Measures of noncompactness in the space of regulated functions on an unbounded interval

In this paper, we formulate a criterion for relative compactness in the space of regulated functions on an unbounded interval and not necessarily bounded. Next we construct measure of noncompactness in this space and investigate its properties. The presented measure is simpler and more convenient to use than all known so far in space of regulated functions on an unbounded interval. Moreover, we show the applicability of the measure of noncompactness in proving the existence of solutions of some Volterra type integral equation.


Introduction
Fixed point theorems are one of the main tools used to study of solvability various kinds of equations. There are many fixed point theorems which use measures of noncompactness. In this approach, very important is to choice the suitable space and define the effective and convenient measure. A detailed discussion of the application of measures of noncompactness in various spaces in the study of the solvability of various types of integral equations can be found in [11]. One of these spaces is the space of regulated functions (consisting of functions having one-side limits at every point). It obviously contains the space of continuous functions, as well as the space of functions of bounded variation. This space is naturally used in theory of measure differential equations (MDEs, for short), also known as differential equations driven by measures, which arise in many areas of applied mathematics such as nonsmooth mechanics, game theory. In the last years, there have investigated several variants of this space introducing in them the structure of Banach space (see [2, 4-7, 10, 12]).
However, when considering regulated functions on an unbounded interval ℝ + , it is better to locate the considerations in the Fréchet space G(ℝ + , E) than in the Banach space BG(ℝ + , E) . In the case of the space BG(ℝ + , E) all known and convenient to use measures of noncompactness have the disadvantage that they do not capture all relatively compact sets, that is, the family of those sets on which these measures are zero is significantly smaller than the family of all relatively compact sets. It is quite different in the case of the space G(ℝ + , E) . As we will show later in this paper, the Fréchet space structure in G(ℝ + , E) allows for the formulation of an elegant compactness criterion. The measure of noncompactness constructed by this criterion does not have this disadvantage as before, i.e. the family of sets on which this measure is zero is exactly equal to the family of all relatively compact sets in G(ℝ + , E) . An even more important feature of the measure in G(ℝ + , E) constructed in this work is that it is simpler, because it is only two-component, while the aforementioned measures of noncompactness in BG(ℝ + , E) have three components and when using such measures to investigate the solvability various kinds of equations require stronger assumptions, enforced by the presence of these three components.
The paper is organized as follow. The second section is devoted to recalling some notions, facts and theorems. In the third section, we introduce in the space G(ℝ + , E) the structure of Fréchet space (using the family of pseudonorm). Moreover, we provide the compactness criterion and using it we define the new measure of noncompactness which is more convenient than usually used in the space BG(ℝ + , E) . Additionally, we study properties of this measure of noncompactness and we provide the example showing its applicability.

Notation, definitions and auxiliary facts
In this section, we recall some facts which are needed further on.
Assume that E is a real Banach space with the norm ‖ ⋅ ‖ and the zero element . Denote by B E (x, r) the closed ball centered at x and with radius r. The symbol B E (r) stands for the ball B E ( , r) . For X being a nonempty subset of E we write X, convX, ConvX to denote the closure, convex hull and the convex closure of a set X, respectively. Moreover, let us denote by E the family of all nonempty and bounded subsets of E and by E its subfamily consisting of all relatively compact sets.
We accept the following definition of the notion of a measure of noncompactness [1].
is said to be a measure of noncompactness in a Banach space E if it satisfies the following conditions: is a sequence of closed sets from E such that X n+1 ⊂ X n for n = 1, 2, ... and if lim n→∞ (X n ) = 0 , then the intersection X ∞ ∶= ⋂ ∞ n=1 X n is nonempty.
In the sequel, we will use measures of noncompactness having some additional properties. Namely, a measure is said to be sublinear if it satisfies the following two conditions: We consider also weak maximum property A sublinear measure of noncompactness satisfying the condition (maximum property), i.e.
and such that ker = E is said to be regular. For a given X ⊂ E , we denote by (X) the so-called Hausdorff measure of noncompactness of X. This quantity is defined by the formula and it is an example of the regular measure of noncompactness in E.
where E is a Banach space, is said to be a regulated function if for every t ∈ J + the right-sided limit x(t + ) ∶= lim s→t + x(s) exists and for every t ∈ J − the left-sided limit Denote by G(J, E) the space consisting of all regulated functions defined on the interval J with values in a Banach space E. Obviously, G(J, E) is a linear space. In Compactness criteria and measures of noncompactness in the space G([0, T], E) were investigated in several research papers (see [2,4,5,10]). We recall below the concept of a equiregulated subset of the space G([0, T], E). Definition 2. 3 We will say that the set X ⊂ G([0, T], E) is equiregulated on the interval J = [0, T] if the following two conditions are satisfied: Now we are going to recall the compactness criterion in G([0, T], E) formulated by Fraňková [10], (see also [2,4,5]).

