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Karapinar et al. established a more general class of contractions, namely, hybrid interpolative Riech Istrǎstescue-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.

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

In this paper, we introduce a new contraction-type mapping and provide a fixed-point theorem which generalizes and improves some existing results in the literature. Thus, we prove that the Boyd and Wong theorem (1969) and, more recently, the fixed-point results due to Wardowski (2012), Turinici (2012), Piri and Kumam (2016), Secelean (2016), Proinov (2020), and others are consequences of our main result. An application in integral equations and some illustrative examples are indicated.

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 θ-contraction IFS, which is a finite collection of generalized θ-contraction functions T1,…,TN from finite Cartesian product space 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 θ-contraction IFSs are Picard. Finally, when the map θ is continuous, we show the relation between the code space and the attractor of θ-contraction IFS.

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: H D 0 + α , β ; ψ y ( t ) = f ( t , y t ) ; t ∈ ( 0 , b ] , I 0 + 1 − γ ; ψ y ( 0 + ) = y 0 , y ( t ) = φ ( t ) ; t ∈ ( − ∞ , 0 ] . Here 0 α 1 , 0 ≤ β ≤ 1 , y 0 ∈ R , and D 0 + α , β ; ψ , I 0 + 1 − γ ; ψ are generalized fractional operators in the concepts Hilfer and Riemann–Liouville, respectively. Some new and recent results of existence and Ulam–Hyers–Mittag-Leffler (UHML) stability of solution for the proposed problem will also be highlighted. The concerned analysis is carried out via using the Banach fixed point theorem, Picard operator method, and generalized Gronwall’s inequality. Finally, an example is given to illustrate the effectiveness of our main results.

Starting from two extensions of the Banach contraction principle due to Ćirić (1974) and Wardowski (2012), in the present paper we introduce the concepts of Ćirić type &psi, F -contraction and &psi, F -quasicontraction on a metric space and give some sufficient conditions under which the respective mappings are Picard operators. Some known fixed point results from the literature can be obtained as particular cases.

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: (a) f is a Picard operator with its unique fixed point x * f ; (b) there exists η > 0 such that d(f(x), g(x)) ≤ η, for every x ∈ M. The problem consists in estimating d(g n (x), x * f ), for x ∈ M and n ∈ 𝕅*. II. Let B be a Banach space and f, g : B → B be two operators. We suppose that f is a Picard operator. The problem is to find sufficient conditions which guarantee that f + g is a Picard operator.

[EN] 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., The authors are thankful to the learned referee for valuable suggestions. The second author is also thankful to AISTDF, DST for the research grant vide project No. CRD/2018/000017.

Following the idea of T.A. Burton, of progressive contractions, presented in some examples (T.A. Burton, \emph{A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions}, Fixed Point Theory, 20 (2019), No. 1, 107-113) and the forward step method (I.A. Rus, \emph{Abstract models of step method which imply the convergence of successive approximations}, Fixed Point Theory, 9 (2008), No. 1, 293-307), in this paper we give some variants of contraction principle in the case of operators with Volterra property. The basic ingredient in the theory of step by step contraction is $G$-contraction (I.A. Rus, \emph{Cyclic representations and fixed points}, Ann. T. Popoviciu Seminar of Functional Eq. Approxim. Convexity, 3 (2005), 171-178). The relevance of step by step contraction principle is illustrated by applications in the theory of differential and integral equations.

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.

Agratini, Octavian/0000-0002-2406-4274; Deniz, Emre/0000-0003-0546-4229 WOS: 000408222400023 The paper aims to study two classes of linear positive operators representing modifications of Picard and Gauss operators. The new operators reproduce both constants and a given exponential function. Approximation properties in polynomial weighted spaces are investigated and the speed of convergence is measured using a certain weighted modulus of smoothness. Also, the asymptotic behavior of the integral operators are established. Finally, aspects on generalized convexity are analyzed.

The concept of Picard operator is one of the most important concept of fixed point theory. As known, a self mapping T of a metric space X is called Picard operator (PO) if it has unique fixed point and every Picard iteration sequence converges to this fixed point. There some weaker forms of PO in the litareture as weakly Picard operator (WPO) and pseudo Picard operator (PPO). In this study, we present a new kind of PO as almost Picard operator (APO) and we show the differences from the others. Then we show that every continuous P-contractive self mapping of a compact metric space is APO. Also we present some open problems.

3rd International Conference of Mathematical Sciences (ICMS) -- SEP 04-08, 2019 -- Maltepe Univ, Istanbul, TURKEY WOS: 000505225800055 The concept of Picard operator is one of the most important concept of fixed point theory. As known, a self mapping T of a metric space X is called Picard operator (PO) if it has unique fixed point and every Picard iteration sequence converges to this fixed point. There are some weaker forms of PO in the literature as weakly Picard operator (WPO) and pseudo Picard operator (PPO). In this study, we present a new kind of PO as almost Picard operator (APO) and we show the differences from the others. Then we show that every continuous P-contractive self mapping of a compact metric space is APO. Also we present some open problems.

Altun, Ishak/0000-0002-7967-0554; Samet, Bessem/0000-0002-6769-3417 WOS: 000448342100007 In this paper, we present a new class of pseudo Picard operators in the setting of generalized metric spaces introduced recently in [M. JLELI AND B. SAMET: A generalized metric space and related fixed point theorems, Fixed Point Theory Appl., (2015) 2015:61]. An example is provided to illustrate the main result. Deanship of Scientific Research at King Saud UniversityDeanship of Scientific Research at King Saud University [RGP-237] The second author extends his appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No RGP-237.

In the present article, we exend the concept of $G$-contraction by introducing a $(p, G)$-contraction in which, we use $w$-distance in the contractive condition instead of a metric. Our work may be used to deduce many results in the area of fixed point theory particularly in partial ordered metric spaces where two comparable elements can be thought of to be connected by an edge of a graph. Applications and examples are produced to justify the usability of our results.

In this paper we present some of my favorite problems, all the time open, in the fixed point theory. Theseproblems are in connection with the following two: Which properties have the fixed point equations for which an iterative algorithm is convergent ? Let us have a fixed point theorem, T, and an operator f (single or multivalued) which does not satisfythe conditions in T. In which conditions the operator f has an invariant subset Y such that the restrictionof f to Y , fY , satisfies the conditions of T ?

The purpose of this paper is to extend Krasnoselskii's fixed point theorem to the case of generalized Banach spaces for singlevalued and multivalued operators. As applications, we will give some existence results for abstract system of Fredholm-Volterra type differential equations and inclusions.

The purpose of this paper is to present a fixed point theory for multivalued φ-contractions using the following concepts: fixed points, strict fixed points, periodic points, strict periodic points, multivalued Picard and weakly Picard operators; data dependence of the fixed point set, sequence of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, well-posedness of the fixed point problem, limit shadowing property of a multivalued operator, set-to-set operatorial equations and fractal operators. Our results generalize some recent theorems given in Petruşel and Rus (The theory of a metric fixed point theorem for multivalued operators, Proc. Ninth International Conference on Fixed Point Theory and its Applications, Changhua, Taiwan, July 16-22, 2009, 161-175, 2010). 2010 Mathematics Subject Classification 47H10; 54H25; 47H04; 47H14; 37C50; 37C70

Altun, Ishak/0000-0002-7967-0554; Minak, Gulhan/0000-0002-3711-9633 WOS: 000359454800003 In this paper we show that on complete metric spaces the class of weakly Picard operators contains some operators which are more general than the class of almost contractions.