Emden–Fowler type dynamic equations on time scales

Nonoscillatory solutions for super-linear

Emden–Fowler type dynamic equations on time scales

Hui Li, Zhenlai Han

and Yizhuo Wang

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P. R. China

Received 6 April 2015, appeared 22 August 2015 Communicated by Ivan Kiguradze

Abstract. In this paper, we consider the following Emden–Fowler type dynamic equa- tions on time scales

a(t)|x^∆(t)|^αsgnx^∆(t)^∆+b(t)|x(t)|^βsgnx(t) =0,

when α < β. The classification of the nonoscillatory solutions are investigated and some necessary and sufficient conditions of the existence of oscillatory and nonoscilla- tory solutions are given by using the Schauder–Tychonoff fixed point theorem. Three possibilities of two classes of double integrals which are not only related to the co- efficients of the equation but also linked with the classification of the nonoscillatory solutions and oscillation of solutions are put forward. Moreover, an important property of the intermediate solutions on time scales is indicated. At last, an example is given to illustrate our main results.

Keywords:Emden–Fowler type dynamic equations, intermediate solutions, time scales, nonoscillatory solutions.

2010 Mathematics Subject Classification: 39A13, 34B18, 34A08.

1 Introduction

Emden–Fowler dynamic equations originated in the early 20th century and they were estab- lished in the early research of gas dynamics in astrophysics [8]. They also occur in the study of fluid mechanics, relativity, nuclear physics and chemical reaction systems, one can see the survey article by Wong [15] for detailed background of the generalized Emden–Fowler equa- tion. With the development of science and technology, the super-linear Emden–Fowler type dynamic equations on time scales have played an important and extensive role in physics and engineering technology. We refer the reader to [16] and the references cited therein. The basic theorems and applications can be found in Agarwal et al.[1]. In the recent years, there have been lots of results for Emden–Fowler type equations in [2,4,7,9].

BCorresponding author. Email: hanzhenlai@163.com

In 2007 and 2011, Cecchi et al. [3] and Naito [13] studied the asymptotic behavior of nonoscillatory solutions for differential equations of the following forms

a(t)φ(x)⁰⁰+b(t)φ(x) =0, and

(p(t)|x⁰|^αsgnx⁰)⁰+q(t)|x|^βsgnx=0,

respectively. In 2008, Cecchiet al.[2] considered the intermediate solutions for Emden–Fowler type equations

a(t)|x⁰(t)|^αsgnx⁰(t)⁰+b(t)|x(t)|^βsgnx(t) =0.

In 2010, Kamo and Usami [11] discussed the slowly decaying positive solutions for the quasi- linear ordinary differential equations

(p(t)|u⁰|^α−1u⁰)⁰+q(t)|u|^λ−1u=0.

In 2011, Jiaet al.[10] discussed oscillatory solutions for the second order super-linear dynamic equations on time scales

x^∆∆(t) +p(t)f x(σ(t)) =0.

In 2011, Erbe et al. [7] considered the asymptotic behavior of solutions for Emden–Fowler equations on time scales

x^∆∆(t) +p(t)x^α(t) =0, α>0, (1.1) where p ∈ C_rd([t₀,∞)_T,R)_, α is the quotient of odd positive integers, andT denotes a time scale which is unbounded from above. This article proposed an important property about the solution of (1.1) under the condition R_∞

t0 t^α|p(t)|_∆t <_∞.

Zhou and Lan gave a classification of nonoscillatory solutions for the second-order neutral delay dynamic equations on time scales

[x(t)−c(t)x(t−τ)]^∆∆+ f(t,x(g₁(t)), . . . ,x(g_m(t))) =0, t∈_T,

and some existence results of each kind of nonoscillatory solutions were also established in [17].

In 2014, Došlá and Marini [4] studied the nonoscillatory solutions for second order Emden–

Fowler type differential equation

a(t)|x⁰(t)|^αsgnx⁰(t)⁰+b(t)|x(t)|^βsgnx(t) =0.

This article has an important and far-reaching influence because it solved the open problem on the possible coexistence of three types of nonoscillatory solutions for super-linear Emden–

Fowler differential equations. However, to the best of our knowledge, the coexistence of nonoscillatory solutions for dynamic equations on time scales has been scarcely investigated.

