Fixed Point Theorems About Expansion And Compression Of Concave Functional Type And Applications

Recently, there are many fixed point theorems and they have been applied to many problems. Among these fixed point theorems are the types of expansion and compression in the forms of various kinds of functions. In chapter1of this thesis, the author gives the summary of these problems.In chapter2, we first construct a retract by the concave functional in a cone. Then with the help of the retract, we prove that the fixed point index of completely continuous operators is0with the expansion type, and the fixed point index is1with the compression type. So we get the fixed point theorem about cone expansion and compression of concave functional type. At last we use our results to consider the boundary value problems of differential equations.In chapter3, we first define the semi-concave functional, and we construct two retracts in Banach spaces by the semi-concave functional. Then we prove that the topological degree of operators is0with the expansion type, and the topological degree is1with the compression type. So we get the fixed point theorems about domain expansion and compression of semi-concave functional type. In the end we give an example of semi-concave functional to show its existence.