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Fixed point theorems of 1-set-contractive operators in Banach spaces. (English) Zbl 1113.47045

The article deals with degree theory and fixed point theorems for 1-set-contractive operators in Banach spaces. The authors present 14 theorems about conditions under which the degree of vector fields with 1-set-contractive operators is equal to 1 or 0 and corresponding fixed point results; all these theorems are modifications for vector fields with 1-set-contractive operators of well-known ones by E. Rothe, J. Leray and J. Schauder, M. Krasnosel’skij, M. Altmann, W. V. Petryshin and other mathematicians concerning vector fields with completely continuous or condensing operators.

Nevertheless, the authors’ results appear doubtful. First of all, some known results are cited in the article in incorrect form; for example, the authors write (as if due to E. Rothe) that a condensing operator A:D ¯E (D is a bounded open subset in a Banach space E), without fixed points on D and satisfying the condition Axx for each xD, has at least one fixed point. This statement is false even for one-dimensional E; nonetheless, later the authors prove it in the case when A is a semi-closed 1-set-contractive A. Second, the authors do not formulate exact definitions of 1-set-contractive operators and this does not allow to check and estimate the authors’ arguments.

47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H11Degree theory (nonlinear operators)
47H09Mappings defined by “shrinking” properties