Abstract
Problem Statement: With respect to our observation in the relevant literature, work on stability and boundedness of solution for certain third order nonlinear differential equations where the nonlinear and the forcing terms depend on certain variables are scare. The objective of this study was to get criteria for stability and boundedness of solutions for these classes of differential equations. Approach: Using Lyapunov second or direct method, a complete Lyapunov function was constructed and used to obtain our results. Results: Conditions were obtained for: (i) Uniform asymptotic stability and, (ii) Uniform ultimate boundedness, of solutions for certain third order non-linear non-autonomous differential equations. Conclusion: Our results do not only bridge the gap but extend some well known results in the literature.
Highlights
On setting x& = y, &x& = z Eq 1 is equivalent to the system of differential equation: x& = y, y& = z, z& = p(t,x,y,z) − f(t,x,y)z− q(t)g(y) − r(t)h(x)
Conditions for uniform asymptotic stability and uniform ultimate boundedness of solutions of the nonlinear differential Eq 1 will be considered with the aid of an effective method for studying stability and ultimate boundedness of solutions namely Lyapunov second or direct method
It is well known that the problem of ultimate boundedness of solutions of nonlinear is very important in the theory and applications of differential equations
Summary
We shall be concerned here, with uniform asymptotic stability of the zero solutions (that is when p(t, x y, z) = 0) and uniform ultimate boundedness of solutions of the third order, non-linear, non-autonomous differential equations:. The studies of qualitative behaviour of solutions have been discussed by many authors in a series of research study. With respect to our observation in the relevant literature, these authors considered stability, asymptotic behaviour, boundedness of solutions of Eq 1, 2 in the case f(t,x,x&) equal any of f(x,x&,x&&),f(x,x&),f(x) and a where a is positive constant and q(t) = r(t) = 1. In[17] Swick discussed conditions for uniform boundedness of Eq 1 when p(t,x,x& ,x&&) ≡ 0 using an incomplete Lyapunov functions.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