Motivated by [4], we consider the second order super-linear dynamic equations on time scales

a(t)|x^∆(t)|^αsgnx^∆(t)^∆+b(t)|x(t)|^βsgnx(t) =0, (1.2) where 0<α< βare constants and a(t)>0, b(t)≥ 0 are rd-continuous functions on[0,∞)_T, and

Ia =

Z _∞

0

1

a^1/α(s)^∆s =_∞, I_b=

Z _∞

0 b(s)_∆s<_∞.

When a(t)≡1, that is equation

|x^∆(t)|^αsgnx^∆(t)^∆+b(t)|x(t)|^βsgnx(t) =0. (1.3) Ifα6= β, then the prototype of (1.3) is the Emden–Fowler equation

x^∆∆+b(t)|x|^βsgnx =_{0. (1.4)} Moreover, the half-linear case of (1.2) is the following form

a(t)|x^∆(t)|^αsgn x^∆(t)^∆+b(t)|x(t)|^αsgn x(t) =0. (1.5) We will consider only the eventually positive solutions of (1.2) in the following section and denote

x^[1](t) =a(t)|x^∆(t)|^αsgnx^∆(t) for convenience.

The main work of this article can be listed as follows. Firstly, we improve the result in [6].

We will show that the case when the solution is a constant asx^[1](t)tends to a constant is im- possible. Secondly, we investigate the necessary and sufficient conditions for the existence of oscillatory and nonoscillatory solutions by methods different from [6]. Thirdly, we present an important property about intermediate solutions on time scales which generalize the related contributions to the subject in [4]. The research about the second order super-linear dynamic equations on time scales unifies the cases of differential equations and difference equations.

The paper is organized as follows. In Section 2, we introduce some definitions and a lemma about oscillatory and nonoscillatory solutions and the Schauder–Tychonoff fixed point theorem. In Section 3, we investigate the classification of the nonoscillatory solutions. Then we give some necessary and sufficient conditions for the existence of some oscillatory and nonoscillatory solutions by the Schauder–Tychonoff fixed point theorem. We propose three possibilities of two classes of double integrals which is related to the coefficients of the equa- tion and an important property of the intermediate solutions. Moreover, an example is given to illustrate our main results.

2 Preliminaries

In this section, we collect some definitions and a lemma about dynamic equations on time scales.

Definition 2.1([14]). We say that a nontrivial solutionx of (1.2) has a generalized zero att, if x(t)x(σ(t))≤0. Ifx(t) =0 we say that solution xhas a common zero att.

Definition 2.2([14]). We say that a solutionxof equation (1.2) is nonoscillatory on T, if there exists τ∈ _Tsuch that there does not exist any generalized zero att fort∈[τ,∞)_T.

A nontrivial solutionxof equation (1.2) is called oscillatory onT, if for everyτ∈_Thasx a generalized zero on[τ,∞)_T.

Definition 2.3([6]). We say that equation (1.2) is super-linear, if there exists a constant γ>₀ such that|v^−γ||b(s)v^β|is nondecreasing in |v|for each fixeds and

Z _∞

M

∆v

v^γ/α <_∞ for any M >0. (2.1)

Lemma 2.4 (Schauder–Tychonoff fixed point theorem [12]). Let X be a locally convex space, K⊂X be nonempty and convex, S⊂ K, S be compact. Given a continuous map F: K→S, then there exists x˜ ∈S such that F(x˜) =x.˜

3 Main results

In this section, we investigate the classification of the nonoscillatory solutions. Then we give some necessary and sufficient conditions for the existence of some oscillatory and nonoscilla- tory solutions by Schauder–Tychonoff fixed point theorem. We also present three possibilities of two classes of double integrals which is related to the coefficients of the equation and an important property of the intermediate solutions. We extend some results of [4] to time scales.

Theorem 3.1. The class Pof all eventually positive solutions of (1.2) can be divided into three sub- classes:

M⁺<sub>∞,`</sub> ={x∈<sub>P</sub>:x(<sub>∞</sub>) =<sub>∞</sub>, x<sup>[1]</sup>(<sub>∞</sub>) =`, 0< ` <_∞}, M_∞,0⁺ ={x∈_P:x(_∞) =_∞, x^[1](_∞) =0},

M<sub>`</sub><sup>+</sup><sub>,0</sub>={x∈<sub>P</sub>:x(<sub>∞</sub>) =`, x^[1](_∞) =0, 0< ` <_∞}.

The superscript symbol “+” means that solutions are eventually positive increasing. We call solutions in M⁺_{∞,`</sub>, M<sup>+</sup><sub>∞,0</sub>, M<sub>`}⁺_,0 dominant solutions, intermediate solutions and subdomi- nant solutions.

