We present a general procedure for constructing exact black hole (BH) solutions with a magnetic charge in the context of nonlinear electrodynamics (NED) theory as well as in the coherent state approach to noncommutative geometry (NCG). In this framework, the Lagrangian density for a noncommutative Hayward BH is obtained and the weak energy condition is satisfied. The noncommutative Hayward solution depends on two kind of charges, without which the Schwarzschild solution is applicable. Moreover, to find a link between the BH evaporation and uncertainty relations, we may calculate the Hawking temperature and find the effect of the Lagrangian density of BHs on the Hawking radiation. Therefore, a generalized uncertainty principle (GUP) emerges from the modified Hawking temperature in Einstein–NED theory. The origin of this GUP is the combined influence of a nonlinear magnetic source and an intrinsic property of the manifold associated with a fictitious charge. Finally, we find that there is an upper bound on the Lagrangian uncertainty of the BHs that is caused by the NED field and (or) the fictitious charge.
Entropy bounds and nonlinear electrodynamics