Picard Operators Research Articles

Existence and Stability of Solutions for p-Proportional ω-Weighted κ-Hilfer Fractional Differential Inclusions in the Presence of Non-Instantaneous Impulses in Banach Spaces

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the p-proportional ω-weighted κ-Hilfer derivative includes an exponential function, Da,λσ,ρ,p,κ,ω, and then we consider a non-instantaneous impulse differential inclusion containing Da,λσ,ρ,p,κ,ω with order σ∈(1,2) and of kind ρ∈[0,1] in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the (p,ω,κ)-generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the (p,ω,κ)-generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator Da,λσ,ρ,p,κ,ω, our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings.

Global existence and stability results for a time-fractional diffusion equation with variable exponents

This paper aims to study the existence and stability results concerning a fractional partial differential equation with variable exponent source functions. The local existence result for α∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1)$$\\end{document} is established with the help of the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-resolvent kernel and the Schauder-fixed point theorem. The non-continuation theorem is proved by the fixed point technique and accordingly the global existence of solution is achieved. The uniqueness of the solution is obtained using the contraction principle and the stability results are discussed by means of Ulam-Hyers and generalized Ulam-Hyers-Rassias stability concepts via the Picard operator. Examples are provided to illustrate the results.

Hybrid fixed point theorems of graphic contractions with applications

In this work, a family of hybrid contractions, termed Jaggi-type hybrid (ℵ-τ)-contractive mapping is proposed in metric space equipped with a graph and new conditions under which the mapping is a Picard operator are studied. The novel ideas proposed in this manuscript are exemplified to display the validity of the presented results and to show how they differ from the existing ones. Additionally, some corollaries which reduce our proposed notion to some recently announced concepts in the existing findings are indicated and examined. Finally, we study Ulam-type stability for the fixed point equations with hybrid contractions.

New Results on the Ulam–Hyers–Mittag–Leffler Stability of Caputo Fractional-Order Delay Differential Equations

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work.

(α,β)-nonexpansive mappings and Picard operators

(α,β)-nonexpansive mappings and Picard operators

Stability and controllability analysis of non–linear Volterra Fredholm Hammerstein impulsive integro–dynamic systems with delay on time scale

The focus of this article is the examination of Volterra Fredholm Hammerstein type impulsive integro-dynamic systems, along with their corresponding fractional order systems, within the framework of a time scale domain that incorporates delays. The Banach Contraction Principle is employed to establish the existence and uniqueness of solutions for both types of problems. In order to address controllability in our proposed scenarios, we employ control functions and the Picard operator. We also verified the Hyers–Ulam stability using Grönwall's inequality on time scale. To demonstrate our findings, we include several illustrative examples.

Ulam Type Stability Results of Nonlinear Impulsive Volterra–Fredholm Integro-Dynamic Adjoint Equations on Time Scale

This paper is dedicated to exploring the existence, uniqueness and Ulam stability analysis applied to a specific class of mathematical equations known as nonlinear impulsive Volterra Fredholm integro-dynamic adjoint equations within finite time scale intervals. The primary aim is to establish sufficient conditions that demonstrate Ulam stability for this particular class of equations on the considered time scales. The research methodology relies on the Banach contraction principle, Picard operator and extended integral inequality applicable to piecewise continuous functions on time scales. To illustrate the applicability of the findings, an example is provided.

Neutral Caputo-Fabrizio Fractional Differential Equation with Hyers-Ulam Stability

In this paper, we will discuss the existence and uniqueness of a neutral Caputo-Fabrizio differential equation which fulfills several required conditions. Furthermore, using the Picard operator, we show that the problem has Hyers-Ulam type stability.

Fixed Points and λ-Weak Contractions

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 m1/2. 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.

The stability of nonlinear delay integro-differential equations in the sense of Hyers-Ulam

Abstract 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.

Analysis of fractional integro causal evolution impulsive systems on time scales

In this article, we study the existence and uniqueness along with Ulam‐type stability of fractional integro causal evolution impulsive systems on time scales. To establish our main results, we use Picard operator along with abstract Grönwall inequality and Gröwall lemma. We constitute some assertions to remove difficulties. At the end, the results are demonstrated via the support of an example.

Fixed Point Results of a New Family of Contractions in Metric Space Endowed with a Graph

In this note, a general class of contractive inequality, namely, admissible hybrid H − α − ϕ contraction, is proposed in metric space endowed with a graph, and new criteria for which the mapping is a Picard operator are examined. The significance of this type of contraction is connected with the possibility that its inequality can be particularized in more than one way, depending on the provided constants. Relevant examples are designed to support the assumptions of our obtained notions and to show how they are different from the known ones. A corollary which reduces our obtained result to a recently published result in the literature is pointed out and discussed.

Fixed point of Hardy-Rogers-type contractions on metric spaces with graph

<abstract><p>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.</p></abstract>

Fractional differential equations with maxima on time scale via Picard operators

In this paper, we prove a result of existence and uniqueness of solutions for the following class of problem of initial value for differential equations with maxima and Caputo?s fractional order on the time scales: c??a u(?) = ?(?, u(?), max ??[a,?] u(?)), ? ? J := [a, b]T, 0 < ? ? 1, u(a) = ?, We used the techniques of the Picard and weakly Picard operators to obtain some data dependency on the parameters results.

Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory

Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory

Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory

Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory

Ulam's stability of conformable neutral fractional differential equations

This article is concerned with the existence and uniqueness of solutions of a nonlinear neutral conformable fractional differential system with infinite delay, involving conformable fractional derivative. Additionally, we study the Ulam--Hyres stability, Ulam--Hyres--Mittag--Leffler stability, Ulam--Hyres--Mittag--Leffler--Rassias stability for the solutions of considered system using Picard operator. For application of the theory, we add an example at the end.

Existence, uniqueness and stability of semilinear nonautonomous impulsive systems on time scales

This primer article focuses on the existence, uniqueness and Ulams stability of semilinear nonautonomous dynamic problem with impulses on time scales, by utilizing fixed point method. We establish our results with the help of abstract Grönwall inequality and Picard operator. We utilize some assertions to overcome difficulties. To demonstrate our main result, we provide an example at the end.

New classes of Picard operators

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 \(x\in X\), the sequence \(\{T^nx\}_{n\in \mathbb {N}}\) converge to u. In this paper, we give new results concerning the existence of Picard operators.

Stability of Nonlinear Fractional Delay Differential Equations

This article discusses several forms of Ulam stability of nonlinear fractional delay differential equations. Our investigation is based on a generalised Gronwall’s inequality and Picard operator theory. Implementations are provided to demonstrate the stability results obtained for finite intervals.