Positive periodic solutions for multiparameter nonlinear differential systems with delays

Abstract

Abstract We establish several criteria for the existence of positive periodic solutions of the multi-parameter differential systems {u′(t)+a1(t)g1(u(t))u(t)=λb1(t)f(u(t−τ1(t)),v(t−ζ1(t))),v′(t)+a2(t)g2(v(t))v(t)=μb2(t)g(u(t−τ2(t)),v(t−ζ2(t))), $$\left \{ \textstyle\begin{array}{l} u'(t)+a_{1}(t)g_{1}(u(t))u(t)=\lambda b_{1}(t)f(u(t-\tau_{1}(t)),v(t-\zeta_{1}(t))), \\ v'(t)+a_{2}(t)g_{2}(v(t))v(t)=\mu b_{2}(t)g(u(t-\tau_{2}(t)),v(t-\zeta_{2}(t))), \end{array}\displaystyle \right . $$ where the functions g1,g2:[0,∞)→[0,∞) $g_{1}, g_{2}:[0,\infty)\to[0,\infty)$ are assumed to be unbounded. The analysis in the paper relies on the classical fixed point index theory. Our main findings improve and complement some existing results in the literature.

Keywords

• Positive periodic solutions
• Existence
• Multiparameter systems
• Fixed point