Fixed point theorems on cone \(S\)-metric spaces using implicit relation

G. S.  Saluja

doi:10.4067/S0719-06462020000200273

DOI:

https://doi.org/10.4067/S0719-06462020000200273

Abstract

In this paper, we establish some fixed point theorems in the framework of cone \(S\)-metric spaces using implicit relation. Our results extend, unify and generalize several results from the current existing literature. Especially, they extend the corresponding results of Sedghi and Dung [24] to the setting of complete cone \(S\)-metric spaces.

Keywords

Fixed point , implicit relation , cone \(S\)-metric space , cone

G. S. Saluja Department of Mathematics, Govt. Kaktiya P. G., College Jagdalpur, Jagdalpur - 494001 (C.G.), India.

Pages: 273–288
Date Published: 2020-08-23
Vol. 22 No. 2 (2020)

A. Aliouche and V. Popa, General common fixed point theorems for occasionally weakly compatible hybmappings and applications, Novi Sad J. Math. 39(1) (2009), 89–109.

V. Berinde, Approximating fixed points of implicit almost contractions, Hacet. J. Math. Stat. 41 (2012), no. 1, 93–102.

V. Berinde and F. Vetro, Common fixed points of mappings satisfying implicit contractive conditions, Fixed Point Theory Appl. 2012, 2012:105, 8 pp.

S. K. Chatterjae, Fixed point theorems compactes, Rend. Acad. Bulgare Sci. 25 (1972), 727- 730

Lj. B. Ä†iriÄ‡, A generalization of Banach‘s contraction principle, Proc. Amer. Math. Soc. 4 (1974), 267–273.

D. Dhamodharan and R. Krishnakumar, Cone S-metric space and fixed point theorems of contractive mappings, Annals of Pure Appl. Math. 14(2) (2017), 237-243.

K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985.

L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), no. 2, 1468–1476.

M. Imdad, S. Kumar and M. S. Khan, Remarks on some fixed point theorems satisfying implicit relations, Rad. Mat. 11 (2002), no. 1, 135–143.

R. Kannan, Some results on fixed point theorems, Bull. Calcutta Math. Soc. 60(1969), 71–78.

J. K. Kim, S. Sedghi and N. Shobkolaei, Common fixed point theorems for the R-weakly commuting mappings in S-metric spaces, J. Comput. Anal. Appl. 19 (2015), no. 4, 751–759.

R. Krishnakumar and D. Dhamodharan, Fixed point theorems in normal cone metric space, Int. J. Math. Sci. Engg. Appl. 10(III) (2016), 213–224.

Nguyen Van Dung, N. T. Hieu and S. Radojevi Ìc, Fixed point theorems for g-monotone maps on partially ordered S-metric spaces, Filomat 28 (2014), no. 9, 1885–1898.

N. Yilmaz Özgür and N. Tas, Some fixed point theorems on S-metric spaces, Mat. Vesnik 69 (2017), no. 1, 39–52.

V. Popa, Fixed point theorems for implicit contractive mappings, Stud. Cercet. S Ì§tiin Ì§t. Ser. Mat. Univ. Bacau No. 7 (1997), 127–133 (1999).

V. Popa, On some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstr. Math. 32(1) (1999), 157–163.

V. Popa, A general fixed point theorem for four weakly compatible mappings satisfying an implicit relation, Filomat No. 19 (2005), 45–51.

V. Popa and A.-M. Patriciu, A general fixed point theorem for pairs of weakly compatible mappings in G-metric spaces, J. Nonlinear Sci. Appl. 5 (2012), no. 2, Special issue, 151–160.

V. Popa and A.-M. Patriciu, Fixed point theorems for two pairs of mappings in partial metric spaces, Facta Univ. Ser. Math. Inform. 31 (2016), no. 5, 969–980.

M. U. Rahman and M. Sarwar, Fixed point results of Altman integral type mappings in S-metric spaces, Int. J. Anal. Appl. 10(1) (2016), 58–63.

S. Reich, Some remarks concerning contraction mappings, Canada. Math. Bull. 14 (1971), 121–124.

Sh. Rezapour and R. Hamlbarani, Some notes on the paper: “Cone metric spaces and fixed point theorems of contractive mappings” [J. Math. Anal. Appl. 332 (2007), no. 2, 1468–1476; by L.-G. Huang and X. Zhang, J. Math. Anal. Appl. 345 (2008), no. 2, 719–724.

S. Sedghi, N. Shobe and A. Aliouche, A generalization of fixed point theorems in S-metric spaces, Mat. Vesnik 64 (2012), no. 3, 258–266.

S. Sedghi and N. V. Dung, Fixed point theorems on S-metric space, Mat. Vesnik 66(1) (2014), 113–124.

S. Sedghi, N. Shobe and T. DošenoviÄ‡, Fixed point results in S-metric spaces, Nonlinear Funct. Anal. Appl. 20(1) (2015), 55–67.

S. Sedghi et al., Common fixed point theorems for contractive mappings satisfying Î¦-maps in S-metric spaces, Acta Univ. Sapientiae Math. 8 (2016), no. 2, 298–311.

N. Tas and N. Yilmaz Ozgur, New generalized fixed point results on Sb-metric spaces, arxiv:1703.01868v2 [math.gn] 17 apr. 2017.

J. S. Vandergraft, Newton‘s method for convex operators in partially ordered spaces, SIAM J. Numer. Anal. 4 (1967), 406–432.

P. P. Zabrejko, K-metric and K-normed linear spaces: survey, Collect. Math. 48 (1997), no. 4-6, 825–859.