On non-expansivity of topical functions by a new pseudo-metric

In this paper, we ﬁrst deﬁne a new pseudo-metric d on a normed linear space X . We do this by introducing two different classes of elementary topical functions. Next, we use this pseudo-metric d to investigate the non-expansivity and some properties of topical functions. Finally, the characterizations of ﬁxed points of topical functions are given, and a relation between the pseudo-metric d and the original norm of the normed linear space X is presented.

of this metric in algebra and analysis. For example, the Perron-Frobenius theorem for non-negative matrices has been proved by an application of the Banach contraction-mapping theorem in suitable metric spaces.
Birkhoff's version of Hilbert's metric is a distance between pairs of rays in a closed cone, and is closely related to Hilbert's classical cross-ratio metric [6]. A version of this metric is discussed by Bushell [5] which could be traced back to the work of Birkhoff [4] and Samelson [16]. It has found numerous applications in mathematical analysis, especially in the analysis of linear, and nonlinear, mappings on cones. Birkhoff's version of Hilbert's metric provides a different perspective on Hilbert geometries and naturally leads to infinitedimensional generalizations. Recently, it has been given a strong convergence theorem for an iterative algorithm that approximates fixed points of those self-mappings of the Hilbert ball which are non-expansive with respect to the hyperbolic metric [10]. Moreover, in [9] an extension of the Banach Contraction Principle for best proximity points of a non-self mapping on the open unit Hilbert ball has been obtained, and this result also was established for best proximity points of non-expansive mappings and firmly non-expansive mappings. This is a motivation for us to define a metric (or, a pseudo-metric) and investigate the non-expansivity, characterizations of fixed points and other properties of topical functions (i.e., plus-homogeneous and increasing). Indeed, we do this, by introducing two different classes of elementary topical functions, in particular, these two classes are of min-type and max-type functions (see, Example 3.1, below).
The structure of the paper is as follows. In Sect. 2, we provide definitions, notations and preliminary results related to topical functions. In Sect. 3, we first introduce two different classes of elementary topical functions, and define a new pseudo-metric d using these two classes. Next, we investigate the properties of the pseudo-metric d. The results on non-expansivity of topical functions and characterizations of fixed points of this class of functions are given in Sect. 4. Furthermore, we present a relation between the pseudo-metric d and the original norm of the normed linear space X.

Preliminaries
Let (X, · ) be a real normed linear space. We assume that X is equipped with a closed convex pointed cone S. The cone S is called pointed if S ∩ (−S) = {0}. We also assume that int S = ∅, where int A denotes interior of a subset A of X. For each x, y ∈ X, we say that x ≤ y or y ≥ x if and only if y − x ∈ S. Also, we say that x < y or y > x if and only if y − x ∈ S\{0}. It is easy to see that " ≤ " is a partial order on X, and so, (X, ≤) is an ordered normed linear space.
Moreover, we assume that S is a normal cone. Recall [14] that the cone S is called normal if there exists a constant m > 0 such that x ≤ m y , whenever 0 ≤ x ≤ y with x, y ∈ X. Let 1 ∈ int S (see Remark 2.5, below) and let It is well known and easy to check that B can be considered as the unit ball of a certain norm · 1 on X, which is equivalent to the initial norm · . Assume without loss of generality that · = · 1 .
In the sequel, we will consider the ordered normed linear space (X, ≤, · ) and the unit ball B as described above unless stated otherwise.
We recall from [11] the following definitions. A function p : Remark 2.5 Since S is a cone, it follows that int S is also a cone. Therefore, there exists 0 = u ∈ int S because int S = ∅, and hence, u u ∈ int S. Let 1 := u u . So, 1 ∈ int S and 1 = 1.
Throughout the paper, let 1 ∈ int S be as in Remark 2.5.

A new pseudo-metric and its properties
In this section, using the elementary topical functions defined in [11,12], we first introduce a new pseudo-metric d, and then we investigate its properties. We start with the definition of the elementary topical functions. In [11,12], the elementary topical function ϕ : It should be noted that, in view of (1), the set {λ ∈ R : λ1 + y ≤ x} is non-empty and bounded from above (by x − y ). Clearly, this set is closed. So, in the definition of ϕ, we can use maximum instead of supremum. It follows from the definition of ϕ that: (2) We enlist some properties of the function ϕ, which have been obtained in [11,12] (therefore, we state them without proof).
Now, for each y ∈ X, we define the function ϕ y : X −→ R by ϕ y (x) := ϕ(x, y) for all x ∈ X. Therefore, it is clear that, for each y ∈ X, the function ϕ y satisfies the relations (2)-(10).
In the sequel, we also consider the elementary topical function (cf. [11,12]) ψ : X × X → R defined by It is worth noting that, in view of (1), the set {λ ∈ R : x ≤ λ1 + y} is non-empty and bounded from below (by − x − y ). Clearly, this set is closed. So, in the definition of ψ, we can use minimum instead of infimum. It follows from the definition of ψ that: We present some properties (without proof) of the function ψ obtained in [11,12].
For each y ∈ X, we define the function ψ y : X −→ R by ψ y (x) := ψ(x, y) for all x ∈ X. Therefore, it is clear that, for each y ∈ X, the function ψ y satisfies the relations (11)-(19). We now give some crucial properties of ϕ y and ψ y (y ∈ X ).

