Kuratowski Measure Of Noncompactness Research Articles

Caputo-Fabrizio type fractional differential equations with non-instantaneous impulses: Existence and stability results

In the current article, we establish the existence and uniqueness of solutions to the initial value problem for nonlinear implicit fractional differential equations with non-instantaneous impulses, including the Caputo-Fabrizio fractional derivative. This derivative encapsulates non-local and memory-dependent effects in the system's behavior. The problem's complexity arises from its implicit nature, where the relationships between variables are not directly specified. This inherent intricacy is further compounded by the introduction of non-instantaneous impulses, signifying abrupt, intermittent changes within the system at specific junctures, which in turn trigger sudden transitions. At its core, the initial value problem is concerned with deciphering a system's behavior based on its initial conditions, necessitating a solution that adheres to the given equation while considering the presence of non-instantaneous impulses. To derive the existence results, the study leverages established mathematical tools, specifically, the standard Banach's fixed point theorem and Darbo's fixed point theorem connected to Kuratowski's measure of noncompactness (KMN). Furthermore, the Hyers-Ulam stability (HU-stability) of the given problem is discussed, which, in turn, enhances the utility and reliability of these solutions in practical applications. In order to demonstrate the applicability of the discovered results for various values of δ, we construct some examples towards the end. The practical implications of these results extend to a wide range of fields, offering improved modeling, control, and understanding of complex systems.

The Existence of Mild Solutions for Hilfer Fractional Stochastic Evolution Equations with Order μ∈(1,2)

In this study, we investigate the existence of mild solutions for a class of Hilfer fractional stochastic evolution equations with order μ∈(1,2) and type ν∈[0,1]. We prove the existence of mild solutions of Hilfer fractional stochastic evolution equations when the semigroup is compact as well as noncompact. Our approach is based on the Schauder fixed point theorem, the Ascoli–Arzelà theorem and the Kuratowski measure of noncompactness. An example is also provided, to demonstrate the efficacy of this method.

Controllability for Fractional Evolution Equations with Infinite Time-Delay and Non-Local Conditions in Compact and Noncompact Cases

The goal of this dissertation is to explore a system of fractional evolution equations with infinitesimal generator operators and an infinite time delay with non-local conditions. It turns out that there are two ways to regulate the solution. To demonstrate the presence of the controllability of mild solutions, it is usual practice to apply Krasnoselskii’s theorem in the compactness case and the Sadvskii and Kuratowski measure of noncompactness. A fractional Caputo approach of order between 1 and 2 was used to construct our model. The families of linear operators cosine and sine, which are strongly continuous and uniformly bounded, are used to achieve the mild solution. To make our results seem to be applicable, a numerical example is provided.

Boundary-Value Problem for Nonlinear Fractional Differential Equations of Variable Order with Finite Delay via Kuratowski Measure of Noncompactness

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. All of the results in this study are established with the help of generalized intervals and piecewise constant functions. We convert the Riemann–Liouville fractional variable-order problem to equivalent standard Riemann–Liouville problems of fractional-constant orders. Finally, two examples are constructed to illustrate the validity of the observed results.

Controllability of fractional measure evolution systems with state-dependent delay and nonlocal condition

<p style='text-indent:20px;'>This paper is concerned with the existence of mild solutions and exact controllability for a class of fractional measure evolution systems with state-dependent delay and nonlocal conditions. We first establish an existence result of mild solutions for the concerned problem by applying an integral equation which is given in terms of probability density and semigroup theory. Then, the exact controllability are obtained by using the fractional calculus theory, Kuratowski measure of noncompactness and Mönch fixed point theorem, without imposing the Lipschitz continuity on nonlinear term. Finally, we give two applications to support the validity of the study.</p>

Levitin-Polyak well-posedness for vector optimization problems in linear spaces

In this paper, we deal with vector optimization problems where the images of the objective maps are given in linear spaces not endowed with any topological structure. Firstly, we propose new concepts of semicontinuity of vector-valued maps acting from metric spaces into linear ones and discuss their properties and characterizations. Next, using Zorn's Lemma and these properties, we formulate existence conditions for such problems. Then, we introduce two notions of Levitin-Polyak well-posedness for the reference problems, and study sufficient conditions for their fulfillment, as well as relationships of these notions. Finally, using the Kuratowski measure of noncompactness, several metric characterizations of such well-posedness properties for vector optimization problems are investigated. Many examples are given to analyze our results.

