• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.869-885
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• 2021

In this article, we define Picard's three-step iteration process for the approximation of fixed points of Zamfirescu operators in an arbitrary Banach space. We prove a convergence result for Zamfirescu operator using the proposed iteration process. Further, we prove that Picard's three-step iteration process is almost T-stable and converges faster than all the known and leading iteration processes. To support our results, we furnish an illustrative numerical example. Finally, we apply the proposed iteration process to approximate the solution of a mixed Volterra-Fredholm functional nonlinear integral equation.

• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.961-969
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• 2021

In this paper, we study the ∆-convergence and strong convergence of SP-iteration scheme involving a nonexpansive mapping in partially ordered hyperbolic metric spaces. Also, we give an example to support our main result and compare SP-iteration scheme with the Mann iteration and Ishikawa iteration scheme. Thus, we generalize many previous results.

• Communications of the Korean Mathematical Society
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• v.13 no.1
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• pp.195-209
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• 1998

In this paper we present a new iteration method for solving optimization problems governed by partial differential equations. We generalize the existing methods such as simple gradient methods and pseudo-time methods to get an efficient iteration method. Numerical tests show that the convergence of the new iteration method is much faster than those of the pseudo-time methods especially when the parameter $\sigma$ in the cost functional is small.

• Communications of the Korean Mathematical Society
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• v.26 no.2
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• pp.229-235
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• 2011

In this paper, a continuous real function on the real line is considered. The necessary and sufficient conditions for the convergence of the Ishikawa iteration with error terms for the functional are obtained.

• Journal of applied mathematics & informatics
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• v.31 no.5_6
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• pp.733-745
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• 2013

In this paper, we first propose a new technique of the functional iterative methods VIM (Variational iteration method) and NHPM (New homotopy perturbation method) for solving two-point boundary value problems, and then we compare their numerical results with those of the finite difference method (FDM).

• Korean Journal of Mathematics
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• v.30 no.2
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• pp.205-229
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• 2022

The aim of this paper is to obtain some results which belong to fixed point theory such as strong convergence, rate of convergence, stability, and data dependence by using the new Jungck-type iteration method for a mapping defined in complex-valued Banach spaces. In addition, some of these results are supported by nontrivial numerical examples. Finally, it is shown that the sequence obtained from the new iteration method converges to the solution of the functional integral equation in complex-valued Banach spaces. The results obtained in this paper may be interpreted as a generalization and improvement of the previously known results.

• Nonlinear Functional Analysis and Applications
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• v.27 no.4
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• pp.903-920
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• 2022

In this paper, we propose and study an implicit iteration process for a finite family of total asymptotically quasi-nonexpansive mappings and a finite family of asymptotically quasi-nonexpansive mappings in the intermediate sense in convex metric spaces and establish some strong convergence results. Also, we give some applications of our result in the setting of convex metric spaces. The results of this paper are generalizations, extensions and improvements of several corresponding results.

• Nonlinear Functional Analysis and Applications
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• v.29 no.1
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• pp.1-14
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• 2024

In this paper, we establish strong and ∆-convergence theorems for new iteration process namely S-R iteration process for a generalized α-nonexpansive mappings in a uniformly convex hyperbolic space and also we show that our iteration process is faster than other iteration processes appear in the current literature's. Our results extend the corresponding results of Ullah et al. [5], Imdad et al. [16] in the setting of uniformly convex hyperbolic spaces and many more in this direction.

• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.995-1009
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• 2021

Many iterative algorithms like that Picard, Mann, Ishikawa and S-iteration are very useful to elucidate the fixed point problems of a nonlinear operators in various topological spaces. The recent trend for elucidate the fixed point via inertial iterative algorithm, in which next iterative depends on more than one previous terms. The purpose of the paper is to establish convergence theorems of new inertial Picard normal S-iteration algorithm for nonexpansive mapping in Hilbert spaces. The comparison of convergence of InerNSP and InerPNSP is done with InerSP (introduced by Phon-on et al. [25]) and MSP (introduced by Suparatulatorn et al. [27]) via numerical example.

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.59-82
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• 2022

In this paper, we present a modified (improved) generalized M-iteration with the inertial technique for three quasi-nonexpansive multivalued mappings in a real Hilbert space. In addition, we obtain a weak convergence result under suitable conditions and the strong convergence result is achieved using the hybrid projection method with our modified generalized M-iteration. Finally, we apply our convergence results to certain optimization problem, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other improved iterative methods (modified SP-iterative scheme) in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.