Evolution Equations in Hilbert Spaces via the Lacunae Method

TL;DR

This paper develops a method for solving evolution equations in Hilbert spaces with special operator conditions, providing existence, uniqueness, and series solutions, including applications to fractional differential equations with various operators.

Contribution

It introduces a novel approach to formulate solutions for evolution equations using root vectors, applicable to fractional and other complex operators.

Findings

Established existence and uniqueness theorems for the class of evolution equations.

Derived solutions in series form based on root vectors.

Applied the method to fractional differential equations involving Riemann-Liouville, Riesz, and difference operators.

Abstract

In this paper we consider evolution equations in the abstract Hilbert space under the special conditions imposed on the operator at the right-hand side of the equation. We establish the method that allows us to formulate the existence and uniqueness theorem and find a solution in the form of a series on the root vectors of the right-hand side. As an application we consider fractional differential equations of various kinds. Such operators as the Riemann-Liouville fractional differential operator, the Riesz potential, the difference operator have been involved.