ON VARIOUS TYPES OF COMPATIBLE MAPS AND COMMON FIXED POINT THEOREMS FOR NON-CONTINUOUS MAPS

Year 2011, Volume: 40 Issue: 4, 503 - 513, 01.04.2011

M.r. Singh Y. Mahendra Singh

Abstract

Concepts of S- and A- compatible of type (E), S- and A- reciprocally continuous are introduced. In this paper, we give a brief discussion on various types of compatible maps and obtain some common fixed point theorems for non-continuous self maps on a metric space. Here the utility of the newly introduced maps viz. S- compatible of type (E) and S- reciprocally continuous (one sided) in lieu of continuity is shown. Our results improve, generalize or extend the results of H. Bouhadjera (General common fixed point theorem for compatible mappings of type (C), Sarajevo J. Math. 1 (14), 261–270, 2005), X. P. Ding (Some common fixed point theorems of commuting mappings II, Math.Seminar Note 11, 301–305, 1983), M. L. Diviccaro and S. Sessa (Some remarks on common fixed points of four mappings, Jnanabha 15, 139– 149, 1985), G. Jungck (Compatible mappings and common fixed points (2), Internat. J. Math. & Math. Sci. 11 (2), 285–288, 1988), S.M. Kang, Y. J. Cho and G. Jungck (Common fixed points of compatible mappings, Internat. J. Math. & Math. Sci. 13 (1), 61–66, 1990), V. Popa (Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstr. Math. 32(1) (1999), 157–163, 1999), and others.

Keywords

Compatible maps, Compatible of type (E), S- and A-compatible of type (E), S- and A-reciprocally continuous, Weakly compatible maps and common fixed points, 2000 AMS Classification: 47 H 10

ON VARIOUS TYPES OF COMPATIBLE MAPS AND COMMON FIXED POINT THEOREMS FOR NON-CONTINUOUS MAPS

Abstract

References

• Aliouche, A. and Djoudi, A. Common fixed point theorems for mappings satisfying an im- plicit relation without decreasing assumption, Hacet. J. Math. Stat. 36 (1), 11–18, 2007.
• Bouhadjera, H. General common fixed point theorem for compatible mappings of type (C), Sarajevo J. Math. 1 (14), 261–270, 2005.
• Cho, Y. J., Murthy, P. P. and Jungck, G. A common fixed point theorems of Meir-Keeler type, Internat. J. Math. & Math. Sci., 16 (4), 669–674, 1993.
• Ding, X. P. Some common fixed point theorems of commuting mappings II, Math. Seminar Note 11, 301–305, 1983. [5] Diviccaro, M. L. and Sessa, S. Some remarks on common fixed points of four mappings, Jnanabha 15, 139–149, 1985.
• Jungck, G. Commuting mappings and fixed points, Amer. Math Monthly 83, 261–263, 1976. [7] Jungck, G. Compatible mappings and common fixed points, Internat. J. Math. & Math. Sci.9 (4), 771–779, 1986. [8] Jungck, G. Compatible mappings and common fixed points (2), Internat. J. Math. & Math. Sci. 11 (2), 285–288, 1988.
• Jungck, G., Murthy, P. P. and Cho, Y. J. Compatible mappings of type (A) and common fixed points, Math. Japonica 36, 381–390, 1993.
• Jungck, G. and Rhoades, B. E. Fixed point for set valued functions without continuity, Indian J. Pure Appl. Math. 29 (3), 227–238, 1998.
• Kang, S. M., Cho, Y. J. and Jungck, G. Common fixed points of compatible mappings, In- ternat. J. Math. & Math. Sci. 13 (1), 61–66, 1990.
• Matkowski, J. Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc. 62, 344–348, 1977.
• Pathak, H. K., Chang, S. S. and Cho, Y. J. Fixed point theorems for compatible mappings of type (P), Indian J. Math. 36 (2), 151–166, 1994.
• Pathak, H. K., Cho, Y. J., Kang, S. M. and Lee, B. S. Fixed point theorems for compatible mappings of type (P) and applications to dynamic programming, Le Mathematche 50 (1), 15–33, 1995.
• Pathak, H. K. and Khan, M. S. Compatible mappings of type (B) and common fixed point theorems of Gregu ¨U type, Czechoslovak Math. J. 45 (120), 685–698, 1995.
• Pathak, H. K., Cho, Y. J., Kang, S. M. and Madharia, B. Compatible mappings of type (C) and common fixed point theorem of Gregu ¨U type, Demonstr. Math. 31 (3), 499–517, 1998. [17] Pant, R. P. A common fixed point theorem under a new condition, Indian. J. Pure Appl. Math. 30 (2), 147–152, 1999.
• Popa, V. Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstr. Math. 32 (1), 157–163, 1999.
• Popa, V. A general fixed point theorem for weakly compatible mappings in compact metric spaces, Turk. J. Math. 25, 465–474, 2001.
• Popa, V. A general common fixed point theorem of Meir-Keeler type for non continuous weak compatible mappings, FILOMAT 18, 33–40, 2004.
• Popa, V. and Mocanu, M. Altering distance and common fixed points under implicit rela- tions, Hacet. J. Math. Stat. 38 (3), 329-337, 2009.
• Sessa, S. On a weakly commutativity condition in a fixed point considerations, Publ. Inst. Math. 32 (46), 149–153, 1986.
• Singh, S. L. and Mishra, S. N. Coincidences and fixed points of reciprocally continuous and compatible hybrid maps, Intenat. J. Math. & Math. Sci. 30 (10), 627–635, 2002.
• Singh, M. R. & Mahendra Singh, Y. Compatible mappings of type (E) and common fixed point theorems of Meir-Keeler type, International J. Math. Sci. & Engg. Appl. 1 (2), 299– 315, 2007.