On Generalized Hardy–Rogers Type α-Admissible Mappings in Cone b-Metric Spaces over Banach Algebras
Abstract
:1. Introduction
2. Preliminaries
- 1.
- ,
- 2.
- ,
- 3.
- ,
- 4.
- for all
- (CbM1) for all , if and only if ;
- (CbM2) for all ;
- (CbM3) there exists , such that for all .
- Then we call d a cone b-metric on W. The space is called a cone b-metric space over a BA with coefficient s (in short CnMs-BA). If , we call is CMS over BA (in short CMS-BA).
- (i)
- We call a c-sequence, if for each , there exists is such that for all
- (ii)
- We call a θ-sequence if as .
- (i)
- b-converges to, ifis a c-sequence.
- (ii)
- is b-Cauchy if for eachwiththere issuch thatfor all.
- (iii)
- is called a b-complete CnMs, if wheneveris b-Cauchy in W, thenis b-convergent.
- (a)
- If and , then
- (b)
- If are such that and , then ,
- (c)
- If and , then for any fixed we have
- 1.
- Let . Then if and only if is a θ-sequence.
- 2.
- Every θ-sequence in is c-sequence.
- 3.
- Each c-sequence in P is a θ-sequence if and only if P is normal.
3. Main Results
- 1.
- there is such that ;
- 2.
- a commute with each other;
- 3.
- .
- 1.
- there is such that ;
- 2.
- h is continuous;
- 3.
- commute with each other;
- 4.
- .
- 1.
- There is such that ;
- 2.
- a commute with each other;
- 3.
- ;
- 4.
- .
- 5.
- is α-regular.
- (i)
- and ;
- (ii)
- and ;
- (iii)
- and Then h possess unique fixed point.
- (i)
- for all with ;
- (ii)
- there exists such that
- (iii)
- either is regular or is continuousThen h possess a fixed point in W.
4. Examples and Applications
- (1)
- For each we have if and only if
- (2)
- with implies has a inverse and
- (3)
- Let in which and then
- (4)
- Consider Assume that if with and is an invertible operator, so
- (5)
- Let be unital and is Hermitian. If for some then u is positive. In reverse direction, for every if and u is positive, then
- (6)
- For every implies
- (7)
- if then for each
- (8)
- (9)
- For all if then
- (10)
- Assume that then implies
- (11)
- Let and . Then for any both and are positive elements and
- (i)
- for all and if and only if
- (ii)
- for all
- (iii)
- for all
- (1)
- and
- (2)
- there exists a continuous function and such that
- (3)
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Shatanawi, W.; D. Mitrović, Z.; Hussain, N.; Radenović, S. On Generalized Hardy–Rogers Type α-Admissible Mappings in Cone b-Metric Spaces over Banach Algebras. Symmetry 2020, 12, 81. https://doi.org/10.3390/sym12010081
Shatanawi W, D. Mitrović Z, Hussain N, Radenović S. On Generalized Hardy–Rogers Type α-Admissible Mappings in Cone b-Metric Spaces over Banach Algebras. Symmetry. 2020; 12(1):81. https://doi.org/10.3390/sym12010081
Chicago/Turabian StyleShatanawi, Wasfi, Zoran D. Mitrović, Nawab Hussain, and Stojan Radenović. 2020. "On Generalized Hardy–Rogers Type α-Admissible Mappings in Cone b-Metric Spaces over Banach Algebras" Symmetry 12, no. 1: 81. https://doi.org/10.3390/sym12010081
APA StyleShatanawi, W., D. Mitrović, Z., Hussain, N., & Radenović, S. (2020). On Generalized Hardy–Rogers Type α-Admissible Mappings in Cone b-Metric Spaces over Banach Algebras. Symmetry, 12(1), 81. https://doi.org/10.3390/sym12010081