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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, we study the existence of periodic solutions to the following delay differential system: { x 1 ′ (t) = ∫ − 1 0 f 1 (x 2 (t + θ)) d θ , x 2 ′ (t) = ∫ − 1 0 f 2 (x 1 (t + θ)) d θ. By transforming the problem of searching for periodic solutions of the above system to the problem of finding periodic solutions of an associated ordinary differential system with boundary value conditions and using pseudoindex theory, some sufficient conditions are obtained to guarantee the existence of multiple W -symmetric 2-periodic solutions. We also present three specific examples to illustrate our results. [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]