A note on boundedness of solutions to a class of non-autonomous differential equations of second order

Cemil Tunç

doi:10.2298/aadm100601026t

Abstract

By defining some appropriate Liapunov functions, we discuss boundedness of solutions to a class of non-autonomous and nonlinear differential equations of second order. By this work, we prove some results established in the literature by Liapunov's second method instead of the integral test. We give six examples to illustrate the theoretical analysis in this work and effectiveness of the method utilized here.

Highlights

AND MAIN RESULTSIn 1972, Kroopnick [3] considered the following nonlinear differential equation of second order (1)x + a(t)b(x) = 0, where a and b are continuous functions on + = [0, ∞) and = (−∞, ∞), respectively
It is assumed that the derivative a (t) exists and is continuous
Kroopnick [3] proved the following theorem by the integral test: Theorem A. (Kroopnick [3, Theorem I]) If a(t) > α > 0, a (t) ≤ 0 on [T, ∞), t ≥ T, b(x) continuous, 2000 Mathematics Subject Classification. 34C10, 34C11, 34D05

Summary

Cemil Tunc

By defining some appropriate Liapunov functions, we discuss boundedness of solutions to a class of non-autonomous and nonlinear differential equations of second order. We prove some results established in the literature by Liapunov’s second method instead of the integral test. We give six examples to illustrate the theoretical analysis in this work and effectiveness of the method utilized here

AND MAIN RESULTS

It follows that

We also notice that

It is also assumed that the derivative x

It also follows that