Liu and Meng Advances in Difference Equations (2016) 2016:291 DOI 10.1186/s13662-016-0983-3

RESEARCH

Open Access

Interval oscillation criteria for second-order nonlinear forced differential equations involving variable exponent Haidong Liu* and Fanwei Meng *

Correspondence: [email protected] School of Mathematical Sciences, Qufu Normal University, Jingxuan West Road, Qufu, 273165, P.R. China

Abstract In this paper, we establish some interval oscillation criteria for a class of second-order nonlinear forced differential equations with variable exponent growth conditions. Our results not only give the sufficient conditions for the oscillation of equations with variable exponent growth conditions, but also they extend some existing results in the literature for equations with a Riemann-Stieltjes integral. Two examples are also considered to illustrate the main results. Keywords: oscillation; Riemann-Stieltjes integral; second-order nonlinear equation; variable exponent

1 Introduction In this paper, we will establish some interval oscillation criteria for the following equation: 

 p(t)u (t) + q(t)u(t) +



b

 γ (t,s)+–β(t) g(t, s)u(t) sgn u(t) dξ (s) = e(t),

t ≥ t ,

()

a

where p, q, β, e ∈ C[t , +∞) with p(t) > , β(t) > , a ∈ R, b ∈ (a, +∞), g ∈ C([t , +∞) × [a, b]), ξ : [a, b] → R is strictly increasing, γ ∈ C([t , +∞) × [a, b]), and γ (t, ·) is strictly increasing on [a, b] such that  < γ (t, a) < β(t) < γ (t, b),

β(t) ≤ γ (t, a) + ,

t ∈ [t , +∞).

b Here a f (s) dξ (s) denotes the Riemann-Stieltjes integral of the function f on [a, b] with respect to ξ . As usual, a nontrivial solution u(t) of equation () is called oscillatory if it has arbitrary large zeroes, otherwise it is called nonoscillatory. Equation () is said to be oscillatory if all its solutions are oscillatory. For the particular case when γ (t, s) = α(s), a = , and β(t) ≡ , equation () reduces to the following equation: 







p(t)u (t) + q(t)u(t) +

b

 α(s) g(t, s)u(t) sgn u(t) dξ (s) = e(t),

()



which have been observed in Sun and Kong []. © Liu and Meng 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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For the particular case when a = , b = l + m + , where l, m ∈ N and for s ∈ [, l + m + ),  ξ (s) = l+m j= χ(s – j) with  χ(s) =

, s ≥ , , s < ,

γ ∈ C([t , +∞) × [, l + m + )) such that γ (t, j) = αj (t),

j = , , . . . , m;

γ (t, m + i) = θi (t),

i = , , . . . , l,

satisfying  < α (t) < · · · < αm (t) < β(t) < θ (t) < · · · < θl (t), β(t) ≤ α (t) + , t ∈ [t , +∞), g(t, j) = Aj (t) ∈ C[t , +∞),

j = , , . . . , m;

g(t, m + i) = Bi (t) ∈ C[t , +∞),

i = , , . . . , l.

Then equation () reduces to the following equation with variable exponent growth conditions: 

m  α (t)+–β(t)  p(t)u (t) + q(t)u(t) + Aj (t)u(t) j sgn u(t) j=

+

l

 θ (t)+–β(t) Bi (t)u(t) i sgn u(t) = e(t).

()

i=

For the particular case when p(t) ≡ , q(t) ≡ , m = , α (t) ≡ α = , β(t) ≡ , Bi (t) ≡ , i = , , . . . , l, equation () reduces to the well-known Emden-Fowler equation,  α u (t) + q(t)u(t) sgn u(t) = .

()

In the past  years, extensive work has been done and great progress has been made on oscillation of equation () and more general equations (see [–] and the references therein). On the other hand, with wide use in the nonlinear elasticity theory and electrorheological fluids (see [, ]), the differential equations and variational problems with variable exponent growth conditions have been investigated by many authors in recent years (see [–]). However, we notice that no criteria were found for equation () even for the special case of equation () to be oscillatory so far in the literature. The purpose of this paper is to establish some interval oscillation results for equation () which involves variable exponent growth conditions. Clearly, our work is of significance because equation () allows an infinite number of nonlinear terms and even a continuum of nonlinearities determined by the function ξ . The organization of this article is as follows. After this introduction, in Section , we establish interval oscillation criteria of both the El-Sayed type and the Kong type for equation (). In Section , we give two examples to illustrate our main results.

