On Nonlinear Systems of BVPs with Positive Green's Functions

Giovanni Vidossich

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 3, page 607-642
  • ISSN: 0392-4041

Abstract

top
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.

How to cite

top

Vidossich, Giovanni. "On Nonlinear Systems of BVPs with Positive Green's Functions." Bollettino dell'Unione Matematica Italiana 6.3 (2013): 607-642. <http://eudml.org/doc/294038>.

@article{Vidossich2013,
abstract = {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.},
author = {Vidossich, Giovanni},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {607-642},
publisher = {Unione Matematica Italiana},
title = {On Nonlinear Systems of BVPs with Positive Green's Functions},
url = {http://eudml.org/doc/294038},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Vidossich, Giovanni
TI - On Nonlinear Systems of BVPs with Positive Green's Functions
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/10//
PB - Unione Matematica Italiana
VL - 6
IS - 3
SP - 607
EP - 642
AB - 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.
LA - eng
UR - http://eudml.org/doc/294038
ER -

References

top
  1. AHMAD, S. and LAZER, A.C., 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. MR502533
  2. ALBRECHT, J., Zur Wahl der Norm beim Iterationsverfahren für Randwertaufgaben, ZAMM52 (1972), 626-628. Zbl0253.65047MR405867DOI10.1002/zamm.19720521012
  3. AMANN, H., On the unique solvability of semi-linear operator equations in Hilbert spaces, J. Math. Pures Appl.61 (1982), 149-175. Zbl0501.47024MR673303
  4. AMANN, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review18 (1976), 620-709. Zbl0345.47044MR415432DOI10.1137/1018114
  5. BREZIS, H., ``Analyse fonctionelle (Théorie et applications), Masson, Paris, 1983. MR697382
  6. COPPEL, W.A., ``Disconjugacy'', LN in Math.220, Springer, New York, 1971. MR460785
  7. DEGLA, G., On the principal eigenvalue of disconjugate BVPs with L 1 -coefficients, Adv. Nonlinear Stud.2 (2002), 19-39. Zbl1013.34009MR1881997DOI10.1515/ans-2002-0102
  8. DUGUNDJI, J., ``Topology'', Allyn and Bacon, Boston, 1966. MR193606
  9. DUISTERMAAT, J.J. and KOLK, J.A.C., ``Multidimensional Real Analysis'', voll. I-II, Cambridge Univ. Press, Cambridge, 2004. Zbl1077.26002MR2121977DOI10.1017/CBO9780511616723.003
  10. ELIAS, U., ``Oscillation theory of Two-Term Differential Equations'', Kluwer Academic Publishers, Dodrecht, 1997. Zbl0878.34022MR1445292DOI10.1007/978-94-017-2517-0
  11. HAI, D.D. and SCHMITT, K., Existence and uniqueness results for nonlinear boundary value problems, Rocky Mountain J. Math.24(1994), 77-91. Zbl0807.34025MR1270028DOI10.1216/rmjm/1181072453
  12. HAMMERSTEIN, A., Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math.54 (1930), 117-176. MR1555304DOI10.1007/BF02547519
  13. HORN, R.A. and JOHNSON, C.R., ``Matrix Analysis'', Cambridge Univ. Press, Cambridge, 1985. Zbl0576.15001MR832183DOI10.1017/CBO9780511810817
  14. KAZDAN, J.L. and WARNER, F.W., Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math.28 (1975), 567-597. Zbl0325.35038MR477445DOI10.1002/cpa.3160280502
  15. KRASNOSELSKI, M.A., On the theory of completely continuous fields (in Russian), Ukrain. Mat. Ž3 (1951), 174-183. MR55677
  16. KRASNOSELSKI, M.A., LIFSHITS, JE.A. and SOBOLEV, A.V., ``Positive Linear Systems (The Method of Positive Operators)'', Heldermann Verlag, Berlin, 1989. MR1038527
  17. KRASNOSELSKI, M.A., ``Topological Methods in the Theory of Nonlinear Integral Equations'', Pergamon Press, New York, 1963. 
  18. LASOTA, A., Une généralisation du premier théorème de Fredholm et ses applications à la théorie des équations différentielles ordinaires, Annales Polon. Math.18 (1966), 65-77. Zbl0139.09201MR194640DOI10.4064/ap-18-1-65-77
  19. LASOTA, A., 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. Zbl0286.34025MR281986
  20. MAWHIN, J., Two point boundary value problems for nonlinear second order differential equations in Hilbert space, Tôhoku Math. J.32 (1980), 225-233. Zbl0436.34057MR580278DOI10.2748/tmj/1178229639
  21. TIPPETT, J., A existence-uniqueness theorem for two point baoundary value problems, SIAM J. Math. Anal.5 (1974), 153-157. Zbl0245.34016MR338496DOI10.1137/0505017
  22. VIDOSSICH, G., A general existence theorem for boundary value problems for ordinary differential equations, Nonlinear Anal. TMA15 (1990), 897-914. Zbl0738.34018MR1081661DOI10.1016/0362-546X(90)90073-P
  23. VIDOSSICH, G., ``Lectures on the Topological Degree'', in the (hopefully!) final preparation. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.