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Search: MSC category 47D06 ( One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] )

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1. CMB Online first

Bu, Shangquan; Cai, Gang
Periodic solutions of second order degenerate differential equations with delay in Banach spaces
We give necessary and sufficient conditions of the $L^p$-well-posedness (resp. $B_{p,q}^s$-well-posedness) for the second order degenerate differential equation with finite delays: $(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where $A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying $D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from $L^p([-2\pi,0];X)$ (resp. $B_{p,q}^s([-2\pi,0];X)$) into $X$.

Keywords:second order degenerate differential equation, Fourier multiplier theorem, well-posedness, Lebesgue-Bochner space, Besov space
Categories:34G10, 34K30, 43A15, 47D06

2. CMB Online first

Bu, Shangquan; Cai, Gang
Hölder continuous solutions of degenerate differential equations with finite delay
Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely characterize the $C^\alpha$-well-posedness of the first order degenerate differential equations with finite delay $(Mu)'(t) = Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$ by the boundedness of the $(M, F)$-resolvent of $A$ under suitable assumption on the delay operator $F$, where $A, M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r, 0]; X)$ to $X$ and $r \gt 0$ is fixed.

Keywords:well-posedness, degenerate differential equation, $\dot{C}^\alpha$-multiplier, Hölder continuous function space
Categories:34N05, 34G10, 47D06, 47A10, 34K30

3. CMB 2010 (vol 54 pp. 527)

Preda, Ciprian; Sipos, Ciprian
On the Dichotomy of the Evolution Families: A Discrete-Argument Approach
We establish a discrete-time criteria guaranteeing the existence of an exponential dichotomy in the continuous-time behavior of an abstract evolution family. We prove that an evolution family ${\cal U}=\{U(t,s)\}_{t \geq s\geq 0}$ acting on a Banach space $X$ is uniformly exponentially dichotomic (with respect to its continuous-time behavior) if and only if the corresponding difference equation with the inhomogeneous term from a vector-valued Orlicz sequence space $l^\Phi(\mathbb{N}, X)$ admits a solution in the same $l^\Phi(\mathbb{N},X)$. The technique of proof effectively eliminates the continuity hypothesis on the evolution family (\emph{i.e.,} we do not assume that $U(\,\cdot\,,s)x$ or $U(t,\,\cdot\,)x$ is continuous on $[s,\infty)$, and respectively $[0,t]$). Thus, some known results given by Coffman and Schaffer, Perron, and Ta Li are extended.

Keywords:evolution families, exponential dichotomy, Orlicz sequence spaces, admissibility
Categories:34D05, 47D06, 93D20

4. CMB 2010 (vol 54 pp. 364)

Preda, Ciprian; Preda, Petre
Lyapunov Theorems for the Asymptotic Behavior of Evolution Families on the Half-Line
Two theorems regarding the asymptotic behavior of evolution families are established in terms of the solutions of a certain Lyapunov operator equation.

Keywords:evolution families, exponential instability, Lyapunov equation
Categories:34D05, 47D06

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