Metrical common fixed point theorems without completeness and closedness

doi:10.1186/1687-1812-2012-18

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Metrical common fixed point theorems without completeness and closedness

Journal Article•DOI•

Metrical common fixed point theorems without completeness and closedness

Dhananjay Gopal¹, Mohammad Imdad², Mujahid Abbas³•Institutions (3)

Sardar Vallabhbhai National Institute of Technology, Surat¹, Aligarh Muslim University², Lahore University of Management Sciences³

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

TL;DR: In this article, the notions of R-weakly commuting property of type and sequential continuity of type were combined to improve the results of Babu and Alemayehu, which is the main result of an article due to Ali and Imdad.

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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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Journal Article•DOI•

A Note on Various Classes of Compatible-Type Pairs of Mappings and Common Fixed Point Theorems

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Zoran Kadelburg¹, Stojan Radenović¹, Naseer Shahzad²•Institutions (2)

University of Belgrade¹, King Abdulaziz University²

06 Nov 2013-Abstract and Applied Analysis

TL;DR: In this paper, it was shown 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.

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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.

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16 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 Article•DOI•

Some discussion on the existence of common fixed points for a pair of maps

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Deepesh Kumar Patel¹, Poom Kumam², Dhananjay Gopal¹•Institutions (2)

Sardar Vallabhbhai National Institute of Technology, Surat¹, King Mongkut's University of Technology Thonburi²

16 Jul 2013-Fixed Point Theory and Applications

TL;DR: In this article, 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 weaklycompatible, subcompatible, pseudo-compatible, etc.) and continuity-type conditions (i.e., reciprocal, weak reciprocal, g-recipient, conditionally reciprocal, subsequential and sequential continuity of type and ) in the context of existence of common fixed points of a pair of maps.

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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.

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8 citations

Journal Article•DOI•

Variants of sub-sequentially continuous mappings and integral-type fixed point results

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Sunny Chauhan, Wasfi Shatanawi¹, Stojan Radenović², Issam Abu-Irwaq³•Institutions (3)

Hashemite University¹, University of Belgrade², Jordan University of Science and Technology³

01 Apr 2014-Rendiconti Del Circolo Matematico Di Palermo

TL;DR: In this paper, the authors prove integral-type common fixed point results in metric space using variants of sub-sequentially continuity with weak commutativity, and introduce an example to support the useability of their results.

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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.

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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 Article•DOI•

More general common fixed point theorems under a new concept

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H. Bouhadjera¹•Institutions (1)

University of Annaba¹

01 Mar 2016-Demonstratio Mathematica

TL;DR: In this article, a general common fixed point theorem for two pairs of weakly subsequentially continuous mappings satisfying a significant estimated implicit function is proved and an extension of this result is thereby obtained.

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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.

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2 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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Common Fixed Point Theorems in Fuzzy Metric Spaces Using Sequential Continuous Maps of Type (A f ) and (A g )

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Saurabh Manro

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).

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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).

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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 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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Journal Article•DOI•

Compatible mappings and common fixed points

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Gerald Jungck

01 Jan 1986-International Journal of Mathematics and Mathematical Sciences

TL;DR: In this paper, a common fixed point theorem of S.L. and S.P. Singh is generalized by weakening commutativity hypotheses and by increasing the number of functions involved.

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Abstract: A common fixed point theorem of S.L. and S.P. Singh is generalized by weakening commutativity hypotheses and by increasing the number of functions involved.

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1,249 citations

Fixed point theorems for occasionally weakly compatible mappings

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Gerald Jungck, B. E. Rhoades

01 Jan 2006

TL;DR: In this article, the authors obtained two fixed point theorems for a class of operators called occasionally weakly compatible maps defined on a symmetric space and established two of the most general fixed point theorem for four maps.

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Abstract: We obtain two fixed point theorems for a class of operators called occasionally weakly compatible maps defined on a symmetric space . These results establish two of the most general fixed point theorems for four maps.

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227 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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Journal Article•DOI•

On Semi-Metric Spaces

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Wallace Alvin Wilson

01 Apr 1931-American Journal of Mathematics

215 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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Journal Article•DOI•

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

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A. Aliouche

15 Oct 2006-Journal of Mathematical Analysis and Applications

TL;DR: Aamri and El Moutawakil as mentioned in this paper proved a common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type and a property.

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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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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....
[...]
...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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Journal Article•DOI•

Fixed point theory in symmetric spaces with applications to probabilistic spaces

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Troy L. Hicks¹, Billy E. Rhoades²•Institutions (2)

Missouri University of Science and Technology¹, Indiana University²

01 May 1999-Nonlinear Analysis-theory Methods & Applications

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]....
[...]