A treatise on generating functions by Hari Srivastava, H. L. Manocha

By Hari Srivastava, H. L. Manocha

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Let X = H be a real Hilbert space (identified with its own dual) and let A be a maximal monotone subset of H × H. , convex, proper function such that D(A) ∩ D(∂ g) = 0/ and, for some h ∈ H, ϕ ((I + λ A)−1 (x + λ h)) ≤ ϕ (x) +Cλ (1 + ϕ (x)), ∀x ∈ D(ϕ ), λ > 0. 47) Then A + ∂ ϕ is maximal monotone and D(A + ∂ ϕ ) = D(A) ∩ D(ϕ ). Proof. 7. Let y be arbitrary but fixed in H. Then, for every λ > 0, the equation xλ + Aλ xλ + ∂ ϕ (xλ ) y has a unique solution xλ ∈ D(∂ ϕ ). 47). This yields Aλ xλ 2 + (Aλ xλ , Jλ (xλ ) − Jλ (xλ + λ h)) ≤ Cλ ( y + h + xλ + ϕ (xλ ) + 1), where Jλ = (I + λ A)−1 .

1. Let X be reflexive and let B be a hemicontinuous monotone and bounded operator from X to X ∗ . Let A ⊂ X × X ∗ be maximal monotone. Then A + B is maximal monotone. 36 2 Maximal Monotone Operators in Banach Spaces Proof. 1 in Chapter 1), we may take an equivalent norm in X such that X and X ∗ are strictly convex. It is clear that after this operation the monotonicity properties of A, B, A + B as well as maximality do not change. Also, without loss of generality, we may assume that 0 ∈ D(A); otherwise, we replace A by u → A(u + u0 ), where u0 ∈ D(A) and B by u → B(u + u0 ).

11. Let X = H be a real Hilbert space (identified with its own dual) and let A be a maximal monotone subset of H × H. , convex, proper function such that D(A) ∩ D(∂ g) = 0/ and, for some h ∈ H, ϕ ((I + λ A)−1 (x + λ h)) ≤ ϕ (x) +Cλ (1 + ϕ (x)), ∀x ∈ D(ϕ ), λ > 0. 47) Then A + ∂ ϕ is maximal monotone and D(A + ∂ ϕ ) = D(A) ∩ D(ϕ ). Proof. 7. Let y be arbitrary but fixed in H. Then, for every λ > 0, the equation xλ + Aλ xλ + ∂ ϕ (xλ ) y has a unique solution xλ ∈ D(∂ ϕ ). 47). This yields Aλ xλ 2 + (Aλ xλ , Jλ (xλ ) − Jλ (xλ + λ h)) ≤ Cλ ( y + h + xλ + ϕ (xλ ) + 1), where Jλ = (I + λ A)−1 .

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