## 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:

 [1] Coppel, W, Stability and asymptotic behavior of differential equations, (1965), Heath Boston · Zbl 0154.09301 [2] Dannan, F, Integral inequalities of Gronwall-Bellman-bihari type and asymptotic behavior of certain second order nonlinear differential equations, J. math. anal. appl., 108, 151-164, (1985) · Zbl 0586.26008 [3] Pinto, M, Asymptotic integration of second order linear differential equations, J. math. anal. appl., 111, 388-406, (1985) · Zbl 0591.34034 [4] Pinto, M, Nonlinear integral inequalities and their application to the study of the asymptotic behavior of nonlinear differential equations, (), 109-123 [5] Trench, W, On the asymptotic behavior of solutions of second order linear differential equations, (), 12-14 · Zbl 0116.29304 [6] Eastham, M.S.P, Asymptotic formulae of Liouville-Green type for higher-order differential equations, J. London math. soc. (2), 28, 507-518, (1983) · Zbl 0532.34038 [7] Pinto, M, Des inégalités fonctionnelles et leurs applications, (), 142, 364/TS-06
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.