Coupled common fixed point results involving a "Equation missing" -contractive condition for mixed g-monotone operators in partially ordered metric spaces

Jain, Manish; Taş, Kenan; Kumar, Sanjay; Gupta, Namita

doi:10.1186/1029-242x-2012-285

Cited by 11 publications

(13 citation statements)

References 8 publications

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“…Lakshmikantham andĆirić [20] proved coupled coincidence and common coupled fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces and extended the results established in [8]. Many authors focused and proved related remarkable results including ( [3], [6], [10], [14], [16], [17], [18], [23], [25], [30], [37]). …”

Section: Let (Xmentioning

confidence: 86%

Common Coupled Fixed Point Theorems for Two Hybrid Pairs of Mappings Satisfying an Implicit Relation

Deshpande¹,

Handa²

2015

SJM

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Abstract. We establish two common coupled fixed point theorems for two hybrid pairs of mappings satisfying an implicit relation under weak commutativity and w−compatibility on a complete metric space, which is not partially ordered. We do not use the condition of continuity of any mapping for finding the coupled coincidence and common coupled fixed point. We improve, extend and generalize several known results.

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Section: Let (Xmentioning

confidence: 86%

Common Coupled Fixed Point Theorems for Two Hybrid Pairs of Mappings Satisfying an Implicit Relation

Deshpande¹,

Handa²

2015

SJM

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“…Let ( , ≤) be a partially ordered set and : × → . The mapping is said to have the mixed monotone property if ( , ) is monotone non-decreasing in and monotone non-increasing in ; that is, for any , ∈ , Jain et al [38], extended Berinde's contraction (1.3) for a pair of compatible mappings and obtained coupled coincidence points under the following contraction: +∞ for all > 0, where is the nth iterate of .…”

Section: Definition 12 ([19]mentioning

confidence: 99%

Coupled Fixed Point Results For Mappings Involving (α,ψ) - Weak Contractions In Ordered Metric Spaces And Applications

Jain¹,

Gupta²,

Kumar³

2014

J. Math. Computer Sci.

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In this paper we introduce the notion of , -weak contractions and use the notion to establish the existence and uniqueness of coupled common fixed points for the mixed monotone operators in partially ordered metric spaces. The obtained results extend, improve, complement and unify many recent coupled fixed point results present in the literature. The theoretic results are accompanied with suitable examples. An application to the existence and uniqueness of the solution of the system of integral equations is also presented. 19] proved the following main result. Theorem 1.2 ([19]). Let , ≤ be a partially ordered set and be a metric on such that ( , ) is a complete metric space. Let : × → be a continuous mapping having the mixed monotone property on . Assume that there exists a in [0,1) withthen there exists , in such that = ( , ) and ( , ).In [19, Theorem 2.2], it has also been shown that the continuity assumption of in Theorem 1.2 can be replaced by an alternative condition imposed on convergent non-decreasing and nonincreasing sequences. Assumption 1.1. has the property that (i) if a non-decreasing sequence ⊂ converges to , then ≤ for all , (ii) if a non-increasing sequence ⊂ converges to , then ≥ for all . M. Jain, N. Gupta, S. Kumar / J. Math. Computer Sci. ( ), -25 Berinde [20, Theorem 3], in an alternative way generalized the results of Bhaskar and Lakshmikantham [19] under a weaker contraction given below: , , , + , , , ≤ ( , ) + , , (1.3)with ≥ and ≤ , where ∈ [0,1).On the other hand, Lakshmikantham and C iric [21] extended the notion of mixed monotone property to mixed -monotone property and generalized the results proved in [19] by establishing the existence of coupled coincidence point results using a pair of commutative mappings. This generated new trends in coupled fixed point theory with applications to partially ordered sets. Since then much work has been done in this direction by different authors. For more details the reader may consult ([22-38]). Definition 1. 3 ([21]). Let ( , ≤) be a partially ordered set and : × → and : × → . We say has the mixed -monotone property if is monotone -nondecreasing in its first argument and is monotone -nonincreasing in its second argument; that is, for any

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“…Recently, there are several coupled fixed point results for single valued mappings established. Some instances of these works are in [6,[9][10][11]14,15,25,34,36].…”

mentioning

confidence: 99%

Common coupled fixed point theorems for hybrid pair of mappings satisfying an implicit relation with application

Deshpande

Handa

2015

Afr. Mat.

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We establish some common coupled fixed point theorems satisfying an implicit relation under newly defined weaker conditions for hybrid pair F : X × X → 2 X and g : X → X on a noncomplete metric space, which is not partially ordered. An example supporting to our results has also been cited. As an application the solution of integral equations are obtain here to illustrate the usability of the obtained results. We improve, extend and generalize several known results.

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Coupled common fixed point results involving a "Equation missing" -contractive condition for mixed g-monotone operators in partially ordered metric spaces

Cited by 11 publications

References 8 publications

Common Coupled Fixed Point Theorems for Two Hybrid Pairs of Mappings Satisfying an Implicit Relation

Common Coupled Fixed Point Theorems for Two Hybrid Pairs of Mappings Satisfying an Implicit Relation

Coupled Fixed Point Results For Mappings Involving (α,ψ) - Weak Contractions In Ordered Metric Spaces And Applications

Common coupled fixed point theorems for hybrid pair of mappings satisfying an implicit relation with application

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