Asymptotic equivalence of second order differential equations. (English) Zbl 0555.34048

Kuben, Jaromír

Czech. Math. J. 34(109), 189-202 (1984).

The asymptotic properties of the equation (1) \([p(t)x']'+q(t)x=0\) and the perturbed equation (2) \([p(t)y']'+q(t)y=f(t,y,y')\) are compared. It is supposed that \(p,q\in C^ 0(j)\), \(p>0\) on j, \(f\in C^ 0(j\times R^ 2)\), \(j=[t_ 0,+\infty)\). Let \(\mu_ 0\), \(\mu_ 1\in C^ 0(j)\). The equations (1) and (2) are called \(\mu_ 0\)-asymptotically equivalent \((\mu_ 0\)-a.e.) if for each solution x(t) of (1) there exists a solution y(t) of (2) defined for large t such that \(x(t)\)-y(t)\(=o[\mu_ 0(t)]\), \(t\to +\infty\), and conversely. Analogously the weak \(\mu_ 1\)- a.e. \((x'\)-y’\(=o(\mu_ 1))\) and the strong \((\mu_ 0,\mu_ 1)\)-a.e. \((x\)-y\(=o(\mu_ 0)\), \(x'\)-y’\(=o(\mu_ 1))\) is defined. It is assumed that there exists a nonnegative function F(t,r,s), \(F\in C^ 0(j\times R^ 2_+)\), nondecreasing in r and s for each fixed \(t\in j\) such that \(| f(t,r,s)| \leq F(t,| r|,| s|)\). Then general theorems on a.e. are proved using Tikhonov’s fixed point theorem. Also some special perturbations are considered as \(F=h(t)| r|\), \(F=k(t)| s|\) and \(F=a(t)g(| r|)\), \(F=b(t)h(| s|)\). In the last part the more detailed results on a.e. are obtained for nonoscillatory equation (1). Here the concept of the principal solution is used. A large list of references is contained.

MSC:

34E10

Perturbations, asymptotics of solutions to ordinary differential equations

Keywords:

asymptotic equivalence; nonoscillatory equation

References:

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