Non-Unique Fixed Point Theorems in Modular Metric Spaces
Abstract
:1. Introduction
2. Preliminaries
- (i)
- if and only if ;
- (ii)
- ; and
- (iii)
- ,
- (ip)
- for all and ,
- (is)
- given , if there exists a non-negative real number λ, possibly depending on x and y, such that , then .
- (iv)
- (i)
- A sequence in is called ω-convergent to if and only if , for some . Then, x is said to be the modular limit of
- (ii)
- A sequence in is called ω-Cauchy if , for some .
- (iii)
- A subset X of is called -complete if every ω-Cauchy sequence in X is ω-convergent to
- (i)
- A map is said to be ω-contractive if there exists and such that
- (ii)
- A map is said to be strong ω-contractive if there exists and such that
3. Extension of Non-Unique Fixed Point of Ćirić on Modular Metric Spaces
4. Extension of Non-Unique Fixed Point of Pachpatte on Modular Metric Spaces
5. Extension of Non-Unique Fixed Point of Achari on Modular Metric Spaces
6. Conclusions
Funding
Acknowledgments
Conflicts of Interest
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Hussain, S. Non-Unique Fixed Point Theorems in Modular Metric Spaces. Symmetry 2019, 11, 549. https://doi.org/10.3390/sym11040549
Hussain S. Non-Unique Fixed Point Theorems in Modular Metric Spaces. Symmetry. 2019; 11(4):549. https://doi.org/10.3390/sym11040549
Chicago/Turabian StyleHussain, Sharafat. 2019. "Non-Unique Fixed Point Theorems in Modular Metric Spaces" Symmetry 11, no. 4: 549. https://doi.org/10.3390/sym11040549