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, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {\lambda }_{1},{\lambda }_{2} , subjected to periodic boundary conditions. We establish the existence of a continuous curve <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> </m:math> \Gamma that separates the first quadrant into two disjoint unbounded open sets <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">O</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {{\mathcal{O}}}_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">O</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {{\mathcal{O}}}_{2} . Specifically, we prove that the periodic system has no positive solutions if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">O</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> \left({\lambda }_{1},{\lambda }_{2})\in {{\mathcal{O}}}_{1} , at least one positive solution if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Γ</m:mi> </m:math> \left({\lambda }_{1},{\lambda }_{2})\in \Gamma , and at least two positive solutions if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">O</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> \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.