Arzela-Ascoli Research Articles

Controllability of a generalized multi-pantograph system of non-integer order with state delay

<abstract><p>This paper presents the dynamical aspects of a nonlinear multi-term pantograph-type system of fractional order. Pantograph equations are special differential equations with proportional delays that are employed in many scientific disciplines. The pantograph mechanism, for instance, has been applied in numerous scientific disciplines like electrodynamics, engineering, and control theory. Because of its key rule in diverse fields, the current study establishes some necessary criteria for its controllability. The main idea of the proof is based on converting the system into a fixed point problem and introducing a suitable controllability Gramian matrix $ \mathcal{G}_{c} $. The Gramian matrix $ \mathcal{G}_{c} $ is used to demonstrate the linear system's controllability. Controllability criteria for the associated nonlinear system have been established in the sections that follow using the Schaefer fixed-point theorem and the Arzela-Ascoli theorem, as well as the controllability of the linear system and a few key assumptions. Finally, a computational example is listed.</p></abstract>

UPPER SEMICONTINUITY OF UNIFORM RANDOM ATTRACTORS FOR DELAY PARABOLIC EQUATION

This paper concentrates on the upper semicontinuity of uniform random attractors for a class of delay parabolic equations with additive noise and nonautonomous external force terms. Firstly, through the uniform estimation of the solution, it is proved that the solution of the equation has a closed uniform pullback absorbing set with respect to the symbolic space. Then, by Arzela-Ascoli theorem, we prove uniformly pullback compactness of solutions as well as the existence and uniqueness of uniform random attractors. Finally, we prove the upper semicontinuity of the uniform random attractors when time delay approaches to zero.

Stochastic controllability of a non-autonomous impulsive system with variable delays in control

In this paper, we study the sufficient conditions for the relative controllability of a nonautonomous stochastic impulsive differential equation with variable delays in control. The evolution family, Arzela-Ascoli theorem, Schauder?s fixed point theorem, and stochastic analysis are used to achieve the main results. Two examples are provided in support of the main results.

Asymptotic behavior of non-autonomous fractional stochastic <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-Laplacian equations with colored noise and delay on <inline-formula><tex-math id="M2">$ \mathbb{R}^n $</tex-math></inline-formula>

This paper deals with the asymptotic behavior of the solutions to a class of non-autonomous fractional stochastic $ p $-Laplacian equations with delay driven by nonlinear colored noise on the entire space $ \mathbb{R}^n $. We firstly considered the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the time delay. We then showed pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform tail-estimates of the solutions for large space variables when time is large enough to surmount the lack of compact Sobolev embeddings on unbounded domains.

THE OPTIMAL CONTROL PROBLEM FOR SYSTEMS OF INTEGRO-DIFFERENTIAL EQUATIONS ON THE HALF-AXIS

This article is devoted to exploring the optimal control problem for a system of integro-differential equations on the infinite interval. Sufficient conditions for the existence of optimal controls and trajectories have been obtained in terms of right-hand sides and the quality criterion function. Integro-differential equation systems are the mathematical models for many natural science processes, such as those in fluid dynamics and kinetic chemistry, among others. Many of these equations have the control that minimizing specific functionals related to the dynamics of these processes. This work specifically focuses on deriving sufficient optimality conditions for integro-differential systems on the half-axis. The complexity of the research is in the following aspects: Firstly, the problem at hand involves optimal control with an infinite horizon, which makes the direct application of compactness criteria like the Arzela-Ascoli theorem impossible. Secondly, the problem is considered up to the moment $\tau$ when the solution reaches the boundary of the domain. This reach moment depends on the control $\tau = \tau(u)$. Hence, the solution to the problem is essentially represented by the triplet $(u^*, x^*, \tau^*)$ — the optimal control, the optimal trajectory, and the optimal exit time. We note that a particular case of this problem is the problem of optimal speed. The main idea of proving the existence of an optimal solution relies on a compactness approach and involves the following steps: identifying a weakly convergent minimizing sequence of admissible controls, extracting a strongly convergent subsequence of corresponding trajectories, and justifying boundary transitions in equations and the quality criterion. The work provides a problem statement, formulates, and proves the main result.

