Fixed point results in generalized suprametric spaces

: We introduce the concept of generalized suprametric spaces, which subsumes some existing abstract metric spaces. Then, we show the existence of ﬁ xed points for maps satisfying nonlinear contractions involving either extended comparison or ρ -subhomogeneous functions. This study was carried out in generalized suprametric spaces as well as partially ordered generalized suprametric spaces. Some related results in JS-metric spaces and in b -suprametric spaces are improved or extended.


Introduction
In 1906, Fréchet [11] introduced the distance function that assigns a nonnegative real number to each pair x y , ( ) of points of a given set X .This distance should always be positive and vanishes if and only if the points coincide.Moreover, the distances between the pair of points x y , ( ) and y x , ( ) are equal.Furthermore, the distance must satisfy the triangle inequality, which says that the distance between the points of the pair x y , ( ) cannot exceed the sum of distances between the points of the pairs x z , ( )and z y , ( )for every z in X .Actually, a set equipped with such a distance is known as a "standard" metric space.The well-known Banach's contraction principle is one of the most famous results established in the context of standard metric spaces.
Due to the importance of this principle in several branches of mathematics, many authors have developed it in spaces for which the triangle inequality is no longer satisfied, such as those obtained by Matthews [17], Czerwik [9], and Branciari [8].This principle has also been extended to the generalized metric spaces of Jleli and Samet [12].Such spaces, known in the literature as JS-metric spaces, recover the standard metric spaces and the b-metric spaces of Czerwik [9] as well as many other topological structures.Since then, several studies have been devoted to developing the concept of JS-metric spaces (see, for example, the following list of references) [1][2][3][4][13][14][15][16]18].
In this study, we introduce a new concept of generalized suprametric spaces, which subsumes both the JSmetric spaces [12] and the b-suprametric spaces, a notion recently introduced by the current author in [5] (see also [6]).Then, we establish new fixed point theorems for mappings satisfying nonlinear contractions via extended comparison or ρ-subhomogeneous functions.Some results in [5][6][7]12] are improved or extended.

Generalized suprametric spaces
In the sequel, we denote by the set of all nonnegative integers.Let x 0 be a point of a given set X , → f X X : be a map, and consider the following notations: , , for all , where × → ∞ X X : 0 , [ ]is a given function.
Definition 2.1.Let X be a nonempty set.A function × → ∞ X X : 0 , [ ]is called a generalized semimetric on X if the following conditions hold: ) for all ∈ x y X , .
A generalized semimetric space is a pair X , ( ), where X is a nonempty set and is a generalized semimetric on X .
2.1 Generalized suprametric spaces and some related notions Definition 2.2.Let X , ( ) be a generalized semimetric space.Then, is called generalized suprametric on X if the following additional condition holds: A generalized suprametric space is a pair X , ( ), where X is a nonempty set and is a generalized suprametric on X .
]be a function given by: ) , which means that d is a general- ized suprametric.
In the next sections, we use the following notions.Definition 2.4.Let X , ( ) be a generalized suprametric space.Let x n { } be a sequence in X and ∈ x X.We say that ) be a generalized suprametric space and let x n { } be a sequence in X .We say that x n { } is a -Cauchy sequence if Definition 2.6.Let X , ( ) be a generalized suprametric space.We say that X , ( ) is complete or X is -complete if every -Cauchy sequence, -converges in X .
) is said to be partially ordered generalized suprametric space if X , ( ) is a generalized suprametric space and ≼ is a partial order on X .We denote by ≼ E the subset of × X X defined by: ) be a generalized suprametric space and → f X X : be a given mapping.We say that f is weak continuous if ) be a partially ordered generalized suprametric space and → f X X : be a given mapping.We say that f is nondecreasing if ) be a partially ordered generalized suprametric space.We say that the pair ), then there exists a subsequence , for every k large enough.
We next show the uniqueness of the limit in generalized suprametric spaces.
then by (D 1 ), we deduce that = x y. □

Examples of generalized suprametric spaces
We first recall the concept of JS-metric spaces, which was introduced by Jleli and Samet in [12].
Definition 2.12.Let X , ( ) be a generalized semimetric space.Then, is called JS-metric on X if the following additional condition holds: (D3′) there exists A JS-metric space is a pair X , ( ), where X is a nonempty set and is a JS-metric on X .
Currently, it is known that many topological structures are indeed JS-metric spaces.
Proposition 2.13.[12] Every standard metric space, respectively, b-metric space, dislocated metric space, and modular space with the Fatou property, is a JS-metric space.
Remark 2.14.Note that every JS-metric space with constant C is a JS-metric space with constant ′ > C C. Clearly, if X , ( ) is a nontrivial JS-metric space, then the set of constants for which (D′ 3 ) holds is nonempty.The infimum of this set is greater than or equal to 1 when C X x , , ( ) is nonempty for every ∈ x X.This can be seen by taking the constant sequence = x x n for all ∈ n in (D′ 3 ) (see [10,Remark 2.14]).
As an immediate consequence, we obtain the following proposition.
We now recall the concept of b-suprametric spaces from [5].
, , the following properties hold: ), where X is a nonempty set and d is a b-suprametric.
), where X is a nonempty set and x y; and the conditions (ii) and (iii) of Definition 2.16.
) be a dislocated b-suprametric space.The conditions (D 1 ) and (D 2 ) are clearly satisfied by (i') and (ii), respectively.It suffices to prove that (D 3 ) is satisfied.Let ∈ x X and For every ∈ y X, by the property (iii) of Definition 2.16, we have for every ∈ n .Thus, we have
3 Fixed point theorems in generalized suprametric spaces In this section, we establish new fixed point theorems in generalized suprametric spaces by using two classes of control functions.The first class is denoted by Φ, and it contains the extended comparison functions ]that satisfy the following two properties: Remark 3.2.Assume that there exists which is a contradiction, and if ψ is not continuous at 0, then there exists a constant > c 0 such that ( ) , and for all ∈ ∞ t 0, ( ), and ψ 1 does not.
Fixed point results in generalized suprametric spaces  5

