Investigation of The Resolvent Kernel of a Higher Order Differential Equation With Normal Operator Coefficients On The Semi-Axis

Year 2024, Volume: 14 Issue: 2, 99 - 108, 31.07.2024

Hamzaga D. Orudzhev Gahire L. Shahbazova

Abstract

Green function of a 2n-th order differential equation with normal coefficients
on the half-axis is studied. We first consider the Green function of our equation with
“frozen” coefficients. Using Levi’s method, we obtain a Fredholm-type integral equation
for the Green function of our problem, whose kernel is a Green function of a problem
with constant coefficients. We prove an existence and uniqueness theorem for this integral
equation in some Banach spaces of operator-valued functions. The main result of this
paper is a theorem stating that the solution of the obtained integral equation is a Green
function of our problem.

Keywords

operator, operator-differential equations, resolvent, Green function, spectrum, integral equation, eigenvalues, eigenfunctions, Hilbert spaces, Banachspaces

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