# Distance between beros of certain differential equations having delayed arguments

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## 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)|^{μ}*sgn* y(x)= =m(x)|y(g(x))|^{v}*sgn* 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## References

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