Some new common fixed point theorems under strict contractive conditions

Aamri, Mohamed; Moutawakil, Driss El

doi:10.1016/s0022-247x(02)00059-8

Cited by 370 publications

(370 citation statements)

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“…In a recent work, Aamri and Moutawakil [1] introduced the property (E.A) and thus generalized the notion of noncompatible maps.…”

Section: Resultsmentioning

confidence: 99%

“…Mappings, in Example 1 above, illustrate the argument as f and g are conditionally commuting but not R-weakly commuting. Recently, Aamri and Moutawakil [1] introduced the property (E.A) and thus generalized the concept of noncompatible maps. The results obtained in the metric fixed point theory by using the notion of noncompatible maps or the property (E.A) are very interesting.…”

Section: Introductionmentioning

confidence: 99%

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Strict contractive conditions and common fixed point theorems with application

Pant¹

2013

Demonstratio Mathematica

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Abstract. The aim of the present paper is to obtain common fixed point theorems under a strict contractive condition by assuming minimal commutativity conditions. Our theorems extend the results due to Pant and Pant [4]. In the setting of our results, we also provide pair of mappings which ensures the existence of a common fixed point; however, both the mappings are discontinuous at the common fixed point. We, thus, provide one more answer to the problem of Rhoades [10]. In the last section, we apply our results to solve an eigenvalue problem. IntroductionIn 1986, Jungck [3] generalized the notion of weak commutativity by introducing the concept of compatible maps. Two selfmaps f and g of a metric space X are called compatible if lim n d(f gx n , gf x n ) = 0, whenever {x n } is a sequence in X such that lim n f x n = lim n gx n = t for some t ∈ X. It follows that the maps f and g are called noncompatible if they are not compatible. Thus f and g will be noncompatible if there exists at least one sequence {x n } such that lim n f x n = lim n gx n = t for some t ∈ X but lim n d(f gx n , gf x n ) is either non-zero or non-existent.In the study of common fixed points of compatible mappings we often require assumptions on the completeness of the space or continuity of the mappings involved, besides some contractive condition, but, the study of fixed points of noncompatible mappings can be extended to the class of nonexpansive or Lipschitz type mapping pairs even without assuming continuity of the mappings involved or completeness of the space.In 1994, Pant [6] defined the notion of R-weakly commuting mappings. Two maps A and S are called R-weakly commuting at a point x if d(ASx, SAx) ≤ Rd(Ax, Sx) for some R > 0. A and S are called pointwise 2010 Mathematics Subject Classification: 54H25.

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“…In a recent work, Aamri and Moutawakil [1] introduced the property (E.A) and thus generalized the notion of noncompatible maps.…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strict contractive conditions and common fixed point theorems with application

Pant¹

2013

Demonstratio Mathematica

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“…F pXˆXq " r0, 1 4 s Ę r0, 1 4 q Y t 1 2 u " T pXq and GpXˆXq " r0, 1 6 s Ę r0, 1 6 q Y t 1 2 u " SpXq. Also SpXq and T pXq are not closed.…”

Section: +¸mentioning

confidence: 99%

“…Since Sa " F pa, bq P F pXˆXq Ď T pXq, there exists a 1 P X such that Sa " T a 1 and Sb " F pb, aq P F pXˆXq Ď T pXq, there exists b 1 P X such that Sb " T b 1 . Now from (14), we have T a 1 " Sa " Sb " T b 1 ,…”

Section: +¸mentioning

confidence: 99%

“…In 2002, Aamri and El-Moutawakil [1] introduced the idea of the property (E.A) for a pair of self mappings defined on a metric space, which contains the classes of non-vacuously compatible as well as non-compatible mappings in metric spaces as proper subclasses and utilized the same to prove common fixed point theorems under strict contractive condition. Although the property (E.A) is a generalization of the concepts of non-compatible as well as non-vacuous compatible pairs, yet such results do require either completeness of the whole space or any one of the range subspaces or continuity conditions on the involved maps.…”

Section: Introduction With Preliminariesmentioning

confidence: 99%

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Common Fixed Point Theorems for Mappings under (CLRS)-Property in Partial Metric Spaces

Rao¹,

Imdad²,

Parvathi³

2016

Demonstratio Mathematica

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Abstract. In this paper, we introduce the concept of (CLRS)-property for mappings F : XˆX Ñ X and S : X Ñ X (wherein X stands for a partial metric space) and utilize the same to prove two common fixed point theorems for two pairs of mappings in partial metric spaces. We also furnish two examples to illustrate our main theorems. Introduction with preliminariesIn 2002, Aamri and El-Moutawakil [1] introduced the idea of the property (E.A) for a pair of self mappings defined on a metric space, which contains the classes of non-vacuously compatible as well as non-compatible mappings in metric spaces as proper subclasses and utilized the same to prove common fixed point theorems under strict contractive condition. Although the property (E.A) is a generalization of the concepts of non-compatible as well as non-vacuous compatible pairs, yet such results do require either completeness of the whole space or any one of the range subspaces or continuity conditions on the involved maps. But, quite contrary to this, the new notion of (CLR S )-property recently given by Sintunavarat and Kumam [8] does not impose such conditions. The importance of (CLR S )-property ensures that one does not require the completeness of the whole space or closedness of range subspaces.Nazir and Abbas [6] introduced the property (E.A) for a pair of self maps in partial metric spaces. In this paper, we introduce the concept of (CLR S )-property for the maps F : XˆX Ñ X and S : X Ñ X in a partial metric space and utilize the same to prove two unique common fixed point theorems for four mappings in partial metric spaces. As usual, let us denote 2010 Mathematics Subject Classification: 54H25, 47H10.

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