Topological and Coincidence Degree based on the Degree of Non-Densifiability and Applications

Abstract

Using the concept of the degree of nondensifiability in Banach spaces, we develop a new version of the topological degree based on retraction mappings. This approach combines the degree of nondensifiability with the Leray--Schauder degree. As applications, we establish Leray--Schauder and Schaefer type fixed point theorems. Moreover, we introduce a version of Mawhin's coincidence degree via this framework for monotone operator contractions with respect to the degree of nondensifiability. Our results extend and unify several classical and recent contributions available in the literature. Finally, we apply the main theoretical results to investigate the existence of solutions for certain classes of differential equations subject to boundary value conditions.