Modified forward-backward splitting midpoint method with superposition perturbations for the sum of two kinds of infinite accretive mappings and its applications

Abstract

In a real uniformly convex and p-uniformly smooth Banach space, a modified forward-backward splitting iterative algorithm is presented, where the computational errors and the superposition of perturbed operators are considered. The iterative sequence is proved to be convergent strongly to zero point of the sum of infinite m-accretive mappings and infinite -inversely strongly accretive mappings, which is also the unique solution of one kind variational inequalities. Some new proof techniques can be found, especially, a new inequality is employed compared to some of the recent work. Moreover, the applications of the newly obtained iterative algorithm to integro-differential systems and convex minimization problems are exemplified.

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

“…Since Theorem 4 tells us that (28) has a unique solution, then we know that {u n } generated by (16) converges strongly to the unique solution of variational inequality (28).…”

“…For projection iterative algorithms such as (16), rare work can be found to show that the limit of the iterative sequences is also the solution of a kind of variational inequalities.…”

“…In addition, some iterative algorithms for approximating common zero points of T i + S i are constructed in [3,[15][16][17]. In particular, Wei et al proposed the following implicit mid-point forward-backward projection algorithm in [17]:…”

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

“…Later, in 2017, the work in ( 6) is extended to approximate the solutions of the systems of monotone inclusions (3). The following is a special case in Hilbert space presented in [17]:…”

Some Operations on Mixed Monotone Operator in Banach Spaces

“…One of them is a monotone function. On the other side, we have known a term called mixed monotone operator, that is a function defined on the Cartesian Product of two subsets of Real Banach Space [1][2][3][4][5][6][7][8][9]13]. It has been well-known that under operation of addition, the sum of two monotone functions is monotone function too, but not for case of multiplication operation [10].…”