Annali di Matematica Pura ed Applicata

, Volume 106, Issue 1, pp 273–291

# Distance between beros of certain differential equations having delayed arguments

• Stanley B. Eliason
Article

## Summary

For the continuous real valued functions, p, m and g, with p(x)≥0, and m(x)≥0, and μ>0, v>0 being reals, the differential equations y″(x)+p(x)|y(x)|μsgn y(x)= =m(x)|y(g(x))|vsgn y(g(x)) is considered. Lyapunov type integral inequalities are established which yield implicit lower bounds on the distance between consecutive zeros of a nontrivial solution of the above equation, and several others. The same is done for a problem involving the distance from a zero of a solution y to the next greater zero of its derivative y′. Special conditions are placed on the corresponding initial functions. They allow for application of results to oscillatory solutions of the given equation, and also to non-trivial solutions having a zero initial function. When p(x) ≡ 0. the results take on a special form; and when in addition m(x)>0, g(x)<x and v=1, one result establishes a necessary condition for the existence of an oscillatory solution having infinitely many small semicycles. This condition is the weak form of a strict integral inequality, due to G. Ladas et al., which establishes a sufficient condition for the oscillation of all bounded solutions.

## Keywords

Differential Equation Lower Bound Special Form Weak Form Nontrivial Solution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
A. Èl’bert,A generalization of a certain inequality of Ljapunov to differential equations with retarded argument, (Russian), Ann. Univ. Sci. Budapest, Eötvös Sect. Math.,12 (1968), pp. 107–111.Google Scholar
2. [2]
S. B. Eliason,Lyapunov type inequalities for certain second order functional differential equations, SIAM J. Appl. Math.,27 (1974), pp. 180–199.
3. [3]
G. B. Gustafson,Bounded oscillations of linear and nonlinear delay-differential equations of even order, J. Math. Anal. Appl.,46 (1974), pp. 175–189.
4. [4]
G. Ladas, G. S. Ladde -J. S. Papadakis,Oscillations of functional-differential equations generated by delays, J. Diff. Eq.,12 (1972), pp. 385–395.
5. [5]
G. Ladas -V. Lakshmikantham -J. S. Papadakis,Oscillations of higher-order retarded differential equations benerated by the retarded argument, inDelay and Functional Differential Equations and Their Applications, Academic Press, K. Schmitt ed., New York, 1972, pp. 219–231.Google Scholar
6. [6]
G. S. Ladde,Oscillations of nonlinear functional differential equations generated by retarded actions, Dept. of Math., Univ. Rhode Island, Kingston, R.I., Academic Press, K. Schmitt ed., New York, 1972, pp. 355–365.Google Scholar
7. [7]
A. D. Myskis,Linear Differential Equations with Retarded Arguments, (German), Deutscher Verlag Der Wissenschaften, Berlin, 1955.Google Scholar
8. [8]
S. B. Norkin,Differential Equations of the Second Order with Retarded Argument, Vol.31, Translations of Mathematical Monographs, Amer. Math. Soc., Providence, R.I., 1972.Google Scholar