Second Order Differential Equations in Banach Space

Travis, Curtis C.; Webb, Gautam

doi:10.1016/b978-0-12-434160-9.50025-4

Cited by 107 publications

(64 citation statements)

References 18 publications

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“…Let A : D(A) ⊆ ϑ → X be the infinitesimal generator of a strongly continuous cosine family of bounded linear operators (C(ù))ù ∈R on Banach space ϑ. We denote by (S(ù))ù ∈R the sine function associated with (C(ù))ù ∈R which is defined by S(ù)ϑ = ù 0 C(s)ϑds, for ϑ ∈ X and u ∈ R. We refer them to [3,19] for the necessary concepts about cosine functions. After that we only mention a few results and notations about this matter needed to establish our results.…”

Section: Preliminariesmentioning

confidence: 99%

A Study of Second Order Impulsive Neutral Evolution Differential Control Systems with an Infinite Delay

Palani¹,

Gunasekar²,

Angayarkanni³

et al. 2018

Preprint

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Section: Preliminariesmentioning

confidence: 99%

A Study of Second Order Impulsive Neutral Evolution Differential Control Systems with an Infinite Delay

Palani¹,

Gunasekar²,

Angayarkanni³

et al. 2018

Preprint

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“…This type of equations have received a lot of attention in recent years [3]. There are many results concerning second-order di erential equations, see for example Fattorini [4], and Travis and Webb [5]. Useful for the study of abstract second order equations is the existence of an evolution system U(t, s) for the homogenous equation…”

Section: Y( ) = G(y) Y ( ) = H(y)mentioning

confidence: 99%

Second order evolution equations with nonlocal conditions

Benchohra

Nieto

Rezoug

2017

Demonstratio Mathematica

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“…The following theorem appears in C. Travis and G. Webb [8] and [9]: Theorem 1. Let C(t), t £ R, be a strongly continuous cosine family with associated sine family S(t), t £ R, and infinitesimal generator A.…”

Section: If C(t + S) + C(t -S) = 2c(t)c(s) For All St £ R C(0) = Imentioning

confidence: 99%

A remark on cosine families

Rankin¹

1980

Proc. Amer. Math. Soc.

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Abstract.Let C(t), t 6 R, be a strongly continuous cosine family and A its def infinitesimal generator. Then the set E = {x e X: C{t)x is once continuously differentiable in t on R} of the Banach space X is contained in the domain of {-Af for 0 < a < 1/2.The purpose of this note is to prove for a strongly continuous cosine family C(t), t £ R, defined on a Banach space X and with infinitesimal generator A, that the def set E = {x £ X: C(t)x is once continuously differentiable in t on R} is containedis the domain of the a power of the operator -A.A one parameter family C(t), t £ R, of bounded linear operators mapping the Banach space X into itself is called a strongly continuous cosine family if and only if C(t + s) + C(t -s) = 2C(t)C(s) for all s,t £ R, C(0) = I, C(t)x is continuous in t £ R for each fixed x £ X. The associated sine family is given by S(t)x = f0 C(s)x ds for x £ X and t £ R. The linear operator A: X ^ X defined by Ax = C"(0)x and with dense domain D(A) = {x £ X: C(t)x is twice continuously differentiable in / on R} is called the infinitesimal generator of C(t), t £ R. For other properties of cosine families used in this paper see [2] or [9].The following theorem appears in C. Travis and G. Webb [8] and [9]: Theorem 1. Let C(t), t £ R, be a strongly continuous cosine family with associated sine family S(t), t £ R, and infinitesimal generator A. The following statements are equivalent:(i) there exists a closed linear operator B on X such that B2 = A and B commutes with every operator in B(X, X) which commutes with A ; S(t) maps X into D(B) for each t £ R; BS(t)x is continuous in t £ Rfor each fixed x £ X;(ii) E = D(B), the domain of B.The conditions stated in part (i) of Theorem 1 have also been considered by H. Fattorini in [2] and [3]. Fattorini has shown in [3] that every strongly continuous cosine family defined on the Banach space Lp, 1

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Second Order Differential Equations in Banach Space

Cited by 107 publications

References 18 publications

A Study of Second Order Impulsive Neutral Evolution Differential Control Systems with an Infinite Delay

A Study of Second Order Impulsive Neutral Evolution Differential Control Systems with an Infinite Delay

Second order evolution equations with nonlocal conditions

A remark on cosine families

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