Proof. Let x(t)be a positive solution of (1.2) for large t. Then there exists a t0 > 0 such that x(t)>0 ast ≥t₀. From (1.2) we have

(x^[1](t))^∆ =−b(t)|x(t)|^βsgnx(t)≤0,

so x^[1](t) is non-increasing for t ≥ t₀, which implies that x^[1](t) is eventually positive or negative.

We conclude that x^[1](t) ≥ 0, t ≥ t₀. Otherwise, if x^[1](t) < 0 for t ≥ t₀, then there is a positive numberc, such thatx^[1](t)≤ −c. Integrating the last inequality from 0 totand letting t→_{∞, we have}x(t)→ −_∞ast →_∞sincex^∆(t)<0, which is contrary to the assumption of the eventually positive solution. Sox^[1](t)≥0, i.e.x^∆(t)≥0.

Thus for any`, 0< ` <_∞, the possible cases ofx^[1](t)andx(t)whent→_∞are as follows:

(i) x(_∞) =`, x<sup>[1]</sup>(<sub>∞</sub>) =`; (ii) x(_∞) =_∞, x^[1](_∞) =`; (iii) x(_∞) =_∞, x^[1](_∞) =_0;

(iv) x(_∞) =`, x^[1](_∞) =0.

Now we prove the case (i) is impossible. If lim_t→_∞x^[1](t) = `, then there exists t<sub>0</sub> > 0 such that x<sup>[1]</sup>(t)≥ <sub>2</sub><sup>`</sup> for t ≥ t0, i.e. x^∆(t) ≥ ₍^`1/α

2a)^1/α. Integrating the last inequality and by the conditionIa =_{∞, we have}x(_∞) =∞. The proof is completed.

Denote J =

Z _∞

0

1 a^1/α(s)

_{Z ∞}

s b(r)_∆r 1/α

∆s, K=

Z _∞

0 b(s) _{Z s}

0

1 a^1/α(r)^∆r

β

∆s.

Jm =

Z _∞

0

1 a^1/α(s)

Z _∞

s b(r)_∆r 1/m

∆s, Km =

Z _∞

0 b(s) Z _s

0

1 a^1/α(r)^∆r

m

∆s.

Now we give some sufficient and necessary conditions for the existence of oscillatory and nonoscillatory solutions.

Theorem 3.2. The following hold for(1.2):

i₁) The classM⁺_{`,0</sub>is nonempty if and only if J <∞. Moreover, for any`, 0 < ` <∞, there exists x∈<sub>M</sub><sup>+</sup><sub>`,0} such thatlim_t→_∞x(t) =`.

i₂) The classM⁺_{∞,`</sub> is nonempty if and only if K< <sub>∞</sub>. Moreover, for any`, 0< ` <<sub>∞</sub>, there exists x∈<sub>M</sub><sup>+</sup><sub>∞,`}such thatlimt→∞x^[1](t) =`.

i₃) Letα< β. Equation(1.2)is oscillatory if and only if J= _∞.

i₄) Letα> β. Equation(1.2)is oscillatory if and only if K=_∞.

Proof. i₁) (The “only if” part) Let x(t) be a nonoscillatory solution in M⁺_`,0, t > t0 > 0. i.e.

x(_∞) =`, x^[1](_∞) =0. Integrating equation (1.2) from tto∞, we get a(t)(x^∆(t))^α =

Z _∞

t b(t)x^β(t)_∆t, i.e.

x^∆(t) = ¹ a^1/α(t)

_{Z ∞}

t b(s)x^β(s)_∆s 1/α

≥ ¹ a^1/α(t)

` 2

β/α_{Z ∞}

t b(s)_∆s 1/α

. Integrating the above inequality from 0 to∞, we obtain

x(_∞)≥x(0) + `

2

β/αZ _∞

0

1 a^1/α(s)

_{Z ∞}

s b(r)_∆r 1/α

∆s.

By contrary, if J =_{∞, then}x(_∞) =∞, which is a contradiction with x(_∞) =`, 0< ` <_{∞. So} J < _∞.

