New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations
Abstract
:1. Introduction and Preliminaries
- 1.
- In the MS , is a fixed point of Υ if and only if .
- 2.
- The metric function is continuous in the sense that if are two sequences in ℑ with for some , as , then as . Similarly, the function Δ is continuous because if as , then as for any .
2. Multivalued Kannan Type F-contraction
3. Multivalued Reich Type F-Contraction
- 1.
- In Theorem 4, if we take , then Theorem 3 is obtained. Thus GMKFC introduced in this paper is a particular case of GMRFC when .
- 2.
- Theorem 4 is more general than Theorem 2.7 in [21] in terms of relaxation of degrees of freedom of the constants involved.
4. An Application to Integral Equations
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Debnath, P.; Srivastava, H.M. New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations. Symmetry 2020, 12, 1090. https://doi.org/10.3390/sym12071090
Debnath P, Srivastava HM. New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations. Symmetry. 2020; 12(7):1090. https://doi.org/10.3390/sym12071090
Chicago/Turabian StyleDebnath, Pradip, and Hari Mohan Srivastava. 2020. "New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations" Symmetry 12, no. 7: 1090. https://doi.org/10.3390/sym12071090
APA StyleDebnath, P., & Srivastava, H. M. (2020). New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations. Symmetry, 12(7), 1090. https://doi.org/10.3390/sym12071090