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Integral equivalence between a nonlinear system and its nonlinear perturbation. (English) Zbl 0598.34030
We consider differential systems of the forms \[ (a)\quad x'=A(t,x)+B(t,x)\quad (b)\quad y'=A(t,y), \] where x,y,A,B, are real- valued n-vectors, and the functions A(t,x),B(t,x) are defined and continuous on \(I\times R^ n\), \(I=<t_ 0,\infty)\), \(t_ 0\geq 0\), \(R^ n\) is the space of all real n-vectors. The problem we deal with in this paper is the integral equivalence of two systems (a) and (b). It is easy to see that if the two systems (a) and (b) are (1,p)-integrally equivalent and some solution y(t) of (b) is \(L^ p\)-bounded, then the corresponding solution x(t) of (a) is also \(L^ p\)-bounded, and conversely. On the other hand, two systems (a) and (b) may be (1,p)- integrally equivalent although no solution of either of them is \(L^ p\)- bounded.

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A34 Nonlinear ordinary differential equations and systems
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