0 Boundary Value Problems

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Oscillatory behavior of second-order nonlinear neutral differential equations with distributed deviating arguments

Tongxing Li1*, Blanka Baculfkova2 and Jozef Dzurina2 Dedicated to Professor Ivan Kiguradze

Correspondence: litongx2007@163.com

1Qingdao TechnologicalUniversity, Feixian, Shandong 273400, P.R. China

Full list of author information is available at the end of the article

Abstract

We study oscillatory properties of a class of second-order nonlinear neutral functional differential equations with distributed deviating arguments. On the basis of less restrictive assumptions imposed on the neutral coefficient, some new criteria are presented. Three examples are provided to illustrate these results. MSC: 34C10; 34K11

Keywords: oscillation; neutral differential equation; second-order equation; distributed deviating argument

ft Spri

ringer

1 Introduction

This paper is concerned with oscillation of the second-order nonlinear functional differential equation

(r(t)\Z(t)\a-1Z(t))' + / q(t,£)\x[g(t,£)]\a-1x[g(t,£)]da(£) = 0, (1.1)

where t > t0 > 0, a > 1 is a constant, and z := * + p ■ x o t. Throughout, we assume that the following hypotheses hold:

(H1) I := [to, to), r,p e C1(I,R), r(t) > 0, andp(t) > 0;

(H2) q e C(I x [a, b], [0, to)) and q(t, £) is not eventually zero on any [tt, to) x [a, b], tt e I; (H3) g e C(I x [a, b], [0, to)), liming«,g(t,£) = TO,andg(t,a) <g(t,£) for £ e [a, b]; (H4) t e C2(I,R), t'(t) > 0, limt^to t(t) = to, andg(t(t),£) = t[g(t,£)]; (H5) a e C([a, b], R) is nondecreasing and the integral of (1.1) is taken in the sense of Riemann-Stieltijes.

By a solution of (1.1), we mean a functionx e C([tx, to), R) forsome t* > t0, which has the properties that z e C1([tx,to),R), r|z'|a-1z' e C1([tx,to),R), and satisfies (1.1) on [tx,to). We restrict our attention to those solutions x of (1.1) which exist on [tx, to) and satisfy sup{|x(t)| : t > T} > 0 for any T > t*. A solution x of (1.1) is termed oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions oscillate.

© 2014 Li 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/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly cited.

As is well known, neutral differential equations have a great number of applications in electric networks. For instance, they are frequently used in the study of distributed networks containing lossless transmission lines, which rise in high speed computers, where the lossless transmission lines are used to interconnect switching circuits; see [1]. Hence, there has been much research activity concerning oscillatory and nonoscillatory behavior of solutions to different classes of neutral differential equations, we refer the reader to [2-30] and the references cited therein.

In the following, we present some background details that motivate our research. Recently, Baculikova and Lackova [6], Dzurina and Hudakova [12], Li etal. [15,18], and Sun et al. [22] established some oscillation criteria for the second-order half-linear neutral differential equation

where z := x + p ■ x o t,

0 < p(t) <1 or p(t) > 1.

Baculikova and Dzurina [4, 5] and Li et al. [17] investigated oscillatory behavior of a second-order neutral differential equation

Ye and Xu [26] and Yu and Fu [27] considered oscillation of the second-order differential equation

Assuming 0 < p(t) < 1, Thandapani and Piramanantham [23], Wang [24], Xu and Weng [25], and Zhao and Meng [30] studied oscillation of an equation

As yet, there are few results regarding the study of oscillatory properties of (1.1) under the conditions p(t) > 1 or limt^œp(t) = œ. Thereinto, Li and Thandapani [19] obtained several oscillation results for (1.1) in the case where (1.2) holds, a (Ç) = £, and

(r(t)\^(t)\a-1z'(t)] ' + q(t)\x( 5(f))\a-1x( 5(f)) = 0,

(r(t)(x(t) + p(t)x [t (t)]) ')' + q(f)x [a (t)] = 0,

0 < p(t) < p0 < œ and t'(t) > t0 > 0.

