Some results on premonotone operators

‎In this paper‎, ‎some properties of pre-monotone operators are proved‎. ‎It is shown that in a reflexive Banach space‎, ‎a full domain multivalued σ-monotone operator with sequentially norm×weak∗ closed graph is norm×weak∗ upper semicontinuous‎. ‎The notion of σ-convexity is introduced and the‎ ‎relations between the σ-monotonicity and σ-convexity is investigated‎. ‎Moreover‎, ‎some results on the sum and difference of two σ-monotone operators is considered.