... inequality with the use of Gronwallinequality. Then, v ≤ u inDT, and the proof of part i of the lemma is completed. The similarargument gives the proof of part ii of the lemma.Advances ... theprinciple of induction, we complete the proof of the lemma.3. Uniform Estimates of {um}, {um}To prove the existence of solutions to 1.1, in this section, we show the uniform estimates of {um}, ... 0,independent of m.Proof. By Hypothesis H, 3.20,and3.27, the proof similar to that of 15, Lemma 4.4 gives3.31,and,by3.20 and 3.28, the proof similar to that of 15, Lemma 4.3...