Arzela-Ascoli Research Articles

Invariant measures of stochastic Schrödinger delay lattice systems

In this paper, we investigate stochastic Schrödinger lattice systems with time delay, whose drift and diffusion coefficientsare locally Lipschitz continuous. Firstly, some uniform estimates of solutions areestablished which include higher-order moment estimates and uniform tail-ends estimates. Then the tightness of a family ofprobability distributions of solutions in $C([-\\rho,0];l^2)$ is proved by the Arzel${\\rm~\\grave{a}}$-Ascoli theorem and the technique of diadicdivision. Finally, the existence of invariant measures for the Markov semigroup of the system is provedby the Krylov-Bogolyubov method.

Existence of Periodic Traveling Wave Solutions for a K-P-Boussinesq Type System

In this paper, via a variational approach, we show the existence of periodic traveling waves for a Kadomtsev-Petviashvili Boussinesq type system that describes the propagation of long waves in wide channels. We show that those periodic solutions are characterized as critical points of some functional, for which the existence of critical points follows as a consequence of the Mountain Pass Theorem and Arzela-Ascoli Theorem.

Existence and Kummer Stability for a System of Nonlinear ϕ-Hilfer Fractional Differential Equations with Application

Using Krasnoselskii’s fixed point theorem and Arzela–Ascoli theorem, we investigate the existence of solutions for a system of nonlinear ϕ-Hilfer fractional differential equations. Moreover, applying an alternative fixed point theorem due to Diaz and Margolis, we prove the Kummer stability of the system on the compact domains. We also apply our main results to study the existence and Kummer stability of Lotka–Volterra’s equations that are useful to describe and characterize the dynamics of biological systems.

Long-time dynamics of fractional nonclassical diffusion equations with nonlinear colored noise and delay on unbounded domains

This paper deals with the long-time dynamics for a class of highly nonlinear fractional nonclassical diffusion equations with nonlinear colored noise and time delay defined on the whole space Rn. The existence and uniqueness of tempered pullback random attractors of the equations are established in C([−ρ,0],Hα(Rn)) (ρ>0 and α∈(0,1)) for polynomial growth drift and diffusion terms as well as Lipschitz time-delay terms. In a special case where the nonlinear diffusion term depends only on the space variable, the approximation of those random attractors is also investigated in C([−ρ,0],Hα(Rn)) when the correlation time of the colored approaches zero. The pullback asymptotical compactness of the solutions in C([−ρ,0],Hα(Rn)) is proved by virtue of the arguments of Arzela-Ascoli theorem, spectral decomposition as well as uniform tail-estimates developed by Wang (1999) [59] in order to surmount several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains as well as the weakly dissipative distinguishing structures of the equations.

Existence, uniqueness and stability of fractional impulsive functional differential inclusions

In the paper, we discuss necessary and sufficient conditions to obtain the existence, uniqueness and stability of solutions of fractional impulsive functional differential equations towards the \(\psi \)-Liouville–Caputo fractional derivative, through fixed point theorem, Arzela–Ascoli theorem and multivalued analysis theory.

Existence of Solutions for a Singular Fractional q-Differential Equations under Riemann–Liouville Integral Boundary Condition

We investigate the existence of solutions for a system of m-singular sum fractional q-differential equations in this work under some integral boundary conditions in the sense of Caputo fractional q-derivatives. By means of a fixed point Arzelá–Ascoli theorem, the existence of positive solutions is obtained. By providing examples involving graphs, tables, and algorithms, our fundamental result about the endpoint is illustrated with some given computational results. In general, symmetry and q-difference equations have a common correlation between each other. In Lie algebra, q-deformations can be constructed with the help of the symmetry concept.

Controllability of a damped nonlinear fractional order integrodifferential system with input delay

A novel nonlinear fractional order damped Pantographs equation involving input delay is formulated. We explored the controllability of the underlying nonlinear damped dynamical Pantograph equation of fractional order with input delay. The solution of the problem under consideration has initially expressed in the form of the Mittag–Leffler function in two different domains. The controllability Grammians matrices for both cases have been defined. Controllability in the linear case has been shown and applied Schaefer’s fixed point theorem as well as the Arzela-Ascoli theorem to derive necessary conditions for the controllability of the nonlinear case.

Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise

<p style='text-indent:20px;'>This paper is concerned with the pullback random attractors of nonautonomous nonlocal fractional stochastic <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian equation with delay driven by multiplicative white noise defined on bounded domain. We first prove the existence of a continuous nonautonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise. We then show pullback asymptotical compactness of solutions and the existence of tempered random attractors by utilizing the Arzela-Ascoli theorem and appropriate uniform estimates of the solutions. Finally, we establish the upper semicontinuity of the random attractors when time delay approaches zero.</p>

A case study of Covid-19 epidemic in India via new generalised Caputo type fractional derivatives.

The first symptomatic infected individuals of coronavirus (Covid‐19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid‐19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor‐Corrector scheme. For the proposed model of Covid‐19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non‐integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's‐Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non‐linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non‐linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non‐integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results.

Multistability for Almost-Periodic Solutions of Takagi–Sugeno Fuzzy Neural Networks With Nonmonotonic Discontinuous Activation Functions and Time-Varying Delays

This article investigates the problem of multistability of almost-periodic solutions of Takagi-Sugeno fuzzy neural networks with nonmonotonic discontinuous activation functions and time-varying delays. Based on the geometrical properties of nonmonotonic activation functions, by using the Ascoli-Arzela theorem and the inequality techniques, it is demonstrated that under some reasonable conditions, the addressed networks have a locally exponentially stable almost-periodic solution in some hyperrectangular regions. We also estimate the attraction basins of the locally stable almost-periodic solutions, which indicates that the attraction basins of the locally exponentially stable almost-periodic solution can be larger than original hyperrectangular regions. These results, which include boundedness, globally attractivity, multiple stability, and attraction basins, generalize and improve the earlier publications, and can be extended to monostability and multistability of Takagi-Sugeno fuzzy neural networks with nonmonotonic discontinuous activation functions. Finally, several numerical examples are given to show the feasibility, the effectiveness, and the merits of the theoretical results.

Existence of attractors for stochastic diffusion equations with fractional damping and time-varying delay

This paper deals with the well-posedness and existence of attractors of a class of stochastic diffusion equations with fractional damping and time-varying delay on unbounded domains. We first prove the well-posedness and the existence of a continuous non-autonomous cocycle for the equations and the uniform estimates of solutions and the derivative of the solution operators with respect to the time-varying delay. We then show pullback asymptotic compactness of solutions and the existence of random attractors by utilizing the Arzelà–Ascoli theorem and the uniform estimates for the derivative of the solution operator in the fractional Sobolev space Hα(Rn), with 0 < α < 1.

Existence of Solutions for Fractional Boundary Value Problem Involving p-Laplacian Operator

In this paper, we investigate the question of existence of nonnegative solution for some fractional boundary value problem involving p-Laplacian operator, The results presented in this thesis are based on fixed point theorem, more precisely, Krasnosilski fixed point theorem, on the cones to prove the existence of a fixed point for a mathematics operator and that fixed point is a solution to the given fractional equation by combining some properties of the associated Green function. We will study the problem of boundary value and find the unique solution. We will present definition and properties of Riemann-Liouville derivatives, the Guo-Krasnoselskii fixed point theorem, the Green Function to find the positive solution of problem involving p-Laplacian. The main idea is to transfer the given problem to some integral equation; from this integral equation we define an operator with Green function. We prove that this operator is completely continuous and uniformly bounded which yields by Arzela Ascoli Theorem that this operator is compact. So by Krasnosilski fixed point theorem can find the result. This type of problem is very interesting in different areas such as biophysics, astronomy and others. Note that this problem can be extended to the fractional Hadamard derivative and to the Leggett Williams fixed point theorem; moreover, it can be the content of a rigorous thesis topic.

Adaptive control for discontinuous variable-order fractional systems with disturbances

This paper deals with the global asymptotic stabilization and finite-time stabilization issues for variable-order fractional systems with partial a priori bounded disturbances by designing an appropriate adaptive controller. Via the inductive method and Arzela-Ascoli theorem, the existence and uniqueness of the solution for the considered system is firstly verified under the proposed control strategy. By applying Lyapunov stability theory, non-smooth analysis and inequality technique, sufficient stabilization criteria are established under the framework of variable-order fractional Filippov differential inclusion. Finally, two numerical simulations are given to demonstrate the validity of the proposed method.