2 Main results In the sequel, we denote by Lξ [a, b] the set of Riemann-Stieltjes integrable functions on [a, b] with respect to ξ . We further assume that for any t ∈ [t , +∞), γ (t, ·), /γ (t, ·) ∈ Lξ [a, b].

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Lemma . Suppose that γ ∈ C([t , +∞) × [a, b]), and for any t ∈ [t , +∞), γ (t, ·) is strictly increasing on [a, b], β ∈ C[t , +∞) such that β(t) ≤ γ (t, a) + ,

 < γ (t, a) < β(t) < γ (t, b),

t ∈ [t , +∞).

Let h = sup{s ∈ (a, b) : γ (t, s) ≤ β(t), t ∈ [t , +∞)}, and 

b

m (t) := h



h

m (t) := a

β  (t) γ (t, s) β  (t) γ (t, s)



–

b

dξ (s) h



dξ (s),

t ∈ [t , +∞),

dξ (s),

t ∈ [t , +∞).

–

h

dξ (s) a

Then, for any function θ satisfying θ (t) ∈ (m (t), m (t)) for t ∈ [t , +∞), there exists η : [t , +∞) × [a, b] → (, +∞) satisfying for any t ∈ [t , +∞), η(t, ·) ∈ Lξ [a, b], such that 

b

γ (t, s)η(t, s) dξ (s) = β  (t),

(t, s) ∈ [t , +∞) × [a, b],

()

a

and 

b

η(t, s) dξ (s) = θ (t),

(t, s) ∈ [t , +∞) × [a, b].

()

a

Proof Define  η (t, s) =

β  (t)  b ( dξ (s))– , γ (t,s) h

(t, s) ∈ [t , +∞) × [h, b],

,

(t, s) ∈ [t , +∞) × [a, h),

,

(t, s) ∈ [t , +∞) × [h, b],

()

and  η (t, s) =

β  (t)  h ( dξ (s))– , γ (t,s) a

(t, s) ∈ [t , +∞) × [a, h).

()

Note that, for any t ∈ [t , +∞), /γ (t, ·) ∈ Lξ [a, b]. Thus, for any t ∈ [t , +∞), ηi (t, ·) ∈ Lξ [a, b], and 

b

γ (t, s)ηi (t, s) dξ (s) = β  (t),

i = , .

()

a

Moreover, by the choice of h, we can easily get, for any t ∈ [t , +∞), –

 b β  (t) dξ (s) dξ (s) h γ (t, s) h

 b –  b β(t) dξ (s) = β(t) dξ (s) h h γ (t, s)

 b –  b < β(t) dξ (s) dξ (s) 

b

m (t) =

h

= β(t),

h

()

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 h – β  (t) dξ (s) dξ (s) a γ (t, s) a

 h –  h β(t) dξ (s) = β(t) dξ (s) a a γ (t, s)

 h –  h > β(t) dξ (s) dξ (s) 

h

m (t) =

a

a

= β(t).

()

Therefore, for any θ (t) ∈ (m (t), m (t)), t ∈ [t , +∞), there exists a function p∗ : [t , +∞) → (, ) such that

 – p∗ (t) m (t) + p∗ (t)m (t) = θ (t).

()

Let

η(t, s) =  – p∗ (t) η (t, s) + p∗ (t)η (t, s),

for (t, s) ∈ [t , +∞) × [a, b],

()

then η(t, s) >  for (t, s) ∈ [t , +∞) × [a, b], and for any t ∈ [t , +∞), η(t, ·) ∈ Lξ [a, b]. From ()-(), we get 

b

a



η(t, s) dξ (s) =  – p∗ (t)



b

η (t, s) dξ (s) + p∗ (t)

a

=  – p∗ (t) m (t) + p∗ (t)m (t)



b

η (t, s) dξ (s) a

= θ (t). Also from () and (), we have 

b



∗

γ (t, s)η(t, s) dξ (s) =  – p (t) a

+ p∗ (t)





b

γ (t, s)η (t, s) dξ (s) a

b

γ (t, s)η (t, s) dξ (s) a



=  – p∗ (t) β  (t) + p∗ (t)β  (t) = β  (t). This completes the proof of Lemma ..



