The sum of of a monotone Lipschitz operator and a Lipschitz operator

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Is there a result which says that the sum of a monotone Lipschitz operator and a Lipschitz operator is a monotone operator?

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No, because the statement is false.

As a counterexample, consider the operators $f,g:\Bbb R\to\Bbb R$ given via $$ f(x)=x, g(x)=-2x. $$