A constructive approach is proposed for the approximate finding of periodic solutions of differential systems with deviating argument and non-linearities satisfying the Lipschitz condition in a bounded region. Under suitable assumptions, the method allows one to prove solvability based on computational results. We describe a computationally less demanding version of the scheme involving trigonometric polynomial interpolation, in which higher-order approximations are constructed without symbolic integration. The practical use of the method is demonstrated with numerical examples.

We present simple techniques which allow one to reduce a wide class of boundary value problems with linear conditions to initial value problems for functional differential equations with finite-dimensional perturbations. Using this approach, we formulate certain general solvability conditions of linear boundary value problems.

Conditions for the existence of a symmetric solution of nonlinear functional differential equations with symmetries are established.

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.

We obtain general conditions sufficient for the solvability of a singular Cauchy problem for functional differential equations with non-increasing nonlinearities.

We show the use of parametrization techniques and successive approximations for the effective construction of solutions of linear boundary value problems for differential systems with multiple argument deviations. The approach is illustrated with a numerical example.

It is shown that a class of symmetric solutions of scalar non-linear functional differential equations can be investigated by using the theory of boundary value problems. We reduce the question to a two-point boundary value problem on a bounded interval and present several conditions ensuring the existence of a unique symmetric solution.

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.

For a class of weakly non-linear ordinary differential equations, the existence of a unique symmetric solution is established and its stability is studied. The symmetry of a solution is understood in the sense of a certain linear functional equality which includes, in particular, the cases of periodic, anti-periodic, even, and odd solutions. Efficient stability conditions in terms of logarithmic norms and spectral stability conditions are obtained. The theory is illustrated by examples.

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.

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.

We establish general conditions for the unique solvability of nonlinear measure functional differential equations in terms of properties of suitable linear majorants.

We consider a two-point boundary value problem for a non-linear system of functional differential equations, for which a scheme of efficient investigation using certain iteration process is constructed. A new convergence condition is established in the case where a certain additional property of the Lipschitz operator is present, which may lead one to weaker assumptions.