Oscillation criteria for second-order neutral equations with distributed deviating argument

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Abstract

We present new oscillation criteria for the second-order neutral delay differential equations with distributed deviating argument(r(t)ψ(x(t))Z(t))+abp(t,ξ)f[x(g(t,ξ))]dσ(ξ)=0,tt0,where Z(t)=x(t)+q(t)x(t-τ). New oscillation criteria are established, which are based on a class of new functions Φ(t,s,l) defined in the sequel. Our results are sharper than some previous results which can be seen by the example at the end of this paper.

Introduction

In this paper, we are concerned with the oscillation problem for the second-order neutral delay differential equations with distributed deviating argument(r(t)ψ(x(t))Z(t))+abp(t,ξ)f[x(g(t,ξ))]dσ(ξ)=0,tt0,where Z(t)=x(t)+q(t)x(t-τ),τ0. In this paper, we assume that

  • (A1)

    r,qC(I,R) and 0q(t)1,r(t)>0 for tI,1r(s)ds=,I=[t0,);

  • (A2)

    ψC1(R,R),ψ(x)>0 for x0;

  • (A3)

    f(R,R),xf(x)>0 for x0;

  • (A4)

    pC(I×[a,b],[0,)) and p(t,ξ) is not eventually zero on any half linear [tu,)×[a,b],tut(0);

  • (A5)

    gC(I×[a,b],[0,)),g(t,ξ)t for ξ[a,b],g(t,ξ) has a continuous and positive partial derivative on I×[a,b] with respect to the first variable t and nondecreasing with respect to the second variable ξ, respectively, and liminftg(t,ξ)= for ξ[a,b];

  • (A6)

    σC([a,b],R) is nondecreasing, and the integral of Eq. (1.1) is in the sense of Riemann–Stieltijes.

We restrict our attention to those solutions x(t) of Eq. (1.1) which exist on some half linear [tx,) and satisfy sup{|x(t)|:ttx}0 for any Tt0. As usual, such a solution of Eq. (1.1) is called oscillatory if the set of its zeros is unbounded from above, otherwise, it is said to be nonoscillatory. Eq. (1.1) is called oscillatory if all its solutions are oscillatory.

In the last decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions for different classes of second-order differential equations [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].

The oscillation problem for nonlinear delay equation such as[r(t)x(t)]+q(t)f(x(σ(t)))=0,t>t0as well as for the linear ordinary differential equation[r(t)x(t)]+p(t)x(t)+q(t)x(t)=0,t>t0and the neutral delay differential equation(x(t)+q(t)x(t-σ))+p(t)x(t-τ)=0have been studied by many authors with different methods. Some results can be found in [1], [2], [3], [4], [5], [6], [7], [8], [10], [14] and references therein. Recently, In [13], by using Riccati technique and averaging functions method, Ruan established some general oscillation criteria for second-order neutral delay differential equation[r(t)(x(t)+q(t)x(t-σ)]+p(t)f(x(t-τ))=0.Sahiner [11] obtained some general oscillation criteria for neutral delay differential equations(r(t)ψ(x(t))Z(t))+p(t)f(σ(t))=0is oscillatory. In [9], by using Riccati technique and averaging functions method, Wang established some general oscillation criteria for second-order neutral delay differential equation with distributed deviating argument(r(t)Z(t))+abp(t,ξ)x(g(t,ξ))dσ(ξ)=0.Very recently, Xu and Weng [12] also have established oscillation criteria for study Eq. (1.1), by using the generalized Riccati transformation.

In this paper, by using a generalized Riccati technique and the integral averaging technique and following the results of Wong [15] and Philos [16], we establish some oscillation criteria for Eq. (1.1), which complement and extend the results in [11], [12]. We will use the function class Y to study the oscillatory of Eq. (1.1), we say that a function Φ=Φ(t,s,l) belongs to the function class Y denoted by ΦY, if Φ(E,R), where E={(t,s,l):t0lst<}, which satisfies Φ(t,t,l)=0,Φ(t,l,l)=0,Φ(t,s,l)0 for l<s<t, and has the partial derivative Φ/s on E such that Φ/s is locally integrable with respect to sE.

