GENERALIZATIONS OF STURM-PICONE THEOREM FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS

Abstract

The goal of this paper is to show a generalization to Sturm--Picone theorem for a pair of second-order nonlinear differential equations \begin{gather*} (p_1(t) x'(t))' + q_1(t) f_{1}(x(t)) = 0. \\ (p_2(t) y'(t))' + q_2(t) f_{2}(y(t)) = 0, \; t_1 \lt t \lt t_2. \end{gather*} This work generalizes well-known comparison theorems [C. Sturm, J. Math. Pures. Appl. 1 (1836), 106--186; M. Picone, Ann. Scoula Norm. Sup. Pisa, 11 (1909),39; W. Leighton, Proc. Amer. Math. Soc.13 (1962), 603--610], which play a key role in the qualitative behavior of solutions. We establish the generalization to a pair of nonlinear singular differential equations and elliptic partial differential equations also. We show generalization via the quadratic functionals associated to the above pair of equations. The celebrated Sturm--Picone theorem for a pair of linear differential equations turns out to be a particular case of our result. We also use these comparison theorems to ensure the oscillatory as well as nonoscillatory behavior of solutions for a class of nonlinear equations.