Inverse potential problem for a semilinear generalized fractional diffusion equation with spatio-temporal dependent coefficients

In this work, we are interested in an inverse potential problem for a semilinear generalized fractional diffusion equation with a time-dependent principal part. The missing time-dependent potential is reconstructed from an additional integral measured data over the domain. Due to the nonlinearity of the equation and arising of a space-time dependent principal part operator in the model, such a nonlinear inverse problem is novel and significant. The well-posedness of the forward problem is firstly investigated by using the well known Rothe’s method. Then the existence and uniqueness of the inverse problem are obtained by employing the Arzelà–Ascoli theorem, a coerciveness of the fractional derivative and Gronwall’s inequality, as well as the regularities of the direct problem. Also, the ill-posedness of the inverse problem is proved by analyzing the properties of the forward operator. Finally a modified non-stationary iterative Tikhonov regularization method is used to find a stable approximate solution for the potential term. Numerical examples in one- and two-dimensional cases are provided to illustrate the efficiency and robustness of the proposed algorithm.

Periodic measures of impulsive stochastic neural networks lattice systems with delays

This paper is concerned with the periodic measures of a class of periodic stochastic neural networks lattice models with delays and nonlinear impulses. First, by employing the idea of uniform estimates on the tails of the solutions, the technique of diadic division, and generalized Ascoli–Arzela theorem, we prove the tightness of a family of distributions of the segment solutions of the lattice systems. Then, the existence of periodic measures is established by using the Krylov–Bogolyubov method.

Controllability and Observability Results of an Implicit Type Fractional Order Delay Dynamical System

Recently, several research articles have investigated the existence of solutions for dynamical systems with fractional order and their controllability. Nevertheless, very little attention has been given to the observability of such dynamical systems. In the present work, we explore the outcomes of controllability and observability regarding a differential system of fractional order with input delay. Laplace and inverse Laplace transforms, along with the Mittage–Leffler matrix function, are applied to the proposed dynamical system in Caputo’s sense, and a general solution is obtained in the form of an integral equation. Then, we set out conditions for the controllability of the underlying model, regarding the linear case. We then expound controllability conditions for the nonlinear case by utilizing the fixed point result of Schaefer and the Arzola–Ascoli theorem. Using the fixed point concept, we prove the observability of the linear case using the observability Grammian matrix. The necessary and sufficient conditions for the nonlinear case are investigated with the help of the Banach contraction mapping theorem. Finally, we add some examples to elaborate on our work.

Study on generalized fuzzy fractional human liver model with Atangana-Baleanu-Caputo fractional derivative.

This study aims to develop a novel fuzzy fractional model for the human liver that incorporates the ABC fractional differentiability, also known as ABC gH-differentiability, based on the generalized Hukuhara derivative. In addition, a novel fuzzy double parametric q-homotopy analysis method with a generalized transform and ABC gH-differentiability is used to deal with the fuzzy mathematical model and examine its convergence analysis. The stability of the unique equilibrium point for the fuzzy fractional human liver model and the existence of a unique solution in the proposed model are investigated using the Arzela-Ascoli theorem and Schauder's fixed-point theory. Some numerical experiments are conducted to visualize better results and test the proposed method's efficacy. The results of the q-HAShTM employing the presented approaches coincide with most of the clinical data, providing it more precise and superior to the generalized Mittag-Leffler function method.