In the subsequent sections, we shall utilize the Riccati substitution technique and some inequalities to establish several new oscillation criteria for (1.1) assuming that (1.3) holds

All functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough.

2 Main results

In what follows, we use the following notation for the convenience of the reader: Q(t,£) := min{q(t,£),q(t(t),£)}, d+(t) := max{0,d(t)j,

ap![h(t)W(t) t "(t) p+(t)

0(t) := —FTT^---77^, z (t) := —^T + ^(t),

p[h(t)] t (t) p(t)

/ M (t)\a+1 pa [h(t)](Z+(t))a+^ p ds

v(t):-{p+w) + —^—, and 5(t):=L r^

where h, p, and n will be specified later.

Theorem2.1 Assume (H1)-(H5), (1.3), andletg(t,a) e C1(I,R),g'(t,a) > 0,g(t,a) < t, and g(t, a) < t (t) for t e I. Suppose further that there exists a real-valued function h e C1(I, R) such thatp[g(t, £)] < p[h(t)] fort e I and £ e [a, b]. If there exists a real-valued function p e C1(I, (0, ()) such that

limsup / p (s)

t X Jt0

r rb Qis. £ ma I rials, a)hn(s)

ds = (, (2.1)

t r/abQ(s, £)do(£) _ r[g(s, a)Ms)

2a-1 (a + 1)a+1(g'(s, a))'

then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a t1 e I such that x(t) > 0, x[t(t)] > 0, and x[g(t, £)] > 0 for all t > t1 and

£ e [a, b]. Then z(t) > 0. Applying (1.1), one has, for all sufficiently large t,

(r(t)|z'(t)|a-1z'(t^' + / q(t,£)xa[g(t,£)]do(£)

+ / q(t(t),£)pa[h(t)]x*[g(t(t),£)] do(£)

+ ^Wr {r[t (t)]Z'[T (t)ir-1z'[T (t)])' = 0.

Using the inequality (see [5, Lemma 1])

(A + B)a < 2a-1(Aa + Ba), for A > 0, B > 0, and a > 1,

the definition of z,g(T(t),£) = t[g(t,£)], andp[g(t,£)] <p[h(t)], we conclude that

(r(t)\z'(t)\a-1z'(t))' + —j Q(t, £)za [g(t, £)] da(£)

+ ^Thr(r[T(t)Hz'[T(t)]\a-1z'[t(t)])'< 0. (2.2)

By virtue of (1.1), we get

(r(t)\z'(t)\a-1z'(t))'< 0, t > t1. (.3)

Thus, r|z'|a-1z' is nonincreasing. Now we have two possible cases for the sign of z': (i)z' < 0 eventually, or (ii) zZ >0 eventually. (i) Assume that Z(t)<0 for t > t2 > t1. Then we have by (2.3)

r(t) \z'(t) \a-V (t) < r(t2) \z'(t2) rVfe) <0, t > t2,

which yields

z(t) < Z(t2)-r1/a (t2)\z'(t2)\ i \-1/a (s) ds.

Then we obtain limt^TO z(t) = -to due to (1.3), which is a contradiction. (ii) Assume that Z(t)>0 for t > t2 > ti. It follows from (2.2) andg(t, £) > g(t, a) that

(r(t)(z'(t))a)' + (r[T (t)](z'[T (t)])a)'

+ — z%(t,a)] J Q(t,£) da(£) < 0. (2.4)

We define a Riccati substitution (A /A r(t)(z'(t))a

«(t) := P(th r ^ Mto , t > t2. (.5)

(z[g(t, a)])a

Then «(t) > 0. From (2.3) andg(t, a) < t, we have

z [g(t, a)] > (r(t)/r[g(t, a)])1/aZ(t). (2.6)

Differentiating (2.5), we get

r(t)(z(t))a ,.(r(t)(Z(t))a)' « (t) = P (t)^^-r^z + P(t)-

(z[g (t, a)])a (zg (t, a)])a

^ r(t)(z(t))a za-1 [g(t, a)]z[g (t, a)]g'(t, a)

- aP (t)-^-. (2.?)

