Home (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
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(α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces

  • Khairul Habib Alam , Yumnam Rohen and Anita Tomar EMAIL logo
Published/Copyright: September 26, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this study, we generalize fuzzy metric-like, non-Archimedean fuzzy metric-like, and all the variants of fuzzy metric spaces. We propose the idea of fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike, respectively. We also propose the idea of ( α , F ) -Geraghty-type generalized F -contraction mappings utilizing fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces. We investigate the presence of unique fixed points using the recently introduced contraction mappings. In order to complement our study, we consider an application to dynamic market equilibrium.

Keywords: fuzzy metric-unlike space; α-admissibility; (α, F)-Geraghty-type generalized F-contraction mapping
MSC 2010: 47H10; 54H25

1 Introduction and preliminaries

In the domain of fixed point theory, the conclusions of Banach’s [1] fixed point theorem have recently demonstrated significance across metric spaces and have been extensively studied in various newly introduced metric spaces with varied generalizations among scholars (for example, see [2–13]). Kramosil and Michalek’s [14] description of the fuzzy metric was published in 1975, followed by George and Veeramani’s [15] definition of the Hausdorff topology via fuzzy metric. Several well-known metric space results have been enhanced to fuzzy metric space. Gregory and Sapena [16] have established the fixed point theorem in the sense of both George and Veeramani, as well as Kramosil and Michalek in complete spaces. Later, for various contractions in fuzzy metric spaces, there are numerous fixed point outcomes in literature (see for instance, [17–20] and [21]). In 2017, Zhao et al. [22] extended the concept of fuzzy metric space to fuzzy metric-like space to establish some common fixed point conclusions. In this study, we generalize fuzzy metric-like, non-Archimedean fuzzy metric-like, and all the fuzzy metric variants and introduce fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike. Furthermore, we propose ( α , F ) -Geraghty-type generalized F -contraction mappings in both spaces to investigate the presence of unique fixed points. To demonstrate the usefulness of our new space, we apply our conclusions to the problem of the dynamic market equilibrium.

The notations in the current document include the following: a collection of natural numbers is designated as N . A collection of real numbers is designated as R . A collection of real numbers that are positive is designated as R + .

Now, we will recall some definitions, results, and examples.

The following is how Edelstein [23] explained Banach’s fixed point principle [1] in 1962:

Theorem 1.1

[23] In a complete metric space ( V , d ) , if a self-mapping H : V ⟶ V is such that

d ( H v 1 , H v 2 ) < d ( v 1 , v 2 ) ,

for all v 1 , v 2 ∈ V , with v 1 ≠ v 2 . Then, H has a unique fixed point in V.

A novel contraction known as the F -contraction was introduced in 2012 by Wardowski [7] to prove Theorem 1.2.

Definition 1.1

[7] If for some τ > 0 and for every v 1 , v 2 ∈ V , d ( H v 1 , H v 2 ) > 0 implies

τ + F ( d ( H v 1 , H v 2 ) ) ≤ F ( d ( v 1 , v 2 ) ) ,

then the self-mapping H : V ⟶ V in the metric space ( V , d ) is referred to as a F -contraction, if F : R + ⟶ R is strictly increasing so that:

  1. lim n → ∞ α n = 0 iff lim n → ∞ F ( α n ) = − ∞ , where { α n } n ∈ N ⊆ R + ,

  2. lim α → 0 + α δ F ( α ) = 0 , for some δ ∈ ( 0 , 1 ) .

Theorem 1.2

[7] An F-contraction H : V ⟶ V in a complete metric space ( V , d ) has a unique fixed point v * ∈ V and the sequence { H n v 0 } n ∈ N converges to v * , for each v 0 ∈ V .

The requirement (F2) in the definition of “ F -contraction” is recently replaced with the continuity of the F -type functions, by Piri and Kumam [13], to prove theorems in different types of generalized complete metric spaces.

In 2012, Samet et al. [24] defined α -admissibility as follows:

Definition 1.2

[24] Let there be two mappings: α : V × V ⟶ [ 0 , + ∞ ] and H : V ⟶ V . The self-mapping H will therefore be referred to as being α -admissible if for all v 1 , v 2 ∈ V ,

α ( H v 1 , H v 2 ) ≥ 1 , whenever α ( v 1 , v 2 ) ≥ 1 .

Now, we state the definition of continuous t-norm before we begin with the terminology of fuzzy metric space.

Definition 1.3

[25] A continuous t-norm is a commutative and associative continuous binary mapping ⋆ : [ 0 , 1 ] × [ 0 , 1 ] ⟶ [ 0 , 1 ] satisfying

  1. α 1 ⋆ 1 = α 1 , for every α 1 ∈ [ 0 , 1 ] ,

  2. α 1 ⋆ α 3 ≤ α 2 ⋆ α 4 , for all α 1 , α 2 , α 3 , α 4 ∈ [ 0 , 1 ] with α 1 ≤ α 2 , α 3 ≤ α 4 .

