Mathematics

The inversion of the fractional polynomial Euler operator

H. Alkadi

Voronezh State University, Voronezh, Russia

Abstract: We consider, for the first time, a fractional differential polynomial operator generalizing a polynomial with the integer Euler differentiation. We study its invertibility in spaces of functions bounded with a power weight on a segment. We establish the existence of a bounded inverse to the operator under consideration in these spaces. The results are applied to prove the correct solvability of the problem without initial conditions for the inhomogeneous generalized Euler equation with fractional differentiation of the Marchaud. We provide an integral representation and an estimate for the solution in terms of the right-hand side.

Keywords: Euler differential equation, strongly continuous semigroups, fractional powers of operators, well-posed problems, orthogonal polynomials.

UDC: 517.956

Received: 03.02.2025
Revised: 09.08.2025

DOI: 10.47475/2500-0101-2025-10-4-634-641