FUNCTIONS WITH BOUNDED E - d -VARIATION ON UNDIRECTED TREE NETWORKS

. In this paper we deﬁne and study functions with bounded E - d -vari- ation on undirected tree networks. For these functions with bounded E - d -varia- tion we establish a Jordan type theorem. We adopt the deﬁnition of network as metric space introduced by P. M. Dearing and R. L. Francis (1974). MSC 2000. 65P40


INTRODUCTION
In [9] are introduced a class of sets and a class of functions called E-convex sets and E-convex functions.This kind of generalized convexity is based on the effect of an operator E on the sets and domain the definition of the functions.In [1] are defined and studied E-monotone functions and functions of bounded E-variation.
In the following lines we will define and study E-d-monotone functions and functions with bounded E-d-variation on undirected tree networks.
We consider an undirected, connected graph G = (W, A), without loops or multiple edges.To each vertex w i ∈ W = {w 1 , . . ., w m } we associate a point v i from an euclidean space X.This yields a finite subset V = {v 1 , . . ., v m } of X, called the vertex set of the network.We also associate to each edge (w i , w j ) ∈ A a rectifiable arc [v i , v j ] ⊂ X called edge of the network.We assume that any two edges have no interior common points.Consider that [v i , v j ] has the positive length l ij and denote by U the set of all edges.We define the network N = (V, U ) by It is obvious that N is a geometric image of G, which follows naturally from an embedding of G in X. Suppose that for each [v i , v j ] ∈ U there exist a continuous one-to-one mapping Any connected and closed subset of an edge bounded by two points x and y of [v i , v j ] is called a closed subedge and is denoted by [x, y].If one or both of x, y are missing we say than the subedge is open in x, or in y or is open and we denote this by (x, y], [x, y) or (x, y), respectively.Using θ ij , it is possible to compute the length of [x, y] as Particularly we have y) linking two points x and y in N is a sequence of edges and at most two subedges at extremities, starting at x and ending at y.If x = y then the path is called cycle.The length of a path (cycle) is the sum of the lengths of all its component edges and subedges and will be denoted by l L (x, y) .If a path (cycle) contains only distinct vertices then we call it elementary.
A network is connected if for any points x, y ∈ N there exists a path L (x, y) ⊂ N .
A connected network without cycles is called tree.In a tree network N there is an unique path between two points x, y ∈ N .
Let L * (x, y) be a shortest path between the points x, y ∈ N .This path is also called geodesic.Definition 1. [2].For any x, y ∈ N , the distance from x to y, d (x, y), in the network N is the length of a shortest path from x to y: and x, y is called the metric segment between x and y.
We consider now a map E : N → N .

E-d-MONOTONE FUNCTIONS ON UNDIRECTED TREE NETWORKS
We consider an undirected tree network N = (V, U ), a map E : N → N and two points x, y ∈ N .
We denote Obviously E x, y E ⊆ E(x), E(y) .Generally, the converse inequality is not true.Now, let us define the following order relation on x, y E .For Obviously x ≤ E y and every z ∈ x, y E satisfies x ≤ E z ≤ E y.
We consider now the function f : N → R and the points x, y ∈ N .Definition 6.
(1) The function f : N → R is said to be E-d-increasing between x and y if for every z 1 , z 2 ∈ x, y E such that z 1 ≤ E z 2 we have (2) The function f : N → R is said to be E-d-decreasing between x and y if for every ( A function that is either E-d-increasing or E-d-decreasing between x and y is said to be E-d-monotone between x and y.If all the inequalities in Definition 6 are strict then f is called strictly E-d-increasing, strictly E-d-decreasing or strictly E-d-monotone.

Remarks. 1. If a function f : N → R is both E-d-increasing and E-d-decreasing between x and y then it is E-d-constant between x and y.
2. If a function f : N → R is E-d-increasing between x and y then it is E-d-decreasing between y and x.
In the following we give an example of E-d-monotone function.
Example 1.We consider a tree network N = (V, U ) with For the points x = v 1 and y = v 2 we have

NETWORKS
We consider an undirected tree network N = (V, U ), a map E : N → N , an E-d-convex set M ⊂ N and the function f : M → R. We also consider the points x, y ∈ M .
For a division (σ) of the set x, y E ∩ M by the points we define the number called the E-d-variation of the function f on the division (σ).We denote by D the set of all divisions (σ) of the set x, y E ∩ M .
Definition 7. The number Definition 14.The function f : M → R satisfies the E-Lipschitz condition on x, y E ∩ M if there is a number k > 0 such that for any pair of points z 1 , z 2 ∈ x, y E ∩ M it is satisfied the relation: Proof.Indeed, if the function f : M → R satisfies the E-Lipschitz condition on x, y E ∩ M then there is a number k > 0 such that for any pair of points z 1 , z 2 ∈ x, y E ∩ M is satisfied the relation (3).Consequently we have: In the following lines we will establish a Jordan type theorem.

and only if there exist two E-d-increasing functions between x and y, g
Proof.The sufficiency of the condition follows from Theorem 9 and Theorem 12.
For the necessity, let us define the function From Theorem 16 follows that the function g is E-d-increasing between x and y.
We define now the function This function is E-d-increasing between x and y.Indeed, if we consider the points z , z ∈ x, y E ∩ M , such that z < E z , we have Consequently h(z ) − h(z ) ≥ 0, that is, h is E-d-increasing between x and y.Hence f = g − h, where the functions g and h are E-d-increasing between x and y.
Remark 1.The representation of a function with bounded E-d-variation as a difference of two E-d-increasing functions is not unique.Indeed, if f = g − h and the functions g and h are E-d-increasing between x and y then we also have f = g + c − (h + c), c being constant on x, y E ∩M .The functions g +c and h+c are E-d-increasing between x and y too.
Remark 2. In [6] we already defined and studied another class of functions with bounded variation on undirected networks.
f ) < ∞ and f is with bounded E-d-variation on x, y E ∩ M .We consider now a pointz ∈ x, y E ∩ M \ t ∈ M | t = E x or t = E y .Theorem 16.If the function f : M → R is with bounded E-d-variation on x, y E ∩ M then it is with bounded E-d-variation on x, z E ∩ M and on z, y E ∩ M and f ).Theorem 17.If the function f : M → R is with bounded E-d-variation on x, z E ∩ M and on z, y E ∩ M then it is with bounded E-d-variation on x, y E ∩ M .