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Zbl 0721.47048
Ĺ eda, Valter
Some remarks to coincidence theory.
(English)
[J] Czech. Math. J. 38(113), No.3, 554-572 (1988). ISSN 0011-4642; ISSN 1572-9141/e

The aim of the present paper is to prove criteria for the existence of solutions of an abstract equation $(*)\quad Lx=Fx$ for F of controlled growth. Well within the frame of the alternative method [see, e.g., the reviewer, in Nonlinear functional analysis and differential equations, East Lansing 1975, 1-197, New York (1976; Zbl 0343.47038)], let $X\subset Z$ be real Banach spaces, let $L: D(L)\subset X\to Z$ be a linear map not necessarily bounded, with finite-dimensional ker L and finite-dimensional Im L, and consider the usual projection operators $P: X\to X,$, $Q: Z\to Z,$ with $Im P=\ker L,\ker Q=Im L,$ $X=\ker L\oplus \ker P,$ $Z=Im Q\oplus Im L$ (direct sums). Then L has a partial inverse $K\sb P: Im L\to D(L)\cap \ker P.$ The following assumptions are often made: \par $(L\sb 1)$ L is a Fredholm map of index zero, i.e., $0=Ind L=\dim \ker L- co\dim Im L;$ \par $(L\sb 2)$ $K\sb P: Im L\subset Z\to X$ is continuous; \par $(L\sb 3)$ $K\sb P$ is completely continuous; \par $(L\sb 4)$ Im $L\cap \ker L=\{0\}.$ \par Other operators and other alternate assumptions are made here yielding a fine spectral analysis of the relevant operators. By the use of index theory the author proves the rather involved criterion. \par (I): Under assumptions $(L\sb 1)$, $(L\sb 2)$, $(L\sb 3)$, $(L\sb 4)$, let $F:X\to Z$ be a continuous map which transforms bounded sets into bounded sets and which satisfies the following two conditions: \par $(F\sb 1)$ there are constants $a,b>0$ such that $a\Vert K\sb P\Vert <1,\Vert Fx\Vert\sb Z\le a\Vert x\Vert\sb X+b$ for all $x=\bar x+\tilde x\in D(L),\bar x\in \ker L,\tilde x\in \ker P;$ and \par $(F\sb 2)$ for $\epsilon =\pm 1$, $R\sb 1=\Vert K\sb P\Vert b/(1-a\Vert K\sb P\Vert),$ there is $R\sb 2>0$ such that $\epsilon F(\bar x+\tilde x)+k\bar x\in Im L$ for all $x=\bar x+\tilde x\in D(L),\Vert \bar x\Vert\sb X\ge R\sb 2,\Vert \tilde x\Vert\sb X<R\sb 1,$ $k\in {\bbfR}$, implies $k\le 0;$ \par then (*) has at least a solution $x\in D(L).$ \par The author then proves other criteria, and extends a statement of Mawhin to Banach lattices. (For X,Z Banach spaces and F of controlled growth existence criteria were established in two papers by the reviewer [J. Differential Equations 28, No.1, 43-59 (1978; Zbl 0395.34034); Nonlinear analysis, 43-67, New York (1978; Zbl 0463.47044)]. For one of these criteria with $X=Z$ Hilbert and F bounded, which is included in (I), {\it R. Kannan} and {\it P. J. McKenna jun.} gave a two line proof [Bol. Un. Mat. Ital., V. Ser. A 14, No.2, 355-358 (1977; Zbl 0352.47030)].)
MSC 2000:
*47J05 Equations involving nonlinear operators (general)
47J10 Nonlinear eigenvalue problems
47A53 (Semi-)Fredholm operators; index theories

Keywords: real Banach spaces; projection operators; Fredholm map of index zero; completely continuous; Banach lattices; controlled growth

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