SciELO - Scientific Electronic Library Online

 
vol.15 número3Composition operators in hyperbolic general Besov-type spacesOn centralizers of standard operator algebras with involution índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados

Revista

Articulo

Indicadores

Links relacionados

  • En proceso de indezaciónCitado por Google
  • No hay articulos similaresSimilares en SciELO
  • En proceso de indezaciónSimilares en Google

Compartir


Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.15 no.3 Temuco  2013

http://dx.doi.org/10.4067/S0719-06462013000300004 

 

Coincidence and common fixed point theorems in Non-Archimedean Menger PM-spaces

 

Sunny Chauhan1, B. D. Pant2 and Mohammad Imdad3

1R.H. Government Postgraduate College, Kashipur-244713, (U.S. Nagar), Uttarakhand, India. sun.gkv@gmail.com

2Government Degree College, Champawat-262523, Uttarakhand, India. badridatt.pant@gmail.com

3Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India. mhimdad@yahoo.co.in


ABSTRACT

The object of this work is to point out a fallacy in the proof of Theorem 1 contained in the recent paper of Khan et al. [Jordan J. Math. Stat. (JJMS) 5(2) (2012), 137-150] proved in Non-Archimedean Menger PM-space by using the notions of sub-compatibility and sub-sequential continuity. We show that the results of Khan et al. [Jordan J. Math. Stat. (JJMS) 5(2) (2012), 137-150] an be recovered in two ways. Further, we establish some illustrative examples to show the validity of the main results. Our results improve a multitude of relevant fixed point theorems of the existing literature.

Keywords and Phrases: t-norm, ompatible mappings, re ipro al ontinuity, sub ompatible mappings, subsequential ontinuity.


RESUMEN

El objetivo de este trabajo es señalar una falacia en la demostración del Teorema 1 contenido en un articulo reciente de Khan et al. [Jordan J. Math. Stat. (JJMS) 5(2) (2012), 137-150] probado en un espacio-PM No-Arquimedeano Menger usando nociones de continuidad subcompatible y sub secuencial. Mostramos que el resultado de Khan et al. [Jordan J. Math. Stat. (JJMS) 5(2) (2012), 137-150] puede recuperarse de dos maneras. Además, establecemos algunos ejemplos ilustrativos que muestran la validez de los resultados principales. Nuestro resultado mejora una gran cantidad de teoremas de punto fijo importantes existentes en la literatura.

2010 AMS Mathematics Subject Classification: 47H10, 54H25.


 

References

[1] M.A. Al-Thagafi and N. Shahzad, Generalized I-nonexpansive selfmaps and invariant approximations, Acta Math. Sin. (Engl. Ser.), 24(5) (2008), 867-876.         [ Links ]

[2] H. Bouhadjera and C. Godet-Thobie, Common fixed theorems for pairs of subcompatible maps, arXiv:0906.3159v1 [math.FA] 17 June (2009) [Old version]         [ Links ].

[3] H. Bouhadjera and C. Godet-Thobie, Common fixed theorems for pairs of subcompatible maps, arXiv:0906.3159v2 [math.FA] 23 May (2011) [New version]         [ Links ].

[4] S.S. Chang, Fixed point theorems for single-valued and multi-valued mappings in Non- Archimedean Menger probabilistic metric spaces, Math. Japonica 35(5) (1990), 875-885.         [ Links ]

[5] S. Chauhan, Z. Kadelburg and S. Dalal, A common fixed point theorem in metric space under general contractive condition, J. Appl. Math. 2013, vol. 2013, Article ID 510691, 7 pages.         [ Links ]

[6] S. Chauhan, B.D. Pant, S. Kumar and A. Tomar, A common fixed point theorem in Non-Archimedean Menger PM-space, Analele Universităţii Oradea Fasc. Matematica XX(2) (2013), in printing.         [ Links ]

[7] S. Chauhan, S. Radenović, M. Imdad and C. Vetro, Some integral type fixed point theorems in Non-Archimedean Menger PM-Spa es with common property (E.A) and application of functional equations in dynamic programming, Revista de la Real A ademia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas (2013), in press.         [ Links ]

[8] Y.J. Cho, K.S. Ha and S.S. Chang, Common fixed point theorems for compatible mappings of type (A) in Non-Archimedean Menger PM-spaces, Math. Japon. 46(1) (1997), 169-179. MR1466131        [ Links ]

[9] B.S. Choudhury, S. Kutukcu and K. Das, On fixed points in Non-Archimedean Menger PM- spaces, Kochi J. Math. 7 (2012), 41-50.         [ Links ]

[10] R.C. Dimri and B.D. Pant, Fixed point theorems in Non-Archimedean Menger spaces, Kyungpook Math. J. 31(1) (1991), 89-95.         [ Links ]

