Nontrivial periodic solutions of nonlinear functional differential systems with feedback control

This paper examines the existence of nontrivial periodic solutions for the nonlinear functional differential system with feedback control:{aligned x (t)=x(t)a(t)-big[sum_{i=1}^n a_i(t)int_0^{+infty} f(t, x(t-theta)) d}varphi_i(theta) +sum_{j=1}^m b_j(t) int_0^{+infty} f(t,x (t-theta)),d}phi_j(theta)+sum_{mu=1}^p c_mu(t) int_0^{infty} u(t-theta),d}delta_mu(theta)big], u (t)=-rho(t)u(t)+sum_{nu=1}^q beta_nu(t) int_0^{infty} f(t, x(t-theta)),d}psi_nu(theta).endaligned Under certain growth conditions on the nonlinearity f, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray-Schauder nonlinear alternative.