On the generalized nonlinear variational inequalities and quasi-variational inequalities
The paper deals with two kinds of variational inequality problems: 1. (Generalized nonlinear variational inequality problem -- GNVI\((T,g,b,X)\)) Find \(\widehat x\in X\) and \(\widehat w\in T\widehat x\) such that \[ \text{Re}\langle\widehat w,g(\widehat x) -g(y)\rangle+ b(\widehat x,\widehat x)- b(\widehat x,y)\geq 0\;\forall y\in X; \] 2) (Generalized nonlinear quasi-variational inequality problem -- GNQVI\((T,g,b,S)\)) Find \(\widehat x\in S\widehat x\) and \(\widehat w\in T\widehat x\) such that \[ \text{Re}\langle\widehat w,g(\widehat x)- g(y)\rangle+ b(\widehat x,\widehat x)- b(\widehat x,y)\geq 0\;\forall y\in S\widehat x. \] The conditions ensuring the existence of solutions to the aforementioned problems have been established. Moreover, by suitable specifications of the functions \(g\) and \(b\) the well-known results on variational inequalities (\textit{X.-P. Ding} and \textit{E. Tarafdar} [Appl. Math. Lett. 7, No. 4, 5-11 (1994; Zbl 0808.49013)]; \textit{S. S. Chang} and \textit{C. J. Zhang} [J. Math. Anal. Appl. 179, No. 1, 250-259 (1993; Zbl 0803.49011)] -- among others) have been derived as special cases.
scientific article; zbMATH DE number 902494