Abstract
Let X, Y be Banach spaces, L: X ⊃ Dom(L)→Y a linear Fredholm operator with Fredholm index zero, and T a asymptotic linear k-set contraction with k∈[0, 1(L)), where 1(L) denotes the lower bound of L with respect to the set-measure of noncampactness. Using coincidence degree we deduce a Fredholm alternative for the equation Lx=Tx+y (y∈Y), which involves the results of [11] and [17] in the case X=Y and L=Id. Applications are given for a functional differential equation of neutral type and for a boundary value problem of a second order differential equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literatur
AMANN, H.: Fixed points of asymptotically linear maps in ordered Banach spaces. J. Funkt. Anal, 14 (1973), 162–171.
AMANN, H.: Existence of multiple solutions. Indiana Univ. Math. 21 (1972), 925–935.
GOLDBERG, S.: Unbounded linear operators. New-York (1966).
HAF, H.: Zur Existenztheorie für ein nichtlineares Randwertproblem der Potentialtheorie bei beliebigem komplexen Parameter. J. Math. Anal. Appl. 42 (1973), 627–638.
HALE, J. K.: Functional differential equations, Springer Verl. (1971).
HALE, J.K.: α-contractions and differential equations. In: equations différentielles et fonctionelles non linéaires, Hrsg. P. Janssens, J. Mahwin, H. Rouche, Paris (1973).
HENRY, D.: The adjoint of a linear functional differential equation and boundary value problems. J. Diff. Equations 9 (1971), 55–66.
HETZER, G.: Some remarks on ø+-Operators and on the coincidence degree for a Fredholm equation with non compact non linear perturbations (erscheint noch).
HETZER, G.: Some applications of the coincidence degree for set-contractions to functional differential equations of neutral type (erscheint noch).
JÖRGANS, K.: Lineare Integraloperatoren. Teubner (1970), Stuttgart.
KACHUROWSKII, R. I.: On Fredholm theory for non linear operator equation. Soviet Math. Dokl. 11 (1970), 751–754.
MAWHIN, J.: Equivalence theorems for non linear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces. J. Diff. Equations 12 (1972), 610–636.
MAWHIN, J.: Non linear perturbations of Fredholm mappings in normed spaces and applications to differential equations. Trabalho de Matematica Nr. 61, Univ. Brasilia (1974).
MAWHIN, J.: Problemes aux limites, du type Neumann pour certaines équations différentielles ou aux dérivées partielles non linéaires. In “Equa-Diff 73”, Hermann, Paris 1973.
NUSSBAUM, R. D.: Degree theory for local condensing maps. J. Math. Anal. Appl. 37 (1972), 741–766.
NUSSBAUM, R. D.: Estimates for the number of solutions of operator equations. Appl. Analysis 1 (1971), 183–200.
PETRYSHYN, W. V.: Fredholm alternative for non linear k-ball-contractive mappings with applications. J. Diff. Equations 17 (1975), 82–95.
SADOVSKII, B. N.: Limit-compact and condensing operators. Russian Math. Surveys 27 (1972), 85–155.
STUART, C. A.: TOLAND, J. F.: The fixed point index of a linear k-set contraction. J. London Math. Soc. 6 (1973), 317–320.
WENDLAND, W.: Bemerkungen über die Predholmschen Sätze. In Meth. u. Verf. der math. Physik, BI-Mannheim 3 (1970), 141–176.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hetzer, G., Stallbohm, V. Eine Existenzaussage für asymptotisch lineare Störungen eines Fredholmoperators mit Index O. Manuscripta Math 21, 81–100 (1977). https://doi.org/10.1007/BF01176903
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01176903