FIXED POINT THEOREMS USING (CLCS) PROPERTY IN COMPLEX VALUED $b$-METRIC SPACES
Abstract
or two pair of mappings using either (CLR) property ([24]), or by taking
one of the range-subspace closed. In this paper, we introduce the notion of
(CLCS)-property i.e., “common limit converging in the range sub-space”. Using this property, we prove common fixed point theorems for two pairs of
weakly compatible mappings in complex valued b-metric spaces satisfying a
collection of contractive conditions. Our notion is meaningful and valid because the required common fixed point will always lie on the range-subspace of the mapping-pair. We give some examples to show that if a mapping pair (f, g) of a closed complex valued b-metric space X satisfy the (CLRf ) property, then it is also (CLRg), and vice-versa.
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A. Aghajani, M. Abbas and J. R. Roshan: Common fixed points of generalized weak contractive mappings in partially ordered b-metric spaces. Math. Slovaka 64 (2)
(2014), 941–960. Doi: 10.2478/s12175-014-0250-6.
J. Ahmad, A. Azam and S. Saejung: Common fixed point results for contractive mappings in complex valued metric spaces. Fixed Point Theory and Application 2014,
(67)(2014).
M. Aamri and D. El. Moutawakil: Some new common fixed point theorems under strict contractive conditions. J. Math. Anal. Appl. 27(1) (2002), 181–188.
A. Azam, B. Fisher and M. Khan: Common fixed point theorems in complex-valued metric spaces. Numer. Funct. Anal. Optim. 32 (3)(2011), 243–253.
S. Bhatt, S. Choukiyal and R. C. Dimri: Common fixed point of mappings satisfying rational inequality in complex-valued metric spaces. Int. J. Pure Appl. Math. 73(2) (2011), 159–164.
S. Bhatt, S. Choukiyal and R. C. Dimri: A common fixed point theorem for weakly compatible maps in complex valued metric spaces. Int. J. Math. Sci. Anal. 1(3) (2011),
–1389.
S. Banach: Sür les op´ erations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math. 3 (1922), 133–181.
I. A. Bakhtin: The contraction principal in quasi-metric spaces. Funct. Anal. 30 (1989), 26–37.
S. Chandok and D. Kumar: Some common fixed point results for rational type contraction mappings in complex valued metric spaces. J. Oper. 2013, Article ID 813707 (2013).
S. Chauhan, W. Sintunavarat and P. Kumam: Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces using (JCLR) property. Applied Mathematics 3(9) (2012), 976–982.
S. Chauhan: Fixed points of weakly compatible mappings in fuzzy metric spaces satisfying common limit in the range property. Indian J. Math. 543(3) (2012), 375–397.
S. Chauhan, M. A. Khan and S. Kumar: Unified fixed point theorems in fuzzy metric spaces via common limit range property. J. Ineql. Appl. 2013, Article ID 182
(2013).
S. Czerwik: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostraviensis 1 (1993), 5–11.
A. K. Dubey, Rita Shukla and R. P. Dubey: Common fixed point theorems in complex valued b metric spaces. Journal of complex system 2015, Article ID 832467,
pages.
M. Hakwadia, R. K. Gujetia and D. K. Mali: Fixed point theorem in complex valued metric spaces for continuity and complexity. Amer. Int. J. Res. Sci. Tech. Engg.
Math. 2014, 217–223.
M. Imdad, S. Chauhan and P. Kumam: Fixed point theorems for two hybrid pairs of non-self mappings under joint common limit range property in metric spaces. Journal of nonlinear and convex analysis 16(2) (2015), 243–254.
M. Imdad, B. D. Pant and S. Chauhan: Fixed point theorems in Menger space using the (CLR ST ) property and Applications. J. Nonlinear Anal. Optim.- Theory and
Applications 3(2) (2012), 225–237.
M. Jain, K. Tas, S. Kumar and N. Gupta: Coupled fixed point theorems for a pair of weakly compatible maps along with CLR g property in fuzzy metric spaces. J. Applied Math. 2012, Article ID 961210, 13 pages.
G. Jungck: Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9(4) (1986), 771–779.
G. Jungck: Common fixed points for non-continuous non-self mappings on a nonnumeric spaces. Far East J. Math. Sci. 4(2) (1996), 199–212.
M. Kumar, P. Kumar and S. Kumar: Common fixed point theorems in complex valued metric spaces. J. Anal. Num. Theory 2(2) (2014), 103–109.
M. Kumar, P. Kumar and S. Kumar: Some common fixed point theorems using (CLR g )-property in cone metric spaces. Advances in Fixed Point Theory 2(3) (2012), 340–356.
M. Kumar, P. Kumar, S. Kumar and S. M. Kang: Common fixed points for weakly compatible mappings in complex valued metric spaces, Int. J. Pure Appl. Math. 92(3) (2014), 403–419.
