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 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 Perturbations of local C-cosine functions

2020 ◽  
Vol 65 (4) ◽  
pp. 585-597
Author(s):  
Chung-Cheng Kuo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Bounded Linear ◽  
Cosine Functions

"We show that $\tA+\tB$ is a closed subgenerator of a local $\tC$-cosine function $\tT(\cdot)$ on a complex Banach space $\tX$ defined by $$\tT(t)x=\sum\limits_{n=0}^\infty \tB^n\int_0^tj_{n-1}(s)j_n(t-s)\tC(|t-2s|)xds$$ for all $x\in\tX$ and $0\leq t<T_0$, if $\tA$ is a closed subgenerator of a local $\tC$-cosine function $\tC(\cdot)$ on $\tX$ and one of the following cases holds: $(i)$ $\tC(\cdot)$ is exponentially bounded, and $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ so that $\tB\tC=\tC\tB$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$; $(ii)$ $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ which commutes with $\tC(\cdot)$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$; $(iii)$ $\tB$ is a bounded linear operator on $\tX$ which commutes with $\tC(\cdot)$ on $\tX$. Here $j_n(t)=\frac{t^n}{n!}$ for all $t\in\Bbb R$, and $$\int_0^tj_{-1}(s)j_0(t-s)\tC(|t-2s|)xds=\tC(t)x$$ for all $x\in\tX$ and $0\leq t<T_0$."

scholarly journals Regularization of some classes of ultradistribution semigroups and sines

2010 ◽  
Vol 87 (101) ◽  
pp. 9-37 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Linear Operator ◽  
Higher Order ◽  
Cosine Function ◽  
Closed Linear Operator ◽  
Cauchy Problems ◽  
Ultradifferentiable Functions ◽  
Injective Operator ◽  
Regularized Semigroups ◽  
Abstract Cauchy Problems

We systematically analyze regularization of different kinds of ultradistribution semigroups and sines, in general, with nondensely defined generators and contemplate several known results concerning the regularization of Gevrey type ultradistribution semigroups. We prove that, for every closed linear operator A which generates an ultradistribution semigroup (sine), there exists a bounded injective operator C such that A generates a global differentiable C-semigroup (C-cosine function) whose derivatives possess some expected properties of operator valued ultradifferentiable functions. With the help of regularized semigroups, we establish the new important characterizations of abstract Beurling spaces associated to nondensely defined generators of ultradistribution semigroups (sines). The study of regularization of ultradistribution sines also enables us to perceive significant ultradifferentiable properties of higher-order abstract Cauchy problems.

scholarly journals Degenerate C-distribution cosine functions and degenerate C-ultradistribution cosine functions in locally convex spaces

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3075-3089
Author(s):  
Daniel Velinov ◽  
Marko Kostic ◽  
Stevan Pilipovic
Keyword(s):  
Linear Operator ◽  
Infinitesimal Generator ◽  
Cosine Function ◽  
Locally Convex Spaces ◽  
Regularizing Operator ◽  
Multivalued Linear Operator ◽  
C Distribution ◽  
Locally Convex ◽  
Cosine Functions ◽  
Convex Spaces

The main purpose of this paper is to investigate degenerate C-(ultra)distribution cosine functions in the setting of barreled sequentially complete locally convex spaces. In our approach, the infinitesimal generator of a degenerate C-(ultra)distribution cosine function is a multivalued linear operator and the regularizing operator C is not necessarily injective. We provide a few important theoretical novelties, considering also exponential subclasses of degenerate C-(ultra)distribution cosine functions.

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 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 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.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

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