\textit{I. Bihari} [Publ. Math. Inst. Hungar. Acad. Sci. 2, 159-172 (1958; Zbl 0089.068)] defined the half-linear second order differential equation (1) \((p(t)x')'+q(t)f(x,p(t)x')=0\) for the unknown function \(x=x(t)\) where the functions p(t), q(t) are continuous on some interval \(I=[a,b)\) \((- \infty 0\) and the function f(x,y) satisfies the following relations: (Bi) f(x,y) is defined on \(R^ 2\) and is Lipschitzian on every bounded domain in \(R^ 2\), (Bii) \(xf(x,y)>0\) if \(x\neq 0\) (consequently \(f(0,y)=0\) for all \(y\in R)\), (Biii) \(f(\lambda x,\lambda y)=\lambda f(x,y)\) for all \(\lambda\in R\), \((x,y)\in R^ 2\). The aim of our paper is to relax the restrictions (Bi)-(Biii) so that the set of equation (1) should cover also the differential equations like \[ (2)\quad (p^{1/n}x')'+\frac{1}{n}qp^{1/n-1}x^ n| x'|^{1- n}=0\quad if\quad x'\neq 0, \] (3) \(u''(\tau)+\mu u^+(\tau)-vu^- (\tau)=0\) with \(u^+=\max \{u,0\}\), \(u^-=\max \{-u,0\}\), \(\mu >0\), \(v>0\).