Remark . We will see from the proof of Lemma . that the function η can be constructed explicitly for any nondecreasing function ξ . Remark . If we take γ (t, s) = α(s), a = , and β(t) ≡ , then Lemma . reduces to Lemma . in []. Lemma . Let functions θ : [t , +∞) → (, +∞), w : [t , +∞) × [a, b] → [, +∞), η : [t , +∞) × [a, b] → (, +∞) satisfy for any t ∈ [t , +∞), ω(t, ·) ∈ Lξ [a, b], η(t, ·) ∈ Lξ [a, b],

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and 

b

(t, s) ∈ [t , +∞) × [a, b].

η(t, s) dξ (s) = θ (t),

()

a

Then, for any t ∈ [t , +∞), 

b

a



 b η(t, s) ln[θ (t)w(t, s)] dξ (s) , η(t, s)w(t, s) dξ (s) ≥ exp a θ (t)

()

where we use the convention that ln  = –∞ and e–∞ = . Proof Without loss of generality we assume that, for any t ∈ [t , +∞), 

b

η(t, s)w(t, s) dξ (s) > . a

For otherwise 

b

η(t, s)w(t, s) dξ (s) = , a

b a

η(t, s) ln[θ (t)w(t, s)] dξ (s) = –∞, θ (t)

and hence () is obviously satisfied. It is easy to check that ln t ≤ t –  for t ≥ , and then, for any (t, s) ∈ [t , +∞) × [a, b],



ln θ (t)w(t, s) – ln

= ln  b a



b

η(t, s)w(t, s) dξ (s)

a



θ (t)w(t, s)

≤ b

η(t, s)w(t, s) dξ (s)

a

θ (t)w(t, s) η(t, s)w(t, s) dξ (s)

– .

()

Multiplying () by η(t, s), we obtain 



η(t, s) ln θ (t)w(t, s) – ln



b

η(t, s)w(t, s) dξ (s) a

≤ η(t, s)  b a

θ (t)w(t, s) η(t, s)w(t, s) dξ (s)

 – ,

()

by integrating the inequality () over dξ (s) and applying (), we get 

b





η(t, s) ln θ (t)w(t, s) – ln

a

 ≤ a

η(t, s)  b 

= θ (t) –

a

θ (t)w(t, s) η(t, s)w(t, s) dξ (s)

dξ (s)

 –  dξ (s)

b

η(t, s) dξ (s) = , a

η(t, s)w(t, s) dξ (s) a



b



b

()

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that is 

b



η(t, s) ln θ (t)w(t, s) dξ (s) ≤

a



b

 η(t, s) ln

a

η(t, s)w(t, s) dξ (s) dξ (s) a

 = ln

 = ln



b

b



b

η(t, s)w(t, s) dξ (s) a

η(t, s) dξ (s) a

b

 η(t, s)w(t, s) dξ (s) θ (t).

()

a

Dividing () by θ (t), we have b a

 b  η(t, s) ln[θ (t)w(t, s)] dξ (s) ≤ ln η(t, s)w(t, s) dξ (s) , θ (t) a

() 

which implies (). This completes the proof of Lemma ..

Following El-Sayed [], for c, d ∈ [t , +∞) with c < d, we define the function class V (c, d) := {v ∈ C  [c, d] : v(c) =  = v(d), v ≡ }. Our first result provides an oscillation criterion for equation () of the El-Sayed type. Theorem . Suppose that for any T > t , there exist T ≤ a < b ≤ a < b such that, for i = , , g(t, s) ≥ ,

for (t, s) ∈ [ai , bi ] × [a, b] and

(–)i e(t) ≥ ,

for t ∈ [ai , bi ].

