J. Korean Math. Soc. 47 (2010), No. 3, pp. 573–583 DOI 10.4134/JKMS.2010.47.3.573 ANTI-PERIODIC SOLUTIONS FOR HIGHER-ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS Tai Yong Chen, Wen Bin Liu, Jian Jun Zhang, and Hui Xing Zhang Abstract. In this paper, the existence of anti-periodic solutions for higher-order nonlinear ordinary differential equations is studied by us- ing degree theory and some known results are improved to some extent. 1. Introduction Anti-periodic problems arise naturally from the mathematical models of a variety of physical processes and have important applications in auto-control, partial differential equations and engineering. Recently, there has been a great deal of research on anti-periodic boundary value problem (see [1], [2], [9], [10], [11], [12] and references therein). In mechanics, the simplest model of oscillation equation is single pendulum equation x ′′ + ω 2 sin x = p(t)= p(t +2π), (1) whose anti-periodic solutions satisfy x (t + π)= -x(t), ∀t ∈ R. In particular, many authors have discussed the existence of anti-periodic so- lutions for the first order or second order nonlinear ordinary differential equa- tion. Mawhin ([9]) generalized equation (1) to the general Duffing equation x ′′ + g(x)= p(t), and obtained the existence results for anti-periodic solutions by using critical point theory. In [1], the author proved the existence of anti-periodic solutions for the following abstract nonlinear second order evolution equation -x ′′ (t)+ ax ′ (t)+ A(t)x(t)= f (t) Received August 30, 2008. 2000 Mathematics Subject Classification. 34B15, 34C25. Key words and phrases. higher-order differential equation, anti-periodic solution, Leray- Schauder principle. This work was financially supported by National Natural Science Foundation of China (10771212) and Science Foundation of China University of Mining and Technology (2008A037). c 2010 The Korean Mathematical Society 573