×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: EuDML
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.