Fredholm alternative and boundary value problems

Sȩdziwy, Stanisław

doi:10.1090/S0002-9939-03-07256-3

Fredholm alternative and boundary value problems
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by Stanisław Sȩdziwy

Proc. Amer. Math. Soc. 132 (2004), 1779-1784

DOI: https://doi.org/10.1090/S0002-9939-03-07256-3

Published electronically: November 4, 2003

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Abstract:

This note presents a simple proof of A. Lasota’s application of the nonlinear Fredholm alternative to the existence proofs of the boundary value problems involving ordinary differential equations. It then uses Lasota’s result to get a stronger version of the theorem of Herzog and Lemmert on the Dirichlet boundary value problem for the second-order systems of ordinary differential equations.

References

Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
Svatopluk Fučík, Jindřich Nečas, Jiří Souček, and Vladimír Souček, Spectral analysis of nonlinear operators, Lecture Notes in Mathematics, Vol. 346, Springer-Verlag, Berlin-New York, 1973. MR 0467421
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
Gerd Herzog and Roland Lemmert, An existence theorem for systems of boundary value problems, Proc. Amer. Math. Soc. 128 (2000), no. 1, 157–160. MR 1622754, DOI 10.1090/S0002-9939-99-05011-X
M. A. Krasnoselski, A. I. Perow, A. I. Powolozki, and P. P. Sabrejko, Vektorfelder in der Ebene, Mathematische Lehrbücher, Band XIII, Akademie-Verlag, Berlin, 1966 (German). Aus dem Russischen übersetzt von Anita Bittner; In deutscher Sprache herausgegeben Leonhard Bittner. MR 0202121
A. Lasota, Sur une généralisation du premier théorème de Fredholm, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 89–94 (French). MR 156225
A. Lasota, Une généralisation du premier théorème de Fredholm et ses applications à la théorie des équations différentielles ordinaires, Ann. Polon. Math. 18 (1966), 65–77 (French). MR 194640, DOI 10.4064/ap-18-1-65-77
I. M. Kamenetzky, Sur l’interpolation au moyen des dérivées et sur les procédés d’interpolation correspondants. I, C. R. (Doklady) Acad. Sci. URSS (N.S.) 25 (1939), 356–358 (French). MR 0002003
Bernhard Ruf, A nonlinear Fredholm alternative for second order ordinary differential equations, Math. Nachr. 127 (1986), 299–308. MR 861733, DOI 10.1002/mana.19861270121
Franz Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Assisted by J. Berkowitz; With a preface by Jacob T. Schwartz. MR 0240668