Multi Fractals of Generalized Multivalued Iterated Function Systems in b-Metric Spaces with Applications
Abstract
:1. Introduction
2. Preliminaries
- (b1)
- if and only if ;
- (b2)
- and
- (b3)
- (triangle inequality).
- (i)
- Convergent iff for each and there exists such that . i.e., as . Here, a is the limit of the sequence and can be written as
- (ii)
- Cauchy iff for each there is some for which i.e., as
- (a)
- Closed iff each sequence of elements of K has a limit, e.g. a, then . (i.e., )
- (b)
- Compact iff every sequence in K has a convergent subsequence in K.
- (G1)
- G is strictly increasing mapping, i.e., implies that
- (G2)
- , i.e., if is a sequence, then and both are equivalent.
- (G3)
- There exists for which .
- (a)
- (b)
- .
- (c)
- .
3. Main Results
- (1)
- maps elements of to elements in .
- (2)
- If for any , then the mapping is a generalized multivalued G-contraction on .
4. Applications
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Kumari, S.; Chugh, R.; Cao, J.; Huang, C. Multi Fractals of Generalized Multivalued Iterated Function Systems in b-Metric Spaces with Applications. Mathematics 2019, 7, 967. https://doi.org/10.3390/math7100967
Kumari S, Chugh R, Cao J, Huang C. Multi Fractals of Generalized Multivalued Iterated Function Systems in b-Metric Spaces with Applications. Mathematics. 2019; 7(10):967. https://doi.org/10.3390/math7100967
Chicago/Turabian StyleKumari, Sudesh, Renu Chugh, Jinde Cao, and Chuangxia Huang. 2019. "Multi Fractals of Generalized Multivalued Iterated Function Systems in b-Metric Spaces with Applications" Mathematics 7, no. 10: 967. https://doi.org/10.3390/math7100967