Definition 1.4

[14,15] An ordered triple ( V , d , ⋆ ) , where V is a non-empty set, d is a fuzzy set on V × V × ( 0 , + ∞ ) , and ⋆ is a continuous triangular norm, is a fuzzy metric space if it satisfies the subsequent hypotheses:

  1. d ( v 1 , v 2 , s 1 ) > 0 (non-negativity),

  2. d ( v 1 , v 2 , s 1 ) = 1 iff v 1 = v 2 (antisymmetric),

  3. d ( v 1 , v 2 , s 1 ) = d ( v 2 , v 1 , s 1 ) (symmetric),

  4. d ( v 1 , v 3 , s 1 + s 2 ) ≥ d ( v 1 , v 2 , s 1 ) ⋆ d ( v 2 , v 3 , s 2 ) (triangular inequality),

  5. with respect to the third variable, d ( v 1 , v 2 , ⋅ ) : ( 0 , + ∞ ) ⟶ [ 0 , 1 ] is continuous,

for all v 1 , v 2 , v 3 ∈ V and s 1 , s 2 > 0 .

Definition 1.5

[15] By substituting any of the following for the criterion (Fd4) into the aforementioned definition, we have a definition of non-Archimedean fuzzy metric space ( V , d , ⋆ ) :

  1. d ( v 1 , v 3 , max { s 1 , s 2 } ) ≥ d ( v 1 , v 2 , s 1 ) ⋆ d ( v 2 , v 3 , s 2 ) ∀ v 1 , v 2 , v 3 ∈ V ; s 1 , s 2 > 0 ,

  2. or,

  3. d ( v 1 , v 3 , s 1 ) ≥ d ( v 1 , v 2 , s 1 ) ⋆ d ( v 2 , v 3 , s 1 ) ∀ v 1 , v 2 , v 3 ∈ V ; s 1 > 0 .

Remark 1.1

The aforementioned definitions make it apparent that non-Archimedean fuzzy metric spaces are likewise fuzzy metric spaces, and hence, inequality (Fd4 ′ ) entails inequality (Fd4).

Definition 1.6

[26] If we make the requirement (Fd2) one-sided in the definition of fuzzy metric space above by

  1. d ( v 1 , v 2 , s 1 ) = 1 implies v 1 = v 2 ,

then ( V , d , ⋆ ) will become a fuzzy metric-like space.

Remark 1.2

It is noticeable from the aforementioned definitions that each fuzzy metric is fuzzy metric-like. Because of this, each non-Archimedean fuzzy metric is a fuzzy metric.

Definition 1.7

[22] An ordered triple ( V , d , ⋆ ) , where V is a non-empty set, d : V 2 × ( 0 , + ∞ ) ⟶ [ 0 , 1 ] and ⋆ is a continuous triangular norm, which satisfies

  1. d ( v 1 , v 2 , s 1 ) > 0 (non-negativity),

  2. d ( v 1 , v 2 , s 1 ) = 1 implies v 1 = v 2 (antisymmetric),

  3. d ( v 1 , v 2 , s 1 ) = d ( v 2 , v 1 , s 1 ) (symmetric),

  4. d ( v 1 , v 3 , s 1 + s 2 ) ≥ d ( v 1 , v 2 , s 1 ) ⋆ d ( v 2 , v 3 , s 2 ) (triangular inequality),

  5. with respect to the third variable, d ( v 1 , v 2 , ⋅ ) : ( 0 , + ∞ ) ⟶ [ 0 , 1 ] is continuous,

for all v 1 , v 2 , v 3 ∈ V and s 1 > 0 , is referred to as a fuzzy metric-like space.

Definition 1.8

[22] An ordered triple ( V , d , ⋆ ) , where V is a non-empty set, d : V 2 × ( 0 , + ∞ ) ⟶ [ 0 , 1 ] and ⋆ is a continuous triangular norm, which satisfies

  1. d ( v 1 , v 2 , s 1 ) > 0 (non-negativity),

  2. d ( v 1 , v 2 , s 1 ) = 1 implies v 1 = v 2 (antisymmetric),

  3. d ( v 1 , v 2 , s 1 ) = d ( v 2 , v 1 , s 1 ) (symmetric),

  4. d ( v 1 , v 3 , s 1 ) ≥ d ( v 1 , v 2 , s 1 ) ⋆ d ( v 2 , v 3 , s 1 ) (triangular inequality),

  5. with respect to the third variable, d ( v 1 , v 2 , ⋅ ) : ( 0 , + ∞ ) ⟶ [ 0 , 1 ] is continuous,

for all v 1 , v 2 , v 3 ∈ V and s 1 > 0 , is referred to as a non-Archimedean fuzzy metric-like space.

