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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 paper, the E + -Conley index theory has been used to study the existence of periodic solutions of nonautonomous delay differential equations (in short, DDEs). The variational structure for DDEs is built, and the existence of periodic solutions of DDEs is transferred to that of critical points of the associated function. When DDEs are 2 π -nonresonant, some sufficient conditions are obtained to guarantee the existence of periodic solutions. When the system is 2 π -resonant at infinity, by making use a second disturbing of the original functional, some sufficient conditions are obtained to guarantee the existence of periodic solutions to DDEs. [ABSTRACT FROM AUTHOR]

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]

In this paper, we study a class of nonlinear differential equations with multiple distributed delays. We reduced the problem of search for periodic solutions of the distributed delay differential equations to that of finding zeros of an operator on some Banach space. Utilizing the equivariant degree theory, we obtain some sufficient conditions to guarantee the existence of multiplicity of periodic solutions. Also, we give some information about the symmetry of the periodic solutions. At last, we give an example to illustrate our result. [ABSTRACT FROM AUTHOR]