Omer Cakir2025-12-122025-12-122011Omer Cakir/ About Existence of Solution of Third-Order Derivative of −y'' + q(x)y = f Type Equations/https://repository.sdu.edu.kz/handle/123456789/2332In this paper, we investigate the existence of solutions for third-order differential equations of the form −y'' + q(x)y = f within a Hilbert space of square-integrable functions on the interval from −π to π. The study focuses on the differential operator defined by −y'' + q(x)y under periodic-type conditions, where the function and its first two derivatives satisfy y(i)(−π) = y(i)(π) for i = 0, 1, 2. Several preliminary lemmas are established, including integral identities and inequalities related to the scalar product involving the operator. Using these lemmas, it is shown that the kernel of the operator is trivial, and a bounded inverse operator exists. Consequently, the boundary value problem has a unique solution for any square-integrable function f, provided that the coefficient function q(x) is continuous. These results contribute to the theoretical understanding of higher-order differential operators in Hilbert spaces.enAttribution-NonCommercial-ShareAlike 4.0 Internationalthird-order differential equationperiodic boundary conditionsdifferential operatorexistence of solutioninverse operatorArticle