Existence of bounded uniformly continuous mild solutions on $\Bbb{R}$ of evolution equations and some applications

Source: arXiv


We prove that there is $x_{\phi}\in X$ for which (*)$\frac{d u(t)}{dt}= A
u(t) + \phi (t) $, $u(0)=x$ has on $\r$ a mild solution $u\in C_{ub} (\r,X)$
(that is bounded and uniformly continuous) with $u(0)=x_{\phi}$, where $A$ is
the generator of a holomorphic $C_0$-semigroup $(T(t))_{t\ge 0}$ on ${X}$ with
sup $_{t\ge 0} \,||T(t)|| < \infty$, $\phi\in L^{\infty} (\r,{X})$ and $i\,sp
(\phi)\cap \sigma (A)=\emptyset$. As a consequence it is shown that if $\n$ is
the space of almost periodic $AP$, almost automorphic $AA$, bounded Levitan
almost periodic $LAP_b$, certain classes of recurrent functions $REC_b$ and
$\phi \in L^{\infty} (\r,{X})$ such that $M_h \phi:=(1/h)\int_0^h \phi
(\cdot+s)\, ds \in \n$ for each $h >0$, then $u\in \n\cap C_{ub}$. These
results seem new and generalize and strengthen several recent Theorems.

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    ABSTRACT: The present monograph (by 4 authors!) is apparently an important and very interesting presentation of the abstract Cauchy problem treated especially by means of the (vector-valued) Laplace and Laplace-Stieltjes transforms. It appears as a worthy continuation of the classical books by it Hille [Am. Math. Soc. (1948; Zbl 0033.06501)] and it Hille and it Phillips [Am. Math. Soc. (1957; Zbl 0078.10004)] about functional analysis and semigroups.par The basic idea reads: If $A$ is a closed linear operator on a Banach space $X$, one considers the Cauchy problem ("initial value problem") $$u'(t)= Au(t),quad tge 0,quad u(0)= x,$$ where $xin X$ is given. If $u(cdot)$ is an exponentially bounded continuous function, which is also a (mild) solution, that is: $$int^t_0 u(s) dsin D(A)quadtextandquad u(t)= x+ Aint^t_0 u(s) ds,quad tge 0,$$ and if one considers the Laplace transform: $$widehat u(lambda)= int^infty_0 e^-lambda tu(t) dt,$$ which converges for large $lambda$, then $(lambda- A)widehat u(lambda)= x$ ($lambda$ large), and conversely. Thus, if $lambdain rho(A)$ -- the resolvent set of $A$ -- then $widehat u(lambda)= (lambda- A)^-1x$.par This fundamental relationship indicates that the Laplace transform is the link between solutions and resolvents, between Cauchy problems and spectral properties of operators.par A further study concerns criteria to decide whether a given function is a Laplace transform. Such results -- in the vector-valued case -- when applied to the resolvent of an operator, would give information on the solvability of the Cauchy problem.par Finally, let us note that our aim here is not to provide a summary -- or appreciation -- of the wealth of concepts contained in this 500 pages book. We invite all interested mathematicians to study this monograph, or any part of it. The reward should be considerable, as always is when reading great mathematics.
    01/2001; Monographs in Mathematics. 96. Basel: Birkhäuser. xi, 523 p. DM 196.00; sFr. 148.00.
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    ABSTRACT: LetC ub ( \mathbbJ\mathbb{J} , X) denote the Banach space of all uniformly continuous bounded functions defined on \mathbbJ\mathbb{J} 2 ε {ℝ+, ℝ} with values in a Banach spaceX. Let ℱ be a class fromC ub( \mathbbJ\mathbb{J} ,X). We introduce a spectrumspℱ(φ) of a functionφ εC ub (ℝ,X) with respect to ℱ. This notion of spectrum enables us to investigate all twice differentiable bounded uniformly continuous solutions on ℝ to the abstract Cauchy problem (*)ω′(t) =Aω(t) +φ(t),φ(0) =x,φ ε ℱ, whereA is the generator of aC 0-semigroupT(t) of bounded operators. Ifφ = 0 andσ(A) ∩iℝ is countable, all bounded uniformly continuous mild solutions on ℝ+ to (*) are studied. We prove the bound-edness and uniform continuity of all mild solutions on ℝ+ in the cases (i)T(t) is a uniformly exponentially stableC 0-semigroup andφ εC ub(ℝ,X); (ii)T(t) is a uniformly bounded analyticC 0-semigroup,φ εC ub (ℝ,X) andσ(A) ∩i sp(φ) = Ø. Under the condition (i) if the restriction ofφ to ℝ+ belongs to ℱ = ℱ(ℝ+,X), then the solutions belong to ℱ. In case (ii) if the restriction ofφ to ℝ+ belongs to ℱ = ℱ(ℝ+,X), andT(t) is almost periodic, then the solutions belong to ℱ. The existence of mild solutions on ℝ to (*) is also discussed.
    Semigroup Forum 12/1997; 54(1):58-74. DOI:10.1007/BF02676587 · 0.37 Impact Factor
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