Theorem 2.4 A nonempty subset X ⊂ G([0, T], E) is relatively compact in G([0, T], E) if and only if X is equiregulated on the interval [0, T] and the sets X(t) are relatively compact in E for t ∈ [0, T].
Let us recall the construction of regular measure of noncompactness in the space G([0, T], E) (see [2,4,5,9,15]). Therefore, let us take a set X ∈ G([0,T],E) . For x ∈ X and > 0 let us denote the following quantities: The above quantities can be interpreted as left-hand and right-hand sided moduli of convergence of the function x at the point t. Further, let us put: Finally, let us define quantity Theorem 2.5 [15] The function T given by formula

Measures of noncompactness in G(ℝ + , E)
Now we define the space mentioned in the title of this paper. Denote by G(ℝ + , E) the space consisting of all regulated functions defined on the interval ℝ + with values in a Banach space E. There is no natural norm in this space, which would provide the structure of Banach space. Therefore, we introduce the notion ‖x‖ T ∶= sup{‖x(t)‖ ∶ t ∈ [0, T]} for T > 0 and we define in G(ℝ + , E) the sequence of pseudonorms {‖ ⋅ ‖ n } n∈ℕ . The space G(ℝ + , E) becomes a Fréchet space furnished with the distance The above defined metric implies the following property: E) if and only if {x k } is uniformly convergent to x on bounded subsets of ℝ + , i.e. lim k→∞ ‖x k − x‖ n → 0 for n = 1, 2, ...

Obviously every relative compact set in
(X(t)).
Definition 3. 1 We will say that the set Now, we can formulate the following compactness criterion for considered space. . Therefore, we obtain the sequence of subsequences {x i,n }, i = 1, 2, ... converging on the interval [0, i] with respect to the pseudonorm ‖ ⋅ ‖ i to some regulated function defined on ℝ + . Next, putting z n ∶= x n,n , n = 1, 2, ... we get the sequence {z n } , uniformly convergent on bounded intervals to some regulated function z ∶ ℝ + → E . In virtue of the condition (A) the sequence {z n } is convergent in the space G(ℝ + , E) and it implies that X is relative compact in G(ℝ + , E) . ◻ Using the above proved compactness criterion, we are going to introduce the measure of noncompactness in the space G(ℝ + , E) . However, in contrast to the Banach space E where the measure of noncompactness is single function ∶ E → ℝ + (satisfying Definition 2.1), in the case of Fréchet space F, it is more convenient to introduce the sequence of functions { n } playing the role of measure of noncompactness (often called a family of measures of noncompactness). Let us provide the appropriate notions. The family of all nonempty and bounded subsets of F will be denoted by F while its subfamily consisting of all relatively compact sets is denoted by F . Now we can present the definition of the family of measures of noncompactness (see [14]).