S. Manro: Some new common fixed point theorems in complex valued metric space. South Asian J. Math. 5, (1)(2015), 13–24.
S. Manro, S. S. Bhatia, S. Kumar, P. Kumam and S. Dalal: Weakly compatible mappings along with CLRS property in Fuzzy metric spaces. J. Nonlinear Anal. Appl.
, Article ID 00206, 12 pages.
S. K. Mohanta and R. Maitra: Common fixed points for φ-pairs in complex valued metric spaces. Int. J. Math. Comp. Res. 1(9) (2013), 251–256.
A. A. Mukheimer: Some common fixed point theorems in complex valued b-metric spaces. Sci. World. Jour. 2014, Article ID 587825.
N. Malhotra and B. Bansal: Some common coupled fixed point theorems for generalized contraction in b-metric spaces. Jour. Nonlinear Sci. Appl. 8 (2015), 8–16.
M. Oztürk: Common fixed point theorems satisfying contractive type conditions in complex valued metric space. Abst. Appl. Anal. 2014, Article ID 598465.
M. Oztürk and N. Kaplan: Common fixed points of f-contraction mappings in complex valued metric spaces. Math. Sci. 2014, 8:129(2014).
R. P. Pant: Common fixed points of noncommuting mappings. J. Math. Anal. Appl. 188 (1994), 436–440.
H. K. Pathak, R. R. Lopéz and R. K. Verma: A common fixed point theorem using implicit relation and property (EA) in metric spaces. Filomat(Niˇ s) 21(2) (2007),
–234.
H. K. Pathak, Y. J. Cho and S. M. Kang: Remarks on R-weakly commuting mappings and common fixed point theorems. Bull. Korean Math. Soc. 34(2) (1997),
–257.
H. K. Pathak, R. R. Lopez and R. K. Verma: A common fixed point theorem of integral type using implicit relation. Nonlinear funct. Anal. Appl. 15(1) (2009), 1–12
K. Rao, P. Swami and J. Prasad: A common fixed point theorem in complex valued b-metric spaces. Bull. Math. Stat. Res. 1(1) (2013), 1–8.
H. Piri and H. Asharfi: Some fixed point theorems in complete partial b-metric spaces. Advances in Fixed Point Theory 4(3) (2014), 444–461.
A. Roldan, E. Karapinar and P. Kumam: Irremisible stimulate on ’Unified fixed point theorems in fuzzy metric spaces via common limit range property’. J. Ineq. Appl.
, 2014:257.
A. Roldan and W. Sintunavarat: Common fixed point theorems in fuzzy metric spaces using the (CLR g ) property. Fuzzy Sets and System 282 (2016), 131–142.
F. Rouzkard and M. Imdad: Some common fixed point theorems on complex valued metric spaces, Comp. Math. Appl. 64(6) (2012), 1866–1874.
S. Shukla and S. S. Pagey: Some common fixed point theorems in complex valued metric spaces satisfying (E.A) property and (CLR) property, Int. J. Sci. Inno. Math.
Res. 2 (4)(2014), 399–407.
Y. R. Sharma: Common fixed point theorems in complex valued metric spaces. Int. J. Inno. Res. Sci. Engg. Tech. 2(12) (2013), 8282–8286.
W. Sintunavarat and P. Kumam: Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. Journal of Applied Mathematics (2011), Article ID 637958, 14 pages.
W. Sintunavarat and P. Kumam: Common fixed points for R-weakly commuting in fuzzy metric spaces. Annali dell’Universita’ di Ferrara 58 (2012), 389–406.
W. Sintunavarat and P. Kumam: Generalized common fixed point theorems in complex valued metric spaces and applications. J. Inequal. Appl. 2012, 84 (2012).
W. Sintunavarat, Y. J. Cho and P. Kumam: Ursyhon integral equations approach by common fixed point theorems in complex valued metric spaces. Advances in Differential equation 49 (2013).
D. Singh, O. P. Chauhan, Naval Singh and Vishal Joshi: Common fixed point theorems in complex valued b-metric spaces. Advances in Fixed Point Theory 5(2)
(2015), 263–280.
D. Singh, O. P. Chauhan, Nanal Singh and Vishal Joshi: Common fixed point theorems in complex valued b−metric spaces. Advances in Fixed Point Theory 5(3)
(2015), 412–429.
K. Sitthikul and S. Saejung: Some fixed point theorems in complex valued metric spaces. Fixed Point Theory and Appllications 2012, 189: 2012. doi:10.1186/1687-1812-2012-189.
N. Wairojjana W. Sintunanavarat and P. Kumam: Common tripled fixed points for W-compatible mappings along with CLR g property in abstract metric spaces. J. ineq. Appl. 2014, 2014:133.
R. K. Verma and H. K. Pathak: Common fixed point theorems using property (E.A) in complex valued metric spaces. Thai. J. Math. 11(2) (2013), 347–355.
X. Q. Hu: Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. Fixed Point Theory and Applications 2011, Article ID 363716, 14 pages, 2011.
DOI: https://doi.org/10.22190/FUMI1703269V
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