COMMON BEST PROXIMITY POINT THEOREMS ON CONE B-METRIC SPACES OVER BANACH ALGEBRAS

Year 2017, Volume: 30 Issue: 2, 159 - 172, 19.06.2017

Seyed Masoud Aghayan Ahmad Zireh , Ali Ebadian

Abstract

In this paper, we obtain the existence of some common best proximity
point theorems for generalized Lipschitz contractive mappings on cone b-metric space over Banach algebra without assumption of normality. Our results generalize the corresponding result by Xu and Radenovic (Fixed Point Theory and Appl. 2014, 2014:102) and by Huang and Radenovic ( J. Computational Anal. and Appl. 2016, 20(3)). Further, we give an example to illustrate that our works are never equivalent with the counterparts in the literature.

Keywords

Common best proximity point theorems, Cone metric spaces over Banach algebras, Generalized Lipschitz mappings, c-sequence.

References

• Huang, L-G, Zhang, X: Cone metric spaces and fixed point theorems of contractive mappings. J. Math.
• Anal. Appl. 332, 1468-1476 (2007)
• Jiang, S, Li, Z: Extensions of Banach contraction principle to partial cone metric spaces over a nonnormal
• solid cone. Fixed Point Theory Appl. 2013, 250 (2013)
• Al-Khaleel, M, Al-Sharifa, S, Khandaqji, M: Fixed points for contraction mappings in generalized cone
• metric spaces. Jordan J. Math. Stat. 5(4), 291-307 (2012)
• Gajic, L, Rako cevi c, V: Quasi-contractions on a nonnormal cone metric space. Funct. Anal. Appl.
• (1), 75-79 (2012)
• Jankovic, S, Kadelburg, Z, Radenovic, S: On the cone metric space: a survey. Nonlinear Anal. 74,
• -2601 (2011)
• Cakall, H, Snmez, A, Gen, : On an equivalence of topological vector space valued cone metric spaces
• and metric spaces. Appl. Math. Lett. 25, 429-433 (2012)
• Kadelburg, Z, Radenovic, S, Rako cevi c, V: A note on the equivalence of some metric and cone
• metric fixed point results. Appl. Math. Lett. 24, 370-374 (2011)
• Feng, Y, Mao, W: The equivalence of cone metric spaces and metric spaces. Fixed Point Theory 11(2),
• -264 (2010)
• Liu, H, Xu, S: Cone metric spaces with Banach algebras and fixed point theorems of generalized
• Lipschitz mappings. Fixed Point Theory Appl. 2013, 320 (2013)
• Rudin, W: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)
• Kadelburg, Z, Radenovic, S: A note on various types of cones and fixed point results in cone metric
• spaces. Asian J. Math. Appl. 2013, Article ID ama0104 (2013)
• Xu S.-Y.,Radenovic S.: Fixed point theorems of generalized Lipschitz mappings on cone metric spaces
• over Banach algebras without assumption of normality, Fixed Point Theory Appl., 2014, 2014: 102.
• Kumam P., Dung N. V., Hang V.-T.-L.: Some equivalence between cone b-metric spaces and b-metric
• spaces, Abstr. Appl. Anal., 2013, Article ID 573740, 8 pages, 2013.
• Huang H.-P.,Xu S.-Y.: Fixed point theorems of contractive mappings in cone b-metric spaces and
• applications, Fixed Point Theory Appl., 2013, 2013: 112.
• Azam A., Mehmood N., Ahmad J., Radenovic S.: Multivalued fixed point theorems in cone b-metric
• spaces, Fixed Point Theory Appl., 2013, 2013: 582.
• Huang H., Radenovic S. :Some fixed point results of generalized Lipschitz mappings on cone b-metric
• spaces over Banach algebras, J. Computational Aanl. and Appl., 20(3), 2016, 566-583.
• Huang H., Xu S., Liu H., Radenovic S: Fixed point theorems and T-stability of Picard iteration for
• generalized Lipschitz mappings in cone metric spaces over Banach algebras, J. Computational Aanl. and
• Appl., 20(5), 2016, 869-888.