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New fixed point theorems for 1-set-contractive operators in Banach spaces. (English) Zbl 1124.47038

This article presents a list of sufficient conditions for the existence of fixed points of semi-closed 1-set-contractive operators in real Banach spaces. All results presented here are evident corollaries of the following analogue for semi-closed 1-set-contractive operators of the Leray-Schauder fixed point principle: If A is a semi-closed 1-set-contractive operator such that

Ax-x 0 k(x-x 0 )forsomex 0 ΩandforallxΩ,1<k<,(*)

then A has at least one fixed point in Ω ¯.

In turn, this result is an evident corollary of the statement that deg(I-A,Ω)=1 provided that A is semi-closed 1-set-contractive operator satisfying the condition Ax-x 0 k(x-x 0 ) for some x 0 Ω and for all xΩ, 1k<. The main part of the article is devoted to inequalities that imply (*); among others, the inequality

Ax-x α Ax α+β -x α ,xΩ,

(α>1, β0) is treated.

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