Spectral properties of cosine operator functions

Abstract

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

Periodic solutions of second order differential equations in Banach spaces

“…We also obtain a characterization in the case where the operator A generates a cosine function on X. As a consequence of the foregoing theorem, we obtain the following characterization in Hilbert spaces (see also [11,Theorem 2]). However, when N = 1, this condition is never satisfied.…”

“…As in the proof of Theorem 4.6 or, alternatively, using the arguments as in [11,Theorem 2] we can deduce that u n (0) and u n (0) converge to some x ∈ E and y ∈ X, respectively, as n → ∞. More precisely to verify that x ∈ E, we make use of the identity…”

“…By a result of Cioranescu-Lizama [11], it is known that if A is the generator of a cosine function {C(t)} t∈R then the existence of a unique mild periodic solution of class C 1 for (4.1) is characterized by the condition 1 ∈ ρ(C(2π)) which , in Hilbert spaces, is equivalent to the boundedness of the set…”

On the mild solutions of higher‐order differential equations in Banach spaces

“…t s S(t − τ) f (τ)dτ, (1.4) has been also studied in [3,8,18]. Recently, Arendt and Batty [1], Schweiker [20], and Schüler and Phóng [19] studied the first-and second-order differential equations, in which A is not the generator of a C 0 -semigroup or of a cosine family, respectively.…”

Some applications of Fejer’s theorem to operator cosine functions in Banach spaces

“…In the case of a Hilbert space, I. Cioranescu and C. Lizama [1] proved the following result: ( * ) 1 ∈ ρ(C(2π)) if and only if − N however, the problem of the validity of ( * ) in general Banach spaces was left open. Results of this type for C 0 -semigroups, as well as applications, were obtained by different authors (see [8], [6] and the references therein).…”