Constructive analysis of periodic solutions with interval halving

Abstract

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.An existence theorem based on properties of approximations is proved. A relation to Mawhin's continuation theorem is indicated. MSC: 34B15

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

“…This closely correlates with the idea of the LyapunovSchmidt reduction (see, e.g., [12]). The solvability of the determining system (41) (47), which can be carried out by analogy to [3,4,13], is not treated here.…”

“…The rigorous analysis confirming the existence ofũ, which consists in the verification of suitable sufficient conditions similar to [3], is omitted here, and we focus on the construction of approximations only. In this case, arguing as shown above and substituting the values from (76) into (69), we obtain the following expression for the first approximation toũ : e U 11 ðtÞ ¼ 0:3923536713 þ 0:0025 t 4 þ 0:004490021967 t 3 À 0:07479600670 t 2 þ 0:02495444725 t; e U 12 ðtÞ ¼ À0:1570525052 þ 0:200075291 t À 0:0009018708475 t 4 À 0:005693773113 t 3 þ 0:05577210445 t 2 for t 2 ½0; 1=2.…”

On the construction of solutions of general linear boundary value problems for systems of functional differential equations

“…However, unlike systems of ordinary differential equations, in the case of boundary value problems for more general systems of functional differential equations, there are still almost no methods for approximate construction of their solutions. We can mention in this relation the parametrization methods (see, e.g., [18,24]) using the ideas close to [17,19,20,23] in the ordinary case.…”

Successive approximations and interval halving for integral boundary value problems

“…R n are a continuous functions in a certain bounded set D and d 2 R n is a given vector. We use an appropriate numerical-analytic approach and a natural interval halving technique which was suggested in [6], [4], [7], [8], [11], [3]. At first, we reduce the given problem (1.4), (1.5) To study the solutions of BVPs (1.6), (1.7) and (1.8), (1.9) we use the special modified form of parameterized successive approximations x m .t;´; / and y m .t; ; Á/ of type (1.3) constructed in analytic form and well defined on the intervals t 2 h a; a C i , respectively.…”