Theories of fixed point index and applications

This thesis is devoted to the study of theories of fixed point index for generalized and weakly inward maps of condensing type and weakly inward A-proper maps. In Chapter 1 we recall some basic concepts such as cones, wedges, measures of noncompactness and theories of fixed point index for compact and gamma-condensing self-maps. We also give some new results and provide new proofs for some known results. In Chapter 2 we study approximatively compact sets giving examples and proving new results. The concept of an approximatively compact set is of importance in defining our index for a generalized inward map since there exists upper semicontinuous multivalued metric projections onto the approximatively compact convex set. We also introduce the concept of an M1-set which will play an important role in defining our fixed point index for generalized inward maps of condensing type since there exists continuous single-valued metric projections onto an Ml-closed convex set. Many examples of M1-closed convex sets are given. Weakly inward sets and weakly inward maps are studied in detail. New properties and examples on such sets and maps are given. We also introduce the new concept of generalized inward sets and generalized inward maps. The class of generalized inward maps strictly contain the class of weakly inward maps. Several necessary and sufficient conditions for a map to be generalized inward and examples of generalized inward maps are given. In Chapter 3 we define a fixed point index for a generalized inward compact map defined on an approximatively compact convex set and obtain many new fixed point theorems and nonzero fixed point theorems. In particular, norm-type expansion and compression theorems for weakly inward continuous maps in finite dimensional Banach spaces are obtained, which have not been considered previously. In Chapter 4 we define a fixed point index for a generalized inward maps of condensing type defined on an M1-closed convex set and obtain many new fixed point theorems and nonzero fixed point theorems. We also apply the abstract theory to some perturbed Volterra equations. In Chapter 5 we define a fixed point index for weakly inward A-proper maps. We obtain new fixed point theorems, nonzero fixed point theorem and results on existence of eigenvalues. We also give an application of the abstract theory to the existence of nonzero positive solutions of boundary value problems for second order differential equations.