Equilibrium positions of nonlinear integral-differential boundary value problems unsolved with respect to the derivative

Abstract

The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boichuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of linear boundary-value problems for ordinary differential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, M.M. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the linear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear oscillations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the nonlinear boundary value problems for the integral-differential boundary value problem unsolved with respect to the derivative, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the nonlinear integral-differential boundary value problem unsolved with respect to the derivative. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the nonlinear integral-differential boundary value problem unsolved with respect to the derivative in the form of equilibrium positions. The case of a nonlinear equation whose dimension does not coincide with the dimension of the unknown has been researched.