Relationship between solutions of families of two-point boundary value problems and Cauchy problems

We carry out a qualitative analysis and suggest a method for the solution of the two-point boundary value problems A _l υ = g, x ? [0, l], l ? (0, L) = B ? R ₊; $$ alpha v'_x left| {_{x = + 0} = F_1 (v,l)} right|_{x = 0} , beta v'_x left| {_{x = l - 0} = F_2 (v,l)} right|_{x = l} , $$ , where α, β ? R, υ is the unknown function, g = g(x) is a given real function, F ₁(y, l) and F ₂(y, l) are known real functions defined on the sets Y ₁ × B and Y ₂ × B, respectively, where Y ₁ ∪ Y ₂ ? R, and A _l is the restriction of A corresponding to the embedding parameter l. (Here A is an operator taking an arbitrary function in the set of function classes defined in the paper to C(B ₁), where B ₁ = [0, L).) The study takes into account the dependence of solutions of various versions of these two-point boundary value problems on the parameter l. We construct algorithms for the reduction of these families of two-point boundary value problems to Cauchy problems for ordinary differential equations and integro-differential equations that contain only first derivatives of the unknown functions with respect to the parameter l.