ON THE GREEN FUNCTION OF HIGHER ORDER DIFFERENTIAL EQUATION WITH NORMAL OPERATOR COEFFICIENTS ON THE SEMI-AXIS

In the paper we study the Green function for 2n-th order differential equation with a normal operator coefficient on the semi-axis. To this end, at first we construct the Green function of the equation with ”frozen” coefficients. Then we use the Levi method and obtain an integral equation for the Green function of the given equation. We prove the solvability of the integral equation. Using the integral equation we establish the main properties of the Green function.

Here y ∈ H 1 and the derivatives are understood in the strong sense.
Let D be a union of all functions of the form p ∑ k=1 ϕ k (x) f k , where ϕ k (x) are finite, 2n-times continuously differentiable scalar functions, and Denote by L an operator generated by differential expression (1.1) and boundary conditions (1.2) with domain of definition D .
We will assume that the operator coefficient Q(x) satisfies the following conditions: 1) The operator Q(x) for almost all x ∈ [0, ∞) is a normal operator in H and for almost all x ≥ 0 has a common domain of definition D(Q) in H.
) For all x ≥ 0 for |x − ξ | > 1 the following inequality is valid: where The main goal of the paper is to study the Green function of the operator L.
We have the following theorem.
Theorem 1.If conditions 1) -4) are fulfilled, then for sufficiently large µ > 0 there exists the inverse operator R µ = (L + µE) −1 that is an integral operator with operator kernel G(x, η; µ) that will be called Green function of the operator L. G(x, η; µ) is an operator function in H and depends on two variables x, η(0 ≤ x, η < ∞), the parameter µ and satisfies the following , and The proof of the theorem is carried out in two stages.At first we construct the Green function G 1 (x, η; µ) of the operator L 1 , generated by the expression and the boundary conditions where ξ is a fixed number.
2. Constructing the Green function of the operator L 1 .
We will look for the Green function G 1 (x, η, ξ ; µ) of the operator L 1 in the form where g(x, η, ξ , µ) is the Green function of the equation l 1 (y) = 0 on the whole axis.
As is known [4], it has the form: Here ω α denotes the roots from the (−1) degree of 2n, lying in the upper halfplane, and The function V (x, η, ξ ; µ) is a bounded solution as x → ∞ of the following problem: Then for V (x, η, ξ ; µ) we have The coefficients A k (η, ξ ; µ) are determined from boundary conditions (2.4).
We will have: Then by the Cramer's formulas we get: Substituting the expression A k (η, ξ ; µ) in equality (2.5), we get Then the Green function of the problems (1.3)-(1.4)will have the form: We can write the obtained formula in the form: (2.9) As Q(ξ ) is an unbounded normal operator, and Re(2iω k K ξ η) < 0, Thus, from (2.9) it follows (2.10) It is proved that under sufficiently large µ, equation (2.10) is solvable and its solution is the Green function of the operator L.
and as µ → ∞ we have r(x, η, ξ ; µ) = o(1) uniformly with respect to (x, η).Now let us construct and study some properties of the Green function of the operator L generated by the differential expression (1.1) and boundary conditions (1.2).The Green function G(x, η; µ) of the problem (1.1), (1.2) is the solution of the following integral equation: