Nonlinear Integrodifferential Boundary-Value Problems with Deviating Argument Unsolved with Respect to the Derivative

The investigations of linear differential-algebraic boundary-value problems are closely connected with extensive applications of the corresponding mathematical models in the theory of nonlinear oscillations, mechanics, biology, radio-engineering, and the theory of stability of motion. Thus, the problem of generalization of the results obtained by S. Campbell, A. M. Samoilenko, and O. A. Boichuk to the case of nonlinear boundary-value problems unsolved with respect to the derivative seems to be quite urgent. In particular, this is true for finding necessary and sufficient conditions for the existence of the required solutions of nonlinear integrodifferential boundary-value problems with deviating argument unsolved with respect to the derivative. We establish conditions for the existence of solutions of a nonlinear integrodifferential boundary-value problem with deviating argument unsolved with respect to the derivative and propose a constructive scheme for finding these solutions.