Asymptotic Behavior Of Mild Solutions Research Articles

Wellposedness and controllability results of stochastic integrodifferential equations with noninstantaneous impulses and Rosenblatt process

The purpose of this work is to investigate a novel class of noninstantaneous impulsive stochastic integrodifferential equations (SIDEs) driven by Brownian motion and Rosenblatt process. We construct a new set of adequate assumptions for the existence and uniqueness of mild solutions using stochastic analysis, analytic semigroup theory, integral equation theory, and a fixed point methodology. Additionally, we study the asymptotic behavior of mild solutions and provide stochastic system controllability results. Finally, we include an example to illustrate the application of our main findings.

Global solution to the Cauchy problem of fractional drift diffusion system with power-law nonlinearity

<abstract><p>In this paper, we consider the global existence, regularizing decay rate and asymptotic behavior of mild solutions to Cauchy problem of fractional drift diffusion system with power-law nonlinearity. Using the properties of fractional heat semigroup and the classical estimates of fractional heat kernel, we first prove the global-in-time existence and uniqueness of the mild solutions in the frame of mixed time-space Besov space with multi-linear continuous mappings. Then, we show the asymptotic behavior and regularizing-decay rate estimates of the solution to equations with power-law nonlinearity by the method of multi-linear operator and the classical Hardy-Littlewood-Sobolev inequality.</p></abstract>

Exponential behavior and upper noise excitation index of solutions to evolution equations with unbounded delay and tempered fractional Brownian motions

In this paper, we investigate stochastic evolution equations with unbounded delay in fractional power spaces perturbed by a tempered fractional Brownian motion \(B_Q^{\sigma ,\lambda }(t)\) with \(-1/2<\sigma <0\) and \(\lambda >0\). We first introduce a technical lemma which is crucial in our stability analysis. Then, we prove the existence and uniqueness of mild solutions by using semigroup methods. The upper nonlinear noise excitation index of the energy solutions at any finite time t is also obtained. Finally, we consider the exponential asymptotic behavior of mild solutions in mean square.

Asymptotic behavior of mild solutions to fractional Cauchy problems in Banach spaces

In this paper we investigate the existence, asymptotic behavior and uniform p-integrability of fractional resolvent families generated by sectorial operators in Banach spaces. As a consequence, we obtain properties on the behavior of mild solutions to abstract fractional Cauchy problems for the Caputo and Riemann–Liouville fractional derivatives.

Determining Nodes for Semilinear Parabolic Evolution Equations in Banach Spaces

We are concerned with the determination of the asymptotic behavior of mild solutions to the abstract initial value problem for semilinear parabolic evolution equations in $L_{p}$ by the asymptotic behavior of these mild solutions on a finite set. More precisely, if the asymptotic behavior of the mild solution is known on an suitable finite set which is called determining nodes, then the asymptotic behavior of the mild solution itself is entirely determined. Not only the asymptotic equivalence but also the rate of monomial or exponential convergence can be clarified. We prove the above properties by the theory of analytic semigroups on Banach spaces.

Asymptotic behavior of mild solutions for a class of abstract nonlinear difference equations of convolution type

We prove the existence and uniqueness of a weighted pseudo asymptotically mild solution to the following class of abstract semilinear difference equations: \t\t\tu(n+1)=A∑k=−∞na(n−k)u(k+1)+∑k=−∞nb(n−k)f(k,u(k)),n∈Z,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u(n+1)= A \\sum_{k=-\\infty }^{n} a(n-k)u(k+1)+ \\sum _{k=-\\infty }^{n} b(n-k)f\\bigl(k,u(k)\\bigr),\\quad n\\in \\mathbb{Z}, $$\\end{document} where A is the generator of a resolvent sequence {S(n)}_{nin mathbb{N}_{0}} of bounded and linear operators defined in a Banach space X, the sequences a, b are complex-valued, and fin l^{1}( mathbb{Z}times X, X).

The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay

We first prove the existence, uniqueness and continuous dependence of mild solutions to stochastic delay evolution equations with a Caputo fractional derivative:DtαCy(t)=Ay(t)+f(t,yt)+g(t,yt)dW(t)dt,12<α<1. Then, we investigate the asymptotic behavior of mild solutions to fractional stochastic delay evolution equations of the formDtαCy(t)=Ay(t)+It1−αf(t,yt)+[It1−αg(t,yt)]dW(t)dt,0<α<1. In particular, the existence of a global forward attracting set in the mean-square topology is established. A general theorem on the existence of mild solutions is obtained by using α-order fractional resolvent operator theory and the Schauder fixed point theorem.

Asymptotic Behavior of Mild Solutions for Nonlinear Fractional Difference Equations

AbstractIn this paper, we establish some sufficient criteria for the existence, uniqueness of discrete weighted pseudo asymptotically periodic mild solutions and asymptotic behavior for nonlinear fractional difference equations in Banach space, where the nonlinear perturbation is Lipschitz type, or non-Lipschitz type. The results are a consequence of application of different fixed point theorems, namely, the Banach contraction mapping principle, Leray-Schauder alternative theorem and Matkowski’s fixed point technique.

