On separation by function of bounded variation

In the present paper we are concerned with the problem of separation of two given functions by an increasing (decreasing) function or by a function of bounded variation. In the context of separation by a function of bounded variation we give two approaches. The first one concerns the separation with respect to the classical order in the set of real numbers, and this allow us to the new concept of variation involving two functions. This approach in fact leads us to the unusual phenomenon, namely the bounded joint variation of two functions forces these functions to be essentially the same. The second approach concerns the separation with respect to the partial order generated by the Lorentz cone.


Introduction
The purpose of this paper is to find (if it is possible) the necessary and sufficient conditions under which two given functions can be separated by an increasing function as well as a decreasing function or by a function of bounded variation. A problem of finding for a given pair of real valued functions a function belonging to a given class which lies between them has a long story. Probably the first result of this type was obtained by Hahn in 1917 (see [5]) who proved that if a function g : X → R is upper semicontinuous and f : X → R is lower semicontinuous (where X stands for a metrizable topological space) and g ≤ f then there exists a continuous function h : X → R such that g ≤ h ≤ f on X . The another very important separation type result is a consequence of a geometrical version of Hahn-Banach theorem proved by Kakutani [7] and says that if f , −g are convex functions and g ≤ f then there always exists an affine function h which separates f and g. The corresponding result for separating a superadditive function g and a subadditive f by an additive one was proved by Kranz [9] (see also [8]). These results give only the necessary conditions for separation.
We already know several results which give the necessary and sufficient conditions for separation by a function from many important classes. For example: by convex function [2], B Andrzej Olbryś andrzej.olbrys@us.edu.pl 1 Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland by affine function [11], by monotonic function [4], by subadditive and sublinear function [10] and many other.

Separation by increasing or decreasing function
We start with separation theorems for increasing and decreasing functions. A corresponding theorem for monotonic functions was proved in [4].
By definition it is clear that To see that h is an increasing function fix x, y ∈ I , x < y arbitrarily. Since which finishes the proof.
By the same arguments as in the previous proof we obtain a corresponding theorem for decreasing functions.

Theorem 2.2 Real functions f and g, defined on a real interval I satisfy the inequality
for all x, y ∈ I , x ≤ y if and only if there exists a decreasing function h : As an application we get the following stability result for increasing functions.

Separation by function of bounded variation
Now, we will consider the separation problem for functions of bounded variation. Let P [a,b] denote the set of partitions of the interval [a, b] i.e.
The concept of bounded variation functions was introduced in 1881 by Camille Jordan [6] for real functions defined on a closed interval [a, b] ⊂ R in the study of Fourier series. He proved that a function is of bounded variation if and only if it can be represented as the difference of two increasing functions. This representation is known as the Jordan decomposition. We refer the interested reader to the comprehensive book on functions of bounded variation by Appell, Banaś and Merentes [1]. Now, we will focus our attention on the problem of separation by a function of bounded variation. Let us observe that if there exists a function h : then for any partition (x 0 , . . . , This leads us to the following definitions. For arbitrary two functions f , g : [a, b] → R, we define Proof The assertions (a)-(c) and (e) follow immediately from the definition of β f ,g . We prove assertion (d). Assume contrary, that is, for some x, y, z ∈ I we have From one side, and on the other side . Consequently, and this contradiction finishes the proof.
The above proposition leads to the following concept of variation involving two functions. Let f , g : [a, b] → R be two given functions. We define As we will see the above defining variation V b a ( f , g) depending on two functions has similar properties to usual variation.
Let (x 0 , . . . , x p ) ∈ P [a,c] and (y 0 , . . . , y q ) ∈ P [c,b] be arbitrary partitions. Clearly, Taking the supremum over all partitions (x 0 , . . . , Let define the new partitions Taking the supremum over all partitions (x 0 , . . . , x n ) ∈ P [a,b] , we obtain the reverse inequality Now, using the concept of bounded variation involving two functions we obtain the following separation theorem for functions of bounded variation.

Remark 3.5 It follows from the above theorem that if
then there always exists a function of bounded variation h : [a, b] → R which lies between f and g. Indeed, it is enough to put h := f or h := g. Then the separation property is trivially fulfilled.
The following example below shows that if the condition V b a ( f , g) < ∞ is not satisfied then it can happen that there is no function of bounded variation between f and g so this condition is sufficient for separation by a function of bounded variation. From this example it also follows that the condition V b a ( f , g) < ∞ is not necessary for separation by a function of bounded variation.

