Sturm-Picone type theorems for second-order nonlinear differential equations

Abstract

The aim of this article is to give Sturm-Picone type theorems for the pair of second-order nonlinear differential equations $$\displaylines{ (p_1(t)|x'|^{\alpha-1}x')'+q_1(t)f_1(x)=0 \cr (p_2(t)|y'|^{\alpha-1}y')'+q_2(t)f_2(y)=0,\quad t_1<t<t_2 }$$ in both regular and singular cases. Our results include some earlier results and generalize the well-known comparison theorems given by Sturm [19], Picone [18] and Leighton [5] which play a key role in the qualitative behaviour of the solutions.

Keywords

• Comparison theorem
• Sturm-Picone theorem
• half-linear
• second order differential equations
• singular equation
• variational lemma