The Generalized Differential Dominated Vector Complementarity Problem of order $\lambda$ $(GDDVCP;\lambda)$ and Generalized $F$-Minty's Lemma

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The aim of this paper is to define some new concept of variational inequality and complementarity problem and to study them in different domain. We establish an uniqueness theorem for the generalized variational inequality problem in real Banach space. We represent the generalized $F$-Minty's lemma in $\eta$-invex set. We introduce some new type of variational inequality problems such as generalized differential dominated variational inequality problem $(GDDVIP)$, generalized differential dominated complementarity problem $(GDDCP)$ and generalized differential inequality problem $(GDIP)$ in real Banach spaces. We also explore the existence theorems of these problem. Next we introduce the generalized differential dominated vector variational inequality problem of order $\lambda$ $(GDDVVIP;\lambda)$ and generalized differential dominated vector complementarity problem of order $\lambda$ $(GDDVCP;\lambda)$.