In the paper,the boundary conditions(Sω,A,a)and(LM,ω)are introduced,the resultson the extension and the computation of fixed point index for k-set-contraction locally properfield are established.Use them with the weak semidifferentiability, the problems concerningthe positive solution,the multiple positive solutions and the set of eigenvectors for a class ofnonlinear equations are studied.And as an application, a two point boundary value problem isstudied in this paper.