On a multivalued nonlinear variational inequality (Q5951207)

The authors state several assertions on the existence and convergence of approximate solutions of the inequality \[ x_0\in X:\qquad \langle w_0-y_0,J(x-x_0)\rangle \geq f(x_0)-f(x), \qquad\forall x\in X, \] where \(X\) is a reflexive Banach space, \(y_0\in X^*\), \(J:X\to X\) is linear and continuous, \(f:X\to (-\infty,+\infty]\) is convex, \(w_0\in Tx_0\) and \(T:X\to 2^{X^*}\) is \(J\)-monotone. The approximation is of the form \[ \langle w_h+\alpha {\mathcal U}Jx-y_\delta,J(x-x_0)\rangle\geq f(x_0)-f(x)-\varepsilon g(\|Jx\|)\|J(x-x_0)\|, \] where \({\mathcal U}\) is the duality mapping, \(g\) is a nonnegative continuous function with linear growth, \(w_h\in T_hx\), \(T_h,y_\delta\) are approximations of \(T,y_0\), and \(\varepsilon,\alpha\to 0\).