# Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions Let A be a set, and let E be the Banach space of bounded functions ξ : A → R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,ξ) : R × E → E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to ξ. Then there exists a solution u: [a,b] × [0,T] → E of the problem ut(x,t) − uxx(x,t) = F(x,t,u(x,t)) ((x,t) ∈ R), u(x,t) = 0 ((x,t) ∈ R ∖ R). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

, Volume 1 (1) – Oct 14, 2004
10 pages

/lp/springer_journal/parabolic-differential-equations-in-ordered-banach-spaces-of-bounded-dhbqjtKuFO
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009722417595
Publisher site
See Article on Publisher Site

### Abstract

Let A be a set, and let E be the Banach space of bounded functions ξ : A → R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,ξ) : R × E → E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to ξ. Then there exists a solution u: [a,b] × [0,T] → E of the problem ut(x,t) − uxx(x,t) = F(x,t,u(x,t)) ((x,t) ∈ R), u(x,t) = 0 ((x,t) ∈ R ∖ R).

### Journal

PositivitySpringer Journals

Published: Oct 14, 2004

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