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In this article, we study the existence of periodic solutions to a class of distributed delay differential equations. We transform the search for periodic solutions with the special symmetry of a delay differential equation to the problem of finding periodic solutions of an associated Hamiltonian system. Using the critical point theory and the pseudo-index theory, we obtain some sufficient conditions for the multiplicity of periodic solutions. This is the first time that critical point theory has been used to study the existence of periodic solutions to distributed delay differential equations.

In this article, we use the Nehari manifold method to study a class of fourth-order even difference systems. First, we show that there exist multiple periodic solutions to the non-autonomous system. Second, we obtain sufficient conditions to guarantee the existence of solutions with prescribed minimal periods to the autonomous system. Our results generalize the results in reference.

In this paper, we study the existence of periodic solutions of the following differential delay equations z ′ ′ (t) = ∑ k = 1 M − 1 (− 1) k f (z (t − k)) , z ′ ′ (t) = ∑ k = 1 M − 1 (− 1) k f (z (t − k)) , where f ∈ C (R N , R N) f ∈ C (R N , R N) , M , N ∈ N M , N ∈ N and M M is odd. By making use of S 1 S 1 -geometrical index theory, we obtain an estimation about the number of periodic solutions in term of the difference between eigenvalues of asymptotically linear matrices at the origin and at infinity. [ABSTRACT FROM AUTHOR]