P-th moment exponential convergence analysis for stochastic networked systems driven by fractional Brownian motion

In this paper, the existence, uniqueness and asymptotic behavior of mild solutions of stochastic neural network systems driven by fractional Brownian motion are investigated. By applying the Banach fixed point theorem, the existence and uniqueness of mild solution are analytically proved in a Hilbert space. Based on the moment inequality of wick-type integral analysis technique, the p-th moment exponential convergence condition of the mild solution is presented. Finally, two numerical examples are presented to demonstrate the validity of the theoretical results.

Existence and asymptotic behavior of solutions for neutral stochastic partial integrodifferential equations with infinite delays

In this work we study the existence, uniqueness and asymptotic behavior of mild solutions for neutral stochastic partial integrodifferential equations with infinite delays. To prove the results, we use the theory of resolvent operators as developed by R. Grimmer [13], as well as a version of the fixed point principle. We establish sufficient conditions ensuring that the mild solutions are exponentially stable in [Formula: see text]th-moment. An example is provided to illustrate the abstract results.

Exponential Behavior of Solutions to Stochastic Integrodifferential Equations with Distributed Delays

In this work, we study the existence, uniqueness, and exponential asymptotic behavior of mild solutions to stochastic integrodifferential delay evolution equations. We assume that the non-delay part generates a C0-semigroup.

Existence and stability results for neutral stochastic delay differential equations driven by a fractional Brownian motion

In this paper we investigate the existence, uniqueness, asymptotic behavior of mild solutions to neutral stochastic differential equations with delays driven by a fractional Brownian motion in a Hilbert space. The cases of finite and infinite delays are analyzed.

Asymptotic Behavior of Neutral Stochastic Partial Functional Integro--Differential Equations Driven by a Fractional Brownian Motion

This paper deals with the existence, uniqueness and asymptotic behavior of mild solutions to neutral stochastic delay functional integro-dierenti

Asymptotic behavior for neutral stochastic partial differential equations with infinite delays

This paper is concerned with the existence and asymptotic behavior of mild solutions to a class of non-linear neutral stochastic partial differential equations with infinite delays. By applying fixed point principle, we present sufficient conditions to ensure that the mild solutions are exponentially stable in $p$th-moment ($p\geq 2$) and almost surely exponentially stable. An example is provided to illustrate the effectiveness of the proposed result.

Asymptotic Behavior of Mild Solutions to Semilinear Fractional Differential Equations

This paper is mainly concerned with the asymptotically almost automorphic mild solutions to a semilinear fractional differential equation. Some asymptotic behavior of mild solutions to this equation has been established by properties and composition theorems of asymptotically almost automorphic functions and fixed point theorems.

Different asymptotic behavior of global solutions for a parabolic system with nonlinear gradient terms

In this paper we study the global existence and the different asymptotic behavior of mild solutions for the nonlinear parabolic system: ∂ t u = Δ u + a | ∇ v | p , ∂ t v = Δ v + b | ∇ u | q , t > 0 , x ∈ R N , where a , b ∈ R , N ⩾ 1 , 1 < p ⩽ q < 2 and p q > q N + 1 + N + 2 N + 1 . We prove, in particular, that if the initial values behave as u ( 0 , x ) ∼ ω 1 ( x / | x | ) | x | − α , v ( 0 , x ) ∼ ω 2 ( x / | x | ) | x | − β as | x | → ∞ , 0 < α , β < N , β + 2 − q q < α , α + 2 − p p < β and under suitable conditions on ω 1 , ω 2 , then the resulting solutions are global. Furthermore, although the scaling invariance properties of these initial values and the system are different, we prove that some of the solutions are asymptotic to self-similar solutions of appropriate asymptotic systems which depend on the values of α and β. The asymptotic behavior estimates are given in the W 1 , ∞ ( R N ) × W 1 , ∞ ( R N ) -norm and are stable under some small perturbations. The results of this paper complete those of Al-Elaiw and Tayachi (2010) [1] known only for β + 2 − q q = α and α + 2 − p p = β .

The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion

In this paper we investigate the existence, uniqueness and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a fractional Brownian motion B Q H ( t ) : d X ( t ) = ( A X ( t ) + f ( t , X t ) ) d t + g ( t ) d B Q H ( t ) , with Hurst parameter H ∈ ( 1 / 2 , 1 ) . We also consider the existence of weak solutions.

Existence, Uniqueness, and Asymptotic Behavior of Mild Solutions to Stochastic Functional Differential Equations in Hilbert Spaces

In this paper we shall consider the existence, uniqueness, and asymptotic behavior of mild solutions to stochastic partial functional differential equations with finite delay r>0:dX(t)=[−AX(t)+f(t, Xt)]dt+g(t, Xt)dW(t), where we assume that −A is a closed, densely defined linear operator and the generator of a certain analytic semigroup. f:(−∞, +∞)×Cα→H, g:(−∞, +∞)×Cα→L02(K, H) are two locally Lipschitz continuous functions, where Cα=C([−r, 0], D(Aα)), L02(K, H) are two proper infinite dimensional spaces, 0<α<1. Here, W(t) is a given K-valued Wiener process and both H and K are separable Hilbert spaces.

A class of semi-linear equations of evolution

In this paper we prove existence, and study the asymptotic behavior of mild solutions of a class of semi-linear equations of evolution which are characterized by the fact that the associated homogeneous linear problem “generates” a strongly continuous semi-group of compact operators. We also prove a regularity result for which the associated homogeneous linear problem generates a holomorphic semi-group.