Generalized well-posedness for parametric fuzzy generalized multiobjective games

The purpose of this article is to study well-posedness for parametric fuzzy generalized multiobjective games. First, we introduce the concepts of the well-posedness and well-posedness in the generalized sense for parametric fuzzy generalized multiobjective games. Then, under suitable conditions, we establish some sufficient conditions for the well-posedness and the well-posedness in the generalized sense for such problem. Further, results on metric characterization of the well-posedness and the well-posedness in the generalized sense for parametric fuzzy generalized multiobjective games in terms of the behavior of the approximate solution sets are also obtained. Finally, we state results on the well-posedness in the generalized sense for the parametric fuzzy generalized multiobjective games by using the Kuratowski measure of noncompactness. Examples are given to illustrate our results. The results presented in the paper are new.

Cauchy problems for Hilfer fractional evolution equations on an infinite interval

It is well known that the classical Ascoli‐Arzelà theorem is powerful technique to give a necessary and sufficient condition for investigating the relative compactness of a family of abstract continuous functions, while it is limited to finite compact interval. In this paper, we shall generalize the Ascoli‐Arzelà theorem on an infinite interval. As its application, we investigate an initial value problem for fractional evolution equations on infinite interval in the sense of Hilfer type, which is a generalization of both Riemann‐Liuoville and Caputo fractional derivatives. Our methods are based on the Hausdorff theorem, classical/generalized Ascoli‐Arzelà theorem, Schauder fixed point theorem, Wright function, and Kuratowski measure of noncompactness. We obtain the existence of mild solutions on an infinite interval when the semigroup is compact as well as noncompact.

On boundary value problems of Caputo fractional differential equation of variable order via Kuratowski MNC technique

In this manuscript, we examine both the existence and the stability of solutions to the boundary value problem of Caputo fractional differential equations of variable order by converting it into an equivalent standard Caputo boundary value problem of the fractional constant order with the help of the generalized intervals and the piece-wise constant functions. All results in this study are established using Darbo’s fixed point theorem combined with the Kuratowski measure of noncompactness. Further, the Ulam–Hyers stability of the given problem is examined; and finally, we construct an example to illustrate the validity of the observed results.

Nonlinear impulsive problems for uncertain fractional differential equations

This paper deals with the impulsive problem for uncertain fractional dynamical system. Firstly, the concept of uncertain fractional impulsive problem involving the Caputo derivative is introduced and the analytic solutions to several linear uncertain fractional impulsive problems are derived with the help of the Mittag-Leffler functions. Then the existence and uniqueness theorems are developed via the Kuratowski measure of noncompactness and fixed point theorems, respectively. Finally, an illustrative example is provided to explain our main results.

Existence results of mild solutions for nonlocal fractional delay integro-differential evolution equations via Caputo conformable fractional derivative

<abstract><p>In this paper, we investigate the existence of mild solutions for nonlocal delay fractional Cauchy problem with Caputo conformable derivative in Banach spaces. We establish a representation of a mild solution using a fractional Laplace transform. The existence of such solutions is proved under certain conditions, using the Mönch fixed point theorem and a general version of Gronwall's inequality under weaker conditions in the sense of Kuratowski measure of non compactness. Applications illustrating our main abstract results and showing the applicability of the presented theory are also given.</p></abstract>

Kuratowski MNC method on a generalized fractional Caputo Sturm\u2013Liouville\u2013Langevin q-difference problem with generalized Ulam\u2013Hyers stability

In this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch’s fixed point theorem along with a theoretical method by terms of Kuratowski measure of noncompactness (MNC) and the Banach contraction principle (BCP). Furthermore, two dynamical notions of Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability are addressed for solutions of the supposed Sturm–Liouville–Langevin quantum boundary value problem (q-FBVP). Two examples are presented to show the validity and also the effectiveness of theoretical results. In the last part of the paper, we conclude our exposition with some final remarks and observations.

