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Weakly nonlinear boundary value problem for a matrix differential equation

“…The proved Theorem 3 generalizes analogous statements for various boundary value problem for the differential equations [5,14,17]. The proved Theorem 3 also generalizes analogous statements for periodic boundary value problem for a matrix Riccati equation [4]. The length " of the segment OE0; " on which the iterative procedure (2.14) converges to the required solution of the boundary value problem (1.1), (1.2) can be estimated by analogy with [7,8,14].…”

“…The proved Lemma 1 generalizes analogous statements for various boundary value problem for the differential equations [5]. The proved Lemma 1 also generalizes analogous statements for periodic boundary value problem for a matrix Riccati equation [4]. Provided P Q ¤ 0; we say that the boundary value problem (1.3) there is a critical case, and the problem (1.3) solvable only for those inhomogeneities of F .t / and A; for which the condition (2.3).…”

“…By setting " D 0 in the last relation, we obtain (2.7). Following [4,5,14,17], we refer to equation (2.7) as an equation for generating constants.…”

“…(1.1) that satisfy the boundary condition [3,4,9] LZ. ; "/ D A; A 2 R ı (1.2) and the construction of this solution.…”

“…The structure of the set of solution of the generating matrix boundary value problem (1.3) was studied by Bellman [1]. In [4] the authors established solvability conditions and construction of the generalized Green operator in case˛Dˇfor the periodical matrix boundary value problem (1.3). Denote by U.t / and V .t / -normal fundamental matrices:…”