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Some remarks to coincidence theory. (English) Zbl 0721.47048
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 $$K_ P: Im L\to D(L)\cap \ker P.$$ The following assumptions are often made:
$$(L_ 1)$$ L is a Fredholm map of index zero, i.e., $$0=Ind L=\dim \ker L- co\dim Im L;$$
$$(L_ 2)$$ $$K_ P: Im L\subset Z\to X$$ is continuous;
$$(L_ 3)$$ $$K_ P$$ is completely continuous;
$$(L_ 4)$$ Im $$L\cap \ker L=\{0\}.$$
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.
(I): Under assumptions $$(L_ 1)$$, $$(L_ 2)$$, $$(L_ 3)$$, $$(L_ 4)$$, let $$F:X\to Z$$ be a continuous map which transforms bounded sets into bounded sets and which satisfies the following two conditions:
$$(F_ 1)$$ there are constants $$a,b>0$$ such that $$a\| K_ P\| <1,\| Fx\|_ Z\leq a\| x\|_ X+b$$ for all $$x=\bar x+\tilde x\in D(L),\bar x\in \ker L,\tilde x\in \ker P;$$ and
$$(F_ 2)$$ for $$\epsilon =\pm 1$$, $$R_ 1=\| K_ P\| b/(1-a\| K_ P\|),$$ there is $$R_ 2>0$$ such that $$\epsilon F(\bar x+\tilde x)+k\bar x\in Im L$$ for all $$x=\bar x+\tilde x\in D(L),\| \bar x\|_ X\geq R_ 2,\| \tilde x\|_ X<R_ 1,$$ $$k\in {\mathbb{R}}$$, implies $$k\leq 0;$$
then (*) has at least a solution $$x\in D(L).$$
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), R. Kannan and 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:
 47J05 Equations involving nonlinear operators (general) 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems 47A53 (Semi-) Fredholm operators; index theories
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References:
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