(The “if” part) SupposeJ < ∞, then there existc>0, t₁>0 such that c^β/α

Z _∞

t1

1 a^1/α(t)

_{Z ∞}

t b(s)_∆s 1/α

∆t≤ ^c

2. (3.1)

DefineX= {x∈C_rd[t₁,∞)_T: ^c₂ ≤ x(t)≤c, t ≥t₁}and Tx(t) =c−

Z _∞

t

1 a^1/α(s)

_{Z ∞}

s b(r)x^β(r)_∆r 1/α

∆s, t≥t₁. Now, we separate the proof into the following two steps.

(i)Tmaps Xinto itself. Ifx ∈X, then from (3.1) we get 0≤

Z _∞

t

1 a^1/α(s)

_{Z ∞}

s b(r)x^β(r)_∆r 1/α

∆s ≤ ^c 2. So ^c₂ ≤ Tx≤c, which implies thatTx∈ X.

(ii) T is continuous. Obviously Tx is compact. Let {x_n} be a sequence of measurable functions ofXconverging to x∈X asn→_∞in the topology ofC_rd[t₁,∞)_T.

Since 0 ≤ x_n(t) ≤ c, we get ₂^c ≤ Tx_n(t) ≤ c. The Lebesgue dominated convergence theorem shows that Tx_n(t₁) → Tx(t₁), which implies Tx_n(t) → Tx(t) uniformly on [t₁,∞)_, that isT is continuous.

Therefore, applying the Schauder–Tychonoff fixed point theorem, we see that there exists an element x ∈ X such that x = Tx, which shows that x(t)is a positive solution of equation

(1.2) fort ≥t₁. Sincex(t)is uniformly bounded, then from Theorem3.1we get thatx(t)is an element inM⁺_`,0.

i₂) (The “only if” part) Let x(t) be a nonoscillatory solution in M⁺_∞,`, t > t₁ > _{0, i.e.}

x(_∞) =_∞, x^[1](_∞) =`. According to the definition of limit, we know a(t)(x<sup>∆</sup>(t))<sup>α</sup> ≥ `

2, i.e. x^∆(t)≥ (^`₂)^1/α a^1/α(t)^. Integrating from 0 tot, we get

x(t)≥ `

2

1/αZ _t

0

1 a^1/α(s)^∆s.

Substituting the above to equation (1.2) and integrating from 0 to ∞, we have Z _∞

0

(x^[1](t))^∆_∆t= −

Z _∞

0 b(t)x^β(t)_∆t

≤ − `

2

1/αZ _∞

0 b(t) Z _t

0

1 a^1/α(s)^∆s

β

∆t, i.e.

x^[1](_∞)≤ x^[1](0)− `

2

1/αZ _∞

0 b(t) Z _t

0

1 a^1/α(s)^∆s

β

∆t.

By contrary, if K = _{∞, then} x^[1](_∞) ≤ −∞, which is a contradiction with the condition x^[1](_∞) =`, 0< ` <_{∞. So}K< _∞.

(The “if” part) SupposeK<_∞holds. We can choose proper` >0 such that Z _∞

t1

b(_t) Z _t

t1

(₂`)^1/α a^1/α(s)^∆s

^β

∆t ≤`_.

Define the subsetXofC_rd[t₁,∞)_Tand the mapping A: X→C_rd[t₁,∞)_T by X=

x∈ C_rd[t₁,∞)_T: Z _t

t1

`^1/α

a^1/α(s)^∆s≤ x(t)≤

Z _t

t1

(2`)^1/α

a^1/α(s)^{∆s, t}≥t₁

and

Ax(t) =

Z _t

t1

(`+R_∞

s b(r)x^β(r)_∆r)^1/α

a^1/α(s) ^{∆s, t}≥t₁. (3.2)

Now, in order to use the Schauder–Tychonoff fixed point theorem in Lemma2.4 we sepa- rate the proof into the following three steps.

(i)Amaps Xinto itself. For any x∈ X, we have 0≤

Z _∞

s b(_r)_x^β(_r)_∆r ≤

Z _∞

s b(_r) Z _t

t1

(₂`)^1/α a^1/α(u)^∆u

^β

∆r ≤`_{, s}≥t1, (3.3) from (3.2) and (3.3) we obtain

Z _t

t1

`^1/α

a^1/α(s)^∆s≤ Ax(t)≤

Z _t

t1

(2`)^1/α

a^1/α(s)^{∆s, t} ≥t₁, which implies thatAx∈ X.

