  In General Relativity coupled to a non-linear electromagnetic field, together
with a negative cosmological constant, we obtain the general static spherical
symmetric black hole solution with magnetic charges, which is asymptotic to
anti-de Sitter (AdS) space-times. In particular, for a degenerate case the
solution becomes a Hayward-AdS black hole, which is regular everywhere in the
full space-time. The existence of such a regular black hole solution preserves
the weak energy condition while the strong energy condition is violated. We
then derive the first law and the Smarr formula of the black hole solution. We
further discuss its thermodynamic properties and study the critical phenomena
in the extended phase space where the cosmological constant is treated as a
thermodynamic variable as well as the parameter associated with the non-linear
electrodynamics. We obtain many interesting results such as: the Maxwell's
equal area law in the $P-V$ (or $S-T$) diagram is violated and consequently the
critical point $(T_*\,,P_*)$ of the first order small-large black hole
transition does not coincide with the inflection point ($T_c\,,P_c$) of the
isotherms; the Clapeyron equation describing the coexistence curve of the Van
der Waals (vdW) fluid is no longer valid; the heat capacity at constant
pressure is finite at the critical point; the various exponents near the
critical point are also different from those of the vdW fluid.
