Conditions for bifurcation of solutions of degenerate boundary-value problems

Abstract

We obtain conditions for the bifurcation of solutions of linear degenerate Noetherian boundary-value problems with small parameter under the assumption that the unperturbed degenerate differential system can be reduced to the central canonical form. Statement of the Problem and Main AssumptionsWe consider the following linear inhomogeneous boundary value-problem with small parameter:where A(t), A 1 (t), B(t), and B 1 (t) are n × n matrices whose components are real functions continuously differentiable sufficiently many times onan n-dimensional column vector from the space C q−1 [a; b] (the value of q is determined according to Theorem 2.1 in [1]), α is an m-dimensional column vector of constants, α ∈ R m , and l and l 1 are linear vector functionals defined on the space of n-dimensional vector functions continuous on [a; b], l = col (l 1 , . . . , l m ) : C[a, b] → R m , l 1 = col l 1 1 , . . . , l 1 m : C[a, b] → R m , l i , l 1 i : C[a, b] → R. Assume that the generating degenerate boundary-value problemwhich is obtained from (1), (2) for ε = 0, does not have solutions for arbitrary inhomogeneities f (t) ∈ C q−1 [a, b] and α ∈ R m . Assume that, by a nondegenerate linear transformation, system (3) can be reduced to a central canonical form [1,2].Let us find conditions for the perturbing coefficients A 1 (t) and B 1 (t) in the differential system (1) and for l 1 in the boundary condition (2) under which the boundary-value problem (1), (2) has a solution.

Periodic solutions of linear degenerate systems of ordinary differential equations of the second order

“…There are many works (see, e.g., [1][2][3][4][5][6]) devoted to the development of methods for the construction of solutions of degenerate differential systems and the investigation of their qualitative behavior. System (1) was studied in [1] under certain assumptions, one of which is the constancy of the rank of the matrix A.t/: In [2], sufficient conditions for the existence of a periodic solution of the scalar equation (1) were obtained on the basis of investigation of the degenerate Riccati equation.…”