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Abstract: Periodic travelling wave solutions of reaction–diffusion equations were studied by many authors. The type reaction–diffusion system is a notable special model that admits explicit periodic travelling wave solutions and was introduced by Kopell and Howard in 1973. There are now similar systems which are investigated by means of autonomous dynamics. In contrast, there are few papers which are concerned with non-autonomous cases. For this reason, we apply Mawhin’s continuation theorem to derive the existence of periodic travelling wave solutions for non-autonomous systems, and we describe the ‘disappearance’ of periodic travelling wave solutions under special situations. Our main result is also illustrated by examples. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of periodic solutions of the second order differential equations . Using continuation lemma, we obtain the existence of periodic solutions provided that () is sublinear when x tends to positive infinity and satisfies a new condition where M, d are two positive constants. [Copyright &y& Elsevier]

Abstract: Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface . If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones. [Copyright &y& Elsevier]

Abstract: Two fixed point theorems for the sum of contractive and compact operators are obtained in this paper, which generalize and improve the corresponding results in [H. Schaefer, Über die methode der a priori-Schranken, Math. Ann. 129 (1955) 415–416; T.A. Burton, Integral equations, implicit functions and fixed points, Proc. Amer. Math. Soc. 124 (1996) 2383–2390; V.I. Istrǎtescu, Fixed Point Theory, an Introduction, Reidel, Dordrecht, 1981; T.A. Burton, K. Colleen, A fixed point theorem of Krasnoselskii–Schaefer type, Math. Nachr. 189 (1998) 23–31; D.R. Smart, Fixed Point Theorems, Cambridge Univ. Press, Cambridge, 1980]. As the applications for the results, we obtain the existence of periodic solutions for some evolution equations with delay, which extend the corresponding results in [T.A. Burton, B. Zhang, Periodic solutions of abstract differential equations with infinite delay, J. Differential Equations 90 (1991) 357–396]. [Copyright &y& Elsevier]

In this paper, we develop Kaplan–Yorke''s method and consider the existence of periodic solutions for delay differential equations with two delays. Especially, we study Hopf and saddle-node bifurcations of periodic solutions for the equation with parameters, and give conditions under which the bifurcations occur. [Copyright &y& Elsevier]