Therefore, by (2.5), (2.6), and (2.7), we see that

P(A /A (r(t)(z(t))T) ag'(t, a) (a+1)/a^

«(t) < PW "(t) + P (t)(№Cza])0r- p1/a (fr^gfe a)] ) (t). (2.§)

Similarly, we introduce another Riccati transformation: iA iA r[T(t)](z'[v(t)])a

U(t):=p(t) / r ^ Mto , t > t2. (9)

(z[g (t, a)])a

Then u(t) > 0. From (2.3) andg(t, a) < t(t), we obtain

4g(t, a)] > (r[T(t)]/r[g(t, a)])1/a^[r(t)]. (.10)

Differentiating (2.9), we have

,(A l(A r[T (t)](z/[r (t)])a (r[r (t)](z'[r (t)])a)' u (t) = p (t) . r . + p(t)-

(z[g(t, a)])a (z[g (t, a)])a

uj[t (t)](z [t (t)])a za-1 [g(t, a)]z; g (t, a)]g/(t, a)

- ap (t)-maoa--. (211)

Therefore, by (2.9), (2.10), and (2.11), we find

u (t) < p^u(t) + p(t)(r[T(tt)](z[X(t)])a)'__Og-M._u("+«/"(t) (2 12)

U (t) < p(t) u(t) + p(t) (z[g(t, a)])a pV° (t)r1/a [g(t, a)] u (t). (2.12)

Combining (2.8) and (2.12), we get

pa [h(t)] o(t) + —THT' u (t)

, (r(t)(z!(t))a)' + p-ahf(r[r(t)](z'[r(t)])a)' p'(t) , ^

< p(t)-, T ) MV,-+ ^ 0J(t)

(z[g(t, a)])a p(t)

ag'^a) o(a+1)/^ . Pa[h(t)] p'(t).

a + ^u (t)

p1/a(t)r1/a [g(t, a)] T (t) p(t)

pa [h(t)] ag'(t, a)

(a+1)/a

t (t) p1/a (t)r1/a [g(t, a)] It follows from (2.4) that

^^ u '» < fa'»* £ > + oW

"g/(t, a) o(a+1)/^ , -a [h(t)] p+ (t).

'(t)+ u (t)

p1/a (t)r1/a [g(t, a)] T (t) p (t)

[h(t)] "g/(t, a) u(a+1)/a

T (t) p1/a (t)r1/a [g(t, a)]

Integrating the latter inequality from t2 to t, we obtain

-a [h(t)^ [h(t2)] , .

o(t)-o(t2) + -—— u (t)--—— u (t2)

T'(t) t'fe)

< -ft pS /bQ(s, £) do (£) ds Jt2 2 Ja

tt Jt2

P+ (s) . P (s)

«(s)-

a g'(s, a)

P1/a(s)r1/a[g (s, a)]

( +1)/

Define A := B :=

pa [h(s)] [ t'(s) 1

ag'(t, a)

P+ (s) P (s)

+ 0(s)

u(s) -

ag'(s, a)

P 1/a(s)r1/a[g(s, a)]

( +1)/

(s) ds. (2.13)

_P1/a(t)r1/a [g(t, a)]_

/( +1)

«(t) and

a P+(t)

_a + 1 p (t) Using the inequality a +1

ag'(t, a)

_p 1/a(t)r1/a[g(t, a)]_

/( +1)

AB1/a - A(a+1)/a < -B(a+1)/a, for A > 0 andB > 0, a a

(2.14)

«(t) - ■

ag'(t, a)

we get P+ (t)

p (t)~w p1/a(t)r1/a[g(t, a)] On the other hand, define

ag'(t, a) - a /(a+1)

( +1)/

1 r[g(t, a)](p+ (t))a (a + 1)a+1 (p (t)g'(t, a))a

_ p1/a(t)r1/a [g(t, a)]_

u (t) and

ag'(t, a)

p1/a(t)r1/a[g (t, a)]

-a /(a+1)-

Then we have by (2.14)

Z+(t)U (t)- p1/a(t)r1/a^ (t, a)] " Thus, from (2.13), we get

(a+1)/a (t) < 1 r[g(U a)](Z+(t))a+1p(t)

(a + 1)'

(g'(t, a))a

(A (f\ p^h® (A p" [h(t2)]

«(t) -«(t2) + --- V (t)--V (t2)

T '(t)

T'(t2)

7J p(s){

/abQ(s, £) da (£)

r[g(s, a)]

(a + 1)a+1(g '(s, a))"

P+ (s)

pa [h(s)](Z+(s))a

P (s) J t '(s)

which contradicts (2.1). This completes the proof. □

Assuming (1.2), where p0 and t0 are constants, we obtain the following result.

