scholarly journals Well-posedness of second order degenerate differential equations in Hölder continuous function spaces

2016 ◽  
Vol 34 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Function Spaces ◽  
Second Order ◽  
Well Posedness ◽  
Hölder Continuous ◽  
Order Degenerate ◽  
Degenerate Differential Equations

Well-posedness of Third Order Differential Equations in Hölder Continuous Function Spaces

2018 ◽  
Vol 62 (4) ◽  
pp. 715-726
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Function Spaces ◽  
Linear Operators ◽  
Well Posedness ◽  
Third Order ◽  
Hölder Continuous ◽  
Vector Valued ◽  
Closed Linear Operators

AbstractIn this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$-Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$-well-posedness for the third order differential equations $au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$, ($t\in \mathbb{R}$), where $A,B$ are closed linear operators on a Banach space $X$ such that $D(A)\subset D(B)$, $a\in \mathbb{C}$ and $0<\unicode[STIX]{x1D6FC}<1$.

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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