BEST PROXIMITY POINT RESULTS VIA SIMULATION FUNCTIONS IN METRIC-LIKE SPACES

In this paper, we discuss the existence of best proximity points of certain mappings via simulation functions in the frame of complete metric-like spaces. Some consequences and examples are given of the obtained results.


Introduction
Khojasteh et al. introduced in [13] the notion of simulation function in order to unify several fixed point results obtained by various authors. These functions were later utilized by Karapinar and Khojasteh in [9] to solve some problems concerning best proximity points.
On the other hand, spaces more general than metric and fixed point and related problems in them have been lately a wide field of interest of huge number of mathematicians. Among them, metric-like spaces, introduced by Amini-Harandi in [2], took a prominent place.
In this paper, we are going to extend these investigations to best proximity points of mappings acting in complete metric-like spaces, using conditions involving simulation functions. The results will be illustrated by several examples, showing the strength of these results compared with others existing in the literature.

Preliminaries
Throughout the paper, R and R + , R + 0 will denote the set of real numbers, the set of positive real numbers and the set of nonnegative real numbers, respectively. Also, N 0 and N will denote the set of nonnegative, resp. positive integers.
Note that, according to the axiom (ζ 2 ), each simulation function ζ satisfies ζ(t, t) < 0 for all t > 0. The family of all simulation functions will be denoted by Z.

Definition 2.2 ([2]
). Let X be a nonempty set, and a mapping σ : X × X → R + 0 is such that, for all x, y, z ∈ X, Then (X, σ) is said to be a metric-like space.
As is well known, each partial metric space is an example of a metric-like space. The converse is not true. The following example illustrates this statement. Example 2.2. Take X = {1, 2, 3} and consider the metric-like σ : X × X → R + 0 given by Since σ(2, 2) = 0, σ is not a metric and since σ(2, 2) > σ(2, 1), σ is not a partial metric.
Every metric-like σ on X generates a topology τ σ whose base is the family of all open σ-balls for all x ∈ X and δ > 0.

Definition 2.3 ([2]
). Let (X, σ) be a metric-like space, let {x n } be a sequence in X and x ∈ X. (iv) a function f : X → X is continuous if for any sequence {x n } in X such that Note that the limit of a sequence in a metric-like space might not be unique.

Lemma 2.1 ([11]
). Let (X, σ) be a metric-like space. Let {x n } be a sequence in X such that x n → x where x ∈ X and σ(x, x) = 0. Then for all y ∈ X, we have lim n→∞ σ(x n , y) = σ(x, y).
Ψ will denote the family of non-decreasing functions ψ : R + 0 → R + 0 satisfying the following conditions: (i) ψ(t) < t, for any t ∈ R + ; (ii) ψ is continuous at 0. Let (X, σ) be a metric-like space, and U and V be two non-empty subsets of X. Recall the following standard notation: Consider now a non-self mapping T : U → V and the equation T u = u (u ∈ U ). As is well known, a solution of this equation, if it exists, is called a fixed point of T . If such solution does not exist, an approximate solution u * ∈ U have the least possible error when σ(u * , T u * ) = σ(U, V ). In this case, u * is called a best proximity point of the mapping T : U → V .
Finally, recall the following useful notions.

Definition 2.4 ([6]
). Let U and V be nonempty subsets of a metric-like space (X, σ), and α : U × U → R + 0 be a function. We say that the mapping T is α-proximal admissible if If σ(U, V ) = 0, then T reduces from α-proximal admissible to α-admissible. Definition 2.5 ([8, 10]). Let T : X → X be a mapping and α : X × X → R + 0 be a function. We say that the mapping T is triangular weakly-α-admissible if

