Normal periodic solutions for the fractional abstract Cauchy problem

Abstract

Abstract We show that if A is a closed linear operator defined in a Banach space X and there exist t 0 ≥ 0 $t_{0} \geq 0$ and M > 0 $M>0$ such that { ( i m ) α } | m | > t 0 ⊂ ρ ( A ) $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$ , the resolvent set of A, and ∥ ( i m ) α ( A + ( i m ) α I ) − 1 ∥ ≤ M for all | m | > t 0 , m ∈ Z , $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ then, for each 1 p < α ≤ 2 p $\frac{1}{p}<\alpha \leq \frac{2}{p}$ and 1 < p < 2 $1< p < 2$ , the abstract Cauchy problem with periodic boundary conditions { D t α G L u ( t ) + A u ( t ) = f ( t ) , t ∈ ( 0 , 2 π ) ; u ( 0 ) = u ( 2 π ) , $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ where D α G L $_{GL}D^{\alpha }$ denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each f ∈ L 2 π p ( R , X ) $f\in L^{p}_{2\pi }(\mathbb{R}, X)$ that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle ϕ A ∈ ( 0 , α π / 2 ) $\phi _{A} \in (0, \alpha \pi /2)$ and ∫ 0 2 π f ( t ) d t ∈ Ran ( A ) $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$ .

Keywords

• Grünwald–Letnikov fractional order derivative
• Periodic solution
• Normal convergence