Existence and multiplicity of positive solutions for multi-parameter three-point differential equations system

In this paper, we study the existence and multiplicity of positive solutions for the differential equations system{x″+λa1(t)f1(x(t),y(t))=0,00λ,μ>0 are parameters, 0<η2⩽η1<10<η2⩽η1<1, α1,α2∈(0,1)α1,α2∈(0,1), a1∈C([0,1],[0,∞))a1∈C([0,1],[0,∞)) and f2∈C([0,∞)×[0,∞),[0,∞))f2∈C([0,∞)×[0,∞),[0,∞)). The system is a semi-positone problem since the nonlinear term f1f1 is allowed to take negative values and a2(t)a2(t) may change sign on [0,1][0,1]. The results are established via fixed point index theory.