scholarly journals Direct and Inverse Fractional Abstract Cauchy Problems

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1016 ◽  
Author(s):  
Mohammed AL Horani ◽  
Angelo Favini ◽  
Hiroki Tanabe
Keyword(s):  
Cauchy Problem ◽  
Direct Problem ◽  
Abstract Cauchy Problem ◽  
Real Interpolation ◽  
Interpolation Spaces ◽  
Cauchy Problems ◽  
Interpolation Theory ◽  
Basic Assumptions ◽  
Abstract Approach ◽  
Abstract Cauchy Problems

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.

scholarly journals Fractional Cauchy Problems for Infinite Interval Case-II

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1165
Author(s):  
Mohammed Al Horani ◽  
Mauro Fabrizio ◽  
Angelo Favini ◽  
Hiroki Tanabe
Keyword(s):  
Existence And Uniqueness ◽  
Direct Problem ◽  
Abstract Cauchy Problem ◽  
Continuous Functions ◽  
Cauchy Problems ◽  
Fractional Powers ◽  
Uniqueness Of Solutions ◽  
Infinite Intervals ◽  
Abstract Cauchy Problems ◽  
Densely Defined

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.

scholarly journals Local K-convoluted C-cosine functions and abstract cauchy problems

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2583-2598 ◽  
Author(s):  
Chung-Cheng Kuo
Keyword(s):  
Linear Operator ◽  
Abstract Cauchy Problem ◽  
Cosine Function ◽  
Equivalence Relations ◽  
Cauchy Problems ◽  
Bounded Linear ◽  
Unique Existence ◽  
Locally Integrable Function ◽  
Cosine Functions ◽  
Abstract Cauchy Problems

Let K : [0,T0)? F be a locally integrable function, and C : X ? X a bounded linear operator on a Banach space X over the field F(=R or C). In this paper, we will deduce some basic properties of a nondegenerate local K-convoluted C-cosine function on X and some generation theorems of local Kconvoluted C-cosine functions on X with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local K-convoluted C-cosine function on X with subgenerator A and the unique existence of solutions of the abstract Cauchy problem: U''(t)=Au(t)+f(t) for a.e. t ? (0, T0), u(0) = x, u'(0) = y when K is a kernel on [0, T0), C : X ? X an injection, and A : D(A) ? X ? X a closed linear operator in X such that CA ? AC. Here 0 < T0 ? ?, x,y ? X, and f ? L1,loc([0,T0),X).

scholarly journals Embedding functions and their role in interpolation theory

1996 ◽  
Vol 1 (3) ◽  
pp. 305-325 ◽  
Author(s):  
Evgeniy Pustylnik
Keyword(s):  
Weak Type ◽  
Real Interpolation ◽  
Interpolation Spaces ◽  
Interpolation Theory ◽  
Type Interpolation ◽  
Intermediate Space ◽  
Banach Couple ◽  
Alpha Beta

The embedding functions of an intermediate spaceAinto a Banach couple(A0,A1)are defined as its embedding constants into the couples(1αA0,1βA1),∀α,β>0. Using these functions, we study properties and interrelations of different intermediate spaces, give a new description of all real interpolation spaces, and generalize the concept of weak-type interpolation to any Banach couple to obtain new interpolation theorems.

scholarly journals New interpolation spaces and strict Hölder regularity for fractional abstract Cauchy problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Md Mansur Alam ◽  
Shruti Dubey ◽  
Dumitru Baleanu
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Analytic Semigroup ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Interpolation Spaces ◽  
Holder Regularity

AbstractWe know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP.

scholarly journals Nonuniqueness and wellposedness of abstract Cauchy problems in a Fréchet space

2001 ◽  
Vol 63 (1) ◽  
pp. 123-131 ◽  
Author(s):  
Peer Christian Kunstmann
Keyword(s):  
Cauchy Problem ◽  
Linear Operator ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Continuous Selection ◽  
Fréchet Space ◽  
Inhomogeneous Equation ◽  
Frechet Space ◽  
Closed Linear Operator ◽  
Cauchy Problems

Suppose that A is a closed linear operator in a Fréchet space X. We show that there always is a maximal subspace Z containing all x ∈ X for which the abstract Cauchy problem has a mild solution, which is a Fréchet space for a stronger topology. The space Z is isomorphic to a quotient of a Fréchet space F, and the part Az of A in Z is similar to the quotient of a closed linear operator B on F for which the abstract Cauchy problem is well-posed. If mild solutions of the Cauchy problem for A in X are unique it is not necessary to pass to a quotient, and we reobtain a result due to R. deLaubenfels.Moreover, we obtain a continuous selection operator for mild solutions of the inhomogeneous equation.

scholarly journals Local k-convoluted c-semigroups and complete second order abstract Cauchy problems

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6789-6797 ◽  
Author(s):  
Chung-Cheng Kuo
Keyword(s):  
Integrable Function ◽  
Abstract Cauchy Problem ◽  
Second Order ◽  
Local Times ◽  
Cauchy Problems ◽  
Bounded Linear ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
Locally Integrable Function ◽  
Abstract Cauchy Problems

Let C : X ? X be a bounded linear operator on a Banach space X over the field F(=R or C), and K : [0,T0)?F a locally integrable function for some 0 < T0 ? ?. Under some suitable assumptions, we deduce some relationship between the generation of a local (or an exponentially bounded) K-convoluted (C 0 0 C)-semigroup on X x X with subgenerator (resp., the generator) (0 I B A) and one of the following cases: (i) the well-posedness of a complete second-order abstract Cauchy problem ACP(A,B,f,x,y): w??(t) = Aw?(t) + Bw(t) + f (t) for a.e. t ?(0,T0) with w(0) = x and w?(0) = y; (ii) a Miyadera-Feller-Phillips-Hille- Yosida type condition; (iii) B is a subgenerator (resp., the generator) of a locally Lipschitz continuous local ?-times integrated C-cosine function on X for which A may not be bounded; (iv) A is a subgenerator (resp., the generator) of a local ?-times integrated C-semigroup on X for which B may not be bounded.

scholarly journals Normal periodic solutions for the fractional abstract Cauchy problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jennifer Bravo ◽  
Carlos Lizama
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Periodic Solution ◽  
Periodic Boundary ◽  
Abstract Cauchy Problem ◽  
Periodic Boundary Conditions ◽  
Sectorial Operator ◽  
Closed Linear Operator ◽  
Resolvent Set ◽  
Spectral Angle

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

Fractional abstract Cauchy problem on complex interpolation scales

2020 ◽  
Vol 23 (4) ◽  
pp. 1125-1140
Author(s):  
Andriy Lopushansky ◽  
Oleh Lopushansky ◽  
Anna Szpila
Keyword(s):  
Cauchy Problem ◽  
Time Derivative ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Sectorial Operator ◽  
Complex Interpolation ◽  
Bounded Domains ◽  
Initial Boundary Value Problems ◽  
Fractional Time Derivative ◽  
Initial Boundary Value

AbstractAn fractional abstract Cauchy problem generated by a sectorial operator is investigated. An inequality of coercivity type for its solution with respect to a complex interpolation scale generated by a sectorial operator with the same parameters is established. An application to differential parabolic initial-boundary value problems in bounded domains with a fractional time derivative is shown.

scholarly journals Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

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