By Hino Y., et al.

Show description

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

Best functional analysis books

The Elements of Operator Theory

"The writer endeavors to provide the options and concepts as a substitute to the computational procedure, attempting to steer clear of lengthy calculations by way of stressing the mathematical strategies at the back of the statements. . . . many difficulties [are] said in the course of the e-book, quite often observed by means of tricks. "—Mathematical studies (review of the 1st edition)"This is a rigorous, logically well-organized textbook offering simple rules and common concept of operators.

Abstract Volterra Integro-Differential Equations

The idea of linear Volterra integro-differential equations has been constructing swiftly within the final 3 many years. This e-book presents a simple to learn concise creation to the idea of ill-posed summary Volterra integro-differential equations. a massive a part of the examine is dedicated to the examine of varied varieties of summary (multi-term) fractional differential equations with Caputo fractional derivatives, basically from their priceless value in modeling of varied phenomena showing in physics, chemistry, engineering, biology and lots of different sciences.

The Complex Analytic Theory of Teichmuller Spaces

An available, self-contained therapy of the advanced constitution of the Teichmüller moduli areas of Riemann surfaces. complicated analysts, geometers, and particularly string theorists (! ) will locate this paintings essential. The Teichmüller area, parametrizing the entire a number of advanced buildings on a given floor, itself includes (in a very typical method) the advanced constitution of a finite- or infinite-dimensional complicated manifold.

Additional resources for Almost periodic solutions of differential equations in Banach spaces

Example text

Then for all invariant subspaces M satisfying condition H, σ(PˆM )\{0} = σ(P )\{0}. Proof. For u, v ∈ M , consider the equation (λ − PˆM )u = v . It is equivalent to the equation (λ − P (t))u(t) = v(t), t ∈ R. If λ ∈ ρ(PˆM )\{0} , for every v the first equation has a unique solution u , and u ≤ R(λ, PˆM ) v . Take a function v ∈ M of the form v(t) = yeiµt , for some µ ∈ R ; the existence of such a µ is guaranteed by the axioms of condition H. Then the solution u satisfies u ≤ R(λ, PˆM ) y . Hence, for every y ∈ X the solution of the equation (λ − P (0))u(0) = y has a unique solution u(0) such that u(0) ≤ sup u(t) ≤ R(λ, PˆM ) sup v(t) ≤ R(λ, PˆM ) y .

E. Q(t) = e−iµ V (t, t− 1) and (Tµh )h≥0 denote the evolution semigroup associated with the evolutionary process (V (t, s))t≥s . Then by the same argument as above we can show that since σ(Tµh ) = e−iµ σ(T h ), σ(Tµh ) ∩ S 1 = . 2, the following equation t V (t, ξ)f (ξ)dξ, ∀t ≥ s y(t) = V (t, s)y(s) + s has a unique almost periodic solution y(·) . Let x(t) := eiµt y(t) . Then 40 CHAPTER 2. 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.

55, Chap. 2]) that if the spectrum of the monodromy operator does not circle the origin (of course, it should not contain the origin), then the evolution operators admit Floquet representation. In the example below, in general, Floquet representation does not exist. For instance, if the sectorial operator A has compact 44 CHAPTER 2. SPECTRAL CRITERIA resolvent, then monodromy operator is compact (see [90] for more details). Thus, if dimX = ∞, then monodromy operators cannot be invertible. However, the above results can apply.

Download PDF sample

Rated 4.85 of 5 – based on 40 votes