Theorem 2.2 Suppose (H1MH5), (1.2), (1.3), and letg(t, a) e C1(I, R), g'(t, a) > 0,g(t, a) < t, and g(t, a) < t (t) for t e I. If there exists a real-valued function p e C1(I,(0,to)) such

limsup I

t^œ Jt0

trP(s)fbQ(s,£)da(£) 1 + ^ r[g(s,a)](p+ (s))-1]

as = œ, (2.15)

2a-1 (a + 1)+1 (p(s)g'(s, a))a

then (1.1) is oscillatory.

Proof As above, let x be an eventually positive solution of (1.1). Proceeding as in the proof of Theorem 2.1, we have z'(t) > 0, (2.3), and (2.4) for all sufficiently large t. Using (1.2), (2.3), and (2.4), we obtain

(r(t)( z'(t))a)' + ^ (r [t (t)] (Z [t (t)])'

+ — Z* [g(t, a)]j Q(s, £) da(£) < 0. (2.16)

The remainder of the proof is similar to that of Theorem 2.1, and hence it is omitted. □

Theorem 2.3 Suppose we have (H1)-(H5), (1.3), and let t(t) < t andg(t, a) > t(t) for t e I. Assume also that there exists a real-valued function h e C1(I, R) such that p[g(t, £)] < p[h(t)] for t e I and £ e [a, b]. If there exists a real-valued function p e C1(I, (0, to)) such that

limsup / p(s)

t^œ Jt0

t ^^ r[T(s)],(s) ds = œ, (2.17)

2a-1 (a + 1)a+1(t '(s))

then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a ti e I such that x(t) > 0, x[t(t)] > 0, and x[g(t, £)] > 0 for all t > ti and £ e [a, b]. As in the proof of Theorem 2.1, we obtain (2.3) and (2.4). In view of (2.3), r|z'|a-1z' is nonincreasing. Now we have two possible cases for the sign of z':(i) Z <0 eventually, or (ii) Z >0 eventually.

(i) Suppose that Z(t) < 0 for t > t2 > t1. Then, with a proof similar to the proof of case (i) in Theorem 2.1, we obtain a contradiction.

(ii) Suppose that z'(t) > 0 for t > t2 > t1. We define a Riccati substitution

w(t) := P(t) , r /Alk , t > t2. (218) (z[t (t)])a

Then a>(t) > 0. From (2.3) and t(t) < t, we have

z'[t(t)] > (r(t)/r[T(t)])1/az(t). (2.19)

Differentiating (2.18), we obtain

r(t)(z'(t))a (r(t)(Z(t))ay v(t) = P (t) , r + P(t)-

(z[T (t)])a rw (z[T (t)])a iAr(t)(z'(t))a z' -1[T (t)]z[T (t)]T '(t)

- ap (t)-(zmr-. (2.20)

Therefore, by (2.18), (2.19), and (2.20), we see that

p'(t) <a u,(r(t)(z(t))a)' ax (t) (a+1)/atA ° (t) < -pit)0(t) + p (t) (z[r(f)])a - p1/a (t)rV«[r (t)] 0 )/ (t). (2.21)

Similarly, we introduce another Riccati substitution:

(t) (t) r[T (t)](z/[r (t)])a t ( 2

u(t):=p(t) (z[T(t)])a , t > t2. (2)

Then u(t) > 0. Differentiating (2.22), we have

r[T(t)](z/[r(t)])a (r[r(t)](z'[r(t)])a)' u (t) = p (t)-, r , ... -+ p(t)-

(z[T (t)])a (z[t (t)])a

M r[T(t)](z/[r (t)])aza-1[t(t)]z'[x(t)]T (t)

- ap(t)-1 r -• (2.23)

(z[r (t)])2a

Therefore, by (2.22) and (2.23), we get

Combining (2.21) and (2.24), we have

'M -a[h(t)] ^ (r(t)(z'(t))a)' + -p^(r[r(t)](z/[r(t)])a)' p'(t)

o (t) + —TTTT u (t) < p(t)-/ rto-+ ~Ta o(t)

T'(t) (z[r (t)])a p (t)

«T'(t) ,ra+1)/^ , Pa [h(t)] p'(t).

