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Cyclic (1-cyclic) operators have been studied in a long list of papers over the last decade. r-cyclic operators were introduced recently [Păcurar, M., Synchronous r-cyclic contractions on metric spaces, Fixed Point Theory, 24 (2023), No. 2, 683-700] and show interesting properties. In the present paper we introduce asynchronous r-cyclic contractions condition and study the conditions under which they are Picard operators. Their behavior is compared to the one of the synchronous r-cyclic contractions. Besides generalizing known results, our approach opens a research direction with some potential, both theoretical and practical. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss some problems of the fixed point theory for non-self operators on Rm+-metric spaces. The results complement and extend some known results given in the paper: A. Chis-Novac, R. Precup, I.A. Rus, Data dependence of fixed points for non-self generalized contractions, Fixed Point Theory, 10(2009), No. 1, 73–87. [ABSTRACT FROM AUTHOR]

Using the Hutchinson–Barnsley approach, we provide another way to proving that a Ćirić–Reich–Rus type iterated function system (IFS) has a unique attractor than that taken by Miculescu and Mihail (J Fixed Point Theory Appl 18(2), 285–296, 2016). [ABSTRACT FROM AUTHOR]

In this paper we introduce a new general multivariate fractal interpolation scheme using elements of the zipper methodology. Under the assumption that the corresponding Read-Bajraktarevic operator is well-defined, we enlarge the previous frameworks occurring in the literature, considering the constitutive functions of the iterated function system whose attractor is the graph of the interpolant to be just contractive in the last variable (so, in particular, they can be Banach contractions, Matkowski contractions, or Meir-Keeler contractions in the last variable). The main difficulty that should be overcome in this multivariate framework is the well definedness of the above mentioned operator. We provide three instances when it is guaranteed. We also display some examples that emphasize the generality of our scheme. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed bounded (not necessary convex) subset of a Banach space X and let T : C → C be an (αβ)-nonexpansive mapping with α > 0, β > 0 and α + β < 1. In this paper, we show that T has a unique fixed point. Moreover, T is a Picard operator if and only if T is asymptotically regular. [ABSTRACT FROM AUTHOR]

In this work, a new class of general contractive mappings, with the name Jaggi-Suzuki-type hybrid (K-α-ϕ)-contractive mapping is discussed in metric space equipped with a graph and new criteria for which the mapping is a Picard operator are studied. The superiority of this type of contractive mapping lies in the fact that its contractive inequality can be fixed in different ways, depending on the specified constants. Substantial illustrations are furnished to validate the axioms of our obtained ideas and to show their difference from the existing concepts. Supplementarily, some corollaries which collapse our obtained notion to recently propounded results in the literature are brought out and analysed.

In this study, we pay attention to a functional differential equation (FDE) of first order including N-variable delays. We construct new sufficient conditions in relation to the Hyers-Ulam stability (HUS) and the generalized Hyers-Ulam-Rassias stability (GHURS) of the FDE of first order including N-variable delays. By using Banach contraction principle (BCP), Picard operator and Gronwall's lemma, we confirm two new theorems with regard to the HUS and the GHURS. The results of this study are new, they extend and improve some earlier results of the HUS and the GHURS. [ABSTRACT FROM AUTHOR]

In this study, an initial-value problem for a nonlinear Volterra functional integro-differential equation on a finite interval was considered. The nonlinear term in the equation contains multiple time delays. In addition to giving some new theorems on the existence and uniqueness of solutions to the equation, the authors also prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the equation. The proofs use several different tools including Banach’s fixed point theorem, the construction of a Picard operator, and an application of Pachpatte’s inequality. An example is provided to illustrate the existence, uniqueness, and stability properties of solutions.

This paper presents a novel concept of G-Hardy-Rogers functional operators on metric spaces endowed with a graph. It investigates sufficient circumstances under which such a mapping becomes a Picard operator. As applications of the principal idea discussed herein, a few important corresponding fixed point results in ordered metric spaces and cyclic operators are pointed out and analyzed. For upcoming research papers in this field, comparative graphical illustrations are created to highlight the pre-eminence of proposed notions with respect to the existing ones.

