Fixed point theorems of 1-set-contractive operators in Banach spaces

Abstract

Petryshyn [W.V. Petryshyn, Remark on condensing and k -set-contractive mappings, J. Math. Anal. Appl. 39 (1972) 717–741] and Nussbaum [R.D. Nussbaum, Degree theory for local condensing maps, J. Math. Anal. Appl. 37 (1972) 741–766] have defined the topological degree of 1-set-contractive fields and studied fixed point theorems of 1-set-contractive operators by virtue of the potential tool. In this paper, we continue to investigate boundary conditions, under which the topological degree of 1-set-contractive fields, deg ( I − A , D , p ) , is equal to unity or zero. Correspondingly, we can obtain some new fixed point theorems of 1-set-contractive operators and existence theorems of solutions for the equation A x = μ x , which improve and extend many famous theorems (e.g., Leray–Schauder’s theorem, Rothe’s theorem, Krasnoselskii’s theorem, Altman’s theorem, Petryshyn’s theorem etc.). On the other hand, this class of 1-set-contractive operators include completely continuous operators, strict set-contractive operators, condensing operators, non-expansive maps, semi-contractive maps, LANE maps and others [L.S. Liu, Approximation theorems and fixed point theorems for various class of 1-set-contractive mappings in Banach spaces, Acta Math. Sinica 17 (2001) 103–112 (in English)]. So the results in this paper remain valid for the above maps. In addition, our conclusions and methods are different from ones in many recent works (e.g., [L.S. Liu, Approximation theorems and fixed point theorems for various class of 1-set-contractive mappings in Banach spaces, Acta Math. Sinica 17 (2001) 103–112 (in English)]).