An infinitedimensional extension of theorems of Hartman and Wintner on monotone positive solutions of ordinary differential equations
Author:
A. F. Izé
Journal:
Proc. Amer. Math. Soc. 110 (1990), 7784
MSC:
Primary 34G20; Secondary 34C35, 58D25
MathSciNet review:
1015679
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Abstract: Consider the equation (1) , a Banach sequence space with a Schauder Basis. It is proved that if is a positive operator and the solution operator is compact for , then system (1) has at least one solution such that , and consequently are monotone nonincreasing for .
 [1]
K. Deimling, Ordinary differential equations in Banach spaces, Lecture Notes in Math., vol. 596, SpringerVerlag, 1977. MR 0463601 (57:3546)
 [2]
P. Hartman and A. Wintner, Linear differential and difference equation with monotone solutions, Amer. J. Math. 75 (1953), 731743. MR 0057404 (15:221f)
 [3]
, On monotone solutions of systems of nonlinear differential equations, Amer. J. Math. (1954), 860866.
 [4]
D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, SpringerVerlag, Berlin and New York. MR 610244 (83j:35084)
 [5]
A. F. Izé, On a fixed point index method for the analysis of the asymptotic behavior and boundary value problems of infinite dimensional dynamical systems and processes. J. Differential Equations 52 (1984). MR 741266 (86a:34074)
 [6]
J. Lindenstrauss and Z. Tzafriri, Classical Banach space I, Sequence spaces, Engebnisse der Mathematik und ihrer Grenzgebiete 92, SpringerVerlag, Berlin, 1977. MR 0500056 (58:17766)
 [7]
T. Kato, Quasilinear equations of evolution, with applications to partial differential equations, Lecture Notes in Math., vol. 448, SpringerVerlag, 1974. MR 0407477 (53:11252)
 [8]
M. A. Krasnoselskii, Positive solutions of operator equations. Noordhoff, Groningen, 1964. MR 0181881 (31:6107)
 [1]
 K. Deimling, Ordinary differential equations in Banach spaces, Lecture Notes in Math., vol. 596, SpringerVerlag, 1977. MR 0463601 (57:3546)
 [2]
 P. Hartman and A. Wintner, Linear differential and difference equation with monotone solutions, Amer. J. Math. 75 (1953), 731743. MR 0057404 (15:221f)
 [3]
 , On monotone solutions of systems of nonlinear differential equations, Amer. J. Math. (1954), 860866.
 [4]
 D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, SpringerVerlag, Berlin and New York. MR 610244 (83j:35084)
 [5]
 A. F. Izé, On a fixed point index method for the analysis of the asymptotic behavior and boundary value problems of infinite dimensional dynamical systems and processes. J. Differential Equations 52 (1984). MR 741266 (86a:34074)
 [6]
 J. Lindenstrauss and Z. Tzafriri, Classical Banach space I, Sequence spaces, Engebnisse der Mathematik und ihrer Grenzgebiete 92, SpringerVerlag, Berlin, 1977. MR 0500056 (58:17766)
 [7]
 T. Kato, Quasilinear equations of evolution, with applications to partial differential equations, Lecture Notes in Math., vol. 448, SpringerVerlag, 1974. MR 0407477 (53:11252)
 [8]
 M. A. Krasnoselskii, Positive solutions of operator equations. Noordhoff, Groningen, 1964. MR 0181881 (31:6107)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199010156797
PII:
S 00029939(1990)10156797
Keywords:
Differential equations,
Banach spaces,
infinite dimensional spaces,
positive solutions,
operator equations,
strongly positive,
solid cone,
egress points,
strict egress points,
trajectory,
orbit,
consequent operator,
left shadow,
process,
retract,
infinitesimal generator,
classical solution
Article copyright:
© Copyright 1990
American Mathematical Society
