A priori estimates of solutions for the boundary value problem of the 4th order with integral conditions.

We consider a fourth order ordinary differential operator with nonlocal boundary conditions and spectral parameter. The boundary conditions are given by Riemann integrals, which contain both the unknown function and the derivatives of the unknown function. In the Sobolev space we introduce an equivalent norm depending on the spectral parameter $\lambda$. In terms of equivalent norms, we obtain a priori estimates for the solutions of the problem for sufficiently large values of the parameter $\lambda$. Using these estimates, the spectral properties of the corresponding operators are studied.