New shrinking iterative methods for infinite families of monotone operators in a Banach space, computational experiments and applications

Abstract

Abstract New shrinking iterative algorithms for approximating common zeros of two infinite families of maximal monotone operators in a real uniformly convex and uniformly smooth Banach space are designed. Two steps of multiple choices can be made in the new iterative algorithms, two groups of interactive containment sets C n $C_{n}$ and Q n $Q_{n}$ are constructed and computational errors are considered, which are different from the previous ones. Strong convergence theorems are proved under mild assumptions and some new proof techniques can be found. Computational experiments for some special cases are conducted to show the effectiveness of the iterative algorithms and meanwhile some inequalities are proved to guarantee the strong convergence. Moreover, the applications of the abstract results on convex minimization problems and variational inequalities are exemplified.

Keywords

• Lyapunov functional
• Monotone operators
• Shrinking iterative methods
• Convex minimization problems
• Variational inequalities
• Visual Basic Six