Lyapunov-type integral inequalities for certain higher order differential equations

Let q(t) be a T-periodic potential such that ∫ T 0 q(t) dt < 0. The classical Lyapunov criterion to stability of Hill's equation -x + q(t)x = 0 is ∥q-∥ 1 = ∫ T 0 |q- (t)|dt < 4/T, where q- is the negative part of q. In this paper, we will use a relation between the (anti-)periodic and the Dirichlet eigenvalues to establish some lower bounds for the first anti-periodic eigenvalue. As a result, we will find the best Lyapunov-type stability criterion using L α norms of q-, 1 < α < ∞. The numerical simulation to Mathieu's equation shows that the new criterion approximates the first stability region very well.

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A Lyapunov-type stability criterion using ^{} norms

For nearly 50 years Lyapunov inequalities have been an important tool in the study of differential equations. In this survey, building on an excellent 1991 historical survey by Cheng, we sketch some new developments in the theory of Lyapunov inequalities and present some recent disconjugacy results relating to second and higher order differential equations as well as Hamiltonian systems.

Lyapunov Inequalities and their Applications

In this work, first we give a survey of the most basic results on Lyapunov-type inequality, and next we sketch some recent developments related to this type of inequalities.

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Recent Developments of Lyapunov-type Inequalities

On inequalities of Lyapunov type

For a second-order differential equation, we obtain from Opial’s inequality lower bounds for the spacing between two zeros of a solution or between a zero of a solution and a zero of its derivative. These bounds are expressed in terms of antiderivatives of the potential, and in particular we derive some new Liapunov type inequalities from them.

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D. Hinton

Papers

Opial’s inequality and oscillation of 2nd order equations