Sentence examples for fixed point of note from inspiring English sources

Thus is eventually constant, so we have a fixed point of (note that ).

Then the sequence { x n } generated by the explicit iterative method (1.2) converges to x ˜ ∈ Fix ( T ), which is the minimum-norm fixed point of T. Note that the idea of the iterative algorithms with perturbations has been extended to the other topics, see, for example, [26].

Moreover, by taking (A=bigl ( {scriptsizebegin{matrix} frac{1}{sqrt{c}} & 0 cr 0 & frac{1}{sqrt{c}} end{matrix}} bigr )), we have (|A|<1), and taking (x_{0}=frac{1}{c}) one can check that all conditions of Theorem 2.5 are satisfied and 0 is fixed point of T. Note that Theorem 1.4 is not applicable here, since the contractive condition (2) does not hold, for example, at (x=0), (y=4).

Remark 5 Note that each largest (resp. least) fixed point of T must be a maximal (resp. minimal) fixed point of T, but the converse is not true.

Thus z is fixed point of T. □.

That is, fixed point results (which are fixed point of 1-order results) can be generated from similar results on coupled, tripled or even quadruple fixed points.

Let H be a real Hilbert space, let F be a κ-Lipschitzian and η-strongly monotone operator on H with k > 0, η > 0, and let T be a quasi-nonexpansive mapping on H, and f is a L-Lipschitzian mapping with coefficient L > 0 for all x,y ∈ H. Assume the set Fix(T) of fixed points of T is nonempty and we note that Fix(T) is closed and convex.

Let H be a real Hilbert space, let A be a bounded linear operator on H, and let T be a quasi-nonexpansive mapping on H, and f is a contraction with coefficient α; that is ||f (x) - f y)|| ≤ α||x - y|| for all x, y ∈ H. Assume the set Fix(T) of fixed points of T is nonempty and we note that Fix T) is closed and convex (see [14] for more general results).

As noted before, fixed points of (6a)–(6b), (mathrm{FP}_{mathrm{av}i}) (yellow diamonds), are given by the intersections of these nullclines, and one can usually determine the stability of the fixed points by considering the nullcline configuration.

Note here that the k − 1 fixed points of ϕ on S 1 are all transversal so that the hypotheses of Theorem 3 hold.

In this note, we establish a general theorem to approximate fixed points of quasi-contractive.