“…We can also state the following properties of D σ , which can be proved easily. The following theorems show the invariance of I σ under the flow of (2), (3) for 0 < E (0) < d and E (0) = d, respectively, and can be proved by contradiction as in [20]. Now, we give a lemma for σ > 1, which states similar results to Lemmas 2.1, 2.2, and can be proved similarly.…”