T. Caraballo

T. Caraballo
Universidad de Sevilla | US · Differential Equations and Numerical Analysis

PhD

About

343
Publications
30,405
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7,099
Citations
Citations since 2017
151 Research Items
3986 Citations
20172018201920202021202220230200400600800
20172018201920202021202220230200400600800
20172018201920202021202220230200400600800
20172018201920202021202220230200400600800
Additional affiliations
October 1984 - present
Universidad de Sevilla
Position
  • Professor (Full)

Publications

Publications (343)
Article
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Two mathematical models of chemotherapy cancer treatment are studied and compared, one modeling the chemotherapy agent as the predator and the other modeling the chemotherapy agent as the prey. In both models constant delay parameters are introduced to incorporate the time lapsed from the instant the chemotherapy agent is injected to the moment it...
Article
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This paper stands for the almost sure practical stability of nonlinear stochastic differential delay equations (SDDEs) with a general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functionals. Furthermore, we provide some numerical examples to validate the effectiveness of the abstract resul...
Article
In this work, we investigate stochastic fractional diffusion equations with Caputo–Fabrizio fractional derivatives and multiplicative noise, involving finite and infinite delays. Initially, the existence and uniqueness of mild solution in the spaces C p ( [ − a , b ] ; L q ( Ω , H ˙ r ) ) ) and C δ ( ( − ∞ , b ] ; L q ( Ω , H ˙ r ) ) ) are establis...
Article
In this paper we investigate four non-autonomous chemostat models with non-monotonic consumption function, where wall growth and nutrient recycling are also taken into account. In each case, we prove the existence and uniqueness of non-negative global solution that generates a non-autonomous dynamical system. In addition, we also prove the existenc...
Article
In this paper, two problems related to FitzHugh–Nagumo lattice systems are analyzed. The first one is concerned with the asymptotic behavior of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise. We obtain a new result ensuring that such a system approximates the corresponding deterministic system when the correlati...
Article
In this paper we investigate stochastic dynamics and invariant measures for stochastic 3D Lagrangian-averaged Navier–Stokes (LANS) equations driven by infinite delay and additive noise. We first use Galerkin approximations, a priori estimates and the standard Gronwall lemma to show the well-posedness for the corresponding random equation, whose sol...
Article
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This paper investigates the dynamics of a class of three-dimensional globally modified Navier–Stokes equations with double delay in the forcing and convective terms. We first prove the well-posedness of solutions of such system, which enables us to establish suitable non-autonomous dynamical systems. We then show the existence and uniqueness of pul...
Article
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In this article, we investigate the existence and uniqueness of solution of controlled hybrid neutral stochastic differential equations with infinite delay (HNSFDEswID). It is known that the time lag generated by the controller in each discrete observation must be different. The controlled HNSFDEswID are affected by the variable delay induced by th...
Article
The Lyapunov approach is one of the most effective and efficient methods for the investigation of the stability of stochastic systems. Several authors analyzed the stability and stabilization of stochastic differential equations via Lyapunov techniques. Nevertheless, few results are concerned with the stability of stochastic systems based on the kn...
Article
Stability of nonlinear delay evolution equation with stochastic perturbations is considered. It is shown that if the level of stochastic perturbations fades on the infinity, for instance, if it is given by square integrable function, then an exponentially stable deterministic system remains to be exponentially stable (in mean square). Applications...
Article
In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from ones used in the previous published literature on the long time behavior of heat equations with...
Preprint
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In this paper, we discuss a class of nonlinear Schr\"odinger equations with the power-type nonlinearity: $(\mathrm{i} \frac{\partial}{\partial t} + \Delta ) \psi = \lambda |\psi|^{2\eta}\psi$ in $\mathbf R^N \times \mathbf R^+$. Based on the Gagliardo-Nirenberg interpolation inequality, we prove the local existence and long-time behavior (continuat...
Article
In this work, we study the continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded random perturbations of autonomous semilinear problems. First, we study the existence and permanence of unstable sets of hyperbolic solutions. Then, we use this to establish the lower sem...
Article
This paper deals with fractional stochastic nonlocal partial differential equations driven by multiplicative noise. We first prove the existence and uniqueness of solution to this kind of equations with white noise by applying the Galerkin method. Then, the existence and uniqueness of tempered pullback random attractor for the equation are ensured...
Article
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In this paper, practical stability with respect to a part of the variables of stochastic differential equations driven by G-Brownian motion (G-SDEs) is studied. The analysis of the global practical uniform p th moment exponential stability, as well as the global practical uniform exponential stability with respect to a part of the variables of G-SD...
