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1
Content available remote Computation of solution of integral equations via fixed point results
Alqudah Manar A., Garodia Chanchal, Uddin Izhar, Nieto Juan J.
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
772--785
EN
The motive of this article is to study a modified iteration scheme for monotone nonexpansive mappings in the class of uniformly convex Banach space and establish some convergence results. We obtain weak and strong convergence results. In addition, we present a nontrivial numerical example to show the convergence of our iteration scheme. To demonstrate the utility of our scheme, we discuss the solution of nonlinear integral equations as an application, which is again supported by a nontrivial example.
2
Content available remote Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications
Pakkaranang Nuttapol, Rehman Habib ur, Kumam Wiyada
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
280--298
EN
The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.
3
Content available remote An iterative approximation of common solutions of split generalized vector mixed equilibrium problem and some certain optimization problems
Mewomo Oluwatosin T., Oyewole Olawale K.
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
335--358
EN
In this paper, we study the problem of finding a common solution of split generalized vector mixed equlibrium problem (SGVMEP), fixed point problem (FPP) and variational inequality problem (VIP). We propose an inertial-type iterative algorithm, which uses a projection onto a feasible set and a linesearch, which can be easily calculated. We prove a strong convergence of the sequence generated by the proposed algorithm to a common solution of SGVMEP, fixed point of a quasi- ϕ -nonexpansive mapping and VIP for a general class of monotone mapping in 2-uniformly convex and uniformly smooth Banach space E1 and a smooth, strictly convex and reflexive Banach space E2 . Some numerical examples are presented to illustrate the performance of our method. Our result improves some existing results in the literature.
4
Convergence theorems for generalized hemicontractive mapping in p-uniformly convex metric space
Ugwunnadi Godwin C., Izuchukwu Chinedu, Mewomo Oluwatosin T.
Journal of Applied Analysis
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2020
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Vol. 26, nr 2
221--229
EN
In this paper, we introduce and study an Ishikawa-type iteration process for the class of generalized hemicontractive mappings in p-uniformly convex metric spaces, and prove both Delta-convergence and strong convergence theorems for approximating a fixed point of generalized hemicontractive mapping in complete p-uniformly convex metric spaces. We give a surprising example of this class of mapping that is not a hemicontractive mapping. Our results complement, extend and generalize numerous other recent results in CAT(0) spaces.
5
Convergence theorem for a finite family of asymptotically demicontractive multi-valued mappings in CAT(0) spaces
Ugwunnadi Godwin C., Mewomo Oluwatosin T., Izuchukwu Chinedu
Journal of Applied Analysis
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2020
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Vol. 26, nr 1
117--130
EN
In this paper, we introduce the class of asymptotically demicontractive multivalued mappings and establish a strong convergence theorem of the modified Mann iteration to a common fixed point of a finite family of asymptotically demicontractive multivalued mappings in a complete CAT(0) space. We also give a numerical example of our iterative method to show its applicability.
6
Content available remote Strong convergence of an inertial extrapolation method for a split system of minimization problems
Gebrie Anteneh Getachew, Wangkeeree Rabian
Demonstratio Mathematica
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2020
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Vol. 53, nr 1
332--351
EN
In this article, we propose an inertial extrapolation-type algorithm for solving split system of minimization problems: finding a common minimizer point of a finite family of proper, lower semicontinuous convex functions and whose image under a linear transformation is also common minimizer point of another finite family of proper, lower semicontinuous convex functions. The strong convergence theorem is given in such a way that the step sizes of our algorithm are selected without the need for any prior information about the operator norm. The results obtained in this article improve and extend many recent ones in the literature. Finally, we give one numerical example to demonstrate the efficiency and implementation of our proposed algorithm.
7
Content available remote A strong convergence theorem for a zero of the sum of a finite family of maximally monotone mappings
Wega Getahun B., Zegeye Habtu, Boikanyo Oganeditse A.
Demonstratio Mathematica
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2020
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Vol. 53, nr 1
152--166
EN
The purpose of this article is to study the method of approximation for zeros of the sum of a finite family of maximally monotone mappings and prove strong convergence of the proposed approximation method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems and provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.
