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Some nonunique common fixed point theorems in symmetric spaces through \(\text{CLR}_{(S, T)}\) property. (English) Zbl 1263.54054

Summary: We introduce a new class of mappings satisfying the “common limit range property” in symmetric spaces and utilize the same to establish common fixed point theorems for such mappings in symmetric spaces. Our results generalize and improve some recent results contained in the literature of metric fixed point theory. Some illustrative examples to highlight the realized improvements are also furnished.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
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[1] G. Jungck, “Compatible mappings and common fixed points,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 4, pp. 771-779, 1986. · Zbl 0613.54029 · doi:10.1155/S0161171286000935
[2] S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,” Institut Mathématique. Publications, vol. 32, pp. 149-153, 1982. · Zbl 0523.54030
[3] R. P. Pant, “Common fixed points of noncommuting mappings,” Journal of Mathematical Analysis and Applications, vol. 188, no. 2, pp. 436-440, 1994. · Zbl 0830.54031 · doi:10.1006/jmaa.1994.1437
[4] R. P. Pant, “Common fixed points of Lipschitz type mapping pairs,” Journal of Mathematical Analysis and Applications, vol. 240, no. 1, pp. 280-283, 1999. · Zbl 0933.54031 · doi:10.1006/jmaa.1999.6559
[5] K. P. R. Sastry and I. S. R. Krishna Murthy, “Common fixed points of two partially commuting tangential selfmaps on a metric space,” Journal of Mathematical Analysis and Applications, vol. 250, no. 2, pp. 731-734, 2000. · Zbl 0977.54037 · doi:10.1006/jmaa.2000.7082
[6] S. L. Singh and A. Kumar, “Fixed point theorems for Lipschitz type maps,” Rivista di Matematica della Università di Parma. Serie 7, vol. 3, pp. 25-34, 2004. · Zbl 1069.54028
[7] M. Imdad and A. H. Soliman, “Some common fixed point theorems for a pair of tangential mappings in symmetric spaces,” Applied Mathematics Letters, vol. 23, no. 4, pp. 351-355, 2010. · Zbl 1213.54066 · doi:10.1016/j.aml.2009.10.009
[8] A. H. Soliman, M. Imdad, and M. Hasan, “Proving unified common fixed point theorems via common property (E-A) in symmetric spaces,” Communications of the Korean Mathematical Society, vol. 25, no. 4, pp. 629-645, 2010. · Zbl 1210.54060 · doi:10.4134/CKMS.2010.25.4.629
[9] Y. Liu, J. Wu, and Z. Li, “Common fixed points of single-valued and multivalued maps,” International Journal of Mathematics and Mathematical Sciences, no. 19, pp. 3045-3055, 2005. · Zbl 1087.54019 · doi:10.1155/IJMMS.2005.3045
[10] M. Abbas and A. R. Khan, “Common fixed points of generalized contractive hybrid pairs in symmetric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 869407, 11 pages, 2009. · Zbl 1185.54038 · doi:10.1155/2009/869407
[11] D. Gopal, M. Imdad, and C. Vetro, “Common fixed point theorems for mappings satisfying common property (E.A.) in symmetric spaces,” Filomat, vol. 25, no. 2, pp. 59-78, 2011. · Zbl 1265.54169 · doi:10.2298/FIL1102059G
[12] N. Hussain, M. A. Khamsi, and A. Latif, “Common fixed points for JH-operators and occasionally weakly biased pairs under relaxed conditions,” Nonlinear Analysis, vol. 74, no. 6, pp. 2133-2140, 2011. · Zbl 1270.47042 · doi:10.1016/j.na.2010.11.019
[13] N. Hussain and M. Abbas, “Common fixed point results for two new classes of hybrid pairs in symmetric spaces,” Applied Mathematics and Computation, vol. 218, no. 2, pp. 542-547, 2011. · Zbl 1225.54018 · doi:10.1016/j.amc.2011.05.098
[14] M. Imdad, J. Ali, and L. Khan, “Coincidence and fixed points in symmetric spaces under strict contractions,” Journal of Mathematical Analysis and Applications, vol. 320, no. 1, pp. 352-360, 2006. · Zbl 1098.54033 · doi:10.1016/j.jmaa.2005.07.004
[15] D. Gopal, M. Hasan, and M. Imdad, “Absorbing pairs facilitating common fixed point theorems for Lipschitzian type mappings in symmetric spaces,” Communications of the Korean Mathematical Society, vol. 27, no. 2, pp. 385-397, 2012. · Zbl 1246.54038 · doi:10.4134/CKMS.2012.27.2.385
[16] D. Gopal, R. P. Pant, and A. S. Ranadive, “Common fixed point of absorbing maps,” Bulletin of the Marathwada Mathematical Society, vol. 9, no. 1, pp. 43-48, 2008.