2 Main results

Before presenting our main theorem, we introduce fuzzy metric-unlike space, which generalizes all the earlier fuzzy-type generalized metric spaces. Also, we will define ( α , F ) -Geraghty-type generalized F -contraction mapping in terms of both fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces.

Definition 2.1

An ordered triple ( V , d , ⋆ ) , where V is a non-empty set, d : V 2 × ( 0 , + ∞ ) ⟶ [ 0 , 1 ] and ⋆ is a continuous triangular norm, which satisfies

  1. d ( v 1 , v 2 , s 1 ) > 0 (non-negativity),

  2. d ( v 1 , v 2 , s 1 ) = d ( v 2 , v 1 , s 1 ) (symmetric),

  3. d ( v 1 , v 3 , s 1 + s 2 ) ≥ d ( v 1 , v 2 , s 1 ) ⋆ d ( v 2 , v 3 , s 2 ) (triangular inequality),

  4. with respect to the third variable, d ( v 1 , v 2 , ⋅ ) : ( 0 , + ∞ ) ⟶ [ 0 , 1 ] is continuous,

for all v 1 , v 2 , v 3 ∈ V and s 1 > 0 , is referred to as a fuzzy metric-unlike space.

Remark 2.1

The aforementioned definitions make it apparent that fuzzy metric-unlike spaces are not any of the fuzzy-type generalized metric spaces because it does not include antisymmetric property.

The following is an example of a fuzzy metric-unlike space. However, it is none of the fuzzy-type generalized metric spaces.

Example 2.1

Let V = [ 0 , + ∞ ) . Define d : V × V × ( 0 , + ∞ ) ⟶ [ 0 , 1 ] as

d ( v 1 , v 2 , t ) = t t + min { v 1 , v 2 }

and ⋆ : [ 0 , 1 ] × [ 0 , 1 ] ⟶ [ 0 , 1 ] by δ 1 ⋆ δ 2 = min { δ 1 , δ 2 } . Then, ( V , d , ⋆ ) is a fuzzy metric-unlike space. Clearly, d does not satisfy the antisymmetric property.

Definition 2.2

An ordered triple ( V , d , ⋆ ) , where V is a non-empty set, d : V 2 × ( 0 , + ∞ ) ⟶ [ 0 , 1 ] and ⋆ is a continuous triangular norm, which satisfies

  1. d ( v 1 , v 2 , s 1 ) > 0 (non-negativity),

  2. d ( v 1 , v 2 , s 1 ) = d ( v 2 , v 1 , s 1 ) (symmetric),

  3. d ( v 1 , v 3 , s 1 ) ≥ d ( v 1 , v 2 , s 1 ) ⋆ d ( v 2 , v 3 , s 1 ) (triangular inequality),

  4. with respect to the third variable, d ( v 1 , v 2 , ⋅ ) : ( 0 , + ∞ ) ⟶ [ 0 , 1 ] is continuous,

for all v 1 , v 2 , v 3 ∈ V and s 1 > 0 , is referred to as a non-Archimedean fuzzy metric-unlike space.

Remark 2.2

The aforementioned definitions make it apparent that non-Archimedean fuzzy metric-unlike spaces are likewise fuzzy metric-unlike spaces, and hence, inequality (nFd3) entails inequality (Fd3).

Remark 2.3

Example 2.1 makes it apparent that fuzzy metric-unlike spaces may not be a non-Archimedean fuzzy metric-unlike space. Indeed, Example 2.1 does not satisfy the condition (nFd3). Also, it does not satisfy the antisymmetric property.

The following is an example of a non-Archimedean fuzzy metric-unlike space.

Example 2.2

Let V = [ 0 , + ∞ ) . Define d : V × V × ( 0 , + ∞ ) ⟶ [ 0 , 1 ] as

d ( v 1 , v 2 , t ) = t t + max { v 1 , v 2 }

and ⋆ : [ 0 , 1 ] × [ 0 , 1 ] ⟶ [ 0 , 1 ] by δ 1 ⋆ δ 2 = min { δ 1 , δ 2 } . Then, ( V , d , ⋆ ) is a non-Archimedean fuzzy metric-unlike space.

Definition 2.3

In ( V , d , ⋆ ) , a fuzzy metric-unlike space (respectively, non-Archimedean fuzzy metric-unlike space), consider any sequence { v n } n ∈ N and ε > 0 is arbitrary.

  1. Then, { v n } n ∈ N is convergent to v ∈ V if there is m ∈ N so that d ( v n , v , s 1 ) < ε for each n ≥ m , s 1 > 0 , in this case, we write

    lim n → + ∞ v n = v .

