By Yoshiyuki Hino, Toshiki Naito, Nguyen VanMinh, Jong Son Shin
This monograph offers fresh advancements in spectral stipulations for the lifestyles of periodic and virtually periodic recommendations of inhomogenous equations in Banach areas. a few of the effects signify major advances during this sector. particularly, the authors systematically current a brand new procedure in keeping with the so-called evolution semigroups with an unique decomposition approach. The publication additionally extends classical concepts, akin to fastened issues and balance equipment, to summary useful differential equations with functions to partial sensible differential equations. virtually Periodic ideas of Differential Equations in Banach areas will attract an individual operating in mathematical research.
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Extra resources for Almost periodic solutions of differential equations in Banach spaces
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1 B anach space X (U(t, s) ) t ? s , (t, s E R) from a i -periodic strongly continuous evolutionary process A family of b ounded linear operators to itself is called if the following conditions are satisfied: i) ii) U(t, t) = I v) t E R, U(t, s)U(s, r) = U(t, r) iii) The map iv) for all U(t + (t, s) U(t, s) x t ;::: s ;::: r , is continuous for every fixed = U(t, s) for all t ;::: s , t s I I U(t, s) 1 1 < New ( - ) for some positive N, w I, s+ f-t for all x E X, 1) independent of t ;::: s . If it does not cause any danger of confusion, for the sake of simplicity, we shall often call I-periodic strongly continuous evolutionary process (evolutionary) process.
Then the following inhomogeneous equation du dt = (-A + B (t) )u + f(t) has a unique almost periodic solution u such that sp(u) C {A + 2rrk , k E Z , A E sPun . We now show Claim 2 Let the conditions of Claim i 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.
We suppose that g( t, x ) is Lipschitz continuous with coefficient k and the Nemystky operator F defined by (Fv) ( t ) = g(t, v(t) ) , Vt E R acts in M. Below we can assume that M is any closed subspace of the space of all bounded continuous functions BC(R, X) . We consider the operator L in BC(R, X) . If (U(t, s)) t > s is strongly continuous, then L is a single-valued operator from D (L) C BC(R�X) to BC(R, X) . (U(t, s)) t > s be strongly continuous and Eq. 4) be uniquely solvable in M . Then for sufficiently small k , Eq.