Strongly continuous operator cosine functions

Lutz, D.

doi:10.1007/bfb0069842

Cited by 22 publications

(9 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let X be a complex Banach space and A the generator of a strongly continuous cosine function R ~ t --~ Ct ~ £~'(X); then A x = C" (O) For fundamental facts on cosine functions we refer the reader to [1], [8], [11], [12].…”

Section: Mild Periodic Solutions In Banach Spacesmentioning

confidence: 99%

Spectral properties of cosine operator functions

Ciorǎnescu

Lizama

1988

Aeq. Math.

View full text Add to dashboard Cite

Let A be the generator of a cosine function C, t • R in a Banach space X; we shall connect the existence and uniqueness of a T-periodic mild solution of the equation u" = Au + f with the spectral property 1 • p(Cr) and, in case X is a Hilbert space, also with spectral properties of A.In [9] the following formula was established (~2 _ A) fo sinh ~(t -s)Csds = ((cosh (t -C,), t~R ,~e C , Spectral properties of cosine operator functions which directly implies 81 (*) # e p(Ct) ~ {(2; cosh (t = /~} c p(A) and sup I[((( 2 -A) -111 < + ~. ch ~t = ltO u r purpose in this paper is to investigate cases when this implication is an equivalence; in our considerations we are inspired by ideas of J. Priil3 in [10]. Namely, as in the case of semigroup generators, it turns out that the spectral condition 1 e p ( C r ) is equivalent to the existence and uniqueness of a T-periodic mild solution for the inhomogenous second order differential equation u" = Au + f, where f is a forcing periodic term (Theorem 1).In the case of a Hilbert space the results are completed by Theorem 2:AEQ. MATH. $1 _,vk(t)dt = f~ AS1 _,Stuk(O)dt-'~ S1-,C, vk(O)dt + S1-t C,_sfk(S)ds dt LJO J

show abstract

Section: Mild Periodic Solutions In Banach Spacesmentioning

confidence: 99%

Spectral properties of cosine operator functions

Ciorǎnescu

Lizama

1988

Aeq. Math.

View full text Add to dashboard Cite

show abstract

“…We use his ideas in the following proof. We note that for cosine operator functions there is a second proof by Lutz (1982), see Theorem 2.18, based on holomorphic functional calculus.…”

Section: The Generator: Basic Properties and Relationsmentioning

confidence: 99%

Sturm–Liouville operator functions

Früchtl

2018

Dissertationes Math.

View full text Add to dashboard Cite

Many special functions are solutions of both, a differential and a functional equation. We use this duality to solve a large class of abstract Sturm-Liouville equations, initiating a theory of Sturm-Liouville operator functions; cosine, Bessel, and Legendre operator functions are contained as special cases. This is part of a general concept of operator functions being multiplicative with respect to convolution of a hypergroupcontaining all representations of (hyper)groups, and further abstract Cauchy problems. Viele spezielle Funktionen lösen sowohl eine Differential-als auch eine Funktionalgleichung. Wir verwenden diese Dualität um eine große Klasse von abstrakten Sturm-Liouville-Gleichungen zu lösen. Hierfür wird eine Theorie von Sturm-Liouville-Operatorfunktionen angestoßen; Kosinus-, Bessel-und Legendre-Operatorfunktionen sind als Spezialfälle enthalten. Dies ist Teil eines allgemeinen Konzepts von Operatorfunktionen, die multiplikativ bezüglich einer Hypergruppen-Faltung sind; alle Darstellungen von (Hyper)gruppen und weitere abstrakte Cauchy-Probleme sind darin enthalten.iii

show abstract

“…Since we deal with values of possibly approaching 2, then we are compelled to assume that −A is the infinitesimal generator of a C 0 − operator cosine function. Even if the class of such generators is wider (see [25]), we restrict here the analysis to the self-adjoint case, which occurs mostly in applications.…”

Section: Fractional Wave Equationsmentioning

confidence: 99%