  2. Then, { v n } n ∈ N is a Cauchy if d ( v n + p , v n , s 1 ) < ε , for all s 1 > 0 and p > 0 .

  3. If each Cauchy sequence in ( V , d , ⋆ ) is convergent, then ( V , d , ⋆ ) is complete.

Definition 2.4

In a non-Archimedean fuzzy metric-unlike space ( V , d , ⋆ ) , let H : V ⟶ V , α : V × V ⟶ [ 0 , + ∞ ) and β : [ 0 , + ∞ ) ⟶ [ 0 , 1 ) . If for some τ > 0 ,

(1) α ( v 1 , v 2 ) ( τ + F ( d ( H v 1 , H v 2 , s 1 ) ) ) ≤ β ( M v 1 , v 2 , s 1 ) F ( M v 1 , v 2 , s 1 ) ,

for all v 1 , v 2 ∈ V and s 1 > 0 with d ( H v 1 , H v 2 , s 1 ) > 0 and α ( v 1 , v 2 ) ≥ 1 , where F is the same as Definition 1.1 and lim n → + ∞ β ( r n ) = 1 ⇒ lim n → + ∞ r n = 0 .

M v 1 , v 2 , s 1 = max d ( v 1 , v 2 , s 1 ) , d ( H v 1 , v 1 , s 1 ) , d ( v 2 , H v 2 , s 1 ) , d ( H v 1 , v 2 , s 1 ) + d ( v 1 , H v 2 , s 1 ) 4 , [ 1 + d ( v 1 , H v 1 , s 1 ) ] d ( v 2 , H v 2 , s 1 ) d ( v 1 , v 2 , s 1 ) + 1 .

Then, H is ( α , F ) -Geraghty-type generalized F -contraction.

Theorem 2.1

An ( α , F ) -Geraghty-type generalized F-contraction self-mapping H : V ⟶ V , in a complete non-Archimedean fuzzy metric-unlike space ( V , d , ⋆ ) has a unique fixed point u * ∈ V , and also, the sequence { H n u } n ∈ N , for any u ∈ V , must converge to u * .

Proof

For an arbitrary point u 0 ∈ V , let us establish a sequence { u n } by

u 1 = H u 0 , u 2 = H u 1 = H 2 u 0 , ⋮ u n = H n u 0 , ∀ n ∈ N .

Now, if for some n ∈ N , u n + 1 = u n , then, H u n = u n and u n is a fixed point of H .

Hence, we can assume that the sequence consists of distinct points. Now, we prove sequence { u n } is Cauchy as follows:

Suppose { u n } is not Cauchy, then we can find subsequences { p n } n ∈ N , { q n } n ∈ N of { n } n ∈ N for every ε > 0 , k ∈ N with p n > q n > k such that

d ( u p n , u q n , r 0 ) > ε and d ( u p n − 1 , u q n , r 0 ) ≤ ε , for all n > k and r 0 > 0 .

Again,

ε ≥ d ( u p n − 1 , u q n , r 0 ) ≥ d ( u p n , u p n − 1 , r 0 ) ⋆ d ( u p n , u q n , r 0 ) > d ( u p n − 1 , u p n , r 0 ) ⋆ ε .

When n is large enough, we have

lim n → + ∞ d ( u p n , u q n , r 0 ) = ε , lim n → + ∞ d ( u p n − 1 , u q n , r 0 ) = ε , lim n → + ∞ d ( u p n , u q n − 1 , r 0 ) = ε , lim n → + ∞ d ( u p n − 1 , u q n − 1 , r 0 ) = ε .

Again,

M u p n − 1 , u q n − 1 , r 0 = max { d ( u p n − 1 , u q n − 1 , r 0 ) , d ( H u p n − 1 , u p n − 1 , r 0 ) , d ( u q n − 1 , H u q n − 1 , r 0 ) , d ( H u p n − 1 , u q n − 1 , r 0 ) + d ( u p n − 1 , H u q n − 1 , r 0 ) 4 , [ 1 + d ( u p n − 1 , H u p n − 1 , r 0 ) ] d ( u q n − 1 , H u q n − 1 , r 0 ) d ( u p n − 1 , u q n − 1 , r 0 ) + 1 = max d ( u p n − 1 , u q n − 1 , r 0 ) , d ( u p n , u p n − 1 , r 0 ) , d ( u q n − 1 , u q n , r 0 ) , d ( u p n , u q n − 1 , r 0 ) + d ( u p n − 1 , u q n , r 0 ) 4 , [ 1 + d ( u p n − 1 , u p n , r 0 ) ] d ( u q n − 1 , u q n , r 0 ) d ( u p n − 1 , u q n − 1 , r 0 ) + 1 .

This implies

lim n → + ∞ M u p n − 1 , u q n − 1 , r 0 = ε .

