Existence and multiplicity of positive solutions for multiparameter periodic systems

Abstract

We deal with the existence and multiplicity of positive solutions for differential systems depending on two parameters, λ1,λ2{\lambda }_{1},{\lambda }_{2}, subjected to periodic boundary conditions. We establish the existence of a continuous curve Γ\Gamma that separates the first quadrant into two disjoint unbounded open sets O1{{\mathcal{O}}}_{1} and O2{{\mathcal{O}}}_{2}. Specifically, we prove that the periodic system has no positive solutions if (λ1,λ2)∈O1\left({\lambda }_{1},{\lambda }_{2})\in {{\mathcal{O}}}_{1}, at least one positive solution if (λ1,λ2)∈Γ\left({\lambda }_{1},{\lambda }_{2})\in \Gamma , and at least two positive solutions if (λ1,λ2)∈O2\left({\lambda }_{1},{\lambda }_{2})\in {{\mathcal{O}}}_{2}. Our approach relies on the fixed point index theory and the method of lower and upper solutions.

Keywords

• positive solution
• non-existence/existence
• periodic systems
• lower and upper solutions
• 34b15
• 34b18