()

We further assume that, for i = , , there exist functions vi ∈ V (ai , bi ) and θ satisfying θ (t) ∈ (m (t), β(t)] for t ∈ [t , +∞), and a continuous function η : [t , +∞) × [a, b] → (, +∞) satisfying () and (), where m (t) is defined as in Lemma . such that 

bi

ai

Q(t)vi (t) – p(t)v i (t) dt ≥ ,

()

where  β  (t)–θ (t)β(t) (β  (t) – θ (t)β(t) + θ (t))|e(t)| β  (t)–θ (t)β(t)+θ (t) Q(t) = q(t) + β  (t) – θ (t)β(t) 

  θ (t) ln β  (t) – θ (t)β(t) + θ (t) · exp  β (t) – θ (t)β(t) + θ (t) b  g(t,s) a η(t, s) ln η(t,s) dξ (s) . + θ (t)

()

Here we use the convention that ln  = –∞ and e–∞ = , and  = . Then equation () is oscillatory. Proof Assume, for the sake of contradiction, that equation () has an extendible solution u(t) which is eventually positive or negative. Without loss of generality, we may assume

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that u(t) >  for all t ≥ t . When u(t) is eventually negative, the proof is carried out in the same way using the interval the interval [a , b ] instead of [a , b ]. Define p(t)u (t) , u(t)

ω(t) = –

t ∈ [a , b ].

Then, for t ≥ t , ω satisfies ω (t) = q(t) +





γ (t,s)–β(t) e(t) ω (t) g(t, s) u(t) dξ (s) – + . u(t) p(t)

b

a

()

(I) We first consider the case when θ (t) ≡ β(t). From () and () we have, for t ∈ [a , b ], ω (t) ≥ q(t) +



b

a



γ (t,s)–β(t) ω (t) g(t, s) u(t) dξ (s) + . p(t)

()

Since η(t, s) satisfying () and () with θ (t) ≡ β(t), it follows that 

b



η(t, s) γ (t, s) – β(t) dξ (s) ≡ ,

for any t ∈ [t , +∞).

()

a

Therefore, by () and Lemma ., we get, for t ∈ [a , b ], 

b



γ (t,s)–β(t) g(t, s) u(t) dξ (s)

a



b

=



γ (t,s)–β(t) η(t, s)η– (t, s)g(t, s) u(t) dξ (s)

a



≥ exp

= exp

 β(t)

 β(t)

 

b



γ (t,s)–β(t) dξ (s) η(t, s) ln β(t)η– (t, s)g(t, s) u(t)



a b

η(t, s) ln a

β(t)g(t, s) dξ (s) η(t, s)

   ln u(t) b η(t, s) γ (t, s) – β(t) dξ (s) β(t) a 

 b β(t)g(t, s)  dξ (s) = exp η(t, s) ln β(t) a η(t, s) 

 b g(t, s)  dξ (s) . η(t, s) ln = exp ln β(t) + β(t) a η(t, s) 

+

()

Substituting () into (), 

 b ω (t)  g(t, s) dξ (s) + η(t, s) ln ω (t) ≥ q(t) + exp ln β(t) + β(t) a η(t, s) p(t) 

= Q(t) +

ω (t) , p(t)

t ∈ [a , b ],

()

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where Q(t) is defined by () with θ (t) ≡ β(t). Multiplying both sides of () by v (t), integrating every term from a to b , and using integration by parts, we find 

b

a

Q(t)v (t) – p(t)v  (t)



b 

dt + a

ω(t)v (t) + p/ (t)v (t) p/ (t)

 dt ≤ .

