On existence and uniqueness of solutions for ordinary differential equations with nonlinear boundary conditions

Alessandro Calamai

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 2, page 469-481
  • ISSN: 0392-4041

Abstract

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We prove an existence and uniqueness theorem for a nonlinear functional boundary value problem, that is, an ordinary differential equation with a nonlinear boundary condition. The proof is based on a Global Inversion Theorem of Ambrosetti and Prodi, which is applied to the boundary operator restricted to the manifold of the global solutions to the equation. Our result is a generalization of an analogous existence and uniqueness theorem of G. Vidossich, as it is shown with some examples.

How to cite

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Calamai, Alessandro. "On existence and uniqueness of solutions for ordinary differential equations with nonlinear boundary conditions." Bollettino dell'Unione Matematica Italiana 7-B.2 (2004): 469-481. <http://eudml.org/doc/195957>.

@article{Calamai2004,
abstract = {We prove an existence and uniqueness theorem for a nonlinear functional boundary value problem, that is, an ordinary differential equation with a nonlinear boundary condition. The proof is based on a Global Inversion Theorem of Ambrosetti and Prodi, which is applied to the boundary operator restricted to the manifold of the global solutions to the equation. Our result is a generalization of an analogous existence and uniqueness theorem of G. Vidossich, as it is shown with some examples.},
author = {Calamai, Alessandro},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {469-481},
publisher = {Unione Matematica Italiana},
title = {On existence and uniqueness of solutions for ordinary differential equations with nonlinear boundary conditions},
url = {http://eudml.org/doc/195957},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Calamai, Alessandro
TI - On existence and uniqueness of solutions for ordinary differential equations with nonlinear boundary conditions
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/6//
PB - Unione Matematica Italiana
VL - 7-B
IS - 2
SP - 469
EP - 481
AB - We prove an existence and uniqueness theorem for a nonlinear functional boundary value problem, that is, an ordinary differential equation with a nonlinear boundary condition. The proof is based on a Global Inversion Theorem of Ambrosetti and Prodi, which is applied to the boundary operator restricted to the manifold of the global solutions to the equation. Our result is a generalization of an analogous existence and uniqueness theorem of G. Vidossich, as it is shown with some examples.
LA - eng
UR - http://eudml.org/doc/195957
ER -

References

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  1. AMBROSETTI, A.- PRODI, G., A Primer of Nonlinear Analysis, Cambridge University Press, Cambridge (1993). Zbl0781.47046MR1225101
  2. SCHWARTZ, J. T., Nonlinear Functional Analysis, Gordon and Breach, New York (1969). Zbl0203.14501MR433481
  3. SMALE, S., An infinite dimensional version of Sard's theorem, American Journal of Mathematics, 87 (1965), 861-866. Zbl0143.35301MR185604
  4. VIDOSSICH, G., Differentiability of solutions of boundary value problems with respect to data, Journal of Differential Equations, 172 (2001), 29-41. Zbl1003.34016MR1824083

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