... :ξ0fk,sds for any ξ ∈ R. Obviously, Φ,J ∈ C1H, R,thatis,Φ andJ are continuously Fr´echet differentiable in H. Using the summation by parts formula and thefact that x0xT 10 for any ... −Tk1ΔφpΔxk − 1zk2.6 for any x, z ∈ H. Noticing the fact that x0xT 10 for any x ∈ H again, we obtainJxzlimt→0Jx tz − JxtTk1fk,xkzk2.7 for any x, z ∈ H.4 ... 1p−12pΦx ≤prT 1p−12p3.12 for any k ∈Z1,T. From the definition of r, it follows thatΦ−1 −∞,r ⊆{x ∈ H : |xk|≤c, ∀k ∈ Z1,T}. 3.13Thus, for any x ∈ H,wehavesupx∈Φ−1−∞,rwJx...