Article
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In this paper, we investigate the pth moment exponential stability of stochastic differential equations driven by G-Brownian motion (G-SDEs) with respect to a part of the variables by means of the G-Lyapunov functions and recently developed Itô's calculus for SDEs driven by G-Brownian motion, as well as Gronwall's inequalities. We establish suffici...
Article
In this paper, we study a stochastic system of differential equations with nonlocal discrete diffusion. For two types of noises, we study the existence of either positive or probability solutions. Also, we analyze the asymptotic behavior of solutions in the long term, showing that under suitable assumptions they tend to a neighborhood of the unique...
Article
We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order [Formula: see text] driven by a tempered fractional Brownian motion (TFBM) [Formula: see text] with [Formula: see text] and [Formula: see text]. First, the global existence and uniqueness of mild solutions are established by using a new t...
Article
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In the present paper, we investigate the existence of positive periodic solutions for an n-species Lotka-Volterra system with distributed delays and nonlinear impulses. In the process, we convert the given system into an equivalent integral equation. Then, we construct appropriate mappings and use Krasnoselskii’s fixed point theorem in a cone of a...
Article
In this article, we first prove some sufficient conditions guaranteeing the existence of invariant sample measures for random dynamical systems via the approach of global random attractors. Then we consider the two-dimensional incompressible Navier-Stokes equations with additive white noise as an example to show how to check the sufficient conditio...
Preprint
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This paper studies the non-autonomous non-Newtonian micropolar fluids in two-dimensional bounded domains. We first establish that the generated continuous process of the solutions operator possesses a pullback attractor. Then we verify the existence of statistical solutions by constructing the invariant Borel probability measures. Further, we prove...
Preprint
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The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators...
Article
The paper provides conditions for the fractional Laplacian and its spectral representation on stationary Gaussian random fields to be well-defined. In addition, we study existence and uniqueness of the weak solution for a stochastic fractional elliptic equation driven by an additive colored noise over an open bounded set. Both spectral and variatio...
Article
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In this paper, the well-posedness of stochastic time fractional 2D-Stokes equations of order \(\alpha \in (0,1)\) containig finite or infinite delay with multiplicative noise is established, respectively, in the spaces \(C([-h,0];L^2(\varOmega ;L^2_{\sigma }))\) and \(C((-\infty ,0];L^2(\varOmega ;L^2_{\sigma }))\). The existence and uniqueness of...
Article
In this paper we consider the nonautonomous semilinear parabolic problems with time-dependent linear operatorsut+A(t)u=f(t,u),t>τ;u(τ)=u0, in a Banach space X. Under suitable conditions, we obtain regularity results for ut(t,x) with respect to its spatial variable x and estimates for ut in stronger spaces (Xα). We then apply those results to a nona...
Article
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In this paper, we study a semilinear parabolic problem $$\begin{aligned} u_t +A u = f(t,u), \;\, t>\tau ; \quad u(\tau ) = u_0 \in X, \end{aligned}$$in a Banach space X, where \(A:D(A) \subset X \rightarrow X\) is an almost sectorial operator. This problem is locally well-posed in the sense of mild solutions. By exploring properties of the semigrou...
Article
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In this paper, we investigate the partial asymptotic stability (PAS) of neutral pantograph stochastic differential equations with Markovian switching (NPSDEwMSs). The main tools used to show the results are the Lyapunov method and the stochastic calculus techniques. We discuss a numerical example to illustrate our main results.
Article
In this paper, the existence of positive periodic solutions of neural networks with time-varying delays is discussed by using the fixed point theory on cones. Some necessary and sufficient conditions guaranteeing the existence of one positive periodic solution of the considered system are established. Finally, we exhibit an example to verify the ap...
Article
In this paper, by using the Gronwall inequality, we show two new results on the Ulam-Hyers and the Ulam-Hyers-Rassias stabilities of neutral stochastic functional differential equations. Two examples illustrating our results are exhibited. ARTICLE HISTORY
Article
In this article, we are interested in investigating the nonlocal nonlinear reaction - diffusion system with final conditions. This problem is called backward in time problem, or terminal value problem which is understood as redefining the previous distributions when the distribution data at the terminal observation are known. There are three main g...
Article
This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions m...