[17] A. Aliouche, “A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 796-802, 2006. · Zbl 1111.47046 · doi:10.1016/j.jmaa.2005.09.068
[18] F. Galvin and S. D. Shore, “Completeness in semimetric spaces,” Pacific Journal of Mathematics, vol. 113, no. 1, pp. 67-75, 1984. · Zbl 0558.54019 · doi:10.2140/pjm.1984.113.67
[19] T. L. Hicks and B. E. Rhoades, “Fixed point theory in symmetric spaces with applications to probabilistic spaces,” Nonlinear Analysis, vol. 36, pp. 331-344, 1999. · Zbl 0947.54022 · doi:10.1016/S0362-546X(98)00002-9
[20] W. A. Wilson, “On semi-metric spaces,” American Journal of Mathematics, vol. 53, no. 2, pp. 361-373, 1931. · Zbl 0001.22804 · doi:10.2307/2370790
[21] S.-H. Cho, G.-Y. Lee, and J.-S. Bae, “On coincidence and fixed-point theorems in symmetric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 562130, 9 pages, 2008. · Zbl 1169.54020 · doi:10.1155/2008/562130
[22] D. K. Burke, “Cauchy sequences in semimetric spaces,” Proceedings of the American Mathematical Society, vol. 33, pp. 161-164, 1972. · Zbl 0233.54015 · doi:10.2307/2038192
[23] R. P. Pant and V. Pant, “Common fixed points under strict contractive conditions,” Journal of Mathematical Analysis and Applications, vol. 248, no. 1, pp. 327-332, 2000. · Zbl 0977.54038 · doi:10.1006/jmaa.2000.6871
[24] M. Aamri and D. El Moutawakil, “Some new common fixed point theorems under strict contractive conditions,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 181-188, 2002. · Zbl 1008.54030 · doi:10.1016/S0022-247X(02)00059-8
[25] S. Chauhan, W. Sintunavarat, and P. Kumam, “Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces using (JCLR) property,” Applied Mathematics, vol. 3, no. 9, pp. 976-982, 2012.
[26] M. Imdad, B. D. Pant, and S. Chauhan, “Fixed point theorems in menger spaces using the (CLRST) property and applications,” Journal of Nonlinear Analysis and Optimization, vol. 3, no. 2, pp. 215-223, 2012. · Zbl 1394.54023
[27] S. L. Singh, B. D. Pant, and S. Chauhan, “Fixed point theorems in non-archimedean menger PM-spaces,” Journal of Nonlinear Analysis and Optimization, vol. 3, no. 2, pp. 153-160, 2012.
[28] W. Sintunavarat and P. Kumam, “Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces,” Journal of Applied Mathematics, vol. 2011, Article ID 637958, 14 pages, 2011. · Zbl 1226.54061 · doi:10.1155/2011/637958
[29] W. Sintunavarat and P. Kumam, “Common fixed points for R-weakly commuting mappings in fuzzy metric spaces,” Annali dell’Universita’ di Ferrara, vol. 58, no. 2, pp. 389-406, 2012. · Zbl 1302.54088 · doi:10.1007/s11565-012-0150-z
[30] G. Jungck, “Common fixed points for noncontinuous nonself maps on nonmetric spaces,” Far East Journal of Mathematical Sciences, vol. 4, no. 2, pp. 199-215, 1996. · Zbl 0928.54043
[31] V. Pant, “Common fixed points under Lipschitz type condition,” Bulletin of the Korean Mathematical Society, vol. 45, no. 3, pp. 467-475, 2008. · Zbl 1155.54027 · doi:10.4134/BKMS.2008.45.3.467
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