Matrix approaches to approximate solutions of variational inequalities in Hilbert spaces

Abstract

A matrix approach to approximating solutions of variational inequalities in Hilbert spaces is introduced. This approach uses two matrices: one for iteration process and the other for regularization. Ergodicity and convergence (both weak and strong) are studied. Our methods combine new or well-known iterative methods (such as the original Mann’s method) with regularized processes involved regular matrices in the sense of Toeplitz.

Keywords:

• variational inequality
• -hybrid
• iterative method
• ergodic
• nonspreading mappings
• Cesaro’s means
• regular matrix

AMS Subject Classifications:

• 47J20
• 47J25
• 49J40
• 65J15

Notes

No potential conflict of interest was reported by the authors.

1 However, we can find a wide study on quasi-nonexpansivity and some convergence results in Dotson [8] and Senter and Dotson [9].

2 for instance nonexpansive mappings, firmly nonexpansive mappings, pseudocontractive mappings, strictly pseudoconctractive mappings and so on.

3 for instance, Hilbert spaces, uniformly convex Banach spaces, uniformly smooth Banach spaces and so on.

Additional information

Funding

This work was supported in part by NSC [102-2115-M-110-001-MY3].