Nonlinear contraction via extended comparison functions
The following result shows the existence of fixed points for mappings satisfying nonlinear contractions involving extended comparison functions.Theorem 3.5.Let X , ( ) be a complete generalized suprametric space and → f X X : be a given mapping.Assume that there exists ∈ ϕ Φ such that and there exists an ∈ x X 0 such that < ∞ δ 0 .Then, the sequence x y , ( ) , then = * * x y .
Proof.First, observing that if ∈ p q , with ≥ p q, we deduce from (2) that , thus it follows from (2), (3), and (i ϕ ) that Since for ≥ k 1, the previous inequality holds for all ≥ ≥ p q k, we conclude that By induction, the monotonicity of ϕ and Remark 3.1, we deduce that ≤ δ ϕ δ k k 0 ( ) for all ≥ k 1.Thus, using (1), we obtain for all ∈ m n , .Using (ii ϕ ) and the fact that < ∞ δ 0 , we deduce that Consequently, the sequence which implies a contradiction if ).We conclude by (D 1 ) that = * * x y .□ As an immediate consequence, we obtain the following result.We next recall a result from [5].
)be a complete b-suprametric space and → f X X : be a mapping.Assume that there exists a comparison function ϕ such that Then, f has a unique fixed point * x and Clearly, we have the following proposition.

Nonlinear contraction via extended subhomogeneous functions
The contractions here are controlled by extended subhomogeneous functions.
Theorem 3.9.Let X , ( ) be a complete generalized suprametric space, → + ρ : 0, 1 ( ) be a given function, and → f X X : be a given mapping.Assume that there exists ∈ ψ Ψ ρ such that and there exists an ∈ x X 0 such that where and * y is a fixed point of f such that < ∞ * * x y , ( ) , then = * * x y .
Proof.Observe first that by ( 4), if , which implies that fx 0 is a fixed point of f , and we are done.Assume next that ≠ ψ δ 0 0 ( ) , so ∈ λ 0, 1 0 ( ), and from (4), we obtain ≤ < δ ψ δ δ 1 0 0 ( ) .Using ( 4) and the properties of ψ, we obtain x X, say.Now, we shall show that * x is a fixed point of f .Using (3), we obtain Taking → ∞ n in the previous inequality, we deduce by Remark 3.2 that 4 Fixed point theorems in partially ordered generalized suprametric spaces In this section, we present several fixed point theorems in partially ordered generalized suprametric spaces.

Nonlinear contractions via extended comparison functions
In the next result, we suppose that the function is weak continuous.
) be a partially ordered generalized suprametric space such that X , ( ) is complete.Let → f X X : be a weak continuous and nondecreasing mapping.Assume that there exists ∈ ϕ Φ that and there exists an ∈ x X ( ) , so using the monotonicity of f , we deduce by induction that Due to the transitivity of the partial order, observe that if for ∈ p q , , ≼ p q, then ≼ f x f x .p q 0 0 Hence, it follows by induction and the symmetry of that for all integer ≥ n 1 and ∈ i j , , which implies that ( ) and by induction, we deduce that ≤ δ ϕ δ n n 0 ( ) for all ∈ n .Hence, using that < ∞ δ 0 and (ii ϕ ), we obtain X. Now, we shall show that * x is a fixed point of f .The mapping f is weak continuous, so there exists a subsequence f x is -convergent to * fx .By Proposition 2.11, it follows that * x is a fixed point of f .Assume now that * y is a fixed point of f such that ≤ * * x y and < ∞ * * x y , ( ) , then by (7), we have which is a contradiction by Remark 3.
and there exists an ∈ x X where ≔ − λ δ ψ δ Using the transitivity of the partial order, then if for ∈ p q , , ≼ p q, then ≼ f x f x .p q 0 0 Hence, by induction and the symmetry of that for all integer ≥ n 1 and ∈ i j , .Thus, By induction, we easily obtain x X, say.The mapping f is weak continuous, so there exists a subsequence f  Question 4.6.The JS-metric is not a b-suprametric, since the JS-metric can reach infinity, but the b-suprametric cannot.However, it is not known whether the b-suprametric and the JS-metric are different.Also, it would be interesting to compare the JS-metric and the generalized suprametric.

Conclusion
In this article, we introduced new concepts of generalized metric space called generalized suprametric spaces and partially ordered generalized suprametric space, which recover various topological structures including JS-metric and b-suprametric spaces.We also established new fixed point theorems in these spaces.The involved maps satisfy nonlinear contractions either via extended comparison functions or via new control functions called ρ-subhomogeneous functions.The first advantage of this work is to offer a unifying frame- work to establish results in both JS-metric and b-suprametric spaces.In future studies, it will be interesting to investigate applications in generalized suprametric spaces.

0 1 and
fx 0 becomes a fixed point of f , and we are done.Assume next that ≠ Using again(9) and the properties of ψ, we obtain
[15,s worth to note that the super metric is different from the generalized suprametric.In fact, the same arguments as in[15, Example 2.2]show that a super metric is not a generalized suprametric.Besides that the supermetric does not reach infinity, the converse is generally not true.To see this, let ∈ there exists ≥ s 1 such that for all ∈ y X, there exist distinct sequences In this subsection, we present some fixed point results for contractions controlled by extended subhomogeneous functions.
Then, the sequence In the next result, we replace the continuity with a regularity condition.Proof.As in the proof of Theorem 4.3, we obtain that □Finally, we end with the following proposition.