# Metrical common fixed point theorems without completeness and closedness

Abstract: In this article, we point out that some recent results proved in Babu and Alemayehu are corollaries of the main result of an article due to Ali and Imdad. However, combining the notions of R-weakly commuting property of type (Ag) (due to Pathak et al. together with sequential continuity of type (Ag) (to be introduced in this note) (also alternately R-weakly commuting property of type (Af) (due to Pathak et al.) together with sequential continuity of type (Af) (to be introduced in this note)) these results are improved which are possibly new results in the literature of metric fixed point theory. Mathematics Subject Classification (2000): 47H06; 47H10.

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##### Citations

15 citations

### Cites background from "Metrical common fixed point theorem..."

...A different kind of conclusions can bemade for the results from [28, 51, 52]....

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### Cites background or methods from "Metrical common fixed point theorem..."

...[16] introduced the variants of sub-sequential continuity (type (i) and type (ii)) and utilized these notions with R-weakly commutative mappings of type (i) and type (ii) for the existence and uniqueness of common fixed points in metric spaces....

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...Integral-type common fixed point results 55 Definition 2 [16] Let A, S : X → X be two self mappings of a metric space (X, d)....

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### Cites background from "Metrical common fixed point theorem..."

...[4] introduced the notions of sequential continuity of type pAf q (resp....

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...[4] A pair pf, gq of self-mappings defined on a metric space pX , dq is said to be sequentially continuous of type pAgq if and only if there exists a sequence txnu in X such that lim nÑ8 fxn “ lim nÑ8 gxn “ t for some t P X and lim nÑ8 gfxn “ gt and lim nÑ8 ffxn “ ft....

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...[4] A pair pf, gq of self-mappings defined on a metric space pX , dq is said to be sequentially continuous of type pAf q if and only if there exists a sequence txnu in X such that lim nÑ8 fxn “ lim nÑ8 gxn “ t for some t P X and lim nÑ8 fgxn “ ft and lim nÑ8 ggxn “ gt....

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### Cites background from "Metrical common fixed point theorem..."

...5: Let (X , M, * ) be fuzzy metric space with usual metric on X where X = [2, 20]....

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...12[2]: A pair of self mappings (f, g ) of a fuzzy metric space (X, M, *) is said to be sequential continuous of type (Af) if there exists a sequence {xn} is a sequence in X such that lim lim n n n n fx gx u for some uX which satisfies lim n n fgx fu and lim ....

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...Gopal, Imdad and Abbas [2] and prove common fixed point theorems in fuzzy metric spaces using R-weakly commuting maps of type (Af) and (Ag)....

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##### References

1,207 citations

221 citations

### "Metrical common fixed point theorem..." refers background in this paper

..., [11,12]) that the notion of occasional weak compatibility relaxes the requirement of completeness as well as closedness condition on underlying space or subspaces in proving common fixed point theorems for contractive type mappings....

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197 citations

### "Metrical common fixed point theorem..." refers background in this paper

...The following axioms are relevant to this note which are available in Aliouche [5], Galvin and Shore [6], Hicks and Rhoades [7], and Wilson [8]....

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...Wilson, WA: On semi-metric spaces....

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...[8]) Given {xn}, x and y in X with d(xn, x) ® 0 and d(xn, y) ® 0 imply x = y....

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132 citations

### "Metrical common fixed point theorem..." refers background in this paper

...[5]) Given {xn}, {yn} and an x in X with d(xn, x) ® 0 and d(yn, x) ® 0 imply d(xn, yn) ® 0....

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...The following axioms are relevant to this note which are available in Aliouche [5], Galvin and Shore [6], Hicks and Rhoades [7], and Wilson [8]....

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...Aliouche, A: A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type....

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101 citations

### "Metrical common fixed point theorem..." refers background in this paper

...Jungck, G, Rhoades, BE: Fixed point theorems for occasionally weakly compatible mappings....

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...Notice that in this example neither X is complete nor A(X) = {2} ∪ ( (8)3 , 20 3 ] or S(X) = [2,7) ∪ {18} is closed (e....

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...Jungck, G, Rhoades, BE: Erratum to fixed point theorems for occasionally weakly compatible mappings....

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...Hicks, TL, Rhoades, BE: Fixed point theory in symmetric spaces with applications to probabilistic spaces....

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...The following axioms are relevant to this note which are available in Aliouche [5], Galvin and Shore [6], Hicks and Rhoades [7], and Wilson [8]....

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