Kuratowski Measure Research Articles

CASE STUDY ON TOTAL CONTROLLABILITY AND OPTIMAL CONTROL OF HILFER NEUTRAL NON-INSTANTANEOUS FRACTIONAL DERIVATIVE

The aim of the study is to discuss the controllability results of Hilfer neutral non-instantaneous impulsive fractional integro-differential equations (HNNIIFIE). Total controllability results were discussed by employing set-contraction theory and Kuratowski measure of non-compactness. Also we derived the results on optimal control using appropriate conditions. An illustration is given to validate the outcomes.

Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator

Abstract In this paper, we study the existence of solutions for the neutral evolution equations with nonlocal conditions and delay in α \alpha -norm, which are more general than in many previous publications. We assume that the linear part generates an analytic semigroup and transforms them into suitable integral equations. By using the Kuratowski measure of noncompactness and fixed-point theory, some existence theorems are established without the assumption of compactness on the associated semigroup. Particularly, our results cover the cases where the nonlinear term F F takes values in different spaces such as X α {X}_{\alpha } . An example of neutral partial differential system is also given to illustrate the feasibility of our abstract results.

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.

Investigation of the Generalized Proportional Langevin and Sturm–Liouville Fractional Differential Equations via Variable Coefficients and Antiperiodic Boundary Conditions with a Control Theory Application Arising from Complex Networks

This article studies the existence theory of an innovation type of generalized proportional fractional differential equations with the assistance of the technique of Kuratowski measure on noncompactness combined with the fixed-point theorem of Mönch. Also, we use Lebesgue’s dominated convergence theorem and Arzelá–Ascoli fixed point theorem on existence and uniqueness results. An application is also presented by employing two illustrative iks, which enrich our outcomes.

Long time behavior of a degenerate NPZ model with spatial heterogeneity

This work proposes a nutrient–phytoplankton–zooplankton (NPZ) model which is governed by a partially degenerate reaction–diffusion equation. Due to the noncompactness of solution map, the Kuratowski measure is introduced to prove the existence of the global attractor. By an innovative construction of Lyapunov functions, sufficient conditions are given to obtain the global attractivity of steady states of the system with spatial parameters.

Propagation Dynamics for Time-Periodic and Partially Degenerate Reaction-Diffusion Systems

This paper is concerned with the propagation dynamics for partially degenerate reaction-diffusion systems with monostable and time-periodic nonlinearity. In the cooperative case, we prove the existence of periodic traveling fronts and the exponential stability of continuous periodic traveling fronts. In the noncooperative case, we establish the existence of the minimal wave speed of periodic traveling waves and its coincidence with the spreading speed. More specifically, when the system is nondegenerate, the existence of periodic traveling waves is proved by using Schauder's fixed point theorem and the regularity of analytic semigroup, while in the partially degenerate case, due to the lack of compactness and standard parabolic estimates, the existence result is obtained by appealing to the asymptotic fixed point theorem with the help of some properties of the Kuratowski measure of noncompactness. It may be the first work to study periodic traveling waves of partially degenerate reaction-diffusion systems with noncooperative and time-periodic nonlinearity.

Generalized fractals in semimetric spaces

The aim of the present paper is to extend the classical fractal theory using condensing maps and generalized contractions of semimetric spaces. Our method is independent from the approach of Hutchinson: It is based on the Kuratowski measure of noncompactness and avoids completely the Blaschke Completeness Theorem.

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.

Analysis of a degenerated reaction–diffusion cholera model with spatial heterogeneity and stabilized total humans

In this paper, we perform a complete analysis on a degenerated reaction–diffusion cholera model with stabilizing total humans and non-mobility of cholera bacteria in a spatially heterogeneous bounded domain. The existence of a global attractor is established through introducing the Kuratowski measure of non-compactness. The basic reproduction number and its equivalent characterizations have been used to analyze the threshold-type results, where the persistence and extinction of cholera can also be characterized by the dispersal rate of infected humans. In the homogeneous case, the global attractivity of the unique positive equilibrium is achieved by the Lyapunov function. Moreover, we compare the basic reproduction numbers for the models without and with considering the mobility of cholera bacteria. Our results suggest that: the basic reproduction numbers attain different values as the dispersal rates of infected humans and cholera approach to infinity, while they attain the same value as the dispersal rates of infected humans and cholera approach to zero. Numerical simulations support our analytical results and discuss the impact of the dispersal rate of infected humans on the transmission of cholera.

Existence Results of Global Solutions for a Coupled Implicit Riemann-Liouville Fractional Integral Equation via the Vector Kuratowski Measure of Noncompactness

The main goal of this study is to demonstrate an existence result of a coupled implicit Riemann-Liouville fractional integral equation. First, we prove a new fixed point theorem in spaces with an extended norm structure. That theorem generalized Darbo’s theorem associated with the vector Kuratowski measure of noncompactness. Second, we employ our obtained fixed point theorem to investigate the existence of solutions to the coupled implicit fractional integral equation on the generalized Banach space C([0,1],R)×C([0,1],R).

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>

Nonlinear fractional differential inclusions with non-singular Mittag-Leffler kernel

<abstract><p>In the existing article, the existence of solutions to nonlinear fractional differential inclusions in the sense of the Atangana-Baleanu-Caputo ($ \mathcal{ABC} $) fractional derivatives in Banach space is studied. The investigation of the main results relies on the set-valued issue of Mönch fixed point theorem incorporated with the Kuratowski measure of non-compactness. A simulated example is proposed to explain the obtained results.</p></abstract>

Existence results for higher order fractional differential equations with integral boundary conditions via Kuratowski measure of noncompactnes

Existence results for higher order fractional differential equations with integral boundary conditions via Kuratowski measure of noncompactnes

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.

On countable determination of the Kuratowski measure of noncompactness

A long-standing question in the theory of measures of noncompactness is that for the Kuratowski measure of noncompactness α defined on a metric space M, and for every bounded subset B⊂M, is there a countable subset B0⊂B such that α(B0)=α(B)? In this paper, we give an affirmative answer to the question above. It is done by showing that for each nonempty set B of a Banach space, there is a countable subset B0⊂B so that B is strongly finitely representable in B0, and that there is a free ultrafilter U so that B is affinely isometric to a subset of the ultrapower [co(B0)]U of co(B0).

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.