Almost periodic solutions of differential equations in by Hino Y., et al. PDF

By Hino Y., et al.

Show description

Read Online or Download Almost periodic solutions of differential equations in Banach spaces PDF

Similar functional analysis books

Download e-book for iPad: An introduction to inverse scattering and inverse spectral by Khosrow Chadan, David Colton, Lassi Päivärinta, William

Inverse difficulties try and receive information regarding buildings via non-destructive measurements. This creation to inverse difficulties covers 3 valuable parts: inverse difficulties in electromagnetic scattering conception; inverse spectral idea; and inverse difficulties in quantum scattering conception.

John Garnett's Bounded Analytic Functions PDF

The publication is a bit turse. the writer can have incorporated extra info within the proofs.

Download e-book for kindle: Triangular and Jordan Representations of Linear Operators by M. S. Brodskii

During this publication we current the principles of the speculation of triangular and Jordan
representations of bounded linear operators in Hilbert area, a topic which has
arisen within the final 10-15 years.
It is celebrated that for each selfadjoint matrix of finite order there eXists
a unitary transformation which consists of it into diagonal shape. Geometrically this
means finite-dimensional Hilbert house, within which there's given a selfad-
joint operator A, is representable within the kind of the orthogonal sum of one-dimen-
sional subspaces invariant relative to A. greater than 60 years in the past David Hilbert
formulated the infinite-dimensional analog of this truth.
Any sq. matrix, based on Schur's theorem, may be decreased by way of
a sure unitary transformation to triangular form.

The first step within the thought of triangular representations of nonselfadjoint
operators working in infinite-dimensional areas used to be taken by way of M. S. Livsic [1]
in 1954. U sing the speculation of attribute features created by means of him, he con-
structed a triangular useful version of a bounded linear operator with nuclear
imaginary part. in a while, because of the investigations of L. A. Sahnovic
[1,2], A. V. Kuzel' [1,2], V. T. PoljackiT[l] and others, triangular practical
models of operators belonging to different sessions have been stumbled on. concurrently, within the
work of the current writer [1- 4], 1. C. Gohberg and M. G. KreIn, [1--6], Ju. 1. Ljubic
and V. 1. Macaev [1,2,3], V. 1. Macaev [1,2], V. M. BrodskiT [1], and V. M. Brod-
skiT and the current writer [1], the idea of summary triangular representations
was formulated. It was once proved specifically that each thoroughly non-stop
operator, and in addition each bounded operator with a very non-stop imaginary
component, whose eigenvalues are inclined to 0 sufficiently swiftly, is representable in
an imperative shape that is the common analog of the ri£ht part of formulation (1). An-
alogously, invertible operators, shut in a undeniable experience to unItary operators,
turned out to be attached with formulation (2).

J. Dieudonne's History of Functional Analysis PDF

Background of sensible research provides sensible research as a slightly advanced combination of algebra and topology, with its evolution stimulated by way of the advance of those branches of arithmetic. The publication adopts a narrower definition―one that's assumed to fulfill a number of algebraic and topological stipulations.

Extra info for Almost periodic solutions of differential equations in Banach spaces

Example text

Then the following inhomogeneous equation du = (−A + B(t))u + f (t) dt has a unique almost periodic solution u such that sp(u) ⊂ {λ + 2πk, k ∈ Z, λ ∈ sp(f )}. We now show Claim 2 Let the conditions of Claim 1 be satisfied except for the compactness of the resolvent of A . 21) dt has an exponential dichotomy if and only if the spectrum of the monodromy operator does not intersect the unit circle. Moreover, if A has compact resolvent, it has an exponential dichotomy if and only if all multipliers have modulus different from one.

SPECTRAL CRITERIA t x(t) = eiµt y(t) = U (t, s)eiµs y(s) + U (t, ξ)eiµξ f (ξ)dξ s t iµξ = U (t, s)x(s) + U (t, ξ)e f (ξ)dξ ∀t ≥ s. 15). The uniqueness of x(·) follows from that of the solution y(·) . We now prove the converse. Let y(t) be the unique almost periodic solution to the equation t U (t, ξ)eiµξ f (ξ)dξ, ∀t ≥ s. 16) s Then x(t) := e−iµt y(t) must be the unique solution to the following equation t x(t) = e−iµ(t−s) U (t, s)x(s) + e−iµ(t−ξ) U (t, ξ)f (ξ)dξ), ∀t ≥ s. 17) s And vice versa. We show that x(t) should be periodic.

Moreover, let f ∈ AP (X) such that σ(P ) ∩ {eiλ , λ ∈ sp(f )} = . 4) has an almost periodic solution which is unique in M(f ). Proof. 2 it follows that the function space M(f ) satisfies condition H. 8) we have σ(S(1)|M(f ) ) = eσ(D|M(f ) ) , where D|M(f ) is the generator of (S(t)|M(f ) )t≥0 . 6 we have σ(D|M(f ) ) = iΛ, where Λ = {λ + 2πk, λ ∈ sp(f ), k ∈ Z}. Hence, since eσ(D|M(f ) ) = eiΛ ⊂ eisp(f ) ⊂ eiΛ , we have σ(S(1)|M(f ) ) = eσ(D|M(f ) ) = eisp(f ) . σ(S(−1)|M(f ) ). σ(S(−1)|M(f ) )\{0} which follows from the commutativeness of the operator PˆM(f ) with the operator S(−1)|M(f ) , the above inclusion implies that 1 ∈ σ(T 1 |M(f ) ).

Download PDF sample

Almost periodic solutions of differential equations in Banach spaces by Hino Y., et al.


by William
4.4

Rated 4.78 of 5 – based on 30 votes