Positive periodic solutions of neutral logistic equations with distributed delays

Abstract

Using a fixed point theorem of strict-set-contraction, we establish criteria for the existence of positive periodic solutions for the periodic neutral logistic equation, with distributed delays, $$ x'(t)= x(t)Big[a(t)-sum_{i=1}^n a_i(t)int_{-T_i}^0 x(t+heta), mathrm{d}mu_i(heta)- sum_{j=1}^m b_j(t) int_{-hat{T}_j}^0 x'(t+heta),mathrm{d} u_j(heta)Big], $$ where the coefficients $a, a_i ,b_j$ are continuous and periodic functions, with the same period. The values $T_i, hat{T}_j$ are positive, and the functions $mu_i, u_j$ are nondecreasing with $int_{-T_i}^0,{ m d} mu_i=1$ and $int_{-hat{T}_j}^0,{ m d} u_j=1$.

Keywords

• Positive periodic solution
• neutral delay logistic equation
• strict-set-contraction.