Perturbed iterative algorithms with errors for completely generalized strongly nonlinear implicit quasivariational inclusions
In this paper, the authors have considered the problem of finding \(u\in H\) such that \[ 0\in N(f(u),p(u))+M(g(u)-m(u),u)\tag{1} \] where \(f,p, g, m, N\) are single-valued mappings and \(M(.,.)\) is a maximal monotone mapping with respect to the first argument. Problem (1) is called the completely generalized strongly nonlinear implicit quasivariational inclusion. The authors have suggested Ishikawa and Mann type perturbed iterative algorithms for the problem (1) using the standard technique. In fact, all the results obtained in this paper are either incorrect or trivial generalizations of the known results. Even the problem (1) is \textit{not} well-defined from mathematical point of view. For example, if the mapping \(g = 0\), then problem (1) reduces to: Find \(u\in H\) such that \[ 0\in N(f(u),p(u)) + M(-m(u),u), \] which does not make any sense from mathematical point of view. In fact, this problem is not even a generalization of any type of the variational inclusion. Similarly, the problems (2.2), (2.3)[ 16], (2.5)[10], (2.6)[18] and (2.7), which can be obtained as special cases from (1) for \((g = 0)\) are no longer variational inclusions and variational inequalities.
completely generalized strongly nonlinear implicit quasivariational inclusion
Ishikawa and Mann type perturbed iterative algorithms
scientific article; zbMATH DE number 1526345