  In this paper, we prove existence of symmetric homoclinic orbits for the
suspension bridge equation $u""+\beta u" + e^u-1=0$ for all parameter values
$\beta \in [0.5,1.9]$. For each $\beta$, a parameterization of the stable
manifold is computed and the symmetric homoclinic orbits are obtained by
solving a projected boundary value problem using Chebyshev series. The proof is
computer-assisted and combines the uniform contraction theorem and the radii
polynomial approach, which provides an efficient means of determining a set,
centered at a numerical approximation of a solution, on which a Newton-like
operator is a contraction.