8
Content available Convergence Analysis of An Improved Extreme Learning Machine Based on Gradient Descent Method
Yusong L., Zhixun S., Bingjie Y., Xiaoling G., Zhaoyang S.
Journal of Applied Computer Science Methods
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2016
|
Vol. 8 No. 1
5--15
EN
Extreme learning machine (ELM) is an efficient algorithm, but it requires more hidden nodes than the BP algorithms to reach the matched performance. Recently, an efficient learning algorithm, the upper-layer-solution-unaware algorithm (USUA), is proposed for the single-hidden layer feed-forward neural network. It needs less number of hidden nodes and testing time than ELM. In this paper, we mainly give the theoretical analysis for USUA. Theoretical results show that the error function monotonously decreases in the training procedure, the gradient of the error function with respect to weights tends to zero (the weak convergence), and the weight sequence goes to a fixed point (the strong convergence) when the iterations approach positive infinity. An illustrated simulation has been implemented on the MNIST database of handwritten digits which effectively verifies the theoretical results.
9
Content available remote Convergence of modified s-iteration process for two asymptotically quasi-nonexpansive type mappings in CAT(0) spaces
Saluja G. S.
Demonstratio Mathematica
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2016
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Vol. 49, nr 1
107--118
EN
The purpose of this paper is to study convergence of a newly defined modified S-iteration process to a common fixed point of two asymptotically quasi-nonexpansive type mappings in the setting of CAT(0) space. We give a sufficient condition for convergence to a common fixed point and establish some strong convergence theorems for the said iteration process and mappings under suitable conditions. Our results extend and improve many known results from the existing literature.
10
Approximation of common fixed points of left Bregman strongly nonexpansive mappings and solutions of equilibrium problems
Shehu Y., Ogbuisi F. U.
Journal of Applied Analysis
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2015
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Vol. 21, nr 2
63--77
EN
In this paper we propose an iterative algorithm based on the hybrid method in mathematical programming for approximating a common fixed point of an infinite family of left Bregman strongly nonexpansive mappings which also solves a finite system of equilibrium problems in a reflexive real Banach space.We further prove that our iterative sequence converges strongly to a common fixed point of an infinite family of left Bregman strongly nonexpansive mappings which is also a common solution to a finite system of equilibrium problems. Our result extends many recent and important results in the literature.
11
Content available remote Approximation of fixed points of some classes of nonlinear mappings
Okeke G A
Fasciculi Mathematici
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2014
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Nr 53
113--127
EN
We introduce a new class of nonlinear mappings, the class of generalized strongly successively Φ- hemicontractive mappings in the intermediate sense and prove the convergence of Mann type iterative scheme with errors to their fixed points. This class of nonlinear mappings is more general than those defined by several authors. In particular, the class of generalized strongly successively Φ- hemicontractive mappings in the intermediate sense introduced in this study is more general than the class defined by Liu et al. [Z. Liu, J. K. Kim and K. H. Kim, Convergence theorems and stability problems of the modified Ishikawa iterative sequences for strictly successively hemicontractive mappings, Bull. Korean Math. Soc. 39 (2002), No. 3, pp. 455-469].
12
Content available remote Viscosity approximation methods for nonexpansive multi-valued nonself mappings and equilibrium problems
Cholamjiak P., Cholamjiak W., Suantai S.
Demonstratio Mathematica
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2014
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Vol. 47, nr 2
382--395
EN
In this paper, strong convergence theorems by the viscosity approximation method for nonexpansive multi-valued nonself mappings and equilibrium problems are established under some suitable conditions in a Hilbert space. The obtained results extend and improve the corresponding results existed in the literature.
13
Content available remote Strong convergence of implicit iteration processes for nonexpansive semigroups in Banach spaces
Kozlowski W. M.
Commentationes Mathematicae
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2014
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Vol 54, No. 2
203--208
EN
Let C be a convex compact subset of a uniformly convex Banach space. Let {Tt}t≥0 be a strongly-continuous nonexpansive semigroup on C. Consider the iterative process defined by the sequence of equations xk+1=ckTtk+1(xk+1)+(1−ck)xk. We prove that, under certain conditions on {ck} and {tk}, the sequence {xk}∞n=1 converges strongly to a common fixed point of the semigroup {Tt}t≥0. There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, and also weak convergence results in Banach spaces that are simultaneously uniformly convex and uniformly smooth. In this paper, we do not assume the Opial property or uniform smoothness of the norm.
