On metric-preserving functions and fixed point theorems

Kirk and Shahzad have recently given fixed point theorems concerning local radial contractions and metric transforms. In this article, we replace the metric transforms by metric-preserving functions. This in turn gives several extensions of the main results given by Kirk and Shahzad. Several examples are given. The fixed point sets of metric transforms and metric-preserving functions are also investigated.


Introduction
The concept of metric transforms is introduced by L. M. Blumenthal [1,2] in 1936 while the concept of metric-preserving functions seems to be introduced by W. A. Wilson [28] in 1935 and is investigated in details by many authors [3,4,5,6,7,8,9,10,11,12,18,19,20,21,23,25,27]. Recently, Petruşel, Rus, and Şerban [18] have shown the role of metric-preserving functions in fixed point theory. In addition, Kirk and Shahzad [16] have given results concerning metric transforms and fixed point theorems. Their main results are as follows: Theorem 1. (Kirk and Shahzad [16,Theorem 2.2]) Let (X, d) be a metric space and g : X → X. Suppose there exists a metric transform φ on X and a number k ∈ (0, 1) such that the following conditions hold: Definition 5. Let (X, d) be a metric space and γ be a path in X, that is, a continuous map γ : [a, b] → X. A partition Y of [a, b] is a finite collection of points Y = {y 0 , . . . , y N } such that a = y 0 ≤ y 1 ≤ y 2 ≤ · · · ≤ y N = b. The supremum of the sums over all the partitions Y of [a, b] is called the length of γ. A path is said to be rectifiable if its length is finite. A metric space is said to be rectifiably pathwise connected if each two points of X can be joined by a rectifiable path.
We will give some auxiliary results in Section 2. Then we will give the results concerning metric-preserving functions, local radial contractions, and uniform local multivalued contractions in Section 3 and Section 4. Finally, we investigate the fixed point sets of metric transforms and metric-preserving functions in Section 5. Proof. The proof is given in [4,Proposition 1.3], [5,Proposition 2.8], and [7, p. 17].
For a metric-preserving function f , let K f denote the set Recall also that we define inf ∅ = +∞. Then we have the following result.
The next lemma is probably well-known but we give a proof here for completeness.
Lemma 11. If f : [0, ∞) → [0, ∞) is amenable and concave, then the func- x is decreasing on (0, ∞) Proof. Let a, b ∈ (0, ∞) and a < b. Since f is concave, we obtain . Assume that f is amenable and there is a periodic function g such that f (x) = x + g(x) for all x ≥ 0. Then f is metric-preserving if and only if f is increasing and subadditive.
Lemma 13. (Hu and Kirk [14]) Let (X, d) be a complete metric space for which each two points can be joined by a rectifiable path, and suppose g : X → X is a local radial contraction. Then g has a unique fixed point x 0 , and lim n→∞ g n (x) = x 0 for each x ∈ X.
As noted by Kirk and Shahzad [16], an assertion in the proof of Lemma 13 given in [14] was based on a false proposition of Holmes [13]. But Jungck [15] proved that the assertion itself is true. Hence the proof given in [14] with minor changes is true. Kirk and Shahzad [16] apply Tan's result [24] to extend some of their theorems. We will also apply Tan's result as well.
Lemma 14. (Tan [24]) Let X be a topological space, let x 0 ∈ X, and let g : X → X be a mapping for which f := g N satisfies lim n→∞ f n (x) = x 0 for each x ∈ X. Then lim n→∞ g n (x) = x 0 for each x ∈ X. (Also if x 0 is the unique fixed point of f , it is also the unique fixed point of g.) We will use Nadler's result concerning set-valued mappings. So let us recall some more definitions. If ε > 0 is given, a metric space (X, d) is said to be ε-chainable if given a, b ∈ X there exist x 1 , x 2 , . . . , x n ∈ X such that a = x 1 , b = x n , and d(x i , x i+1 ) < ε for all i ∈ {1, 2, . . . , n − 1}. The result of Nadler that we need is the following.