Boundary value problem for nonlinear fractional differential equations of variable order via Kuratowski MNC technique

In the present research study, for a given multiterm boundary value problem (BVP) involving the Riemann-Liouville fractional differential equation of variable order, the existence properties are analyzed. To achieve this aim, we firstly investigate some specifications of this kind of variable-order operators, and then we derive the required criteria to confirm the existence of solution and study the stability of the obtained solution in the sense of Ulam-Hyers-Rassias (UHR). All results in this study are established with the help of the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of our observed results.

Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order

A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads to the necessity of obtaining existence criteria for a boundary value problem for Hadamard fractional differential equations of variable order. Also, the stability in the sense of Ulam–Hyers–Rassias is investigated. The results are obtained based on the Kuratowski measure of noncompactness. An example illustrates the validity of the observed results.

On the Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order via Kuratowski MNC Technique

In this manuscript, we examine both the existence and the stability of solutions of the boundary value problems of Hadamard-type fractional differential equations of variable order. New outcomes are obtained in this paper based on the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of the observed results.

Investigation of the neutral fractional differential inclusions of Katugampola-type involving both retarded and advanced arguments via Kuratowski MNC technique

A class of the boundary value problem is investigated in this research work to prove the existence of solutions for the neutral fractional differential inclusions of Katugampola fractional derivative which involves retarded and advanced arguments. New results are obtained in this paper based on the Kuratowski measure of noncompactness for the suggested inclusion neutral system for the first time. On the one hand, this research concerns the set-valued analogue of Mönch fixed point theorem combined with the measure of noncompactness technique in which the right-hand side is convex valued. On the other hand, the nonconvex case is discussed via Covitz and Nadler fixed point theorem. An illustrative example is provided to apply and validate our obtained results.

Controllability for nonlocal stochastic integrodifferential evolution equations with the lack of compactness

In this work, we study the controllability of a class of stochastic integrodifferential equations with noncompact semigroups and nonlocal condition on Hilbert spaces.We suppose that the linear part admits a resolvent operator in the sense of Grimmer. Our main results are obtained by utilizing stochastic analysis theory, Kuratowski measure of noncompactness and Mönch fixed point Theorem. As a result, we obtain a generalization of a host of important results in the literature, without assuming the compactness of the resolvent operator. Finally, we provide an example to illustrate the efficiency of our results.

Successive approximations for a differential equation in a Banach space via Constantin condition

Abstract We give some sufficient conditions for the convergence of the sequence of successive approximations to the unique solution of first order Cauchy problem in a Banach space. Our approach is based on a generalized Nagumo condition due to A. Constantin and the properties of the Kuratowski measure of noncompactness.

Well-Posedness for Set Optimization Problems Involving Set Order Relations

In this paper, we investigate set optimization problems with three types of set order relations. Various kinds of well-posedness for these problems and their relationship are concerned. Then, sufficient conditions for set optimization problems to be well-posed are established. Moreover, Kuratowski measure of noncompactness is applied to survey characterizations of well-posedness for set optimization problems. Furthermore, approximating solution maps and their stability are researched to propose the link between stability of the approximating problem and well-posedness of the set optimization problem.

Second Order Semilinear Volterra-Type Integro-Differential Equations with Non-Instantaneous Impulses

We consider a non-instantaneous system represented by a second order nonlinear differential equation in a Banach space E. We use the family of linear bounded operators introduced by Kozak, Darbo fixed point method and Kuratowski measure of noncompactness. A new set of sufficient conditions is formulated which guarantees the existence of the solution of the non-instantaneous system. An example is also discussed to illustrate the efficiency of the obtained results.