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a version of zhong's coercivity result for a general class of ...

a version of zhong's coercivity result for a general class of ...

610 A

610 A version of Zhong’s coercivity resultIn addition, by means of equality u k+11 = w k ,(3.26), and the triangle inequality,we have∥ w k − w k+1∥ ∥ = limn→+∞≤ limn→+∞j=1∥ u k+11 − u k+1∥nn−1∑∥ u k+1j − u k+1∥j+1+∞∑= ∥ u k+1j − u k+1∥ j+1 .j=1(3.37)Making use of monotonicity of h in conjunction with (3.35) and(3.36), theinequality above leads toɛ1+h (∥∥ ∥) ∥∥ ∥ ∑+∞w k w k − w k+1 ≤j=1ɛ1+h (∥∥ ∥) ∥∥ w k u k+1j − u k+1∥j+1+∞∑ ɛ≤j=11+h (∥∥ u k+1∥ ) ∥∥ u k+1j − u k+1∥(3.38)j+1j≤ I ( ) ( )w k − I wk+1 .This means that if (3.35) holds, inequality (3.32)ischecked.Property (3.35) istrueforj = 1 being verified with equality. Thus, for completingthe proof it is sufficient to verify∥ u k+1∥ < ∥∥ ∥w k = rk , ∀j ≥ 2. (3.39)jThe proof of (3.39) is done by recurrence. Considering first the case j = 2, wepoint out that the set{u ∈ M k+1 : I(u)

D. Motreanu et al. 611thatI ( u k+1j) (

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