Global attractivity in delay differential equations using a mixed monotone technique

Abstract

We derive new sufficient conditions for global attractivity in nonlinear delay differential equations using a mixed monotone technique. The equations considered include the equation of the form d dt [x(t) − ax(t − τ)] = − μx(t) − bx(t − σ) + f(x(t − γ)) , where a, b, μ, −, σ, and γ are nonnegative numbers such that a ϵ [0, 1), b + μ > 0 and λ(1 − ae − λτ ) = − μ − be − λσ has a negative root; moreover f( x) is a mixed monotone function, that is, f( x) = ω( x, x), where ω( x, y) is monotone decreasing in x and increasing in y. Our results are applied to some delay differential equations from mathematical biology.