New Inertial Forward-Backward Mid-Point Methods for Sum of Infinitely Many Accretive Mappings, Variational Inequalities, and Applications

Abstract

Some new inertial forward-backward projection iterative algorithms are designed in a real Hilbert space. Under mild assumptions, some strong convergence theorems for common zero points of the sum of two kinds of infinitely many accretive mappings are proved. New projection sets are constructed which provide multiple choices of the iterative sequences. Some already existing iterative algorithms are demonstrated to be special cases of ours. Some inequalities of metric projection and real number sequences are widely used in the proof of the main results. The iterative algorithms have also been modified and extended from pure discussion on the sum of accretive mappings or pure study on variational inequalities to that for both, which complements the previous work. Moreover, the applications of the abstract results on nonlinear capillarity systems are exemplified.

Iterative Design for the Common Solution of Monotone Inclusions and Variational Inequalities

“…From Theorems 1 and 2, we may find that the limit q 0 of the iterative sequence {x n } is not only the solution of the system of monotone inclusions (3) but also the solution of variational inequalities ( 14) and (15). That is, the study on iterative construction of the solution of ( 14) in [18] and the solution of (15) in [17] are unified in our paper.…”

“…In this section, one kind capillarity system discussed in [18] is employed again to demonstrate the application of Theorems 1 and 2.…”

“…In 2019, Wei et al, proposed some new iterative algorithms. The inertial forwardbackward iterative algorithm for approximating the solution of monotone inclusions (3) in [18] is as follows:…”

“…The inertial forward-backward iterative algorithm for approximating common solution of monotone inclusions and one kind variational inequalities, where T : C → H is maximal monotone and τ-Lipschitz continuous, is presented as follows in [18]:…”

A new shrinking iterative scheme for d-accretive mappings with applications to capillarity systems

“…m-accretive mappings and capillarity systems. The capillarity equation is an important equation appeared in the capillarity phenomenon (see [20]) and the following capillarity systems were studied in [21] as an example of m-accretive mappings in the Hilbert space L 2 (Ω) : The study on (3.1) was based on the following assumptions in [21].…”