()

From (), we see that ω(t)v (t) + p/ (t)v (t) ≡ , p/ (t)

t ∈ [a , b ],

which implies from the definition of w that v (t)/v (t) ≡ u (t)/u(t) and hence v (t) ≡ u(t), t ∈ [a , b ] for some constant c = . This contradicts the assumption that v (a ) = v (b ) =  and u(t)is positive on [a , b ]. (II) Next we consider the case when θ (t) ∈ (m (t), β(t)). From (), we have 



γ (t,s)–β(t) e(t) g(t, s) u(t) dξ (s) – u(t) a   b

γ (t,s)–β(t) e(t) η(t, s) = dξ (s) g(t, s) u(t) – u(t) θ (t) a   b

γ (t,s)–β(t) |e(t)| η(t, s) g(t, s) u(t) dξ (s) + = u(t) θ (t) a    b

γ (t,s)–β(t) |e(t)| η(t, s) θ (t) g(t, s) u(t) dξ (s). = + θ (t) η(t, s) u(t) a b

()

If we let p= A=

θ (t) β  (t) – θ (t)β(t) + θ (t)

,

q=

β  (t) – θ (t)β(t) β  (t) – θ (t)β(t) + θ (t)



γ (t,s)–β(t) β  (t) – θ (t)β(t) + θ (t) g(t, s) u(t) , η(t, s)

B=

,

 |e(t)| , q u(t)

() ()

then from the Young inequality (pA + qB ≥ Ap Bq , where p + q = , p, q > , A ≥ , B ≥ ), we get

γ (t,s)–β(t) |e(t)| θ (t) g(t, s) u(t) + η(t, s) u(t) 

 

γ (t,s)–β(t) p  |e(t)| q β (t) – θ (t)β(t) + θ (t) ≥ g(t, s) u(t) η(t, s) q u(t) p

q



(γ (t,s)–β(t))p–q |e(t)| β (t) – θ (t)β(t) + θ (t) g(t, s) = u(t) η(t, s) q

 p



γ (t,s)θ (t)–β  (t) |e(t)| q β (t) – θ (t)β(t) + θ (t) u(t) β  (t)–θ (t)β(t)+θ (t) . = g(t, s) η(t, s) q

()

By () and (), we get  a

b



η(t, s) γ (t, s)θ (t) – β  (t) dξ (s) ≡ ,

for any t ∈ [t , +∞).

()

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From (), (), (), (), (), and Lemma ., we see that, for t ∈ [a , b ], 



γ (t,s)–β(t) e(t) g(t, s) u(t) dξ (s) – u(t) a

 p

  b

γ (t,s)θ (t)–β  (t) |e(t)| q η(t, s) β (t) – θ (t)β(t) + θ (t) u(t) β  (t)–θ (t)β(t)+θ (t) dξ (s) ≥ g(t, s) θ (t) η(t, s) q a b

 b η(t, s) ln[( β  (t)–θ(t)β(t)+θ(t) g(t, s))p ( |e(t)| )q [u(t)] β  (t)–θ (t)β(t)+θ (t) ] dξ (s)  γ (t,s)θ (t)–β  (t)

≥ exp

= exp

a

q

η(t,s)

θ (t)

 b η(t, s) ln[( β  (t)–θ(t)β(t)+θ(t) g(t, s))p ( |e(t)| )q ] dξ (s)  a

q

η(t,s)

θ (t)

 b η(t, s)[ γ (t,s)θ(t)–β  (t) ] ln u(t) dξ (s)  a β  (t)–θ(t)β(t)+θ(t) · exp θ (t)   b

η(t, s) ln[( β (t)–θ(t)β(t)+θ(t) g(t, s))p ( |e(t)| )q ] dξ (s)  a η(t,s) q = exp θ (t) b

ln u(t) η(t, s)[γ (t, s)θ (t) – β  (t)] dξ (s)  β  (t)–θ(t)β(t)+θ(t) a · exp θ (t) b 

b )q a η(t, s) dξ (s)  g(t, s)] dξ (s) ln( |e(t)| p a η(t, s) ln[ β (t)–θ(t)β(t)+θ(t) q η(t,s) + = exp θ (t) θ (t)

