Summary: The differential equation \(u'(t)+ Au(t)= f(t)\) \((-\infty< t<\infty)\) in a general Banach space \(E\) with the strongly positive operator \(A\) is ill-posed in the Banach space \(C(E)= C(\mathbb{R},E)\) with norm \(\|\varphi\|_{C(E)}= \sup_{-\infty< t< \infty}\|\varphi(t)\|_E\). In the present paper, the well-posedness of this equation in the Hölder space \(C^\alpha(E)= C^\alpha(\mathbb{R},E)\) with norm \[ \|\varphi\|_{C^\alpha(E)}= \sup_{-\infty< t<\infty}\|\varphi(t)\|_E+ \sup_{-\infty< t< t+ s< \infty}(\|\varphi(t+ s)- \varphi(t)\|_E/s^\alpha),\;0< \alpha< 1, \] is established. The almost coercivity inequality for solutions of the Rothe difference scheme in \(C(\mathbb{R}_\tau,E)\) spaces is proved. The well-posedness of this difference scheme in \(C^\alpha(\mathbb{R}_\tau,E)\) spaces is obtained.