Main Results
Definition 3.1. Let (X, σ) be a metric-like space, U and V be two non-empty subsets of X, ψ ∈ Ψ, α : X × X → R + 0 and ζ ∈ Z. We say that T : Definition 3.2. Let (X, σ) be a metric-like space, U and V be two non-empty subsets of X, α : X × X → R + 0 and ζ ∈ Z. We say that T : Notice that Definition 3.2 is not a special case of Definition 3.1 since the function ψ(t) = t does not belong to Ψ.
The following lemma provides a standard step in proving that the given sequence is Cauchy in a certain space. Lemma 3.1 (See, e.g., [14]). Let (X, σ) be a metric-like space and let {x n } be a sequence in X such that σ(x n+1 , x n ) is non-increasing and that lim n→∞ σ(x n+1 , x n ) = 0. If {x n } is not a Cauchy sequence, then there exist an > 0 and two sequences {m k } and {n k } of positive integers such that the following four sequences tend to when k → ∞: Now we present the main results of this article.
(1) T is triangular weakly-α-admissible; (2) U is closed with respect to the topology τ σ ; Recursively, a sequence {x n } ⊂ U can be chosen satisfying meaning that x k is the required best proximal point. Hence, we will further assume that Using relations (3.3) and (3.4), we get that σ(x n , T x n−1 ) = σ(x n+1 , T x n ) = σ(U, V ), for all n ∈ N. Furthermore, by (3.1) since T : U → V is an α-ψ-ζ-contraction. Regarding (3.4) and (ζ 2 ), the inequality (3.5) implies that Thus, {σ(x n , x n+1 )} is a non-increasing sequence bounded from below and there exists L ∈ R + 0 such that σ(x n , x n+1 ) → L as n → ∞. We shall prove that L = 0. Suppose, on the contrary, that L > 0. Taking the upper limit in (3.5) as n → ∞, regarding (ζ 3 ), property (i) of ψ ∈ Ψ and that ζ is non-decreasing with respect to the second argument, we deduce 0 ≤ lim sup which is a contradiction. We conclude that lim n→∞ σ(x n , x n+1 ) = 0.
We shall now prove that the sequence {x n } is Cauchy. Suppose that it is not. Then, there exist > 0 and subsequences {x m k } and {x m k } of {x n }, so that n k > m k > k and (3.6) σ(x m k , x n k ) ≥ and σ(x m k , x n k −1 ) < .
Since T is triangular weakly-α-admissible, from (3.3), we get that Since T is an α-ψ-ζ-contraction, the obtained relations (3.7) yield the following inequality: Letting k → ∞, using (3.6) and (ζ 3 ), and regarding properties of ψ ∈ Ψ and that ζ is non-decreasing with respect to the second argument, we obtain which is a contradiction. Thus, we conclude that the sequence {x n } is Cauchy in U . Since U is a closed subset of a complete metric-like space (X, σ), there exists z ∈ U such that (3.8) lim n→∞ σ(x n , z) = 0. Since T is continuous, we deduce that The continuity hypothesis in Theorem 3.1 can be omitted if we assume the following additional condition on U : (P ) if a sequence {u n } in U converges to u ∈ U and is such that α(u n , u n+1 ) ≥ 1 for n ≥ 1, then there is a subsequence {u n(k) } of {u n } with α(u n(k) , u) ≥ 1 for all k. (5) is replaced by (5') (P ) holds. Then T has a best proximity point.

Theorem 3.2. Let all the conditions of Theorem 3.1 hold, except that the condition
Proof. As in the proof of Theorem 3.1 we conclude that there exists a sequence {x n } in U 0 which converges to z ∈ U 0 . Using (3), we note that T z ∈ V 0 and hence Notice that from (P ), we have α(x n k , z) ≥ 1 for all k ∈ N. Since T is α-proximal admissible and we obtain that α(x n k +1 , u 1 ) ≥ 1 for all k ∈ N and Then, (ζ 2 ) implies that and so lim k→∞ σ(u 1 , x n k +1 ) → 0. Thus, u 1 = z and by (3.10) we have σ(z, T z) = σ(U, V ).

