(ε, δ) contractive condition and common fixed points

• Autorzy: Pant V. , Jha K.
• Języki publikacji: EN

Abstrakty

EN

In the present paper we prove a common fixed point theorem (Theorem 1) for four mappings under the (ε, δ) contractive condition, however, without either imposing any additional restriction on δ or assuming the ∅-contractive condition together with. While proving the theorem, neither the completeness of the metric space is assumed nor any of the mappings is required to be continuous. Thus we also provide one more answer to the problem of Rhoades [24] which ensures the existence of common fixed point, however, does not force the maps to be continuous at the common fixed point. Theorem 2 generalizes further the result obtained in Theorem 1.

Słowa kluczowe

EN

common fixed point; compatible maps; noncompatible maps; weak compatible maps of type (A); contractive condition and reciprocal continuity

PL

punkt stały

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2009
• Tom: Nr 42
• Strony: 73--84
• Opis fizyczny: Bibliogr. 26 poz.

Twórcy

autor

Pant V.

autor

Jha K.

• A-24, J.K. Puram, Choti Mukhani Haldwani - 263 139 Nainital Uttarakhand, India,

Bibliografia

• [1] Boyd D.W., Wong J.S., On nonlinear contractions, Proc. Amer. Math. Soc., 20(1969), 458-464.
• [2] Carbone A., Rhoades B.E., Singh S.P., A fixed point theorem for generalized contraction map, Indian J. Pure Appl. Math., 20(1989), 543-548.
• [3] Chugh R., Kumar S., Common fixed points of weak compatible mappings of type (A), J. Indian Math. Soc., 67(2000), 137-143.
• [4] Jachymski J., Common fixed point theorems for some families of maps, Indian J. Pure Appl. Math., 25(1994), 925-937.
• [5] Jachymski J., Equivalent conditions and Meir-Keeler type theorems, J. Math. Anal. Appl., 194(1995), 293-303.
• [6] Jungck G., Moon K.B., Park S., Rhoades B.E., On generalizations of the Meir-Keeler type contraction maps: corrections, J. Math. Anal. Appl., 180(1993), 221-222.
• [7] Jungck G., Compatible mappings and common fixed points, Internat. J. Math. Math. Sci., 9(1986), 771-779.
• [8] Kannan R., Some results on fixed point, Bull. Cal. Math. Soc., 60(1968), 71-76.
• [9] Matkowski J., Integrable Solutions of Functional equations, Diss. Math., 127(1975).
• [10] Meir A., Keeler E., A theorem on contraction mappings, J. Math. Anal. Appl., 28(1969), 326-329.
• [11] Maiti M., Pal T.K., Generalization of two fixed point theorems, Bull. Cal. Math. Soc., 70(1978), 57-61.
• [12] Pant R.P., Common fixed points of two pairs of commuting mappings, Indian J. Pure Appl. Math., 17(1986), 187-192.
• [13] Pant R.P., Common fixed points of weakly commuting mappings, Math. Student, 62(1993), 97-102.
• [14] Pant R.P., Pant V., A unified fixed point theorem, Bull. Cal. Math. Soc., 91(1999), 227-232.
• [15] Pant R.P., Joshi P.C., Gupta V., A Meir-Keeler type fixed point theorem, Indian J. Pure. Appl. Math., 32(2001), 1-9.
• [16] Pant R.P., A common fixed point theorem under a new condition, Indian J. Pure Appl. Math., 30(1999), 147-152.
• [17] Pant R.P., A new common fixed point principle, Soochow J. Math., 27(2001), 287-297.
• [18] Pant R.P., Meir-Keeler type fixed point theorems and dynamics of functions, Demonstratio Math., 1(2003), 199-206.
• [19] Pant R.P., Pant V., Pandey V.P., Generalization of Meir-Keeler type fixed point theorems, Tamkang Journal of Mathematics, 35(3)(2004), 179-187.
• [20] Park S., Rhoades B.E., Extension of some fixed point theorems of Hegedus and Kasahara, Math. Seminar Notes, 9(1981), 113-118.
• [21] Park S., Bae J.S., Extension of fixed point theorem of Meir and Keeler, Arkiv. Math., 19(1981), 232-228.
• [22] Pathak H.K., Kang S.M., Baek J.H., Weak compatible mappings of type (A) and common fixed points in Menger spaces, Comm. Korean Math. Soc., 10(1)(1995), 63-67.
• [23] Pathak H.K., Kang S.M., Baek J.H., Weak compatible mappings of type (A) and common fixed points, Kyungpook Math. J., 35(2)(1995), 345-359.
• [24] Rhoades B.E., Contractive definitions and continuity, Contemporary Math., 72(1988), 233-245.
• [25] Rhoades B.E., Park S., Moon K.B., On generalizations of the Meir-Keeler type contractive maps, J. Math. Anal. Appl., 146(1990), 482-494.
• [26] Singh S.L., Kasahara S., On some recent results on common fixed points, Ind. J. Pure. Appl. Math., 13-17(1982), 757-761.