$\left( \Phi ,\rho \right)$-MONOTONICITY AND GENERALIZED $\left(\Phi ,\rho \right)$-MONOTONICITY

Abstract

In this paper, new concepts of monotonicity, namely $\left( \Phi, \rho \right)$-monotonicity, $\left( \Phi, \rho \right)$-pseudo-monotonicity and $\left( \Phi, \rho \right)$-quasi-monotonicity are introduced for functions defined in Banach spaces. Series of necessary conditions are also given that relate $\left( \Phi, \rho \right)$-invexity and generalized $\left( \Phi, \rho \right)$-invexity of the function with $\left( \Phi, \rho \right)$-monotonicity and generalized $\left( \Phi, \rho \right)$-monotonicity of its gradient.