In this article, we first introduce a modified inertial mann algorithm and an inertial CQ-algorithm by combining the accelerated mann algorithm and the CQ-algorithm with the inertial extrapolation, respectively. This ...
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In this article, we first introduce a modified inertial mann algorithm and an inertial CQ-algorithm by combining the accelerated mann algorithm and the CQ-algorithm with the inertial extrapolation, respectively. This strategy is intended to speed up the convergence of the given algorithms. Then we established the convergence theorems for two provided algorithms. For the inertial CQ-algorithm, the conditions on the inertial parameters are very weak. Finally, the numerical experiments are presented to illustrate that the modified inertial mann algorithm and inertial CQ-algorithm may have a number of advantages over other methods in computing for some cases.
The purpose of this paper is to prove two Delta-convergence theorems of the mann algorithm to a common fixed point for a countable family of mappings in the case of a complete geodesic space with curvature bounded abo...
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The purpose of this paper is to prove two Delta-convergence theorems of the mann algorithm to a common fixed point for a countable family of mappings in the case of a complete geodesic space with curvature bounded above by a positive number. The first one for nonexpansive mappings improves the recent result of He et al. (Nonlinear Anal. 75: 445-452, 2012). The last one is proved for quasi-nonexpansive mappings and applied to the problem of finding a common fixed point of a countable family of quasi-nonexpansive mappings.
The purpose of this paper is to present accelerations of the mann and CQ algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepes...
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The purpose of this paper is to present accelerations of the mann and CQ algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerate the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the mann and CQ algorithms. Under certain assumptions, we show that the new algorithms converge to a fixed point of a nonexpansive mapping. Finally, we show the efficiency of the accelerated mann algorithm by numerically comparing with the mann algorithm. A numerical example is provided to illustrate that the acceleration of the CQ algorithm is ineffective.
In this paper, we prove strong convergence theorem of the general inertial mann-Halpern algorithm for nonexpansive mappings in the setting of Hilbert spaces. We also prove weak convergence theorem of the general inert...
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In this paper, we prove strong convergence theorem of the general inertial mann-Halpern algorithm for nonexpansive mappings in the setting of Hilbert spaces. We also prove weak convergence theorem of the general inertial mann algorithm for k-strict pseudo-contractive mappings in the setting of Hilbert spaces. These convergence results extend and generalize some existing results in the literature. Finally, we provide examples to verify our main results.
This paper presents a weak convergence residual algorithm for finding a fixed point of a nonexpansive mapping in a real Hilbert space. To study the numerical behavior of the algorithm it is included an extensive serie...
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This paper presents a weak convergence residual algorithm for finding a fixed point of a nonexpansive mapping in a real Hilbert space. To study the numerical behavior of the algorithm it is included an extensive series of numerical experiments. Our computational experiments show that the new algorithm is computationally efficient.
The purpose of this paper is to investigate the problem of finding a common element of the set of zero points of the sum of two operators and the fixed point set of a quasi-nonexpansive mapping. We introduce modified ...
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The purpose of this paper is to investigate the problem of finding a common element of the set of zero points of the sum of two operators and the fixed point set of a quasi-nonexpansive mapping. We introduce modified forward-backward splitting methods based on the so-called inertial forward-backward splitting algorithm, mann algorithm and viscosity method. We establish weak and strong convergence theorems for iterative sequences generated by these methods. Our results extend and improve some related results in the literature.
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