Fixed points for reciprocally continuous mappings and variants of compatible mappings
Abstract
In this paper, we introduce $(\psi,\phi)$-weak contraction condition involving cubic terms of distance function and prove some fixed point theorems for pairs of compatible mappings of type $(E)$, type $(K)$ and subcompatible mappings satisfying a newly introduced contraction condition. We also provide examples in support of our results and give an application for the mappings satisfying an integral contractive type $(\psi,\phi)$-weak contraction condition.
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References
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[24] Y. Rohen and M. R. Singh, {\it Common fixed point of compatible mappings of type $(R)$ in complete metric spaces}, Int. J. Math. Sci. Eng. Appl., {\bf 2} (2008), no. 4, 295--303.
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[26] S. L. Singh and S. N. Mishra, {\it Coincidences and fixed points of reciprocally continuous and compatible hybrid maps}, Int. J. Math. Math. Sci., {\bf 30} (2002), no. 10, 627--635.
[27] M. R. Singh and Y. M. Singh, {\it Compatible mappings of type $(E)$ and common fixed point theorem of Meir-Keeler type}, Int. J. Math. Sci. Eng. Appl., {\bf 1} (2007), 299--315.
[28] M. R. Singh and Y. M. Singh, {\it On various types of compatible maps and common fixed point theorems for non-continuous maps}, Hacettepe J. Math. Stat., {\bf 40} (2011), no. 4, 503--513.
[29] Q. Zhang and Y. Song, {\it Fixed point theory for generalized $\phi-$weak contractions}, Appl. Math. Lett., {\bf 22} (2009), 75--78.
[2] S. Banach, {\it Surles operations dans les ensembles abstraites et leaurs applications}, Fund. Math., {\bf 3} (1922), 133--181.
[3] S. Benchabane and S. Djebali, {\it Common fixed point for multivalued $(\psi$-$G)$-contraction mappings in partial metric spaces}, Bol. Soc. Parana. Mat. {\bf 40} (2022), Paper No. 59.
[4] H. Bouhadjera and C. Godet-Thobie, {\it Common fixed point theorems for pairs of subcompatible maps}, arxiv:0906.3159v1/[math.FA] (2009).
[5] H. Bouhadjera and C. Godet-Thobie, {\it Common fixed point theorems for pairs of subcompatible maps}, arxiv:0906.3159v2/[math.FA] (2011).
[6] D. W. Boyd and J. S. W. Wong, {\it On nonlinear contractions}, Proc. Am. Math. Soc., {\bf 20} (1969), no. 2, 458--464.
[7] A. Branciari, {\it A fixed point theorem for mappings satisfying a general contractive condition of integral type}, Int. J. Math. Math. Sci., {\bf 29} (2002), no. 9, 531--536.
[8] D. Jain, S. Kumar and C. Y. Jung, {\it Common fixed point theorems for weakened compatible mappings satisfying the generalized $\phi-$weak contraction condition}, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., {\bf 26} (2019), no. 2, 99--110.
[9] D. Jain, S. Kumar and S. M. Kang, {\it Weak contraction condition for compatible mappings involving cubic terms of the metric function}, Far East J. Math. Sci., {\bf 103} (2018), no. 4, 799--818.
[10] D. Jain, S. Kumar and S. M. Kang, {\it Generalized weak contraction condition for compatible mappings of types involving cubic terms under the fixed point consideration}, Far East J. Math. Sci., {\bf 103} (2018), no. 8, 1363--1377.
[11] K. Jha, V. Popa and K. B. Manandhar, {\it Common fixed point theorem for compatible of type $(K)$ in metric space}, Int. J. Math. Sci. Eng. Appl., {\bf 8} (2014), 383--391.
[12] C. Y. Jung, D. Jain, S. Kumar and S. M. Kang, {\it Generalized weak contraction condition for compatible mappings of types involving cubic terms of the metric function}, Int. J. Pure. Appl. Math., {\bf 119} (2018), 9--30.
[13] G. Jungck, {\it Compatible mappings and common fixed points}, Int. J. Math. Math. Sci., {\bf 9} (1986), 771--779.
[14] G. Jungck, P. P. Murthy and Y. J. Cho, {\it Compatible mappings of type $(A)$ and common fixed points}, Math. Japon., {\bf 38} (1993), 381--390.
[15] M. S. Khan, M. Swalek and S. Sessa, {\it Fixed point theorems by altering distances between two points}, Bull. Austral. Math. Soc., {\bf 30} (1984), 1--9.
[16] M. Kumar and S. Arora, {\it Fixed point theorems for modified generalized $F$-contraction in $G$-metric spaces}, Bol. Soc. Parana. Mat. {\bf 40} (2022), Paper No. 97.
[17] P. P. Murthy and K. N. V. V. V. Prasad, {\it Weak contraction condition involving cubic terms of $ d(x, y)$ under the fixed point consideration}, J. Math., {\bf 2013} (2013), Article ID 967045. doi: 10.1155/2013/967045
[18] R. P. Pant, {\it A common fixed point theorem under a new condition}, Indian J. Pure Appl. Math., {\bf 30} (1999), no. 2, 147--152.
[19] H. K. Pathak, Y. J. Cho, S. M. Kang and B. S. Lee, {\it Fixed point theorems for compatible mappings of type $(P)$ and applications to dynamic programming}, Matematiche, {\bf 50} (1995), 15--33.
[20] H. K. Pathak, Y. J. Cho, S. M. Kang and B. Madharia, {\it Compatible mappings of type $(C)$ and common fixed point theorems of Gregus type}, Demonstr. Math., {\bf 31} (1998), 499--518.
[21] H. K. Pathak and M. S. Khan, {\it Compatible mappings of type $(B)$ and common fixed point theorems of Gregus type}, Czechoslov. Math. J., {\bf 45} (1995), 685--698.
[22] G. S. M. Reddy, V. S. Chary, D. S. Chary, S. Radenovi\'{c} and S. Mitrovic, {\it Coupled fixed point theorems fof $JS$-$G$-contraction on $G$-metric spaces}, Bol. Soc. Parana. Mat. {\bf 41} (2023), Paper No. 72.
[23] B. E. Rhoades, {\it Some theorems on weakly contractive maps}, Nonlinear Anal., {\bf 47} (2001), no. 4, 2683--2693.
[24] Y. Rohen and M. R. Singh, {\it Common fixed point of compatible mappings of type $(R)$ in complete metric spaces}, Int. J. Math. Sci. Eng. Appl., {\bf 2} (2008), no. 4, 295--303.
[25] S. Sessa, {\it On a weak commutativity conditions of mappings in fixed point consideration}, Publ. Inst.Math. Beograd, {\bf 32} (46) (1982), 146--153.
[26] S. L. Singh and S. N. Mishra, {\it Coincidences and fixed points of reciprocally continuous and compatible hybrid maps}, Int. J. Math. Math. Sci., {\bf 30} (2002), no. 10, 627--635.
[27] M. R. Singh and Y. M. Singh, {\it Compatible mappings of type $(E)$ and common fixed point theorem of Meir-Keeler type}, Int. J. Math. Sci. Eng. Appl., {\bf 1} (2007), 299--315.
[28] M. R. Singh and Y. M. Singh, {\it On various types of compatible maps and common fixed point theorems for non-continuous maps}, Hacettepe J. Math. Stat., {\bf 40} (2011), no. 4, 503--513.
[29] Q. Zhang and Y. Song, {\it Fixed point theory for generalized $\phi-$weak contractions}, Appl. Math. Lett., {\bf 22} (2009), 75--78.
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2025-12-05
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