Levin’s comparison theorems for second order nonlinear differential equations and inequalities. (English) Zbl 0736.34021

Yeh, Cheh-Chih

Math. Japon. 36, No. 4, 703-710 (1991).

Comparison theorems of Levin’s type [A. Yu. Levin, Sov. Math., Dokl. 1, 1313–1316 (1961); translation from Dokl. Akad. Nauk SSSR 135, 783–786 (1960; Zbl 0102.07801)] are proved for \(C^2\)-functions \(u\), \(v\) satisfying
\[ -[A(t)g(u(t))u'(t)]'\geq B(t)f(u(t)), \quad - [a(t)g(v(t))v'(t)]'=b(t)f(v(t)), \]
respectively, on a closed interval \(I=[\alpha,\beta]\), where \(a(t)\geq A(t)\) and \(B\) majorizes \(b\) on \(I\) in a technical sense. If \(u(t)>0\) on \(I\), \(v(\alpha)\geq u(\alpha)\), and \(a\), \(A\), \(b\), \(B\), \(f\), \(g\) satisfy appropriate conditions, a typical theorem states that \(u'(t)<0\), \(v(t)\geq u(t)\), and
\[ -A(t)g(u(t))u'(t)/f(u(t)) > | a(t)g(v(t))v'(t)/f(v(t))|\quad\text{for all }t\in I. \]
Additional theorems apply to the cases (i) \(v(\beta)\geq u(\beta)\); (ii) \([\alpha,\infty)\) replaces \([\alpha,\beta]\); and (iii) \(u(t)<0\) on \(I\).

Reviewer: Charles A. Swanson (Vancouver)

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems
34A40	Differential inequalities involving functions of a single real variable

Keywords:

second order nonlinear differential equation; second order nonlinear differential inequalities; comparison theorems of Levin’s type

Citations:

Zbl 0102.07801