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

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 $$I$$.

### 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

Zbl 0102.07801