Existence and multiplicity of positive solutions for multiparameter periodic systems - Wei Liping; Dai Yongqiang; Su Shunchang (Open Mathematics)
We deal with the existence and multiplicity of positive solutions for differential systems depending on two parameters, λ1,λ2{λ }_{1},{λ }_{2}, subjected to periodic boundary conditions. We establish the existence of a continuous curve ΓΓ 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({λ }_{1},{λ }_{2})\in {{\mathcal{O}}}_{1}, at least one positive solution if (λ1,λ2)∈Γ\left({λ }_{1},{λ }_{2})\in Γ , and at least two positive solutions if (λ1,λ2)∈O2\left({λ }_{1},{λ }_{2})\in {{\mathcal{O}}}_{2}. Our approach relies on the fixed point index theory and the method of lower and upper solutions.