Nonconvex Retracts And Computation For Fixed Point Index In Cones

In this paper, we construct three nonconvex retracts in cones. With nonconvex retracts, we calculate the fixed point index in cones.Firstly, the development and present situation of the expansion and compression fixed point theorem are introduced briefly.Then, three retracts are constructed in a cone by nonnegative functionals of convex and concave types, and some examples are given to illustrate that the retracts are nonconvex(because closed convex sets are the retracts).Finally, the nonconvex retracts are used to compute the fixed point index for the completely continuous operator in the domains D1∩D2and D1∪D2, where D1and D2are bounded open sets in the cones. The computation for fixed point index can be applied to the existence and the more precise location of positive fixed points.