Semigroups of continuous module homomorphisms on complex complete random normed modules*

Thang, Dang Hung; Son, Ta Cong; Thinh, Nguyen An

doi:10.1007/s10986-019-09442-z

Cited by 6 publications

(26 citation statements)

References 19 publications

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“…Then for any x ∈ D(A), ∨ t>0 ||A t x|| belongs to L 0 + ( ). In 2019, in order to generalize Proposition 2 in Thang et al, 4…”

Section: The Homogeneous Initial Value Problem In Complete Rn Modulesmentioning

confidence: 99%

“…Definition Let

S

be an

R N

module. A mapping

f : false[a, b false] \to S

is said to be

L^{0}

‐Lipschitz on a finite real interval

false[a, b false]

if there exists a

ξ \in L_{+}^{0} false(scriptF false)

such that

‖ f (t_{1}) - f (t_{2}) ‖ \leq ξ | t_{1} - t_{2} | \forall t_{1}, t_{2} \in [a, b] .

…”

Section: Preliminariesmentioning

confidence: 99%

“…The following example shows that it is necessary to require the

L^{0}

‐Lipschitz assumption in Proposition ; one can also refer to Thang et al and Zhang et al…”

Section: Preliminariesmentioning

confidence: 99%

“…It is well known that the classical theory of

C_{0}

semigroups of bounded linear operators is a powerful tool in the study of the first order Cauchy problem . Recently, the theory of

C_{0}

semigroups of almost surely (a.s.) bounded linear operators has obtained some advances, and we found that the Cauchy initial value problem in the random setting plays an important role in the study of random operator equations and random integral equations, so it is natural to study the Cauchy initial value problem in the random setting.…”

Section: Introductionmentioning

confidence: 99%

“…Later on, random metric spaces, random normed modules (briefly,

R N

modules), and random locally convex modules formed the three basic frameworks of random functional analysis. In particular, the theory of

R N

modules, together with applications to Lebesgue‐Bochner function spaces, conditional convex risk measures, and a.s. bounded random linear operators, has undergone a deep and systematic development . The purpose of this paper is devoted to studying, for the first time in the literature on the theory of

R N

modules, two kinds of Cauchy initial value problems with respect to an a.s. bounded

C_{0}

semigroup of continuous module homomorphisms.…”

Section: Introductionmentioning

confidence: 99%

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The Cauchy initial value problem in complete random normed modules

Zhang

Wang

et al. 2019

Math Meth Appl Sci

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We study, for the first time in the literature on the theory of random functional analysis, the Cauchy initial value problem in complete random normed modules. Under the L0‐Lipschitz assumption on the solution, we prove that two kinds of Cauchy initial value problems with respect to almost surely bounded C0 semigroups of continuous module homomorphisms are well‐posed. Moreover, the counterexample also shows that it is necessary to require the almost sure boundedness for such C0 semigroups.

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“…Then for any x ∈ D(A), ∨ t>0 ||A t x|| belongs to L 0 + ( ). In 2019, in order to generalize Proposition 2 in Thang et al, 4…”

Section: The Homogeneous Initial Value Problem In Complete Rn Modulesmentioning

confidence: 99%

“…Definition Let