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The method of upper, lower solutions and monotone iterative scheme for higher order hyperbolic partial differential equations

Part of: Partial differential equations

Published online by Cambridge University Press:  09 April 2009

Ravi P. Agarwal
Ravi P. Agarwal
Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge CrescentSingapore0511
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Abstract

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Uniformly monotone convergent iterative methods for obtaining multiple solutions of (n + m)th order hyperbolic partial differential equations together with initial conditions are discussed. Appropriate partial differential inequalities which connect upper and lower solutions, and variation of parameters formula is developed.

MSC classification

Secondary: 35A35: Theoretical approximation to solutions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 1 , August 1989 , pp. 153 - 170
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Agarwal, R. P., ‘On integrodifferential inequalities in two independent variables’, J. Math. Anal. Appl. 89 (1982), 581–597.CrossRefGoogle Scholar
[2]Agarwal, R. P., ‘Inequalities involving partial derivatives’, J. Math. Anal. Appl. 89 (1982), 628–638.CrossRefGoogle Scholar
[3]Agarwal, R. P., Boundary value problems for higher order differential equations, (World Scientific, Singapore and Philadelphia, 1986).CrossRefGoogle Scholar
[4]Agarwal, R. P., ‘Existence-uniqueness and iterative methods for higher order hyperbolic partial differential equations’, submitted.Google Scholar
[5]Agarwal, R. P. and Usmani, R. A., ‘ Monotone convergence of iterative methods for right focal point boundary value problems’, J. Math. Anal. Appl. (to appear).Google Scholar
[6]Bellen, A., ‘Monotone methods for periodic solutions of second order scalar functional differential equations’, Numer. Math. 42 (1983), 15–30.CrossRefGoogle Scholar
[7]Devasahayam, M. Paul, ‘Monotone iterative scheme for nonlinear hyperbolic boundary value problem’, Applicable Anal 20 (1985), 49–55.Google Scholar
[8]Kovac, I. Ju., ‘Application of the theorem of differential inequalities to the Goursat problem for a linear system of partial differential equations’, Differencial'nye Uravnenija 1 (1965), 411–420 (in Russian).Google Scholar
[9]Ladde, G. S., Lakshmikantham, V. and Vatsala, A. S., Monotone iterative techniques for nonlinear differential equations, (Pitman Advanced Publishing Program, Boston, London, Melbourne, 1985).Google Scholar
[10]Lakshmikantham, V. and Pandit, S. G., ‘The method of upper, lower solutions and hyperbolic partial differential equations’, J. Math. Anal Appl 105 (1985), 466–477.CrossRefGoogle Scholar
[11]Lakshmikantham, V. and Pandit, S. G., ‘Periodic solutions of hyperbolic partial differential equations’, Comput. Math. Appl. 11 (1985), 249–259.CrossRefGoogle Scholar
[12]De Lucia, P., ‘Su un problem al contorno per un'equazione di tipo iperbolico di ordine 2m’, Ricerche Mat. 16 (1967), 264–302.Google Scholar
[13]Szarski, J., Differential inequalities, (Monograf. Mat. 43, Warsaw, 1965).Google Scholar
[14]Volpato, M., ‘Sugli elementi uniti di transformazioni funzionali: un problema ai limiti per una classe di equazioni alle derivate parziali di tipo iperbolico’, Ann. Univ. Ferrara 2 (1952/1953), 95–109.CrossRefGoogle Scholar
[15]Walter, W., Differential and integral inequalities, (Springer-Verlag, New York, 1970).CrossRefGoogle Scholar