Functional Analysis

Yosidå, Kôsaku

doi:10.1007/978-3-662-11791-0

Cited by 548 publications

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“…The same arguments as in the last paragraph of the proof of Theorem 5 now show that v' = Cv, v(0) =fi This means that C generates a strongly continuous semigroup. The well-known theory of fractional powers of closed operators developed by Bochner, Phillips and Balakrishnan (see [15] for a readable exposition) assures us that -C has a square root which generates a holomorphic semigroup. If we let iA = ( -C)1'2, then clearly A2 = C, and A is closed and densely defined with nonempty resolvent.…”

Section: Now \[(T(t) + T(-t))f+(t(t)-t(-t))g]mentioning

confidence: 99%

Theory of Random Evolutions with Applications to Partial Differential Equations

Griego¹,

Hersh²

1971

Transactions of the American Mathematical Society

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Abstract. The selection from a finite number of strongly continuous semigroups by means of a finite-state Markov chain leads to the new notion of a random evolution. Random evolutions are used to obtain probabilistic solutions to abstract systems of differential equations. Applications include one-dimensional first order hyperbolic systems. An important special case leads to consideration of abstract telegraph equations and a generalization of a result of Kac on the classical 7¡-dimensional telegraph equation is obtained and put in a more natural setting. In this connection a singular perturbation theorem for an abstract telegraph equation is proved by means of a novel application of the classical central limit theorem and a representation of the solution for the limiting equation is found in terms of a transformation formula involving the Gaussian distribution.In a little-known section of an out-of-print book [7], Mark Kac has considered a particle which moves on a line at speed v, and reverses direction according to a Poisson process with intensity a. After showing that such a motion is governed by a pair of partial differential equations, Kac comments, "The amazing thing is that these two equations can be combined into a hyperbolic equation"-namely, the telegraph equation, 1W =

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Section: Now \[(T(t) + T(-t))f+(t(t)-t(-t))g]mentioning

confidence: 99%

Theory of Random Evolutions with Applications to Partial Differential Equations

Griego¹,

Hersh²

1971

Transactions of the American Mathematical Society

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“…REMARK 3.1. By virtue of (3.3), (3.6), and Corollary 2, p. 241 of [9] we have that D{F) is dense in D(A …”

Section: Jo IImentioning

confidence: 85%

Exponential representation of solutions to an abstract semi-linear differential equation

Webb¹

1977

Pacific J. Math.

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“…Thus x * ∈ Ker A * ∩ Im A * . But because of (2.18), Ker A * ∩ Im A * = {0} by the Yosida mean ergodic theorem [22]. So x * = 0, and this implies ( * ).…”

Section: Lemma 5 (A Markus [15]) Let a Be A Bounded Linear Operatomentioning

confidence: 88%

Metric domains, holomorphic mappings and nonlinear semigroups

Reich

Shoikhet²

1998

Abstract and Applied Analysis

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Abstract. We study nonlinear semigroups of holomorphic mappings on certain domains in complex Banach spaces. We examine, in particular, their differentiability and their representations by exponential and other product formulas. In addition, we also construct holomorphic retractions onto the stationary point sets of such semigroups. Introductionwhere the limit is taken with respect to the topology of D.A subset W of D is said to be the stationary point set of S if it consists of all the points a ∈ D such that F t (a) = a for all t ∈ (0, T).In other words,1991 Mathematics Subject Classification. Primary: 32H15, 34G20, 46G20, 47H17, 47H20.

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Functional Analysis

Cited by 548 publications

References 0 publications

Theory of Random Evolutions with Applications to Partial Differential Equations

Theory of Random Evolutions with Applications to Partial Differential Equations

Exponential representation of solutions to an abstract semi-linear differential equation

Metric domains, holomorphic mappings and nonlinear semigroups

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