 

b g(t,s) p a η(t, s)[ln(β  (t) – θ (t)β(t) + θ (t)) + ln η(t,s) ] dξ (s) |e(t)| q + ln = exp θ (t) q   b b  



g(t,s) dξ (s) p ln(β  (t) – θ (t)β(t) + θ (t)) a η(t, s) dξ (s) + p a η(t, s) ln η(t,s) |e(t)| q = exp q θ (t)  β  (t)–θ (t)β(t) (β  (t) – θ (t)β(t) + θ (t))|e(t)| β  (t)–θ (t)β(t)+θ (t) = β  (t) – θ (t)β(t) b  

g(t,s)    θ (t) a η(t, s) ln η(t,s) dξ (s) ln β (t) – θ (t)β(t) + θ (t) + · exp  . β (t) – θ (t)β(t) + θ (t) θ (t)

Then from () and the above inequality, we have  β  (t)–θ (t)β(t) (β  (t) – θ (t)β(t) + θ (t))|e(t)| β  (t)–θ (t)β(t)+θ (t) ω (t) ≥ q(t) + β  (t) – θ (t)β(t) 

  θ (t) ln β  (t) – θ (t)β(t) + θ (t) · exp  β (t) – θ (t)β(t) + θ (t) b  g(t,s) ω (t) a η(t, s) ln η(t,s) dξ (s) + + θ (t) p(t) 



= Q(t) +

ω (t) , p(t)

()

where Q(t) is defined by () with θ (t) ∈ (m (t), β(t)). The rest of the proof is similar to that of part (I) and hence is omitted. This completes the proof of Theorem .. 

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Following Philos [] and Kong [], we say that a function H = H(t, s) belongs to a function class H, denoted by H ∈ H, if H ∈ C(D, [, ∞)), where D = {(t, s) : t ≥ s ≥ t }, and H satisfies for t ≥ t

H(t, t) = ,

and

H(t, s) > ,

for t > s ≥ t ,

()

and has continuous partial derivatives ∂H/∂t and ∂H/∂s on D such that  ∂H = h (t, s) H(t, s) and ∂t

 ∂H = –h (t, s) H(t, s), ∂s

()

where h , h ∈ Lloc (D, R). Next, we use the function class H to establish an oscillation criterion for equation () of the Kong type. Theorem . Suppose that for any T > , there exist nontrivial subintervals [a , b ] and [a , b ] of [T, +∞) such that () holds for i = , . We further assume that, for i = , , there exist a constant ci ∈ (ai , bi ) and functions H ∈ H and θ satisfying θ (t) ∈ (m (t), β(t)] for t ∈ [t , +∞), and a continuous function η : [t , +∞) × [a, b] → (, +∞) satisfying () and (), where m (t) is defined as in Lemma . such that  H(ci , ai )



ci 

Q(t)H(t, ai ) – ai

 + H(bi , ci )



 p(t)h (t, ai ) dt 

 p(t)h (bi , t) dt ≥ , Q(t)H(bi , t) – 

bi 

ci

()

where Q(t) is defined by (). Then equation () is oscillatory. Proof Proceeding as in the proof of Theorem ., we get ω (t) ≥ Q(t) +

ω (t) , p(t)

t ∈ [a , b ];

()

see () and () for the cases when θ (t) ≡ β(t) and θ (t) ∈ (m (t), β(t)), respectively. Let ci ∈ (ai , bi ) be such that () holds. Multiplying both sides of () by H(t, a ), integrating it from a to c , and using integration by parts we have  H(c , a )ω(c ) ≥

c 

Q(t)H(t, a ) + ω(t) a

 ∂H H(t, a )ω (t) (t, a ) + dt. ∂t p(t)

()

It follows from (), (), and () that c 

 p(t)h (t, a ) Q(t)H(t, a ) – dt H(c , a )ω(c ) ≥  a    c   p(t)h (t, a ) H(t, a ) + ω(t) dt. +  p(t) a 

()

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Similarly, multiplying both sides of () by H(b , t) and integrating it from c to b , we get b 

 p(t)h (b , t) dt  c    b   p(t)h (b , t) H(b , t) + + ω(t) dt.  p(t) c

 –H(b , c )ω(c ) ≥

Q(t)H(b , t) –

()

By dividing () and () by H(c , a ) and H(b , c ), respectively, and then adding them together, from () we have  H(c , a )



c 



a

 p(t)h (t, a ) + 

H(t, a ) ω(t) p(t)

 dt = 

()

and  H(b , c )

b 



c

   p(t)h (b , t) H(b , t) + ω(t) dt = .  p(t)

()

We can reach a contradiction from either of the above. For instance, () implies that 

 p(t)h (t, a ) + 

H(t, a ) ω(t) ≡ , p(t)

t ∈ [a , c ].

