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

Medina, Rigoberto; Pinto, Manuel

J. Math. Anal. Appl. 135, No. 2, 399-405 (1988).

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

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