Modified semi-implicit midpoint rule for nonexpansive mappings

Abstract

The purpose of the paper is to construct iterative methods for finding the fixed points of nonexpansive mappings. We present a modified semi-implicit midpoint rule with the viscosity technique. We prove that the suggested method converges strongly to a special fixed point of nonexpansive mappings under some different control conditions. Some applications are also included.MSC: 47J25; 47N20; 34G20; 65J15

The modified viscosity implicit rules for uniformly L-Lipschitzian asymptotically pseudocontractive mappings in Banach spaces

“…Since every nonexpansive mapping is asymptotically nonexpansive and an asymptotically nonexpansive mapping is both asymptotically pseudocontractive and uniformly L-Lipschitzian, in Theorems 2.1 and 2.2, if T is a nonexpansive mapping in Hilbert spaces, then it is the main results of Yu et al [15]. Thus, Theorem 2.1 improves and extends the Yu et al's theorem in several respects and improves some other results (see [1,5,7,8,11,12,14]). Remark 2.6.…”

“…Therefore, we extend the main results of Yu et al (see [15]) from Hilbert spaces to Banach spaces when T is an asymptotically nonexpansive respective asymptotically pseudocontractive map. Further, some other results are also improved (see [1,5,7,8,11,12,14]). …”

“…In 2016, Yu et al [15] extended the work of Yao et al [14] and gave the following generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces:…”

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

“…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:…”

Strong convergence of a new iterative algorithm for fixed points of asymptotically nonexpansive mappings

“…Lately, in order to find a common element of the fixed points of nonexpansive mappings or asymptotically nonexpansive mappings, many authors established many viscosity iterative algorithms, and they solved many practical problems, such as the equilibrium problems, variational inequality problems, and split feasibility problems and so on. See [1,2,4,[9][10][11][12][13][14][21][22][23][24] and the references therein.…”