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Results tagged with fixed-point-theorems
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user 165275
A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.
1
vote
The Tarski-Lindenbaum theorem of the middle value
Not a very satisfactory answer, but some considerations to the proof of the MV theorem:
One might think that analogously to the proof of CBS (see e.g. Joel David Hamkins answer how to use KT for that …
- 1,869
1
vote
Accepted
Continuity of Kakutani fixed points
I assume that you mean that $F$ is upper semicontinuous on the product space. Then in particular (since $X$ is compact, Hausdorff and $F$ has closed values), $F$ has a closed graph. This implies that …
- 1,869
1
vote
How do I apply Brouwer fixed-point theorem in this claim?
Only now I realize the condition that $\zeta$ is nonnegative. (Was it really there in the first formulation of the question?)
With this condition, it is possible to get the required a-priori bound req …
- 1,869
1
vote
How do I apply Brouwer fixed-point theorem in this claim?
What is needed is an a-priori $L_\infty$ bound for the solution $v_k$. If you know such an a-priori bound, you can modify $\zeta$ outside of this bound, and you can assume without of generality that $ …
- 1,869