  We report constraints on the sources of ultra-high-energy cosmic ray (UHECR)
above $10^{9}$ GeV, based on an analysis of seven years of IceCube data. This
analysis efficiently selects very high energy neutrino-induced events which
have deposited energies from $\sim 10^6$ GeV to above $10^{11}$ GeV. Two
neutrino-induced events with an estimated deposited energy of $(2.6 \pm 0.3)
\times 10^6$ GeV, the highest neutrino energies observed so far, and $(7.7 \pm
2.0) \times 10^5$ GeV were detected. The atmospheric background-only hypothesis
of detecting these events is rejected at 3.6$\sigma$. The hypothesis that the
observed events are of cosmogenic origin is also rejected at $>$99% CL because
of the limited deposited energy and the non-observation of events at higher
energy, while their observation is consistent with an astrophysical origin. Our
limits on cosmogenic neutrino fluxes disfavor the UHECR sources having
cosmological evolution stronger than the star formation rate, e.g., active
galactic nuclei and $\gamma$-ray bursts, assuming proton-dominated UHECRs.
Constraints on UHECR sources including mixed and heavy UHECR compositions are
obtained for models of neutrino production within UHECR sources. Our limit
disfavors a significant part of parameter space for active galactic nuclei and
new-born pulsar models.
