ðT 2 ÞÞÞ ¼ > 0: g ðxðT 2 ÞÞ
limt→∞ inf 4
Zt
3 Z ðsÞds5 ≥ 0 for all large
T;
T
α ρðt Þ r ðt Þψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ ≥ C T 2 þ g ðxðt ÞÞ Zt −
x′ ðsÞg ′ ðxðsÞÞ
T2
Zt
þ
Zt
ð2:11Þ
ρðsÞ½qðsÞ − pðsÞds
Zt
T2
Zs
lim
ρðuÞ½qðuÞ−pðuÞduds
t→∞
0 @
T0
T2
ρ′ ðsÞr ðsÞψ ðxðsÞÞ −f x′ ðsÞ
α
ds ≤ C T 2 ;
1 ρðsÞr ðsÞ
Z∞
11 =α ds¼∞; Z ðuÞ d A
ð2:12Þ
s
and
T2
∀t ≥ T 1 :
Therefore, α ρðt Þ r ðt Þψ ðxðt ÞÞ −f x′ ðt Þ ≥ CT 2 From (2.1) and (2.2), we find CT 2 ; r ðt Þρðt Þ 1
1 CT 2 α; ′ ðψ ðxðt ÞÞÞα ð−f ðx ðt ÞÞÞ ≥ r ðsÞρðsÞ 1
1 Zt Zt −C T α ds; 2 k 2 ðψ ðxðsÞÞÞα x′ ðsÞds ≤ r ðsÞρðsÞ α
ψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ ≥
T2
ZxðtÞ xð T 2 Þ
T2
1 k 2 ðψ ðyÞÞα dy ≤
1 −C T 2 α ds: r ðsÞρðsÞ
Zt T2
From (2.3) and 0 < x(t) ≤ x(T2), this implies that ZxðtÞ 1 k 2 ðψ ðyÞÞα dy is lower bounded, but the right side xð T 2 Þ
of it tends to mines infinity. Then, this is a contradiction. Example 2.2. Consider the following differential equation
Z∞ ε
1 ψ ðyÞ α dy < ∞ g ðyÞ
for every
ε > 0:
ð2:13Þ
Thus all solutions of Eq. (1.1) are oscillatory. Proof. Let x(t) be a non-oscillatory solution on [T, ∞), T ≥ T0 of Eq. (1.1). Let us assume that x(t) is positive on [T, ∞) and consider the following three cases for the behavior of x′(t). Case 1: x′(t) > 0 for T1 ≥ T for some t ≥ T1; then from (2.10), we obtain Zt T1
α−1 r ðT 1 ÞρðT 1 Þψ ðxðT 1 ÞÞf ðx′ ðT 1 ÞÞ f ðx′ ðT 1 ÞÞ Z ðsÞds ≤ g ðxðT 1 ÞÞ α
−
ρðt Þr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ : g ðxðt ÞÞ
From (2.1) and (2.2), we obtain 1 r ðt Þρðt Þ
Zt
α
Z ðsÞds ≤ T1
Hence, for all t ≥ T1
ψ ðxðt ÞÞf ðx′ ðt ÞÞ g ðxðt ÞÞ
Salhin et al. SpringerPlus 2014, 3:300 http://www.springerplus.com/content/3/1/300
0 B @ Zt T1
1 r ðt Þρðt Þ 0
Z∞
Z∞
≤ k1 xðT 1 Þ
" !#′
5 ′ x4 ð t Þ ð x ð t Þ Þ 1 7x′ ðt Þ þ þ 3 x3 ðt Þ t 4 4 ′ x ðt Þ þ 1 t ðx ðt ÞÞ þ 1 x3 cosx sin2x′ ðt Þ ¼ ; t > 1; t4
11=α C Z ðsÞdsA
T1
@ 1 r ðsÞρðsÞ
Page 4 of 7
Z∞
≤
ψ ðxðt ÞÞ
s
ψ ðy Þ g ðy Þ
f ðx ðt ÞÞ
Zt ds ≤ k 1
ψ ðxðsÞÞ1=α x′ ðsÞ g ðxðsÞÞ1=α
T1
1=α
′
g ðxðt ÞÞ1=α
11=α Z ðuÞduA
1=α
ds;
dy:
Evidently, if we take pðt Þ ¼ t14 ; ρðt Þ ¼ t and α ¼ 1: Then the equation given in Example 2.2 is oscillatory by Theorem 2.2. Z∞ Remark 2.1. Condition (2.10) implies that Z ðsÞ ≥ 0
Using (2.13), we obtain Zt T1
0 @ 1 r ðsÞρðsÞ
Z∞
11 =α ds < ∞;
Z ðsÞds ¼
and lim inf t→∞
Z ðuÞduA
T
Z∞
s
T
Z∞
Z∞
Z ðsÞds; hence (2.11) takes T
Z ðsÞ ≥ 0; for all large T.
