Unbounded Solutions of Asymmetric Oscillator

and Applied Analysis 3 Theorem 3. Consider (19) with f(t, x, y) = f(t) ∈ L[0, T] is T-periodic. Let λ(θ) be given in Corollary 2 and assume λ(θ) ≡ 0. Define τ-periodic functions λ(θ) and μ(θ) as

Liu [6] discussed the boundedness of all solutions of the following nonlinear equation: where with   = 2/ sin(/),  ∈ N. Let  = 1; he defined a 2-periodic function () as where with the initial value (V(0), V  (0)) = (0, 1).Under some smoothness conditions, he showed that all solutions of (11) are bounded if the function () has no zero for all  ∈ R.
Recently, Li and Zhang [7] studied the unboundedness of solutions of the following asymmetric oscillator: where ,  are positive constants satisfying nonresonance condition: and  are bounded and  is 2-periodic in .By using the similar method used in [4], they obtained some sufficient conditions for the existence of unbounded solutions of (15).For more recent results on the boundedness or unboundedness of solutions of differential equations of second order, we refer to [6,[8][9][10][11][12][13][14][15][16][17][18] and the references therein.Assume ,  are positive constants.Let () be the solution of the following initial value problem: then it is well-known that  is -periodic with In this paper, we consider the following asymmetric oscillator: where   () = || −2 ,  > 1,  ± = max{±, 0},  is continuous, bounded, and -periodic in , and the limits lim  → ±∞, → ±∞ (, , ) = (, ±∞, ±∞) ∈  ∞ [0, ] exist.We assume that  satisfies the following resonance condition: A solution () of ( 19) is called to have large initial value if We will give some sufficient conditions for the unboundedness of solutions of (19).Especially, if (, , ) = (), we will obtain higher order approximation and corresponding sufficient conditions for the unboundedness of solutions of (19) with large initial values.The results of this paper are new which improve some relative results on the literature in some sense.Throughout this paper, we assume / = / with ,  ∈ N.
The main results of this paper are the following. where Then all solutions of (19) with large initial values are unbounded provided that the function () has only finite number of zeros in [0, ] and all its zeros are simple.( Then all solutions of (19) with large initial values are unbounded provided one of the following conditions holds:

Generalized Polar Coordinate Transformation
Let  = () be the solution of the following initial value problem: and define a function () as then it is easy to verify that  ∈  2 ; is -periodic and satisfies the identity: Moreover, one can verify that  is the solution of (17).

Unboundedness Motions of Planar Mappings
In this section, we adopt the notations used in [9].Given  > 0, let the set   be the exterior of the open ball   centered at the origin with radius ; that is, then Define  1 = R\Z; then the points and the group distance in  1 can be described, respectively, by Let  :   → R 2 be a one to one and continuous mapping.We denote its lift by  in the form where  ∈  1 [0, ] is -periodic,  ∈ N,   ̸ = 0 is a constant, and ,  are continuous, -periodic and satisfy (, ) = (1), (, ) = ( −1 ).
Given a point ( 0 ,  0 ) ∈   , let {(  ,   )} ∈ be the unique solution of the initial value problem for the differential equation: This solution is defined in a maximal interval where   ,   are certain numbers in the set Z ∪ {+∞, −∞} satisfying The solution {(  ,   )} is said to be defined in the future if   = +∞ and is said to be defined in the past if   = −∞.
Let {  }  =1 be the ordered sequence of zeros of () in [0, ) such that Lemma 7. If  has a simple zero   ; that is,   (  ) ̸ = 0,   ∈ Ω, then there exists orbits of (54) which are to be defined in the future and satisfy or they are defined in the past and satisfy Moreover, if Ω ̸ = 0 and all zeros of () are simple, then every orbit of (54) with large initial value is either to be defined in the future satisfying or is defined in the past satisfying The proof of the above lemma is similar to that of Proposition 3.1 in [9].
Proof.Let () > 0, for all  ∈ R, then 2 0 = max ∈R () ≥ min ∈R () = 2 0 > 0 and it follows from (63) that for  0 ≫ 1, By induction, we get from (65) and replacing  0 by   ,  1 by  +1 we get from (64) Obviously, (66) implies that the orbit   is defined in the future.Next we claim that the orbit   is unbounded in the future.In fact, (67) implies that   is monotone increasing, hence the limit lim  → +∞   =  * ≤ ∞ exists.If  * < ∞, taking the limit on both sides of (67) we obtain which is a contradiction.Hence the orbit of ( 63) is defined in the future and satisfies lim  → +∞   = ∞.Similarly, if () < 0, for all  ∈ R, we can prove that the orbit of (63) is defined in the past and satisfies lim  → −∞   = ∞.
If  ̸ = 2, then it follows from Lemma 6 that Now, Lemma 7 implies that all orbits of (37) with  0 ≫ 1 go to infinity either in the future or in the past, which means that all solutions of (19) with large initial values are unbounded.
If  = 2, then by assumption of Theorem 3, () ≡ 0 and the function () has no sign-changing.Lemma 8 implies that all orbits of (37) with  0 ≫ 1 go to infinity either in the future or in the past, which implies that all solutions of (19) with large initial values are unbounded.
and it is not difficult to show that   ⋅  ̸ = 0. Then () has only finite number of zeros in [0, 2  ] and all of them are simple.Corollary 2 implies that all solutions of (19) with large initial values are unbounded.