# Quadruple random common fixed point results of generalized Lipschitz mappings in cone $b$-metric spaces over Banach algebras

Volume 11, Issue 1, pp 131--149

Publication Date: 2018-01-12

http://dx.doi.org/10.22436/jnsa.011.01.10

### Authors

Chayut Kongban - KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
Poom Kumam - KMUTT-Fixed Point Theory and Applications Research Group (KMUTT-FPTA), Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

### Abstract

In this paper, we introduce the concept of cone $b$-metric spaces over Banach algebras and present some quadruple random coincidence points and quadruple random common fixed point theorems for nonlinear operators in such spaces.

### Keywords

Quadruple random fixed point, quadruple common random fixed point, quadruple random coincidence point, cone $b$-metric space over Banach algebra

### References

[1] A. Alotaibi, S. M. Alsulami, Coupled coincidence points for monotone operators in partially ordered metric spaces, Fixed Point Theory Appl., 2011 (2011), 13 pages.
[2] I. Altun, B. Damjanović, D. Djorić, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett., 23 (2010), 310–316.
[3] A. D. Arvanitakis, A proof of the generalized Banach contraction conjecture, Proc. Amer. Math. Soc., 131 (2003), 3647– 3656.
[4] I. A. Bakhtin, The contraction mapping principle in almost metric space, (Russian) Functional analysis, Ul’yanovsk. Gos. Ped. Inst., Ul’yanovsk, 30 (1989), 26–37.
[5] D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458–464.
[6] L. Ćirić, V. Lakshmikantham, Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces, Stoch. Anal. Appl., 27 (2009), 1246–1259.
[7] Y. Han, S.-Y. Xu, Some new theorems of expanding mappings without continuity in cone metric spaces, Fixed point Theory Appl., 2013 (2013), 9 pages.
[8] C. J. Himmelberg, Measurable relations, Fund. Math., 87 (1975), 53–72.
[9] H.-P. Huang, S. Radenović, Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications, J. Nonlinear Sci. Appl., 8 (2015), 787–799.
[10] L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468–1476.
[11] N. Hussain, A. Latif, N. Shafqat, Weak contractive inequalities and compatible mixed monotone random operators in ordered metric spaces, J. Inequal. Appl., 2012 (2012), 20 pages.
[12] N. Hussain, M. H. Shah, KKM mappings in cone b-metric spaces, Comput. Math. Appl., 62 (2011), 1677–1684.
[13] B.-H. Jiang, Z.-L. Cai, J.-Y. Chen, H.-P. Huang, Tripled random coincidence point and common fixed point results of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras, J. Nonlinear Sci. Appl., 10 (2017), 465–482.
[14] E. Karapınar, P. Kumam, W. Sintunavarat, Coupled fixed point theorems in cone metric spaces with a c-distance and applications, Fixed Point Theory Appl., 2012 (2012), 19 pages
[15] E. Karapınar, N. V. Luong, Quadruple fixed point theorems for nonlinear contractions, Comput. Math. Appl., 64 (2012), 1839–1848.
[16] E. Karapınar, N. V. Luong, N. X. Thuan, Coupled coincidence points for mixed monotone operators in partially ordered metric spaces, Arab. J. Math. (Springer), 1 (2012), 329–339.
[17] P. Kumam, Random common fixed points of single-valued and multivalued random operators in a uniformly convex Banach space, J. Comput. Anal. Appl., 13 (2011), 368–375.
[18] P. Kumam, S. Plubtieng, Random coincidence and random common fixed points of nonlinear multivalued random operators, Thai J. Math., 5 (2007), 155–163.
[19] P. Kumam, S. Plubteing, Random common fixed point theorems for a pair of multi-valued and single-valued nonexpansive random operators in a separable Banach space, Indian J. Math., 51 (2009), 101–115.
[20] T.-C. Lin, Random approximations and random fixed point theorems for non-self-maps, Proc. Amer. Math. Soc., 103 (1988), 1129–1135.
[21] H. Liu, S.-Y. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory Appl., 2013 (2013), 10 pages.
[22] E. J. McShane, R. B. Warfield Jr., On Filippov’s implicit functions lemma, Proc. Amer. Math. Soc., 18 (1967), 41–47.
[23] J. Merryfield, B. Rothschild, J. D. Stein Jr., An application of Ramsey’s theorem to the Banach contraction principle, Proc. Amer. Math. Soc., 130 (2002), 927–933.
[24] Z. Mustafa, H. Aydi, E. Karapınar, Mixed g-monotone property and quadruple fixed point theorems in partially ordered metric spaces, Fixed Point Theory Appl., 2012 (2012), 19 pages.
[25] S. Phiangsungnoen, Ulam-Hyers stability and well-posedness of the fixed point problems for contractive multi-valued operator in b-metric spaces, Commun. Math. Appl., 7 (2016), 241–262.
[26] S. Radenović, Common fixed points under contractive conditions in cone metric spaces, Comput. Math. Appl., 58 (2009), 1273–1278.
[27] H. Rahimi, G. Soleimani Rad, P. Kumam, Coupled common fixed point theorems under weak contractions in cone metric type spaces, Thai. J. Math., 12 (2014), 1–14.
[28] W. Rudin, Functional analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw- Hill, Inc., New York, (1991).
[29] I. Şahni, M. Telci, Fixed points of contractive mappings on complete cone metric spaces, Hacet. J. Math. Stat., 38 (2009), 59–67.
[30] W. Shatanawi, F. Awawdeh, Some fixed and coincidence point theorems for expansive maps in cone metric spaces, Fixed point Theory Appl., 2012 (2012), 10 pages.
[31] W. Shatanawi, Z. Mustafa, On coupled random fixed point results in partially ordered metric spaces, Mat. Vesnik, 64 (2012), 139–146.
[32] Y.-H. Shen, D. Qiu, W. Chen, Fixed point theorems in fuzzy metric spaces, Appl. Math. Lett., 25 (2012), 138–141.
[33] L. Shi, S.-Y. Xu, Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces, Fixed Point Theory Appl., 2013 (2013), 11 pages.
[34] W. Sintunavarat, Y. J. Cho, P. Kumam, Common fixed point theorems for c-distance in ordered cone metric spaces, Comput. Math. Appl., 62 (2011), 1969–1978.
[35] W. Sintunavarat, P. Kumam, Y. J. Cho, Coupled fixed point theorems for nonlinear contractions without mixed monotone property, Fixed Point Theory Appl., 2012 (2012), 16 pages.
[36] A. Špaček, Zufällige Gleichungen, (German) Czechoslovak Math. J., 5 (1955), 462–466.
[37] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., 136 (2008), 1861–1869.
[38] S.-Y. Xu, S. Radenović, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed point Theory Appl., 2014 (2014), 12 pages.