Here it is assumed that the spaces are all real spaces to simplify the presentation.
Definition 23.7.1 Let A : D
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Denote by G
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is onto where F is the duality map with p = 2.
Actually, it is more usual to say that the graph is maximal monotone if the graph is monotone and there is no monotone graph which properly contains the given graph. However, the two conditions are equivalent and I am more used to using the version in the above definition.
There is a fundamental result about these which is given next.
Theorem 23.7.2 Let X, X′ be reflexive and have strictly convex norms. Let A be a monotone set valued map as just described. Then if λF + A is onto for some λ > 0, then whenever
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it follows that z ∈ Au and u ∈ D
Proof: Suppose that for all
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Does it follow that z ∈ At? By assumption, z + λF
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and so
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which implies from Theorem 23.2.5 that t =
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Note that this would have worked with no change if the duality map had been for arbitrary p > 1.