Since d ( u p n , u q n , r 0 ) > ε > 0 , we have

τ + F ( d ( u p n , u q n , r 0 ) ) ≤ α ( u p n − 1 , u q n − 1 ) ( τ + F ( d ( u p n , u q n , r 0 ) ) ) ≤ β ( M u p n − 1 , v q n − 1 , r 0 ) F ( M u p n − 1 , v q n − 1 , r 0 ) , by ( 1 ) .

Since β ( r ) < 1 , limiting n → + ∞ , we have

τ + F ( ε ) < F ( ε ) , for all r ∈ [ 0 , + ∞ ) , a contradiction to the hypothesis  τ > 0 .

Hence, the sequence { u n } n ∈ N becomes Cauchy and ( V , d , ⋆ ) is complete space. So the sequence { u n } must be convergent in ( V , d , ⋆ ) , i.e., for some u * ∈ V , we have

lim n → + ∞ u n = u * .

Next, we demonstrate that u * is our point such that H u * = u * . Let { u q n } be any subsequence of { u n } such that lim n → ∞ u q n = H u * , so that H u * = u * and we are done. Suppose H u q n ≠ H u * for any subsequence { u q n } . Then, H u * ≠ u * , i.e., d ( H u q n , H u * , r 0 ) > 0 and d ( H u * , u * , r 0 ) > 0 for any r 0 > 0 .

Utilizing (1), we obtain

τ + F ( d ( H u q n , H u * , r 0 ) ) ≤ α ( u q n , H u * , r 0 ) ( τ + F ( d ( u q n + 1 , H u * , r 0 ) ) ) ≤ β ( M u q n , u * , r 0 ) F ( M u q n , u * , r 0 ) ,

where

M u q n , u * , r 0 = max { d ( u q n , u * , r 0 ) , d ( u q n + 1 , u q n , r 0 ) , d ( u * , H u * , r 0 ) , d ( u q n + 1 , u * , r 0 ) + d ( u q n , H u * , r 0 ) 4 , [ 1 + d ( u q n , u q n + 1 , r 0 ) ] d ( u * , H u * , r 0 ) d ( u q n , u * , r 0 ) + 1 .

As d ( H u * , u * , r 0 ) > 0 , we have lim n → + ∞ M u q n , u * , r 0 = d ( H u * , u * , r 0 ) and

τ + F ( d ( u * , H u * , r 0 ) ) ≤ β ( d ( H u * , u * , r 0 ) ) F ( d ( H u * , u * , r 0 ) ) < F ( d ( H u * , u * , r 0 ) ) , a contradiction to the hypothesis τ > 0 .

Hence, H u * = u * .

We shall now demonstrate the uniqueness of point u * . Instead, imagine that we have two distinct fixed points u * and u * * . In this case, sequence u n converges to u * as well as u * * . Also, H u * = u * * and H u * * = u * * .

Since u * ≠ u * * , we have d ( H u * , H u * * , r 0 ) = d ( u * , u * * , r 0 ) > 0 for r 0 > 0 , and this implies

τ + F ( d ( H u * , H u * * , r 0 ) ) ≤ α ( u * , u * * ) ( τ + F ( d ( H u * , H u * * , r 0 ) ) ) ≤ β ( M u * , u * * , r 0 ) F ( M u * , u * * , r 0 ) , using (1) ,

where

M u * , u * * , r 0 = max d ( u * , u * * , r 0 ) , d ( H u * , u * , r 0 ) , d ( u * * , H u * * , r 0 ) , d ( H u * , u * * , r 0 ) + d ( u * , H u * * , r 0 ) 4 , [ 1 + d ( u * , H u * , r 0 ) ] d ( u * * , H u * * , r 0 ) d ( u * , u * * , r 0 ) + 1 = max d ( u * , u * * , r 0 ) , d ( u * , u * * , r 0 ) 2 = d ( u * , u * * , r 0 ) .

Thus,

τ + F ( d ( u * , u * * , r 0 ) ) ≤ β ( d ( u * , u * * , r 0 ) ) F ( d ( u * , u * * , r 0 ) ) < F ( d ( u * , u * * , r 0 ) ) , since  β ( d ( u * , u * * , r 0 ) ) < 1 ,

which implies τ ≤ 0 , a contradiction to the hypothesis τ > 0 . Hence, u * = u * * implies that H has a unique fixed point u * .□

Example 2.3

Let V = 0 , 1 3 , 1 2 . Then, V , with the usual metric in R , is a complete metric space. Define d : V 2 × ( 0 , + ∞ ) ⟶ [ 0 , 1 ] by

d ( v 1 , v 2 , r ) = e − ∣ v 1 − v 2 ∣ r , ∀ v 1 , v 2 ∈ V , r > 0 ,

and ⋆ : [ 0 , 1 ] × [ 0 , 1 ] ⟶ [ 0 , 1 ] by

δ 1 ⋆ δ 2 = min { δ 1 , δ 2 } , ∀ δ 1 , δ 2 ∈ [ 0 , 1 ] .

