In this work, we investigate the solvability and the approximate construction of solutions of certain types of regular non-linear boundary value problems for systems of ordinary differential equations on a compact interval. For this purpose, we construct analytically a uniformly convergent parametrised sequence of functions depending on the properties of the concrete boundary conditions and non-linearities in the given systems. The value of the parameter introduced artificially into the scheme is to be determined by solving a certain system of algebraic or transcendental equations. The text is divided into 10 sections. Sections 1 and 2 contain the notation used in what follows and provide a short introduction. In Section 3, the successive approximation techniques are treated for the investigation of periodic solutions of non-autonomous periodic systems. In Section 4, we apply the method for the study the periodic solutions of autonomous systems by using the appropriate reduction to a non-autonomous system. In Section 5, we establish conditions under which a system of non-linear non-autonomous ordinary differential equations has a family of solutions that are periodic with a common period and possess a certain symmetry property. Sections 6 and 7 deal with the investigation of non-linear two-point boundary value conditions by using a parametrisation that leads one to a family of problem with linear two-point conditions considered together with certain additional algebraic or transcendental equations with respect to certain parameters. In Section 8, we use the parametrisation approach to study some three-point non-linear boundary value problem which, as a result, can be investigated through auxiliary two-point problems. Most of theoretical results are illustrated by examples. Section 9 contains some historical remarks concerning the development and application of the method. Finally, in Section 10, we give several exercises concerning the successive approximation technique under consideration.

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

A new approach to non-local boundary value problems for ordinary differential systems

For a constructive analysis of the periodic boundary value problem for systems of non-linear non-autonomous ordinary differential equations, a numerical-analytic approach is developed, which allows one to both study the solvability and construct approximations to the solution. An interval halving technique, by using which one can weaken significantly the conditions required to guarantee the convergence, is introduced. The main assumption on the equation is that the non-linearity is locally Lipschitzian.

/pdf/constructive-analysis-of-periodic-solutions-with-interval-p32er8n4xk.pdf

Constructive analysis of periodic solutions with interval halving

We show how a suitable interval halving and parametrisation technique can help to es- sentially improve the sufficient convergence condition for the periodic successive approximations dealing with periodic solutions of nonlinear non-autonomous systems of ordinary differential equations.

/pdf/periodic-successive-approximations-and-interval-halving-3actr3pudc.pdf

Periodic successive approximations and interval halving

The article deals with approximate solutions of a nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions. We provide a scheme of numerical-analytic method based upon successive approximations constructed in analytic form. We give sufficient conditions for the solvability of the problem and prove the uniform convergence of the approximations to the parameterized limit function. We provide a justification of the polynomial version of the method with several illustrating examples. 2000 Mathematics Subject Classification: 34B15; 65L10.

/pdf/numerical-analytic-technique-for-investigation-of-solutions-d3xk2218ce.pdf

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

A three-point boundary-value problem for a system of nonlinear differential equations is reduced to a family of two-point problems, whose solutions are investigated by using the numerical-analytic method.

On the parametrization of three-point nonlinear boundary-value problems

We obtain exact, in a sense, conditions sufficient for the unique solvability of the Cauchy problem for systems of linear functional-differential equations of the general form. Efficient criteria for the unique solvability of the initial-value problem for systems of equations with deviating argument are given.

Some Exact Conditions for the Solvability of the Initial-Value Problem for Systems of Linear Functional-Differential Equations

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On the parametrization of three-point nonlinear boundary-value problems

Some Exact Conditions for the Solvability of the Initial-Value Problem for Systems of Linear Functional-Differential Equations