In this paper, we introduce a new type of contractions on a metric space (X , d) in which the distance d (x , y) is replaced with a function, depending on a parameter λ , that is not symmetric in general. This function generalizes the usual case when λ = 1 / 2 and can take bigger values than m 1 / 2. We call these new types of contractions λ-weak contractions and we provide some of their properties. Moreover, we investigate cases when these contractions are Picard operators. [ABSTRACT FROM AUTHOR]

Motivated by the existence of cyclic phenomena in which some characteristics are mapped into corresponding ones over more than one phase, we introduce the r-cyclic operators with respect to a covering of a metric space and investigate their behavior. We study the convergence of the Picard iteration to a fixed point of such an operator under a Banach type generalized contraction condition. The obtained results may have interesting practical applications in various research areas. [ABSTRACT FROM AUTHOR]

Karapinar et al. established a more general class of contractions, namely, hybrid interpolative R iech I strǎstescue-type contractions, and presented some results on the platform of metric spaces. This research uses the domain of b -metric spaces to modify this class proficiently. Several interesting fixed-point results are presented by using this new class defined on b -metric space, where the symmetric condition is preserved in this study. Examples are provided for the authentication of proved results. Eventually, an application is also provided in order to comprehend our extensive effort in a better way. [ABSTRACT FROM AUTHOR]

This paper presents a novel concept of G -Hardy-Rogers functional operators on metric spaces endowed with a graph. It investigates sufficient circumstances under which such a mapping becomes a Picard operator. As applications of the principal idea discussed herein, a few important corresponding fixed point results in ordered metric spaces and cyclic operators are pointed out and analyzed. For upcoming research papers in this field, comparative graphical illustrations are created to highlight the pre-eminence of proposed notions with respect to the existing ones. [ABSTRACT FROM AUTHOR]

In this paper, we present new variants of some known fixed point theorems and new fixed point results for cyclic operators on ordered sets, on distance spaces, and on ordered distance spaces. Some results in terms of the fixed point structures are also presented. Our results include existence, uniqueness, convergence of successive approximations, well posedness of fixed point problems and the Ostrowsky property. We also identify some of the main research directions regarding cyclic operators and formulate several open problems for each of them. [ABSTRACT FROM AUTHOR]

An operator T on metric space (X, d) is called a Picard operator if T has a unique fixed point u in X and for any x ∈ X , the sequence { T n x } n ∈ N converge to u. In this paper, we give new results concerning the existence of Picard operators. [ABSTRACT FROM AUTHOR]

This paper concerns the study of a class of fractional functional differential equations (FFDEs) involving Hilfer-Hadamard fractional derivative. The existence, uniqueness, and Ulam-Hyers-Mittag-Leffler (UHLM) stability of solutions to the problem at hand are investigated. Our discussion is based upon a known fixed point theorem of Banach, Picard operator technique, and Gronwall inequality. An example is also given to show the applicability of our results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to improve some fixed point results given by I.A. Rus in (On a fixed point theorem of Maia, Studia Univ. Babes¸-Bolyai Math., 22(1977), 40-42), (Basic problem for Maia's theorem, Sem. on Fixed Point Theory, Preprint 3(1981), Babes¸-Bolyai Univ. Cluj- Napoca, 112-115) and (Data dependence of the fixed points in a set with two metrics, Fixed Point Theory, 8(2007), no. 1, 115-123), by applying, instead of contraction principle, some variants of Maia's fixed point theorem. 2010 Mathematics Subject Classification: 47H10, 47H09, 34K05, 34K12, 45D05, 45G10, [ABSTRACT FROM AUTHOR]

In this paper, we define weak θm− contraction mappings and give a new class of Picard operators for such class of mappings on a complete metric space. Also, we obtain some new results on the existence and uniqueness of attractor for a weak θm− iterated multifunction system. Moreover, we introduce (α, β, θm)− contractions using cyclic (α, β)− admissible mappings and obtain some results for such class of mappings without the continuity of the operator. We also provide an illustrative example to support the concepts and results proved herein.

In this short note, we propose a fixed point theorem in the setting of a Banach space without using a Picard operator.