Fractional delay integrodifferential equations of nonsingular kernels: Existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials

This paper considers linear and nonlinear fractional delay Volterra integrodifferential equation of order [Formula: see text] in the Atangana–Beleanu–Caputo (ABC) sense. We used continuous Laplace transform (CLT) to find equivalent Volterra integral equations that have been used together with the Arzela–Ascoli theorem and Schauder’s fixed point theorem to prove the local existence solution. Moreover, the obtained Volterra integral equations and the contraction mapping theorem have been successfully applied to construct and prove the global existence and uniqueness of the solution for the considered fractional delay integrodifferential equation (FDIDE). The Galerkin algorithm instituted within shifted Legendre polynomials (SLPs) is applied in the approximation procedure for the corresponding delay equation. Indeed, by this algorithm, we get algebraic system models and by solving this system we gained the approximated nodal solution. The reliability of the method and reduction in the size of the computational work give the algorithm wider applicability. Linear and nonlinear examples are included with some tables and figures to show the effectiveness of the method in comparison with the exact solutions. Finally, some valuable notes and details extracted from the presented results were presented in the last part, with the sign to some of our future works.

Hilfer fractional stochastic evolution equations on infinite interval

Abstract This paper concerns the global existence of mild solutions for a class of Hilfer fractional stochastic evolution equations on infinite interval (0, +∞), while the existing work were considered on finite interval. The main difficulties here are how to construct suitable Banach spaces, proper operator relations, and then how to formulate the new criteria to guarantee the global existence of mild solutions on the previous constructed spaces under non-Lipschitz conditions. We mainly rely on the generalized Ascoli–Arzela theorem we established, Wright function, Schauder’s fixed point principle, and Kuratowski’s measure of noncompactness to handle with the infinite interval problems. Moreover, we give two examples to demonstrate the feasibility and utility of our results.

The Dirichlet condition for the conformable fractional sub-elliptic equation on the Heisenberg group

In this paper, the following conformable fractional sub-elliptic equation is investigated: 0, & {\\rm in} \\ \\Omega,\\\\ u=0, & {\\rm on} \\ \\partial \\Omega. \\end{cases} \\] ]]> { − Δ H α u = f ( ξ , u , ∇ H α u ) , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω . Ω represents the bounded domain in space H n , α ∈ ( 0 , 1 ) , and f>0 is a bounded function. This paper is organized into three sections. In the first section, we present some of the features of the Heisenberg Laplacian. To prove the solution to the above equation, we use the Arzelà–Ascoli theorem and Bolzano–Weierstrass theorem in the second section. We then prove the uniqueness and cylindrical symmetry of the solution in the third section.

Global Mittag-Leffler Stability and Global Asymptotic ω-Period for Fractional-Order Cohen–Grossberg Neural Networks with Time-Varying Delays

The dynamic behaviors for fractional-order Cohen–Grossberg neural networks with time-varying delays (FCGNND) are studied in this paper. By introducing the Mittag-Leffler (ML) function, based on properties of fractional calculus, the differential mean-value theorem and Arzela–Ascoli theorem, we give some sufficient theorems to determine the boundedness, global Mittag-Leffler stability (GMLS) and global asymptotical [Formula: see text]-periodicity (GAP) for FCGNND. Finally, a numerical example is given to verify the effectiveness of the theorems.

Fractional crossover delay differential equations of Mittag-Leffler kernel: Existence, uniqueness, and numerical solutions using the Galerkin algorithm based on shifted Legendre polynomials

In the present work, we consider a class of fractional delay differential equations of order ρ with Atangana-Baleanu fractional derivatives in the Caputo sense. We convert our fractional delay problem to Volterra integral equation and used them to establish the local existence theorem using the Arzela-Ascoli theorem and Schauder’s fixed point theorem. After that, the contraction mapping theorem was used to prove the global existence and uniqueness theorem. For a numerical solution, we develop the Galerkin algorithm based on shifted Legendre polynomials to solve the utilized fractional delay problem. This algorithm is based on reducing such problems to those of solving a system of algebraic equations, also we state and prove the convergence and error estimate theorems. Four numerical examples, two linear and two nonlinear are presented to test the efficiency and accuracy of the proposed algorithm with some tables and figures to compare our results with the exact solutions. Several, conclusions, recommendations, inductions, and highlights were formulated in the last chapter to extinguish the perfection of the work.

