This paper is concerned with periodic solutions of first-order nonlinear functional differential equations with deviating arguments. Some new sufficient conditions for the existence of periodic solutions are obtained. The paper extends and improves some well-known results.

We study the question of the unique solvability of the periodic type problem for the second order linear integro-differential equation with distributed argument deviation $$u^{\text{'}\text{'}}\left(t\right)=p_0\left(t\right)u\left(t\right)+{\int }_0^{\omega }p(t,s)u\left(\tau (t,s)\right)\mathrm{d}s+q\left(t\right),$$ and on the basis of the obtained results by the a priori boundedness principle we prove the new results on the solvability of periodic type problem for the second order nonlinear functional differential equations, which are close to the linear integro-differential equations. The proved results are optimal in some sense.

By using the well-known Leggett–Williams multiple fixed point theorem for cones, some new criteria are established for the existence of three positive periodic solutions for a class of n-dimensional functional differential equations with impulses of the form ⎧y’(t) = A(t)y(t) + g(t,yt), $t\ne t_j$, j ∈ ℤ, ⎨ ⎩$y\left(t{⁺}_j\right)=y\left(t{¯}_j\right)+I_j\left(y\left(t_j\right)\right)$, where $A\left(t\right)={\left(a_{ij}\left(t\right)\right)}_{n\times n}$ is a nonsingular matrix with continuous real-valued entries.