Hybrid fixed point theorems in symmetric spaces via common limit range property

• Autorzy: Imdad M. , Chauhan S. , Soliman A. H. , Ahmed M. A.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we point out that some recent results of Vijaywar et al. (Coincidence and common fixed point theorems for hybrid contractions in symmetric spaces, Demonstratio Math. 45 (2012), 611–620) are not true in their present form. With a view to prove corrected and improved versions of such results, we introduce the notion of common limit range property for a hybrid pair of mappings and utilize the same to obtain some coincidence and fixed point results for mappings defined on an arbitrary set with values in symmetric (semi-metric) spaces. Our results improve, generalize and extend some results of the existing literature especially due to Imdad et al., Javid and Imdad, Vijaywar et al. and some others. Some illustrative examples to highlight the realized improvements are also furnished.

Słowa kluczowe

EN

symmetric space; multivalued mappings; quasi-coincidentally commuting mappings; common limit range property; coincidence point; common fixed point

PL

przestrzeń symetryczna; odwzorowania wielowartościowe; punkt wspólny; wspólny punkt stały

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2014
• Tom: Vol. 47, nr 4
• Strony: 949--962
• Opis fizyczny: Bibliogr. 34 poz.

Twórcy

autor

Imdad M.

• Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India

autor

Chauhan S.

• R.H. Government Postgraduate College, Kashipur, 244713, (U.S. Nagar) Uttarakhand, India

autor

Soliman A. H.

• Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut-71524, Egypt

autor

Ahmed M. A.

• Department of Mathematics, Faculty of Science, Assiut University, Assiut-71516, Egypt

Bibliografia

• [1] M. Aamri, D. El. Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270(1) (2002), 181–188. MR1911759 (2003d:54057)
• [2] M. Aamri, D. El. Moutawakil, Common fixed points under contractive conditions in symmetric spaces, Appl. Math. E-notes 3 (2003), 156–162.
• [3] J. Ali, M. Imdad, Common fixed points of nonlinear hybrid mappings under strict contractions in semi-metric spaces, Nonlinear Anal. Hybrid Syst. 4 (2010), 830–837.
• [4] A. Aliouche, A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type, J. Math. Anal. Appl. 322(2) (2006), 796–802. MR2250617 (2007c:47066)
• [5] D. K. Burke, Cauchy sequences in semi-metric spaces, Proc. Amer. Math. Soc. 33 (1972), 161–164.
• [6] S. H. Cho, G. Y. Lee, J. S. Bae, On coincidence and fixed-point theorems in symmetric spaces, Fixed Point Theory Appl., Article ID 562130, 9 pages, 2008.
• [7] F. Galvin, S. D. Shore, Completeness in semi-metric spaces, Pacific. J. Math. 113(1) (1984), 67–75.
• [8] D. Gopal, M. Hasan, M. Imdad, Absorbing pairs facilitating common fixed point theorems for Lipschitzian type mappings in symmetric space, Commun. Korean Math. Soc. 27(2) (2012), 385–397.
• [9] D. Gopal, M. Imdad, C. Vetro, Common fixed point theorems for mappings satisfying common property (E.A.) in symmetric spaces, Filomat 25(2) (2011), 59–78.
• [10] T. L. Hicks, B. E. Rhoades, Fixed point theory in symmetric spaces with applications to probabilistic spaces, Nonlinear Anal. 36 (1999), 331–344.
• [11] M. Imdad, A. Ahmad, S. Kumar, On nonlinear non-self hybrid contractions, Rad. Mat. 10(2) (2001), 243–254.
• [12] M. Imdad, J. Ali, Common fixed point theorems in symmetric spaces employing a new implicit function and common property (E.A), Bull. Belg. Math. Soc. Simon Stevin 16 (2009), 421–433.
• [13] M. Imdad, J. Ali, Jungck’s common fixed point theorem and E.A property, Acta Math. Sinica (English Ser.) 24(1) (2008), 87–94.
• [14] M. Imdad, J. Ali, L. Khan, Coincidence and fixed points in symmetric spaces under strict contractions, J. Math. Anal. Appl. 320 (2006), 352–360.
• [15] M. Imdad, A. H. Soliman, Some common fixed point theorems for a pair of tangential mappings in symmetric spaces, Appl. Math. Lett. 23(4) (2010), 351–355.
• [16] S. Itoh, W. Takahashi, Single-valued mappings, multivalued mappings and fixed-point theorems, J. Math. Anal. Appl. 59(3) (1977), 514–521. MR0454752 (56 #13000)
• [17] J. Jachymski, J. Matkowski, T. Swiatkowski, Nonlinear contractions on semimetric spaces, J. Appl. Anal. 1(2) (1995), 125–134. MR1395268
• [18] G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9(4) (1986), 771–779. MR0870534 (87m:54122)
• [19] G. Jungck, Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci. 4(2) (1996), 199–215.
• [20] T. Kamran, Coincidence and fixed points for hybrid strict contractions, J. Math. Anal. Appl. 299(1) (2004), 235–241. MR2091284 (2005e:54042)
• [21] H. Kaneko, S. Sessa, Fixed point theorems for compatible multi-valued and single-valued mappings, Int. J. Math. Math. Sci. 12(2) (1989), 257–262. MR0994907 (90i:54097)
• [22] E. Karapınar, D. K. Patel, M. Imdad, D. Gopal, Some nonunique common fixed point theorems in symmetric spaces through CLRST property, Int. J. Math. Math. Sci. Article ID 753965, 8 pages, 2013. DOI: 10.1155/2013/753965
• [23] D. Miheµ, A note on a paper of Hicks and Rhoades, Nonlinear Anal. 65 (2006), 1411–1413.
• [24] D. El Moutawakil, A fixed point theorem for multivalued maps in symmetric spaces, Appl. Math. E-Notes 4 (2004), 26–32.
• [25] S. B. Jr. Nadler, Multivalued contraction mappings, Pacific J. Math. 20(2) (1969), 457–488.
• [26] R. P. Pant, Common fixed points of non-commuting mappings, J. Math. Anal. Appl. 188 (1994), 436–440.
• [27] R. P. Pant, V. Pant, Common fixed points under strict contractive conditions, J. Math. Anal. Appl. 248 (2000), 327–332.
• [28] H. K. Pathak, Fixed point theorems for weak compatible multi-valued and single-valued mappings, Acta Math. Hungar. 67(1–2) (1995), 69–78. MR1316710
• [29] S. Sessa, On a weak commutativity condition in fixed point considerations, Publ. Inst. Math. (Beograd) (N.S.) 34(46) (1982), 149–153.
• [30] S. L. Singh, A. M. Hashim, New coincidence and fixed point theorems for strictly contractive hybrid maps, Aust. J. Math. Anal. Appl. 2(1) (2005), Art. 12, 7 pages.
• [31] W. Sintunavarat, P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces J. Appl. Math. Article ID 637958, 14 pages, 2011. MR2822403
• [32] D. Turkoglu, I. Altun, A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying an implicit relation, Bol. Soc. Mat. Mexicana 13 (2007), 195–205.
• [33] Y. K. Vijaywar, N. P. S. Bawa, P. K. Shrivastava, Coincidence and common fixed point theorems for hybrid contractions in symmetric spaces, Demonstratio Math. 45(3) (2012), 611–620.
• [34] W. A. Wilson, On semi-metric spaces, Amer. J. Math. 53 (1931), 361–373.