Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An infinite-dimensional 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), 77-84
MSC: Primary 34G20; Secondary 34C35, 58D25
MathSciNet review: 1015679
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider the equation (1) $ \dot x + A\left( t \right)x = - f\left( {t,x} \right)\;x\left( 0 \right) = {x^0},{x^0} \in X$, a Banach sequence space with a Schauder Basis. It is proved that if $ f\left( {t,0} \right) = 0,A\left( t \right)\left( \cdot \right) + f\left( {t, \cdot } \right)$ is a positive operator and the solution operator $ K\left( {t,0} \right){x^0} = {x^0} - \int_0^t {A\left( s \right)ds - \int_0^t {f\left( {s,x\left( s \right)} \right)ds} } $ is compact for $ t > 0$, then system (1) has at least one solution $ x\left( t \right),x\left( t \right)\not\equiv 0$ such that $ x\left( t \right) \geq 0, - \dot x\left( t \right) \leq 0$, and consequently $ x\left( t \right)$ are monotone nonincreasing for $ t \geq 0$.

References [Enhancements On Off] (What's this?)

  • [1] Klaus Deimling, Ordinary differential equations in Banach spaces, Lecture Notes in Mathematics, Vol. 596, Springer-Verlag, Berlin-New York, 1977. MR 0463601
  • [2] Philip Hartman and Aurel Wintner, Linear differential and difference equations with monotone solutions, Amer. J. Math. 75 (1953), 731–743. MR 0057404,
  • [3] -, On monotone solutions of systems of nonlinear differential equations, Amer. J. Math. (1954), 860-866.
  • [4] Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
  • [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), no. 2, 162–174. MR 741266,
  • [6] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056
  • [7] Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Springer, Berlin, 1975, pp. 25–70. Lecture Notes in Math., Vol. 448. MR 0407477
  • [8] M. A. Krasnosel′skiĭ, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964. MR 0181881

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34G20, 34C35, 58D25

Retrieve articles in all journals with MSC: 34G20, 34C35, 58D25

Additional Information

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

American Mathematical Society