Specific Approach of the Noetherization Theory For an Integro-Differential Operator A in the Space $$C_{-1}^{(m)}[-1,1]$$


Authors: Abdourahman Haman Adji, Shankishvili Lamara Dmitrievna

Abstract: In this article, our primary objective is to construct a noetherian theory for a singular linear integro-differential operator. This operator is defined by an integral equation of the third kind, the principal part of which is expressed as a higher-order differential operator. The choice of a methodology requiring specific approaches proved justified, as it stems from solving the linear differential equation under study—featuring a specific, well-known non-homogeneous term—thereby allowing for the systematic identification of solvability conditions. In this research, we employ integration techniques to establish the noetherian property of the operator, while highlighting the solvability conditions for the integral equation of the third kind that defines this noetherian integro-differential operator. Ultimately, we establish the noetherian properties for both the principal part operator and its associated, subsequently developing the noetherian theory for the operator under study in parallel with that of its associated. A concrete illustrative example of the noetherian theory, applied to an integro-differential operator A, is presented at the conclusion of the work.

Pages: 8-17

DOI: 10.46300/91019.2026.13.2

International Journal of Pure Mathematics, E-ISSN: 2313-0571, Volume 13, 2026, Art. #2

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