1. Introduction

The aim of this paper is to study existence of periodic solution for a damped and forced Duffing oscillator of the form

(1.1)

with boundary conditions

(1.2)

In equation (1.1) are real constants and is continuous. Also, is periodic in . Duffing oscillator is a second order nonlinear differential equation used to model dynamics of special types of mechanical and electrical systems. This differential equation has been named after the studies of Duffing in [1] which has a cubic nonlinearity and describes an oscillator. It is the simplest oscillator displaying catastrophic jumps of amplitude and phase when the frequency of the forcing term is taken as a gradually changing parameter. The main application have been in electronics and biology. For example, the brain is full of oscillators at micro and macro levels [2]. Several techniques have been used by many authors to study the existence of periodic solution of the Duffing type of equation (1.1) such as polar coordinates, the method of upper and lower solution, coincidence degree theory and a series of existence results of nontrivial solution of equation (1.1). We refer to [3-5] and reference therein. However, some methods of proving existence have some limitations and in fact for practical purposes serious difficulties arise frequently in the search for fixed point of Duffing equation with cubic nonlinearity.

In this paper, we chose another strategy of proof which rely essentially on a fixed point theorem due to Krasnoselskii for a set that is closed, bounded and convex subset of a Banach space [6]. This result has been extensively employed in the related literature in the study of several kinds of separated boundary value problems (see for instance in [7, 8, 9, 10, 11] and their references); while for the periodic problem, it is more difficult to find references [12]. The reason for this contrast may be the fact that in order to apply this fixed point theorem, it is necessary to study the semigroup operator for linear equation, contraction and compactness of solution which are relatively difficult to study. To overcome this problem, Gronwalls inequality and Eberlein Simultian theorem were employed to obtain uniqueness and compactness of solution of Duffing equation.

2. Preliminaries

Definition 2.1. (Boundedness of a function): A function is bounded if

Definition 2.2. (Convex Set): Suppose X is a vector space. A subset is said to be convex if whenever and , it follows that The closure of a set is again convex.

Definition 2.3. (Hilbert Space): A pre-Hilbert space which is complete (considered as a normed linear space) is called Hilbert space.

Definition 2.4. (Pre-Hilbert Space): A linear space X is said to be pre-Hilbert space if for every ordered pair of elements there is associated real number where X is a real linear space and complex number where X is a complex linear space such that

Definition 2.5. (Banach space): A normed linear space is called Banach space if it is complete in the sense of a metric given by the norm. Completeness means that every Cauchy sequence is convergent. Let be any Cauchy sequence that is a sequence for which as independently, then an element such that

(1.3)

Definition 2.6. Let be a complete metric space. Then is called a contraction mapping if there exists a constant such that

(1.4)

for each and where

Theorem 2.9. (Contraction Mapping Principle) Let be a complete metric space and let be a contraction, then T has a unique fixed point . Furthermore, for each

(1.6)

From this, one draws three conclusion in which this paper is written

(i) T has a unique fixed point, say

(ii) For each the picard sequence converges to and also sequentially compact and converges to

(iii) The convergence is uniform if X is bounded.

Theorem 2.10. (Gronwalls-Bellman’s Inequality) Let f and g be continuous real-valued functions on some interval then

(1.7)

for some implies that

(1.8)

Proof. Multiplying both sides of equation (2.9) by we have

By hypothesis then

Hence,

Remark: There is a generalization of this inequality. Its statement and proof is all about technicality.

Definition 2.11. (Compactness) The subset A of a topological space X i.e is said to be compact if every open cover of A has a finite subcover.

Note: A subset is pre-compact if is compact.

Definition 2.12. (Contraction Semigroup) Let E be a Banach space. A one-parameter family of bounded linear operators on E into itself is called a contraction semigroup of class or simply a contraction semigroup if it satisfies the following conditions

Note: (i) is called the semigroup property.

Examples:

(1) is a semigroup.

(2) Let , with and is a semigroup.

Theorem 2.13. A subset of a Banach space X is relatively weak compact if only if it is relatively weakly sequentially compact. In particular, a subset of a Banach space X is weakly compact if and only if it is weakly sequentially compact.

Theorem 2.14. (Krasnoselskii’s Fixed Point Theorem) Assume that F is a closed bounded convex subset of a Banach space X. Futhermore, assume that and are mappings from F into X such that the following conditions hold:

Theorem 2.15. If is a then and such that for each

Proof: Since is continuous. Suppose by contraction, Let there exist a sequence such that , then by the uniform boundedness principle, there exist such that is unbounded contradicting the fact that is continuous at

3. Main Result

We consider the Duffing equation of the form

(1.9)

Equation (1.9) can be re-written in the following form

(1.10)

Vectorization of equation (1.10) is as follows

Let then equation (1.10) gives

(1.11)

The equivalent system of equation (1.11) is given by

(1.12)

In matrix form equation (1.12) is written as

The above is of the form of non-autonomous equation given by

(1.13)

where

General solution of equation (1.13) is

(1.14)

(1.15)

where is a semigroup operator.

To generate sequence of solutions in equation (1.14) we have that for we have

(1.16)

Now assuming that and are solutions of equation (1.14) then we have that

(1.17)

Furthermore, we will show that that is uniqueness.

Suppose that where X is a Banach space and F is a subspace and consider convexity of its solutions. For say, then

Claim 1: is convex,

Proof:

Case I

Take then 0.

Case II

Take then

Case III, take we have

Claim 2: Is

Proof:

Define a ball, is convex. Then

(1.18)

Hence the solution of the equation is convex.

Next, we verified the boundedness property of our solution.

Let we show that

This is true by uniform boundedness principle. Applying theorem 2.15 we have

(1.19)

By semigroup property,

(1.20)

We have

(1.21)

where

. So bounded for each where as

Next, we show that that is Uniqueness of Solution.

But where is a semigroup

(1.22)

Hence,

(1.23)

by theorem 2.15 and theorem 2.10 we have

(1.24)

Therefore and since uniqueness of solution are satisfied, the closure is trivial. Hence

Furthermore, we will show that is a contraction that is

From equation (1.15) we have that

(1.25)

where is a semigroup operator, and

Claim: Equation (1.25) is a contraction

Proof:

(1.26)

(1.27)

Using equation (1.20) which have the same idea with Banach-Mazur distance in supper multiplicative metric space we have

(1.28)

where

By theorem 2.15

(1.29)

By theorem 2.10 equation (1.25) becomes

(1.30)

Now take and equation (1.30) becomes

(1.31)

Hence equation (1.31) is a contraction.

To show that is a continuous, we proceeds as follows. Recall that that is uniqueness of solution, then is also a contraction.

Given any and such that . If then

(1.32)

By equation (1.32) is continuous.

Finally we use theorem 2.13 to establish the compactness of . Assume that we are in finite dimensional space, Hein-Borel guarantees our result that is since is closed and bounded hence compact. But since we are in infinite dimensional space (Banach space), by theorem 2.13, is sequentially compact hence compact. Therefore has a fixed point in F.