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Abstract
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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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