Hölder Continuous Solutions of Second Order Degenerate Differential Equations with Finite Delay

2019 ◽  
Vol 74 (1) ◽  
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Finite Delay ◽  
Order Degenerate ◽  
Degenerate Differential Equations

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].

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.

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