Bulletin of the American Mathematical Society

MathSciNet review: 1319817
Full text of review: PDF   This review is available free of charge.
Book Information:

Author: Hal L. Smith
Title: Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems
Additional book information: Mathematical Surveys and Monographs, vol. 41, Amer. Math. Soc., Providence, RI, 1995, x + 174 pp., ISBN 0-8218-0393-X, $49.00$

[1]

R. Courant and D. Hilbert, Methoden der Mathematischen Physik, vol. 2, Springer-Verlag, Berlin, 1937.

Peter Hess, Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics Series, vol. 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. MR 1100011

Morris W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Nonlinear partial differential equations (Durham, N.H., 1982) Contemp. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1983, pp. 267–285. MR 706104

Morris W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 1–64. MR 741723, DOI 10.1090/S0273-0979-1984-15236-4

Morris W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), no. 3, 423–439. MR 783970, DOI 10.1137/0516030

Morris W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1–53. MR 921986, DOI 10.1515/crll.1988.383.1

[7]

E. Kamke, Zur Theorie der Systeme gewöhnlicher Differentialgleichungen II, Acta Math. 58 (1932), 57--85.

[8]

A. Leung, Systems of nonlinear partial differential equations, Kluwer Academic Publishers, Boston, 1989.

Hiroshi Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 645–673. MR 731522

C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. MR 1212084

Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861

Hal L. Smith, Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM Rev. 30 (1988), no. 1, 87–113. MR 931279, DOI 10.1137/1030003

Hal L. Smith and Horst R. Thieme, Monotone semiflows in scalar non-quasi-monotone functional-differential equations, J. Math. Anal. Appl. 150 (1990), no. 2, 289–306. MR 1067429, DOI 10.1016/0022-247X(90)90105-O

Hal L. Smith and Horst R. Thieme, Strongly order preserving semiflows generated by functional-differential equations, J. Differential Equations 93 (1991), no. 2, 332–363. MR 1125223, DOI 10.1016/0022-0396(91)90016-3

[15]

J. Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Springer, New York, 1994. CMP 95:03

[1]

R. Courant and D. Hilbert, Methoden der Mathematischen Physik, vol. 2, Springer-Verlag, Berlin, 1937.

[2]

P. Hess, Periodic-parabolic boundary value problems and positivity, Longman Scientific and Technical, New York, 1991.MR 1100011

[3]

M. Hirsch, Differential equations and convergence almost everywhere in strongly monotone flows, Contemporary Mathematics vol. 17 (J. Smoller, ed.), Amer. Math. Soc., Providence, RI, 1983, pp. 267--285. MR 0706104

[4]

------, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 1--64. MR 0741723

[5]

------, Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), 423--439.MR 0783970

[6]

------, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1--53.MR 0921986

[7]

E. Kamke, Zur Theorie der Systeme gewöhnlicher Differentialgleichungen II, Acta Math. 58 (1932), 57--85.

[8]

A. Leung, Systems of nonlinear partial differential equations, Kluwer Academic Publishers, Boston, 1989.

[9]

H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo 30 (1984), 645--673.MR 0731522

[10]

C.-V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.MR 1212084

[11]

M. Protter and H. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, NJ, 1967.MR 0219861

[12]

H. Smith, Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM Rev. 30 (1988), 87--113.MR 0931279

[13]

H. Smith and H. Thieme, Monotone semiflows in scalar non-quasi-monotone functional differential equations, J. Math. Anal. Appl. 150 (1990), 289--306.MR 1067429

[14]

------, Strongly order preserving semiflows generated by functional differential equations, J. Diff. Equations 93 (1991), 332--363. MR 1125223

[15]

J. Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Springer, New York, 1994. CMP 95:03