Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T00:45:20.022Z Has data issue: false hasContentIssue false

Oscillation Criteria for Quasilinear Equations

Published online by Cambridge University Press:  20 November 2018

W. Allegretto*
Affiliation:
University of Alberta, Edmonton, Alberta
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.

Several authors have recently considered the problem of establishing sufficient criteria to guarantee the oscillation or non-oscillation of all solutions of a second order elliptic equation or system. We mention in particular the papers of C. A. Swanson, [15; 16], K. Kreith [9], Kreith and Travis [10], Noussair and Swanson [13], Allegretto and Swanson [3], Allegretto and Erbe [2] and the references therein.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Agmon, S., Unicité et convexité dans les problèmes différentiels (University of Montreal Press, Montreal, 1966).Google Scholar
2. Allegretto, W. and Erbe, L., Oscillation criteria for matrix differential inequalities, Can. Math. Bull. 16 (1973), 510.Google Scholar
3. Allegretto, W. and Swanson, C. A., Oscillation criteria for elliptic systems, Proc. Amer. Math. Soc. 27 (1971), 325330.Google Scholar
4. Coffman, C. V. and Wong, J. S., Oscillation and non-oscillation of solutions of generalized Emden-Fowler equations, Trans. Amer. Math. Soc. 167 (1972), 399434.Google Scholar
5. Erbe, L., Oscillation criteria for nonlinear delay equations, Can. Math. Bull. 16 (1973), 4956.Google Scholar
6. Izyumova, D. V., On the conditions for the oscillation and non-oscillation of nonlinear secondorder differential equations, Differencial'nye Uravnenija 2 (1966), 15721586.Google Scholar
7. Jackson, L. K., Subfunctions and second-order ordinary differential inequalities, ‘'Lectures on Ordinary Differential Equations” (Academic Press, New York, 1970).Google Scholar
8. Kneser, A., Untersuchungtn über die reelen Nullstellen der Intégrale linear en Differ entialgleichungen, Math. Ann. 42 (1893), 409435.Google Scholar
9. Kreith, K., Oscillation theorems for elliptic equations, Proc. Amer. Math. Soc. 15 (1964), 341344.Google Scholar
10. Kreith, K. and Travis, C., Oscillation criteria for self-adjoint elliptic differential equations, Pacific J. Math. 41 (1972), 743753.Google Scholar
11. Leighton, W., The detection of the oscillation of solutions of a second order linear differential equation, Duke Math. J. 17 (1950), 5761.Google Scholar
12. Noussair, E., Elliptic equations of order 2m, J. Differential Equations 10 (1971), 100111.Google Scholar
13. Noussair, E. and Swanson, C. A., Oscillation criteria for differential systems, J. Math. Anal. Appl. 36 (1971), 575580.Google Scholar
14. Protter, M. and Weinberger, H., Maximum principles in differential equations (Prentice- Hall, New York, 1967).Google Scholar
15. Swanson, C. A., Oscillation criteria for nonlinear matrix differential inequalities, Proc. Amer. Math. Soc. 24 (1970), 824827.Google Scholar
16. Swanson, C. A., Comparison theorems for quasilinear elliptic differential inequalities, J. Differential Equations 7 (1970), 243249.Google Scholar
17. Swanson, C. A., A comparison theorem for eigenfunctions, Proc. Amer. Math. Soc. 37 (1973), 537540.Google Scholar
18. Wong, J. S., On second order nonlinear oscillation, Funkcial. Ekvac. 11 (1968), 207234.Google Scholar
19. Zlâmal, M., Oscillation criterions, Časopis Pěst. Mat. 75 (1950), 213217.Google Scholar