Theorem 3.3.
Let (X, σ) be a metric-like space, U and V be two non-empty subsets of X, ζ ∈ Z and α : X × X → R + 0 . Suppose that T : U → V is an α-ζ-contraction and that conditions (1)-(4) of Theorem 3.1 are satisfied, as well as (5 ) T is continuous or (P) holds. Then, T has a best proximity point.
Proof. By following the lines in the proof of Theorem 3.1, we easily construct a sequence {x n } in U which converges to some z ∈ U , moreover (3.11) lim n→∞ σ(x n , z) = 0.
Suppose first that T is continuous. Then In other words, z ∈ U is a best proximity of the mapping T . Suppose now that (P) holds. Regarding (3), we note that T z ∈ V 0 and hence Notice that from (P ), we have α(x n k , z) ≥ 1 for all k ∈ N. Since T is α-proximal admissible, and σ(u 1 , T z) = σ(x n k +1 , T x n k ) = σ(U, V ), we get that α(x n k +1 , u 1 ) ≥ 1 for all k ∈ N and Thus, u 1 = z and by (3.13) we have σ(z, T z) = σ(U, V ) and the proof is completed.
Notice that Theorem 3.3 cannot be obtained by combining Theorems 3.1 and 3.2, since the function ψ(t) = t does not belong to Ψ. Furthermore, in Theorems 3.1 and 3.2, we have an additional condition that ζ is non-decreasing in its second argument. Definition 3.3. Let (X, σ) be a metric-like space, U and V be two non-empty subsets of X, α : X × X → R + 0 and ζ ∈ Z. We say that T : U → V is a generalized α-ζcontraction if T is α-proximal admissible and (3.14) α(x, y) ≥ 1 and σ(u, Theorem 3.4. Let (X, σ) be a metric-like space, U and V be two non-empty subsets of X and α : X × X → R + 0 , ζ ∈ Z. Suppose that T : U → V is a generalized α-ζ-contraction and conditions (1)-(5) of Theorem 3.1 are satisfied. Then T has a best proximity point.
In order to prove that {x n } is a Cauchy sequence, suppose the contrary. Then, as in the proof of Theorem 3.1, there exist > 0 and subsequences {x m k } and {x m k } of {x n }, so that for n k > m k > k we have Also, in the same way, the following inequalities hold: Since T is triangular weakly-α-admissible, we derive that α(x n , x m ) ≥ 1, for all n, m ∈ N 0 with n > m.
Letting k → ∞ and keeping (3.16) and (ζ 3 ) in mind, we get which is a contradiction. Thus, we conclude that the sequence {x n } is Cauchy in U . The final step of the proof is the same as for Theorem 3.1.

Corollaries and Examples
Using Example 2.1, it is possible to get a number of consequences of our main results by choosing the simulation function ζ and α(x, y) in a proper way. We skip making such a list of corollaries since they seem clear. We just state the following one as a sample Corollary 4.1. Let (X, σ) be a metric-like space, U and V be two non-empty subsets of X and α : X × X → R + 0 , ψ ∈ Ψ. Suppose that T : U → V is a given α-proximal admissible mapping such that for all x, y, u, v ∈ U . Suppose also (a) T is triangular weakly-α-admissible; (b) U is closed with respect to the topology induced by τ σ ; (e) T is continuous or (P) holds. Then, T has a best proximity point.
In particular, if the given space (X, σ) is also endowed with a partial order , by taking α(x, y) ≥ 1 ⇔ x y, one can get standard variations of the given results in a partially ordered space.
The following illustrative examples show how our results can be used for certain mappings acting in metric-like spaces. (P G ) if a sequence {u n } in X converges to u ∈ A such that (u n , u n+1 ) ∈ E(G), then there is a subsequence {u n(k) } of {u n } with (u n(k) , u) ∈ E(G) for all k.
Definition 5.1. Let U and V be two non-empty subsets of X and α : X × X → R + 0 . We say that T : U → V is a G-proximal mapping if for all x, y, u, v ∈ U .
Definition 5.2 ([8,10]). Let U and V be two non-empty subsets of X, let T : U → V be a mapping and α : X × X → R + 0 be a function. We say that T is triangular weakly-G-admissible if α(x, y) ∈ E(G) and α(y, z) ∈ E(G) ⇒ α(x, z) ∈ E(G).
Corollary 5.1. Let U and V be two non-empty subsets of X and ψ ∈ Ψ. Suppose that T : U → V is a mapping such that σ(T x, T y) ≤ ψ(σ(x, y)), for all x, y ∈ U such that (x, y) ∈ E(G). Suppose also: (a) T is triangular weakly-G-admissible; (b) T (U 0 ) ⊂ V 0 ; (c) there exist x 0 , x 1 ∈ U such that σ(x 1 , T x 0 ) = σ(U, V ) and (x 0 , x 1 ) ∈ E(G); (d) T is continuous or (R G ) holds. Then, T has a best proximity point.
All the hypotheses of Corollary 4.1 are satisfied.
In this way, we can derive all results and consequences of the paper [15], extending them to partially ordered metric-like spaces. Similarly, we can extend the frame of several other existing results from, e.g., [3,10,12,16].