0(a+1)/a(t) + £_^u(t)

p1/a (t)r1/a [T(t)] T (t) p(t)

[h(t)] aT/(t) u (a+1)/a

T (t) p1/a (t)r1/a [t (t)] It follows from (2.4) andg(t, a) > t (t) that

^ pa[h(t)] p(t) f* ... . p+ (t) .. 0(t) + "T^Tu(t)<^L Q(t,£)do(£) + Wo(t)

aT'(t) ,ra+1)/^ , Pa [h(t)] p+(t).

o(a+1)/a(t) + ^u(t)

p1/a (t)r1/a [T(t)] T (t) p(t)

[h(t)] aT/(t) u(a+1)/a

T (t) p1/a (t)r1/a [t (t)] Integrating the latter inequality from t2 to t, we obtain

[h(t)^ [h(t2)]

o(t)-o(t2) + --- u (t)---- u (t2)

T'(t) T'(t2)

< -ft pri /bQ(s, £) do (£) ds + it

J t2 2 J a J t2

T+M o(s)__^_«<*+!>/* J ds

L p(s) 0(s) p1/a (s)r1/a [t (s)] ° ^

Define

at'(t)

p1/a(t)r1/a [t (t)] a p+(t)

a/(a+1)

«(t) and

a + 1 p (t)

a t '(t)

p 1/a(t)r1/a[T (t)]

-a/(a+1)-

Using inequality (2.14), we have

P+ (t)

«(t) - ■

aT '(t)

( +1)/

p (t) p1/a(t)r1/a [t (t)]

On the other hand, define

a T'(t) n a/(a+1)

r[T (t)](p+ (t))a

(a + 1)a+1 (p(t)T '(t))a

_ p1/a(t)r1/a [t (t)]_ a

u (t) and

rt+(t)

at'(t)

Then, by (2.14), we obtain at'(t)

_ p1/a(t)r1/a [t (t)]_

/( +1)

Z+(t)u (t)--

p1/a(t)r1/a [t (t)] Thus, from (2.25), we get

( +1)/

(a + 1)a

r[T (t)](Z+(t))a+1p(t) " (T'(t)Y .

Pa [h(t)^ pa [h(t2)] , .

«(t)-«(t2) + --- U (t)--—— U (t2)

T '(t)

T'(ta)

"î2t p(s){

/abQ(s, £ ) da (£ )

r[T (s)]

(a + 1)a+1(T '(s))a ds,

p^V+1 + Pa [h(s)](Z+(s))a+1 p (s) J T '(s)

which contradicts (2.17). This completes the proof. □

Assuming we have (1.2), where p0 and t0 are constants, we get the following result.

Theorem 2.4 Suppose we have (Hi)-(H5), (1.2), (1.3), and let t(t) < t andg(t, a) > t(t) for t e I. If there exists a real-valued function p e C1(I, (0, to)) such that

limsup

t^œ Jt0

p (s)fbQ(s, £ ) da (£ )

(a +1)'

1 +■

P0a \r[T (s)](p+ (s))

(T0p (s))a

ds = œ, (2.26)

then (1.1) is oscillatory.

Proof Assume again that x is an eventually positive solution of (1.1). As in the proof of Theorem 2.1, we have z'(t) > 0, (2.3), and (2.4) for all sufficiently large t. Byvirtue of (1.2),

(2.3), and (2.4), we have (2.16) for all sufficiently large t. The rest of the proof is similar to that of Theorem 2.3, and so it is omitted. □

In the following, we present some oscillation criteria for (1.1) in the case where (1.4) holds.

