  We study the relationship between a homological capacity
$c_{\mathrm{SH}^+}(W)$ for Liouville domains $W$ defined using positive
symplectic homology and the existence of periodic orbits for Hamiltonian
systems on $W$: If the positive symplectic homology of $W$ is non-zero, then
the capacity yields a finite upper bound to the $\pi_1$-sensitive Hofer-Zehnder
capacity of $W$ relative to its skeleton and a certain class of Hamiltonian
diffeomorphisms of $W$ has infinitely many non-trivial contractible periodic
points. En passant, we give an upper bound for the spectral capacity of $W$ in
terms of the homological capacity $c_{\mathrm{SH}}(W)$ defined using the full
symplectic homology. Applications of these statements to cotangent bundles are
discussed and use a result by Abbondandolo and Mazzucchelli in the appendix,
where the monotonicity of systoles of convex Riemannian two-spheres in $\mathbb
R^3$ is proved.
