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.

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].

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.

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.

Hölder Continuous Solutions of Degenerate Differential Equations with Finite Delay

2018 ◽  
Vol 61 (2) ◽  
pp. 240-251
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Differential Equations ◽  
Linear Operators ◽  
Hölder Continuous ◽  
Bounded Linear ◽  
Finite Delay ◽  
Order Degenerate ◽  
Degenerate Differential Equations ◽  
Vector Valued ◽  
Closed Linear Operators ◽  
Delay Operator

AbstractUsing known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces Cα(ℝ; X), we completely characterize the Cα-well-posedness of the first order degenerate differential equations with finite delay (Mu)′(t) = Au(t) + Fut + f(t) for t ∊ 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) İ D(M) ≠ = ﹛0﹜, the delay operator F is a bounded linear operator from C([−r, 0]; X) to X, and r > 0 is fixed.

scholarly journals Generalized derivation, SVEP, finite ascent, range closure

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3473-3482
Author(s):  
Farida Lombarkia ◽  
Sabra Megri
Keyword(s):  
Sufficient Conditions ◽  
Generalized Derivation ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Multiplication Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
The Right ◽  
Necessary And Sufficient ◽  
Algebraic Operators

Let X be an infinite complex Banach space and consider two bounded linear operators A,B ? L(X). Let LA ? L(L(X)) and RB ? L(L(X)) be the left and the right multiplication operators, respectively. The generalized derivation ?A,B ? L(L(X)) is defined by ?A,B(X) = (LA-RB)(X) = AX-XB. In this paper we give some sufficient conditions for ?A,B to satisfy SVEP, and we prove that ?A,B-?I has finite ascent for all complex ?, for general choices of the operators A and B, without using the range kernel orthogonality. This information is applied to prove some necessary and sufficient conditions for the range of ?A,B-?I to be closed. In [18, Propostion 2.9] Duggal et al. proved that, if asc(?A,B-?)? 1, for all complex ?, and if either (i) A* and B have SVEP or (ii)?* A,B has SVEP, then ?A,B-? has closed range for all complex ? if and only if A and B are algebraic operators, we prove using the spectral theory that, if asc(?A,B-?) ? 1, for all complex ?, then ?A,B-? has closed range, for all complex ? if and only if A and B are algebraic operators, without the additional conditions (i) or (ii).

Complete Positivity of Two Class of Maps Involving Depolarizing and Transpose-Depolarizing Channels

2021 ◽  
Vol 28 (02) ◽  
Author(s):  
Xiuhong Sun ◽  
Yuan Li
Keyword(s):  
Sufficient Conditions ◽  
Linear Operators ◽  
Complete Positivity ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Bounded Linear ◽  
Positive Map ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Finite Dimensional Hilbert Space

In this note, we mainly study the necessary and sufficient conditions for the complete positivity of generalizations of depolarizing and transpose-depolarizing channels. Specifically, we define [Formula: see text] and [Formula: see text], where [Formula: see text] (the set of all bounded linear operators on the finite-dimensional Hilbert space [Formula: see text] is given and [Formula: see text] is the transpose of [Formula: see text] in a fixed orthonormal basis of [Formula: see text] First, we show that [Formula: see text] is completely positive if and only if [Formula: see text] is a positive map, which is equivalent to [Formula: see text] Moreover, [Formula: see text] is a completely positive map if and only if [Formula: see text] and [Formula: see text] At last, we also get that [Formula: see text] is a completely positive map if and only if [Formula: see text] with [Formula: see text] for all [Formula: see text] where [Formula: see text] are eigenvalues of [Formula: see text].

scholarly journals On Approximate Controllability of Second-Order Neutral Partial Stochastic Functional Integrodifferential Inclusions with Infinite Delay and Impulsive Effects

2015 ◽  
Vol 2015 ◽  
pp. 1-26 ◽  
Author(s):  
Zuomao Yan
Keyword(s):  
Stochastic Analysis ◽  
Approximate Controllability ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Operators ◽  
Bounded Linear ◽  
Approximation Techniques ◽  
Approximately Controllable ◽  
Stochastic Functional

We discuss the approximate controllability of second-order impulsive neutral partial stochastic functional integrodifferential inclusions with infinite delay under the assumptions that the corresponding linear system is approximately controllable. Using the fixed point strategy, stochastic analysis, and properties of the cosine family of bounded linear operators combined with approximation techniques, a new set of sufficient conditions for approximate controllability of the second-order impulsive partial stochastic integrodifferential systems are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result.

QUADRATIC EQUATIONS IN HILBERTIAN OPERATORS AND APPLICATIONS

2009 ◽  
Vol 20 (11) ◽  
pp. 1431-1454
Author(s):  
VICTOR J. MIZEL ◽  
M. M. RAO
Keyword(s):  
Random Fields ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Infinite Dimensions ◽  
Bounded Linear ◽  
Quadratic Equations ◽  
Weak Ordering ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Adjoint Operators

In this paper bounded linear operators in Hilbert space satisfying general quadratic equations are characterized. Necessary and sufficient conditions for sets of operators satisfying two such equations to compare relative to a weak ordering are presented. In addition, averaging operators in finite dimensional spaces are determined, and in this case it is shown that they are unitary models for all projections. It is pointed out, by an example, that the latter result does not hold in infinite dimensions. A key application to certain second order random fields of Karhunen type is given. The main purpose is to present the structure of bounded non-self adjoint operators solving quadratic equations, and indicate their use.

scholarly journals Solutions of the system of operator equations $BXA=B=AXB$ via the *-order

2017 ◽  
Vol 32 ◽  
pp. 172-183 ◽  
Author(s):  
Mehdi Vosough ◽  
Mohammad Sal Moslehian
Keyword(s):  
Hilbert Space ◽  
General Solution ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Operator Equations ◽  
Bounded Linear ◽  
Mild Conditions ◽  
Necessary And Sufficient ◽  
System Of Operator Equations

In this paper, some necessary and sufficient conditions are established for the existence of solutions to the system of operator equations $BXA=B=AXB$ in the setting of bounded linear operators on a Hilbert space, where the unknown operator $X$ is called the inverse of $A$ along $B$. After that, under some mild conditions, it is proved that an operator $X$ is a solution of $BXA=B=AXB$ if and only if $B \stackrel{*}{ \leq} AXA$, where the $*$-order $C\stackrel{*}{ \leq} D$ means $CC^*=DC^*, C^*C=C^*D$. Moreover, the general solution of the equation above is obtained. Finally, some characterizations of $C \stackrel{*}{ \leq} D$ via other operator equations, are presented.

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