半线性微分方程的概自守与伪概自守解

在Banach空间中,利用发展系统的算子半群理论和Banach压缩原理,在半线性微分方程x′（t）=A（t）x（t）＋f（t,x（t））满足一定的条件下,证明了其概自守与伪概自守mild解的存在性与唯一性．     

Integral Inequalities and Pseudo Almost Automorphic Solutions to Semilinear Differential Equations in Banach Spaces

Mediterranean Journal of Mathematics - This work aims to investigate the existence and uniqueness of pseudo almost automorphic solutions to semilinear differential equations in Banach space. By...     

Almost Automorphic Solutions for Partial Functional Differential Equations with Infinite Delay

In this work, we study the existence of almost automorphic solutions for partial functional differential equations with infinite delay. We assume that the undelayed part is not necessarily densely defined and satisfies the Hille-Yosida condition. We use the so-called reduction principle developed recently in [3], to show the existence of an almost automorphic solution under minimal condition. More precisely, the existence of an almost automorphic solution is proved when there is at least one bounded solution in the positive real half line. We give an application to the Lotka-Volterra model with diffusion.     

Existence and Uniqueness of Almost Automorphic Mild Solutions to Some Semilinear Abstract Differential Equations

We consider the semilinear differential equation x(t) = Ax(t) + f(t, x(t)), t in a Banach space X, where A is the in.nitesimal generator of an exponentially stable C₀-semigroup. Under suitable conditions on f, we prove the existence and uniqueness of an almost automorphic mild solution to the equation.     

Pseudo Almost Automorphic Solutions for Non-autonomous Stochastic Differential Equations with Exponential Dichotomy

In this paper, we consider the existence and uniqueness of the solutions which are pseudo almost automorphic in distribution for a class of non-autonomous stochastic differential equations in a Hilbert space. In conclusion, we use the Banach contraction mapping principle and exponential dichotomy property to obtain our main results.     

Pseudo Almost Automorphic Solutions for Non-autonomous Stochastic Differential Equations With Exponential Dichotomy

In this paper, we consider the existence and uniqueness of the solutions which are pseudo almost automorphic in distribution for a class of non-autonomous stochastic differential equations in a Hilbert space. In conclusion, we use the Banach contraction mapping principle and exponential dichotomy property to obtain our main results.     

抽象空间中无穷时滞微分方程概自守解的存在性

研究抽象空间中无穷时滞微分方程概自守解的存在性,证明了在正实轴上存在有界解蕴含存在概自守解,并给出了结论在L otka-V o lterra型方程中的应用.我们的结果推广了经典的关于非齐次线性概周期微分方程概周期解存在性的结论.     

Spaces of Quasi-Periodic Functions and Oscillations in Differential Equations

We build spaces of q.p. (quasi-periodic) functions and we establish some of their properties. They are motivated by the Percival approach to q.p. solutions of Hamiltonian systems. The periodic solutions of an adequatez partial differential equation are related to the q.p. solutions of an ordinary differential equation. We use this approach to obtain some regularization theorems of weak q.p. solutions of differential equations. For a large class of differential equations, the first theorem gives a result of density: a particular form of perturbated equations have strong solutions. The second theorem gives a condition which ensures that any essentially bounded weak solution is a strong one.     

Almost Automorphic Solutions for Hyperbolic Semilinear Evolution Equations

In the present paper, we deal with mild solutions for the semilinear evolution equation \begin{displaymath} \frac{d}{dt}x(t)=Ax(t)+f(t,x(t)),\qquad t\in \R, \end{displaymath} under the sectoriality of $A$, a linear operator with not necessarily dense domain, in a Banach space $X$ and $\sigma(A)\cap i\R=\emptyset$. We prove the existence and uniqueness of an almost automorphic solution in an intermediate space $X_{\alpha}$, when the function $f\colon \ \R\times X_{\alpha}\longrightarrow X$ is almost automorphic. An example illustrating the obtained result is given.     

Almost Frechet Differentiability of Lipschitz Mappings Between Infinite-Dimensional Banach Spaces

We give several sufficient conditions on a pair of Banach spacesX and Y under which each Lipschitz mapping from a domain inX to Y has, for every > 0, a point of -Fréchet differentiability.Most of these conditions are stated in terms of the moduli ofasymptotic smoothness and convexity, notions which have appearedin the literature under a variety of names. We prove, for example,that for > r > p 1, every Lipschitz mapping from a domainin an l_r-sum of finite-dimensional spaces into an l_p-sum offinite-dimensional spaces has, for every > 0, a point of-Fréchet differentiability, and that every Lipschitzmapping from an asymptotically uniformly smooth space to a finite-dimensionalspace has such points. The latter result improves, with a simplerproof, an earlier result of the second and third authors. Wealso survey some of the known results on the notions of asymptoticsmoothness and convexity, prove some new properties, and presentsome new proofs of existing results. 2000 Mathematical Subject Classification: 46G05, 46T20.     

光滑Banach空间上扩张值函数的Frechet次微分

首先证明了Frechet光滑Banach空间上齐次函数的次微分的一个有用定理,然后利用下半连续函数和的次微分规则把Clarke-Ledyaev多方向中值不等式推广到多个函数的情形.     

Pseudo-almost Periodic and Pseudo-almost Automorphic Solutions to Evolution Equations in Hilbert Spaces

In this work we prove some existence and uniqueness results for pseudo-almost periodic and pseudo-almost automorphic solutions to a class of semi-linear differential equations in Hilbert spaces using theoretical measure theory. The main technique is based upon some appropriate composition theorems combined with the Banach contraction mapping principle and the method of the invariant subspaces for unbounded linear operators. A few illustrative examples will be discussed at the end of the paper.     

Almost p-Regular Spaces and Some Functions

In this paper, we define a new class of sets called rgp-closed and use the sets to obtain further characterizations of almost p-regular spaces and almost regular spaces. The main result of this paper is that almost p-regularity is preserved under M-preopen pre rgp-closed surjective R-maps.     

Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

Let A be a set, and let E be the Banach space of bounded functions : A R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,) : R × E E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to . Then there exists a solution u: [a,b] × [0,T] E of the problem ut(x,t) – uxx(x,t) = F(x,t,u(x,t)) ((x,t) R), u(x,t) = 0 ((x,t) R R).     

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In this paper, we introduce a concept of Poisson $p$-mean almost automorphy for stochastic processes and give the composition theorems for (Poisson) $p$-mean almost automorphic functions under non-Lipschitz conditions. Our abstract results are, subsequently, applied to study a class of neutral stochastic evolution equations driven by L\'evy noise, and we present sufficient conditions for the existence of square-mean almost automorphic mild solutions. An example is provided to illustrate the effectiveness of the proposed result.     

二次概周期系数微分方程的概周期解

利用指数二分性,Schauder不动点定理和Grownwall不等式证明了一类概周期系数微分方程的概周期解、有界解的存在唯一性及概周期解的全局吸引性.     

Almost Periodic Solutions to a Stochastic Differential Equation in Hilbert Spaces

In this paper, we prove the existence and uniqueness of quadratic mean almost periodic mild solutions for a class of stochastic differential equations in a real separable Hilbert space. The main technique is based upon an appropriate composition theorem combined with the Banach contraction mapping principle and an analytic semigroup of linear operators.     

Differential Calculus in Riesz Spaces and Applications to g-Calculus

We apply the theory of Differential and Integral Calculus in Riesz Spaces introduced in [1] and [4] to investigate some properties of the g-calculus and to solve some types of differential, functional and stochastic equations.