A priori estimates and solvability of problems with nonlinear functional boundary conditions (Q2703956)

The author studies the solvability of the boundary value problem NEWLINE\[NEWLINEx^{(m)}(t)=F(x(\cdot))(t),\qquad t\in [a,b], \tag{1}NEWLINE\]NEWLINE with the nonlinear functional boundary conditions NEWLINE\[NEWLINEB^{k}_{i}(x^{(k)}_{i}(\cdot), x(\cdot))=0. \tag{2} NEWLINE\]NEWLINE The author obtains results that permit one to use a priori estimates for proving the solvability of problems with boundary conditions of a complicated structure and shows that if an a priori estimate holds, then there exists a solution.