Spectral problem for second-order ordinary differential equation with integral conditions

Ordinary differential equations with nonlocal and integral boundary conditions have attracted significant attention due to their theoretical importance and wide range of applications. In this work, we investigate a spectral problem for an ordinary differential operator subject to integral conditions involving both the first and second derivatives of the unknown function.

The novelty of this work lies in the formulation of more general integral conditions that extend those studied by K. A. Darovskaya and A. L. Skubachevskii, where the integral conditions involve either the first or the second derivative of the unknown function. The proposed framework allows for a unified treatment of such conditions and leads to a broader class of differential operators.

A priori estimates for the solutions of the problem are obtained for sufficiently large values of the spectral parameter λ. These estimates play a key role in the analysis and provide a basis for studying the corresponding operator. In particular, they are used to investigate the spectral properties of the associated operator and to ensure the well-posedness of the problem.

The approach is based on methods of functional analysis and the theory of Fredholm operators. The obtained results generalize several known results in the literature and contribute to the development of the theory of differential equations with nonlocal conditions.