Generalized variational inequalities with generalized coercivity conditions (Q1701781)

This communication extends some results obtained for \(\mathbb{R}^n\) to infinite-dimensional spaces. The author considers a real reflexive Banach space \(X\) with Fréchet differentiable norm, \(K\subset X\) a nonempty convex subset and \(T:K\mapsto 2^{X^*}\) a set-valued mapping with nonempty values. A generalized variational inequality associated with the pair \((K,T)\) is stated as: find \(x^*\in K\) for which there exists \(y^*\in T(x^*)\) such that \[ \langle y^*,x-x^*\rangle \geq 0 \text{ for all } x\in K. \] Theorems on the existence of solutions of such inequality under generalized coercivity conditions are obtained.