Fixed Point Theorems On Product Cone And Applications To Differential Systems

In this doctoral dissertation, we mainly consider the existence of positive solutions for second-order ordinary differential systems and second-order elliptic systems which consist of two boundary value problems with Dirichlet data. Here, so-called positive solutions are such solutions whose components are both positive. It is obviously different from usual positive solutions (one of which is positive, while the other is nonnegative). In order to deal with these problems better, we develop a new idea that it is to change problems into the fixed point problems of the corresponding compactly continuous mappings on product cone K₁ × K₂ (where K₁,K₂ (?)X) instead of single cone in product space X × X. As to the computation of fixed point index for compactly continuous mappings on product cone, we establish a product formula;by means of the formula, the fundamental propositions of fixed point index and some results about the computation of fixed point index, we obtain some fixed point theorems;moreover, as applications, we consider the existence of positive solutions for "super-sublinear" second-order ordinary differential systems and "super-sublinear" second-order elliptic systems.In Chapter 2, we recapitulate some fundamental definitions and propositions of topological degrees and some fundamental results about the computation of topological degrees;in Chapter 3, applying relative definitions and propositions of topological degrees, we establish product formulae of topological degrees in product space, in particular, the product formula of fixed point index on product cone, moreover, combining with the fundamental propositions and the computation of fixed point index, we obtain some fixed point theorems on product cone.As applications, in Chapters 4 and 5, we consider the existence of positive solutions for "super-sublinear" second-order ordinary differential systems and "super-sublinear" second-order elliptic systems, respectively. By choosing proper open sets in product cone, verifying that corresponding compactly continuous mappings satisfy necessary conditions on the boundary of open setsand applying the fixed point theorems in Chapter 3 we establish the existence of positive solutions.