On the asymptotic analysis of bounded solutions to nonlinear differential equations of second order

Sizar Abid Mohammed,Cemil Tunç

doi:10.1186/s13662-019-2384-x

Abstract

In this paper, we consider two different models of nonlinear ordinary differential equations (ODEs) of second order. We construct two new Lyapunov functions to investigate boundedness of solutions of those nonlinear ODEs of second order. By using the Lyapunov direct or second method and inequality techniques, we prove two new theorems on the boundedness solutions of those ODEs of second order as t to infty . When we compare the conditions of the theorems of this paper with those of Meng in (J. Syst. Sci. Math. Sci. 15(1):50–57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002), we can see that our theorems have less restrictive conditions than those in (Meng in J. Syst. Sci. Math. Sci. 15(1):50–57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002) because of the two new suitable Lyapunov functions. Next, in spite of the use of the Lyapunov second method here and in (Meng in J. Syst. Sci. Math. Sci. 15(1):50–57, 1995; Sun and Meng in Ann. Differ. Equ. 18(1):58–64 2002), the proofs of the results of this paper are proceeded in a very different way from that used in the literature for the qualitative analysis of ODEs of second order. Two examples are given to show the applicability of our results. At the end, we can conclude that the results of this paper generalize and improve the results of Meng in (J. Syst. Sci. Math. Sci. 15(1):50–57, 1995), Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002), and some other that can be found in the literature, and they have less restrictive conditions than those in these references.

Highlights

1 Introduction It is well known that linear and nonlinear ordinary differential equations (ODEs) of second order can arise during many applications in various scientific areas such as physics, biology, chemistry, biophysics, mechanics, medicine, aerodynamics, economy, atomic energy, control theory, information theory, population dynamics, electrodynamics of complex media, and so on
The boundedness of solutions of nonlinear ODEs of second order are of particular interest in applications
This paper investigates the problem of asymptotic boundedness of solutions of ODEs (6) and (7) as t → ∞

Summary

Introduction

It is well known that linear and nonlinear ODEs of second order can arise during many applications in various scientific areas such as physics, biology, chemistry, biophysics, mechanics, medicine, aerodynamics, economy, atomic energy, control theory, information theory, population dynamics, electrodynamics of complex media, and so on. The boundedness of solutions of nonlinear ODEs of second order are of particular interest in applications (see [1,2,3, 6, 8, 16,17,18, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46, 48,49,50,51,52,53,54]). Of solutions of the following ODEs of second order: x + q1(t) + q2(t) x = 0, (1)

Results

Conclusion