Common new fixed point results on b-cone Banach spaces over Banach algebras

Hojjat Afshari; Hadi Shojaat; Andreea Fulga

doi:10.4995/agt.2022.15571

Authors

Hojjat Afshari University of Bonab https://orcid.org/0000-0003-1149-4336
Hadi Shojaat Farhangian University https://orcid.org/0000-0002-6838-072X
Andreea Fulga Transilvania University of Brasov https://orcid.org/0000-0002-6689-0355

DOI:

https://doi.org/10.4995/agt.2022.15571

Keywords:

common fixed point, ϕ-operator, cone b-norm, cone b-Banach space

Abstract

Recently Zhu and Zhai studied the concepts of cone b-norm and cone b-Banach space as generalizations of cone b-metric spaces and theygave a definition of ϕ-operator and obtained some new fixed point theorems in cone b-Banach spaces over Banach algebras by usingϕ-operator. In this paper we propose a notion of quasi-cone over Banach algebras, then by utilizing some new conditions andfollowing their work with introducing two mappings $\mathcal{T}$ and $\mathcal{S}$ we improve the fixed point theorems to the commonfixed point theorems. An example is given to illustrate the usability of the obtained results.

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Author Biographies

Hojjat Afshari, University of Bonab

Department of Mathematics, Faculty of Sciences

Hadi Shojaat, Farhangian University

Department of Mathematics

Andreea Fulga, Transilvania University of Brasov

Department of Mathematics and Computer Sciences

References

M. A. Alghamdi, S. Gulyaz-Ozyurt and E. Karapinar, A note on extended Z-contraction, Mathematics 8 (2020), Paper No. 195. https://doi.org/10.3390/math8020195

T. Abdeljawad, D. Turkoglu and M. Abuloha, Some theorems and examples of cone Banach spaces, J. Comput. Anal. Appl. 12, no. 4 (2010), 739-753.

H. Afshari, H. Aydi and E. Karapinar, On generalized α-ψ-Geraghty contractions on b-metric spaces, Georgian Math. J. 27 (2020), 9-21. https://doi.org/10.1515/gmj-2017-0063

H. Afshari, Sh. Rezapour and N. Shahzad, Absolute retract of the common fixed points set of two multifunctions, Top. Method in Nonlinear Analysis 40 (2012), 42936.

H. Afshari, S. Kalantari and D. Baleanu, Solution of fractional differential equations via α-ϕ-Geraghty type mappings, Adv. Difference Equ. 2018, Paper No. 347. https://doi.org/10.1186/s13662-018-1807-4

H. Afshari, Solution of fractional differential equations in quasi-b-metric and b-metric-like spaces, Adv. Differ. Equ. 2018, Paper No. 285. https://doi.org/10.1186/s13662-018-1807-4

A. Ahmed and J. N. Salunke, Algebra cone generalized b-metric space over Banach algebra and fixed point theorems of generalized lipschitz mappings, Asian-Eur. J. Math. 11, no. 5 (2018), 1850068. https://doi.org/10.1142/S1793557118500687

A. Amini-Harandi, Fixed point theory for quasi-contraction maps in b-metric spaces, Appl. Math. Lett. 24, no. 11 (2011), 1791-1794. https://doi.org/10.1016/j.aml.2011.04.033

H. Aydi, E. Karapinar, M. F. Bota and S. Mitrovic, A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl. 2012, 2012:88. https://doi.org/10.1186/1687-1812-2012-88

H. Aydi, M.F. Bota, E. Karapinar and S. Moradi, A common fixed point for weak ϕ-contractions on b-metric spaces, Fixed Point Theory 13, no. 2 (2012), 337-346. https://doi.org/10.1186/1687-1812-2012-88

C. Chifu, E. Karapinar and G. Petrusel, Fixed point results in varepsilon-chainable complete b-metric spaces, Fixed Point Theory 21, no. 2 (2020), 453-464. https://doi.org/10.24193/fpt-ro.2020.2.32

S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Semin. Mat. Fis. Univ. Modena. 46, no. 2 (1998), 263-276.

S. Czerwik, Contraction mappings in b-metric spaces. Acta Math. Inf. Univ. Ostrav. 1 (1993), 5-11.

A. Fulga, H. Afshari and H. Shojaat, Common fixed point theorems on quasi-cone metric space over a divisible Banach algebra, Adv. Differ. Equ. 2021, Paper No. 306. https://doi.org/10.1186/s13662-021-03464-z

Z. M. Fadail and A. G. B. Ahmad, Coupled coincidence point and common coupled fixed point results in cone b-metric spaces, Fixed Point Theory Appl. 2013, 2013:177. https://doi.org/10.1186/1687-1812-2013-177

H. P. Huang and S. Y. Xu, Fixed point theorems of contractive mappings in cone b-metric spaces and applications, Fixed Point Theory Appl. 2013, 2013:122. https://doi.org/10.1186/1687-1812-2013-112

N. Hussain and M. H. Shah, KKM mappings in cone b-metric spaces, Comput. Math. Appl. 62, no. 4 (2011), 1677-1684. https://doi.org/10.1016/j.camwa.2011.06.004

E. Karapinar and W.-S. Du, A note on b-cone metric and its related results: Generalizations or equivalence?, Fixed Point Theory Appl. 2013, 2013:210. https://doi.org/10.1186/1687-1812-2013-154

E. Karapinar, Fixed point theorems in cone Banach spaces, Fixed Point Theory Appl. 2009, Art. ID 609281. https://doi.org/10.1155/2009/609281

E. Karapinar, Couple fixed point theorems for nonlinear contractions in cone metric spaces, Computers and Mathematics With Applications 59, no. 12 (2010), 3656-3668. https://doi.org/10.1016/j.camwa.2010.03.062

E. Karapinar, Some nonunique fixed point theorems of Ciric type on cone metric spaces, Abstr. Appl. Anal. 2010, Art. ID 123094. https://doi.org/10.1155/2010/123094

E. Karapinar and A. D. Turkoglu, Best approximations theorem for a couple in cone Banach space, Fixed Point Theory Appl. 2010, Art. ID 784578. https://doi.org/10.1155/2010/784578

H. Liu and S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl. 2013, 2013:320. https://doi.org/10.1186/1687-1812-2013-320

X. Y. Lv and Y. Q. Feng, Some fixed point theorems for Reich type contraction in generalized metric spaces, J. Math. Anal. 9, no. 5 (2018), 80-88.

A. Petrusel, G. Petrusel and J. C. Yao, Fixed point and coincidence point theorems in b-metric spaces with applications, Appl. Anal. Discrete Math. 11 (2017), 199-215. https://doi.org/10.2298/AADM1701199P

J. Mathuraiveerana and S. Mookiah, Common fixed point theorems in M-fuzzy cone metric spaces, Results in Nonlinear Analysis 4 (2021) no. 1, 33-46. https://doi.org/10.53006/rna.748994

H. Shah, S. Simić, N. Hussain, Sretenović and A. Radenović, Common fixed points for occasionally weakly compatible pairs on cone metric type spaces, J. Comput. Anal. Appl. 14, no. 2 (2012), 290-297.

H. Shojaat, H. Afshari and M. S. Asgari, A new class of mixed monotone operators with concavity and applications to fractional differential equations, TWMS J. App. and Eng. Math. 11, no. 1 (2021), 122-133.

K. Yosida, Functional analysis, Beijing World Publishing Corporation, 1999.

X. Zhu and C. Zhai, Some extension results on cone b-metric spaces over Banach space via φ-operator, J. Anal. 29 (2021), 281-295. https://doi.org/10.1007/s41478-020-00262-w