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For the nonlinear periodic boundary-value problem posed for an ordinary differential equation in the critical and noncritical cases, we obtain constructive conditions of its solvability and propose a scheme for finding its solutions by using the Adomian decomposition method. [ABSTRACT FROM AUTHOR]

We establish constructive necessary and sufficient conditions of solvability and propose a scheme for the construction of solutions to a nonlinear boundary-value problem unsolved with respect to the derivative. We also suggest convergent iterative schemes for finding approximate solutions of this problem. As an example of application of the proposed iterative scheme, we find approximations to the solutions of periodic boundary-value problems for a Rayleigh-type equation unsolved with respect to the derivative, in particular, in the case of a periodic problem for the equation used to describe the motion of satellites on elliptic orbits. [ABSTRACT FROM AUTHOR]

The investigation of differential-algebraic boundary-value problems was originated in the works by Weierstrass, Luzin, and Gantmakher. The works by Campbell, Boyarintsev, Chistyakov, Samoilenko, Perestyuk, Yakovets, Boichuk, Ilchmann, and Reis were devoted to the systematic study of differential-algebraic boundary-value problems. At the same time, the investigation of differential-algebraic boundary-value problems is closely connected with the analysis of linear boundary-value problems for ordinary differential equations originated in the works by Poincaré, Lyapunov, Krylov, Bogolyubov, Malkin, Myshkis, Grebenikov, Ryabov, Mitropolskii, Kiguradze, Samoilenko, Perestyuk, and Boichuk. The investigation of linear differential-algebraic boundary-value problems is connected with numerous applications of the corresponding mathematical models to 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 Campbell, Samoilenko, and Boichuk to the case of nonlinear integrodifferential boundary-value problems unsolved with respect to the derivative is quite actual. In particular, this is true for the problem of finding necessary and sufficient conditions for the existence of solutions of nonlinear integrodifferential boundary-value problems unsolved with respect to the derivative. In the present paper, we establish conditions for the existence of solutions of the nonlinear integrodifferential boundary-value problem unsolved with respect to the derivative and present a constructive scheme for their finding. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the existence of solutions of a weakly nonlinear Noetherian boundary-value problem for a system of second-order ordinary differential equations in the critical case. [ABSTRACT FROM AUTHOR]

We establish constructive conditions for the existence of solutions of an autonomous Noetherian weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case and develop a modified iterative procedure for finding its solutions. [ABSTRACT FROM AUTHOR]