Distribution of zeros of solutions of functional differential equations
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:where , , g(t) and h(t) are nondecreasing, and . This is accomplished by finding bracketing intervals for roots of the solutions of a companion functional inequality of the form:where , , τ(t) is nondecreasing, and . 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 and inductively define the iterates of by:The iterates and are similarly defined. As in [6], we let be a sequence of functions defined by , andwhere . The properties of have been discussed in [6]. Roughly, if , then either is nondecreasing and or
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 , where .
- (H1)
, and , when , or , when .
- (H2)
.
- (H3)
.
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 equationwhere , , , , . We haveIt is easy to see , by Theorem 3.1 in [6] we have
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).
References (6)
The distribution of zeros of solution of first order delay differential equations
J. Math. Anal. Appl.
(1994)- et al.
The distribution of zeros of solutions of differential equations with a variable delay
J. Math. Anal. Appl
(2001) - et al.
The distribution of zeros of solutions of neutral differential equations
Appl. Math. Comput.
(2004)
Cited by (11)
The distribution of zeros of all solutions of first order neutral differential equations
2015, Applied Mathematics and ComputationOn the distance between consecutive zeros of solutions of first order delay differential equations
2013, Applied Mathematics and ComputationCitation Excerpt :The use of lower estimates for distance between zeros of solutions in the analysis of asymptotic properties of second order equations is presented in [12], and in exponential stability of the first order delay equation in [13]. In recent years, there has been an increasing interest in obtaining upper bounds for the distance between consecutive zeros of solutions of functional differential equations, see e.g. [14–23]. Theorem 2.6 improves the results of Theorem 2.4.
Upper bounds for the distances between adjacent zeros of solutions of delay differential equations
2011, Applied Mathematics and ComputationCitation Excerpt :In case p(t) ⩽ 0, it is known that a nonoscillatory solution of (1) is eventually monotonically increasing or eventually monotonically decreasing, and the distances between adjacent zeros of an oscillatory solution of (1) cannot exceed the delay τ. If p(t) > 0, some upper bounds have also been derived for the distances between adjacent zeros of oscillatory solutions of (1), see e.g. [4–10]. Motivated by the ideas in [3] and others in [4–10], in the present paper we shall consider p(t) that satisfies the condition p(t) ⩾ p > 0 on each interval of the form [t0 − (n + 3)τ, T] where T ⩾ t0 (and n is a nonnegative integer which will be specified later).
On the distribution of zeros of solutions of first order delay differential equations
2011, Nonlinear Analysis, Theory, Methods and ApplicationsNew results for the upper bounds of the distance between adjacent zeros of first-order differential equations with several variable delays
2023, Journal of Inequalities and ApplicationsOn the distribution of adjacent zeros of solutions to first-order neutral differential equations
2023, Turkish Journal of Mathematics