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

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:

 3.4e+11 Perturbations, asymptotics of solutions to ordinary differential equations

### Keywords:

asymptotic equivalence; nonoscillatory equation
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### References:

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