On a class of generalized variational inequalities
Let \(H\) be a real Hilbert space, \(2^H\) a power set of \(H\), \(\langle\;,\;\rangle\) denote the inner product of \(H\), \(T,A:H\to 2^H\), \(g:H\to H\) a nonlinear mapping and \(j:H\to R\cup\{\infty\}\) a proper convex and lower semicontinuous function with \(I_mg\cap\text{dom }\partial j\neq\emptyset\), where \(\partial j\) denotes the subdifferential of \(j\). The authors consider the following generalized variational inequality problem (GVIP): GVIP: Find \(u\in H\), \(x\in T(u)\), \(y\in A(u)\) such that \(g(u)\cap\text{dom }\partial j\neq\emptyset\) and \(\langle x-y,v-g(u)\rangle\geq j(g(u))- j(\nu)\) for all \(\nu\in H\). If \(T\) and \(A\) are single-valued mappings, then GVIP reduces to the problem which was considered and studied by Hassouni and Moudafi. If \(j=\delta_k\), the indicator function of the closed convex set \(K\) of \(H\), then GVIP reduces to the problem considered by Ding. If \(j=\delta_k\), the indicator function of the closed convex set \(K\) of \(H\), \(T\) and \(A\) are single-valued mapping, then GVIP reduces to the problem considered by Noor. If \(j=\delta_k+m(u)\), where \(m:H\to H\) is a single-valued map, and \(T\) and \(A\) are two single-valued maps, then GVIP reduces to the problem considered by Siddiqi and Ansari. The authors develop some iterative algorithms for finding the approximating solution of GVIP, which are more general and powerful than the iterative algorithms developed by Hassouni and Moudafi, and Ding. The authors prove some existence results for GVIP and discuss the convergence criteria for sequence generalized by iterative algorithms.
generalized variational inequality problem
scientific article; zbMATH DE number 1014154