Article
In this paper we consider a model describing the evolution of a nematic liquid crystal flow with delay external forces. We analyze the evolution of the velocity field \begin{document}$ {\boldsymbol u} $\end{document} which is ruled by the 3D incompressible Navier-Stokes system containing a delay term and with a stress tensor expressing the coupling...
Article
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In this paper we consider the singularly nonautonomous evolution problem \begin{document}$ u_t +A(t) u = f(t), \mbox{ } \tau<t<\tau+T; \quad u(\tau) = u_0 \in X, $\end{document} associated with a family of uniformly almost sectorial linear operators \begin{document}$ A(t):D\subset X \rightarrow X $\end{document}, that is, a family for which a secto...
Article
In this article, we study the-stability in qth moment (.s.q.m) of hybrid neutral stochastic differential equations with infinite delay (HNSDEID) using the Lyapunov techniques and the method of M-matrix. Finally, we apply the main result to some examples. K E Y W O R D S-stability, hybrid neutral stochastic differential equations, infinite delay, Ly...
Article
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In this paper, fractionally dissipative 2D quasi-geostrophic equations with an external force containing infinite delay is considered in the space \(H^s\) with \(s\ge 2-2\alpha \) and \(\alpha \in (\frac{1}{2},1)\). First, we investigate the existence and regularity of solutions by Galerkin approximation and the energy method. The continuity of sol...
Article
Full-text available
The approach of Lyapunov functions is one of the most efficient ones for the investigation of the stability of stochastic systems, in particular, of singular stochastic systems. The main objective of the paper is the analysis of the stability of stochastic perturbed singular systems by using Lyapunov techniques under the assumption that the initial...
Preprint
In this work, we study continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded random perturbations of autonomous semilinear problems. First, we study existence and permanence of unstable sets of hyperbolic solutions. Then, we use this to establish lower semicontinuity...
Article
This paper investigates the dynamics of a model of two chemostats connected by Fickian diffusion with bounded random fluctuations. We prove the existence and uniqueness of non-negative global solution as well as the existence of compact absorbing and attracting sets for the solutions of the corresponding random system. After that, we study the inte...
Article
This paper studied a stochastic epidemic model of the spread of the novel coronavirus (COVID-19). Severe factors impacting the disease transmission are presented by white noise and compensated Poisson noise with possibly infinite characteristic measure. Large time estimates are established based on Kunita's inequality rather than Burkholder-Davis-G...
Article
Full-text available
This paper is concerned with the almost sure partial practical stability of stochastic differential equations with general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functions. Finally, we provide a numerical example to demonstrate the efficiency of the obtained results.
Article
This paper addresses the stability study for nonlinear neutral differential equations. Thanks to a new technique based on the fixed point theory, we find some new sufficient conditions ensuring the global asymptotic stability of the solution. In this work we extend and improve some related results presented in recent works of literature. Two exampl...
Article
We first verify the global well-posedness of the impulsive reaction-diffusion equations on infinite lattices. Then we establish that the generated process by the solution operators has a pullback attractor and a family of Borel invariant probability measures. Furthermore, we formulate the definition of statistical solution for the addressed impulsi...
Article
In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild...
Article
In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturba...
Article
Full-text available
In this paper, we investigate the partial practical exponential stability of neutral stochastic functional differential equations with Markovian switching. The main tool used to prove the results is the Lyapunov method. We analyze an illustrative example to show the applicability and interest of the main results.
Article
This paper is concerned with the asymptotic behavior of solutions to nonlocal stochastic partial differential equations with multiplicative and additive noise driven by a standard Brownian motion, respectively. First of all, the stochastic nonlocal differential equations are transformed into their associated conjugated random differential equations...
Preprint
Full-text available
In this work, we investigate the IVP for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn-Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the...
Article
20 0 0 MSC: 60H10 34K20 34K50 Keywords: Neutral pantograph stochastic differential equations with Markovian switching Lévy noise h-stability pth moment a b s t r a c t In this paper we investigate the h-stability in pth moment of neutral pantograph stochastic differential equations with Markovian switching driven by Lévy noise. The main tool used t...
Article
This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of d...
Article
Full-text available
In this paper, we investigate the problem of stability of time-varying stochastic perturbed singular systems by using Lyapunov techniques under the assumption that the initial conditions are consistent. Sufficient conditions on uniform exponential stability and practical uniform exponential stability in mean square of solutions of stochastic pertur...
Article
In this paper, we study two terminal value problems (TVPs) for stochastic bi-parabolic equations perturbed by standard Brownian motion and fractional Brownian motion with Hurst parameter h ∈ ( 1 2 , 1 ) separately. For each problem, we provide a representation for the mild solution and find the space where the existence of the solution is guarantee...
