Near-coincidence point results in metric interval space and hyperspace via simulation functions

Recently, Wu (Mathematics 6(11):219, 2018; Mathematics 6(6):90, 2018) introduced the concept of a near-fixed point and established some results on near fixed points in a metric interval space and hyperspace. Motivated by these papers, we studied the near-coincidence point theorem in these spaces via a simulation function. To show the authenticity of the established results and definitions, we also provide some examples.


Introduction and preliminaries
In mathematics, fixed point theory plays the role of bridge between pure and applied mathematics. Therefore this field has great importance among the branches of pure mathematics and especially in nonlinear analysis. It has many applications to the existence of a solution of a nonlinear system; see, for example, the recent works [3][4][5][6][7][8][9] and many more. Metric fixed point theory is the celebrated area in fixed point theory based on the Banach contraction principle (BCP). This principle is given by Banach [10]. After him, this principle was generalized in different forms. This principle is studied in different structures such as dislocated quasimetric spaces [11], cone metric spaces [12], generalized metric spaces [13] and so on. On the other side, the contraction condition is modified in different forms such as the Kannan contraction condition, Chatterjee contraction condition. For details, see [14][15][16]. Sarwar [17] studied fixed point theorems under rational-type contractions in the setting of complex-valued metric spaces. These results generalized some important results in the present literature. De la Sen [18] used a new approach of (sq)-graphic contraction in b-metric-like spaces. These results generalize and improve several approaches in the existing literature by using this new approach for the proof that a Picard sequence is a Cauchy one.
Recently, Khojasteh [19] introduced the concept of a simulation function ζ : [0, +∞) × [0, +∞) − → R and the concept of Z-contraction, which modifies the contraction condition in the Banach contraction principle. Using a simulation function, he proved some fixed-point results. Then Hiero et al. [20,21] extended the stated concept of a simulation function given by Khojasteh and investigated some coincidence point results. Argubi [22] used this concept to study the results on coincidence and common fixed point in partially ordered metric space. Alharbi [23] combined the concept of a simulation function with admissible function to generalize some existing results in the related literature. Chanda [24] surveyed many of the recent works related with simulation functions and Z-contractions in the existing literature after the publication of Khojasteh et al. Recently, Alsubaie, Alqahtani, and Karapinar [25] proved some interesting results on common fixed points in metric spaces via a simulation function. In 2019, Alqathani and Karapinar [26] introduced the concept of a bilateral contraction, which combines the ideas of Ćirić-type contraction and Caristi-type contraction with the help of simulation function in complete metric spaces. Alghamdi [27] studied common fixed point results in the setting of b-metric space via extended Z-contraction with respect to a ψ-simulation function. His work evaluated and merged as-scattered-as-possible results in fixed point theory from general framework. Karapinar and Agarwal [28] introduced the concept of an interpolative Rus-Reich-Ćirićtype Z-contraction in the setting of a complete metric space.
Recently, Wu [1,2] raised the idea about near-fixed points in metric interval spaces and hyperspaces. He studied some results on near-fixed points in metric interval spaces and hyperspaces. Due to the nonexistence of the inverse of each element, the metric interval spaces and hyperspaces are not conventional metric spaces and normed spaces, respectively. For more detail of interval spaces, we refer the reader to [29,30] Inspired by the works [19, 25-28, 31, 32], we study near-coincidence points in metric interval spaces and hyperspaces via simulation functions. We also provide some examples.

Preliminaries
We state some basic definitions and fundamental results in this framework.

Interval space
Let I be the collection of all closed bounded intervals [l, u], where l, u ∈ R and l ≤ u; we consider l ∈ R as the element [l, l] ∈ I.
The addition and the scaler multiplication are defined as

Null set
The null set is defined as In the interval spaces, the following observations are remarkable: • The distributive law in I is not true in general.
• The distributivity holds for scaler addition if both scalers are positive or both are negative, that is,

Metric interval space
A metric interval space is a pair (I, d), where I is the collection of all closed bounded intervals in R with the null set Ω, and d is a mapping from I × I to nonnegative real numbers that satisfies the following axioms: • If only conditions (ii) and (iii) hold, then the space (I, d) is called a pseudometric interval space. If the following condition (iv) is satisfied for d, then d is said to satisfy the null equalities: (iv) for any ω 1 , ω 2 ∈ Ω and [l, u], [x, y] ∈ I, the following equalities hold: For more detail, see [1]. Then (I, d) is a metric interval space with the null equalities satisfied by d (see [1]).     ii.

