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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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##### References:
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