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On the correctness of nonlinear boundary value problems for systems of generalized ordinary differential equations. (English) Zbl 0876.34021
The concept of a strongly isolated solution of the nonlinear boundary value problem \[ dx(t)= dA(t)f(t,x(t)), \qquad h(x)=0, \] is introduced, where \(A:[a,b]\to \mathbb{R}^{n\times n}\) is a matrix-function of bounded variation, \(f:[a,b]\times \mathbb{R}^n\to\mathbb{R}^n\) is a vector-function belonging to a Carathéodory class \(K([a,b]\times \mathbb{R}^n;\mathbb{R}^n)\), and \(h\) is a continuous operator from the space of \(n\)-dimensional vector-functions of bounded variations \(BV_s([a,b], \mathbb{R}^n)\) into \(\mathbb{R}^n\). It is stated that the problems with strongly isolated solutions are correct. Sufficient conditions for the correctness of these problems are given. As an application for the system of ordinary differential equations \[ \frac{dx(t)}{dt}= f(t,x(t)) \] the author considers the boundary value problem \(h(x)=0\), where \(f\in K([a,b]\times \mathbb{R}^n;\mathbb{R}^n)\), and \(h:BV_\rho ([a,b],\mathbb{R}^n)\to \mathbb{R}^n\) is a continuous operator.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
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