the form of T
which contradicts to the condition (2.13). Case 2: If x′(t) is oscillatory, then there exists a sequence {αn} → ∞ on [T, ∞) such that x′(αn) < 0. Let us assume that N is sufficiently large so that Z∞ Z ðsÞds ≥ 0:
Remark 2.2. when α = 1, ψ(x(t)) = 1 and f(x′(t)) = x′(t), Theorem 2.1 and 2.2 reduce to Theorem 1 and 2 Remili (2010) and Theorems 2.1 and 2.3 are obtained by analogy with Theorems 2.1 and 2.2 from (Temtek and Tiryaki 2013). Theorem 2.5. Assume that f ðyÞ ≥ by for all y ∈ ℝ and for some constant b > 0;
αN
ð2:14Þ
Then, from (2.1), (2.2) and (2.7), we have Zt −C αN − αN
Zt
C αN þ αN
Zε
ρðt Þr ðt Þψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ Z ðsÞds ≥ − g ðxðt ÞÞ ρðt Þr ðt Þψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ Z ðsÞds ≤ g ð xð t Þ Þ
α
α
0< 0
ψ ðu Þ du < ∞ for all ε > 0: g ðu Þ
Furthermore, assume that there exist a constant A such that lim supRðt Þ ¼ A < ∞; Zt ds where Rðt Þ ¼ ; and r ðsÞ t→∞
Thus α
≤ lim inf t→∞
ρðt Þr ðt Þψ ðxðt ÞÞð−f ðx′ ðt ÞÞÞ ≥ C αN g ðxðt ÞÞ Zt
þ lim inf
Zt lim sup
t→∞
t0
Z ðsÞds > 0;
t→∞
αN
which contradicts to the assume that x′(t) oscillates. Case 3: Let x′(t) < 0 for t ≥ T1. Condition (2.11) implies that for any t0 ≥ T1 such that Z∞ ρðsÞ½qðsÞ−pðsÞds ≥ 0 for all t ≥ T1. t
The remaining part of the proof is similar to that of Theorem 2.1 then will be omitted. Example 2.4. Let us consider the following equation
ð2:15Þ
ð2:16Þ
t0
Zs 1 ½qðuÞ−pðuÞduds ¼ ∞: r ðsÞ
ð2:17Þ
t0
Then the differentia Eq. (1.2) is oscillatory. Proof. Without loss of generality, let assume that there exists a solution x(t) of (1.2) such that x(t) > 0 on [T, ∞) for some T ≥ t0. A similar argument holds also for the case when x(t) < 0. Let w(t) be defined by the Riccati Transformation wðt Þ ¼
r ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ ; t ≥ T: g ðxðt ÞÞ
Derivation this equality we have
Salhin et al. SpringerPlus 2014, 3:300 http://www.springerplus.com/content/3/1/300
ðr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞÞ w ðt Þ ¼ g ðxðt ÞÞ ′
−
Page 5 of 7
function ρ : [t0, ∞) → (0, ∞) such that ρ′(t) ≥ 0 for all t ≥ t0, and 0 s 1 Zt Z 1 @ ρðuÞ½qðuÞ − pðuÞduAds ¼ ∞: lim sup t→∞ ρðsÞr ðsÞ
′
r ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞg ′ ðxðt ÞÞx′ ðt Þ : g 2 ðxðt ÞÞ
t0
t0
ð2:18Þ
This, and (1.2) imply w′ ðt Þ ≤ pðt Þ − qðt Þ
t ≥ T:
Integrating this inequality from T to t( ≥ T), we obtain Zt wðt Þ ≤ wðT Þ−
½qðsÞ − pðsÞds T
Then the differential equation (1.2) is oscillatory. Proof. Without loss of generality, let assume that there exists a solution x(t) of (1.2) such that x(t) > 0 on [T, ∞) for some T ≥ t0. Let w(t) be defined by the Riccati Transformation wðt Þ ¼ ρðt Þ
By condition (2.14), we get
r ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ ; t ≥ T: g ðxðt ÞÞ
Derivation this equality we have ′
b
′
r ðt Þψ ðxðt ÞÞx ðt Þ r ðt Þψ ðxðt ÞÞf ðx ðt ÞÞ ≤ g ð xð t Þ Þ g ðxðt ÞÞ Zt ≤ wðT Þ− ½qðsÞ − pðsÞds; b > 0
w ′ ðt Þ ¼
′
ρðt Þðr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞÞ ρ′ ðt Þr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞ þ g ðxðt ÞÞ g ðxðt ÞÞ −
