CHARACTERIZATION OF APPROXIMATE MONOTONE OPERATORS
Abstract
Results concerning local boundedness of operators have a
long history. In 1994, Vesel´y connected the concept of approximate
monotonicity of an operator with local boundedness of that. It is our
desire in this note to characterize an approximate monotone operator.
Actually, we show that a well-known property of monotone operators,
namely representing by convex functions, remains valid for the larger
subclass of operators. In this general framework we establish the similar
results by Fitzpatrick. Also, celebrated results of Mart´ inez-Legaz and
Th´era inspired us to prove that the set of maximal ε-monotone operators
between a normed linear space X and its continuous dual X∗ can be
identified as some subset of convex functions on X × X∗.
long history. In 1994, Vesel´y connected the concept of approximate
monotonicity of an operator with local boundedness of that. It is our
desire in this note to characterize an approximate monotone operator.
Actually, we show that a well-known property of monotone operators,
namely representing by convex functions, remains valid for the larger
subclass of operators. In this general framework we establish the similar
results by Fitzpatrick. Also, celebrated results of Mart´ inez-Legaz and
Th´era inspired us to prove that the set of maximal ε-monotone operators
between a normed linear space X and its continuous dual X∗ can be
identified as some subset of convex functions on X × X∗.
Keywords
θ′-monotone, θ-Fitzpatrick function, convex, θ-subdifferential, maximal θ′-monotone, θ-conjugate.
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