Almost periodic solutions for nonlinear delay evolutions with nonlocal initial conditions

作者：Vrabie Ioan I^*

来源：Journal of Evolution Equations, 2013, 13(3): 693-714.

DOI：10.1007/s00028-013-0198-y

• Differential delay evolution equation
• nonlocal delay initial condition
• almost periodic solutions
• nonresonance condition
• nonlinear hyperbolic equations

We consider the nonlinear delay differential evolution equation
{u'(t) is an element of Au(t) + f(t, u(i)), t is an element of R+, {u'(t) = g(u)(t), t is an element of [-tau,0]
where iota >= 0,X is a real Banach space, A is the infinitesimal generator of a nonlinear semigroup of contractions whose Lipschitz seminorm decays exponentially as t -> e(-omega t) when t -> +infinity and f : R+ X C([-tau, 0]; D (A) over bar -> X is jointly continuous. We prove that if f Lipschitz with respect to its second argument and its Lipschitz constant l satisfies the condition le(omega tau) < omega,g : ([-tau, +infinity);D<(A)over bar> -> C([-tau, 0]; D (A) over bar is nonexpansive and (I - A)(-1) is compact, then the unique C (0)-solution of the problem above is almost periodic.

• 出版日期2013-9