Article
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In this paper, we investigate stochastic evolution equations with unbounded delay in fractional power spaces perturbed by a tempered fractional Brownian motion BQσ,λ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \set...
Article
In this paper we focus on the p-th moment exponential stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEwMS). By means of the Lyapunov method, we develop some sufficient conditions on the p-th moment exponential stability for NSPDEwMS. We analyze two examples to show the interest of the main results.
Article
In this paper, we study a nonlinear time-fractional Volterra equation with nonsingular Mittag-Leffler kernel in Hilbert spaces. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we give a mild solution of our problem. Our main tool here is using some Sobolev embeddings.
Article
In this article, the authors investigate the system of Schrödinger and Klein-Gordon equations with Yukawa coupling. They first prove the existence of pullback attractor and construct a family of invariant Borel probability measures. Then they establish that this family of probability measures satisfies a Liouville type theorem and is indeed a stati...
Article
Full-text available
In this work we present results to ensure a weak upper semicontinuity for a family of impulsive cocycle attractors of nonautonomous impulsive dynamical systems, as well as an example of nonautonomous dynamical system generated by an ODE in the real line to illustrate our results. Moreover, we present theoretical results regarding lower semicontinui...
Article
In this work, we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlin...
Article
We prove an existence and uniqueness result of mild solution for a system of stochastic semilinear differential equations with fractional Brownian motions and Hurst parameter H < 1∕2. Our approach is based on Perov’s fixed point theorem, and we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild...
Preprint
Full-text available
In this paper, we investigate the problem of stability of time-varying stochastic perturbed singular systems by using Lyapunov techniques under the assumption that the initial con- ditions are consistent. Sucient conditions on uniform exponential stability and practical uniform exponential stability in mean square of solutions of stochastic perturb...
Article
This paper investigates a chemostat model with wall growth and Haldane consumption kinetics. In addition, bounded random fluctuations on the input flow, which are modeled by means of the well-known Ornstein-Uhlenbeck process, are considered to obtain a much more realistic model fitting in a better way the phenomena observed by practitioners in real...
Article
The paper addresses a kind of non-autonomous nonlocal parabolic equations when the external force contains hereditary characteristics involving bounded and unbounded delays. First, well-posedness of the problem is analyzed by the Galerkin method and energy estimations in the phase space Cρ(X). Moreover, some results related to strong solutions are...
Article
In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in \begin{document}$ H^{2\alpha+s}(\mathbb{T}^2) $\end{document} with \begin{document}$ \alpha>\frac{1}{2} $\end{document} and \begin{document}$ s>1. $\end{document} We prove the...
Article
A non-autonomous stochastic delay wave equation with linear memory and nonlinear damping driven by additive white noise is considered on the unbounded domain \begin{document}$ \mathbb{R}^n $\end{document}. We establish the existence and uniqueness of a random attractor \begin{document}$ \mathcal{A} $\end{document} that is compact in \begin{document...
Article
In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70's. Thanks to this Dafermos transformation, the original problem with memory is transformed into a non-delay...
Preprint
In this work we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonline...
Preprint
Full-text available
This work is devoted to the study of the asymptotic behavior of nonautonomous reaction-diffusion equations in Dumbbell domains $\Omega_{\varepsilon} \subset \mathbb{R}^{N}$. Each $\Omega_{\varepsilon}$ is the union of a fixed open set $\Omega$ and a channel $R_{\varepsilon}$ that collapses to a line segment $R_0$ as $\varepsilon \rightarrow 0^{+}$....
Article
Stability of nonlinear delay evolution equation with stochastic perturbations is considered. It is shown that if the level of stochastic perturbations fades on the infinity then an exponentially stable deterministic system remains to be exponentially stable (in mean square). This idea has already been checked for ordinary linear stochastic differen...
Article
This work discloses an epidemiological mathematical model to predict an empirical treatment for dogs infected by Pseudomonas aeruginosa. This dangerous pathogen is one of the leading causes of multi-resistant infections and can be transmitted from dogs to humans. Numerical simulations and appropriated codes were developed using Matlab software to g...
Article
In this corrigendum we correct an error in our paper [T. Caraballo, R. Colucci, J. López-de-la-Cruz and A. Rapaport. A way to model stochastic perturbations in population dynamics models with bounded realizations, Commun Nonlinear Sci Numer Simulat, 77 (2019) 239–257]. We present a correct way to model real noisy perturbations by considering a slig...