The aim of this paper is to study the existence and uniqueness of solutions for some Fredholm integral equation systems by applying the vectorial form of Maia's fixed point theorem. Some abstract Gronwall lemmas and an abstract comparison lemma are also obtained. [ABSTRACT FROM AUTHOR]

Let a < c < b real numbers, (B, |·|) a (real or complex) Banach space, H ∈ C([a, b] x [a, c] x B,B), K ∈ C([a,b]² x B,B), g ∈ C([a,b],B), A : C([a,c],B) → C([a,c],B) and B : C([a,b],B) → C([a, b],B). In this paper we study the following functional integral equation, x(t) = ∫ac H(t,s,A(x)(s))ds + ∫at, K(t,s B(x)(s))ds + g(t), t ∈ [a,b]. By a new variant of fibre contraction principle (A. Petrusel, I.A. Rus, M.A. Serban, Some variants of fibre contraction principle and applications: from existence to the convergence of successive approximations, Fixed Point Theory, 22 (2021), no. 2, 795-808) we give existence, uniqueness and convergence of successive approximations results for this equation. In the case of ordered Banach space B, Gronwall-type and comparison-type results are also given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate Ulam stability of firstorder nonlinear dynamic equations on time scales. Based on the method of the Picard operator and using dynamic inequalities, we obtain four types of stability. In addition, as applications of our main result, we obtain new Ulam stability results for other nonlinear dynamic equations. An example is also provided to illustrate our main result. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of solutions and Ulam's type stabilities including the well‐known Ulam–Hyers stability and the newly extended Ulam–Hyers conformable exponential stability for two classes of fractional differential equations with the conformable fractional derivative and the time delay. The Banach contraction principle, the technique of Picard operator, the Grönwall integral inequalities, and generalized iterated integral inequality in the sense of conformable fractional integral are the main tools for deriving our main results. Finally, several illustrative examples will be presented to demonstrate our work. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove some fixed point theorems for ψF-contractions in the framework of quasi-metric spaces generalizing and improving several similar results in metric spaces. At the same time, we consider iterated function systems consisting of ψF-contractions on quasi-metric spaces, and we give some sufficient conditions for the existence and uniqueness of their attractor which is, generally, a fractal. Some illustrative examples are provided.

The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized q-contraction IFS, which is a finite collection of generalized q-contraction functions T1, . . ., TN from finite Cartesian product space X x ... x X into X, where (X, d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued q-contraction IFSs are Picard. Finally, when the map q is continuous, we show the relation between the code space and the attractor of q-contraction IFS. [ABSTRACT FROM AUTHOR]

In this paper we give conditions in which the integral equation... [ABSTRACT FROM AUTHOR]

Let (X , ·) be a Hilbert space and T : X → X be a decreasing operator. Under a metric condition involving the convex combination of x and T(x), we will prove some fixed point theorems which generalize and complement several results in the theory of nonlinear operators. Our results are closely related to the admissible perturbations approach in fixed point theory. [ABSTRACT FROM AUTHOR]

In this paper we provide another characterization of hyperbolic diameter diminishing to zero iterated function systems that were studied in [R. Miculescu, A. Mihail, Diameter diminishing to zero IFSs, arXiv:2101.12705]. The primary tool that we use is an operator HS, associated to the iterated function system S, which is inspired by the similar one utilized in Mihail (Fixed Point Theory Appl., 2015:75, 2015). Some fixed point results are also obtained as byproducts of our main result. [ABSTRACT FROM AUTHOR]

T. A. Burton presented in some examples of integral equations a notion of progressive contractions on C([a, ∞): In 2019, I. A. Rus formalized this notion (I. A. Rus, Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle, Advances in the Theory of Nonlinear Analysis and its Applications, 3 (2019) No. 3, 111-120), put ";step by step"; instead of ";progressive"; in this notion, and give some variant of step by step contraction principle in the case of operators with Volterra property on C([a, b], B) and C([a, ∞; B) where B is a Banach space. In this paper we use the abstract result given by I. A. Rus, to study some classes of functional differential equations with maxima. [ABSTRACT FROM AUTHOR]

The main aim of this article is to investigate some new sufficient conditions of the existence and uniqueness results of ψ -Hilfer fractional functional differential equations with nonlocal condition and finite delay. Further, Ulam–Hyers–Mittag-Leffler stability of solutions to a proposed problem is discussed. Krasnoselskii fixed point theorem, Picard operator method and Gronwall's inequality lemma are the main tools of our analysis. An illustrative example is provided in support of the results obtained. [ABSTRACT FROM AUTHOR]

