# Fixed point theorems in Menger spaces with an application to metric spaces

### Abstract

Recently Fang [19] dened and used the algebraic sum in Menger spaces and proved the common fixed point theorems. Clubbing this concept of algebraic sum, some common fixed point theorems satisfying contractive conditions using occasionally weakly compatible maps for rational terms in Menger spaces are obtained. This idea give rise to new xed point theorems in the setting of similar abstract spaces. Our result partially generalize many results in existing literature.

### Metrics

### References

[2] A. Bhatt, H. Chandra, D.R. Sahu, Common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions, Nonlinear Analysis 73 (2010), 176-182.

[3] A. Razani, M. Shirdaryazdi, A common fixed point theorem of compatible maps in Menger space, Chaos Solitons Fractals 32 (2007) 26-34.

[4] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983.

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

[6] B. Singh, S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl. 301 (2005) 439-448.

[7] C. Vetro, Some fixed point theorems for occasionally weakly compatible mappings in probabilistic semi-metric spaces, Int. J. Modern Math. 4 (2009), 277-284.

[8] D. O'Regan, R. Saadati. Nonlinear contraction theorems in probabilistic metric spaces. Appl Math Comput 195 (2008) 86-93.

[9] D. Singh, A. Ahmed, M. Singh and N. Singh, A contraction theorem containing rational term in Menger spaces, Journal of Advanced Studies in Topology, Vol. 3, No. 3,2012,110- 117.

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

[11] G. Jungck, B.E. Rhoades, Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory 7 (2006), 286-296.

[12] G. Jungck, B.E. Rhoades, Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory 9 (2008), 383-384. (Erratum)

[13] G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (1986) 771-779.

[14] G. L. Cain, R. H. Kasreil. Fixed periodic points of local contraction mappings on probabilistic metric spaces. Math. Syst. Theory 9 (1975/1976) 289-297

[15] H. Bouhadjera, A. Djoudi, B. Fisher, A unique common fixed point theorem for occasionally weakly compatible maps, Surv. Math. Appl. 3 (2008), 177-182.

[16] H. Chandra, A. Bhatt, Fixed point theorems for occasionally weakly compatible maps in probabilistic semimetric space, Int. J. Math. Anal. 3 (2009), 563-570.

[17] H. Sherwood, complete probabilistic metric spaces. Z Wahr Verw 20 (1971)117-128.

[18] Javed Ali, M. Imdad, D. Bahuguna, Common fixed point theorems in Menger spaces with common property (E.A.), Comput. Math. Appl. 60(12) (2010), 3152-3159. MR2739482

[19] Jin-Xuan Fang, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Non-Linear Analysis 71(2009) 1833-1843.

[20] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 535-537.

[21] Lj. Ciric, B. Samet, C. Vetro, Common fixed point theorems for families of occasionally weakly compatible mappings, Math. Comp. Model. 53 (2011), 631-636.

[22] M. Abbas, B.E. Rhoades, Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition, Math. Commun. 13 (2008),

295-301

[23] M.A. Al-Thagafi, N. Shahzad, A note on occasionally weakly compatible maps, Int. J. Math. Anal. (Ruse) 3 (2009), 55-58.

[24] M.A. Al-Thagafi, N. Shahzad, Generalized I-nonexpansive selfmaps and invariant approximations, Acta Math. Sinica (English Series) 24 (2008), 867-876.

[25] M.A. Khan, Sumitra, Common fixed point theorems for occasionally weakly compatible maps in fuzzy metric spaces, Far East J. Math. Sci. 41 (2010), 285-293.

[26] M. Imdad, J. Ali, and M. Tanveer, Coincidence and common fixed point theorems for nolinear contractions in Menger PM spaces,Chaos, Solitons and Fractals 42 (2009) 31213129

[27] O. Hadzic, Fixed point theorems for multivalued mappings in probabilistic metric spaces, Mat. Vesnik 3(16)(31)(2)(1979),125-133, MR0613901(82g:54068)

[28] O. Hadzic, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, Dordrecht, 2001

[29] R. A. Rashwan, A. Hedar, On common fixed point theorems of compatible maps in Menger spaces, Demonst. Math. 31 (1998) 537-546.

[30] R.P. Pant, Common fixed points of contractive maps, J. Math. Anal. Appl. 226 (1998) 251-258.

[31] R.P. Pant, R-weak commutativity and common fixed points, Soochow J. Math. 25 (1999) 37-42.

[32] R.P. Pant, V. Pant, Common fixed point under strict contractive conditions, J. Math. Anal. Appl. 248 (2000) 327-332.

[33] R.P. Pant,Common fixed points of non commuting mappings, J. Math. Anal. Appl. 188(1994)436-440.

[34] S. Chauhan, B.D. Pant, Common fixed point theorems for occasionally weakly compatible mappings using implicit relation, J. Indian Math. Soc. (N.S.) 77 (2010), 13-21.

[35] S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 149-153.

[36] S.N. Mishra, Common fixed points of compatible mappings in PM-spaces, Math. Japon. 36 (1991) 283-289.

[37] S. Sharma and B. Deshpande, On compatible mappings satisfying an implicit relation in common fixed point consideration, Tamkang J. Math. 33 (2002), no. 3, 245-252.

[38] T. L. Hicks, Fixed point theory in probabilistic metric spaces, Univ. u Novom Sadu Zb. Rad. Prirod.Mat. Fak. Ser. Mat. 13 (1983) 63-72.

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

[40] V. M. Sehgal. Some fixed point theorem in functional analysis and probability. Ph. D. Dissertation, Wayne State Univ Michigan; 18 (1966).

[41] V. Popa and D. Turkoglu, Some fixed point theorems for hybrid contractions satisfying an implicit relation, Stud. Cercet. S?int? Ser. Math. Univ. Bacau 1998 (1998), no. 8,75-86.

[42] V. Popa, A general coincidence theorem for compatible multivalued mappings satisfying an implicit relation, Demonstratio Math. 33 (2000), no. 1, 159-164.

[43] V.Popa, A generalization of Meir-Keeler type common fixed point theorems for four noncontinuous mappings, Sarajevo J. Math. 1(13)2005,135-142

[44] V. Popa, Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstratio Math. 32 (1999), no. 1, 157-163.

*Journal of Advanced Studies in Topology*,

*5*(1), 39-46. https://doi.org/10.20454/jast.2014.712

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