Spectral symmetry of solutions of boundary value problems in Banach algebras

For Banach algebras ${\cal A}$ and solutions $u:[0,1] \to {\cal A}$ of $u”(t)+f(u(t))+ \lambda u’(t)^2=0$, $u(0)=0$, $u(1)=0$, we prove symmetry of the spectrum $\sigma (u(t))$, that is, $\sigma (u(t)) = \sigma (u(1-t))$ for all $t \in [0,1]$, whenever $\sigma (u([0,1]))$ lies in a cone of the complex plane.