The research article concerned with developing the qualitative theory for nonlinear fractional functional differential equations (FFDEs) of arbitrary order with infinite delay involving generalized Hilfer fractional derivative of the form: HD0+α,β;ψy(t)=f(t,yt);t∈(0,b],I0+1−γ;ψy(0+)=y0,y(t)=φ(t);t∈(−∞,0].Here 0

In this paper we study the following problems: I. Let (M, d) be a complete metric space and f, g : M → M be two operators. We suppose that:

In this paper, we introduce the notion of orthogonal relation on a soft set (F, A) and some related concepts. This notion allows us to consider fixed point theorem in SO-complete instead of complete soft metric spaces introduced by Yazar et.al. (Filomat 30:2 (2016), 269-279). Then, the existence and uniqueness of soft fixed points for a generalized soft contractive mapping are proved. Also, some examples are given to support that our main theorem is a real extension of Yazar et.al. [ABSTRACT FROM AUTHOR]

This article investigates the existence and uniqueness of solutions to the second-order Volterra integrodifferential equations with nonlocal and boundary conditions through its equivalent integral equations and fixed point of Banach. Furthermore, utilizing the Picard operator theory, we obtain the dependence of solutions on the initial nonlocal data and on functions involved on the right-hand side of the equations. [ABSTRACT FROM AUTHOR]

An open problem proposed by Rhoades is the following. Is there a contractive condition which guarantees the existence of a fixed point, but does not require the mapping to be continuous at the point? In this paper, we generalize a celebrated result of Eshaghi et al., [On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 18 (2017), 569–578], which allows us to find a new solution to this open problem. Furthermore we show that a claim of the aforementioned paper, that Banach's fixed point theorem cannot be applied in their application, is incorrect. Finally, as an application, we prove that a multivalued function satisfying a general linear functional inclusion admits a unique selection fulfilling the corresponding functional equation. [ABSTRACT FROM AUTHOR]

In this paper we introduce the concept of Hardy–Rogers type iterated function system and we prove that the fractal operator associated to such a system is a Picard operator. Some examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we achieve some fixed point results for a newly furnished wider class of contraction mappings called (p, F, G)-contraction. Such a class is defined over a metric space where the concept of w-distance is used in place of a metric. Many previous results pertaining to G-contractions, Fw-contractions, F-G-contractions in the field of metric fixed point theory are subsumed in our main result. Various examples are constructed to justify the generality of our class of contraction mapping and extended results. [ABSTRACT FROM AUTHOR]

This article is about Ulam's type stability of nth order nonlinear differential equations with fractional integrable impulses. It is a best procession to the stability of higher order fractional integrable impulsive differential equations in quasi–normed Banach space. Different Ulam's type stability results are obtained by using the definitions of Riemann–Liouville fractional integral, Hölder's inequality and the beta integral inequality. [ABSTRACT FROM AUTHOR]

In this paper, we define weak Θm- contraction mappings and give a new class of Picard operators for such class of mappings on a complete metric space. Also, we obtain some new results on the existence and uniqueness of attractor for a weak Θm- iterated multifunction system. Moreover, we introduce (α β Θm)- contractions using cyclic (α β)- admissible mappings and obtain some results for such class of mappings without the continuity of the operator. We also provide an illustrative example to support the concepts and results proved herein. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of existence and uniqueness of a solution and Ulam type stabilities for Volterra delay integro-differential equations on a finite interval. Our analysis is based on the Pachpatte's inequality and Picard operator theory. Examples are provided to illustrate the stability results obtained in the case of a finite interval. Also, we give an example to illustrate that the Volterra delay integro-differential equations are not Ulam-Hyers stable on the infinite interval. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of a class of higher order nonlinear delay differential equations with multiple bounded variable delays on a compact interval. Result of existence and uniqueness of solution is obtained by fixed point approach. Meanwhile, for the first time, integral inequality of Grönwall-Bellman-Bihari's type with delay is applied to prove the stability theorem which made our results more generalized and interesting. [ABSTRACT FROM AUTHOR]