(ii)AX is compact. SinceAmaps Xinto itself, we only need to illustrateXis compact. Let y(t) = ^x(t)

Rt t₁ 1

a^1/α(s)∆s. Then

`<sup>1/α</sup> ≤y(t)≤ (2`)^1/α. For any x_n∈ X, since

yn = ^xn(t) Rt

t1

1 a^1/α(s)∆s

is bounded, from the compactness theorem, we can know that there exists a convergent sub- sequence y_nk. So for anyx_n∈ Xthere exists a convergent subsequence x_nk, which shows that Xis compact.

(iii) A is continuous. Let{x_n}be a sequence of measurable functions ofX converging to x∈ Xasn→_∞in the topology ofC_rd[t₁,∞)_T.

From 0≤ xn(t)≤ Rt t1

(2`)^1/α

a^1/α(s)∆s, we obtain 0≤

Z _∞

t1

b(t)x_n^β(t)_∆t≤

Z _∞

t1

b(t) _{Z t}

t1

(2`)^1/α a^1/α(s)^∆s

β

∆t

≤ (2`)^1/α

Z _∞

0 b(t) Z _t

0

1 a^1/α(s)^∆s

β

∆t= (2`)^1/αK<_∞.

The Lebesgue dominated convergence theorem shows that Z _∞

t1

b(s)x_n^β(s)_∆s →

Z _∞

t1

b(s)x^β(s)_∆s asn→_∞,

i.e. Ax_n(_∞) → Ax(_∞), which implies Ax_n(t) → Ax(t) uniformly on [t₁,∞)_{, that is} A is continuous.

Therefore, applying the Schauder–Tychonoff fixed point theorem, we see that there exists an element x ∈ Xsuch that x = Ax, which shows that x(t)is a positive solution of equation (1.2) fort ≥t₁. Sox(t)is an element inM⁺_∞,`.

i₃) The “only if” part follows fromi₁).

To prove the “if” part. Assume for contradiction that (1.2) has a nonoscillatory solution x(t). We may assume without loss of generality that x(t)> 0 fort≥ t0> 0. Integrating (1.2) fromt to∞and noting that lim_t→_∞a(t)(x^∆(t))^α ≥0, we have

a(t) x^∆(t)^α ≥

Z _∞

t b(s)x^β(s)_∆s, t≥ t₀, which implies

x^∆(t)≥ ¹ a^1/α(t)

_{Z ∞}

t b(s)x^β(s)_∆s 1/α

, t ≥t₀. Dividing the above byx^γ/α(t), we obtain

x^∆(t)

x^γ/α(t) ≥ ¹ a^1/α(t)

_{Z ∞}

t

b(s)x^β(s) x^γ(t) ^∆s

1/α

≥ ¹ a^1/α(t)

_{Z ∞}

t

b(s)x^β(s) x^γ(s) ^∆s

1/α

.

Sincex(t)is an eventually positive solution, there exists a positive constantc₀such thatx(t)≥ c0fort≥ t0. We have

b(t)x^β(t)

x^γ(t) ≥ ^b(t)c₀^β

c^γ₀ , t≥t₀, hence

x^∆(t) x^γ/α(t) ≥

1 c^γ₀

1 a(_t)

Z _∞

t b(s)c^β₀∆s 1/α

≥ ¹ c^γ/α₀

1 a(_t)

Z _∞

t b(s)c₀^β∆s 1/α

, t≥t₀. Integrating above over[t₀,t]and by condition (2.1), we have

1 c^γ/α₀

Z _t

t0

1 a(t)

Z _∞

t b(s)c^β₀∆s 1/α

≤

Z _x(t) x(t0)

∆v v^γ/α <_∞, which implies

Z _∞

t0

1 a(t)

Z _∞

t b(s)c₀^β∆s 1/α

∆t< _∞.

But this contradicts J =_∞.

The proof ofi₄) is similar to that ofi₃), so it is omitted here. The proof is completed.

Motivated by [5], now we give the following proof.

Lemma 3.3. If0< m≤1, then J_m =_∞⇒K_m= _∞.

Proof. Let p=1/m. Obviously p≥1. The integrals J_m,K_m can be written as J_m =

Z _∞

0

1 a^1/α(t)

_{Z ∞}

t b(s)_∆s p

∆t, K_m =

Z _∞

0 b(t) _{Z t}

0

1 a^1/α(s)^∆s

1/p

∆t.

