Contemporary Theory of Functional Differential Equations and Some Classical Problems

Proceedings, pp. 154—157 A survey of the recent trends and ideas of the Perm Seminar Let D be a linear manifold of elements x, B be a Banach space of elements z, Rn be the space of vectors α = col{α1,... αn} with real elements. Further assume that there exists a pair of linear operators δ: D → B and r: D → Rn such that the systems of equations δx = z, rx = α, (1) has a unique solution x ∈ D for any z ∈ B, α ∈ Rn. We will write down the solution of (1) in the form x = Wz + Uα. (2) The “W−substitution ” (2) puts an element x ∈ D in correspondence to any pair {z, α} ∈ B × Rn and to any x ∈ D it puts in correspondence a pair z ∈ B and α ∈ Rn by (1). Thus the manifold D is isomorphic to the direct product B×Rn (D ' B×Rn). The linear isomorphism J = {W,U} : B × Rn → D is defined by the operators W: B → D and U: Rn → D. The spaceD is a Banach one under the norm kxkD = kδxkB + krxkRn. An example of such a D is the space of absolutely continuous functions (the functions x: [a, b] → Rn such that x(t) = bZ a ẋ(s) ds + x(a)). Here (Wz)(t) = tZ a z(s) ds, Uα = α, B is the space L of summable z: [a, b] → Rn