“fixed point index”造句，怎么用fixed point index造句

By employing the fixed point index of condensing mapping, the existence results of positive solution are obtained under two cases, the conclusions extend the results recently achieved in this field.

By using a well-known fixed point index theorem, we obtain the existence, multiplicity and nonexistence of positive periodic solution(s) to this equation.

By using fixed point index theory in a cone, we study the existence of positive solutions of boundary value problems for systems of nonlinear second order singular differential equations.