Proposition 3.3
For each y ∈ X, the functions ϕ y and ψ y are topical.

Lemma 3.4
For each x , y ∈ X, Proof The result follows from the definitions of ϕ y and ψ y .
Proof This is an immediate consequence of the relations (3) and (12).
Proof Assume if possible that there exist x, y ∈ X such that ϕ y (x) > ψ y (x). This together with 0 = 1 ∈ S and the fact that S is a cone implies that and hence, ψ y (x)1 < ϕ y (x)1. This contradicts Proposition 3.5 (see page 3, for the definition of the strict order " < "). Now, using the elementary topical functions ϕ y and ψ y (y ∈ X ), we introduce a new pseudo-metric on X.
From now on, we consider the pseudo-metric space (X, d) given by Theorem 3.8.
Example 3.12 Let X be as in Example 3.10, and let E : The following example shows that the pseudo-metric d may be a metric on some subspace W of X, and moreover, (W, d) is a complete metric space.
We only show that if u, v ∈ E m are such that d(u, v) = 0, then u = v. This together with Theorem 3.8 implies that d is a metric on E m , and hence, (E m , d) is a metric space. To this end, assume that u, v ∈ E m are such that d(u, v) = 0. Therefore, u = (x, mx) and v = (y, my) for some x, y ∈ R. Since d(u, v) = 0, it follows from (24) that there exists λ ∈ R such that u = v + λ1. This implies that We claim that λ = 0. Otherwise, in view of (26), we obtain m = 1, which is a contradiction because m ≥ 2. So, it follows from (26) that x = y, and so, u = v. Thus, by Theorem 3.8, d is a metric on E m . Now, let {u k } k≥1 be a Cauchy sequence in E m . Then there exists a sequence {x k } k≥1 in R such that u k = (x k , mx k ) (k = 1, 2, . . . ). So, for every ε > 0, there exists N ∈ N such that d(u k , u n ) < ε for all k, n ≥ N . In view of (25), we have This implies that Therefore, the sequence {x k } k≥1 is Cauchy in R, and so, there exists x ∈ R such that x k −→ x with respect to the Euclidean metric. Let u := (x, mx) ∈ E m . But, This together with the fact that x k −→ x with respect to the Euclidean metric implies that d(u k , u) −→ 0 as k −→ +∞, i.e., u k −→ u with respect to the metric d. Then, (E m , d) is a complete metric space.

Non-expansivity and characterizations of fixed points of topical functions
In this section, using the pseudo-metric d, we obtain some results on non-expansivity of topical functions. In fact, we show that each topical function is non-expansive with respect to the pseudo-metric d. Also, we present the characterizations of fixed points of topical functions. We denote by Fi x T the set of all fixed points of a function T : X −→ X, and is defined by (see, [1,2]). It should be noted that a topical function T : X −→ X has not necessarily a fixed point. In the following, we give an example. We first give the following definition of a topical function.
for all x ∈ X and all λ ∈ R).
Example 4.2 Let 0 = a ∈ R be fixed, and let 1 : It is easy to show that T a is a topical function in the sense of Definition 4.1. Also, by Lemma 3.9 (1), we have So, T a is also non-expansive with respect to the pseudo-metric d. Clearly, T a has no fixed point, i.e., Fi x T a = ∅.

Definition 4.3
A subset G of X is called a generator for X, if for each x ∈ X, there exist y ∈ G and λ ∈ R such that x = y + λ1, and we write X = G .
Theorem 4.5 Let T : X −→ X be an arbitrary function, and let G ⊆ X be a generator for X. Suppose that T is plus-homogeneous on G (see Definition 4.1). If T : G −→ G is non-expansive, then T : X −→ X is also non-expansive.
Proof Let x 1 , x 2 ∈ X be arbitrary. Since G is a generator for X, in view of Definition 4.3, there exist y 1 , y 2 ∈ G and λ 1 , λ 2 ∈ R such that x 1 = y 1 + λ 1 1 and x 2 = y 2 + λ 2 1. This together with Lemma 3.9 and the fact that T is non-expansive and plus-homogeneous on G implies that This completes the proof.
We now show that if a topical function T : X −→ X has at least one fixed point, then T has infinitely many fixed points. We first give the following definition.

