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 (p1(t)x'(t))' + q1(t)f1(x)(t)) = 0. (p2(t)y'(t))' + q2(t)f2(y(t)) = 0, t1 < t < t2. 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.