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Nonlinear boundary-value problems for the Lyapunov equation in the space \(L_p\). (English. Ukrainian original) Zbl 1505.34094

J. Math. Sci., New York 246, No. 3, 394-409 (2020); translation from Neliniĭni Kolyvannya 21, No. 4, 523-536 (2018).
This paper focuses primarily on studying boundary-value problems for a Lyapunov-type equation in the space \(Lp(I;\mathcal{L}_{H})\). Necessary and sufficient conditions for the solvability of the corresponding boundary-value problem are established both in linear and nonlinear cases.
In Section 2, first linear boundary value problem is introduced in the space \(L_p\). The solutions of the linear boundary-value problem are constructed by using the generalized Green operator. Section 3 is devoted to studying the non-linear boundary value problem. It is proposed to find the approximate solutions of the nonlinear equation by using iterative Newton-Kantorovich-type algorithms. Theorem 2 proves the necessary condition and Theorem 3 shows the sufficient condition.

MSC:

34G10 Linear differential equations in abstract spaces
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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