Vidossich: On Nonlinear Systems of BVPs with Positive Green's Functions

Vidossich, Giovanni:
On Nonlinear Systems of BVPs with Positive Green's Functions
Bollettino dell'Unione Matematica Italiana Serie 9 6 (2013), fasc. n.3, p. 607-642, (English)
pdf (429 Kb), djvu (311 Kb). | MR 3202843

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This paper provides some existence and uniqueness theorems for nonlinear systems of BVPs where the Green's functions for the linearization have constant sign (hence these results apply, e.g., to Dirichlet problems for elliptic PDEs as well as to various multipoint BVPs for higher order ODEs). Proofs are based on an original way of using the Linear Functional Analysis of ordered Banach spaces in connection with the traditional topological methods of Nonlinear Functional Analysis.

Referenze Bibliografiche

[1] S. AHMAD and A.C. LAZER, Positive operators and Sturmian theory of non selfadjoint second-order sysstems, pp. 25-42 in: V. LAKSHMIKANTHAM, ``Nonlinear Equa-tions in Abstract Spaces'', Academic Press, New York, 1978. | MR 502533

[2] J. ALBRECHT, Zur Wahl der Norm beim Iterationsverfahren für Randwertaufgaben, ZAMM 52 (1972), 626-628. | fulltext (doi) | MR 405867 | Zbl 0253.65047

[3] H. AMANN, On the unique solvability of semi-linear operator equations in Hilbert spaces, J. Math. Pures Appl. 61 (1982), 149-175. | MR 673303 | Zbl 0501.47024

[4] H. AMANN, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review 18 (1976), 620-709. | fulltext (doi) | MR 415432 | Zbl 0345.47044

[5] H. BREZIS, ``Analyse fonctionelle (Théorie et applications), Masson, Paris, 1983. | MR 697382

[6] W.A. COPPEL, ``Disconjugacy'', LN in Math. 220, Springer, New York, 1971. | MR 460785

[7] G. DEGLA, On the principal eigenvalue of disconjugate BVPs with $L^1$-coefficients, Adv. Nonlinear Stud. 2 (2002), 19-39. | fulltext (doi) | MR 1881997 | Zbl 1013.34009

[8] J. DUGUNDJI, ``Topology'', Allyn and Bacon, Boston, 1966. | MR 193606

[9] J.J. DUISTERMAAT and J.A.C. KOLK, ``Multidimensional Real Analysis'', voll. I-II, Cambridge Univ. Press, Cambridge, 2004. | fulltext (doi) | MR 2121977 | Zbl 1077.26002

[10] U. ELIAS, ``Oscillation theory of Two-Term Differential Equations'', Kluwer Academic Publishers, Dodrecht, 1997. | fulltext (doi) | MR 1445292 | Zbl 0878.34022

[11] D.D. HAI and K. SCHMITT, Existence and uniqueness results for nonlinear boundary value problems, Rocky Mountain J. Math. 24(1994), 77-91. | fulltext (doi) | MR 1270028 | Zbl 0807.34025

[12] A. HAMMERSTEIN, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math. 54 (1930), 117-176. | fulltext (doi) | MR 1555304

[13] R.A. HORN and C.R. JOHNSON, ``Matrix Analysis'', Cambridge Univ. Press, Cambridge, 1985. | fulltext (doi) | MR 832183 | Zbl 0576.15001

[14] J.L. KAZDAN and F.W. WARNER, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567-597. | fulltext (doi) | MR 477445 | Zbl 0325.35038

[15] M.A. KRASNOSELSKI, On the theory of completely continuous fields (in Russian), Ukrain. Mat. Ž 3 (1951), 174-183. | MR 55677

[16] M.A. KRASNOSELSKI, JE.A. LIFSHITS and A.V. SOBOLEV, ``Positive Linear Systems (The Method of Positive Operators)'', Heldermann Verlag, Berlin, 1989. | MR 1038527

[17] M.A. KRASNOSELSKI, ``Topological Methods in the Theory of Nonlinear Integral Equations'', Pergamon Press, New York, 1963.

[18] A. LASOTA, Une généralisation du premier théorème de Fredholm et ses applications à la théorie des équations différentielles ordinaires, Annales Polon. Math. 18 (1966), 65-77. | fulltext EuDML | fulltext (doi) | MR 194640 | Zbl 0139.09201

[19] A. LASOTA, Sur l'existence et l'unicité des solutions du problème aux limites de Nicoletti pour un système d'équations différentielles ordinaires, Zeszyty Nauk. UJ, Prace Mat. 11 (1966), 41-48. | MR 281986 | Zbl 0286.34025

[20] J. MAWHIN, Two point boundary value problems for nonlinear second order differential equations in Hilbert space, Tôhoku Math. J. 32 (1980), 225-233. | fulltext (doi) | MR 580278 | Zbl 0436.34057

[21] J. TIPPETT, A existence-uniqueness theorem for two point baoundary value problems, SIAM J. Math. Anal. 5 (1974), 153-157. | fulltext (doi) | MR 338496 | Zbl 0245.34016

[22] G. VIDOSSICH, A general existence theorem for boundary value problems for ordinary differential equations, Nonlinear Anal. TMA 15 (1990), 897-914. | fulltext (doi) | MR 1081661 | Zbl 0738.34018

[23] G. VIDOSSICH, ``Lectures on the Topological Degree'', in the (hopefully!) final preparation.

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Referenza completa

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Referenze Bibliografiche