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Complex Analysis and Operator Theory

Nonlocal Fractional Differential Equations and Applications Veli Shakhmurov1 Received: 27 December 2019 / Accepted: 5 May 2020 © Springer Nature Switzerland AG 2020

Abstract The regularity properties of nonlocal fractional differential equations in Banach spaces are studied. Uniform L p -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. Particularly, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, maximal regularity properties of nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractıonal differential equations and the system of nonlocal fractıonal differential equations are studied. Keyword Fractional-differential equations · Sobolev-Lions spaces · Abstract differential equations · Maximal L p regularity · Abstract parabolic equations · Operator-valued multipliers Mathematics Subject Classification 47GXX · 34L30 · 34A12 · 34A40 · 47DXX · 43AXX

1 Introduction, Notations and Background In the last years, the maximal regularity properties of boundary value problems (BVPs) for abstract differential equations (ADEs) have found many applications in PDE and pseudo DE with applications in physics (see [1,2,6–9,15–19,22,23] and the references therein). ADEs have found many applications in fractional differential equations (FDEs), pseudo-differential equations (PsDE) and PDEs. FDEs were treated e.g. in [4,7,10–12,14,21]. The regularity properties of FDEs have been studied e.g. in [5,11,12,20]. The existence and uniqueness of solution to fractional ADEs were stud-

Communicated by Daniel Aron Alpay.

B 1

Veli Shakhmurov [email protected] Department of Mechanical Engineering, Istanbul Okan University, Akfirat, Tuzla, Istanbul 34959, Turkey 0123456789().: V,-vol

49

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V. Shakhmurov

ied e.g. in [3,11,12]. Regularity properties of nonlocal ADEs were investigated e.g. in [15–17]. The main objective of the present paper is to discuss the L p (R; H )-maximal regularity of the fractional ADE with parameter a ∗ D γ u + A ∗ u + λu = f (x) , x ∈ R = (−∞, ∞) ,

(1.1)

where a is complex valued function, λ is a complex parameter, A = A (x) is a linear operator function in a Hilbert space H , and D γ is Riemann-Liouville type fractional derivatives of order γ ∈ ( 1, 2] , i.e. d2 1 D u=  (2 − γ ) d x 2 γ

x 0

u (y) dy (x − y)γ −1

,

(1.2)

here  (γ ) is Gamma function for γ > 0 (see e.g. [7,10]) and the convolutions a ∗ D γ u , A ∗ u are defined in the distribution sense (see e.g. [1, Section 3]). For αi ∈ [0, ∞) and = (α1 , α2 , . . . , αn ). Here, D α = D1α1 D2α2 , ., Dnαn . Let E be a Banach space. Here, L p (; E) denotes the space of strongly measurable E-valued functions that are defined on the measurable subset  ⊂ Rn with the norm given by ⎛  f  L p (;E)

=⎝

⎞1



p

f

p (x) E

d x ⎠ , 1 ≤ p < ∞.



Let E 1 and