×

The method of upper, lower solutions and hyperbolic partial differential equations. (English) Zbl 0569.35056

The authors extend the method of sub- and super-solutions to some initial value problems for certain nonlinear hyperbolic equations. They prove existence theorems and obtain minimal solutions.
{Reviewer’s remark: It seems to the reviewer that their requirements on sub- and supersolutions are very restrictive.}
Reviewer: E.Dancer

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agarwal, R. P.; Thandapani, E., Existence and uniqueness of solutions of hyperbolic delay differential equations, (Math. Sem. Notes Kobe Univ., 8 (1980)), 531-541 · Zbl 0449.35100
[2] Bernfeld, S.; Lakshmikantham, V., Monotone methods for nonlinear boundary value problems in Banach spaces, Nonlinear Anal. TMA, 3, 303-316 (1979) · Zbl 0423.34087
[3] Deimling, K.; Lakshmikantham, V., Existence and comparison theorems for differential equations in Banach spaces, Nonlinear Anal. TMA, 3, 569-575 (1979) · Zbl 0418.34062
[4] S. G. Deo and G. S. Ladde; S. G. Deo and G. S. Ladde · Zbl 0215.15101
[5] Kannan, R.; Lakshmikantham, V., Existence of periodic solutions of semilinear parabolic equations and the method of upper and lower solutions, J. Math. Anal. Appl., 97, 291-299 (1983) · Zbl 0542.35044
[6] Keller, H., Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rational Mech. Anal., 35, 363-381 (1969) · Zbl 0188.17102
[7] Ladde, G. S.; Lakshmikantham, V.; Pachpatte, B. G., The method of upper and lower solutions and Volterra integral equations, J. Integral Equations, 4, 353-360 (1982) · Zbl 0489.45004
[8] Lakshmikantham, V.; Leela, S., Existence and monotone method for periodic solutions of first order differential equations, J. Math. Anal. Appl., 91, 237-243 (1983) · Zbl 0525.34031
[9] Lakshmikantham, V.; Vatsala, A. S., Quasi-solutions and monotone method for systems of nonlinear boundary value problems, J. Math. Anal. Appl., 79, 38-47 (1981) · Zbl 0453.34020
[10] Leela, S., Monotone Technique for Periodic Solutions of Differential Equations, University of Texas at Arlington Technical Report No. 190 (1982) · Zbl 0557.34037
[11] Sattinger, D., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, 979-1000 (1972) · Zbl 0223.35038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.