On the Positive Almost Periodic Solutions of a Class of Nonlinear Lotka-Volterra Type System with Feedback Control

Denote x t x1 t , x2 t , . . . , xn t , xt x1t, x2t, . . . , xnt , xit s xi t s i 1, 2, . . . , n , s ∈ −τ, 0 , τ is a positive constant or τ ∞, the norm of a bounded continuous function space C −τ, 0 , R is defined as ‖φ‖ maxs∈ −τ,0 |φ s |, where |φ| max{|φi|, i 1, 2, . . . , n}, C −τ, 0 , R {φ ∈ C −τ, 0 , R : for all θ ∈ −τ, 0 , φ θ > 0}. We call an almost periodic function is positive if and only if each component has its positive infimum. DenoteAP R {x t ∈ R : x t is a continuous almost periodic function on Rn}, AP R {x t ∈ R : x t is a continuous almost periodic function on R}, Tαf t g t refers to limn→∞ f t αn g t , where α {αn}, {αn} is a sequence of real numbers, T f, ε {τ : |f t τ −f t | < ε, t ∈ R}, for more almost periodic monographs, see references 1, 2 . In 3 , Teng first studied the existence of the almost periodic solutions for the following scalar equation: