Direct and Inverse Fractional Abstract Cauchy Problems

Mohammed Al Horani,Hiroki Tanabe,Angelo Favini

doi:10.3390/math7111016

Mohammed Al Horani, Hiroki Tanabe + Show 1 more

Open Access

https://doi.org/10.3390/math7111016

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Abstract

We are concerned with a fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Moreover, we succeeded in handling related inverse problems, extending the treatment given by Alfredo Lorenzi. Some basic assumptions on the involved operators are also introduced allowing application of the real interpolation theory of Lions and Peetre. Our abstract approach improves previous results given by Favini–Yagi by using more general real interpolation spaces with indices θ , p, p ∈ ( 0 , ∞ ] instead of the indices θ , ∞. As a possible application of the abstract theorems, some examples of partial differential equations are given.

Highlights

Consider the abstract equation BMu − Lu = f (1)where B, M, L are closed linear operators on the complex Banach space E, the domain of L is contained in domain of M, i.e., D ( L) ⊆ D ( M ), 0 ∈ ρ( L), the resolvent set of L, f ∈ E and u is the unknown
Our abstract approach improves previous results given by Favini–Yagi by using more general real interpolation spaces with indices θ, p, p ∈
BMu − Lu = f where B, M, L are closed linear operators on the complex Banach space E, the domain of L is contained in domain of M, i.e., D ( L) ⊆ D ( M ), 0 ∈ ρ( L), the resolvent set of L, f ∈ E and u is the unknown

Summary

Introduction

Where B, M, L are closed linear operators on the complex Banach space E, the domain of L is contained in domain of M, i.e., D ( L) ⊆ D ( M ), 0 ∈ ρ( L), the resolvent set of L, f ∈ E and u is the unknown. By using real interpolation spaces, see [3,4], suitable assumptions on the operators B, M, L guarantee that (1) has a unique solution. Such a result was improved by Favini, Lorenzi and Tanabe in [5], see [6,7,8]. Let ( E, D ( B))θ,∞ , 0 < θ < 1, denote the real interpolation space between E and D ( B). We refer to Guidetti [10] and Bazhlekova [11]

Fundamental Results

Fractional Derivative

Inverse Problems

Application

Conclusions