Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T11:14:53.726Z Has data issue: false hasContentIssue false
  • English
  • Français

Growth and oscillation properties of solutions of a fourth order linear difference equation

Published online by Cambridge University Press:  17 February 2009

John W. Hooker  and
William T. Patula
William T. Patula
Affiliation:
Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901, U.S.A.
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.

For the fourth-order linear difference equation Δ4un−2 = bn un, with bn > 0 for all n, generalized zeros are defined, following Hartman [5], and two theorems are proved concerning separation of zeros of linearly independent solutions. Some preliminary results deal with non-oscillation and asymptotic behavior of solutions of this equation for various types of initial conditions. Finally, recessive solutions are defined, and results are obtained analogous to known results for recessive solutions of second-order difference equations.

Type
Research Article
Information
The ANZIAM Journal , Volume 26 , Issue 3 , January 1985 , pp. 310 - 328
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Dahlquist, G., “Stability and error bounds in the numerical integration of ordinary differential equations”, Kungl. Tekn. Högsk. Handl. Stockholm 130 (1959).Google Scholar
[2]Fort, T., Finite differences and difference equations in the real domain (Oxford University Press, London, 1948).Google Scholar
[3]Freedman, H. I., Deterministic mathematical models in population ecology (Marcel Dekker, New York, 1980).Google Scholar
[4]Greenspan, D., Discrete modesl (Addison-Wesley, Reading, Massachusetts, 1973).Google Scholar
[5]Hartman, P., “Difference equations: disconjugacy, principal solutions, Green's functions, complete monotonicity”, Trans. Amer. Math. Soc. 246 (1978), 1–30.Google Scholar
[6]Hartman, P. and Wintner, A., “On linear difference equations of the second order”, Amer. J. Math. 72 (1950), 124–128.CrossRefGoogle Scholar
[7]Henrici, P., Discrete variable mathods in ordinary differential equations (Wiley, New York, 1962).Google Scholar
[8]Leighton, W. and Nehari, Z., “On the oscillation of solutions of self-adjoint linear differential equations of the fourth order”, Trans. Amer. Math. Soc. 89 (1958), 325–377.CrossRefGoogle Scholar
[9]Olver, F. W. J. and Sookne, D. J., “Note on backward recurrence algorithms”, Math. Comp. 26 (1972), 941–947.CrossRefGoogle Scholar
[10]Patula, W. T., “Growth and oscillation properties of second order linear difference equations”, SIAM J. Math. Anal. 10 (1979), 55–61.CrossRefGoogle Scholar
[11]Patula, W. T., “Growth, oscillation and comparison theorems for second order linear difference equations”, SIAM J. Math. Anal. 10 (1979), 1272–1279.CrossRefGoogle Scholar
[12]Van Lint, J. H., Introduction to coding theory (Springer-Verlag, New York, 1982).CrossRefGoogle Scholar