Generalized Common Best Proximity Point Results in Fuzzy Metric Spaces with Application
Abstract
:1. Introduction
2. Preliminaries
- (1)
- andfor all
- (2)
- * is continuous;
- (3)
- σ for all
- (4)
- when and with
- (i)
- (ii)
- if and only if
- (iii)
- (iv)
- (v)
- (s1)
- is nondecreasing,
- (s2)
- for any
- (s3)
- for any
- (s4)
- if is such that then
3. Main Results
3.1. Fuzzy -Proximal Contraction
- (i)
- for every
- (ii)
- for any
- (iii)
- implies that
- (1)
- for any
- (i)
- Υ dominates Γ and are fuzzy -proximal,
- (ii)
- Γ and Υ are compact proximal,
- (iii)
- is non-decreasing function and for any ,
- (iv)
- Γ and Υ are continuous,
- (v)
- and
- (i)
- Υ dominates Γ and are fuzzy -proximal.
- (ii)
- Γ and Υ are compact proximal.
- (iii)
- is non-decreasing and and are convergent sequences such that , then .
- (iv)
- Γ and Υ are continuous.
- (v)
- and
3.2. Fuzzy -Interpolative Reich–Rus–Ciric-Type Proximal Contractions
- (i)
- Υ dominates Γ and are fuzzy -interpolative Riech–Rus–Ciric-type proximal;
- (ii)
- Γ and Υ are compact proximal;
- (iii)
- is a nondecreasing function, and for any ;
- (iv)
- Γ and Υ are continuous;
- (v)
- and
- (i)
- Υ dominates Γ and are fuzzy -interpolative Riech–Rus–Ciric-type proximal;
- (ii)
- Γ and Υ are compact proximal;
- (iii)
- is non-decreasing and and are convergent sequences such that , then ;
- (iv)
- Γ and Υ are continuous;
- (v)
- and
3.3. Fuzzy -Kannan Type Proximal Contraction
- (i)
- Υ dominates Γ and are fuzzy -interpolative Kannan type proximal;
- (ii)
- Γ and Υ are compact proximal;
- (iii)
- is non-decreasing function and for any ;
- (iv)
- Γ and Υ are continuous;
- (v)
- and
- (i)
- Υ dominates Γ and are fuzzy -interpolative Kannan-type proximal;
- (ii)
- Γ and Υ are compact proximal;
- (iii)
- is non-decreasing and and are convergent sequences such that , then ;
- (iv)
- Γ and Υ are continuous;
- (v)
- and
3.4. Fuzzy -Interpolative Hardy–Rogers-Type Proximal Contraction
- (i)
- Υ dominates Γ and are fuzzy -interpolative Hardy–Rogers-type proximal;
- (ii)
- Γ and Υ are compact proximal;
- (iii)
- is nondecreasing function and for any ;
- (iv)
- Γ and Υ are continuous;
- (v)
- and
- (i)
- Υ dominates Γ and are fuzzy -interpolative Hardy–Rogers-type proximal;
- (ii)
- Γ and Υ are compact proximal;
- (iii)
- is non-decreasing and and are convergent sequences such that , then ;
- (iv)
- Γ and Υ are continuous;
- (v)
- and
4. Application
- (A1)
- The Kernal satisfies Carthodory conditions, and
- (A2)
- The function is continuous and bounded on IR.
- (A3)
- There exists a positive constant C such that
- (A4)
- Let . Since guarantees the existence of an element such that, . Also, we have
- (A5)
- There exists a nonnegative and measurable function such that
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Ishtiaq, U.; Jahangeer, F.; Kattan, D.A.; Argyros, I.K. Generalized Common Best Proximity Point Results in Fuzzy Metric Spaces with Application. Symmetry 2023, 15, 1501. https://doi.org/10.3390/sym15081501
Ishtiaq U, Jahangeer F, Kattan DA, Argyros IK. Generalized Common Best Proximity Point Results in Fuzzy Metric Spaces with Application. Symmetry. 2023; 15(8):1501. https://doi.org/10.3390/sym15081501
Chicago/Turabian StyleIshtiaq, Umar, Fahad Jahangeer, Doha A. Kattan, and Ioannis K. Argyros. 2023. "Generalized Common Best Proximity Point Results in Fuzzy Metric Spaces with Application" Symmetry 15, no. 8: 1501. https://doi.org/10.3390/sym15081501
APA StyleIshtiaq, U., Jahangeer, F., Kattan, D. A., & Argyros, I. K. (2023). Generalized Common Best Proximity Point Results in Fuzzy Metric Spaces with Application. Symmetry, 15(8), 1501. https://doi.org/10.3390/sym15081501