Hyperspace
Let S(V ) be the collection of all nonempty convex subsets of V , where V is a topological vector space. The two binary operations of addition and scalar multiplication are defined as follows: The subtraction is defined by Here in S(V ) the inverse does not exist for a nonempty and nonsingleton set, that is, if Φ = U = {0 V }, then U U is not the zero element of S(V ), and so it cannot be a conventional vector space (for more detail, see [2]).

Null set
The null set is defined as Ω = {U U; U ∈ S(V )}. This set can be regarded as the zero element of S(V ) (see [2]).
4 If U is a convex subset of V and λ 1 and λ 2 have the same sign, then The sets U and U * are said to be almost identical. Clearly, U = U * implies U Ω = U * If U, U * , and W are not a singleton set, then U U * = W implies U Ω = U * ⊕ W .

Normed hyperspace
Let V be a vector space, and let S(V ) be the collection of all nonempty convex subsets of V . Then S(V ) is called a normed hyperspace if there exists a length function · : S(V ) → R satisfying the following axioms: then we say that · satisfies the null condition (for more detail, see [2]).
Let S(V ) be the collection of all nonempty convex subsets of V . Then the norm defined on S(V ) is given as

Results and discussion
is called a nearcoincidence point for G and g.
Example 3.2 Let us consider two mappings G and g from I to itself defined by By taking the functions G and g defined before and any sequence, we can easily verify that these mappings are compatible.     In such a case, d g[l n , u n ], g[l n+1 , u n+1 ] > 0 for all n ≥ 0.

Definition 3.14 A mapping
We will prove the result in three steps.
Using condition (2) of a simulation function and condition of a (Z d , g)-contraction, we have which implies that Note that the sequence d(g[l n , u n ], g[l n+1 , u n+1 ]) is a nonnegative decreasing sequence in R, so it converges to a point l, that is, We will show that l = 0. Assume that l = 0, so l > 0.
Using S 3 by taking the sequences Step 2 We will show that {g[l n , u n ]} is a Cauchy sequence in (I, d). On the contrary, suppose that g[l n , u n ] is not Cauchy. So there exists o > 0 such that for all N ∈ N, there exist positive integers m, n such that We can construct two subsequences by giving successive values g[l n k , u n k ] and g[l m k , u m k ] to N such that Hence m(k) = n(k) + 1 is not possible by taking into account (3.1) and (3.2), and therefore we conclude that m(k) ≥ n(k) + 2 for any k ∈ N.
It follows that n k+1 < m k < m k+1 for all k ∈ N. From (3.2) and (3.3) we have As G is a (Z d , g)-contraction associated with S, we get Thus To illustrate the theorem, we consider the following example. in I and the simulation function S(t, s) = λst with λ ≥ 2. As we have proved before, the mappings G and g are compatible, and also G is a (Z d , g)-contraction in (I, d). We will just prove that the sequence {[- is a Picard sequence for G and g. Hence by the theorem the limit of g[x n , y n ] = [-1 n -1, 1 n + 1], which is [-1, 1], is a near-coincidence point for G and g.
If G and g are commuting, then we have So we can state the following corollary.

Corollary 3.18 Let G be a Z-contraction in the complete metric interval space (I, d) and
suppose there exists a Picard sequence for the mappings G and g at the point [l, u] ∈ I, that is, Also, assume that the mappings G and g are continuous and commuting. Then there exists a near-coincidence point for G and g.

Corollary 3.19 Let G and g be the two self-mappings on the complete metric interval space
. Hence the class of near-coincidence point is unique. ∈ [0, 1), then G and g has a near-coincidence point.