Existence of solutions for impulsive hybrid boundary value problems to fractional differential systems

In this paper, the existence of solutions for impulsive mixed boundary value problems involving Caputo fractional derivatives is obtained. And then, our conclusions are based on Krasnoselskii's fixed point theorem and Arzela-Ascoli theorem. Finally, some examples are given to illustrate the main results.

Existence and Uniqueness of Solutions for Fractional Neutral Volterra-Fredholm Integro Differential Equations

The topic fractional calculus can be measured as an old as well as a new subject. In the fractional calculus the various integral inequalities plays an important role in the study of qualitative and quantitative properties of solution of differential and integral equations. In this paper, we study the existence and uniqueness of solutions for the neutral Caputo fractional Volterra-Fredholm integro differential equations with fractional integral boundary conditions by means of the Arzela-Ascoli's theorem, Leray-Schauder nonlinear alternative and the Krasnoselskii fixed point theorem. New conditions on the nonlinear terms are given to pledge the equivalence. An example is provided to illustrate the results.

Metrization of probabilistic metric spaces. Applications to fixed point theory and Arzela-Ascoli type theorem

This work is devoted to the metrization of probabilistic spaces. More precisely, given such a space (G,D,⋆) and provided that the triangle function ⋆ is continuous, we exhibit an explicit and canonical metric σD on G such that the associated topology is homeomorphic to the so-called strong topology. As applications, we make advantage of this explicit metric to present some fixed point theorems on such probabilistic metric structures and we prove a probabilistic version of the Arzela-Ascoli theorem.

On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

We study a family of geometric variational functionals introduced by Hamilton, and considered later by Daskalopulos, Sesum, Del Pino and Hsu, in order to understand the behavior of maximal solutions of the Ricci flow both in compact and noncompact complete Riemannian manifolds of finite volume. The case of dimension two has some peculiarities, which force us to use different ideas from the corresponding higher-dimensional case. Under some natural restrictions, we investigate sufficient and necessary conditions which allow us to show the existence of connected regions with a connected complementary set (the so-called “separating regions”). In dimension higher than two, the associated problem of minimization is reduced to an auxiliary problem for the isoperimetric profile (with the corresponding investigation of the minimizers). This is possible via an argument of compactness in geometric measure theory valid for the case of complete finite volume manifolds. Moreover, we show that the minimum of the separating variational problem is achieved by an isoperimetric region. The dimension two requires different techniques of proof. The present results develop a definitive theory, which allows us to circumvent the shortening curve flow approach of the above mentioned authors at the cost of some applications of the geometric measure theory and of the Ascoli-Arzela's Theorem.

Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay

This paper is concerned with the asymptotic behavior of the solutions to a class of non-autonomous nonlocal fractional stochastic parabolic equations with delay defined on bounded domain. We first prove 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 delay time and noise intensity. We then show 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 estimates of solutions in fractional Sobolev space \begin{document}$ H^\alpha(\mathbb{R}^n) $\end{document} with \begin{document}$ \alpha\in (0,1) $\end{document} as well as their time derivatives in \begin{document}$ L^2(\mathbb{R}^n) $\end{document} . Finally, we establish the upper semi-continuity of the random attractors when noise intensity and time delay approaches zero, respectively.

Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives.

In this manuscript, we solve a model of the novel coronavirus (COVID-19) epidemic by using Corrector-predictor scheme. For the considered system exemplifying the model of COVID-19, the solution is established within the frame of the new generalized Caputo type fractional derivative. The existence and uniqueness analysis of the given initial value problem are established by the help of some important fixed point theorems like Schauder’s second and Weissinger’s theorems. Arzela-Ascoli theorem and property of equicontinuity are also used to prove the existence of unique solution. A new analysis with the considered epidemic COVID-19 model is effectuated. Obtained results are described using figures which show the behaviour of the classes of projected model. The results show that the used scheme is highly emphatic and easy to implementation for the system of non-linear equations. The present study can confirm the applicability of the new generalized Caputo type fractional operator to mathematical epidemiology or real-world problems. The stability analysis of the projected scheme is given by the help of some important lemma or results.

THE EXISTENCE OF THE EXTREMAL SOLUTION FOR THE BOUNDARY VALUE PROBLEMS OF VARIABLE FRACTIONAL ORDER DIFFERENTIAL EQUATION WITH CAUSAL OPERATOR

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results.