Strongly nonlinear variational inequalities and generalized pseudocontractions
Let \(K\) be a closed convex set in a Hilbert space \(H\). For given nonlinear operators \(T,S:H\to H\), find \(u\in K\) such that \[ \langle Su- Tu,v-u\rangle\geq 0,\quad\text{for all }v\in K.\tag{1} \] (1) is called the strongly nonlinear variational inequality, which was considered and studied by \textit{M. A. Noor} [C. R. Math. Acad. Sci., Soc. R. Can. 4, 213-218 (1982; Zbl 0502.49008)]. In this paper, the author has suggested and analyzed an iterative method. It is worth mentioning that Theorem 2.1 was proved by Noor in 1982. Proofs of the main results, Theorem 2.2 and its Corollary 2.1, are wrong and incorrect. In fact, all the results are misleading. It can be proved easily that Lipschitz continuity implies the so-called generalized pseudocontractivity. Consequently, this paper does not contain any new and original result.
strongly nonlinear variational inequality
scientific article; zbMATH DE number 1339479