Sentence examples for greatest fixed point from inspiring English sources
Further if the cone in is normal, then the equation has at least fixed point and a greatest fixed point.
The above proof shows that the operator G = (G1, G2) defined by (6.8) satisfies the hypotheses of Lemma 2.1, whence G has the smallest fixed point x* = (u*, v*) and a greatest fixed point x* = (u*, v*).
Obviously, (sup W1, sup W2) is the supremum of W in P. Similarly one can show that each inversely well-ordered chain of the range of G has the infimum in P. The above proof shows that the operator G = (G1, G2) defined by (5.7) satisfies the hypotheses of Lemma 2.1, and therefore G has the smallest fixed point x* = (u*,v*) and the greatest fixed point x* = (u*, v*).
We know that has in the order interval of least fixed point and greatest fixed point.
Similar to the above discussion, we can prove the existence of the greatest fixed point (u^) of F. This proof is complete.
The algorithm is a greatest fixed point iteration that incrementally constructs a state-action table SA, which indicates which action has to be executed in certain state of D in order to reach a goal state.
From Theorem 5.1, it then follows that G has the smallest and greatest fixed points u ∗ and u ∗, and they are increasing with respect to D and f.
Because x* = (u*, v*) and x* = (u*, v*) are the smallest and greatest fixed points of G, respectively, then (u*, v*) ≤ (u, v) ≤ (u*, v*).
Because x* = (u*, v*) and x* = (u*, v*) are the smallest and greatest fixed points of G, then (u*, v*) ≤ (u, v) ≤ (u*, v*).