Definition 4.6
Let y 0 ∈ X be fixed. A subset C of X is called pseudo-plus cone in the direction y 0 , if x ∈ C, then x + λy 0 ∈ C for all λ ∈ R + .

Proof If Fi x T = ∅, then it is clear that Fi x T is a pseudo-plus cone in the direction 1. Assume that Fi x T = ∅.
Let x 0 ∈ Fi x T and λ ∈ R + be arbitrary. Since T (x 0 ) = x 0 and T is topical, it follows that This implies that x 0 + λ1 ∈ Fi x T for all λ ∈ R + , and hence, Fi x T is a pseudo-plus cone in the direction 1. Consequently, if Fi x T = ∅, then Fi x T has infinitely many points.
The following result exerts that there does not exist any topical function which is contractive with respect to the pseudo-metric d.

Theorem 4.8
There does not exist any topical function T : X −→ X that is contractive, i.e., there does not exist a topical function T : X −→ X, which satisfies the following strict inequality: Proof Assume if possible that there exists a topical function T : X −→ X, which satisfies the strict inequality (28). Now, let x 0 ∈ X be arbitrary, and let α, β ∈ R\{0} be such that α = β. Put x := x 0 +α1 and y := x 0 +β1 in (28). Then, This together with the fact that T is topical implies that Therefore, by Lemma 3.9, we obtain which is a contradiction because d(z, z) = 0 for all z ∈ X. This completes the proof. Note that if p = 1 in Definition 4.9, then T is topical. Proof Let x, y ∈ X be arbitrary. By (21), we have This together with the fact that T is p-topical implies that Now, in view of the definitions of ϕ y and ψ y and (30), we get Therefore, it follows from (23) and (31) that pd(x, y). Remark 4.11 In view of Theorem 4.10, every p-topical function ( p > 0) is continuous with respect to the pseudo-metric d. Remark 4.12 In Theorem 4.10, if p = 1, then T is topical, and so, by (29), we conclude that T is nonexpansive. Consequently, every topical function is non-expansive with respect to the pseudo-metric d. Remark 4.13 In Theorem 4.10, if 0 < p < 1, then it follows from (29) that T is a contraction with respect to the pseudo-metric d.
Example 4.14 Let X := R 2 and 1 := (1, 1) ∈ R 2 . Define the function T : It is easy to check that the function T is 1 2 -topical, and so, in view of Remark 4.13, T is a contraction with the unique fixed point zero. Now, we give the following result on p-topical functions.

Theorem 4.15
Assume that (X, d) is complete. Let T : X −→ X be a p-topical function (0 < p < 1), and let x 0 ∈ X be fixed. Define x n+1 := T (x n ), n = 0, 1, 2, . . . . Then the sequence {x n } n≥0 converges to some point x ∈ X (it should be noted that x is not necessarily unique, and also, x is not necessarily a fixed point of T, see Example 4.16,below). Furthermore, Proof We first show that the sequence {x n } n≥0 converges to some point x ∈ X. To this end, by the hypothesis and Theorem 4.10, we have Thus, for each m, n ∈ N with m > n, we obtain So, the sequence {x n } n≥0 is Cauchy in X. Since X is a complete pseudo-metric space, then there exists x ∈ X such that x n −→ x with respect to the pseudo-metric d. Furthermore, by Theorem 4.10 and the fact that x n+1 = T (x n ), n = 0, 1, 2, . . . , it follows that x n −→ T (x) with respect to the pseudo-metric d. Therefore, for every ε > 0, we conclude that for all sufficiently large n. This implies that d(T (x), x) = 0 (note that in this case, in view of (24), there exists λ ∈ R such that T (x) = x + λ1), and hence, by Theorem 4.10, we deduce that d(T n (x), T m (x)) = 0 for all m, n ∈ N. Then, (1) (X, d) is a complete pseudo-metric space.
Solution (1). Assume that {(t 1 n , . . . , t k n )} n≥0 is a Cauchy sequence in X with respect to the pseudo-metric d. So, for each ε > 0, there exists N ∈ N such that This implies (not difficult to check) that Hence, {t i n } n≥0 is a Cauchy sequence in R with respect to the Euclidean metric for each i = 1, . . . , k. Then, there exists t i ∈ R such that t i n −→ t i (i = 1, . . . , k). Now, we show that {(t 1 n , . . . , t k n )} n≥0 converges to (t 1 , . . . , t k ) with respect to the pseudo-metric d. In view of (32), one has which completes the solution of (1). Solution (2). In view of the definition of the sequence {z n } n≥0 , one has

