Fixed point theorems of cone expansion and compression of concave functional type in product space

A product space of normed linear spaces is considered, and the cone in the product space is produced by the cones in its factor spaces. A completely continuous operator is in the product space defined on the bounded closed set which is the intersection of the cone with two closed balls, and the range is in the cone. Under the mixed cone expansion and compression conditions that are expressed through a class of nonnegative, positively homogeneous, concave functionals on the cone, some fixed point theorems about the completely continuous operator are proved by constructing methods and converting them into the problems of Schauder type. It is illustrated by example that the concave functionals needed here are easily constructed in a common space and on a cone in it. ©, 2015, Northeastern University. All right reserved.