Example 3.6
Let ε > 0 and let f , g : [0, 2] → R be given by the formulas Consider a sequence of partitions Clearly, g ≤ f , moreover, for any h : g ≤ h ≤ f and ε ∈ (0, 2) we have On the other hand if ε ∈ [2, ∞) then the function h ≡ −1 is of bounded variation, and As an immediate consequence of Theorem 3.4 we get The following example shows that the converse to the assertion (a) from the previous corollary does not hold in general.

Example 3.8 Consider two constant functions
On the other hand for an arbitrary sequence of partitions Consequently, The above example shows that if V b a ( f , g) < ∞ then the functions f and g can not be different on a big set. The next theorem gives a characterization of functions f and g for which V b a ( f , g) < ∞. In order to present this theorem we need the following notation: for a function f : [a, b] → R let D f denotes the set of all discontinuity points of f , i.e.
It is well known that if f is a function of bounded variation then D f is at most countable set. Moreover, each function of bounded variation has one sides limits at any point of [a, b].
The following theorem gives necessary and sufficient conditions under which two functions f , g : [a, b] → R enjoy the property if and only if they are both of bounded variation and in the case where D f ∪ D g = {x k : k ∈ N} is an infinite sequence of discontinuity points of f and g.

Proof If
V b a ( f , g) < ∞, then by Corollary 3.7 we have hence f and g are both of bounded variation. Fix an arbitrary point We show that Let {x n } n∈N be an arbitrary sequence of points of interval [a, b] such that Clearly, By the assumption, and since both functions f and g are continuous at x 0 we obtain In the case where is an infinite set, using the inequalities Conversely, assume that Moreover, in the case where the set of discontinuity points of f and g: It is easy to prove that the inequality holds. Using this inequality we get Taking the supremum over all partitions , which finishes the proof.
The above theorem shows that the first approach proposed in the present paper, although natural, leads us to the unusual phenomenon, namely, the bounded joint variation of f and g forces these functions to be essentially the same. This behavior seems unnatural. Therefore we are going to propose an another approach to the separation problem by a function of bounded variation. The starting point will be the following theorem which gives a necessary and sufficient condition that a given map has a bounded variation which is true even in the case where the map takes values in metric space.
Conversely, assume that there exists an increasing function k : [a, b] → R such that for all x, y ∈ [a, b], x ≤ y the inequality holds. For arbitrary (x 0 , . . . , and taking the supremum over all partitions (x 0 , . . . , x n The above theorem allow us to consider the partial order in the spaceX := X × R where (X , d) is a metric space defined in the following manner A slightly different order was used by Ekeland in [3] to prove his famous variational principle which has a number applications in convex analysis. In the case where (X , · ) is a normed space the above mentioned partially order is generated by so called Lorentz cone: This partial order is compatible with the linear structure ofX , i.e.
• x C y ⇒ x + z C y + z for x, y, z ∈X , where the addition and scalar multiplication are defined coordinatewise. The partial order generated by the Lorentz cone plays a fundamental role in the optimization theory and Jordan algebras. In this paper we will consider a very particular case where a normed space is (R, | · |). By holds for all x, y ∈ [a, b], x ≤ y. By analogy to the notion of delta-convexity (see [12]) we say that F is a delta-increasing function with a control function f . Thus, the set of delta-increasing functions on [a, b] forms the smallest linear space containing the increasing functions on [a, b].
Note that, defining for given maps F : we can rewrite the above inequality by the formula The above remark shows that the notion of bounded variation generalizes the notion of usual monotonicity by replacing the classic inequality by the relation of partial order induced by the Lorentz cone. The results for usual monotonicity are obtained by putting F = 0.
The separation theorem for functions of bounded variation with respect to a partial order generated by the Lorentz cone reads as follows. a, b ∈ R, a < b. Then the functions f , g, F, G : [a, b]

Theorem 3.11 Let
Proof Suppose that conditions (a)-(c) hold. Fix x, y ∈ [a, b], x ≤ y arbitrarily. By (a), (b), (c) and the triangle inequality we get Conversely, suppose that the inequality (3.3) holds. Then Let define the functions H , h : [a, b] → R by the formulas We will show that conditions (a)-(c) hold true.
To prove a), by adding the expression to both sides of the inequality we get Analogously, by subtracting the term h 1 (x)+h 2 (x) 2 from both sides of inequality we get and, consequently, For (b), subtracting the expression h 1 (x)+h 2 (x) 2 from both sides of the inequality we get Similarly, by subtracting the expression h 1 (x)−h 2 (x) 2 from both sides of the inequality we get Therefore, Part (c) follows immediately from the definition of H and h. Indeed, for x, y ∈ [a, b], x ≤ y we have The proof of the theorem is finished.
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