Existence of anti-periodic solutions with symmetry for some high-order ordinary differential equations
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RESEARCH
Open Access
Existence of anti-periodic solutions with symmetry for some high-order ordinary differential equations Hai Pu1,2* and Jinyun Yang3 *
Correspondence: [email protected] 1 State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China 2 School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China Full list of author information is available at the end of the article
Abstract The existence of anti-periodic solutions with symmetry for high-order Duffing equations and a high-order Duffing type p-Laplacian equation has been studied by using degree theory. The results obtained enrich some known works to some extent. MSC: 34B15; 34C25 Keywords: anti-periodic solution with symmetry; high-order ordinary differential equation; p-Laplacian operator; Leray-Schauder degree theory
1 Introduction Anti-periodic problems arise naturally from the mathematical models of various physical processes (see [, ]) and also appear in the study of partial differential equations and abstract differential equations (see [–]). For instance, electron beam focusing system in traveling-wave tube theories is an anti-periodic problem (see []). In mechanics, the simplest model of oscillation equation is a single pendulum equation x + ω sin x = e(t) ≡ e(t + π) , whose anti-periodic solutions satisfy x(t + π) = –x(t),
∀t ∈ R.
During the past twenty years, anti-periodic problems have been studied extensively by numerous scholars. For example, for first-order ordinary differential equations, a Massera’s type criterion was presented in [] and the validity of the monotone iterative technique was shown in []. Moreover, for higher-order ordinary differential equations, the existence of anti-periodic solutions was considered in [–]. Recently, existence results were extended to anti-periodic boundary value problems for impulsive differential equations (see []), and anti-periodic wavelets were discussed in []. It is well known that higher-order p-Laplacian equations are derived from many fields such as fluid mechanics and nonlinear elastic mechanics. In the past few decades, many important results on higher-order p-Laplacian equations with certain boundary conditions have been obtained. We refer the readers to [–] and the references cited therein. © 2012 Pu and Yang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Pu and Yang Boundary Value Problems 2012, 2012:108 http://www.boundaryvalueproblems.com/content/2012/1/108
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In [], the authors considered the existence of anti-periodic solutions for the high-order Duffing equation as follows: x(n) +
n–
ai x(i) + g(t, x) = e(t).
(.)
i=
Moreover, in [] the authors discussed the
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