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

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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