Maximal regularity for degenerate differential equations with infinite delay in periodic vector-valued function spaces

2013 ◽  
Vol 56 (3) ◽  
pp. 853-871 ◽  
Author(s):  
Carlos Lizama ◽  
Rodrigo Ponce
Keyword(s):  
Infinite Delay ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Function Spaces ◽  
Linear Operators ◽  
Image Position ◽  
Vector Valued Function ◽  
Degenerate Differential Equations ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractLet A and M be closed linear operators defined on a complex Banach space X and let a ∈ L1(ℝ+) be a scalar kernel. We use operator-valued Fourier multipliers techniques to obtain necessary and sufficient conditions to guarantee the existence and uniqueness of periodic solutions to the equationwith initial condition Mu(0) = Mu(2π), solely in terms of spectral properties of the data. Our results are obtained in the scales of periodic Besov, Triebel–Lizorkin and Lebesgue vector-valued function spaces.

Periodic Solutions of Second Order Degenerate Differential Equations with Delay in Banach Spaces

2018 ◽  
Vol 61 (4) ◽  
pp. 717-737 ◽  
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Second Order ◽  
Linear Operators ◽  
Bounded Linear ◽  
Order Degenerate ◽  
Degenerate Differential Equations ◽  
Necessary And Sufficient ◽  
Equations With Delay ◽  
Closed Linear Operators

AbstractWe give necessary and sufficient conditions of the Lp-well-posedness (resp. -wellposedness) for the second order degenerate differential equation with finite delayswith periodic boundary conditions (Mu)(0) = (Mu)(2π), (Mu)′ (0) = (Mu)′ (2π), where A, B, and M are closed linear operators on a complex Banach space X satisfying D(A) ∩ D(B) ⊂ D(M), F and G are bounded linear operators from into X.

Well-posedness of fractional degenerate differential equations in Banach spaces

2019 ◽  
Vol 22 (2) ◽  
pp. 379-395
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Differential Equation ◽  
Fourier Multiplier ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Well Posedness ◽  
Multiplier Theorems ◽  
Degenerate Differential Equations ◽  
Necessary And Sufficient ◽  
Closed Linear Operators

Abstract We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces $\begin{array}{} B_{p,q}^s \end{array}$ (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 < β < α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and $\begin{array}{} B_{p,q}^s \end{array}$-well-posedness of above equation.

Necessary and Sufficient Conditions for the Discreteness of the Spectrum of Certain Singular Differential Operators

1981 ◽  
Vol 33 (1) ◽  
pp. 229-246 ◽  
Author(s):  
Calvin D. Ahlbrandt ◽  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Differential Operators ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Inner Product ◽  
Dimensional Vector ◽  
Selfadjoint Extension ◽  
Image Position ◽  
Vector Valued Function ◽  
Necessary And Sufficient ◽  
Vector Valued

1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l bywhere y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfyingwith inner productwhere . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.

On the invertibility of length two elementary operators

2015 ◽  
Vol 30 ◽  
pp. 916-913
Author(s):  
Janko Bracic ◽  
Nadia Boudi
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Elementary Operator ◽  
Bounded Linear Operators ◽  
Existence Of A Solution ◽  
Bounded Linear ◽  
Elementary Operators ◽  
Necessary And Sufficient

Let X be a complex Banach space and L(X) be the algebra of all bounded linear operators on X. For a given elementary operator P of length 2 on L(X), we determine necessary and sufficient conditions for the existence of a solution of the equation YP=0 in the algebra of all elementary operators on L(X). Our approach allows us to characterize some invertible elementary operators of length 2 whose inverses are elementary operators.

scholarly journals Jointly separating maps between vector-valued function spaces

2020 ◽  
Vol 70 (3) ◽  
pp. 707-718
Author(s):  
Ziba Pourghobadi ◽  
Masoumeh Najafi Tavani ◽  
Fereshteh Sady
Keyword(s):  
Banach Algebras ◽  
Function Algebra ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Linear Operators ◽  
Hausdorff Spaces ◽  
Absolutely Continuous Functions ◽  
Spaces Of Continuous Functions ◽  
Vector Valued Function ◽  
Vector Valued

AbstractLet X and Y be compact Hausdorff spaces, E be a real or complex Banach space and F be a real or complex locally convex topological vector space. In this paper we study a pair of linear operators S, T : A(X, E) → C(Y, F) from a subspace A(X, E) of C(X, E) to C(Y, F), which are jointly separating, in the sense that Tf and Sg have disjoint cozeros whenever f and g have disjoint cozeros. We characterize the general form of such maps between certain classes of vector-valued (as well as scalar-valued) spaces of continuous functions including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions. The results can be applied to a pair T : A(X) → C(Y) and S : A(X, E) → C(Y, F) of linear operators, where A(X) is a regular Banach function algebra on X, such that f ⋅ g = 0 implies Tf ⋅ Sg = 0, for all f ∈ A(X) and g ∈ A(X, E). If T and S are jointly separating bijections between Banach algebras of scalar-valued functions of this class, then they induce a homeomorphism between X and Y and, furthermore, T−1 and S−1 are also jointly separating maps.

Well-Posedness of Second-Order Degenerate Differential Equations with Finite Delay

2016 ◽  
Vol 60 (2) ◽  
pp. 349-360 ◽  
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Second Order ◽  
Well Posedness ◽  
Bounded Linear ◽  
Finite Delay ◽  
Order Degenerate ◽  
Image Position ◽  
Degenerate Differential Equations

AbstractWe give necessary and sufficient conditions for theLp-well-posedness of the second-order degenerate differential equations with finite delaywith periodic boundary conditions (Mu)(0) = (Mu)(2π), (Mu)′(0) = (Mu)′(2π). HereAandMare closed operators on a complex Banach spaceXsatisfyingD(A) ⊂D(M),α ∈ℂ is fixed,Fis a bounded linear operator fromLp([−2π,0],X) intoX, andutis given byut(s) =u(t+s) whens ∈[−2π,0].

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