Fixed points of an F-contraction on metric spaces with a graph

Batra, Rakesh; Vashistha, Sachin

doi:10.1080/00207160.2014.887700

Cited by 34 publications

(22 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let (E • , . E ) be a Banach space and (S, T) be a pair of new multi-valued generalized F-contractions (12). Assume that c is algebraically closed with respect to the difference.…”

Section: Ppf-dependent Coincidence Pointmentioning

confidence: 99%

“…Define the function F : R + → R by F(α) = ln(α) for all α ∈ R + , τ > 0. By Condition (12) with H E (Tζ, Sξ) > 0, we have…”

Section: Ppf-dependent Coincidence Pointmentioning

confidence: 99%

“…After that, a generalization of the notion of F-contraction to obtain certain fixed point results was given by Abbas et al [11]. Batra et al [12,13] provided a remarkable generalization of F-contraction on graphs and altered distances. Recently, some fixed point results for Hardy-Rogers-type self mappings on abstract spaces have been discussed by Cosentino and Vetro [14].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

PPF-Dependent Fixed Point Results for New Multi-Valued Generalized F-Contraction in the Razumikhin Class with an Application

Hammad

Sen

2019

Mathematics

View full text Add to dashboard Cite

In this paper, a new multi-valued generalized F-contraction mapping is given. Using it, the existence of PPF-dependent fixed point for such mappings in the Razumikhin class is obtained. Moreover, an application for nonlinear integral equations with delay is presented here to illustrate the usability of the obtained results.Definition 1 ([10]). A nonlinear self-mapping T on a metric space (X, d) is said to be an F-contraction, if there exist F ∈ Γ and τ ∈ (0, +∞) such that d(Tx, Ty) > 0 ⇒ τ + F(d(Tx, Ty)) ≤ F(d(x, y)) ∀x, y ∈ X.(1)where Γ is the set of functions F : (0, +∞) → R such that the following axioms hold: (F 1 ) F is strictly increasing, i.e., for all a, b ∈ R + such that a < b, F(a) < F(b); (F 2 ) for every sequence {a n } n∈N of positive numbers lim n→∞ a n = 0 iff lim n→∞ F(a n ) = −∞;The following functions F i : (0, +∞) −→ R for i ∈ {1, 2, 3, 4}, are all the elements of Γ. Furthermore, substituting in Condition (1) these functions, we obtain the following contractions known in the literature, for all x, y ∈ X with α > 0 and Tx = Ty,From the axiom (F 1 ) and Condition (1), one can conclude that every F-contraction T is a contractive mapping and hence automatically continuous.Hence, T is a contraction with a constant 1 2 . Let ξ(t) = t 2 + 1 for all t ∈ [0, 1]. Since T(ξ) = 1 2 sup t∈[0,1] |ξ(t)| = 1 = ξ(0), we have: ξ is a PPF fixed point with dependence of T. Definition 4 ([1]). Let T, S : E • → E be two operators. A point ξ ∈ E • is called a PPF-dependent common fixed point or a common fixed point with PPF-dependentence of T and S if T(ξ) = S(ξ) = ξ(c) for some c ∈ I.Clearly, if we take T = S, then a PPF-dependent common fixed point of T and S collapses to a PPF-dependent fixed point.Definition 5 ([26]). Let P : E • → E and Q : E • → E • . A point ξ ∈ E • is called a PPF-dependent coincidence point or coincidence point with PPF-dependentence of P and Q if P(ξ) = Q(ξ)(c) for some c ∈ I.Let CB(E) be a collection of all non-empty closed bounded subsets of E, and H be the Hausdorff metric determined by . E . Then, for all G, V ∈ CB(E),In 1989, Mizoguchi and Takahashi [27] extended Banach fixed point theorem in a complete metric space. After that, Farajzadeh et al. [28] extended the above results by introducing the following definitions:Please note that if S : E 0 → E is a single-valued mapping, then a multivalued mapping T : E • → CB(E) can be obtained by T(ξ) = {S(ξ)}, for all ξ ∈ E • . Hence, the set of PPF-dependent fixed points of S coincides with the set of PPF-dependent fixed point of T.

show abstract

“…Let (E • , . E ) be a Banach space and (S, T) be a pair of new multi-valued generalized F-contractions (12). Assume that c is algebraically closed with respect to the difference.…”

Section: Ppf-dependent Coincidence Pointmentioning

confidence: 99%

“…Define the function F : R + → R by F(α) = ln(α) for all α ∈ R + , τ > 0. By Condition (12) with H E (Tζ, Sξ) > 0, we have…”

Section: Ppf-dependent Coincidence Pointmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

PPF-Dependent Fixed Point Results for New Multi-Valued Generalized F-Contraction in the Razumikhin Class with an Application

Hammad

Sen

2019

Mathematics

View full text Add to dashboard Cite

show abstract

“…Recently, Wardowski [2] gave a new generalization of Banach contraction to show the existence of the fixed point for such contraction by a more simple method of proof than the Banach's one. After that, several authors studied different variations of Wardowski contraction for single-valued and multivalued mappings-for example, see [3][4][5][6][7][8]. On the other hand, Aydi et al [9] studied a common fixed point for generalized multi-valued contractions.…”

Section: Introductionmentioning

confidence: 99%

Common Fixed Point Results for Generalized Wardowski Type Contractive Multi-Valued Mappings

et al. 2019

View full text Add to dashboard Cite

show abstract

“…Shukla et al [34] obtained some common fixed point results for F -contraction type mappings in the framework of 0-complete partial metric spaces. Batra et al [13] proved fixed point theorems for F -contraction on a metric space endowed with a graph.…”

Section: Introductionmentioning

confidence: 99%

Coincidence best proximity point of Fg -weak contractive mappings in partially ordered metric spaces

Latif¹,

Abbas²,

Hussain³

2016

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

show abstract

Fixed points of an F-contraction on metric spaces with a graph

Cited by 34 publications

References 6 publications

PPF-Dependent Fixed Point Results for New Multi-Valued Generalized F-Contraction in the Razumikhin Class with an Application

PPF-Dependent Fixed Point Results for New Multi-Valued Generalized F-Contraction in the Razumikhin Class with an Application

Common Fixed Point Results for Generalized Wardowski Type Contractive Multi-Valued Mappings

Coincidence best proximity point of Fg -weak contractive mappings in partially ordered metric spaces

Contact Info

Product

Resources

About