Definition 4.17
Let T : X −→ X be a function. Suppose that p ∈ R and n ∈ N. We say that T is a p n -topical function if T n is increasing and Proposition 4.18 Let T : X −→ X be a p-topical function ( p ∈ R), and let n ∈ N. Then, T is a p n -topical function.
Proof By induction on n. For n = 1, by the assumption T is p-topical, so the result follows. Assume that for n = k, the function T is p k -topical. We show that the result holds for n = k + 1. To this end, since T is p-topical, we have Also, for each x, y ∈ X with x ≤ y, by the hypothesis of induction, one has T k (x) ≤ T k (y). As T is increasing, so we conclude that and hence, T is a p n -topical function.

Theorem 4.19
Let the function T : X −→ X be p n -topical ( p > 0, n ∈ N). Then, Proof This follows by an argument similar to the proof of Theorem 4.3.
By Theorem 4.19, the proof of the following theorem is similar to that of Theorem 4.4, and therefore, we omit its proof.

Theorem 4.20
Assume that (X, d) is complete. Let the function T : X −→ X be p-topical (and hence, a p n -topical function) (0 < p < 1, n ∈ N), and let x 0 ∈ X be fixed. Define x n+1 := T n (x 0 ), n = 0, 1, 2, . . . . Then the sequence {x n } n≥0 converges to some point x ∈ X (it should be noted that x is not necessarily unique, and also, x is not necessarily a fixed point of T ).

Lemma 4.21
Let T : X −→ X be a p n -topical function (0 < p < 1, n ∈ N). Then the set of fixed points of T (Fi x T ) is a null set in the sense of Definition 3.2.
Proof Let x, y ∈ Fi x T be arbitrary. Then, T (x) = x and T (y) = y, and so, for each n ∈ N, one has T n (x) = x, T n (y) = y. This together with Theorem 4.19 implies that d(x, y) ≤ p n d(x, y), (n ∈ N).
Since 0 < p < 1, we conclude that d(x, y) = 0. So, diam(Fi x T ) = 0, and hence, Fi x T is a null set.
The definition of a firmly non-expansive operator on a Hilbert space has been given in [2,Chapter 4]. We adopt it in the pseudo-metric space (X, d) as follows.

Definition 4.22
Let T : X −→ X be a function. We say that T is firmly non-expansive if Remark 4.23 It should be noted that in view of Definition 4.22, T is firmly non-expansive if and only if I d − T is firmly non-expansive.

Theorem 4.24
Let T : X −→ X be an arbitrary function, and let G ⊆ X be a generator for X. Suppose that T is plus-homogeneous on G (see Definition 4.1). If T : G −→ G is firmly non-expansive, then T : X −→ X is also firmly non-expansive.
Proof Let x 1 , x 2 ∈ X be arbitrary. Since G is a generator for X, in view of Definition 4.3, there exist y 1 , y 2 ∈ G and α 1 , α 2 ∈ R such that x 1 = y 1 + α 1 1 and x 2 = y 2 + α 2 1. This together with Lemma 3.9 and the fact that T is plus-homogeneous on G implies that and also, Moreover, by Lemma 3.9, we have Hence, (33), (34) and (35) together with the fact that T is firmly non-expansive on G imply that So, T is firmly non-expansive on X, and hence, the proof is complete.
In the following, we give the relation between the norm of X and the pseudo-metric d.

Theorem 4.25
The following inequality holds.
Proof As B is the unit ball of X, so it follows from (1) that Now, let y ∈ X be arbitrary. Since the function ϕ y : X −→ R is increasing, we conclude from (37) that On the other hand, the function ϕ y : X −→ R is plus-homogeneous, so it follows from (38) that By an argument similar to the above and the fact that the function ψ y : X −→ R is topical (increasing and plus-homogeneous), we obtain In view of the definition of the pseudo-metric d, it follows from (41) that |d(x, y) − d(0, y)| ≤ 2 x , ∀ x, y ∈ X.
By the definition of the pseudo-metric d, and the definitions of ϕ y and ψ y , it is easy to check that This together with (43) implies that d(x, y) ≤ 2 x − y , ∀ x, y ∈ X. Proof As B is the unit ball of X, so it follows from (1) that This implies that y − x − y 1 ≤ x ≤ y + x − y 1, ∀ x, y ∈ X.
Since T is a topical function, it follows from (44) that This together with (1) implies that This together with Theorem 4.26 implies that d(T n (x), T n (y)) ≤ 2 x − y , ∀ x, y ∈ X, ∀ n ∈ N, i.e., for each n ∈ N, T n : (X, · ) −→ (X, d) is a Lipschitz continuous function with the Lipschitz constant 2.