Well‐posedness of second order degenerate integro‐differential equations with infinite delay in vector‐valued function spaces

Abstract

We study the well-posedness of the second order degenerate differential equations with infinite delay:where A, B and M are closed linear operators in a Banach space satisfying D( A) ∩ D(B) = {0}, D( A) ∩ D(B) ⊂ D(M), a, b ∈ L 1 (R + ). Using operator-valued Fourier multiplier techniques, we give necessary and sufficient conditions for the well-posedness of this problem in Lebesgue-Bochner spaces L p (T; X ), periodic Besov spaces B s p,q (T; X ) and periodic Triebel-Lizorkin spaces F s p,q (T; X ).

Besov maximal regularity for a class of degenerate integro‐differential equations with infinite delay in Banach spaces

“…We observe here that periodic versions of (3.1) were studied in [5]. Variants of this problem in the periodic case have appeared in [8,10,11,12,13,14,33,34,35,36], many of them in the nondegenerate case.…”

“…In [8] by Arendt-Batty-Bu was used operator-valued Fourier multipliers to study maximal regularity of evolutionary differential equations in Banach spaces in the scale of vector valued Hölder spaces in both evolutionary problem and periodic boundary conditions. Subsequently for non-degenerate integro-differential equations both in the periodic and non periodic cases, operator-valued Fourier multipliers have been used by various authors to obtain well-posedness in various scales of function spaces: see [12,13,14,33,34,35,36,44] and the corresponding references. An overview of maximal regularity using the theory of operator-valued Fourier multipliers is contained in the survey paper [7].…”

Well-posedness for degenerate third order equations with delay and applications to inverse problems

“…In [12] S. Bu investigates, using the same method just described, the second-order degenerate equation [17,14] for second order degenerate differential equations with delay, and the paper by Chill and Srivastava [18] where L p -well-posedness for second-order differential equations in case M = Id is studied.…”

Maximal ‐regularity for fractional differential equations on the line

“…Time fractional differential equations with delay on periodic Lebesgue spaces L p 2π (X ), (X being a U M D space and 1 < p < ∞) have been treated in [8], [9], [10], [11], [27], [34]. See also [12], [13], [14], [18], [35], [36], [40], [42], [43].…”