We define the operator A[.;l,t] byA[θ;l,t]=ltΦ2(t,s,l)θ(s)dsfortslt0andθ(s)C[t0,)and the function ϕ=ϕ(t,s,l) is defined byΦ(t,s,l)s=ϕ(t,s,l)Φ(t,s,l).It is easy to verify that A[.;l,t] is a linear operator and satisfiesA[θ;l·t]=-2A[θϕ;l·t]forθ(s)C1[t0,).In addition, we will make use of the following conditions:

  • (S1)

    There exists a positive real number M such that |f(±uv)|Mf(u)f(v) for uv>0;

  • (S2)

    uψ(u)>0 for u0.

Lemma 1.1

If a>0,b0, then-ax2+bx-a2x2+b22a.

Section snippets

When f(x) is monotone

In this section, we shall deal with the oscillation for Eq. (1.1) under the assumptions (A1)(A6) and the following assumption.

(A7)f(x) exists, f(x)k1 and ψ(x)L-1 for x0.

Theorem 2.1

Let (S1),(A1)(A7) hold. Eq. (1.1) is oscillatory provided that for each lt0, there exist a function ΦY and ρ(t)C1([t0,),R) such thatlimtsupAρ(s)Q1(s)-2ρ(s)r[g(t,a)]k1Lg(t,a)ϕ2;l,t>0,whereQ1(t)=Mabp(t,ξ)f[1-q(g(t,ξ))]dσ(ξ)-(ρ(t))2)r(t)k1Lρ2(t)g(t,a)and the operator A is defined by (1.8) and the function ϕ=ϕ(t,s,l)

When f(x) is not monotone

In this section, we shall deal with the oscillation for Eq. (1.1) under the assumptions (A1)(A6) and the following assumption.

(A8)f(x)xk3 and ψ(x)L-1 for x0.

Theorem 3.1

Let (A1)(A6),(A8) and (S1)(S2) hold. Eq. (1.1) is oscillatory provided that for each lt0, there exists functions ΦY and ρ(t)C1([t0,),R) such thatlimsuptAρ(s)Q2(s)-2ρ(s)r[g(s,a)]Lg(s,a)ϕ2;l,t0,whereQ2(t)=k2ρ(t)abp(t,ξ){1-q(g(t,ξ))}dσ(ξ)-(ρ(t))2r(t)Lρ(t)g(t,a)and the operator T is defined by (1.8) and the function ϕ=ϕ(t,s,l)

Interval oscillation criteria

In this section, we will establish several interval oscillation criteria for Eq. (1.1).

Theorem 4.1

Let (S1),(A1)(A7) hold. Eq. (1.1) is oscillatory provided that for each Tt0, there exist a function ΦY, and two constants d>cT such thatAρ(s)Q1(s)-2ρ(s)r[g(s,a)]k1Lg(s,a)ϕ2;c,d>0,where the operator A is defined by (1.8) and the function ϕ=ϕ(d,s,c) is defined by (1.9).

Proof

With the proof of Theorem 2.1, when t and l are replaced by d and c, respectively, we easily see that every solution of Eq. (1.1) has

Examples

Example 5.1

Consider the nonlinear differential equation12(1+x2(t))x(t)+1t+2x(t-1)+01γ(t+ξ+2)t2(t+ξ+1)x(t+ξ)dξ=0,wherer(t)=1,ψ(x)=12(1+x2(t)),q(t)=1t+2,p(t)=γ(t+ξ+2)t2(t+ξ+1),γ>14,g(t,ξ)=t+ξ,f(x)=x.If we take L=k1=M=1,b=2,a=1,limtsup1t2m+1lt(t-s)2m(s-l)2γ2s2ds>γ2m(2m-1)(2m+1).Hence, by Corollary (2.1) we have Eq. (5.1) is oscillatory for γ>1/4.

Example 5.2

Consider the nonlinear differential equationsin11+x2(t)x(t)+1t+1x(t-1)+01m(t+ξ+1)(t+ξ)x(t+ξ)dξ=0,wherer(t)=1,ψ(x)=sin211+x2(t),q(t)=1t+1,p(t)=m(t+ξ+1)t2(t+ξ)

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This research was partial supported by the NNSF of China (10771118).

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