Journal of the
Korean Mathematical Society JKMS

In this paper, a new class of $(h, \eta)$-proximal mappings for proper functionals in Hilbert spaces is introduced. The existence and Lipschitz continuity of the $(h, \eta)$-proximal mappings for proper functionals are proved. A class of generalized nonlinear quasi-variational-like inclusions in Hilbert spaces is introduced. A perturbed three-step iterative algorithm with errors for the generalized nonlinear quasi-variational-like inclusion is suggested. The existence and uniqueness theorems of solution for the generalized nonlinear quasi-variational-like inclusion are established. The convergence and stability results of iterative sequence generated by the perturbed three-step iterative algorithm with errors are discussed.

Keywords: generalized nonlinear quasi-variational-like inclusion, $(h, \eta)$-proximalmapping, perturbed three-step iterative algorithm with errors, strongly monotone mapping, generalized pseudocontractive mapping, mixed Lipschitz mapping, relaxed coercive mapping

MSC numbers: 47J20, 49J40