Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces

Shangquan Bu; Yi Fang

Shangquan Bu ; Yi Fang

Studia Mathematica, Tome 187 (2008), p. 103-119 / Harvested from The Polish Digital Mathematics Library

Résumé

We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $(P) u^{''} (t) + α u^{'} (t) + d / d t (\int_{- \infty}^{t} b (t - s) u (s) d s) = A u (t) - \int_{- \infty}^{t} a (t - s) A u (s) d s + f (t)$ (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), u’(0) = u’(2π), where A is a closed operator in a Banach space X, α ∈ ℂ, and a,b ∈ L¹(ℝ₊). We use Fourier multipliers to characterize maximal regularity for (P). Using known results on Fourier multipliers, we find suitable conditions on the kernels a and b under which necessary and sufficient conditions are given for the problem (P) to have maximal regularity on $L^{p} (, X)$ , periodic Besov spaces $B_{p, q}^{s} (, X)$ and periodic Triebel-Lizorkin spaces $F_{p, q}^{s} (, X)$

Publié le : 2008-01-01

Zbl 1140.45015

EUDML-ID : urn:eudml:doc:286397

@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-1,
     author = {Shangquan Bu and Yi Fang},
     title = {Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces},
     journal = {Studia Mathematica},
     volume = {187},
     year = {2008},
     pages = {103-119},
     zbl = {1140.45015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-1}
}

Shangquan Bu; Yi Fang. Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces. Studia Mathematica, Tome 187 (2008) pp. 103-119. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm184-2-1/