Improving explicit bounds for the solutions of second order linear differential equations (original) (raw)
Abstract and Applied Analysis, 2012
This paper presents two methods to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second-order half-linear differential equation(p(x)Φ(y'))'+q(x)Φ(y)=0, withp(x)andq(x)piecewise continuous andp(x)>0,Φ(t)=|t|r-2tandrbeing real such thatr>1. It also compares between them in several examples. Lower bounds (i.e., Lyapunov inequalities) for such a distance are also provided and compared with other methods.
Error bounds for asymptotic solutions of differential equations. 2. The general case
Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences, 1966
The result s of the preceding pape r are extend ed to the general system of n first-order differential eq uations having an irregular singu larit y of arbitrary rank at infinity. Formal solutions are explicitly constructed for th e syste m in c anonical form. Proofs of existence and uniqu e ness of solutions of integral equation s defining th e e rror are given. As an example, the case n = 2 is solved completely, and a flow chart of the transformations of this case to canonical form is included. .
The boundedness of solutions of nonlinear differential equations of even order 2s
Journal of Differential Equations, 1985
In this paper we reduce the "problem" of the boundedness of the solutions of a certain system of ordinary differential equations of order 2s > 2 in the real domain to the corresponding problem for a certain differential inequality of order 2. The system of differential equations is (5.1.1), and the differential inequality is (2.1.1). We carry out the reduction of the problem for (5.1.1) by first carrying out the reduction of the problem for the following differential equation of order 2, and then generalising: i+kfi'"-'+g=Eh (f = dx/dt), (1.1.1) where the basic hypotheses are (i) k > 0, E > 0 are parameters; (ii) n > 1 is a rational number whose denominator is odd and positive; (iii) f =f (x, 1, I) > 1; (iv) (hi = Ih(x, 2, t)l < 1 + iznP2; (v) g=&!(x); (vi) f, g, h are continuous functions of their arguments shown in (iii), (iv), and (v); ((1.1.2) G>O, where G = G(x) = 1: g(y) du.
In this paper, we investigate by means of second method of Lyapunov, sufficient conditions that guarantee uniform-asymptotic stability of the trivial solution and ultimate boundedness of all solutions to a certain second order differential equation. We construct a complete Lyapunov function in order to discuss the qualitative properties mentioned earlier. The boundedness result in this paper is new and also complement some boundedness results in literature obtained by using an incomplete Lyapunov function together with a signum function. Finally, we demonstrate the correctness of our results with two numerical examples and graphical representation of the trajectories of solutions to the examples using Maple software.
Error bounds for asymptotic solutions of differential equations. I: The distinct eigenvalue case
Journal of research of the National Bureau of Standards
The method of Olver for bounding the error term in the asymptotic so lutions of a second-order equation having a n irregular singul arity at infinity is extended to the general system of n first-order equations in the case when the eige nvalu es of the lead coefficient matrix are distinct. Vector and norm bounds are given for th e difference between an actual solution vector and a partial su m of a for mal so luti on vector. Two cases are distinguished geometricall y: In one it is possible to exp ress the error vec tor by a s in gle Volterra vec tor integral equat.ion; in the other it is necessa ry to use a simultaneous pair of Volterra vector integral equatio ns. Some ne w inequaliti es for integral equations are given in an append ix .
New Qualitative Results for Solutions of Functional Differential Equations of Second Order
Discrete Dynamics in Nature and Society
In this paper, we are concerned with the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of nonlinear functional differential equations of second order by the second method of Lyapunov. We obtain sufficient conditions guaranteeing the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of the equations considered. We give an example for illustrations by MATLAB-Simulink, which shows the behaviors of the orbits. The findings of this paper extend and improve some results that can be found in the literature.