Then T has a largest fixed point and a least fixed point in [ x 0, y 0 ].
In fact, 0 is the largest fixed point and 1 2 is the least fixed point in [ 1 2, 0 ].
By the least property of x ∗ corresponding to ⪯1, we have x ∗ ⪯ 1 x, and hence x ⪯ x ∗, i.e., x ∗ is a largest fixed point of T corresponding to ⪯.
In fact, 0 is the largest fixed point and 1 2 is the least fixed point in [ 1 2, + ∞ ) ⪯. Remark 3.5 In Example 3.4, the considered function η is not a decreasing function and η − 1 { 0 } = { 0, 1 3 }. Thus, Theorem 1.1 cannot be applied in this situation.
T has a minimal fixed point x ∗ ∈ ( − ∞, x 0 ], i.e., let x ∈ ( − ∞, x 0 ] be a fixed point of T, then x ⪯ x ∗ implies x = x ∗ ; T has a largest fixed point x ∗ ∈ ( − ∞, x 0 ], i.e., let x ∈ ( − ∞, x 0 ] be a fixed point of T, then x ⪯ x ∗. Proof Let ⪯1 be the inverse partial order of ⪯ and φ 1 ( x ) = − φ ( x ).
Assume that there exists x 0 ∈ X such that T x 0 ⪯ x 0. Then (i) T has a minimal fixed point x ∗ ∈ ( − ∞, x 0 ], i.e., let x ∈ ( − ∞, x 0 ] be a fixed point of T, then x ⪯ x ∗ implies x = x ∗ ; (ii) T has a largest fixed point x ∗ ∈ ( − ∞, x 0 ], i.e., let x ∈ ( − ∞, x 0 ] be a fixed point of T, then x ⪯ x ∗. .
This highly depolarized (i.e., large V) fixed point corresponds to the physiological state of depolarization block in the MLT system (Figure 8e).
Although the four models treat the workings of the immune system in slightly different ways, they all predict that adoptive immunotherapy can be successful to move a patient from the large tumor fixed point to an equilibrium with little or no tumor.
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.
Since then, a large number of papers on fixed point results in the setting of b-metric spaces have appeared (see [17 27]).
A nonempty subset L of a convex space X is called a c-compact set [13] if for each finite subset S ⊂ X there is a compact convex set L S ⊂ X such that L ∪ S ⊂ L S. Lassonde [13] presented a simple and unified treatment of a large variety of minimax and fixed point problems.
