The implicit midpoint rule for nonexpansive mappings

In this paper, the implicit midpoint rule of asymptotically nonexpansive mapping in CAT (0) spaces is introduced. By the viscosity approximation method, we prove that the proposed implicit iteration converges strongly to a fixed point of asymptotically nonexpansive mapping under certain assumptions imposed on the sequence of parameters. The results presented in the paper improve and extend various results in the existing literature.

Section: Introductionmentioning

confidence: 99%

Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mapping in complete CAT(0) spaces

Li¹,

Liu²

2017

“…The implicit midpoint rule is a powerful method for solving ordinary differential equations; see [1,2,4,6,12,18,20] and the references therein. Recently, Xu et al [18] considered the following viscosity implicit midpoint rule:…”

Section: Proposition 12 ([15]mentioning

confidence: 99%

Viscosity implicit iterative algorithms based on generalized contractions for strictly pseudo-contractive mappings in Banach spaces

Cheng¹,

Su²

2017

In this manuscript, we construct three viscosity implicit iteration schemes based on generalized contractions for strictly pseudo-contractive mappings. The first scheme is used to approximate a fixed point of a single strictly pseudo-contractive mapping, the second scheme is used to approximate a common fixed point of a finite family of strictly pseudo-contractive mappings, the third scheme is used to approximate a common fixed point of a countable family of strictly pseudo-contractive mappings. Furthermore, three strong convergence theorems based on the purposed iterative schemes are established in the framework of Banach spaces. Finally, three numerical examples are also given to show the efficiency and implementation of our schemes. The main results of this paper modify and improve many important recent results in the literature.

“…Based on the above fact, Alghamdi et al [2] presented the following semi-implicit midpoint rule for nonexpansive mappings in the setting of Hilbert space H:…”

Section: Introductionmentioning

confidence: 99%

Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT(0) Spaces

Zhao¹,

Chang²,

Wang³

et al. 2017

The purpose of this paper is to introduce the implicit midpoint rule of nonexpansive mappings in CAT(0) spaces. The strong convergence of this method is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit of the sequence generated by the implicit midpoint rule solves an additional variational inequality. Applications to nonlinear Volterra integral equations and nonlinear variational inclusion problem are included. The results presented in the paper extend and improve some recent results announced in the current literature.