A Study of Periodic Solution of a Duffing ’ s Equation Using Implicit Function Theorem

In this paper, the well known implicit function theorem was applied to study existence and uniqueness of periodic solution of Duffing-type equation. Under appropriate conditions around the origin, a unique periodic solution was obtained.


Introduction
The well-known implicit function theorem has been employed by many authors to study existence of solution to non-linear differential equations of various types.[1] [2] [3] investigated the existence of solution to ordinary differential equations using implicit function theorem.Other researchers [4]- [10] where ( ) h t is continuous and 2π-periodic in t R ∈ and ( )

Preliminaries
Definition 2.1.Consider the general non-linear differential equation of the form ( ) ( ) where : be a mapping of U into F. f is said to be Frechet differentiable at x 0 if there exists a continuous linear mapping; : Frechet differentiable with respect to the first variable x at ( ) if the following conditions hold.
1) There exists a continuous linear mapping 1 : 3) The mapping , x y , it is Frechet differential with respect to this variable at ( ) 0 0 , x y with the same L 1 .Moreover, this is unique.L 1 is called the strong partial Frechet derivative with respect to the first variable ( ) and denoted by ( ) Lemma 2.7.(The Banach fixed point theorem) Let E be a Banach space and : f E E → be a contraction mapping, then f has a unique fixed point in E, i.e. there exists a unique x E ∈ such that ( ) Lemma 2.8.(The implicit function theorem) Let E, F, G be Banach spaces and U U U = × .For arbitrary ( ) be a mapping satisfying the following conditions.1) 2) f is Frechet differentiable with respect to the first variable at ( ) Then there exists a neighborhood 1 2 and a unique mapping Lemma 2.9.If X and Y are Banach spaces and ( ) where N(A) is the Null spaces of A and R(A) is the range space of A. B(X, Y) is the space of bounded linear transformations from X to Y.

Main Result
We present in this section, the main result of this paper.
x is continous} with the usual norm, [ ] Then, Equation (1.2) is equivalent to where Proof: We first remark that with the norm defined above, 2 2π C is a Banach space.The strategy for the proof involves application of the implicit function theorem to the function f defined in Equation (3.1).We split the proof into steps.
Step 3: ( ) 2 2π 0, 0 : The mapping ( ) 0, 0 : used implicit function theorem to show the existence of periodic solution for non-linear partial differential equations.The Duffing equation (oscillator): b, c are real constants and h(t) is continuous, has been widely used in physics, economics, engineering, and many other physical phenomena.Given its characteristic of oscillation and chaotic nature, many scientists are inspired by this nonlinear differential equation given its nature to replicate similar dynamics in our natural world.This equation together with Van der Pol's equation has become one of the most common examples of nonlinear oscillation in textbooks and research articles.See for instance [11] [12] [13] [14] and the references therein.Due to the importance of the Duffing equation in real world problems, the study of existence of solution of the equation has continued to attract the attention of many researchers.[15] [16] [17] [18] have proposed independently, the existence of periodic solution of Duffing equation of the general form: is a class of C 2 } and equipped with the usual uniform norm linear and continuous and hence bounded.It is also an onto mapping.Linear homeomorphism would have been established if the mapping is shown to be one to one.This is equivalent to replace appropriate conditions on the constants a, c such that Equation (3.5) is solvable.The auxiliary equation of (3.5) is 2 0 a ≠ , only the trivial solution exists.Most generally, put of these conditions imposed, one deduces the one to oneness of of a unique solution is now assured by the implicit function theorem.