Geodesic
Bounded solutions of families of systems of differential equations and their approximation
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 5, pp. 29-47

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In the paper, we consider the problem of finding a bounded solution of a one-parametric family of systems of ordinary differential equations. Using the parametrization method, we prove necessary and sufficient conditions for the existence of a unique solution of the problem considered that is bounded on the whole axis in terms of a two-sided infinite block-band matrix composed with respect to integrals over intervals of length $h>0$ the matrix of the system of differential equations. Also, we construct a family of two-point boundary-value problems on a finite interval that approximate the problem of finding the bounded solution and finds an interconnection between the correct solvability of the initial and approximating problems. 
@article{FPM_2006_12_5_a3,
     author = {D. S. Dzhumabaev},
     title = {Bounded solutions of families of systems of differential equations and their approximation},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {29--47},
     publisher = {mathdoc},
     volume = {12},
     number = {5},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_5_a3/}
}
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D. S. Dzhumabaev. Bounded solutions of families of systems of differential equations and their approximation. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 5, pp. 29-47. http://geodesic.mathdoc.fr/item/FPM_2006_12_5_a3/