Displaying similar documents to “Laplace transform generation theorems and local Cauchy problems.”

On α-times integrated C-semigroups and the abstract Cauchy problem

Chung-Cheng Kuo, Sen-Yen Shaw (2000)

Studia Mathematica

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This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: u ' ( t ) = A u ( t ) + t α - 1 Γ ( α ) x , t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every x...

Moment sequences and abstract Cauchy problems

Claus Müller (2003)

Commentationes Mathematicae Universitatis Carolinae

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We give a new characterization of the solvability of an abstract Cauchy problems in terms of moment sequences, using the resolvent operator at only one point.

On an integral transform by R. S. Phillips

Sten Bjon (2010)

Open Mathematics

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The properties of a transformation f ↦ f ˜ h by R.S. Phillips, which transforms an exponentially bounded C 0-semigroup of operators T(t) to a Yosida approximation depending on h, are studied. The set of exponentially bounded, continuous functions f: [0, ∞[→ E with values in a sequentially complete L c-embedded space E is closed under the transformation. It is shown that ( f ˜ h ) k ˜ = f ˜ h + k for certain complex h and k, and that f ( t ) = lim h → 0 + f ˜ h ( t ) , where the limit is uniform in t on compact subsets of the positive real line. If...