Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-15T12:27:56.242Z Has data issue: false hasContentIssue false
  • English
  • Français

An Existence Theorem for Nonlinear Boundary Value Problems

Published online by Cambridge University Press:  20 November 2018

W. L. McCandless
W. L. McCandless*
Affiliation:
Department of Mathematics, Mount Allison University, Sackville, New Brunswick
Rights & Permissions [Opens in a new window]

Extract

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.

Boundary value problems for ordinary differential equations have long been the subject of extensive research activity. In particular, questions concerning the existence and uniqueness of solutions for these problems have received much attention, and algebraic fixed-point theorems have served as important tools in such investigations. For example Picard [8] based his pioneering work in this area on the use of successive approximation techniques, and recently his classical methods have been refined and extended to more general nonlinear problems (see [1] and [4]). The standard procedure for applying these techniques requires that the boundary value problem under consideration first be converted into an equivalent integral equation through the choice of a suitable Green’s function. The resulting theory is consequently limited to problems for which such a formulation is possible.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 5 , February 1975 , pp. 693 - 701
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Bailey, P. B., Shampine, L. F., and Waltman, P. E., Nonlinear two point boundary value problems, Academic Press, New York, 1968.Google Scholar
2. Conti, R., Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital. (3) XXII (1967), 149, 163–165.Google Scholar
3. Dennis, J. E. Jr, Toward a unified convergence theory for Newton-like methods, in L. B. Rail (ed.), Nonlinear functional analysis and applications, Academic Press, New York, 1971, 425–472.Google Scholar
4. Falb, P. L. and de Jong, J. L., Some successive approximation methods in control and oscillation theory, Academic Press, New York, 1969.Google Scholar
5. McCandless, W. L., Newton’s method and nonlinear boundary value problems, J. Math. Anal. Appl. (to appear).Google Scholar
6. Noble, B., Applied linear algebra, Prentice-Hall, Englewood Cliffs, 1969.Google Scholar
7. Ortega, J. M. and Rheinboldt, W. C., Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970.Google Scholar
8. Picard, E., Sur Vapplication des méthodes d’approximations successives à létude de certaines équations différentielles ordinaires, J. Math. 8 (1893), 217–271.Google Scholar
9. Rail, L. B., Computational solution of nonlinear operator equations, Wiley, New York, 1969.Google Scholar
10. Urabe, M., The Newton method and its application to boundary value problems with nonlinear boundary conditions, in W. Harris/Y. Sibuya (eds.), Proceedings United States-Japan seminar on differential and functional equations, Benjamin, New York, 1967, 383–410.Google Scholar