Successive Approximation Techniques in Non-Linear Boundary Value Problems for Ordinary Differential Equations

Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions

“…The limit function x* satisfies the initial condition x*(0, z, l) = z as well as the boundary conditions in (6), since these conditions are satisfied by all functions x m = x m (t, z, l) of the sequence {x m }. Passing to the limit in the recurrence relation (14) for x m , we show that the limit function x* satisfies the identity (15). If we differentiate this identity, we obtain that x* is a unique solution of the Cauchy problem (16).…”

“…Let us establish the uniform convergence of the sequence (14) and the relation of the limit function to the solution of some additively modified boundary value problem.…”

“…In order to make the computations of x m possible or more easier, we give a justification of a polynomial version of the iterative scheme (14). At first, we recall some results of the theory of approximations in [20].…”

“…At the beginning, we follow the ideas presented by Rontó and Rontó [14] and by Rontó and Shchobak [15], which contains existence results for a system of two nonlinear differential equations with separated boundary conditions. In order to avoid some technical difficulties, we deal in this article, for simplicity, with nonlinear differential equations with homogeneous Dirichlet boundary conditions.…”

“…In Section 6, we illustrate the applicability of our approach to three Dirichlet boundary value problems: the linear one, the semilinear one, and the quasi-linear one containing the p-Laplace operator. Finally, let us note that presented polynomial version of the numericalanalytic method in this article can be extended to more general nonlinear boundary value problems studied in [14].…”