Well-posedness of degenerate differential equations with fractional derivative in vector-valued functional spaces

Abstract

In this paper, we study the well‐posedness of the degenerate differential equations with fractional derivative Dαfalse(Mufalse)false(tfalse)=Aufalse(tfalse)+ffalse(tfalse),false(0≤t≤2πfalse) in Lebesgue–Bochner spaces Lpfalse(double-struckT;Xfalse), periodic Besov spaces Bp,qsfalse(double-struckT;Xfalse) and periodic Triebel–Lizorkin spaces Fp,qsfalse(double-struckT;Xfalse), where A and M are closed linear operators in a complex Banach space X satisfying D(A)⊂D(M), α>0 and Dα is the fractional derivative in the sense of Weyl. Using known operator‐valued Fourier multiplier results, we completely characterize the well‐posedness of this problem in the above three function spaces by the R‐bounedness (or the norm boundedness) of the M‐resolvent of A.

Well‐posedness of degenerate fractional differential equations with finite delay in complex Banach spaces

“…We notice that the well-posedness of the equations (𝑃) was studied in [10] in the special case when 𝐹 = 0. Thus, our results also generalize the previous known results obtained in [4,5,8,9,15]. This paper is organized as follows.…”

Normal periodic solutions for the fractional abstract Cauchy problem

“…More precisely, assuming that 1 < α ≤ 2 and X satisfy a geometrical hypothesis, in [22,Theorem 3.15] the authors showed that for all f ∈ L p 2π (R, X) there exists unique u ∈ H α,p 2π (R, X) ∩ L p 2π (R, D(A)) satisfying (1) if and only if {(im) α } m∈Z ⊆ ρ(A) and the set {(im) α (A + (im) α I) -1 } m∈Z is Rademacher bounded (or R-bounded). See also [3,Theorem 3.3] for the analogous result in the case 0 < α ≤ 1 and [5][6][7]18] for extensions of this result to more general models.…”

“…Therefore, we can conclude that GL D α t u N (t) + Au N (t) = f N (t) i.e. u N is a 2π -periodic solution of (5). Now, we study the convergence normal of the series.…”

“…We say that a sequence of functions u m : R → X, m ∈ ⊆ Z is a normal 2π -periodic solution of (5) if u m is 2π -periodic, u m (t) ∈ D(A) for all t ∈ R and satisfies (5) for each m ∈ and the series n∈Z 1 (n)u n (t) is normally convergent.…”