Perturbed three-step approximation process with errors for a generalized implicit nonlinear quasivariational inclusions

Variational inequality theory has become a rich source of inspiration in the pure and applied mathematics. Variational inequalities not only have stimulated new results dealing with nonlinear partial differential equations, but also have been used in a large variety of problems arising in mechanics, physics, optimization and control nonlinear programming, economics and transportation equilibrium and engineering sciences, and so forth. In recent years variational inequalities have been generalized and applied in various directions. For details we refer to [2, 5, 6, 20]. Recently Huang [10, 11] constructed some new perturbed Ishikawa andMann iterative algorithms to approximate the solution of some generalized implicit quasivariational inclusions (inequalities), which includes many iterative algorithms for variational and quasivariational inequality problems as special cases. On the other hand, Xu [19] revised the definition of Ishikawa and Mann iterative processes with errors and studied the convergence problem of Ishikawa and Mann iterative processes with errors for approximating the solutions of the generative strongly accretive operator equations. Inspired and motivated by recent research works [4, 7, 12, 14–16], in this paper we initiate and construct some perturbed three-step approximation processes with errors for solving a class of generalized implicit nonlinear quasivariational inclusions. We also discuss the convergence and stability of the iterative sequences generated by algorithms.