Theorem 2.5 Suppose we have (H1)-(H5), (1.2), (1.4), andletg(t,a) e C1(I, R), g'(t, a) > 0, g(t, a) < t (t) < tfor te I, andg(t, £) < g(t, b) for £ e [a, b]. Assume further that there exists a real-valuedfunction p e C*(I, (0, to)) such that (2.15) is satisfied. If there exists a real-valued function n e C*(I, R) such that n(t) > t, n(t) > g(t, b), n'(t) > 0for t e I, and

limsup I

t—Jt0

(2.27)

j;Q(s, £ )da (£ )

S" (s)-(l + P—

S(s)rl/a [n(s)]

ds = to,

then (l.l) is oscillatory.

Proof Let x be a nonoscillatory solution of (l.l). Without loss of generality, we assume that there exists a ti e I such that x(t) > O, x[t(t)] > O, and x[g(t, £)] > O for all t > ti and £ e [a, b]. Then z(t) > O. As in the proof of Theorem 2.l, we get (2.2). By virtue of (l.l), we have (2.3). Thus, r\Z\a-1z' is nonincreasing. Now we have two possible cases for the sign of z': (i) zZ < O eventually, or (ii) zZ > O eventually.

(i) Suppose that z'(t) > O for t > t2 > tl. Then, by the proof of Theorem 2.2, we obtain a contradiction to (2.l5).

(ii) Suppose that z'(t) < O for t > t2 > tl. It follows from (2.2), (2.3), and g(t, £) < g(t, b) that

(_r(t)(_z'(t))a)' + (-r[r (t)](-z'[r (t)])7

+ 2l-I za [g(t, b)][ b Q(t, £ ) da (£ ) < O. (2.28)

We define the function u by

r(t)(-z'(t))a , x

u(t) := - ()a(r ).1)) , t > t2. (2.29)

za [n(t)]

Then u(t) < O. Noting that r(-Z)a is nondecreasing, we get rl/ (t)

z (s) < raOsjz (t), s >t > t2.

Integrating this inequality from n(t) to l, we obtain

Cl ds z(l) < z[n(i)} + rl/a(t)z'(t) /n(t)

Letting l —^ to, we have

O < z[n(t)]+ rlla(t)Z(t)S(t).

That is,

-s<«^TTST <

z[n(t)] Thus, we get by (2.29)

-Sa(t)u(t) < 1. (2.30)

Similarly, we define another function v by r[T (t)](-z'[T (t)])a

v(t) :=--^-, t > t2. (1)

za [n(t)]

Then v(t) < 0. Noting that r(-z')a is nondecreasing and t(t) < t, we get

r(t)(-z'(t))a > r[T(t)](-z'[T(t)]f. Thus, 0 < -v(t) < -u(t). Hence, by (2.30), we see that

-Sa(t)v(t) < 1. (2.32)

Differentiating (2.29), we obtain

_ (-r(t)(-z'(t))a)'za[n(t)] + ar(t)(-z(t)Yza-1 [n(t)]z'fo(t)]n'(t) U (t)= z2a [n(t)] .

By (2.3) and n(t) > t, we have ¿[q(t)] < (r(t)/r[n(t)])1/az'(t), and so

(-r(t)(-z'(t))a) j(t) ( Wa+D/a

U (t) <-rm--a Mr , M (-u(tn . (2.33)

za [n(t)] r1/a [n(t)]v y

Similarly, we see that

(-r[T(t)](-^[T(t)])a) n\t) ( /A)(a+1)/a

v (t) <-zaW]--. (Z34)

Combining (2.33) and (2.34), we get

Poa ^ (-r(t)(-z'(t)T)' poa (-r[T(t)](-z'[T(t)])a)' u (t) +-v (t) <-——-+

to w- za n(t)] to za r(t)]

n'(t) ( „OV^D/a aP0a (t) ^/.^(a+D/a /9 orN (-u(t)) - ,A1(-V(t^ . (2.35)

r1/an(t)]v ' to r1/a[n(t)]

Using (2.28), (2.35), andg(t, b) < n(t), we obtain

Poa IgQ(t,t)da(g) n'(t) (

u (t)+—v (t) < —2o-i—a riant)] (-u(t))