Put _a1/α¹_(t,s) =0 fors <tand _a1/α¹_(t,s) = ¹

a^1/α(t) fors≥t. Then we obtain pp

Jm = ^p s

Z _∞

0

1 a^1/α(t)

Z _∞

t b(s)_∆s p

∆t = ^p s

Z _∞

0

Z _∞

t

( ¹

a^1/α(t))^1/pb(s)_∆s p

∆t

= ^p s

Z _∞

0

_{Z ∞}

0

( ¹

a^1/α(t,s))^1/pb(s)_∆s p

∆t≤

Z _∞

0

p

s Z _∞

0

1

a^1/α(t,s)^b(s)^p_∆t∆s

=

Z _∞

0 b(s)_∆s ^p s

Z _∞

0

1

a^1/α(t,s)^∆t =

Z _∞

0 b(s)_∆s ^p s

Z _s

0

1 a^1/α(t)^∆t

=

Z _∞

0 b(s)_∆s ^p s

Z _s

0

1

a^1/α(t,s)^∆s+

Z _∞

s

1 a^1/α(t,s)^∆s

=

Z _∞

0

b(s)_∆s ^p s

Z _s

0

1

a^1/α(t,s)^∆s≤

Z _∞

0

b(s)_∆s ^p s

Z _∞

0

1

a^1/α(t)^∆t =K_m. The proof is completed.

Lemma 3.4. Ifα< β, then J= _∞⇒K =_∞.

Proof. Consider two cases: (i)α≤1; (ii)α>1.

Case (i): Let t0 ≥ 0 be such that Rt 0

1

a^1/α(s)∆s > 1 for t ≥ t0. Since J = ∞, by Lemma 3.3 Kα =_{∞. Since}α<β, we haveK> Kα =_∞.

Case (ii): We have J=

Z _∞

0

1 a^1/α(t)

_{Z ∞}

t b(s)_∆s 1/α

∆t fort≥t₀. Integrating by parts we have

J =

Z _∞

0

_{Z t}

0

1 a1/α(s)^∆s

_{∆Z ∞}

t b(s)_∆s 1/α

∆t

= ¹ α

Z _∞

0 b(t)

Z _σ(t)

0

1 a^1/α(s)^∆s

Z _∞

t b(s)_∆s ¹⁻_α^α

∆t.

Since 1<α<β, by the Hölder inequality, we obtain

J ≤ ¹ α

Z _∞

0

_{Z σ(t)}

0

1 a^1/α(s)^∆s

β

b(t)_∆t

!¹_β

×



 Z _∞

0

b(t)^(1−1/β)2 R_∞

t b(s)_∆s

(α−1)β (β−1)α

∆t





β−1 β

.

SinceR_∞

0 b(s)_∆s< _∞,(1−1/β)²<1 and ⁽₍^α_β⁻₋¹₁⁾₎^β_α <1, we get



 Z _∞

0

b(t)^(1−1/β)2 R_∞

t b(s)_∆s

(α−1)β (β−1)α

∆t





β−1 β

≤



 Z _∞

0

b(t) R_∞

t b(s)_∆s

(α−1)β (β−1)α

∆t





β−1 β

< M,

where M is a finite positive constant. So we can choose properMsuch that J ≤ ^M_αK^1/β. Since J = ∞, this inequality yields the assertion. The proof is completed.

Theorem 3.5. The possible cases of mutual behavior of integrals J,K, whenα<βare as follows:

C₁) J= _{∞, K}=_∞;

C₂) J< _{∞, K}=_∞whenα< β;

C₃) J< _{∞, K}<_∞.

Proof. The theorem can be easily proved by applying Lemmas3.3, and3.4.

Lemma 3.6. Letµ>1,λµ>1and f,g be nonnegative rd-continuous functions on[t₂,∞). Then Z _t

t2

g(s) Z _t

s f(r)_∆r λ

∆s

!µ

≤ λ^µ

µ−1 λµ−1

µ−1Z _t

t2

f(r) Z _r

t2

g(s)_∆s µ

∆r ^{Z t}

t2

f(r)_∆r λµ−1

.

(3.4)

Proof. Consider two cases: (i) f >0; (ii) f has zeros.

Case (i): Integrating by parts, we have Z _t

t2

f(s) Z _t

s f(r)_∆r

λ−1Z _s

t2

g(u)_∆u

∆s = ¹ λ

Z _t

t2

g(s) Z _t

σ(s) f(r)_∆r λ

∆s.