T
ρðt Þr ðt Þψ ðxðt ÞÞf ðx′ ðt ÞÞg ′ ðxðt ÞÞx′ ðt Þ : g 2 ðxðt ÞÞ
This, and (1.2) imply Integrating the above inequality multiplied by from T to t( ≥ T), we have Zt b T
ψ ðxðsÞÞx′ ðsÞ ds ≤ g ðxðsÞÞ
Zt T
1 r ðt Þ
ψ ðxðsÞÞf ðx′ ðsÞÞ ds g ðxðsÞÞ Zt
≤ wðT ÞRðt Þ− T
1 r ðsÞ
Zs ½qðuÞ−pðuÞduds: T
From condition (2.16) and (2.17), we get that Zt
w′ ðt Þ ≤ − ρðt Þ½qðt Þ−pðt Þ þ
Hence for all t ≥ T, we obtain Zt
Zt ρðsÞ½qðsÞ − pðsÞ ds ≤ − T
Zt −
Now, if x(t) ≥ x(T) for large t then θ(t) ≥ 0, which is a contradiction. Hence for large t, x(t) ≤ x(T), so
Zt
θ ðt Þ ¼ − xð t Þ
ψ ðuÞ du > − g ðu Þ
ZxðT Þ 0
ψ ðu Þ du > −∞; g ðuÞ
d ρðsÞ ds
wðsÞ ds: ρðsÞ
T0
−
d ρðsÞ ds
wðt Þ wðT 0 Þ þ ρðt Þ ρðt Þ ρðT 0 Þ
wðsÞ wðT 0 Þ ds ¼ −wðt Þ þ Bρðt Þ ; B ¼ : ρðsÞ ρðT 0 Þ ð2:20Þ
By (2.19) and (2.20) we get Zt ρðsÞ½qðsÞ−pðsÞ ds ¼ −wðt Þ þ Bρðt Þ:
which is again a contradiction. This completes proof the Theorem 2.3. Theorems 2.6. Suppose that conditions (2.14), (2.15) and (2.16) hold. Furthermore, suppose that, there exist a
ð2:19Þ
Zt wðsÞ d wðsÞ ds ¼ −ρðt Þ ds ρðsÞ ds ρðsÞ ¼ −ρðt Þ
T
ZxðT Þ
d ds
By the Bonnet’s Theorem that for each t ≥ T, there exist a T0 ∈ [T, t] such that
T
T
ρðsÞ T
ψ ðxðsÞÞx′ ðsÞ ds→−∞ as t→∞: g ðxðsÞÞ
θ ðt Þ ¼
ρ′ ðt Þ wðt Þ: ρðt Þ
ð2:21Þ
T
Integrating the above inequality multiplied by from T to t( ≥ T), we obtain
1 r ðt Þρðt Þ
Salhin et al. SpringerPlus 2014, 3:300 http://www.springerplus.com/content/3/1/300
Zt b T
ψ ðxðsÞÞx′ ðsÞ ds ≤ g ðxðsÞÞ
Page 6 of 7
Zt
ψ ðxðsÞÞf ðx′ ðsÞÞ ds g ðxðsÞÞ T 0 1 Zs Zt 1 ρðuÞ½qðuÞ−pðuÞduAds: ≤ BRðt Þ − @ r ðsÞρðsÞ T
T
Zt t0
0
Zs
1 ρðuÞ½qðuÞ − pðuÞduAds r ðsÞρðsÞ 0 ts0 1 Zt Z 2u 1 e þ sinu − sinu duAds ¼ ∞; ¼ @ s e @
T
From (2.16) and (2.18), we have Zt θ ðt Þ ¼ T
Competing interests The authors declare that they have no competing interests.
Now, if x(t) ≥ x(T) for large t, then θ(t) ≥ 0, which is a contradiction. Hence for large t, x(t) ≤ x(T), so
θ ðt Þ ¼ − xð t Þ
ψ ðuÞ du > − g ðu Þ
ZxðT Þ 0
x4 ð t Þ x′ ð t Þ 4 x ðt Þ þ 1
et ¼
x7 ðt Þ sint
ψ ðu Þ du > −∞; g ðuÞ
Received: 4 April 2014 Accepted: 16 May 2014 Published: 18 June 2014
′ þ ðe2t þ sint Þx3 ðt Þ 2
ð x′ ð t Þ Þ
ð1 þ x4 ðt ÞÞ2 ðx′ ðt ÞÞ2 þ 1
;
t ≥ 0:
Here, r ðt Þ ¼ et ; qðt Þ ¼ e2t þ sint; ψ ðxðt ÞÞ ¼
x4 ð t Þ ; þ1
x4 ð t Þ
2 x7 ðt Þ sint ð x′ ð t Þ Þ ; g ðxÞ ¼ x3 ; H t; xðt Þ; x′ ðt Þ ¼ ð1 þ x4 ðt ÞÞ2 ðx′ ðt ÞÞ2 þ 1 2
H ðt; xðt Þ; x′ ðt ÞÞ x7 ðt Þ sint ð x′ ð t Þ Þ ¼ g ðxÞ ð1 þ x4 ðt ÞÞ2 ðx′ ðt ÞÞ2 þ 1
1 ≤ sint ¼ pðt Þ: x3
So, can note that Zt limt→∞ supRðt Þ ¼ limt→∞ sup Zε 0
ds < ∞; es
t0
u ðu2 Þ2
þ1
du ¼
1 tan−1 ε2 < ∞: 2
Let us take ρ(t) = 1 we have
Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgement This research has been completed with the support of these grants: DIP2012-31, FRGS/2/2013/SG04/UKM/02/3 and FRGS/1/2012/SG04/UKM/01/1.
which is again a contradiction. This completes proof the Theorem 2.6. Example 2.7. Consider the differential equation
T
then, Theorem 2.4 ensures that every solution of the equation given oscillates.
ψ ðxðsÞÞx′ ðsÞ ds→−∞ as t→∞: g ðxðsÞÞ
ZxðT Þ
1
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