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Well-posedness of the Cauchy-Nicoletti boundary value problem for systems of nonlinear generalized ordinary differential equations. (English. Russian original) Zbl 0853.34018

Differ. Equations 31, No. 3, 352-362 (1995); translation from Differ. Uravn. 31, No. 3, 382-392 (1995).
The paper is concerned with the Cauchy-Nicoletti boundary value problem for the generalized ordinary differential equation \(dx(t) = dA(t) \cdot f(t,x(t))\), \(t \in [a,b]\), \(x \in\mathbb{R}^n\), \(x_i (t_i) = C_i\) \((i = 1, \dots, n)\), where \(A\) is a matrix function of bounded variation, and \(f\) is a vector function belonging to the Carathéodory class. Moreover, the sequence of auxiliary problems \(dx(t) = dA_k (t) \cdot f_k (t,x (t))\), \(x_i (t_{ik}) = C_{ik}\) \((k = 1, 2, \dots)\) is considered. The author finds sufficient conditions providing the solvability of the Cauchy-Nicoletti problem and the convergence, as \(k \to \infty\), of solutions of these auxiliary problems to the original one. Additionally, a method of successive approximations is proposed to reduce the solution of the original Cauchy-Nicoletti problem to the sequence of the related Cauchy problems.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
65J99 Numerical analysis in abstract spaces
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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