  The Georgi-Machacek model is one of many beyond Standard Model scenarios with
an extended scalar sector which can group under the custodial $\rm SU(2)_C$
symmetry. There are 5-plet, 3-plet and singlet Higgs bosons under the
classification of such symmetry in addition to the Standard Model Higgs boson.
Here we study the prospects for detecting the doubly-charged Higgs boson ($\rm
H_5^{\pm\pm}$) through the vector boson fusion production at the
electron-proton colliders. Typically, we concentrate on our analysis through
$\mu$-lepton pair production via pair of same-sign W bosons decay. The
discovery significance are calculated as the functions of the triplet vacuum
expectation value and necessary luminosity.