Definition 3.3 A family of functions
, is said to be a measure of noncompactness in real Fréchet space F if it satisfies the following conditions: (P 1 ) The family ker{ n } ∶= {X ∈ F ∶ n (X) = 0 for n ∈ ℕ} is nonempty and 1, 2, ...) and if lim i→∞ n (X i ) = 0 for each n ∈ ℕ , then the intersection set For any fixed n ∈ ℕ , we introduce the mapping n ∶ G(ℝ + ,E) → ℝ + defined by the formula The main properties of the family of function { n } are contained in the below given theorem.

Theorem 3.4
The family { n } n∈ℕ of functions n ∶ G(ℝ + ,E) → ℝ + , n ∈ ℕ given by the formula (3.1) is a measure of noncompactness in the Fréchet space G(ℝ + , E) . Moreover, ker{ n } = G(ℝ + ,E) and it satisfies the following conditions Proof If X ⊂ G(ℝ + , E) is nonempty, then in virtue of Theorem 3.2, we obtain that X is relatively compact if and only if n (X) = 0 for n ∈ ℕ . Hence, ker{ n } = G(ℝ + ,E) and (P 1 ) is proved. Let us note that properties (P 2 )-(P 4 ) and (P 6 )-(P 8 ) follow immediately from theorem 2.5. Now we show that (P 5 ) is satisfied. Let the sequence {X i } is as in the condition (P 5 ) . Choose arbitrarily x i ∈ X i and let us define the sets Hence, we obtain Y i ∈ ker{ n } = G(ℝ + ,E) . This fact and closedness of Y i means that Y i is compact and nonempty for i ∈ ℕ . Since sequence It can be shown (using simple examples) that the sequence { n } does not satisfy the strong maximum property, i.e. there exists the sets X, Y ∈ G(ℝ + ,E) such that n (X ∪ Y) ≠ max{ n (X), n (Y)}, n ∈ ℕ.

Remark 3.5
Let us notice that the family { n } n∈ℕ given above is simpler than the measure of noncompactness appearing in the the paper [9]. Namely, the measure of noncompactness (3.1) is composed of only two components. Hence, this measure requires weaker assumptions when we applying fixed point theorems than other measures.

An applications
At the beginning of this section, we start with recall some necessary facts.
where is the Hausdorff measure of noncompactness.   V 1 (X) ∶= V(Conv(X)),Ṽ n (X) ∶= V(Conv(Ṽ n−1 (X))), n = 2, 3, ... n (Ṽ m n (X)) ≤ k n n (X) for X ⊂ Ω, n ∈ ℕ Now we will show the applicability of the family of measures of noncompactness constructed in the previous section in the Frechet space G(ℝ + , E) . We will study the following Volterra type integral equation We will impose the following conditions on the functions appearing in this equation.
(H 3 ) There exist nondecreasing functions p 1 , p 2 ∶ ℝ + → ℝ + such that (H 4 ) There are nondecreasing functions , ∶ ℝ + → ℝ + with lim →0 + ( ) = 0 , such that (H 5 ) There exists a function q ∈ L 1 loc (ℝ + , ℝ + ) such that (v(t, s, X)) ≤ q(s) (X) for 0 ≤ s ≤ t and bounded X ⊂ E. Now we formulate our existence theorem as: Proof First, we define operator V such that for any x ∈ G(ℝ + , E) The assumptions (H 1 ) and (H 2 ) guarantee that Vx ∈ G(ℝ + , E) . Moreover we put Next we define a function (4.1)   = 0 , then for any fixed n ∈ ℕ there exists m n ∈ ℕ such that Keeping in mind (2.1) and (4.3) we obtain n (Ṽ m n (X)) ≤ k n n (X) . This fact and Theorem 4.4 complete the proof. ◻ Remark 4. 6 Let us notice that many existence theorems, proved using the classical version of Darbo theorem, need some inconvenient assumption (existence of a solution of some artificial inequalities). Using theorem 4.4, we do not have this problem.
The assumption of this type is not necessary.