Then, V with respect to defined d and ⋆ forms a complete non-Archimedean fuzzy metric-unlike space.

Suppose a self-mapping H on V is defined by

H ( 0 ) = H 1 3 = 0 , H 1 2 = 1 3 .

Now, considering the mappings α : V × V ⟶ [ 0 , + ∞ ) so that

α ( v 1 , v 2 ) = 1 , if v 1 , v 2 ∈ V , 0 , otherwise ,

F : R + ⟶ R so that F ( u ) = − 1 u ,

and

β : ( 0 , + ∞ ) ⟶ [ 0 , 1 ) , given by β ( r ) = 1 1 + r ,

we found H as an ( α , F ) -Geraghty-type generalized F -contraction in the complete non-Archimedean fuzzy metric-unlike space ( V , d , ⋆ ) for 0 < τ ≤ 1 2 . And from the definition of H , we see that the self-mapping H possesses 0 as the only fixed point.

Remark 2.4

Since fuzzy metric-like or non-Archimedean fuzzy metric-like spaces are neither fuzzy metric-unlike nor non-Archimedean fuzzy metric-unlike spaces, our results cannot hold in standard variants of fuzzy spaces. This fact is also supported by Example 2.3 since d does not satisfy the antisymmetric property.

Remark 2.5

In a fuzzy metric-unlike space, we can show a different version of Theorem 2.1, using the same notion of ( α , F ) -Geraghty-type generalized F -contraction.

Theorem 2.2

An ( α , F ) -Geraghty-type generalized F-contraction H : V ⟶ V , in a complete fuzzy metric-unlike space ( V , d , ⋆ ) has a unique fixed point u * ∈ V ; also, the sequence { H n u } n ∈ N , for any u ∈ V , converges to u * .

Proof

Since non-Archimedean fuzzy metric-unlike spaces are fuzzy metric-unlike spaces, proceeding analogous to the proof of Theorem 2.1, we can demonstrate this theorem.□

Remark 2.6

In a non-Archimedean fuzzy metric space (respectively, non-Archimedean fuzzy metric-like space), we can show a different version of Theorem 2.1, using the same notion of ( α , F ) -Geraghty-type generalized F -contraction.

Theorem 2.3

An ( α , F ) -Geraghty-type generalized F-contraction H : V ⟶ V , in a complete non-Archimedean fuzzy metric space (respectively, non-Archimedean fuzzy metric-like space) ( V , d , ⋆ ) has a unique fixed point u * ∈ V ; also, the sequence { H n u } n ∈ N , for any u ∈ V , converges to u * .

Remark 2.7

In a fuzzy metric space (respectively, fuzzy metric-like space), we can show a different version of Theorem 2.2, using the same notion of ( α , F ) -Geraghty-type generalized F -contraction.

Theorem 2.4

An ( α , F ) -Geraghty-type generalized F-contraction H : V ⟶ V , in a complete fuzzy metric space (respectively, fuzzy metric-like space) ( V , d , ⋆ ) has a unique fixed point u * ∈ V ; also, the sequence { H n u } n ∈ N , for any u ∈ V , converges to u * .

Corollary 2.1

In a complete non-Archimedean fuzzy metric-unlike space ( V , d , ⋆ ) , let H : V ⟶ V , α : V × V ⟶ [ 0 , + ∞ ) , and β : [ 0 , + ∞ ) ⟶ [ 0 , 1 ] . If for some τ > 0

(2) α ( v 1 , v 2 ) ( τ + F ( d ( H v 1 , H v 2 , s 1 ) ) ) ≤ β ( M v 1 , v 2 , s 1 ′ ) F ( M v 1 , v 2 , s 1 ′ ) ,

for all v 1 , v 2 ∈ V , and s 1 > 0 with d ( H v 1 , H v 2 , s 1 ) > 0 and α ( v 1 , v 2 ) ≥ 1 , where F is the same as Definition 1.1, lim n → + ∞ β ( r n ) = 1 ⇒ lim n → + ∞ r n = 0 and

M v 1 , v 2 , r ′ = max { d ( v 1 , v 2 , r ) , d ( H v 1 , v 1 , r ) , d ( v 2 , H v 2 , r ) } .

Then, there exists unique u * ∈ V so that H u * = u * . Moreover, the sequence { H n u } converges to u * , for every u ∈ V .

Proof

Continuing analogously to the proof of Theorem 2.1 here, we have

M u p n − 1 , u q n − 1 , r 0 ′ = max { d ( u p n − 1 , u q n − 1 , r 0 ) , d ( H u p n − 1 , u p n − 1 , r 0 ) , d ( u q n − 1 , H u q n − 1 , r 0 ) } = max { d ( u p n − 1 , u q n − 1 , r 0 ) , d ( u p n , u p n − 1 , r 0 ) , d ( u q n − 1 , u q n , r 0 ) } .

