Periodic and subharmonic solutions for duffing equation with a singularity

Cheng, Zhibo; Ren, Jingli

doi:10.3934/dcds.2012.32.1557

Cited by 29 publications

(29 citation statements)

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“…The periodic solution problem for the Duffing equation has attracted much attention, see, eg, previous studies. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] However, the study on quasilinear equation is relatively infrequent. The main difficulty lies in the quasilinear operator ( p (x ′ )) ′ that typically possesses more uncertainty than the linear operator x ′′ .…”

Section: Cheng and Renmentioning

confidence: 99%

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Existence of multiplicity harmonic and subharmonic solutions for second‐order quasilinear equation via Poincaré‐Birkhoff twist theorem

Cheng

Ren

2017

Math Methods in App Sciences

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In this paper, we investigate the existence and multiplicity of harmonic and subharmonic solutions for second-order quasilinear equationwhere p ∶ R → R, p (u) = |u| p−2 u, p > 1, g satisfies the superlinear condition at infinity. We prove that the given equation possesses harmonic and subharmonic solutions by using the phase-plane analysis methods and a generalized version of the Poincaré-Birkhoff twist theorem. 6802 CHENG AND RENNote that when p = 2, the differential operator x  → ( p (x ′ )) ′ reduces to the linear operator x  → x ′′ and then Equation 1 is of the Duffing equation formx ′′ + g(x) = e(t).(2)The periodic solution problem for the Duffing equation has attracted much attention, see, eg, previous studies. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] However, the study on quasilinear equation is relatively infrequent. The main difficulty lies in the quasilinear operator ( p (x ′ )) ′ that typically possesses more uncertainty than the linear operator x ′′ . In recent years, there also appears some results on second-order quasilinear equation, see other works. [17][18][19][20][21][22] In literature, 18 del Pino, Elgueta, and Manásevich considered the Dirichlet boundary value problemand the corresponding eigenvalue problemon a set of positive measure, then (3) has a nontrivial solution. Afterwards, del Pino, Manásevich, and Murúa 20 deal with T-periodic solutions for the nonlinear ordinary differential Equation 3, and f ∶ R × R → R is continuous and T-periodic in t, T > 0. When the nonlinearity f interacts with the Fučík spectrum for the operator u  → (|u ′ | p−2 u ′ ) ′ , existence and multiplicity results are proved for (3). In a study, 21 Liu established some existence and multiplicity results for periodic solutions of the equationby applying the Poincaré-Birkhoff fixed-point theorem, where s ∈ R is a parameter, (t) is a continuous periodic function, and g(x) is locally Lipschitz on R and satisfies some asymptotic conditions. According to the growth speed of g, (1) can be classified into the following 3 cases,Lemma 2.1. (The Poincaré-Birkhoff twist theorem (see Theorem 1.2 of Fonda and Ureña 23 )) Under the above assumptions, the HS has at least 2 distinct mT-periodic solutions z (0) (t), z (1) (t), starting with z (k) (0) ∈ , such that Rot(z (k) (t); [0, mT]) = , for every k = 0, 1.Moreover, if H is twice continuously differentiable with respect to z and all mT-periodic solutions with initial condition on  are nondegenerate, then there are at least 2 of them.Lemma 2.1 generalized some previous versions of the Poincaré-Birkhoff Theorem for planar annuli with star-shaped boundaries. [24][25][26] Consider the equivalent HS of (1), x ′ = q (y), y ′ = −g(x) + e(t),corresponding to the Hamiltonian H(t, x, y

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Section: Cheng and Renmentioning

confidence: 99%

“…It is well known that time map plays an important role in studying the existence and multiplicity of harmonic and subharmonic solutions of (1). We now introduce the time map.…”

Section: Introductionmentioning

confidence: 99%

Existence of multiplicity harmonic and subharmonic solutions for second‐order quasilinear equation via Poincaré‐Birkhoff twist theorem

Cheng

Ren

2017

Math Methods in App Sciences

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“…The authors found a new method for estimating a lower a priori bounds of the periodic solutions to the given equation. Besides, many articles have been published about Liénard equation with repulsive singularity (see [4][5][6][7][8][9][10][11][12][13]). Recently, some good deal of works have been performed on the existence of periodic solutions of Rayleigh equations with singularity (see [14][15][16]).…”

Section: (T) + F X(t) X (T) -G X(t) + ϕ(T)x(t) = H(t)mentioning

confidence: 99%

“…an attractive singularity at x = 0, that is, 4) and σ ∈ C 1 (R, R) is a T-periodic function such that σ (t) < 1. Obviously, the attractivity condition lim x→0 + 1 x g 0 (s) ds = +∞ contradicts the repulsive singularity.…”

Section: (T) + F X(t) X (T) -G X(t) + ϕ(T)x(t) = H(t)mentioning

confidence: 99%