Assuming that the function $f$ is continuous (and so $\
abla f$ is defined), then the Armijo rule can't be satisfied as an equality for a **_global_** minimum. That is you can rewrite the rule as:
$$f(\mathbf{x}_k+\alpha_k\mathbf{p}_k)\leq f(\mathbf{x}_k)+c_1\alpha_k\mathbf{p}_k^{\mathrm T}\
abla f(\mathbf{x}_k) = f(\mathbf{x}_k)$$
since $\
abla f(\mathbf{x}_k)=0$, but the LHS is larger than the RHS because the point $\mathbf{x}_k$ is a minimum. But if the minimum is only **_local_** , then you could meet the condition by arranging for the step to be to a lower minimum.