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Well-posedness of second order degenerate differential equations in vector-valued function spaces

Shangquan Bu — 2013

Studia Mathematica

Using known results on operator-valued Fourier multipliers on vector-valued function spaces, we give necessary or sufficient conditions for the well-posedness of the second order degenerate equations (P₂): d/dt (Mu’)(t) = Au(t) + f(t) (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu’)(0) = (Mu’)(2π), in Lebesgue-Bochner spaces L p ( , X ) , periodic Besov spaces B p , q s ( , X ) and periodic Triebel-Lizorkin spaces F p , q s ( , X ) , where A and M are closed operators in a Banach space X satisfying D(A) ⊂ D(M). Our results...

Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces

Shangquan BuYi Fang — 2008

Studia Mathematica

We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay ( P ) u ' ' ( t ) + α u ' ( t ) + d / d t ( - t b ( t - s ) u ( s ) d s ) = A u ( t ) - - t a ( t - s ) A u ( s ) d s + f ( t ) (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), u’(0) = u’(2π), where A is a closed operator in a Banach space X, α ∈ ℂ, and a,b ∈ L¹(ℝ₊). We use Fourier multipliers to characterize maximal regularity for (P). Using known results on Fourier multipliers, we find suitable conditions on the kernels a and b under which necessary and sufficient conditions...

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