In this paper, we introduce an iterative method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of common fixed points of a countable family of nonexpansive mappings in Hi...
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In this paper, we introduce an iterative method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of common fixed points of a countable family of nonexpansive mappings in Hilbert spaces. Using the result we consider a strong convergence theorem in variational inequalities and equilibrium problems. The result present in this paper extend and improve the corresponding result of Qin et al. (Nonlinear Anal 69:3897-3909, 2008), Plubtieng and Punpaeng (J Math Anal Appl 336:455-469, 2007) and many others.
The aim of this paper is to establish four groups of equivalent theorems of convergence between Ishikawa-Halpern iteration and viscosity approximation method, respectively. Furthermore, the authors consider the viscos...
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The aim of this paper is to establish four groups of equivalent theorems of convergence between Ishikawa-Halpern iteration and viscosity approximation method, respectively. Furthermore, the authors consider the viscosity approximation method with weakly contractive mapping. The results improve and extend the results announced by many others. (C) 2010 Elsevier Ltd. All rights reserved.
To address the split best proximity point and monotone variational inclusion problems in real Hilbert spaces, we present and investigate projection and viscosity approximation methods. Under a few reasonable assumptio...
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To address the split best proximity point and monotone variational inclusion problems in real Hilbert spaces, we present and investigate projection and viscosity approximation methods. Under a few reasonable assumptions, we prove some weak and strong convergence theorems for the aforementioned methods. The efficiency of the proposed method is demonstrated by some numerical examples. Some well-known recent results in this area have been improved, generalized, and extended as an outcome of this paper.
In a real Hilbert space, an iterative scheme is considered to obtain a common fixed point for a countable family of nonexpansive mappings. In addition, strong convergence to the common fixed point of this sequence is ...
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In a real Hilbert space, an iterative scheme is considered to obtain a common fixed point for a countable family of nonexpansive mappings. In addition, strong convergence to the common fixed point of this sequence is investigated. As an application, an equilibrium problem is solved. We also state more applications of this procedure to obtain a common fixed point of W-mappings.
Using the viscosity approximation method introduced by Moudafi (J Math Anal Appl 241:46-55, 2000), we can obtain strong convergence theorems for monotone increasingG-nonexpansive mappings in Hadamard spaces endowed wi...
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Using the viscosity approximation method introduced by Moudafi (J Math Anal Appl 241:46-55, 2000), we can obtain strong convergence theorems for monotone increasingG-nonexpansive mappings in Hadamard spaces endowed with graphs. We also give sufficient conditions for the existence of solutions of the variational inequality problem in this setting. Our results generalize and improve many results in the literature.
In this paper, we introduce a new viscosity approximation method by using the shrinking projection algorithm to approximate a common fixed point of a countable family of nonlinear mappings in a Banach space. Under qui...
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In this paper, we introduce a new viscosity approximation method by using the shrinking projection algorithm to approximate a common fixed point of a countable family of nonlinear mappings in a Banach space. Under quite mild assumptions, we establish the strong convergence of the sequence generated by the proposed algorithm and provide an affirmative answer to an open problem posed by Mainge (Comput. Math. Appl. 59: 74-79, 2010) for quasi-nonexpansive mappings. In contrast with related processes, our method does not require any demiclosedness principle condition imposed on the involved operators belonging to the wide class of quasi-nonexpansive operators. As an application, we also introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of quasi-nonexpansive mappings and the set of solutions of a mixed equilibrium problem in a real Banach space. We prove a strong convergence theorem by using the proposed algorithm under some suitable conditions. Our results improve and generalize many known results in the current literature.
We study the viscosity approximation method for a sequence of quasinonexpansive mappings with contraction-like mappings. We establish a strong convergence theorem and then we apply our result to approximate a solution...
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We study the viscosity approximation method for a sequence of quasinonexpansive mappings with contraction-like mappings. We establish a strong convergence theorem and then we apply our result to approximate a solution of a split feasibility problem and a fixed point of a Lipschitz continuous pseudo-contraction.
For finding a zero of an m-accretive operator in a real Banach space, we consider an implicit hybrid viscosity approximation method and show that its explicit variant converges strongly to a zero under weaker assumpti...
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For finding a zero of an m-accretive operator in a real Banach space, we consider an implicit hybrid viscosity approximation method and show that its explicit variant converges strongly to a zero under weaker assumptions than the ones used recently by Ceng et al. (Numer. Func. Anal. Opt. 35(2), 142-165, 2012). First, we examine the case of spaces which are uniformly smooth or reflexive and strictly convex with a uniformly Gateaux differentiable norm. Afterwards, we improve our convergence results when the space is uniformly convex. Furthermore, we show that the strong convergence of some modified Halpern and Krasnosel'skii-Mann type methods can also be deduced from our results. Finally, a numerical example is also given to illustrate the convergence analysis of the considered method.
In this paper, we consider a common solution of three problems in real Hilbert spaces including the split generalized equilibrium problem, the variational inequality problem and the fixed point problem for nonexpansiv...
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In this paper, we consider a common solution of three problems in real Hilbert spaces including the split generalized equilibrium problem, the variational inequality problem and the fixed point problem for nonexpansive multivalued mappings. For finding the solution, we present a modified viscosity approximation method and prove a strong convergence theorem under mild conditions. Moreover, we also provide a numerical example to illustrate the convergence behavior of the proposed iterative method.
Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a new approach to visualize Mandelbrot and Julia sets for complex polynomials of the form W(z) = z(n) + mz + r;n ...
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Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a new approach to visualize Mandelbrot and Julia sets for complex polynomials of the form W(z) = z(n) + mz + r;n >= 2 where m, r is an element of C, and biomorphs for any complex function through a viscosity approximation method which is among the most widely used iterative methods for finding fixed points of non-linear operators. We derive novel escape criterion for generating Julia and Mandelbrot sets via proposed viscosity approximation method. Moreover, we visualize the sets using the escape time algorithm and the proposed iteration. Then, we discuss the shape change of the obtained sets depending on the parameters of the iteration using graphical and numerical experiments. The presented examples reveal that this change can be very complex, and we are able to obtain a great variety of shapes.
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