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A fixed point theorem. (English) Zbl 0606.47064

Using the maximal principle the author has proved the following fixed point theorem: Let X be a uniformly convex Banach lattice, \(0\in D\) a closed, bounded subset of the positive cone \(C^+\) of X. Let \(f: D\to D\) be a map such that, if \(x,y\in D\), \(x\leq y\), then \(f(x)\leq f(y)\). Then f has a fixed point.
The existence of a solution to the Cauchy problem of the differential equation in the Banach lattice \(X:\) \[ \dot x=f(t,x),\quad x(0)=x_ 0 \] where \(f:[0,1]\times X\to X\) satisfies the Carathéodory condition, is established as a consequence.
Reviewer: I.Beg

MSC:

47H10 Fixed-point theorems
47J05 Equations involving nonlinear operators (general)
47J25 Iterative procedures involving nonlinear operators
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