- —-nr^(-v(t))(a+1)/a. (2.36)

to r1/a[r(t)]

Multiplying (2.36) by Sa (t) and integrating the resulting inequality from t2 to t, we have

u(t)Sa(t) -u(t2)Sa(t2) + a I -, , ,,— ds + a

t2 r1/a[n(s)]

n'(s)Sa (s) t2 r1/a[n(s)]

(-u(s))

(a +1)/a

P0" (As a,A PO" u\sa ap0a /t 5a 1(s)^'(s)v(s)

+ -v(t)5a(t)--v(t2)Sa(t2)+ - -1/ar , -

To To To Jt2 r1/a [n(s)]

apo" ff n'(s)S" (s) ..,„^(a+1)/a To Jt2 r1/a[n(s)]

(-v(s)f

A ltf:Q(s, f )da (f „

ds +/ —---Sa (s)ds < o.

t 2 a-1 _

\n'(t)sa (t)l ( +1)/ u(t) and

_ r1/a[n(t)] _

a Sa-1(t)n'(t) r n'(t)sa (t) 1 -a/(a+1)-

a + 1 r1/a[n(t)] _r1/a[n(t)] _

Using inequality (2.14), we get

8a-1(tW(t)u(t) + n'(t)Sa (t) r1/a[n(t)] r1/a[n(t)]

Similarly, we set

" n'(t)Sa (t) l(a+1)/a

(-u(t))

( +1)/

a V a + 1

5(t)r1/a[n(t)]

i/ r / m v(t) and

r1/a[n(t)]J

a 5a-1(t)n'(t) r n'(t)Sa(t)1-a /(a+1V

_a + 1 r1/a[n(t)] Then we have by (2.14)

8a-1(tW(t)v(t) n'(t)Sa (t)

_r1/a[n(t)] _

(-v(t))

( +1)/

r1/a[n(t)] r1/a[n(t)] Thus, from (2.3o) and (2.32), we find

tt Jt2

XfQi^Kf) ^ , poa

a V a + 1

5(t)r1/a[n(t)]

S(s)r1/a[n(s)]_

< u(t2)Sa (t2) + — V(t2)5a (t2) + 1 +

which contradicts (2.27). This completes the proof.

With a proof similar to the proof ofTheorems 2.4 and 2.5, we obtain the following result.

Theorem 2.6 Suppose we have (H1)-(H5), (1.2), (1.4), and let t(t) < t, g(t, a) > t(t) for t e I, and g(t, f) < g(t, b) for f e [a, b]. Assume also that there exists a real-valued function p e C1(I,(o,to)) such that (2.26) is satisfied. If there exists a real-valued function n e C1(I, R) such that n(t) > t, n(t) > g (t, b), n'(t) >o fort e I, and (2.27) holds, then (1.1) is oscillatory.

3 Applications and discussion

In this section, we provide three examples to illustrate the main results. Example 3.1 Consider the second-order neutral functional differential equation

[x(t)+x(t -2n)]" + x[t + g]dg = 0, t > 10. (3.1)

Let a = 1, a = -5n/2, b = n/2, r(t) = 1,p(t) = 1, t(t) = t-2n, q(t, g) = 1,g(t, g) = t + g, a(g) = g, andp(t) = 1. Then Q(t,g) = min{q(t,g),q(T(t),g)} = 1,g'(t,a) = 1,g(t,a) = t-5n/2 < t + g for g e [-5n/2, n/2], and g(t, a) < t (t) < t. Moreover, letting to = 1, then

limsup I

t—TO Jto

p (s) fOb Q(s, g) da (g)

1 ^ + P0a \ r[g(s, a)](p+ (s))a+1

(a + 1)a+^ to J (p(s)g'(s, a))

= 3n limsuW ds = to.

t—J10

Hence, by Theorem 2.2, (3.1) is oscillatory. As a matter of fact, one such solution is x(t) = sin t.

