The distribution of zeros of solutions for a class of third order differential equation

Clemente Cesarano,Tareq M Al-Shami,Mohammed A Arahet

doi:10.22199/issn.0717-6279-4002

Clemente Cesarano, Tareq M Al-Shami + Show 1 more

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https://doi.org/10.22199/issn.0717-6279-4002

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Abstract

For third order linear differential equations of the form r(t)x'(t)''+ p(t)x'(t) + q(t)x(t) = 0; we will establish lower bounds for the distance between zeros of a solution and/or its derivatives. The main results will be proved by making use of Hardyís inequality, some generalizations of Opialís inequality and Boydís inequality.

Highlights

The distribution of boundary conditions is the distribution of zeros of solutions of differential equations has been started by Picard [15, 16], who derived some uniqnueness results for solutions of the second order nonlinear differential equation with two-points boundary conditions when the nonlinear function satisfies the Lipschitz condition
The distribution of zeros of nth order differential equations with more than two points has been considered by Niccoletti [14]
The first work that was published by C. de la Vallee Poussin in 1929 was on the evaluation of the length of the interval [0, h] in which the boundary value problem y (t1) = y(t2) = ... = y = 0, (0 ≤ t1 < t2 < ... < tn ≤ h) for the linear differential equation (1.1) only admits the null solution

Summary

Introduction

The distribution of boundary conditions is the distribution of zeros of solutions of differential equations has been started by Picard [15, 16], who derived some uniqnueness results for solutions of the second order nonlinear differential equation with two-points boundary conditions when the nonlinear function satisfies the Lipschitz condition. The equation (1.1) is said to be (k, n − k)− disconjugate on an interval I if no nontrivial solution has a zero of order k followed by a zero of order n − k. This means that, for every pair of points α, β ∈ I, α < β, there does not exist a nontrivial solution of (1.1) which satisfies (1.2). The first work that was published by C. de la Vallee Poussin in 1929 was on the evaluation of the length of the interval [0, h] in which the boundary value problem y (t1) = y(t2) = ... Coming to the equation (1.1), C. de la Vallee Poussin proved that this equation is disconjugate in [a, b) , satisfies the inequality

Pj hj j

Rt α

ÃZ β

ÃZ β α

Cancelling the

Integrating by