According to the above equality and Hölder’s inequality, we obtain Z _t

t2

g(s) _{Z t}

σ(s) f(r)_∆r λ

∆s

=λ Z _t

t₂ f^1/p(s)f^1/q(s) _{Z t}

s f(r)_∆r

λ−1_{Z s}

t₂ g(u)_∆u

∆s

≤λ Z _t

t2

f(s) Z _s

t2

g(u)_∆u p

∆s

1/p Z _t

t2

f(s) Z _t

s f(r)_∆r

(λ−1)q

∆s

!1/q

,

where p>1, 1/p+1/q=1. Letting p=µ,q= _µ−^µ₁, we get Z _t

t2

g(s) Z _t

σ(s)f(r)_∆r λ

∆s

≤λ _{Z t}

t₂ f(s) _{Z s}

t₂ g(u)_∆u µ

∆s

1/µ_{Z t}

t₂ f(s) _{Z t}

s f(r)_∆r γ

∆s

(µ−1)/µ

,

(3.5)

whereγ= (λ−1)µ/(µ−1)>−1. Moreover, we have Z _t

t₂ f(s) _{Z t}

s f(τ)_∆τ γ

∆s= ¹ γ+1

_{Z t}

t₂ f(s)_∆s γ+₁

. Hence, from (3.5) we obtain

Z _t

t2

g(s) _{Z t}

σ(s) f(r)_∆r λ

∆s

≤λ

µ−1 λµ−1

(µ−1)/µ_{Z t}

t2

f(s) _{Z s}

t2

g(u)_∆u µ

∆s

1/µ_{Z t}

t2

f(r)_∆r

(λµ−1)/µ

. Case (ii): If f has zeros fort ≥ t₂, for anys ∈ [t₂,t)_T, let I(s) =cl{r ∈ (s,t) : f(r) > 0}. SinceRt

s f(r)_∆r =R

I(s) f(r)_∆r, the conclusion is true as before. The proof is completed.

Theorem 3.7. Let 1 < α < β, andR_∞

0 s^βb(s)_∆s < _∞. Then any intermediate solution x of (1.3) satisfieslim inft→_∞ ^tx

∆(t) x(t) >0.

Proof. Integrating equation (1.3), we obtain x^∆(t) = R_∞

s b(r)x^β(r)_∆r1/α

. We can choose t₃ large enough so thatx(t)>0,x^∆(t)>0 andR_∞

t1 r^βb(r)_∆r <1 fort ≥t₁. Chooset₄ large enough such that

k Z _∞

t2

r^βb(r)_∆r

(β−α)/α

<1, (3.6)

wherek= ¹

α

α(β−1) β−α

(β−1)/β

. Letη=max{t₃,t₄}. Integrating (1.3) twice, we obtain

x(t)−x(η) =

Z _t

η

_{Z t}

σ(s)b(r)x^β(r)_∆r+

Z _∞

t b(r)x^β(r)_∆r 1/α

∆s fort ≥_η.

From the inequality

(X+Y)^1/α ≤ X^1/α+Y^1/α, where X,Yare positive numbers, we obtain

x(t)−x(η)≤

Z _t

η

_{Z t}

σ(s)b(r)x^β(r)_∆r 1/α

∆s+t _{Z ∞}

t b(r)x^β(r)_∆r 1/α

∆s.

Let f(r) =b(r)x^β(r),g(s)≡1,λ=1/αandµ=βand by Lemma3.6we have x(t)−x(η)≤ k

Z _t

η

r^βb(r)x^β(r)_∆r

1/βZ _t

η

b(r)x^β(r)_∆r

(β−α)/α

+t Z _∞

t b(r)x^β(r)_∆r 1/α

, from (3.6),

x(t)−x(η)≤ _{Z t}

η

r^βb(r)x^β(r)_∆r 1/β

+t _{Z ∞}

t b(r)x^β(r)_∆r 1/α

= _{Z t}

η

r^βb(r)x^β(r)_∆r 1/β

+tx^∆(t). Since

_{Z t}

η

r^βb(r)x^β(r)_∆r _1/β

≤ x(t) _{Z ∞}

η

r^βb(r)_∆r _1/β

, we obtain

1− ^x(η) x(t) ≤

_{Z ∞}

η

r^βb(r)_∆r 1/β

+ ^tx

∆(t) x(t) ^, i.e.

tx^∆(t)

x(t) ≥1− ^x(η) x(t) −

_{Z ∞}

η

r^βb(r)_∆r 1/β

. We obtain the assertion from (3.6). The proof is completed.

4 Examples

In this section, we will present an example to illustrate our main results.

