Relation-Theoretic Fixed Point Theorems for Generalized Weakly Contractive Mappings
Abstract
:1. Introduction and Preliminaries
- reflexive if for all ,
- symmetric if implies ,
- transitive if , implies ,
- dichotomous if for all
- trichotomous if or for all ,
- equivalence if is reflexive, symmetric and transitive.
- (1)
- The inverse or dual relation of , denoted by , is defined by
- (2)
- The reflexive closure of , denoted by , is defined as , where .
- (3)
- The symmetric closure of , denoted by , is defined to be .
2. Main Results
- (i)
- is -complete,
- (ii)
- (iii)
- is T-
- (iv)
- is -
- (v)
- either T is - or is d-,
- (vi)
- T is a generalized weakly contractive map on , i.e.,there exist two continuous functions ϕ, , such that if and only if , and if with then
- (vii.a)
- is -connected and if is a fixed point of T, then .
- (vii.b)
- is -connected and is T-transitive.
- (vii.c)
- is -connected and is T-symmetric.
3. Illustration
- (i)
- If and , then.or(since or , according to or respectively )andmax max .Then.
- (ii)
- If and , then,max max or(Since or , according to or respectively)andmax max .Then.
- (iii)
- If , then,max max maxandmax max .Then,.
- (iv)
- If , then,max max maxandmax max max .Then.
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Chakraborty, P.; Choudhury, B.S.; De la Sen, M. Relation-Theoretic Fixed Point Theorems for Generalized Weakly Contractive Mappings. Symmetry 2020, 12, 29. https://doi.org/10.3390/sym12010029
Chakraborty P, Choudhury BS, De la Sen M. Relation-Theoretic Fixed Point Theorems for Generalized Weakly Contractive Mappings. Symmetry. 2020; 12(1):29. https://doi.org/10.3390/sym12010029
Chicago/Turabian StyleChakraborty, Priyam, Binayak S. Choudhury, and Manuel De la Sen. 2020. "Relation-Theoretic Fixed Point Theorems for Generalized Weakly Contractive Mappings" Symmetry 12, no. 1: 29. https://doi.org/10.3390/sym12010029
APA StyleChakraborty, P., Choudhury, B. S., & De la Sen, M. (2020). Relation-Theoretic Fixed Point Theorems for Generalized Weakly Contractive Mappings. Symmetry, 12(1), 29. https://doi.org/10.3390/sym12010029