American Journal of Mathematics and Statistics 2019, 9(3): 131-135
DOI: 10.5923/j.ajms.20190903.03
On the Krasnoselskii’s Fixed Point Theorem and the
Existence of Periodic Solution for a Damped and Forced
Duffing Oscillator
E. O. Eze
*
, U. E. Obasi, C. O. D. Udaya, F. Daniel
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Umuahia, Abia State, Nigeria
Abstract This paper is devoted to study the existence of periodic solution for a damped and forced Duffing oscillator
using the Krasnoselskii’s fixed point theorem in Banach space. As an application, uniqueness and compactness of solution of
Duffing oscillator was achieved using Gronwall’s Inequality and Eberlein Simultan theorem which extends some results in
literature.
Keywords Krasnoselskii’s Fixed Point Theorem, Banach Space, Compactness, Analytic Semigroup, Duffing Oscillator
1. Introduction
The aim of this paper is to study existence of periodic
solution for a damped and forced Duffing oscillator of the
form
̈ + ̇ + +
2
+
3
= ℎ() (1.1)
with boundary conditions
(0) = (2)
̇ (0) = ̇ (2) (1.2)
In equation (1.1) , , , are real constants and ℎ()
is continuous. Also, ℎ: [0 2] →ℝ
+
is periodic in ∈ℝ
+
.
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
* Corresponding author:
obinwanneeze@gmail.com (E. O. Eze)
Published online at http://journal.sapub.org/ajms
Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing
This work is licensed under the Creative Commons Attribution International
License (CC BY). http://creativecommons.org/licenses/by/4.0/
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 ∃ >0 ℎ ℎ ‖()‖≤
⟹ − ≤ () ≤⟹() ∈ [−, ].
Definition 2.2. (Convex Set): Suppose is a vector
space. A subset ⊂ is said to be convex if whenever
, ∈ and ∈ [0,1], it follows that + (1 −)∈.
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.