Conditional reciprocal continuity and common fixed points

• Autorzy: Singh J. , Bisht R K , Joshi M C
• Języki publikacji: EN

Abstrakty

EN

The aim of the present paper is to obtain common fixed point theorems by employing the recently introduced notion of conditional reciprocal continuity. We demonstrate that conditional reciprocal continuity ensures the existence of fixed points under contractive conditions which otherwise do not ensure the existence of fixed points. Our results generalize and extend several well-known fixed point theorems in the setting of metric spaces. We also provide more answers to the open problem posed by B. E. Rhoades [Contractive Definitions and Continuity, Contemporary Math. 72 (1988), 233-245] regarding existence of a contractive condition which is strong enough to generate a fixed point, but which does not force the map to be continuous at the fixed point.

Słowa kluczowe

EN

fixed point theorem; compatible maps; noncompatible mappings; reciprocal continuity; conditional reciprocal continuity

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2013
• Tom: Nr 51
• Strony: 149--158
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Singh J.

• Department of Mathematics D.S.B. Campus Kumaun University Nainital 263002, India

autor

Bisht R K

• Department of Mathematics D.S.B. Campus Kumaun University Nainital 263002, India

autor

Joshi M C

• Department of Mathematics D.S.B. Campus Kumaun University Nainital 263002, India

Bibliografia

• [1] Bisht R.K., Common fixed points of conditionally commuting mappings satisfying weaker continuity conditions, Fasc. Math., 49(2012), 33-41.
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• [6] Kumar S., Pant B.D., A common fixed point theorem in probabilistic metric space using implicit relation, Filomat, 22(2)(2008), 43-52.
• [7] Meir A., Keeler E., A theorem on contraction mappings, J. Math. Anal. Appl., 28(1969), 326-329.
• [8] Muralisankar S., Kalpana G., Common fixed point theorem in intuitionistic fuzzy metric space using general contractive condition of integral type, Int. J. Contemp. Math. Sciences, 4(11)(2009), 505-518.
• [9] Pant R.P., Common fixed points of four mappings, Bull. Cal. Math. Soc., 90(1998), 281-286.
• [10] Pant R.P., Common fixed points of Lipschitz type mapping pairs, J. Math. Anal. Appl., 240(1999), 280-283.
• [11] Pant R.P., , Discontinuity and fixed points, J. Math. Anal. Appl., 240(1999), 284-289.
• [12] Pant R.P., Bisht R.K., Common fixed point theorems under a new continuity condition, Ann. Univ. Ferrara, 58(1)(2012), 127-141.
• [13] Pant R.P., Joshi P.C., Gupta V., A Meir-Keeler type fixed point theorem, Indian J. Pure Appl. Math., 32(6)(2001), 779-787.
• [14] Pant R.P., Pant V., Some fixed point theorem in fuzzy metric space, J. Fuzzy Math., 16(3)(2008), 599-611.
• [15] Rhoades B.E., Contractive definitions and continuity, Contemporary Math., 72(1988), 233-245.
• [16] Singh S.L., Mishra S.N., Coincidences and fixed points of reciprocally con¬tinuous and compatible hybrid maps, Int. J. Math. and Math. Sci., 30(10) (2002), 627-635.