Corollary 3.20 Let (I, d) be a complete metric interval space and let G and g be self mappings on (I, d) such that the CLR(G, g) property holds in I and d(G[l, u], G[w, x])
Proof We will show that G is a Z d -contraction by taking the simulation function S ∈ Z defined by S(l, u) = λul for all l, u ∈ [0, +∞) and λ ∈ [0, 1).
Since by the given condition we have where Φ is a lower semicontinuous function on [0, +∞), and Φ -1 (0) = 0. Then G and g has a near-coincidence point in I.
Proof By taking the simulation function S ∈ Z defined by S(l, u) = u -Φ(u)l for all l, u ∈ [0, +∞).
We can show that G is a Z-contraction, and hence by Theorem 3.16 G and g have a nearcoincidence point.

Lemma 3.22
If F is a (Z · , g)-contraction in the hyperspace S(V ) and U and U * are two near-coincidence points of F and g, then Further, if F or g is injective and if they have a near-coincidence point, then it is unique in the sense of equivalence class.
Proof As U and U * are near-coincidence points for F and g, we have We have to show that g(U) Ω = g(U * ). On the contrary, suppose g(U) Further, if F or g is injective and if they have a near-coincidence point, then it is unique in the sense of equivalence class. Let F be injective, and let U and U * be two different near-coincidence points for F and g. Then from which we have This implies that F(U) Ω = F(U * ),which in turn implies U Ω = U * because F is injective. So the near-coincidence point is unique in the sense of equivalence class. Theorem 3.23 Let (S(V ), · ) be a Banach hyperspace such that · satisfies the null equality, and let F be a (Z · , g)-contraction. Also, assume that the functions F and g are compatible and continuous and there exists a Picard sequence {A n } for F and g. Then F and g have at least one near-coincidence point. Also, we cannot take m k = n k and m k = n k+1 because (3.4) becomes zero for m k = n k , and for m k = n k+1 , we have so m k ≥ n k+2 for all k ∈ N . From Eqs. (3.5) and (3.7) we have As lim n→+∞ g[A n k ] g[A m k ] = 0, by the inequalities we have Similarly, Now as F is a (Z · , g)-contraction, we have which implies that Now from the last inequality consider the two sequences As t k and s k have the same limit and t k < s n , by applying ξ 3 we have which is a contradiction because ξ (t k , s k ) > 0. Hence {g[A n ]} is a Cauchy sequence.
Step 3. In this step, we will show that the limit point of {g[A n ]} is a near-coincidence point for F and g. As the space (S(V ), · ) is complete, the sequence converges to some limit, say A. As F and g are continuous and compatible and A is the limit of . So we have proved that A is a nearcoincidence point of F and g.
As commuting of F and g implies compatibility, we have the following corollary.

Corollary 3.24
Let (S(V ), · ) be a Banach hyperspace such that · satisfies the null equality, and let F be a (Z · , g)-contraction. Also, assume that the functions F and g are commuting and continuous and there exists a Picard sequence {A n } for F and g. Then F and g have at least one near-coincidence point. Proof We will show that F is a Z-contraction by taking the simulation function S ∈ Z defined by S(u, v) = λvu for all l, u ∈ [0, +∞) and λ ∈ [0, 1). According to the given condition, we have The last inequality shows that F is a Z d -contraction, so by Theorem 3.23 it has a nearcoincidence point in S(V ).

Corollary 3.26
Let (S(V ), · ) be a Banach hyperspace such that · satisfies the null equality, and let F and g be self-mappings on S(V ) that satisfy the following condition: where Φ is a lower semicontinuous function on [0, +∞), and Φ -1 (0) = 0. Then G and g have a near-coincidence point in S(V ).
Proof By taking the simulation function S ∈ Z defined by S(l, u) = u -Φ(u)l for all l, u ∈ [0, +∞).
We can show that F is a Z-contraction, and hence by Theorem 3.16 F and g have a nearcoincidence point.

Conclusion
Nowadays, the researchers in the subject area are working to produce more effective and generalized fixed point results. Recently, Wu [1,2] introduced the concept of a near-fixed point and established some results on near-fixed points in metric interval spaces and hyperspaces. Motivated by these papers, we studied the near-coincidence point theorem in these spaces via a simulation function. To illustrate the established results and definitions, we included some examples.