C-Cosine Functions and the Abstract Cauchy Problem, II

For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A₁ = AD(A²) generates aC₁-cosine function onX₁(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC₁ = CX₁. Under the assumption thatAis densely defined andC⁻¹AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫^t₀ Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L¹_locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L¹_locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator.     

C-cosine Operator Functions and the Second Order Abstract Cauchy Problem

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局部C半群与抽象Cauchy问题

在本文中，我们讨论由算子A生成的局部C半群对应的Cauchy问题，将初值x的取值范围从CD（A）推广到（λ-A）~(-1)C（E）．     

n次积分C半群与非齐次抽象柯西问题的强解

引入了主算子为n次积分C半群生成元的线性非齐次抽象柯西问题强解的概念,讨论了相应抽象柯西问题存在强解的一些充分必要条件及强解的表示式.并给出了一个例子验证结果.     

On Integrated Solutions to Abstract Cauchy Problem

51.IntroductionSinceArendt[2j,Kellermann[3j,KellermannandHieder[4j,andNeubrarder[5]es-tablishedthetheoryof"integratedsemigroups",theexponentiallyboundedintegratedsemi-groupshavebeenstudiedintensivly-Butthereexistintegratedsemigroupswicharenotexpo-nentiallybounded.ItiswellknownthatmanyresultsforexponentiallyboundedintegratedsemigroupsarefoundbyusingLapIacetransformtechniques.ltshouldbenotedthatLaplacetransformtechniguesarenotavailableforintegratedsemigroupswhicharenotexponentiallybounded.It…     

α次积分C—存在族与抽象Cauchy问题

本我们讨论弱α次积分C－存在族、α次积分C－存在族、强α次积分C－存在族及其对抽象Cauchy问题的应用。     

局部C半群与抽象Cauchy问题的弱解

利用局部C半群的基本理论,得到了抽象Cauchy问题存在唯一弱解的等价定理及推论.     

完全二阶抽象Cauchy问题的C-适定性及其应用

在本文中,我们给出了算子矩阵生成指数有界的半群的充分必要条件,其中A0,A1为完全二阶抽象柯西问题（ACP2）的系数算子,并讨论了它与（ACP2）在某种意义下的C-适定性之间的关系．最后,我们应用此结果及［3］中的结论讨论了一类具有应用背景的完全二阶抽象柯西问题的C-适定性．     

The Kneser Property for the Second Order Functional Abstract Cauchy Problem

In this work we establish existence of mild solutions for a class of abstract semi-linear functional differential equations of second order and we prove that the set consisting of the mild solutions for these problems is connected.     

m次积分半群逼近及在抽象Cauchy问题中的应用

研究了指数有界的m次积分半群的离散逼近问题,利用可积的离散参数半群,获得了相关离散逼近结果.另外,给出了该逼近理论在非齐次抽象Cauchy问题中的应用.     

Complex Cauchy Problem in a Scale of Analytic Functions with Power-Law Singularities

We consider the Cauchy problem for systems of complex linear partial differential equations and obtain necessary and sufficient conditions for this problem to be well posed in a scale of Banach spaces of analytic functions having power-law singularities as the independent variable tends to the lateral surface of a cone.     

A Noncharacteristic Cauchy Problem for Linear Parabolic Equations I: Solvability

Noncharacteristic Cauchy problems for parabolic equations arc frequently encountered in many areas of heat transfer. These problems are well known to be severely ill-posed. In this paper a solvability criterion for a class of such problems is established. It is proved that a weak solution of a noncharacteristic Cauchy problem for linear parabolic equations in divergence form with coefficients in a Holmgren class 2 in time exists if and only if the Cauchy data arc functions of a Holmgren class 2! A function g(t) defined on (α, β) is said to be of a Holmgren class 2, if g ?C^∞ (α, β) and for all nonnegative integers n there exist positive constants c and s such that |g⁽ⁿ⁾| < csⁿ(2n)!.     

Mild Integrated C-existence Families and Abstract Cauchy Problems

in this paper, we first give a Hille-Yosida type sufficient and necessary conditionfor n-times integrated mild C-existence families. Then, we present a Laplace type sufficientand necessary condition for exponentially bounded n-times integrated C-semigrougs, andstudy the relationship between integrated regularized semigroups and general regularizedsemigroups. Finally, we offer a characterization of integrated C-semigroups in terms of thesolvability of abstract Cauchy problems.     

Special Uniform Approximations of Continuous Vector-Valued Functions. Part I: Special Approximations in CX(T)

In this paper, we give special uniform approximations of functions u from the spaces C_X(T) and C_∞(T,X), with elements of the tensor products C_Γ(T)X, respectively C₀(T,Γ)X, for a topological space T and a Γ-locally convex space X. We call an approximation special, if satisfies additional constraints, namely supp vu⁻¹(X\{0}) and (T) co(u(T)) (resp. co(u(T){0})). In Section 3, we give three distinct applications, which are due exactly to these constraints: a density result with respect to the inductive limit topology, a Tietze–Dugundji's type extension new theorem and a proof of Schauder–Tihonov's fixed point theorem.