New exact solutions to the Eddington-inspired-Born-Infeld equations for a relativistic perfect fluid in a Bianchi type I spacetime

We consider a Bianchi type I physical metric g, an auxiliary metric q and a density matter ρ in Eddington-inspired-Born-Infeld theory. We first derive a system of second order nonlinear ordinary differential equations. Then, by a suitable change of variables, we arrive at a system of first order nonlinear ordinary differential equations. Using both the solution-tube concept for the first order nonlinear ordinary differential equations and the nonlinear analysis tools such as the Arzelá–Ascoli theorem, we prove an existence result for the nonlinear system obtained. The resolution of this last system allows us to obtain new exact solutions for the model considered. Finally, by studying the asymptotic behaviour of the exact solutions obtained, we conclude that this solution is the counterpart of the Friedman–Lemaître–Robertson–Walker spacetime in Eddington-inspired-Born-Infeld theory.

Existence of Attractive Solutions for Hilfer Fractional Evolution Equations with Almost Sectorial Operators

The purpose of this paper is to investigate the existence of attractive solutions for a Cauchy problem of fractional evolution equations with Hilfer fractional derivative, which is a generalization of both the Riemann–Liuoville and Caputo fractional derivatives. Our methods are based on the generalized Ascoli–Arzela theorem, Schauder’s fixed point theorem, the Wright function and Kuratowski’s measure of noncompactness. The symmetric structure of the spaces and the operators defined by us plays a crucial role in showing the existence of fixed points. We obtain the global existence and attractivity results of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as noncompact.

Dynamic analysis of fractional-order neural networks with inertia

<abstract><p>The existence and the S-asymptotic $ \omega $-periodic of the solution in fractional-order Cohen-Grossberg neural networks with inertia are studied in this paper. Based on the properties of the Riemann-Liouville (R-L) fractional-order derivative and integral, the contraction mapping principle, and the Arzela-Ascoli theorem, sufficient conditions for the existence and the S-asymptotic $ \omega $-period of the system are achieved. In addition, an example is simulated to testify the theorem.</p></abstract>

Periodic Traveling Wave Solutions of Time-Periodic SIR Epidemic Models With External Supplies

The existence and non-existence of periodic traveling wave solutions of a class of time-periodic SIR epidemic models with external supplies are considered. Firstly, by constructing appropriate upper and lower solutions of the auxiliary system and defining a closed convex cone, the existence of periodic traveling waves is transformed into a fixed point problem of the non-monotone operator defined on the closed convex cone. The existence of periodic solutions for the auxiliary system is established by Schauder fixed point theorem, and Arzela-Ascoli theorem is used to prove the existence of periodic traveling waves for the original model. Secondly, the non-existence of periodic traveling waves is obtained by analysis techniques.

A new result for fractional differential equation with nonlocal initial value using Caputo-Fabrizio derivative

In this paper, using Caputo-Fabrizio fractional derivative, we obtain some new results for the existence and uniqueness of solutions of differential equation with nonlocal initial value about fractional order 0 < ? < 1 . These results are applied with the help of Arzela-Ascoli theorem and Schauder fixed point. Also based on some ?-contractive maps for such problems, some new unique theorem has been introduced and proved. Finally, some illustrative example is considered to show the effectiveness of the results.

A fractional-order mathematical model for analyzing the pandemic trend of COVID-19.

Many countries worldwide have been affected by the outbreak of the novel coronavirus (COVID‐19) that was first reported in China. To understand and forecast the transmission dynamics of this disease, fractional‐order derivative‐based modeling can be beneficial. We propose in this paper a fractional‐order mathematical model to examine the COVID‐19 disease outbreak. This model outlines the multiple mechanisms of transmission within the dynamics of infection. The basic reproduction number and the equilibrium points are calculated from the model to assess the transmissibility of the COVID‐19. Sensitivity analysis is discussed to explain the significance of the epidemic parameters. The existence and uniqueness of the solution to the proposed model have been proven using the fixed‐point theorem and by helping the Arzela–Ascoli theorem. Using the predictor–corrector algorithm, we approximated the solution of the proposed model. The results obtained are represented by using figures that illustrate the behavior of the predicted model classes. Finally, the study of the stability of the numerical method is carried out using some results and primary lemmas.