Sentence examples for minimal fixed point from inspiring English sources

In Section 3, we state and prove existence of fixed points, also, maximal and minimal fixed point theorem is presented for discontinuous multivalued increasing operators which are our main results.

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.

If the conditions in Theorem 2.1 are satisfied, then has the minimal fixed point and the maximal fixed point in ; that is, and are fixed points of, and for any fixed point of in, one has.

A technical note: Kripke defined being grounded as appearing in the minimal fixed point.

Indeed, by (3.3), x ∗ is also the minimal fixed point of A in [ y, x ˆ ∗ ].

Then the operator T has a minimal fixed point (x^) and a maximal fixed point (y^).

Thereby, has maximal and minimal fixed points on.

Moreover, the maximal and minimal fixed points can be achieved by monotone iterative method under some conditions.

Assume that the operator is increasing and satisfies conditions (H1)–(H3) and (h1)–(h3), then has maximal and minimal fixed points on.

We obtain some new existence theorems of the maximal and minimal fixed points for discontinuous monotone operator on an order interval in an ordered normed space.

Recently, Li [9] proved the existence of maximal and minimal fixed points of increasing mappings by using the partial order introduced by (1).