Distribution of zeros of solutions of functional differential equations

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Abstract

By improving the bracketing techniques for a functional differential inequality in a previous paper [H.W. Wu, Y.T Xu, The distribution of zeros of solutions of neutral differential equations, Appl. Math. Comput. 156 (3) (2004) 665–677], we are able to derive sharper upper bounds on the distance between zeros of solutions of a class of neutral functional differential equations with delays, and hence improve many known bounds in the literature.

Introduction

The distribution of zeros of solutions of differential equations has long been an important subject of investigation, see, e.g., [1], [2], [3], [4], [5], [6] and the references cited therein. Indeed, such information together with others will yield vital information about the properties of the real systems modelled by the corresponding differential equations.

In [6], the authors investigate the distribution of zeros of the solutions of neutral differential equations of the form:[x(t)+P(t)x(g(t))]+Q(t)x(h(t))=0,tt0,where P,hC([t0,),[0,)), Q,gC([t0,),(0,)), g(t) and h(t) are nondecreasing, and limtg(t)=limth(t)=. This is accomplished by finding bracketing intervals for roots of the solutions of a companion functional inequality of the form:x(t)+P(t)x(τ(t))0,tt0,where P,τC([t0,),[0,)), τ(t)t, τ(t) is nondecreasing, and limtτ(t)=. The techniques in [6], however, can further be generalized. Since the results obtained in [6] generalize and improve many known ones in the literature [1], [2], [3], [4], [5], it is of interest to explain how this can be done. Some of the details, specially those that can be transplanted from [6], will be sketched or omitted.

Section snippets

Bracketing interval for (2)

We will need the iterates of the inverses of the functions τ, g and h. We use the notation τ0(t)=t and inductively define the iterates of τ-1 by:τ-i(t)=τ-1(τ-(i-1)(t)),i=1,2,.The iterates g-i and h-i are similarly defined. As in [6], we let {fn}n=0 be a sequence of functions defined by f0(ρ)=1, f1(ρ)=1/(1-ρ) andfn+2(ρ)=fn(ρ)(fn(ρ)+1-eρfn(ρ)),n=0,1,2,,where ρ(0,1). The properties of {fn} have been discussed in [6]. Roughly, if ρ>1/e, then either fn(ρ) is nondecreasing and limnfn(ρ)= or fn(

First order neutral differential equation

Once the above set up is accomplished, we may easily modify the main results in [6] which provide upper bound on the distance between zeros of Eq. (1). Indeed, as in [6], we first make a list of assumptions on the function R(t)=P(h(t))Q(t)/Q(G(t)), where G(t)=h-1(g(h(t))).

  • (H1)

    h(t)g(t)t,R(t)C([t0,),[0,)], and G(t)1, when R(t)0, or R(t)-(G(t)-1)Q(t)0, when R(t)>0.

  • (H2)

    g-1(h(t))tQ(s)/(1+R(g-1(h(s))))dsρ,ρ>1e,tt1.

  • (H3)

    g-1(h(t))tQ(s)/(1+R(g-1(h(s))))dsρ,0<ρ1e,tt1.

Then by slightly modifying the

Remarks and examples

Let us consider the following examples for illustrating our results, which show that the conclusions do not appear to follow from the known oscillation criteria in the literature [1], [2], [3], [4], [5], [6].

Example 4.1

Consider the delay differential equationx(t)+1+2t5tx(t-1)=0,where P(t)=0, g(t)=t, h(t)=t-1, Q(t)=1+2t5t, R(t)=0. We haveτ(t)tP(s)ds=t-1t1+2s5sds=15lntt-1+25.It is easy to see ρ=μ=25, by Theorem 3.1 in [6] we haveφ1=7.5000,φ3=5.3571,φ4=5.1667,φ5=5.0806f1=1.6667,f7=4.1839,f8=5.8467,f

Acknowledgements

This project was supported by the NNSF of China (Nos. 10571183, 10471155, and 10501014), the NSF of South China University of Technology, the Foundation of Zhongshan University Advanced Research Centre (No. 05M12), the Doctoral Programme Foundation of Ministry of Education of China (No. 20020558092) and the NSF of Guangdong Province of China (No. 031608).

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