scholarly journals Complex powers of nondensely defined operators

2011 ◽  
Vol 90 (104) ◽  
pp. 47-64 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Mild Solutions ◽  
Acute Angle ◽  
Analytic Semigroups ◽  
Cauchy Problems ◽  
Growth Order ◽  
Fractional Powers ◽  
Complex Powers ◽  
Cosine Functions ◽  
Abstract Cauchy Problems ◽  
Densely Defined

The power (?A)b, b ? C is defined for a closed linear operator A whose resolvent is polynomially bounded on the region which is, in general, strictly contained in an acute angle. It is proved that all structural properties of complex powers of densely defined operators with polynomially bounded resolvent remain true in the newly arisen situation. The fractional powers are considered as generators of analytic semigroups of growth order r > 0 and applied in the study of corresponding incomplete abstract Cauchy problems. In the last section, the constructed powers are incorporated in the analysis of the existence and growth of mild solutions of operators generating fractionally integrated semigroups and cosine functions.

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 A Parameter Property of Classical Solutions of Cauchy Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Min He
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Semigroup Theory ◽  
Classical Solutions ◽  
Parabolic Partial Differential Equation ◽  
Cauchy Problems ◽  
Integrated Semigroup ◽  
The Cauchy Problem ◽  
Abstract Cauchy Problems ◽  
Densely Defined

This work is concerned with the abstract Cauchy problems that depend on parameters. The goal is to study continuity in the parameters of the classical solutions of the Cauchy problems. The situation considered in this work is when the operator of the Cauchy problem is not densely defined. By applying integrated semigroup theory and the results on continuity in the parameters ofC0-semigroup and integrated semigroup, we obtain the results on the existence and continuity in parameters of the classical solutions of the Cauchy problems. The application of the obtained abstract results in a parabolic partial differential equation is discussed in the last section of the paper.

scholarly journals Complex powers of operators

2008 ◽  
Vol 83 (97) ◽  
pp. 15-25
Author(s):  
Marko Kostic
Keyword(s):  
Complex Plane ◽  
Analytic Semigroup ◽  
Fractional Power ◽  
Growth Order ◽  
Fractional Number ◽  
Densely Defined Operator ◽  
Complex Powers ◽  
Densely Defined

We define the complex powers of a densely defined operator A whose resolvent exists in a suitable region of the complex plane. Generally, this region is strictly contained in an angle and there exists ? ? [0,?) such that the resolvent of A is bounded by O((1 + |?|)?) there. We prove that for some particular choices of a fractional number b, the negative of the fractional power (-A)b is the c.i.g. of an analytic semigroup of growth order r > 0.

scholarly journals Existence of Mild Solutions to Nonlocal Fractional Cauchy Problems via Compactness

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Rodrigo Ponce
Keyword(s):  
Banach Spaces ◽  
Fractional Derivatives ◽  
Mild Solutions ◽  
Cauchy Problems ◽  
Families Of Operators ◽  
Resolvent Families

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases0<α<1and1<α<2.

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