It follows from the definition of w and () that ∂H (t, a ) h (t, a ) u (t) = √ , = ∂t u(t)  H(t, a ) H(t, a )

√ and hence u(t) ≡ c H(t, a ) on [a , c ] for some constant c = . This contradicts the assumption that H(a , a ) =  and u(a ) > . This completes the proof of Theorem .. 

3 Examples In this section, we will work out two numerical examples to illustrate our main results. Here we use the convention that ln  = –∞ and e–∞ = . Example . We consider the following equation: u (t) + q(t)u(t) +



b

 γ (t,s)+–β(t) g(t, s)u(t) sgn u(t) dξ (s) = e(t),

t ≥ ,

()

a

where q(t) = λ sin t, a = , b = , γ (t, s) = se–t , g(t, s) = cos t, β(t) = e–t , ξ (s) = s, e(t) = –f (t) cos t, and λ >  is a constant and f (t) ∈ C[, ∞) is any nonnegative function. For any T ∈ R, we choose k ∈ Z large enough for kπ ≥ T and let a = kπ , a = b = kπ + π , and b = kπ + π . Then m (t) = ln e–t and () holds. Set θ (t) = δe–t ,

δ ∈ (ln , ],

η(t, s) =

δ –δ s δ– e–t . δ – 

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It is easy to verify that () and () are valid, and for t ∈ [ai , bi ], i = , , Let v(t) = sin t. Note that, for i = , , 



bi

π 



Q(t)v (t) dt =

F(λ, δ, t) sin t dt,



ai

where 

  –δ t –δ+δe δet + f (t + kπ) cos(t) –δ

 t   δe · exp ln e–t – δe–t + δe–t t  – δ + δe   et  + ln cos t – η(t, s) ln η(t, s) ds δ 

F(λ, δ, t) = λ sin t +

and 

bi

v (t) dt =



π 

 cos t dt = π.



ai

Thus, by Theorem . we see that equation () is oscillatory for π .



π 



F(λ, δ, t) sin t dt ≥

Example . We consider the following equation: u (t) + q(t)u(t) +



b

 γ (t,s)+–β(t) g(t, s)u(t) sgn u(t) dξ (s) = e(t),

t ≥ ,

()

a

where q(t) = λ sin t, a = , b = , γ (t, s) = s(cos t ), g(t, s) = cos t, β(t) = cos t , ξ (s) = s, λ >  is a constant. For any T ∈ R, we choose k ∈ Z large enough for kπ ≥ T and let a = kπ , a = b = kπ + π , b = kπ + π , c = kπ + π , and c = kπ + π . Assume that e(t) ∈  C[, ∞) is any function satisfying (–)i e(t) ≥  on [ai , bi ] for i = , . Then () holds. Set t θ (t) = δ cos , 

δ ∈ (ln , ],

η(t, s) =

δ t –δ s δ– cos . δ –  

It is easy to verify that () and () are valid, and for t ∈ [ai , bi ], i = , ,   cos  –δ cos    cos t –δ cos t +δ δ e(t) Q(t) = λ sin t +  + cos t – δ cos t 



t δ  t  t ln cos – δ cos + δ cos · exp    cos t – δ cos t + δ   sec t  η(t, s) ln η(t, s) ds . + ln cos t – δ  

t

t

We choose H(t, s) = (t – s) , then by Theorem . we see that equation () is oscillatory if 

kπ + π

kπ

 

Q(t)(t – kπ) dt +

kπ + π

kπ + π

Q(t)(kπ + π/ – t) dt ≥

π 

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and 

kπ + π 

kπ + π

 Q(t)(t – kπ – π/) dt +

kπ + π

kπ + π 

Q(t)(kπ + π/ – t) dt ≥

π . 