14
Strong convergence theorems for the approximation of fixed points of demicontinuous pseudocontractive mappings
Chidume C. E, Shehu Y.
Journal of Applied Analysis
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2013
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Vol. 19, nr 2
213--229
EN
In this paper, we introduce a new explicit iterative scheme for approximation of fixed points of demicontinuous pseudocontractive mappings in uniformly smooth Banach spaces and prove strong convergence of our proposed iterative scheme. Furthermore, we modify our explicit iterative scheme for approximation of zeroes of bounded demicontinuous accretive mappings in uniformly smooth Banach spaces. Our result improves, extends and unifies most of the results that have been proved for this class of mappings.
15
Content available Implicit random iteration process with errors for asymptotically quasi-nonexpansive in the intermediate sense random operators
Saluja G. S.
Opuscula Mathematica
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2012
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Vol. 32, no. 2
327-340
EN
In this paper, we give a necessary and sufficient condition for the strong convergence of an implicit random iteration process with errors to a common fixed point for a finite family of asymptotically quasi-nonexpansive in the intermediate sense random operators and also prove some strong convergence theorems using condition (C) and the semi-compact condition for said iteration scheme and operators. The results presented in this paper extend and improve the recent ones obtained by S. Plubtieng, P. Kumam and R. Wangkeeree, and also by the author.
16
Modified Mann iterative algorithms by hybrid projection methods for nonexpansive semigroups and mixed equilibrium problems
Katchang P., Kumam P.
Journal of Applied Analysis
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2012
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Vol. 18, nr 2
259-273
EN
In this paper, we propose a modified Mann iterative algorithm by two hybrid projection methods for finding a common element of the set of fixed points of nonexpansive semigroups and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we obtain interesting and new strong convergence theorems for the sequences generated by these processes by using the hybrid projection methods in the mathematical programming. The results presented in this paper extend and improve the corresponding one by Nakajo and Takahashi [J. Math. Anal. Appl. 279 (2003), 372-379].
17
Content available Convergence theorems for strictly asymptotically pseudocontractive mappings in Hilbert spaces
Saluja G. S.
Opuscula Mathematica
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2010
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Vol. 30, no. 4
485-494
EN
In this paper, we establish the weak and strong convergence theorems for a k-strictly asymptotically pseudo-contractive mapping in the framework of Hilbert spaces. Our result improve and extend the corresponding result of Acedo and Xu, Liu, Marino and Xu, Osilike and Akuchu, and some others.
18
Content available remote Generalized three-step iteration schemes and common fixed points of three asymptotically quasi-nonexpansive mappings
Rashwan R., Hakim A.
Demonstratio Mathematica
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2008
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Vol. 41, nr 4
895-908
EN
In this paper, we study strong convergence theorems for a generalized three-step iterative scheme with errors to approximate common fixed points of three asymptotically quasi-nonexpansive mappings in real Banach spaces. Our results generalize and improve upon the corresponding results in [1], [2], [3], [4], [5], [6], [7], [9], [13] and [16]. As an application of our results, we give and prove strong convergence theorems in uniformly convex Banach spaces.
19
Content available remote Strong convergence of an implicit iteration process for a finite family of quasi-contractive operators
Rafiq A.
Demonstratio Mathematica
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2007
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Vol. 40, nr 1
203-208
EN
The purpose of this paper is to establish a strong convergence of an implicit iteration process to a common fixed point for a finite family of quasi-contractive operators. Our theorems give an armative response to a question raised by [Xu and Ori, Numer. Funct. Anal. Optim. 22 (2001) 767-773 ].
20
Content available remote A note on the theorem of V. Berinde
Rafiq A.
Demonstratio Mathematica
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2006
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Vol. 39, nr 4
863-867
EN
V. Berinde presented convergence theorem [1] in normed spaces for Ishikawa iterations associated with Zamfirescue operators, which is an extension of the theorem of Rhoades [11]. In this note, we point out some misprints in the calculations and established new inequalities involved in the proof of the theorem. Consequently we give better estimation in proving the convergence theorem.
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