Local radial contractions and metric-preserving functions
In this section, we will give a generalization of Theorem 1 where the metric transform φ is replaced by a metric-preserving function. In fact, we obtain a more general result as follows: Theorem 16. Let (X, d) be a metric space and let g : X → X. Assume that there exists k ∈ (0, 1) and a metric-preserving function f satisfying the following conditions: (a) for each x ∈ X, there exists ε > 0 such that for every u ∈ X Then g is a local radial contraction.
We know from Lemma 10 that f ′ (0) always exists in R ∪ {+∞}. So condition (b) in Theorem 16 makes sense. To prove this theorem, we will first show that g is continuous in the following lemma.
Lemma 17. Suppose that the assumptions in Theorem 16 hold. Then the function g is continuous.
As a consequence of Theorem 16, we can replace the metric transform φ in Theorem 1 by a metric-preserving function and obtain an extension of Theorem 1.
Theorem 18. With the same assumptions in Theorem 16 except that condition (b) is replaced by (b ′ ): there exists c ∈ (0, 1) such that f (ct) ≥ kt for all t > 0 sufficiently small. Then g is a local radial contraction.

Proof of Theorem 16
To show that g is a local radial contraction with the contraction constant c, let x ∈ X. By Lemma 17, g is continuous at x. So there exists δ 2 > 0 such that for every u ∈ X, By condition (a), there exists δ 3 > 0 such that for every u ∈ X, Now let ε = min{δ 1 , δ 2 , δ 3 } and u ∈ X be such that d(x, u) < ε. We need to show that d(g(x), g(u)) ≤ cd(x, u). If d(g(x), g(u)) = 0, then we are done. So assume that d(g(x), g(u)) > 0. Then 0 < d(x, u) < ε and we obtain by (7) that The left hand side of (8) is where the above inequality is obtained from (6) and (5). From (8) and (9), we obtain k c which implies the desired result. This completes the proof.

