## On the half-linear second order differential equations.(English)Zbl 0656.34008

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 $$I=[a,b)$$ $$(- \infty <a<b\leq \infty)$$, $$p(t)>0$$ 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 $$u^+=\max \{u,0\}$$, $$u^-=\max \{-u,0\}$$, $$\mu >0$$, $$v>0$$.

### MSC:

 34A34 Nonlinear ordinary differential equations and systems

### Keywords:

half-linear second order differential equation

Zbl 0089.068
Full Text:

### References:

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