On some set-valued iteration semigroups generated by interval-valued functions

Let X be an arbitrary set. We characterize all interval-valued functions $${A:X\to 2^\mathbb{R}}$$A:X→2R for which a multifunction $${F:(0,\infty)\times X\to 2^X}$$F:(0,∞)×X→2X of the form $${F(t,x)=A^{-}\big(A(x)+\min \{t,q-\inf A(x)\}\big)}$$F(t,x)=A-(A(x)+min{t,q-infA(x)}), where $${q=\sup A(X)}$$q=supA(X), is an iteration semigroup. The multifunction F is the set-valued counterpart of the fundamental form of continuous iteration semigroups of single-valued functions on an interval.


Introduction
Let X be an arbitrary set X. A multifunction F : (0, ∞) × X → 2 X is said to be a set-valued iteration semigroup if F (s + t, x) = F t, F (s, x) for x ∈ X and s, t ∈ (0, ∞).
This notion was introduced and investigated by Smajdor [11] (see also e.g. [12]), studied by J. Olko (see e.g. [9,10]) and by Zdun [15]. In [4,5] we introduced and studied a family of set-valued functions which now will be denoted by (A) (see Sect. 1) and we showed (see [5,Remarks 1 and 3]) that F given by (A) is a set-valued counterpart of the fundamental form of iteration semigroups for single-valued functions which can be found in [ [1,Theorem 1]). In [7] we studied a lower semicontinuity of F given by (A).
In the present paper we will consider the case when the values of the generator A of the multifunction F are intervals. The main aim of this paper is to

Preliminaries
Fix a set X and a set-valued function A : X → 2 R with non-empty values. Put S := A(X) and q := sup S.
Throughout this paper we will always assume that A satisfies the following condition: exists y ∈ X satisfying the conditions and In the following remark we give some properties related to (H For every x ∈ X define Consider the following condition: (H1) for every x, z ∈ X and s, t ∈ (0, ∞) with s + t τ (x) if (1) and (2) hold for a y ∈ X then [A(x) + s + t] ∩ A(z) = ∅. Notice that if A is single-valued then (H1) holds (see also [6,Remark 1]).
Prove the following easy remark.
Remark 3. Assume that the condition (H1) holds. Let x, z ∈ X and u < w for u ∈ A(x) and w ∈ A(z).
and suppose that card A(y) > 1. Let a, b ∈ A(y) and a < b. We can find s ∈ (0, ∞) satisfying the conditions and Observe that, by (3) and (5), we have Moreover, due to (4) and (6), we get Hence, according to the inequality (7) and the condition (H1), On the other hand, by (3) and (5), we obtain This contradiction completes the proof.
Define the following sets: Assume that A and B are arbitrary families of subsets of R. We will write Let F : (0, ∞) × X → 2 X be given by the following formula where In [5, Lemma 3] we proved the following fact.
Now we present three theorems which was proved in [8]. We will use them in the proof of the main result of this paper.
and L S.
Then F is an iteration semigroup if and only if the condition (H1) and all the following conditions hold: (1), then the condition holds for every P ∈ P and t τ (y).
Since in this paper we are interested in multifunctions F given by (A) which are generated by interval-valued functions A, below we prove two properties of the multifunction A under this assumption.
Remark 4. Assume that the values of A are intervals and the condition (H1) holds. Let x, y, z ∈ X.
Proof. (i) Assume (8) and suppose that card A(y) > 1 and inf A(y) < inf A(z). Let Of course s, t ∈ (0, ∞) and Hence, by (10), Therefore, since the values of A are intervals, we get On the other hand, by (11) [ which, by the following inequality contradicts condition (H1) and completes the proof of (i). The proof of (ii) is similar. It is enough to take Vol. 90 (2016) On some set-valued iteration semigroups 387 Notice that if A is interval-valued then for every x ∈ X we get for some a x < q < ∞,

Lemma 1. Assume that the values of A are intervals, (H1) holds and q = ∞.
Then all the following conditions are satisfied Proof. Notice that τ (x) = ∞ for every x ∈ X. The condition (i) follows immediately from Remark 2.
Pass to the proof of (ii). Suppose that there exist x, y, z ∈ X such that A(x) ∈ S ∪ P ∞ , A(y) ∈ L −∞ and sup A(y) > inf A(x). Take and t ∈ (0, ∞), t > s. Of course s, t ∈ (0, ∞) and s + t < τ(x). Since the values of A are intervals, notice that

A(x) ∈ S and A(y) ∈ P ∞ and inf A(y) < sup A(x)
or

A(x) ∈ S and A(y) ∈ R.
Let s ∈ (0, ∞) and t ∈ 0, sup A(x) − inf A(y) . By our assumptions On the other hand which contradicts (H1).
To prove (v) suppose that there exist x, y, z ∈ X such that A(z) ∈ L −∞ , A(y) ∈ R and A(x) ∈ P ∞ . By (ii) we get For every s, t ∈ (0, ∞) we obtain Due to (12) which contradicts (H1) and completes the proof of (v).

Main result
Let A, B ⊂ 2 R . We will say that A has values of type A, if A(x) ∈ A for every x ∈ X. We will say that A has values of type AB, if A(x) ∈ A ∪ B for every x ∈ X and A = ∅ and B = ∅. Similarly for three classes of sets.
We will say that A has property W and write W (A), if int P = int R, for every P, R ∈ A, (where int P denotes the interior of the set P ). Now we present the main result of this paper. The conditions relating to the cardinality of sets and property W follow immediately from Remark 4. Now pass to the proof of the contrary implication. At first notice that if (viii) holds then A is single-valued and, by Fact 2, F is an iteration semigroup. In the cases (i)-(iii), (vi), (ix), (xi)-(xiii) and (xvi)-(xvii) the condition

Theorem. Assume that the values of
is satisfied for all x, z ∈ X and s ∈ 0, τ(x) ∩ R. Hence (H1) holds. Moreover in this cases we have F ≡ X, so the multifunction A generates an iteration semigroup F . It easy to observe that also in other cases condition (H1) is satisfied. Of course each of the conditions (xiv)-(xv) and (xviii)-(xix) defines the multifunction A for which q = ∞. Therefore by Fact 3 in all these cases F is an iteration semigroup. In the cases (iv), (v) and (vii) we obtain that q ∈ S and the conditions (a)-(g) of Fact 5 hold, thus F is an iteration semigroup. Now assume (x). Then q / ∈ S and q = ∞, so due to Fact 4 we obtain that F is an iteration semigroup.
Since in our paper we always assume that A satisfies condition (H), we can ask what it means if one of conditions (i)-(xix) of the Theorem holds. Notice that for an arbitrary interval-valued function A we obtain: On some set-valued iteration semigroups 391