This implies

lim n → + ∞ M u p n − 1 , u q n − 1 , r 0 ′ = ε .

Again,

M u q n , u * , r 0 ′ = max { d ( u q n , u * , r 0 ) , d ( u q n + 1 , u q n , r 0 ) , d ( u * , H u * , r 0 ) }

implies lim n → + ∞ M u q n , u * , r 0 ′ = d ( H u * , u * , r 0 ) and

M u * , u * * , r 0 ′ = max { d ( u * , u * * , r 0 ) , d ( H u * , u * , r 0 ) , d ( u * * , H u * * , r 0 ) } = max { d ( u * , u * * , r 0 ) } = d ( u * , u * * , r 0 ) .

This eventually proves that u * is the only fixed point of H .□

Corollary 2.2

In a complete fuzzy metric-unlike space ( V , d , ⋆ ) , let H : V ⟶ V , α : V × V ⟶ [ 0 , + ∞ ) , and β : [ 0 , + ∞ ) ⟶ [ 0 , 1 ] . If for some τ > 0

(3) α ( v 1 , v 2 ) ( τ + F ( d ( H v 1 , H v 2 , s 1 ) ) ) ≤ β ( M v 1 , v 2 , s 1 ′ ) F ( M v 1 , v 2 , s 1 ′ ) ,

for all v 1 , v 2 ∈ V , and s 1 > 0 with d ( H v 1 , H v 2 , s 1 ) > 0 and α ( v 1 , v 2 ) ≥ 1 , where F is the same as Definition 1.1, lim n → + ∞ β ( r n ) = 1 ⇒ lim n → + ∞ r n = 0 and

M v 1 , v 2 , r ′ = max { d ( v 1 , v 2 , r ) , d ( H v 1 , v 1 , r ) , d ( v 2 , H v 2 , r ) } .

Then, there exists unique u * ∈ V so that H u * = u * . Moreover, the sequence { H n u } converges to u * , for every u ∈ V .

3 Application

In order to complement our study, this section provides an application to the dynamic market equilibrium [27]. Current pricing and price patterns (i.e., whether prices are increasing or decreasing at an increasing or decreasing rate) have an impact on supply and demand in many marketplaces. Then, for the given supply and demand functions:

S = ζ 1 + ζ 2 u ( t ) + ζ 3 u ′ ( t ) + ζ 4 u ″ ( t )

and

D = η 1 + η 2 u ( t ) + η 3 u ′ ( t ) + η 4 u ″ ( t ) ,

the economist is curious about u ( t ) , u ′ ( t ) and u ″ ( t ) , where u ( t ) is current price at time t and ζ 1 , ζ 2 , ζ 3 , ζ 4 , η 1 , η 2 , η 3 , η 4 are the constants. In equilibrium, S = D , putting ζ 1 − η 1 = − d , ζ 2 − η 2 = c , ζ 3 − η 3 = b , ζ 4 − η 4 = a we have the initial value problem

(4) u ″ ( t ) + b a u ′ ( t ) + c a u ( t ) = − d a , u ( 0 ) = 0 , u ′ ( 0 ) = 0 ,

and the corresponding equivalent integral equation

u ( t ) = ∫ 0 1 G ( t , s ) f ( t , s , u ( t ) ) d s ,

where Green’s function

G ( t , s ) = s e c 2 b ( t − s ) , if 0 ≤ s ≤ t ≤ 1 t e c 2 b ( s − t ) , if 0 ≤ t ≤ s ≤ 1 .

Let us consider the complete non-Archimedean fuzzy metric-unlike space ( V , d , ⋆ ) , where V = C [ 0 , 1 ] , the set of all real-valued continuous functions defined on [ 0 , 1 ] ,

d ( u 1 , u 2 , r ) = max t ∈ [ 0 , 1 ] inf { u 1 , u 2 } + r sup { u 1 , u 2 } + r , ∀ r > 0 , u 1 , u 2 ∈ C [ 0 , 1 ] ,

and ⋆ is continuous t-norm given by u 1 ⋆ u 2 = u 1 u 2 .

Consider H : C [ 0 , 1 ] ⟶ C [ 0 , 1 ] as

H u ( t ) = ∫ 0 1 G ( t , s ) f ( t , s , u ( t ) ) d s .

Theorem 3.1

The initial value problem defined in (4) has a unique solution if for τ > 0 , the following conditions are fulfilled:

  1. G : [ 0 , 1 ] × [ 0 , 1 ] ⟶ R + is continuous and ∫ 0 1 G ( t , s ) d s ≤ 1 .

  2. f : [ 0 , 1 ] × [ 0 , 1 ] × C [ 0 , 1 ] ⟶ R + is continuous.