[11] D. Doric , Z. Kadelburg and S. Radenović, A note on occasionally weakly compatible mappings and common fixed point, Fixed Point Theory, 13(2) (2012), 475-479.         [ Links ]

[12] D. Gopal and M. Imdad, Some new common fixed point theorems in fuzzy metric spaces, Ann. Univ. Ferrara Sez. VII Sci. Mat. 57(2) (2011), 303-316        [ Links ]

[13] O. Hadžić, A note on IstratescuŠs fixed point theorem in Non-Archimedean Menger spaces, Bull. Math. Soc. Sci. Math. Rep. Soc. Roum. 24(72) (1980), 277-280.         [ Links ]

[14] M. Imdad, J. Ali and M. Tanveer, Remarks on some recent metrical fixed point theorems, Appl. Math. Lett. 24(7) (2011), 1165-1169        [ Links ]

[15] M. Imdad, D. Gopal and C. Vetro, An addendum to: A common fixed point theorem in intuitionistic fuzzy metric space using subcompatible maps, Bull. Math. Anal. Appl. 4(1) (2012), 168-173        [ Links ]

[16] I. Istrătescu, On some fixed point theorems with applications to the non-Archimedean Menger spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)58(3) (1975), 374-379        [ Links ]

[17] I. Istrătescu, Fixed point theorems for some classes of contraction mappings on Non-Archimedean probablistic metri space, Publ. Math. Debrecen 25(1-2) (1978), 29-34.         [ Links ]

[18] I. Istrătescu and G. Babescu, On the completion on Non-Archimedean probabilistic metri spaces, Seminar de spatii metrice probabiliste, Universitatea Timisoara, Nr. 17, 1979.         [ Links ]

[19] I. Istrătescu and N. Crivat, On some classes of Non-Archimedean probabilistic metric spaces, Seminar de spatii metrice probabiliste, Universitatea Timisoara, Nr. 12, 1974.         [ Links ]

[20] I. Istrătescu u and G. Palea, On Non-Archimedean probabilistic metric spaces, An. Univ. Timişoara Ser. Şti. Mat. 12(2) (1974), 115-118 (1977).         [ Links ]

[21] G. Jungck and B.E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math. 29(3) (1998), 227-238.         [ Links ]

[22] M.A. Khan, Common fixed point theorems in Non-Archimedean Menger PM-spaces, Int. Math. Forum 6(40) (2011), 1993-2000.         [ Links ]

[23] M.A. Khan and Sumitra, A common fixed point theorem in Non-Archimedean Menger PM- space, Novi Sad J. Math. 39(1) (2009), 81-87.         [ Links ]

[24] M.A. Khan and Sumitra, Common fixed point theorems in Non-Archimedean Menger PM- space, JP J. Fixed Point Theory Appl. 5(1) (2010), 1-13.         [ Links ]

[25] M.A. Khan, Sumitra and R. Kumar, Sub-compatible and and sub-sequential continuous maps in Non-Archimedean Menger PM-space, Jordan J. Math. Stat. (JJMS) 5(2) (2012), 137-150.         [ Links ]

[26] S. Kutukcu and S. Sharma, A common fixed point theorem in Non-Archimedean Menger PM- spaces, Demonstratio Math. 42(4) (2009), 837-849.         [ Links ]

[27] R.P. Pant, Common fixed points of four mappings, Bull. Cal. Math. So . 90(4) (1998), 281-286        [ Links ]

[28] R.P. Pant and R.K. Bisht, Common fixed point theorems under a new continuity condition, Ann. Univ. Ferrara Sez. VII Sci. Mat. 58(1) (2012), 127-141        [ Links ]

[29] K.P.R. Rao and E.T. Ramudu, Common fixed point theorem for four mappings in Non- Archimedean Menger PM-spaces, Filomat 20(2) (2006), 107-113.         [ Links ]

[30] F. Rouzkard, M. Imdad and H.K. Nashine, New common fixed point theorems and invariant approximation in convex metric spaces, Bull. Belg. Math. Soc. Simon Stevin 19 (2012), 311-328.         [ Links ]

[31] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 313-334.         [ Links ]

[32] V.M. Sehgal and A.T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Systems Theory 6 (1972), 97-102        [ Links ]

[33] S.L. Singh and B.D. Pant, Common fixed points of weakly commuting mappings on Non-Archimedean Menger PM-spaces, Vikram J. Math. 6 (1985/86), 27-31.         [ Links ]

34 S.L. Singh, B.D. Pant and S. Chauhan, Fixed point theorems in Non-Archimedean Menger PM-spaces, J. Nonlinear Anal. Optim. Theory Appl. 3(2) (2012), 153-160.         [ Links ]


Received: June 2012 / Accepted: September 2013.

Creative Commons License Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons