A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces

Abstract

We propose a new general type of splitting methods for accretive operators in Banach spaces. We then give the sufficient conditions to guarantee the strong convergence. In the last section, we apply our results to the minimization optimization problem and the linear inverse problem including the numerical examples.

A generalized forward-backward method for solving split equality quasi inclusion problems in Banach spaces

“…The strong convergence theorems of the sequences generalized by the algorithm to a solution of the quasi inclusion problem are proved under certain assumptions. The results presented in this paper seem to be the first outside Hilbert space which extend and improve the main results of Chen and Rockafellar [3], Cholamjiak [4], Lions and Mercier [10], López et al [11], Moudafi [15] and Takahashi et al [20]. At the end of the paper, some applications to monotone variational inequalities, convex minimization problem and convexly constrained linear inverse problem are presented also.…”

“…The quasi inclusion problem for monotone and the maximal monotone mappings in the setting of Hilbert space has been considered by many authors (see, for example [3,4,6,7,10,11,17]). This problem includes, as special cases, convex programming, variational inequalities, split feasibility problem, and minimization problem.…”

“…In 2012, López et al [11], and Cholamjiak [4] introduced and studied the quasi inclusion problem for accretive and the maccretive mappings in Banach spaces. By using the Halpern-type forward-backward method and under some appropriate conditions they proved that the sequence generated by the algorithm converges strongly to a solution of the quasi inclusion problem.…”

The viscosity approximation forward-backward splitting method for solving quasi inclusion problems in Banach spaces

“…In 2016, Cholamjiak [8] studied a generalized forward-backward method for solving the inclusion problem (1.1) for an accretive and an m-accretive operator in Banach spaces. They then proved its strong convergence under some mild conditions.…”

The viscosity approximation forward-backward splitting method for the implicit midpoint rule of quasi inclusion problems in Banach spaces

“…For more details, see [11,14,22,29,30,36,37,39,42]. In 2015, Cholamjiak [12] studied a generalized forward-backward method for solving the inclusion problem (1.1) for an accretive and m-accretive operators in Banach spaces.…”