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements The authors thank the reviewers for their helpful and valuable suggestions and comments on this paper. This research was supported by the National Natural Science Foundation of China (No. 11171178) and the Science and Technology Project of High Schools of Shandong Province (No. J14LI09) (China). Received: 15 March 2016 Accepted: 26 September 2016 References 1. Sun, YG, Kong, QK: Interval criteria for forced oscillation with nonlinearities given by Riemann-Stieltjes integrals. Comput. Math. Appl. 62, 243-252 (2011) 2. Agarwal, RP, Grace, SR, Regan, DO: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic, Dordrecht (2002) 3. Butler, GJ: Oscillation theorems for a nonlinear analogue for Hill’s equation. Q. J. Math. 27, 159-171 (1976) 4. Butler, GJ: Integral averages and oscillation of second order nonlinear differential equations. SIAM J. Math. Anal. 11, 190-200 (1980) 5. Kartsatos, AG: On the maintenance of oscillation of nth order equations under the effect of a small forcing term. J. Differ. Equ. 10, 355-363 (1971) 6. Kartsatos, AG: Maintenance of oscillations under the effect of a periodic forcing term. Proc. Am. Math. Soc. 33, 377-383 (1972) 7. Keener, MS: Solutions of a certain linear nonhomogeneous second order differential equations. Appl. Anal. 1, 57-63 (1971) 8. Kwong, MK, Wong, JSW: Linearization of second order nonlinear oscillation theorems. Trans. Am. Math. Soc. 279, 705-722 (1983) 9. Ou, CH, Wong, JSW: Forced oscillation of nth order functional differential equations. J. Math. Anal. Appl. 262, 722-731 (2001) 10. Rankin, SM: Oscillation theorems for second order nonhomogeneous linear differential equations. J. Math. Anal. Appl. 53, 550-553 (1976) 11. Agarwal, RP, Anderson, DR, Zafer, A: Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities. Comput. Math. Appl. 59, 977-993 (2010) 12. Liu, HD, Meng, FW, Liu, PC: Oscillation and asymptotic analysis on a new generalized Emden-Fowler equation. Appl. Math. Comput. 219, 2739-2748 (2012) 13. Nazr, AH: Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential. Proc. Am. Math. Soc. 126, 123-125 (1998) 14. Li, C, Chen, S: Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials. Appl. Math. Comput. 210, 504-507 (2009) 15. Zheng, ZW, Wang, X, Han, HM: Oscillation criteria for forced second order differential equations with mixed nonlinearities. Appl. Math. Lett. 22, 1096-1101 (2009) 16. Sun, YG, Saker, SH: Forced oscillation of higher-order nonlinear differential equations. Appl. Math. Comput. 173, 1219-1226 (2006) 17. Wong, JSW: Second order nonlinear forced oscillations. SIAM J. Math. Anal. 19, 667-675 (1988) 18. Yang, QG: Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential. Appl. Math. Comput. 136, 49-64 (2003) 19. Agarwal, RP, Grace, SR: Forced oscillation of nth order nonlinear differential equations. Appl. Math. Lett. 13(7), 53-57 (2003) 20. Došlý, O, Veselý, M: Oscillation and non-oscillation of Euler type half-linear differential equations. J. Math. Anal. Appl. 429, 602-621 (2015) 21. Došlý, O, Yamaoka, N: Oscillation constants for second-order ordinary differential equations related to elliptic equations with p-Laplacian. Nonlinear Anal. 113, 115-136 (2015) 22. Tunc, E, Avci, H: Oscillation criteria for a class of second order nonlinear differential equations with damping. Bull. Math. Anal. Appl. 4, 40-50 (2012) 23. Hasil, P, Veselý, M: Oscillation of half-linear differential equations with asymptotically almost periodic coefficients. Adv. Differ. Equ. 2013, 122 (2013) 24. Vítovec, J: Critical oscillation constant for Euler-type dynamic equations on time scales. Appl. Math. Comput. 243, 838-848 (2014) 25. Zhikov, VV: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 29, 33-36 (1987) 26. R˚užiˇcka, M: Electro-Rheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2000)

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