乘积空间中凹泛函型锥拉伸与压缩不动点定理

Fixed Point Theorems of Cone Expansion and Compression of Concave Functional Type in Product Space

考虑赋范线性空间的乘积空间,由因子空间中的锥生成乘积空间中的锥.全连续算子定义在乘积空间中锥与两个闭球相交得到的有界闭集上,并且值域在锥中.在由锥上一类非负正齐次凹泛函表示的混合型锥拉伸与压缩条件下,利用构造性方法将其转化为Schauder型问题,证明了几个全连续算子的不动点定理.通过例子说明这里所需要的凹泛函在常用的空间及其锥上是容易构造的.

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.