  3. inf { f ( t , s , u 1 ( t ) ) , f ( t , s , u 2 ( t ) ) } + r sup { f ( t , s , u 1 ( t ) ) , f ( t , s , u 2 ( t ) ) } + r ≤ e − τ inf { u 1 ( t ) , u 2 ( t ) } + r sup { u 1 ( t ) , u 2 ( t ) } + r 1 1 + M u 1 , u 2 , r ′ ,

where M u 1 , u 2 , r ′ is defined in Corollary 2.1.

Proof

We have

inf { H u 1 , H u 2 } + r sup { H u 1 , H u 2 } + r = inf ∫ 0 1 G ( t , s ) f ( t , s , u 1 ( t ) ) d s , ∫ 0 1 G ( t , s ) f ( t , s , u 2 ( t ) ) d s + r sup ∫ 0 1 G ( t , s ) f ( t , s , u 1 ( t ) ) d s , ∫ 0 1 G ( t , s ) f ( t , s , u 2 ( t ) ) d s + r = ∫ 0 1 inf { G ( t , s ) f ( t , s , u 1 ( t ) ) , G ( t , s ) f ( t , s , u 2 ( t ) ) } d s + r ∫ 0 1 sup { G ( t , s ) f ( t , s , u 1 ( t ) ) , G ( t , s ) f ( t , s , u 2 ( t ) ) } d s + r ≤ inf { f ( t , s , u 1 ( t ) ) , f ( t , s , u 2 ( t ) ) } ∫ 0 1 G ( t , s ) d s + r sup { f ( t , s , u 1 ( t ) ) , f ( t , s , u 2 ( t ) ) } ∫ 0 1 G ( t , s ) d s + r ≤ inf { f ( t , s , u 1 ( t ) ) , f ( t , s , u 2 ( t ) ) } + r sup { f ( t , s , u 1 ( t ) ) , f ( t , s , u 2 ( t ) ) } + r ≤ e − τ inf { u 1 ( t ) , u 2 ( t ) } + r sup { u 1 ( t ) , u 2 ( t ) } + r 1 1 + M u 1 , u 2 , r ′

(5) ⇒ max t ∈ [ 0 , 1 ] inf { H u 1 , H u 2 } + r sup { H u 1 , H u 2 } + r ≤ e − τ max t ∈ [ 0 , 1 ] inf { u 1 ( t ) , u 2 ( t ) } + r sup { u 1 ( t ) , u 2 ( t ) } + r 1 1 + M u 1 , u 2 , r ′ .

Now, if F ( r ) = ln ( r ) , β ( r ) = 1 1 + r , and α : V × V ⟶ R defined by

α ( u 1 , u 2 ) = 1 , if u 1 ∈ V , 0 , otherwise ,

then from inequality (5), we have

α ( u 1 , u 2 ) ( τ + F ( d ( H u 1 , H u 2 , r ) ) ) = τ + F ( d ( H u 1 , H u 2 , r ) ) = ln e τ max t ∈ [ 0 , 1 ] inf { H u 1 , H u 2 } + r sup { H u 1 , H u 2 } + r ≤ 1 1 + M u 1 , u 2 , r ′ ln max t ∈ [ 0 , 1 ] inf { u 1 ( t ) , u 2 ( t ) } + r sup { u 1 ( t ) , u 2 ( t ) } + r = β ( M u 1 , u 2 , r ′ ) F ( d ( u 1 , u 2 , r ) ) ≤ β ( M u 1 , u 2 , r ′ ) F ( M u 1 , u 2 , r ′ ) .

Consequently, we have our conclusion from Corollary 2.1.□

Remark 3.1

We can utilize the non-Archimedean fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces in a variety of related domains, such as nonlinear analysis, differential equations, and fractional calculus models, in the future. Our outcomes can be utilized for further expansions.

Acknowledgments

The authors express their gratitude to the reviewers for their constructive feedback on their work. Khairul Habib Alam thanks UGC, New Delhi, for financial support as a Senior Research Fellowship through the Student ID 191620045843.

  1. Funding information: Not applicable.

  2. Author contributions: Conceptualization, K.H.A.; formal analysis, K.H.A., A.T., and Y.R.; supervision, Y.R.; investigation, A.T., and Y.R.; writing – original draft preparation, K.H.A.; writing – review and editing, K.H.A., A.T., and Y.R. All authors have read and agreed to the published version of the manuscript.

  3. Conflict of interest: The authors declare no conflict of interest.

  4. Ethical approval: The research performed does not involve human or animal subjects.

  5. Data availability statement: Data sharing does not apply to this article since no data sets were generated or analyzed during the study.

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Received: 2023-06-02
Accepted: 2024-06-14
Published Online: 2024-09-26

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2024-0046
Keywords for this article
fuzzy metric-unlike space; α-admissibility; (α, F)-Geraghty-type generalized F-contraction mapping
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  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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