Fractional Cauchy Problems for Infinite Interval Case-II

Mohammed Al Horani,Angelo Favini,Mauro Fabrizio,Hiroki Tanabe

doi:10.3390/math7121165

Mohammed Al Horani, Angelo Favini + Show 2 more

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https://doi.org/10.3390/math7121165

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Abstract

We consider fractional abstract Cauchy problems on infinite intervals. A fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces is considered. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Required conditions on spaces are also given, guaranteeing the existence and uniqueness of solutions. The fractional powers of the involved operator B X have been investigated in the space which consists of continuous functions u on [ 0 , ∞ ) without assuming u ( 0 ) = 0 . This enables us to refine some previous results and obtain the required abstract results when the operator B X is not necessarily densely defined.

Highlights

In recent years, many studies were devoted to the problem of recovering the solution u to BMu − Lu = f (1)where B, M, and L are closed linear operators on the complex Banach space E with D ( L) ⊆ D ( M ),0 ∈ ρ( L), f ∈ E and u is unknown
Suppose that A is a not necessarily densely defined closed linear operator in a Banach space X satisfying (i) ρ(− A) ⊃ {λ; |argλ| < π − ω }, 0 < ω < π; (ii) λ(λ + A)−1 is uniformly bounded in each smaller sector {λ; |argλ| < π − ω − e}, 0 < e < π − ω; and (iii) kλ(λ + A)−1 kL(X ) ≤ M for λ > 0 with some M > 0
By virtue of Proposition 1, the fractional power ( BX + e)α of BX + e is defined for α > 0, and the followings hold: ( BX + e)α ( BX + e) β = ( BX + e)α+ β, α > 0, β > 0, ( BX + e)1 = BX + e, ( BX + e)−α = (( BX + e)α )−1 is bounded, (25)

Summary

Introduction

D ( BX ) = {u ∈ C1 ([0, ∞); X ); u and u0 are uniformly continuous and bounded in [0, ∞), u(0) = 0 = u0 (0)}, BX u = u0 for u ∈ D ( BX ) , and M, L are two closed linear operators in the complex Banach space E satisfying k M(λM − L)−1 k ≤ Suppose that A is a not necessarily densely defined closed linear operator in a Banach space X satisfying (i) ρ(− A) ⊃ {λ; |argλ| < π − ω }, 0 < ω < π; (ii) λ(λ + A)−1 is uniformly bounded in each smaller sector {λ; |argλ| < π − ω − e}, 0 < e < π − ω; and (iii) kλ(λ + A)−1 kL(X ) ≤ M for λ > 0 with some M > 0.

Results

Conclusion