Proof of Theorem 18
By Lemma 10, we know that f ′ (0) exists in R ∪ {+∞} and by Theorem 16, it suffices to show that f ′ (0) > k. So we can assume further that Since the limits involved in the following calculation exist, we obtain Therefore f ′ (0) > k, as desired.
As noted earlier, we will show that the class of metric-preserving functions and the class of metric-preserving functions satisfying condition (b) in Theorem 1 are, respectively, larger than the class of metric transforms and the class of metric transforms satisfying condition (b) in Theorem 1.
Proof. Let f be a metric transform. Since f (0) = 0 and f is strictly increasing, f is amenable. Since f is amenable and concave, we obtain by Lemma 6 (ii) that f is metric-preserving. Proof. This follows immediately from Proposition 20 and Theorem 18.
Since f (x) ∈ [1,2] for all x > 0, f is tightly bounded. Therefore by Lemma 8, f is metric-preserving. It is easy to see that f is not increasing (and is not concave either). So f is not a metric transform. It is easy to see that g is amenable and concave, so it is metric-preserving, by Lemma 6 (ii). In addition, if c = k = 1 2 ∈ (0, 1), then g(ct) ≥ kt for all t ∈ [0, 1]. So g satisfies condition (b) in Theorem 1. But g is not a metric transform because it is not strictly increasing. For h, we proved in [22,Example 14] that h is metric-preserving. Similar to g, the function h satisfies the condition (b) in Theorem 1. It is easy to see that h is neither strictly increasing nor concave. Therefore h is not a metric transform.
We can generate more functions similar to g given in Example 22 as follows.
Then f a,b is amenable and concave. So by Lemma 6 (ii), f a,b is metricpreserving. We also have f ′ a,b (0) = a ≥ 1. So it satisfies condition (b) in Theorem 16. However, f a,b is not a metric transform because it is not strictly increasing. Now that we have obtained two extensions of Theorem 1, we give two generalizations of Theorem 2 as follows.
Theorem 25. The following statements hold: (a) Suppose, in addition to the assumptions in Theorem 16, X is complete and rectifiably pathwise connected. Then g has a unique fixed point x 0 , and lim n→∞ g n (x) = x 0 for each x ∈ X.
(b) Suppose, in addition to the assumptions in Theorem 18, X is complete and rectifiably pathwise connected. Then g has a unique fixed point x 0 , and lim n→∞ g n (x) = x 0 for each x ∈ X.
Proof. Part (a) follows immediately from Theorem 16 and Lemma 13. Part (b) follows immediately from Theorem 18 and Lemma 13.
Finally, we remark that Kirk and Shahzad use Tan's result (Lemma 14) to extend Theorem 2 further [16, Theorem 2.3 and Theorem 2.8]. We similarly apply their argument to obtain the following.
Theorem 26. Let X be a metric space which is complete and rectifiably pathwise connected, and suppose g : X → X is a mapping for which (a) g N satisfies the assumptions in Theorem 16 for some N ∈ N, or (b) g M satisfies the assumptions in Theorem 18 for some M ∈ N.
Then g has a unique fixed point x 0 , and lim n→∞ g n (x) = x 0 for each x ∈ X.
Kirk and Shahzad's results on set-valued mappings which will be extended are as follows: Theorem 28. (Kirk and Shahzad [16,Theorem 3.4]) Let (X, d) be a metric space and T : X → CB(X). Suppose there exists a metric transform φ and k ∈ (0, 1) such that the following conditions hold: (a) For each x, y ∈ X, φ(H(T x, T y)) ≤ kd(x, y).
Theorem 29. (Kirk and Shahzad [16,Theorem 3.6]) If, in addition to the assumptions of Theorem 28, X is complete and connected, then T has a fixed point.
Our aim is to replace the metric transform φ in Theorem 28 by a metricpreserving function. We obtain the following theorem.
Theorem 30. Let (X, d) be a metric space and T : X → CB(X). Suppose there exists a metric-preserving function f and k ∈ (0, 1) such that the following conditions hold: (a) For each x, y ∈ X, f (H(T x, T y)) ≤ kd(x, y).
Corollary 31. With the same assumptions in Theorem 30 except that condition (b) is replaced by (b ′ ): there exists c ∈ (0, 1) such that for t > 0 sufficiently small, kt ≤ f (ct). Then for ε > 0 sufficiently small, T is an (ε, c)-uniform local multivalued contraction on (X, d).
Theorem 32. If, in addition to the assumptions of Theorem 30 or Corollary 31, X is complete and ε-chainable, then T has a fixed point. In particular, if X is complete and connected, then T has a fixed point.
The proof of these results are similar to those in Section 3.

Proof of Theorem 30
We define c = 1 2 k f ′ (0) + 1 as in the proof of Theorem 16. Then 0 ≤ k f ′ (0) < c < 1 and there exists δ 1 > 0 such that for every z ∈ [0, ∞) To show that T is an (ε, c)-uniform local multivalued contraction for ε > 0 sufficiently small, we let 0 < ε < δ 1 2 and let x, y ∈ X be such that d(x, y) < ε. By Lemma 9 and (10), we have for every By condition (a), we havve f (H(T x, T y)) ≤ kd(x, y) < kε. Therefore we obtain by (11) that If d(x, y) = 0 or H(T x, T y) = 0, then it is obvious that H(T x, T y) ≤ cd(x, y) and we are done. So assume that H(T x, T y) > 0 and d(x, y) > 0. Then where the first inequality is obtained by applying (12) and (10) and the last inequality is merely the condition (a). This implies H(T x, T y) ≤ cd(x, y), as desired.

Proof of Corollary 31
We can imitate the proof of Theorem 18 to obtain f ′ (0) > k. So Corollary 31 follows immediately from Theorem 30.

