GENERAL FIXED POINT THEOREMS OF GENERALIZED GREGU_S TYPE IN SYMMETRIC SPACES AND APPLICATIONS

Year 2014, Volume: 2 Issue: 1, 34 - 43, 01.06.2014

VELERIU Popa ALINAMIHAELAPATRICIU

Abstract

In this paper a general fixed point theorem of Greguˇs type insymmetric space is proved, which generalize Theorems 1 and 3 [19], Theorem1.2 [23], Theorems 3.1 and 4.1 [28] and we obtain new similar results forstrict expansive mappings. In the last part of this paper, as applications, weobtain new results for mappings satisfying contractive (expansive) conditionof integral type

Keywords

Fixed point , occasionally weakly compatible , Greguˇs type theorem , implicit relation , condition of integral type

References

• Alghamdi, M. A., Radenovi´c, S., Shahzad, N., On some generalizations of commuting map- pings. Abstr. Appl. Anal., Volume 2012, Article ID 952052, 6 pages.
• Aliouche, A., Popa, V., General common fixed point for occasionally weakly compatible hybrid mappings and applications. Novi Sad J. Math. 39 (2009), No.1, 89-109.
• Al-Thagafi, M. A., Shahzad, N., Generalized I-nonexpansive maps and invariant approxima- tions. Acta Math. Sinica 24 (2008), No.5, 867-876.
• Bisht, R. K., Pant, R. P., A critical remark on fixed point theorems for occasionally weakly compatible mappings. J. Egypt. Math. Soc. 21 (2013), 273-275.
• Branciari, A., Fixed point theorems for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 9 (2002), No. 2, 531-536.
• ´Ciri´c, L., A generalization of Greguˇs fixed point theorem. Czechoslovak Math. J. 50 (2000), No.3, 449-458.
• Divicaro, M. L., Fisher, B., Sessa, S., A common fixed point theorem of Greguˇs type. Publ. Math. Debrecen 84 (1987), 83-89.
• Djoudi, A., Nisse, L., Greguˇs type fixed points for weakly compatible maps. Bull. Belg. Math. Soc. Simon Stevin 10 (2003), 369-378.
• Djoudi, A., Aliouche, A., A common fixed point theorem of Greguˇs type for weakly compatible mappings satisfying contractive condition of integral type. J. Math. Anal. Appl. 329 (2007), 31-45.
• Dori´c, D., Kadelburg, Z., Radenovi´c, S., A note on occasionally weakly compatible mappings. Fixed Point Theory, 13 (2012), No.2, 475-480.
• Fisher, B., Sessa, S., On fixed point theorem of Greguˇs type. Int. J. Math. Math. Sci. 9 (1986), 23-28.
• Greguˇs, M., A fixed point theorem in Banach spaces. Boll. Un. Mat. Ital. 17 (1980), No.5, 193-198. [13] Hicks, T. L., Rhoades, B. E., Fixed point theorems in symmetric spaces with applications to probabilistic spaces. Nonlinear Anal. 36 (1999), 331-344.
• Imdad, M., Chauhan, S., Employing common limit range property to prove unified metrical common fixed point theorems. International of Analysis Volume 2013, Article ID 763261, 10 pages.
• Imdad, M., Hasan, M., Nashine, H. K., Murthy, P. P., Employing an implicit function to prove unified common fixed point theorems for expansive type mappings in symmetric spaces. J. Nonlinear Anal. Appl., Volume 2013 (2013), Article ID jnaa-00132, 13 Pages.
• Jain, M. K., Rhoades, B. E., Sabya, A. S., Fixed point theorems for occasionally weakly compatible expansive mappings. J. Adv. Math. Stud. 5 (2012), No. 2, 54-58.
• Jungck, G., Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 9 (1986), 771-779.
• Jungck, G., Common fixed points for noncontinuous nonself maps on a nonnumeric space. Far. East J. Math. Sci. 4 (1996), 192-215.
• Jungck, G., Rhoades, B. E., Fixed points for occasionally weakly compatible mappings. Fixed Point Theory 7 (2006), 287-296.
• Mocanu, M., opa, V., Some fixed point theorems for mappings satisfying implicit relations in symmetric spaces, Libertas Math. 28 (2008), 1-13.
• Pant, R. P., Common fixed points for noncommuting mappings. J. Math. Anal. Appl. 188 (1999), 436-440.
• Pant, R. P., Common fixed points for four mappings. Bull. Calcutta Math. Soc. 9 (1999), 281-287. [23] Pant, R. P., Bisht, R. K., Occasionally weakly compatible mappings and fixed points. Bull. Belg. Math. Soc. Simon Stevin 19 (2012), 655-661.
• Pathak, H. K., Khan, M. S., Compatible mappings of type (B) and common fixed point theorems of Greguˇs type. Czechoslovak Math. J. 45 (1995), no. 4, 685-698.
• Pathak, H. K., Cho, Y. J., Kang, S. M., Madharia, B., Compatible mappings of type (C) and common fixed point theorems of Greguˇs type. Demonstratio Math. J. 31 (1998), 499-518.
• Popa, V., Fixed point theorems for implicit contractive mappings. Stud. Cerc. St. Ser. Math. Univ. Bac˘au 7 (1997), 129-133.
• Popa, V., Some fixed point theorem for compatible mappings satisfying an implicit relation. Demonstratio Math. 32 (1999), 157-163.
• Popa, V., Symmetric spaces and fixed point theorems for occasionally weakly compatible mappings satisfying contractive conditions of integral type. Bull. Inst. Politehn. Ia¸si, Math., Theoret. Mech. Phys. 55 (59) (2009), No.3, 11-21.
• Popa, V., Mocanu, M., A new viewpoint in the study of fixed points for mappings satisfying a contractive condition of integral type. Bull. Inst. Politeh. Ia¸si, Math., Theoret. Mech. Phys. 53 (57) (2007), No.5, 269-286.
• Popa, V., Mocanu, M., Altering distance and common fixed points under implicit relations, Hacettepe J. Math. Statistics 38 (2009), No.3, 329-337.
• Rhoades, B. E., Two fixed point theorems for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 63 (2003), 4007-4019.
• “Vasile Alecsandri” University of Bac˘au, ROMANIA
• E-mail address: [e-mail], [e-mail]