Harmonic Solutions of Duffing Equation with Singularity via Time Map

This paper is devoted to the study of second-order Duffing equation ( ) ( ) ′′ x g x p t + = with singularity at the origin, where ( ) g x tends to positive infinity as x → +∞ , and the primitive function ( ) ( ) ( ) ∫ x G x g s s 1 d = → +∞ as x 0 → . By applying the phase-plane analysis methods and Poincaré-Bohl theorem, we obtain the existence of harmonic solutions of the given equation under a kind of nonresonance condition for the time map.


Introduction
We deal with the second-order Duffing equation ( ) ( ), where : g + →   is locally Lipschitzian and has singularity at the origin, : p →   is continuous and 2π periodic.Our purpose is to establish existence result for harmonic solution of Equation (1).Arising from physical applications (see [1] for a discussion of the Brillouin electron beam focusing problem), the periodic solution for equations with singularity has been widely investigated, referring the readers to [2]- [6] and their extensive references.
As is well known, time map is the right tool to build an approach to the study of periodic solution of Equation (1) (see [7]- [9]).However, the work mainly focused on the equations without singularity.Our goal in this paper is to study the periodic solution of Equation (1) with singularity via time map.There is a little difference between our time map and the time map in [7] [9].We now introduce the time map.
Consider the auxiliary autonomous system ( ) and suppose that ( ) Obviously, the orbits of system (2) are curves c Γ determined by the equation ( ) where c is an arbitrary constant.
In view of the assumptions (g 0 ), (g 1 ) and (G 0 ), there exists a 0 0 c > , such that for ) where ( ) ( ) We recall an interesting result in [7].Ding and Zanolin [7] proved that Equation (1) without singularity possesses at least one T-periodic solution provided that ( ) ( ) lim sgn , Now naturally, we consider the question whether Equation (1) has harmonic solution when we permit ( ) x cross resonance points and use a kind of nonresonance condition for time map.In the following we will give a positive answer.In order to state the main result of this paper, set and assume that Our main result is following.In this case, we generalize the result in [7] to Equations (1) with singularity.The remainer of the paper is organized as follows.In Section 2, we introduce some technical tools and present all the auxiliary results.In Section 3, we will give the proof of Theorem 1.1 by applying the phase-plane analysis methods and Poincaré-Bohl fixed point theorem.

Some Lemmas
we assume throughout the paper that ( ) g x is locally Lipschitz continuous.In order to apply the phase-plane analysis methods conveniently, we study the equation : 1, g − +∞ →  is continuous and has a singularity at 1 x = − .In fact, we can take a parallel translation 1 x u = + to achieve the aim.Then the conditions ( ) Dropping the hats for simplification of notations, we assume that ( ) and ( ) h c and ( ) Consider the equivalent system of ( 6): , ; ,  x t y t x t x y y t x y = be the solution of (8) satisfying the initial condition ( ) ( ) We now follow a method which was used by [4] [6] and shall need the following result.P x y x y x x y y x y =  It is obvious that the fixed points of the Poincaré map P correspond to 2π -periodic solutions of system (8).We will try to find a fixed point of P .To this end, we introduce a function  For the convenience, two lemmas in [6] will be written and the proof can be found in [6].
Proof.Without loss of generality, we may assume that 0 0 x = .From Lemma 2. x

t y t t t t x t y t t t t x t y t t t t x t y t t t t
Throughout the lemma, we always assume that ( ) 0 0 , l x y is large enough.
(1) We shall first estimate 3 2 t t − and 4 3 t t − .We can refer to Lemma 2.6 in [6] and obtain ( ) (2) We now estimate 1 0 t t − and 2 1 t t − .According to conditions ( ) 0 G′ and ( ) Then, By (10) and (11), we have Integrating both sides of the above inequality from 0 t to b t , we obtain ( Recalling the conditions ( ) g and ( 11), we know that there is 0 ς > , such that a b ς − ≤ .Applying Lemma 2.8 in [6], we can derive

Lemma 2 . 1
Assume that conditions ( ) 0 G′ and ( ) 1 g hold.They every solution of system (8) exists uniquely on the whole t-axis.By Lemma 2.1, we can define Poincaré map