Taiwanese Journal of Mathematics

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

Jagmohan Tyagi

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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.

Article information

Source
Taiwanese J. Math., Volume 17, Number 1 (2013), 361-378.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499705893

Digital Object Identifier
doi:10.11650/tjm.17.2013.2074

Mathematical Reviews number (MathSciNet)
MR3028875

Zentralblatt MATH identifier
1271.34037

Subjects
Primary: 34C10: Oscillation theory, zeros, disconjugacy and comparison theory 35J15: Second-order elliptic equations
Secondary: 35B105 34C15: Nonlinear oscillations, coupled oscillators

Keywords
singular equation elliptic partial differential equations zeros comparison theorem oscillatory as well as nonoscillatory behavior

Citation

Tyagi, Jagmohan. GENERALIZATIONS OF STURM-PICONE THEOREM FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS. Taiwanese J. Math. 17 (2013), no. 1, 361--378. doi:10.11650/tjm.17.2013.2074. https://projecteuclid.org/euclid.twjm/1499705893


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