On Solutions of Nonlinear Differential Equations with Deviating Arguments

Abstract

1. Consider the second order differential equation with deviating argument (1)x"(t) f (t, x(t), x(h(t))), a <= <= b, where h(t)<= b. Let qg(t) be a function which is continuously differentiable on [c, a], c min {inf,_<,_<b (h(t)), a} (if c a, q9 is to be interpreted as a constant), and let B be a real number. Grimm and Schmitt in [13 and [23 have obtained sufficient conditions which guarantee the existence of a function x(t) which has the property that it satisfies the boundary conditions (2)Boundary value problems of the type (1)-(2) arise in a very natural way in studying variational problems in control theory where the problem is complicated by the effect oftime delays in signal transmission. One of the simplest such problems is to determine extrema of the functional under the conditions f F(t, x(t), x(h(t)), .) dt x(t) =-(p(t), <= a, x(b) B. Under suitable hypotheses on F this problem leads to a boundary value problem of type (1)-(2). For an account of such variational problems we refer the interested reader to the monograph by El'sgol'ts [3] and the survey papers [4], [5], [6]. These survey papers, the paper by Kamenskii [7] and the paper [2] in turn reveal most of the literature concerning such boundary value problems. Functions x(t) which are solutions of the boundary value problem (BVP) (1)-(2) in general have the defect that their left derivative x'(aand their right derivative x'(a +) at a do not agree as the following simple example illustrates. The BVP x"(t)=x(t)-x(t-1), O_<_t__< 1, x(t)=-O, -1 S O, x(1)= 1 has the unique solution -l=<t_<_0, x(t)= sinht 0<t< 1 sinh 1'

A numerical method to boundary value problems for second order delay differential equations

“…We like to quote only the papers I- 7,8] and [9]. Based on a theory of differential inequalities developed in [1][2][3][4][5][6][7] and [8], Schmitt [12] has found comparison theorems for linear delay equations.…”

Analytic Solutions of a Neutral Differential Equation Near a Singular Point

“…A great deal of progress has been made in the general theory of functional differential equations in the past decade. However, the analytic theory has received little attention, because in the case where the initial set is nontrivial, i.e., does not reduce to a single point, analyticity and growth conditions alone will not in general guarantee the existence of an analytic solution to the fundamental initial value problem, see [5], [6]. Indeed, an arbitrary initial value problem will have an analytic solution if and only if the analytic extension of the initial function satisfies the differential equation.…”

Προβληματα Συνοριακων Τιμων Για Συναρτησιακες Διαφορικες Εξισωσεις

“…(31) jy(t)-Asiny(t-l+|sin(t-i + f>|)}' = y(t-l)+cos|| φ || , 0 <^t <^2 όπου λ είναι μιά σταθερά με 0 < λ < 1. Θα δείξουμε δτι υπάρχει λύση y της (31) που να ικανοποιεί τις συνοριακές συνθήκες: (32) y 0 = Î y 2' y o3 y 2 ecC-i,o].…”