Best Proximity Point Results for Geraghty Type Ƶ-Proximal Contractions with an Application
Abstract
:1. Introduction
- ;
- for all ;
- If are sequences in so that then
2. Preliminaries
- (1)
- is -proximal admissible;
- (2)
- for all
3. Main Results
- (i)
- P is closed and ;
- (ii)
- is triangular -proximal admissible;
- (iii)
- There are so that and ;
- (iv)
- is a continuous Geraghty type -proximal contraction.
- (C)
- If a sequence in P is convergent to so that , then for all
- (i)
- P is closed and ;
- (ii)
- is triangular -proximal admissible;
- (iii)
- there are so that and ;
- (iv)
- the condition holds and is a Geraghty type -proximal contraction.
4. Some Consequences
- (i)
- P is closed and ;
- (ii)
- is ⪯-proximal increasing;
- (iii)
- There are so that and ;
- (iv)
- is continuous or, for every sequence in P is convergent to so that , we have for all
- (v)
- There exist and , such that for all
- (1)
- for all ,
- (2)
- for all
- (i)
- P is closed and ;
- (ii)
- is triangular G-proximal;
- (iii)
- There are so that and ;
- (iv)
- is continuous or, for every sequence in P is convergent to so that , we have for all
- (v)
- There exist and such that for all
5. A Variational Inequality Problem
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Işık, H.; Aydi, H.; Mlaiki, N.; Radenović, S. Best Proximity Point Results for Geraghty Type Ƶ-Proximal Contractions with an Application. Axioms 2019, 8, 81. https://doi.org/10.3390/axioms8030081
Işık H, Aydi H, Mlaiki N, Radenović S. Best Proximity Point Results for Geraghty Type Ƶ-Proximal Contractions with an Application. Axioms. 2019; 8(3):81. https://doi.org/10.3390/axioms8030081
Chicago/Turabian StyleIşık, Hüseyin, Hassen Aydi, Nabil Mlaiki, and Stojan Radenović. 2019. "Best Proximity Point Results for Geraghty Type Ƶ-Proximal Contractions with an Application" Axioms 8, no. 3: 81. https://doi.org/10.3390/axioms8030081