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On the theory of remediability

Hassan Emamirad (2003)

Banach Center Publications

Suppose G ₁ ( t ) t ≥ 0 and G ₂ ( t ) t ≥ 0 are two families of semigroups on a Banach space X (not necessarily of class C₀) such that for some initial datum u₀, G₁(t)u₀ tends towards an undesirable state u*. After remedying by means of an operator ρ we continue the evolution of the state by applying G₂(t) and after time 2t we retrieve a prosperous state u given by u = G₂(t)ρG₁(t)u₀. Here we are concerned with various properties of the semigroup (t): ρ → G₂(t)ρG₁(t). We define (X) to be the space of remedial operators for...

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

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

Studia Mathematica

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 in ( λ - A ) - 1 C ( X ) .

α-times integrated semigroups: local and global

Miao Li, Quan Zheng (2003)

Studia Mathematica

We investigate the relations between local α-times integrated semigroups and (α + 1)-times integrated Cauchy problems, and then the relations between global α-times integrated semigroups and regularized semigroups.

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