  Isobaric single charge-exchange reactions, changing nuclear charges by one
unit but leaving the mass partitions unaffected, have been for the first time
investigated by peripheral collisions of $^{112}$Sn ions accelerated up to
1\textit{A} GeV at the GSI facilities. The high-resolving power of the FRS
spectrometer allows us to obtain $(p, n)$-type isobaric charge-exchange cross
sections with an uncertainty of $3.5\%$ and to separate quasi-elastic and
inelastic components in the missing-energy spectra of the ejectiles. The
inelastic component is associated to the excitation of the $\Delta$(1232)
isobar resonance and the emission of pions in s-wave both in the target and
projectile nucleus, while the quasi-elastic contribution is associated to the
nuclear spin-isospin response of nucleon-hole excitations. An apparent shift of
the $\Delta$-resonance peak of $\sim$63 MeV is observed when comparing the
missing-energy spectra obtained from the measurements with proton and carbon
targets. A detailed analysis, performed with a theoretical model for the
reactions, indicates that this observation can be simply interpreted as a
change in the relative magnitude between the contribution of the excitation of
the resonance in the target and in the projectile.
