Generalizations of sturm-picone theorem for second-order nonlinear differential equations

The goal of this paper is to show a generalization to Sturm-Picone theorem for a pair of second-order nonlinear differential equations (p<inf>1</inf>(t)x '(t))' + q1(t)f1(x(t)) = 0. (p<inf>2</inf>(t)y '(t))' + q<inf>2</inf>(t)f<inf>2</inf>(y(t)) = 0, t<inf>1</inf> < t < t<inf>2</inf>. This work generalizes well-known comparison theorems [C. Sturm, J. Math. Pu res.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.