Periodic solutions of degenerate differential equations in vector-valued function spaces

2016

Math. Nachr.

In this paper, we study the well‐posedness of the degenerate differential equations with fractional derivative Dαfalse(Mufalse)false(tfalse)=Aufalse(tfalse)+ffalse(tfalse),false(0≤t≤2πfalse) in Lebesgue–Bochner spaces Lpfalse(double-struckT;Xfalse), periodic Besov spaces Bp,qsfalse(double-struckT;Xfalse) and periodic Triebel–Lizorkin spaces Fp,qsfalse(double-struckT;Xfalse), where A and M are closed linear operators in a complex Banach space X satisfying D(A)⊂D(M), α>0 and Dα is the fractional derivative in the sense of Weyl. Using known operator‐valued Fourier multiplier results, we completely characterize the well‐posedness of this problem in the above three function spaces by the R‐bounedness (or the norm boundedness) of the M‐resolvent of A.

“…Our results recover the previous known results by Lizama and Ponce when

α = 1

, and Bu when

M = I_{X}

. Thus our results also recover the previous known results obtained by Arendt and Bu in , .…”

Section: Introductionsupporting

confidence: 92%

“…See e.g. –, –, – and references therein. For instance, the first order degenerate equations:

\begin{matrix} true \\ right {false(M u false)}^{'} (t) = A u (t) + f (t), (0 \leq t \leq 2 π), \end{matrix}

with periodic boundary condition

(M u) (0) = (M u) (2 π)

, have been recently studied by Lizama and Ponce, where A and M are closed linear operators in a complex Banach space X .…”

Section: Introductionmentioning

confidence: 99%

“…–, –, – and references therein. For instance, the first order degenerate equations:

\begin{matrix} true \\ right {false(M u false)}^{'} (t) = A u (t) + f (t), (0 \leq t \leq 2 π), \end{matrix}

with periodic boundary condition

(M u) (0) = (M u) (2 π)

L^{p} false(double-struckT; X false)

, periodic Besov spaces

B_{p, q}^{s} false(double-struckT; X false)

and periodic Triebel–Lizorkin spaces

F_{p, q}^{s} false(double-struckT; X false)

, where

T : = [0, 2 π]

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Well-posedness of degenerate differential equations with fractional derivative in vector-valued functional spaces

2016

Math. Nachr.

“…Recently, the first order degenerate equations

\begin{matrix} true \\ right {false(M u false)}^{'} (t) = A u (t) + f (t), t \in [0, 2 π], \end{matrix}

have been studied by Lizama and Ponce, where A and M are closed linear operators on a complex Banach space X satisfying

D (A) \subset D (M)

. Under suitable assumptions on the modified resolvent operator determined by , they gave necessary and sufficient conditions to ensure the well‐posedness of in Lebesgue–Bochner spaces

L^{p} false(double-struckT; X false)

, periodic Besov spaces

B_{p, q}^{s} false(double-struckT; X false)

and periodic Triebel–Lizorkin spaces

F_{p, q}^{s} false(double-struckT; X false)

, where

T : = [0, 2 π]

. Lizama studied the first order differential equations with finite delay

\begin{matrix} true \\ right u^{'} (t) = A u (t) + F u_{t} + f (t), t \in [0, 2 π], \end{matrix}

where A is a closed linear operator on a complex Banach space X , the delay operator

F : L^{p} false([- 2 π, 0]; X false) \to X

is a bounded linear operator,

u_{t}

is defined by

…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness of fractional degenerate differential equations with finite delay on vector‐valued functional spaces

Mathematische Nachrichten

2017

We study the well‐posedness of the fractional degenerate differential equations with finite delay false(Pαfalse):Dαfalse(Mufalse)false(tfalse)=Aufalse(tfalse)+Fut+ffalse(tfalse),false(0≤t≤2π,0.33emα>0false) on Lebesgue–Bochner spaces Lpfalse(double-struckT;Xfalse), periodic Besov spaces Bp,qsfalse(double-struckT;Xfalse) and periodic Triebel–Lizorkin spaces Fp,qsfalse(double-struckT;Xfalse), where A and M are closed linear operators on a Banach space X satisfying D(A)⊂D(M), F is a bounded linear operator from Lpfalse([−2π,0];Xfalse) (resp. Bp,qsfalse([−2π,0];Xfalse) and Fp,qsfalse([−2π,0];Xfalse)) into X, where ut is given by utfalse(sfalse)=ufalse(t+sfalse) when s∈[−2π,0] and t∈[0,2π]. Using known operator‐valued Fourier multiplier theorems, we give necessary or sufficient conditions for the well‐posedness of (Pα) in the above three function spaces.

“…, –, – and references therein. Recently, the first order degenerate equations:

{(M u)}^{'} (t) = A u (t) + f (t), (0 \leq t \leq 2 π),

with periodic boundary condition

(M u) (0) = (M u) (2 π)

, has been studied by Lizama and Ponce , under suitable assumptions on the modified resolvent operator determined by (1.1), they gave necessary and sufficient conditions to ensure the well‐posedness of (1.1) in Legesgue‐Bochner spaces

L^{p} (double-struckT; X)

, Besov spaces

B_{p, q}^{s} (double-struckT; X)

and Triebel‐Lizorkin spaces

F_{p, q}^{s} (double-struckT; X)

, where

T : = [0, 2 π]

. The well‐posedness in Hölder continuous function space

C^{α} (double-struckR; X)

of the same equations on the real line

double-struckR

{(M u)}^{'} (t) = A u (t) + f (t), (t \in double-struckR),

was independently characterized by Ponce and Bu using operator‐valued Fourier multiplier theorem established by Arendt, Batty and Bu .…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness of second order degenerate integro‐differential equations with infinite delay in vector‐valued function spaces

Mathematische Nachrichten

2015

We study the well-posedness of the second order degenerate differential equations with infinite delay:where A, B and M are closed linear operators in a Banach space satisfying D( A) ∩ D(B) = {0}, D( A) ∩ D(B) ⊂ D(M), a, b ∈ L 1 (R + ). Using operator-valued Fourier multiplier techniques, we give necessary and sufficient conditions for the well-posedness of this problem in Lebesgue-Bochner spaces L p (T; X ), periodic Besov spaces B s p,q (T; X ) and periodic Triebel-Lizorkin spaces F s p,q (T; X ).