We introduce a new concept of general
$G$-$eta$-monotone operator generalizing the general
$(H,eta)$-monotone operator cite{arvar2, arvar1}, general
$H-$ monotone operator cite{xiahuang} in Banach spaces, and also
generalizing $G$-$eta$-monotone operator cite{zhang}, $(A,
eta)$-monotone operator cite{verma2}, $A$-monotone operator
cite{verma0}, $(H, eta)$-monotone operator cite{fanghuang},
$H$-monotone operator cite{fanghuang1, {fanghuangthompson}},
maximal $eta$-monotone operator cite{fanghuang0} and classical
maximal monotone operators cite{zeid} in Hilbert spaces. We provide
some examples and study some properties of general
$G$-$eta$-monotone operators. Moreover, the generalized proximal
mapping associated with this type of monotone operator is defined
and its Lipschitz continuity is established. Finally, using
Lipschitz continuity of generalized proximal mapping under some
conditions a new system of variational inclusions is solved.
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