An Implicit Function Implies Several Contraction Conditions
DOI:
https://doi.org/10.5644/SJM.04.2.12Keywords:
Common fixed points, common property $(E.A)$, weakly compatible mappings and implicit functionAbstract
In this paper, we define a new implicit function which includes a majority of contractions of the existing literature of metric fixed point theory and then utilize the same to prove a general common fixed point theorem for two pairs of weakly compatible mappings satisfying the common property $(E.A)$. In the process, a host of previously known results are improved and generalized. Some related results are derived besides furnishing illustrative examples.
2000 Mathematics Subject Classification. 47H10, 54H25
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M. Aamri and D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl., 270 (2002), 181–188.
A. Ahmad and M. Imdad, A common fixed point theorem for four mappings satisfying a rational inequality, Publ. Math. Debrecen, 41 (3–4) (1992), 181–187.
R. Chugh and S. Kumar, Common fixed points for weakly compatible maps, Proc. Indian Acad. Sci. (Math. Sci.), 111 (2) (2001), 241–247.
Lj. B. Ciri´c, ´ Generalized contractions and fixed point theorems, Publ. Inst. Math., 12 (26) (1971), 19–26.
Lj. B. Ciri´c, ´ A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267–273.
J. Danes, Two fixed point theorems in topological and metric spaces, Bull. Austral. Math. Soc., 14 (2) (1976), 259–265.
B. Fisher, Common fixed point and constant mappings satisfying a rational inequality, Math. Sem. Notes, 6 (1978), 29–35.
B. Fisher, Mappings satisfying rational inequalities, Nanta Math., 12 (2) (1979), 193–199.
G. E. Hardy and T. D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull., 16 (1973), 201–206.
S. A. Husain and V. M. Sehgal, On common fixed points for a family of mappings, Bull. Austral. Math. Soc., 13 (2) (1975), 261–267.
M. Imdad and Javid Ali, Pairwise coincidentally commuting mappings satisfying a rational inequality, Italian J. Pure Appl. Math., 20 (2006), 87–96.
M. Imdad and Q. H. Khan, Six mappings satisfying a rational inequality, Rad. Mat., 9 (1999), 251–260.
M. Imdad and Q. H. Khan, A common fixed point theorem for six mappings satisfying a rational inequality, Indian J. Math., 44 (2002), 47–57.
M. Imdad and T. I. Khan, On common fixed points of pairwise coincidentally commuting noncontinuous mappings satisfying a rational inequality, Bull. Calcutta Math. Soc., 93 (4) (2001), 263–268.
M. Imdad, T. I. Khan and Javid Ali, Noncontinuous noncommuting mappings satisfying a rational inequality, Nonlinear Anal. Forum, 11 (2006), 175–183.
M. Imdad, S. Kumar and M. S. Khan, Remarks on some fixed point theorems satisfying implicit relations, Rad. Mat., 11 (2002), 135–143.
G. S. Jeong and B. E. Rhoades, Some remarks for improving fixed point theorems for more than two maps, Indian J. Pure Appl. Math., 28 (9) (1997), 1177–1196.
G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci., 9 (1986), 771–779.
G. Jungck, Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci., 4 (2) (1996), 199–215.
S. M. Kang and Y. P. Kim, Common fixed points theorems, Math. Japonica, 37 (6) (1992), 1031–1039.
R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71–76.
M. S. Khan and M. Imdad, A common fixed point theorem for a class of mappings, Indian J. Pure Appl. Math., 14 (1983), 1220–1227.
Y. Liu, Jun Wu and Z. Li, Common fixed points of single-valued and multi-valued maps, Internat. J. Math. Math. Sci., 19 (2005), 3045–3055.
S. Mudgal and R. K. Vats, Common fixed points for intimate maps, South East Asian J. Math. Math. Sci., 4 (1) (2005), 53–60.
P. P. Murthy, Important tools and possible applications of metric fixed point theory, Nonlinear Anal.(TMA), 47 (2001), 3479–3490.
R. P. Pant, Common fixed points of four mappings, Bull. Calcutta Math. Soc., 90 (1998), 281–286.
V. Popa, Some fixed point theorems for weakly compatible mappings, Rad. Mat., 10 (2001), 245–252.
B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), 257–290.
S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math., 32 (46) (1982), 149–153.
N. Shahzad, Invariant approximations, generalized I-contractions and R-subweakly commuting maps, Fixed Point Theory Appl., 2 (1) (2005), 79–86.
S. L. Singh and S. N. Mishra, Remarks on Jachymski’s fixed point theorems for compatible maps, Indian J. Pure Appl. Math., 28 (5) (1997), 611–615.
S. P. Singh and B. A. Meade, On common fixed point theorems, Bull. Austral. Math. Soc., 16 (1977), 49–53.
