Well-Posedness of Third Order Degenerate Differential Equations with Finite Delay in Banach Spaces

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Well Posedness ◽  
Third Order ◽  
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].

COERCIVE SOLVABILITY CONDITIONS OF THE THREE-ORDER DEGENERATE DIFFERENTIAL EQUATIONS

2020 ◽  
pp. 62-67
Author(s):  
Zh.B. Yeskabylova ◽  
◽  
K.N. Оspanov ◽  
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Third Order ◽  
Existence Of A Solution ◽  
Entire Axis ◽  
Solvability Conditions ◽  
Weighted Integral ◽  
Order Degenerate ◽  
Degenerate Differential Equations

In this paper, we consider one class of the singular nonlinear third-order differential equations given on the entire axis. We show sufficient conditions for the existence of a solution to this equation and the satisfiability of the coercive estimate for solution. The considered equation has the following features. Its intermediate coefficient is not bounded and does not obey to a lower coefficient. In the literature, such equations are called the degenerate differential equations. Further, the corresponding differential operator is not semi-bounded: its energy space may not belong to the Sobolev classes. Previously, the solvability questions of the third-order singular differential equations was studied only in the case that their intermediate coefficients are equal to zero. The main result of this work is proved on the basis of one separability theorem for the linear third-order degenerate differential operators, Schauder's fixed point theorem and some Hardy type weighted integral inequalities.

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