Singular boundary-value problems for ordinary second-order differential equations

This article gives an exposition of the fundamental results of the theory of boundary-value problems for ordinary second-order differential equations having singularities with respect to the independent variable or one of the phase variables. In particular criteria are given for solvability and unique solvability of two-point boundary-value problems and problems concerning bounded and monotonic solutions. Several specific problems are considered which arise in applications (atomic physics, field theory, boundary-layer theory of a viscous incompressible fluid, etc.)Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 105–201, 1987.     

Existence of nonnegative solutions of singular boundary-value problems for second-order ordinary differential equations

It is proved that the boundary-value problem

Nonlinear boundary-value problems for systems of ordinary differential equations

We consider nonlinear boundary-value problems (with Noetherian operator in the linear part) for systems of ordinary differential equations in the neighborhood of generating solutions. By using the Lyapunov — Schmidt method, we establish conditions for the existence of solutions of these boundary-value problems and propose iteration algorithms for their construction. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 162–171, February, 1998.     

Mixed boundary-value problems for singular second-order ordinary differential equations

It is proved that the boundary-value problem

Modified quasilinearization technique for the solution of boundary-value problems for ordinary differential equations

A new quasilinearization algorithm is presented which essentially eliminates the necessity for computer storage. The representation theorem for the standard quasilinearization procedure is reformulated in terms of the initial value of the solution to a final-value problem, leading to a modification of the successive approximations. Several theorems establishing the convergence properties are proved; as in the original procedure, these convergence properties are both quadratic and monotonic. Finally, the modified approximation scheme is illustrated through several examples.     

On Warner's algorithm for the solution of boundary-value problems for ordinary differential equations

The paper discusses the solution of boundary-value problems for ordinary differential equations by Warner's algorithm. This shooting algorithm requires that only the original system of differential equations is solved once in each iteration, while the initial conditions for a new iteration are evaluated from a matrix equation. Numerical analysis performed shows that the algorithm converges even for very bad starting values of the unknown initial conditions and that the number of iterations is small and weakly dependent on the starting point. Based on this algorithm, a general subroutine can be realized for the solution of a large class of boundary-value problems.     

Averaging of boundary-value problems for ordinary differential equations with slow and fast variables

The system x=X(t, x, y), y=Y(t, x, y) with slow and fast variables is considered. A theorem on the justification of the averaging method for the boundary-value problem R(x(0), x(T))=0, y(0)=y₀, is proved.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 42, No. 9, pp. 1275–1278, September, 1990.     

Positive solutions for a multi-parameter system of second-order ordinary differential equations

In this paper, we establish the product formula for the fixed point index on product cone, andthen, as applications, consider the existence, nonexistence and multiplicity of positive solutions for a second-order ordinary differential system with parameters. The discussion is based on the product formula and thefundamental properties of the fixed point index.     

Some experiments with interval methods for two-point boundary-value problems in ordinary differential equations

Methods of interval mathematics are used to find upper and lower bounds for the solution of two-point boundary-value problems at discrete mesh points. They include interval versions of shooting and of finite-difference techniques for linear and non-linear differential equations of second order, and of finite-difference methods for Sturm-Liouville eigenvalue problems.Good results are obtained whenever the difficulties of dependency-width can be avoided, and particularly for the finite-difference method when the associated matrix is anM matrix.     

有序Banach空间常微分方程的正周期解

依据凝聚锥映射的一个krasnoselskii型不动点定理，在有序Banach空间中获得了二阶常微分方程．-u^n(t) Mu(t)＝f(t,u(t))正ω—周期解的存在性结果。     

Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations

In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form

Under some conditions, we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones.     

Existence of positive solutions for boundary-value problems of second-order delay differential equations

The paper is concerned with the existence of positive solutions for the nonlinear eigenvalue problem with singularity and the superlinear semipositone problem of second-order delay differential equations. The main results in this paper extend some of the existing literature. The proofs are based on a fixed-point theorem in cones.