Turkish Journal of Mathematics

Abstract: 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=1ⁿ a_i(t)\int₀^+\infty f(t, x(t-q)) d}j_i(q) +\sum_j=1^m b_j(t) \int₀^+\infty f(t,x'(t-q))\,d}f_j(q)+\sum_m=1^p c_\mu(t) \int₀^\infty u(t-q)\,d}d_\mu(q)\big], u'(t)=-r(t)u(t)+\sum_n=1^q b_\nu(t) \int₀^\infty f(t, x(t-q))\,d}y_\nu(q).\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.

Key Words: Nonlinear functional differential equations with feedback control; Nontrivial periodic solutions; Leray-Schauder nonlinear alternative; Fixed point.