For the constructive analysis of locally Lipschitzian system of non-linear differential equations with mixed periodic and two-point non-linear boundary conditions, a numerical-analytic approach is developed, which allows one to study the solvability and construct approximations to the solution. The values of the unknown solution at the two extreme points of the given interval are considered as vector parameters whose dimension is the same as the dimension of the given differential equation. The original problem can be reduced to two auxiliary ones, with simple separable boundary conditions. To study these problems, we introduce two different types of parametrized successive approximations in analytic form. To prove the uniform convergence of these series, we use the appropriate technique to see that they form Cauchy sequences in the corresponding Banach spaces. The two parametrized limit functions and the given boundary conditions generate a system of algebraic equations of suitable dimensions, the so-called system of determining equations, which give the numerical values of the introduced unknown parameters. We prove that the system of determining equations define all possible solutions of the given boundary value problems in the domain of definition. We established also the existence of the solution based on the approximate determining system, which can always be produced in practice. The theory was presented in detail in the case of a system of differential equations consisting of two equations and having two different solutions.

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We show how an appropriate parametrization technique and special successive approximations can help to control unknown jumps in the case of nonlinear boundary-value problems with state-dependent impulses. The practical application of the proposed technique is shown on a numerical example.

We discuss the application of periodic successive approximations to the investigation of periodic boundary-value problems for a class of linear functional-differential equations. We describe a version involving a kind of interpolation by trigonometric polynomial. The application of the proposed technique is shown for a numerical example.

A constructive technique of analysis involving parametrisation and polynomial interpolation is suggested for general non-local problems for ordinary differential systems with locally Lipschitzian transcendental non-linearities. The practical application of the approach is shown on a numerical example.

We show how an appropriate parametrization technique and successive approximations can help to investigate nonlinear boundary-value problems for systems of differential equations under the condition that the components of solutions vanish at certain unknown points. The technique can be applied to nonlinearities involving the signs of the absolute value and positive or negative parts of functions under boundary conditions of various types.

We describe a reduction technique allowing one to combine an analysis of the existence of solutions with an efficient construction of approximate solutions for a state-dependent multi-impulsive boundary value problem which consists of non-linear system of differential equationsu'(t) = f(t, u(t)) for a.e. t epsilon [a, b],subject to the state-dependent impulse conditionu(t+) - u(t-)= gamma t(u(t-)) for t epsilon (a, b) such that g(t, u(t-)) = 0,and the non-linear two-point boundary conditionV(u(a), u(b)) = 0.

We suggest a new constructive approach for the solvability analysis and approximate solution of certain types of partially solved Lipschitzian differential systems with two-point nonlinear boundary conditions. The practical application of the suggested technique is shown on a numerical example.

We show how appropriate parametrisation technique and successive approximations can help to investigate solutions of Emden-Fowler type equations with a given number of zeroes. The technique can be efficiently applied for more general equations with non-linearities involving absolute value signs and various types of boundary conditions.

We investigate the non-linear system of ordinary differential equationsu'(t)=f(t,u(t)), a.e.t is an element of[a,b],subject to the state-dependent impulse conditionu(t+)-u(t-)=gamma(u(t-)) for t is an element of(a,b) such that g(t,u(t-))=0and the linear two-point boundary conditionAu(a)+Cu(b)=d.Here, -infinity<a<b<infinity, f and gamma are given continuous vector-functions, g is a continuous scalar function, A, C are constant matrices, and d is a constant vector. The instants of time t where the jump occurs are determined by the equation g(t,u(t-))=0 and, thus, are unknown a priori and essentially depend on the solution u. We discuss a reduction technique allowing one to combine the analysis of existence of solutions with an efficient construction of approximate solutions. At present, according to the authors' knowledge, no numerical results for boundary value problems with state-dependent impulses are available in the literature. (C) 2015 Elsevier Inc. All rights reserved.

We show how a suitable interval division and parametrisation technique can help to essentially improve the convergence conditions of the successive approximations for solutions of systems of non-linear ordinary differential equations under non-local boundary conditions. The application of the technique is shown on an example of a problem with non-linear integral boundary conditions involving values of the unknown function and its derivative.

Constructive Methods for Non-Linear Boundary-Value Problems" had taken place in Miskolc, Hungary.The list of participants and the program of the meeting can be found on the workshop web site: http://

Abstract We give a new approach for the investigation of existence and construction of an approximate solutions of nonlinear non-autonomous systems of ordinary differential equations under nonlinear integral boundary conditions depending on the derivative. The constructivity of a suggested technique is shown on the example of non-linear integral boundary value problem with two solutions.

We give a new approach for the investigation of existence and construction of an approximate solutions of nonlinear non-autonomous systems of ordinary differential equations under nonlinear integral boundary conditions depending on the derivative. The constructivity of a suggested technique is shown on the example of non-linear integral boundary value problem with two solutions.

We show how a suitable interval halving and parametrization technique can help to essentially improve the sufficient convergence condition for the successive approximations dealing with solutions of nonlinear non-autonomous systems of ordinary differential equations under integral boundary conditions.

We continue our study of constructive numerical-analytic schemes of investigation of boundary problems. We simplify and improve the recently suggested interval halving technique allowing one to essentially weaken the convergence conditions.

We suggest a new constructive approach for the solvability analysis and approximate solution of general non-local boundary value problems for non-linear systems of ordinary differential equations with locally Lipschitzian non-linearities. The practical application of the techniques is explained on a numerical example.