Example 3.2 Consider the second-order neutral functional differential equation "1 g-

a r g + 1

[x(t) + tx(t - P^ + --x[t + g] dg =0, t > 1,

where P > 0 isa constant. Let a = 1, a = 0, b = 1, r(t) = 1,p(t) = t, t(t) = t - P, q(t, g) = (g + 1)/t, g(t, g) = t + g, a(g) = g, and p(t) = 1. Then Q(t, g) = min{q(t, g), q(T(t), g)} = (g + 1)/t, g(t, a) =g(t, 0) = t < t + g for g e [0,1], t (t) = t - P < t, andg(t, a) > t (t) for t > 1. Further, setting h(t) = t + 1,

0(t) =

ap'[h(t)]h'(t) t "(t) 1

p[h(t)]

t'(t) t +1'

f (t) = PP+t)+ ^(t) = -+-, p (t) t +1

m /P+ (t)Y+L pa [h(t)](f+(t))a+1 1

^(t)H p+t)] + —t^— = t+t

Therefore, we have

limsuW p (s)

t—TO ,/t0

fa Q(s, g) da (g) r[T (s)]^(s)

(a + 1)a+1(T '(s))a

= limsup

t—TO J1 \_Jo s

tr r1 g + 1 .. 1

■dg -

4(s +1) _

ds = limsup

t—to j1

_2s 4(s + 1)_

ds = to.

Hence, (3.2) is oscillatory due to Theorem 2.3.

Example 3.3 Consider the second-order neutral functional differential equation

[t2(x(t)+p(t)x(t - p))']' + f (g + 1)x[t + g]dg = 0, t > 1, (3.3)

where 0 < p(t) < p0, p0 and p are positive constants. Let a = 1, a = 0, b = 1, r(t) = t2, r (t) = t - p, q(t, g )=g + 1, g(t, g ) = t + g, a (g ) = g, p (t) = 1, and n(t) = t + 1. Then Q(t, g) = min{q(t, g), q(r (t), g)} = g +1, r0 = 1,g(t, a) = g(t, 0) = t < t + g for g e [0,1], r (t) = t- p < t, g(t,a) > r(t) for t > 1, and 5(t) = 1/t. Further,

limsup I

t—TO Jto

p(s)/0bQ(s,g)da(g) 1 ^ + p0a\ r[r(s)](p+ (s))a+1

2a-1 (a + 1)a+1V r^ (r0p (s))a

= - limsup / ds = to

2 t—J1

limsup I

t—TO Jt0

^/abQ(s, g) da(g) ^a(s)/1 + a X+1 n'(s)

2a-1 \ r0 J\a + V 5(s)r1/a[n(s)]_

3 1+p^V ft ds

- | limsup i —s_ ) t—TO h s + 1

= to, ifp0 < 5.

,2 4 ) t—

Hence, by Theorem 2.6, (3.3) is oscillatory when 0 < p(t) < p0 < 5.

Remark 3.1 In this paper, we establish some new oscillation theorems for (1.1) in the case where p is finite or infinite on I. The criteria obtained extend the results in [22] and improve those reported in [19]. Similar results can be presented under the assumption that 0 < a < 1. In this case, using [5, Lemma 2], one has to replace Q(t, g) := min{q(t, g), q(r(t), g)} with Q(t,g) := 2a-1 min{q(t, g), q(r(t), g)} and proceed as above. It would be interesting to find another method to investigate (1.1) in the case where g(r(t), g) ^ r g(t, g)].

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Allauthors contributed equally to this work. They allread and approved the finalversion of the manuscript. Author details

'Qingdao TechnologicalUniversity, Feixian, Shandong 273400, P.R. China. 2Department of Mathematics, Faculty of ElectricalEngineering and Informatics, TechnicalUniversity of Kosice, Letna 9, Kosice, 042 00, Slovakia.

Acknowledgements

The authors express their sincere gratitude to the anonymous referees for the carefulreading of the originalmanuscript and usefulcomments that helped to improve the presentation of the results and accentuate important details.

Received: 16 January 2014 Accepted: 6 March 2014 Published: 24 March 2014

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doi:10.1186/1687-2770-2014-68

Cite this article as: Li et al.: Oscillatory behavior of second-order nonlinear neutral differential equations with distributed deviating arguments. Boundary Value Problems 2014 2014:68.