Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-05T02:39:58.767Z Has data issue: false hasContentIssue false
  • English
  • Français

Systems of differential equations with fully nonlinear boundary conditions

Published online by Cambridge University Press:  17 April 2009

H.B. Thompson
H.B. Thompson
Affiliation:
Department of Mathematics, The University of Queensland, Queensland 4072, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give sufficient conditions involving f, g and ω in order that systems of differential equations of the form y″ = f(x, y, y′), x in [0, 1] with fully nonlinear boundary conditions of the form g((y(0), y(1)), (y′(0), y′(1))) = 0 have solutions y with (x, y) in . We use Schauder degree theory in a novel space. Well known existence results for the Picard, the periodic and the Neumann boundary conditions follow as special cases of our results.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 2 , October 1997 , pp. 197 - 208
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Bebernes, J.W. and Schmitt, K., ‘Periodic boundary value problems for systems of second order ordinary differential equations’, J. Differential Equations 13 (1973), 32–47.Google Scholar
[2]Bebernes, J.W., ‘A simple alternative problem for finding periodic solutions of second order ordinary differential systems’, Proc. Amer. Math. Soc. 42 (1974), 121–127.CrossRefGoogle Scholar
[3]Gaines, R.E. and Mawhin, J.L., Coincidence degree and nonlinear differential equations, Lecture Notes in Mathematics 568 (Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
[4]Granas, A., Guenther, R.B. and Lee, J.W., Nonlinear boundary value problems for ordinary differential equations (Dissertationes Mathematicae 244, Warsaw, 1985).Google Scholar
[5]Habets, P. and Schmitt, K., ‘Nonlinear boundary value problems for systems of differential equations’, Arch. Math. 40 (1983), 441–446.Google Scholar
[6]Knobloch, H.-W., ‘On the existence of periodic solutions for second order vector differential equations’, J. Differential Equations 9 (1971), 67–85.CrossRefGoogle Scholar
[7]Knobloch, H.-W. and Schmitt, K., ‘Non-linear boundary value problems for systems of differential equations’, Proc. Roy. Soc. Edinburgh 78 (1977), 139–159.Google Scholar
[8]Lan, Chin-Chin, ‘Boundary value problems for second and third order differential equations’, J. Differential Equations 18 (1975), 258–274.Google Scholar
[9]Mawhin, J.L., Topological degree methods in nonlinear boundary value problems, Regional Conference Series in Mathematics 40 (American Mathematical Society, Providence R.I., 1979).CrossRefGoogle Scholar
[10]Thompson, H.B., ‘Second order ordinary differential equations with fully nonlinear two point boundary conditions’, Pacific J. Math. 172 (1996), 255–277.CrossRefGoogle Scholar
[11]Thompson, H.B., ‘Existence of periodic solutions for first and second order systems of ordinary differential equations’, Bull. Austral. Math. Soc. 54 (1996), 397–410.CrossRefGoogle Scholar