Browsing by Author "Omer Cakir"

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    About Existence of Solution of Third-Order Derivative of −y'' + q(x)y = f Type Equations
    (Suleyman Demirel University, 2011) Omer Cakir
    In 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.
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    About kernel of inverse operator L' to third-order derivative's Ly = −y+q(x)y operator
    (Suleyman Demirel University, 2011) Omer Cakir
    This article investigates the kernel of the inverse operator associated with the third-order differential operator defined by Ly = −y'' + q(x)y under periodic-type boundary conditions. Using Kolmogorov widths and s-number theory, we establish estimates for the operator’s properties and demonstrate that the inverse operator is a kernel operator when q(x) is continuous and satisfies q(x) ≥ 1. Several supporting lemmas are proven, including inequalities for the operator and relationships between s-numbers and Kolmogorov widths. The results provide a theoretical framework for understanding the behavior of higher-order differential operators in Hilbert and Banach spaces.