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Journal ArticleDOI

Metrical common fixed point theorems without completeness and closedness

20 Feb 2012-Fixed Point Theory and Applications (Springer International Publishing)-Vol. 2012, Iss: 1, pp 18

AbstractIn 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.

Topics: Fixed-point theorem (51%)

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Citations
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Journal ArticleDOI
Abstract: Several authors have introduced various conditions which can be used in order to prove common fixed point results. However, it became clear recently that some of these conditions, though formally distinct from each other, actually coincide in the case when the given mappings have a unique point of coincidence. Hence, in fact, new common fixed point results cannot be obtained in this way. We make a review of such connections and results in this paper.

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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Journal ArticleDOI
Abstract: In this paper, we prove some new integral-type common fixed point results in metric space using variants of sub-sequentially continuity with \(R\)-weak commutativity. We introduce an example to support the useability of our results. Our results improve and generalize several well known results existing in the literature.

4 citations


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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Journal ArticleDOI
Abstract: In this paper, the concepts of conditionally sequential absorbing and pseudo-reciprocal continuous maps are introduced in connection to giving a brief discussion on the role of various types of commutativity (e.g., weakly compatible, occasionally weakly compatible, subcompatible, pseudo-compatible, etc.) and continuity-type conditions (e.g., reciprocal, weak reciprocal, g-reciprocal, conditionally reciprocal, subsequential and sequential continuity of type and ) in the context of existence of common fixed points of a pair of maps. Here, the utility of newly introduced maps (i.e., conditionally sequential absorbing and pseudo-reciprocal continuous) in view of common fixed points for a pair of maps satisfying contractive as well as nonexpansive Lipschitz-type conditions is shown. MSC:47H10, 54H25.

4 citations


Journal ArticleDOI
Abstract: Abstract A general common fixed point theorem for two pairs of weakly subsequentially continuous mappings (recently introduced) satisfying a significant estimated implicit function is proved. An extension of this result is thereby obtained. Our results assert the existence and uniqueness of common fixed points in several cases.

1 citations


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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01 Jan 2014
TL;DR: This paper introduces sequential continuity of type (Af) and (Ag) in fuzzy metric space which was recently introduced by Gopal, Imdad and Abbas (2012) and proves common fixed point theorems in fuzzy imperial metric spaces using R- weakly commuting maps of types (Af and Ag).
Abstract: In this paper, we introduce sequential continuity of type (Af) and (Ag) in fuzzy metric space which was recently introduced by Gopal, Imdad and Abbas (2012) and prove common fixed point theorems in fuzzy metric spaces using R- weakly commuting maps of type (Af) and (Ag).

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 uX 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
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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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Abstract: In this paper, we proved a common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type and a property (E.A) introduced in [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181–188]. Our theorem generalizes Theorem 2.2 of [M. Aamri, D. El Moutawakil, Common fixed points under contractive conditions in symmetric spaces, Appl. Math. E-Notes 3 (2003) 156–162] and Theorem 2 of [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181–188].

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"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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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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  • ...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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