Proof of Theorem 32
This follows from Theorem 30, Corollary 31, and Lemma 15. The other part follows from the fact that a connected metric space is ε-chainable for every ε > 0.

Conclusion:
We replace the metric transform φ by a metric-preserving function. Therefore we obtain theorems more general than those of Kirk and

Fixed point set of metric transforms and metric-preserving functions
Recall that for a function f : X → X, we denote by Fix f the set of all fixed points of f . We begin this section with the following lemma.
Proof. Since f is amenable and concave, the function Proof. Let (a n ) be a sequence in Fix f and a n → a. If a = 0 or a = a n for some n ∈ N, then a ∈ Fix f and we are done. So assume that a > 0 and a = a n for any n ∈ N. Since a > 0 and a n → a, a n > 0 for all large n. By passing to the subsequence, we can assume that a n > 0 for every n ∈ N. It is well-known that every sequence of real numbers has a monotone subsequence (see e.g. [26, p. 62]). By passing to the subsequence again, we can assume that (a n ) is monotone. Now suppose that (a n ) is increasing. Then by Lemma 33, [a 1 , a n ] ⊆ [a 1 , a 2 ] ∪ [a 2 , a 3 ] ∪ · · · ∪ [a n−1 , a n ] ⊆ Fix f for every n ∈ N.
Since (a n ) is increasing and a n → a, if a 1 ≤ x < a, then there exists N ∈ N such that a 1 ≤ x < a N , which implies that x ∈ Fix f , by Lemma 33. This shows that [a 1 , a) ⊆ Fix f . Since f is increasing and a n < a, a n = f (a n ) ≤ f (a) for every n ∈ N. Since a n ≤ f (a) for every n ∈ N and a n → a, we have In addition, we obtain by Lemma 11 and the fact that a ≥ a 1 that that there exists 0 < y < a such that f (y) = y. Since 0 < x < a and x, a ∈ Fix f , we obtain by Lemma 33 that y / ∈ [x, a]. So y < x. By Lemma Since f (y) = y, f (y) > y. Since y < x < a, there exists t ∈ (0, 1) such that x = (1 − t)y + ta. By the concavity of f , we obtain a contradiction. This completes the proof.
Since every metric transform is metric-preserving, we immediately obtain that each set of the form Proof. This follows immediately from Theorem 36 and Proposition 20.
Example 38. Let f, g, h : [0, ∞) → [0, ∞) be given by (Recall that ⌈x⌉ is the smallest integer which is larger or equal to x) It is easy to verify that f is amenable, increasing, and subadditive. So by Lemma 7, f is metric-preserving. Since g and h are amenable and tightly bounded, we obtain by Lemma 8 that g and h are metric-preserving. It is easy to see By generating a function similar to h we obtain a more general result as follows: Proposition 39. Let A ⊆ [u, 2u] for some u > 0. Then A ∪ {0} is a fixed point set of a metric-preserving function.
Proof. We define f : and if u / ∈ A, then define f (u) = 2u. Then f is amenable and tightly bounded. Therefore, by Lemma 8, f is metric-preserving. It is easy to see that Fix f = A ∪ {0}. This completes the proof. We see that the fixed point sets of metric-preserving functions are quite difficult to be completely characterized. We leave this to the interested reader. Now we end this article by giving continuous metric-preserving functions which do not satisfy the results in Lemma 33 and Lemma 35.
Example 40. Let f, g : [0, ∞) → [0, ∞) be given by f (x) = ⌊x⌋ + x − ⌊x⌋ and g(x) = x + | sin x|. (Recall that ⌊x⌋ is the largest integer which is less than or equal to x). We will use Lemma 12 to show that f and g are metricpreserving. First, the function x → | sin x| is periodic with period π.
So the function x → | sin x| is also subadditive. From this, we easily see that g satisfies the condition in Lemma 12. So g is metric-preserving. It is not difficult to verify that f is also satisfies the assumption in Lemma 12 and we will leave the details to the reader. It is also easy to see that