Example 4.1. LetT=R. Consider the following Emden–Fowler equation x⁰⁰+ ¹

(t+2)³|x|²sgnx=0, t ≥0. (4.1) We haveα=1,β=2, a(t) =1 andb(t) = _(t+¹₂₎₃ Thus

J =

Z _∞

0

1 a^1/α(t)

_{Z ∞}

t b(s)ds _1/α

dt

=

Z _∞

0

Z _∞

t

1

(s+2)^{3ds dt}

=0<_∞.

(4.2)

From Theorem3.2we can get that (4.1) above has subdominant solutions for t≥0.

5 Conclusion

At the end of this paper, let us suggest the further possible research in the theory of dynamic equations, concretely for Emden–Fowler dynamic equations. First, the coexistence of three classes of nonoscillatory solutions can be studied. Second, the sufficient and necessary con- ditions for the existence of intermediate solutions may be established. Third, the cases of sub-linear and half-linear about the corresponding conclusions can also be considered.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.

This research is supported by the Natural Science Foundation of China (61374074, 61374002), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003)

References

[1] R. P. Agarwal, M. Bohner, D. O’Regan, A. Peterson, Dynamic equations on time scales:

a survey,J. Comput. Appl. Math.141(2002), 1–26.MR1908825;url

[2] M. Cecchi, Z. Došlá, M. Marini, Intermediate solutions for Emden–Fowler type equa- tions: continuous versus discrete,Adv. Dyn. Syst. Appl.3(2008), 161–176.MR2547667 [3] M. Cecchi, Z. Došlá, M. Marini, On intermediate solutions and the Wronskian for

half-linear differential equations,J. Math. Anal. Appl.336(2007), 905–918.MR2352988;url [4] Z. Došlá, M. Marini, On super-linear Emden–Fowler type differential equations,J. Math.

Anal. Appl.416(2014), 497–510.MR3188719;url

[5] Z. Došlá, I. Vrko ˇc, On an extension of the Fubini theorem and its applications in ODEs, Nonlinear Anal.57(2004), 531–548.MR2062993;url

[6] Á. Elbert, T. Kusano, Oscillation and nonoscillation theorems for a class of second order quasilinear differential equations,Acta Math. Hungar. 56(1990), 325–336.MR1111319;url [7] L. Erbe, B. Jia, A. Peterson, On the asymptotic behavior of solutions of Emden–Fowler

equations on time scales,Ann. Mat. Pura Appl. (4)191(2012), 205–217.MR2909796;url [8] R. H. Fowler, Emden’s equation: The solution of Emden’s and similar differential equa-

tions,Mon. Not. R. Astron. Soc.91(1930), 63–91. url

[9] Z. Han, S. Sun, B. Shi, Oscillation criteria for a class of second order Emden–Fowler delay dynamic equations on time scales,J. Math. Anal. Appl.334(2007), 847–858.MR2338632;url [10] B. Jia, L. Erbe, A. Peterson, Kiguradze-type oscillation theorems for second order superlinear dynamic equations on time scales, Canad. Math. Bull. 54(2011), 580–592.

MR2894509;url

[11] K. Kamo, H. Usami, Characterization of slowly decaying positive solutions of second- order quasilinear ordinary differential equations with sub-homogeneity,Bull. Lond. Math.

Soc.42(2010), 420–428.MR2651937;url

[12] Y. Liu, J. Zhang, J. Yan, Existence of oscillatory solutions of second order delay differen- tial equations,J. Comput. Appl. Math.277(2015), 17–22. MR3272160;url

[13] M. Naito, On the asymptotic behavior of nonoscillatory solutions of second order quasi- linear ordinary differential equations,J. Math. Anal. Appl.381(2011), 315–327.MR2796212;

url

[14] J. Vítovec, Critical oscillation constant for Euler-type dynamic equations on time scales, Appl. Math. Comput.243(2014), 838–848.MR3244531;url

[15] J. S. W. Wong, On the generalized Emden–Fowler equation,SIAM Rev.17(1975), 339–360.

MR0367368

[16] J. Yu, Z. Guo, Boundary value problems of discrete generalized Emden–Fowler equation, Sci. China Ser. A49(2006), 1303–1314.MR2287260;url

[17] Y. Zhou, Y. Lan, Classification and existence of nonoscillatory solutions of the second- order neutral delay dynamic equations on time scales, Journal of Mathematical Sciences 198(2014), 279–295.url