## On the asymptotic behavior of certain second order nonlinear differential equations.(English)Zbl 0668.34056

Consider the ODE(*): $$(py')'+qy=g(t,y,y')$$, where $$p=p(t)$$ is a continuously differentiable positive function on $$I_ a=[a,\infty)$$ such that $$p(a)=1$$, $$q=q(t)$$ is a continuous function on $$I_ a$$ and $$g=g(t,u,v)$$ is a continuous function on $$I_ a\times R^ 2$$. Let $$z_ 1$$, $$z_ 2$$ be a fundamental system of solutions of $$(pz')'+qz=0$$. A function $$\omega: [0,\infty)\to [0,\infty),$$ $$\omega\in H$$ means that it is continuous and nondecreasing for $$u\geq 0$$ and positive for $$u>0$$, and there exists a continuous function r (multiplier function) on [0,$$\infty)$$ such that $$\omega (\alpha u)=r(\alpha)\omega (u)$$ for $$\alpha >0$$, $$u>0$$. For $$\omega_ i\in H$$, $$i=1,2$$, we write $$\omega_ 1\propto \omega_ 2$$ if $$\omega_ 2/\omega_ 1$$ is nondecreasing on (0,$$\infty)$$. Let $$W_ i(u)=\int^{u}_{u_ i}ds/\omega_ i(s),$$ $$u>0$$, $$i=1,2$$, and $$W_ i^{-1}$$ are their inverse functions. The main result: Theorem. Suppose the following hypotheses: $$(i)\quad | g(t,u,v)| \leq \lambda_ 1(t)\omega_ 1(u)+\lambda_ 2(t)\omega_ 2(v);$$ (ii) $$\omega_ i\in H$$, with corresponding multiplier functions $$r_ i$$, $$i=1,2$$, and $$\omega_ 1\propto \omega_ 2$$ or $$\omega_ 2\propto \omega_ 1$$; (iii) $$\lambda_ i$$ $$(i=1,2)$$ are continuous and nonnegative functions on $$I_ a$$ such that $$\lambda_ i\tilde z_ i\in L_ i(I_ a)$$, $$i=1,2$$, where $$\tilde z_ 1=(| z_ 1| +| z_ 2|)r_ 1(| z_ 1| +| z_ 2|),$$ $$\tilde z_ 2=(| z_ 1| +| z_ 2|)r_ 2(| z_ 1'| +| z_ 2'|);$$ (iv) there exists a constant $$c>0$$ such that $$\alpha_ i:=\int^{\infty}_{0}\lambda_ i\tilde z_ i(s)ds\leq \int^{\infty}_{\phi (c)}\frac{ds}{\omega_ i(s)},$$ $$i=1,2$$ (the inequality is strict for $$i=j)$$, where $$\phi (u)=W_ j^{-1}(W_ j(u)+\alpha_ j)$$ if $$\omega_ k\propto \omega_ j$$; $$k=1,2$$. Then any solution y of (*) such that $$| y(a)| +| y'(a)| \leq c$$ is defined over all of $$I_ a$$ and satisfies: $$y=(\delta_ 1+o(1))z_ 1+(\delta_ 2+o(1))z_ 2,$$ $$y'(\delta_ 1+o(1))z_ 1'+(\delta_ 2+o(1))z_ 2',$$ where $$\delta_ i$$, $$i=1,2$$ are constants.
Reviewer: Chungyou He

### MSC:

 34E05 Asymptotic expansions of solutions to ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems

### Keywords:

multiplier function
Full Text:

### References:

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