Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces

Petryshyn, W. V.

doi:10.1090/S0002-9947-1973-0328688-2

Fixed point theorems for various classes of $1$-set-contractive and $1$-ball-contractive mappings in Banach spaces
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Trans. Amer. Math. Soc. 182 (1973), 323-352 Request permission

Abstract:

Let X be a real Banach space, D a bounded open subset of X, and $\bar D$ the closure of D. In §1 of this paper we establish a general fixed point theorem (see Theorem 1 below) for 1-set-contractions and 1-ball-contractions $T:\bar D \to X$ under very mild conditions on T. In addition to classical fixed point theorems of Schauder, Leray and Schauder, Rothe, Kransnoselsky, Altman, and others for T compact, Theorem 1 includes as special cases the earlier theorem of Darbo as well as the more recent theorems of Sadovsky, Nussbaum, Petryshyn, and others (see §1 for further contributions and details) for T k-set-contractive with $k < 1$, condensing, and 1-set-contractive. In §§2, 3, 4, and 5 of this paper Theorem 1 is used to deduce a number of known, as well as some new, fixed point theorems for various special classes of mappings (e.g. mappings of contractive type with compact or completely continuous perturbations, mappings of semicontractive type introduced by Browder, mappings of pseudo-contractive type, etc.) which have been recently extensively studied by a number of authors and, in particular, by Browder, Krasnoselsky, Kirk, and others (see §1 for details),

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