Article
This paper is concerned with the asymptotic behaviour of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space $\mathbb {R}^n$ . The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustn...
Preprint
In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturba...
Article
In this paper, we analyze a stochastic coronavirus (COVID-19) epidemic model which is perturbed by both white noise and telegraph noise incorporating general incidence rate. Firstly, we investigate the existence and uniqueness of a global positive solution. Then, we establish the stochastic threshold for the extinction and the persistence of the di...
Article
We revisit the chemostat model with Haldane growth function, here subject to bounded random disturbances on the input flow rate, as often met in biotechnological or waste-water industry. We prove existence and uniqueness of global positive solution of the random dynamics and existence of absorbing and attracting sets that are independent of the rea...
Article
Full-text available
In this paper we consider a non-autonomous Navier-Stokes-Voigt model including a variety of delay terms in a unified formulation. Firstly, we prove the existence and uniqueness of solutions by using a Galerkin scheme. Next, we prove the existence and eventual uniqueness of stationary solutions, as well as their exponential stability by using three...
Article
This article investigates the three-dimensional globally modified Navier-Stokes equations with unbounded variable delays. Firstly, we prove the global well-posedness of the solutions, and give the existence of the pullback attractor for the associated process. Then, we construct a family of invariant Borel probability measures, which is supported b...
Article
Full-text available
In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the sta...
Article
In this article, we first prove, from the viewpoint of infinite dynamical system, sufficient conditions ensuring the existence of trajectory statistical solutions for autonomous evolution equations. Then we establish that the constructed trajectory statistical solutions possess invariant property and satisfy a Liouville type equation. Moreover, we...
Article
A non-autonomous free boundary model for tumor growth is studied. The model consists of a nonlinear reaction diffusion equation describing the distribution of vital nutrients in the tumor and a nonlinear integro-differential equation describing the evolution of the tumor size. First the global existence and uniqueness of a transient solution is est...
Article
Full-text available
In this paper, practical stability with respect to a part of the variables of nonlinear stochastic differential equations is studied. The analysis of the global practical uniform asymptotic stability, the global practical uniform pth moment exponential stability, as well as the global practical uniform exponential stability with respect to a part o...
Article
In this paper, we consider fractional nonclassical diffusion equations under two forms: initial value problem and terminal value problem. For an initial value problem, we study local existence, uniqueness, and continuous dependence of the mild solution. We also present a result on unique continuation and a blow-up alternative for mild solutions of...
Article
In this work, we investigate a stochastic epidemic model with relapse and distributed delay. First, we prove that our model possesses and unique global positive solution. Next, by means of the Lyapunov method, we determine some sufficient criteria for the extinction of the disease and its persistence. In addition, we establish the existence of a un...
Article
Full-text available
We investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka-Volte...
Article
Full-text available
Some results concerning a stochastic 2D Navier–Stokes system when the external forces contain hereditary characteristics are established. The existence and uniqueness of solutions in the case of unbounded (infinite) delay are first proved by using the classical technique of Galerkin approximations. The local stability analysis of constant solutions...
Preprint
We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining an initial condition appearing in a general class of time fractional wave equations with Caputo derivative from knowledge of solution at the final time. We are concerned with the existence, regularity upon the ter...
Article
We randomize the following class of linear differential equations with delay, x τ′ (t)=ax τ (t)+bx τ (t−τ), t > 0, and initial condition, x τ (t)=g(t), −τ≤t≤0, by assuming that coefficients a and b are random variables and the initial condition g(t) is a stochastic process. We consider two cases, depending on the functional form of the stochastic p...
Article
In this paper we establish a strong comparison principle for a nonautonomous differential inclusion with a forcing term of Heaviside type. Using this principle, we study the structure of the global attractor in both the autonomous and nonautonomous cases. In particular, in the last case we prove that the pullback attractor is confined between two s...
Article
In this paper, we prove some existence, uniqueness and Hyers–Ulam stability results for the coupled random fixed point of a pair of contractive type random operators on separable complete metric spaces. The approach is based on a new version of the Perov type fixed point theorem for contractions. Some applications to integral equations and to bound...
Article
The main aim of this letter is to use the strong compact strong trajectory attractor to construct the strong trajectory statistical solutions for two-dimensional dissipative Euler equations. Further, it is established that the constructed trajectory statistical solutions possess an invariant property and satisfy a Liouville type equation.
Article
The existence and uniqueness of global solutions for a fractional functional differential equation is establi