Hölder Continuous Solutions of Degenerate Differential Equations with Finite Delay

Bu, Shangquan ; Cai, Gang

Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 240-251

Résumé

Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces ${{C}^{\alpha }}(\mathbb{R};\,X)$ , we completely characterize the ${{C}^{\alpha }}$ -well-posedness of the first order degenerate differential equations with finite delay $(Mu{)}'(t)\,=\,Au(t)\,+\,F{{u}_{t}}\,+\,f(t)$ for $t\,\in \,\mathbb{R}$ by the boundedness of the $(M,\,F)$ -resolvent of A under suitable assumption on the delay operator $F$ , where $A,M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\,\cap \,D(M)\,\ne \,\{0\}$ , the delay operator $F$ is a bounded linear operator from $C([-r,0];X)$ to $X$ , and $r\,>\,0$ is fixed.

Détail

DOI : 10.4153/CMB-2017-014-2

Mots-clés : 34N05, 34G10, 47D06, 47A10, 34K30, well-posedness, degenerate differential equation, C α-multiplier, Hölder continuous function space