  We construct several new classes of black hole (BH) solutions in the context
of the mimetic Euler-Heisenberg theory. We separately derive three differently
charged BH solutions and their relevant mimetic forms. We show that the
asymptotic form of all BH solutions behaves like a flat spacetime. These BHs,
either with/without cosmological constant, have the non constant Ricci scalar,
due to the contribution of the Euler-Heisenberg parameter, which means that
they are not solution to standard or mimetic $f(R)$ gravitational theory
without the Euler-Heisenberg non-linear electrodynamics and at the same time
they are not equivalent to the solutions of the Einstein gravity with a
massless scalar field. Moreover, we display that the effect of the
Euler-Heisenberg theory makes the singularity of BH solutions stronger compared
with that of BH solutions in general relativity. Furthermore, we show that the
null and strong energy conditions of those BH solutions are violated, which is
a general trend of mimetic gravitational theory. The thermodynamics of the BH
solutions are satisfactory although there appears a negative Hawking
temperature under some conditions. Additionally, these BHs obey the first law
of thermodynamics. We also study the stability, using the geodesic deviation,
and derive the stability condition analytically and graphically. Finally, for
the first time and under some conditions, we derived multi-horizon BH solutions
in the context of the mimetic Euler-Heisenberg theory and study their related
physics.
