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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)].)

47J05 Equations involving nonlinear operators (general)
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
47A53 (Semi-) Fredholm operators; index theories
Full Text: EuDML
[1] S. Ahmad: A resonance problem in which the nonlinearity may grow linearly. Proc. Amer. Math. Soc. 92 (1984), 381-384. · Zbl 0562.34011
[2] L. Cesari, R. Kannan: An abstract existence theorem at resonance. Proc. Amer. Math. Soc. 81 (1977), 221-225. · Zbl 0361.47021
[3] K. Deimling: Nonlinear Functional Analysis. Springer-Verlag, Berlin-Heidelberg, 1985. · Zbl 0559.47040
[4] C. Fabry, C. Franchetti: Nonlinear equations with growth restrictions on the nonlinear term. J. Differential Equations 20 (1976), 283-291. · Zbl 0326.47060
[5] S. Fučík: Surjectivity of operators involving linear noninvertible part and nonlinear compact perturbation. Funkcialaj Ekvacioj, 17 (1974), 73 - 83.
[6] S. Fučík A. Kufner: Nelinearni diferenciâlnî rovnice. Praha, SNTL, 1978.
[7] М. А. Красносельский Є. А. Лифшиц А. В. Соболев: Позитивные линейные скстемы-Метод положительных операторов. Hauka, Москва, 1985. · Zbl 1223.81132
[8] J. Mawhin: Topological Degree Methods in Nonlinear Boundary Value Problems. Regional Confer. Series in Math. No. 40, Amer. Math. Soc., Previdence, 1979. · Zbl 0414.34025
[9] J. Mawhin: Boundary value problems with nonlinearities having infinite jumps. Commentat. Math. Univ. Carol. 25 (1984), 401-414. · Zbl 0562.34010
[10] A. Taylor: Introduction to Functional Analysis. John Wiley and Sons, Inc. New York 1958 · Zbl 0081.10202
[11] J. R. Ward, Jr.: Existence theorems for nonlinear boundary value problems at resonance. J. Differential Equations 35 (1980), 232-247. · Zbl 0447.34015
[12] K. Yosida: Functional Analysis. Springer-Verlag, Berlin, 1980. · Zbl 0435.46002
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