Entire solutions of the abstract cauchy problem

We introduce a family of operators that we will callentire C-groups, and apply them to the first and second order abstract Cauchy problem, for a large class of linear operators on a Banach space. This produces unique solutions, for all initial data in a large (often dense) set, eachof which extends to an entire function, with continuous dependence on the initial data. Applications include the backward heat equation and the Cauchy problem for the Laplace equation.     

n次积分C半群与抽象柯西问题的强解

给出了n次积分C半群的几个性质及其证明,讨论了它与一类抽象柯西问题存在强解的关系及强解的表示式.     

Maximal subspaces for solutions of the second order abstract cauchy problem

For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.     

C-wellposedness of the complete second order abstract cauchy problem and applications

In this paper, we first give a sufficient and necessary condition for to generate an exponentially bounded -semigroup and discuss its relations to the C-wellposedness of the complete second order abstract Cauchy problem ((ACP₂) for short) in some sense. Then we use these results and those in [1] to discuss the C-(exponential) wellposedness of a kind of (ACP₂) with application backgrounds, and develop the results in [2]. This project is supported by the NNSF of China, and the Youth Science and Technique Foundation of Shanxi Province, China     

Harmonic analysis and asymptotic behavior of solutions to the abstract cauchy problem

LetC _ub ( $\mathbb{J}$ , X) denote the Banach space of all uniformly continuous bounded functions defined on $\mathbb{J}$ 2 ε {?⁺, ?} with values in a Banach spaceX. Let ? be a class fromC _ub( $\mathbb{J}$ ,X). We introduce a spectrumsp?(φ) of a functionφ εC _ub (?,X) with respect to ?. This notion of spectrum enables us to investigate all twice differentiable bounded uniformly continuous solutions on ? to the abstract Cauchy problem (*)ω′(t) =Aω(t) +φ(t),φ(0) =x,φ ε ?, whereA is the generator of aC ₀-semigroupT(t) of bounded operators. Ifφ = 0 andσ(A) ∩i? is countable, all bounded uniformly continuous mild solutions on ?⁺ to (*) are studied. We prove the bound-edness and uniform continuity of all mild solutions on ?⁺ in the cases (i)T(t) is a uniformly exponentially stableC ₀-semigroup andφ εC _ub(?,X); (ii)T(t) is a uniformly bounded analyticC ₀-semigroup,φ εC _ub (?,X) andσ(A) ∩i sp(φ) = Ø. Under the condition (i) if the restriction ofφ to ?⁺ belongs to ? = ?(?⁺,X), then the solutions belong to ?. In case (ii) if the restriction ofφ to ?⁺ belongs to ? = ?(?⁺,X), andT(t) is almost periodic, then the solutions belong to ?. The existence of mild solutions on ? to (*) is also discussed.     

On the abstract nonautonomous parabolic cauchy problem in the case of constant domains

Summary We study existence, uniqueness and regularity of the strict, classical and strong solution u C([0,T],E) of the non-autonomous evolution equation u(t)–A(t)u(t)= f(t), with the initial datum, u(0)=x, in a Banach space E, where {A(t)} is a family of infinitesimal generators of analytic semi-groups whose domains are constant in t and possibly not dense in E. We prove necessary and sufficient conditions for existence and Hölder regularity of the solutions and their first derivative.     

A note on an integrated cauchy functional equation

In characterizing the semistable law, [Shimizu reduced the problem to solving the equation ,x0 where andv are given positive measures on [0, ). In this note, we obtain a simple proof and show that some of his conditions can be weakened.     

A note on the dimension of the global attractor for an abstract semilinear hyperbolic problem

We study a semilinear hyperbolic problem, written as a second-order evolution equation in an infinite-dimensional Hilbert space. Assuming existence of the global attractor, we estimate its fractal dimension explicitly in terms of the data. Despite its elementary character, our technique gives reasonable results. Notably, we require no additional regularity, although nonlinear damping is allowed.     

A non-symmetric singular cauchy problem

In this paper we replace uniformly convex (or reflexive and normal structure) as required by Browder and Kirk, by uniformly normal structure to obtain a fixed point theorem for non-expansive self mappings. Examples are given to show that spaces with uniformly normal structure are not all uniformly convex and spaces with normal structure do not all have uniformly normal structure.

AMS (MOS) subject classification (1970) Primary 47410.     

A noncharacteristic cauchy problem for the heat equation

The noncharacteristic Cauchy problem for the heat equation:% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadwhadaWgaaWcbaGaamiEaiaadIhaaeqaaOGaaiikaiaadIha% caGGSaGaamiDaiaacMcacqGH9aqpcaWG1bWaaSbaaSqaaiaadshaae% qaaOGaaiikaiaadIhacaGGSaGaamiDaiaacMcacaGGSaqefeKCPfgB% aGqbbiaa-bcacaaIWaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDOb% cv39gaiyqacqGFKjcHcaWG4bGae4hzIqOae4ha3hJaaeymaiaabYca% caqGTaGaeuOhIuQaeuipaWJaaeiDaiabfYda8iabf6HiLkaacYcaca% WG1bGaaiikaiaaicdacaGGSaGaamiDaiaacMcacqGH9aqpcqqHvpGA% caGGOaGaamiDaiaacMcacaGGSaGaamyDamaaBaaaleaacaWG4baabe% aakiaacIcacaaIWaGaaiilaiaadshacaGGPaGaeyypa0JaaGiYdiaa% cIcacaWG0bGaaiykaiaacYcacaWFGaGaeuOhIuQaeuipaWJaamiDai% abfYda8iabf6HiLcaa!82F8!\[u_{xx} (x,t) = u_t (x,t), 0 \le x \le {\rm{1, - }}\infty < {\rm{t}} < \infty ,u(0,t) = \varphi (t),u_x (0,t) = \psi (t), \infty < t < \infty \]is considered. This problem is well-known to be ill-posed. The well-posedness class of the problem is described and some approximation schemes are proposed. For the case of inexactly given data, a mollification method is suggested.     

A note on the syzygy problem

In the theorem of Evans and Griffith on the syzygy problem. The restriction on the ring being Cohen Macaulay domain is completely eliminated.     

A note on the Waring-Goldbach problem

The focus of this paper will be the extension of the Waring-Goldbach problem to all sufficiently large integers, without congruence restrictions. By reintroducing the effect of small primes, we are able to consider questions which more naturally resemble Waring's problem and the Goldbach conjecture. We extend the results of S.S. Pillai by considering the problem without the use of zero as an addend and we give a small improvement on the number of additional terms required.