关键词：
By applying the fixed point theorem in a cone of Banach space;we obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solutions of two kinds of generalized n -species competition systems with multiple delays and impulses as follows: x i ′ ( t ) = x i ( t ) [ a i ( t ) - b i ( t ) x i ( t ) - ∑ j = 1 n c i j ( t ) x i ( t - τ i j ( t ) ) - ∑ j = 1 n d i j ( t ) x j ( t - γ i j ( t ) ) - ∑ j = 1 n e i j ( t ) ∫ - σ i j 0 f i j ( s ) x j ( t + s ) d s ]; a . e .; t > 0;t ≠ t k; k ∈ Z +; i = 1;n; x i ( t k + ) - x i ( t k - ) = θ i k x i ( t k ); i = 1;n; k ∈ Z +;and x i ′ ( t ) = x i ( t ) [ a i ( t ) - b i ( t ) x i ( t ) + ∑ j = 1 n c i j ( t ) x i ( t - τ i j ( t ) ) - ∑ j = 1 n d i j ( t ) x j ( t - γ i j ( t ) ) - ∑ j = 1 n e i j ( t ) ∫ - σ i j 0 f i j ( s ) x j ( t + s ) d s ]; a . e .; t > 0; t ≠ t k; k ∈ Z +; i = 1;n; x i ( t k + ) - x i ( t k - ) = θ i k x i ( t k ); i = 1;n; k ∈ Z + . It improves and generalizes a series of the well-known sufficiency theorems in the literature about the problems mentioned previously. Published: 2013 First available in Project Euclid: 26 February 2014 zbMATH: 1294.34078 MathSciNet: MR3093763 Digital Object Identifier: 10.1155/2013/980974
摘要：
By applying the fixed point theorem in a cone of Banach space, we obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solutions of two kinds of generalized.. -species competition systems with multiple delays and impulses as follows: x(i)(t)(t) = x(i)(t)[a(i)(t) - b(i)(t)x(i)(t) - Sigma(n)(j=1) c(ij)(t)x(i)(t - tau(ij)(t)) - Sigma(n)(j=1) d(ij)(t)x(i)(t - gamma(ij)(t)) - Sigma(n)(j=1) e(ij)(t)integral(0)(-sigma ij) f(ij)(s)x(j)(t + s)ds], a.e., t > 0, t not equal t(k), k is an element of Z(+), i = 1,2,..., n; x(i)(t(k)(+)) - x(i)(t(k)(-)) - theta(ik)x(i)(t(k)), i = 1,2,..., n, k is an element of Z(+); and x(ji)(t) - x(i)(t)[a(i)(t) - b(i)(t)x(i)(t) + Sigma(n)(j=1) c(ij)(t)x(i)(t - tau(ij)(t)) - Sigma(n)(j=1) d(ij)(t)x(j)(t - gamma(ij)(t)) - Sigma(n)(j=1) e(ij)(t) integral(0)(sigma ij) f(ij)(s)x(j)(t + s)ds, a.e., t > 0, t not equal t(k), k is an element of Z(+), i = 1,2,..., m; x(i)(t(k)(+)) - x(i)(t(k)(-)) = theta(ik)x(i)(t(k)), i = 1,2,...,n, k is an element of Z(+). It improves and generalizes a series of the well-known sufficiency theorems in the literature about the problems mentioned previously.
