Some PPF Dependent Fixed Point Theorems for Generalized α-F-Contractions in Banach Spaces and Applications

Cho, Yeol Je; Kang, Shin Min; Salimi, Peyman

doi:10.3390/math6110267

Cited by 2 publications

(2 citation statements)

References 31 publications

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“…As a consequence, this theory attracted a lot of contributors in the area; see, for instance, Dhage [7,8], Hussain et al [10], Kaewcharoen [12], Kutbi and Sintunavarat [15], as well as the references therein. However, as proved in a recent paper by Cho, Rassias, Salimi and Turinici [4], the starting conditions [imposed by the problem setting] relative to the ambient Razumikhin class R c may be converted into starting conditions relative to the constant class K; so, ultimately, we may arrange for these PPF dependent fixed point results holding over K. In this exposition we bring the discussion a step further, by establishing that Fact-1) the algebraic closeness assumption [used in all these references] imposed upon the Razumikhin class R c yields, in a direct way, R c = K Fact-2) the PPF dependent fixed point problem attached to the constant class K is reducible to a (standard) fixed point problem in the (complete) metrical structure induced by our initial Banach one, under no regularity assumption about the ambient Razumikhin class.…”

Section: Introductionmentioning

confidence: 85%

PPF Dependent Fixed Points in Razumikhin Metrical Chains

Turinici

2019

Mathematical Analysis and Applications

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The PPF dependent fixed point result in algebraically closed Razumikhin classes due to Agarwal et al [Fixed Point Theory Appl., 2013, 2013 is identical with its constant class counterpart; and this, in turn, is reducible to a fixed point principle involving SVV type contractions (over the subsequent metric space of initial Banach structure), without any regularity conditions about the Razumikhin classes. The conclusion remains valid for all PPF dependent fixed point results founded on such global conditions. 2010 Mathematics Subject Classification. 47H10 (Primary), 54H25 (Secondary).

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Section: Introductionmentioning

confidence: 85%

PPF Dependent Fixed Points in Razumikhin Metrical Chains

Turinici

2019

Mathematical Analysis and Applications

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“…This idea seemed to be a very useful and powerful method in the study of functional and integral equations (see [17]). We refer the reader to, for example [18][19][20][21][22][23][24], and references therein for more information on different aspects of fixed point theorems via F-contractions. Theorem 1 ([10]).…”

Section: Introductionmentioning

confidence: 99%

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

Hammad

Sen

2019

Mathematics

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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.

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Some PPF Dependent Fixed Point Theorems for Generalized α-F-Contractions in Banach Spaces and Applications

Cited by 2 publications

References 31 publications

PPF Dependent Fixed Points in Razumikhin Metrical Chains

PPF Dependent Fixed Points in Razumikhin Metrical Chains

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

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