On the method of alternating resolvents

Boikanyo, Oganeditse A.; Moroşanu, Gheorghe

doi:10.1016/j.na.2011.05.009

Cited by 10 publications

(11 citation statements)

References 16 publications

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“…And the problem studied by Theorem 4.13 is one to find a common solution to some variational inclusion problems. Theorem 4.7 and Theorem 4.12 generalize many known results in the literature, for example, [3,4,9,11,12,16,17,19,22,24,25] and references therein.…”

Section: Applications To Multilevel Split Variational Inclusion Problemssupporting

confidence: 65%

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On the multilevel nonlinear problem and its convergence algorithms

He¹

2016

J. Nonlinear Sci. Appl.

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In this paper, applying the geometrical knowledge of Hilbert spaces, we investigate and analyze a system of multilevel split fixed point problems (MSFP). New split solution algorithms are introduced and strong convergence theorems for (MSFP) are established. At the end of this paper, as an application of our results, we investigate and analyze a system of multilevel split variational inclusion problems (MSVIP) and some strong convergence solution for (MSVIP) are obtained. These results obtained by this paper improve and develop some known ones in the literature.

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Section: Applications To Multilevel Split Variational Inclusion Problemssupporting

confidence: 65%

“…For more detail, see the References [9,11,19,25]. Besides of the proximal point algorithm, some other iterative algorithms are also introduced in [4,16,17], which are used to find the approximation solution of the (CVIP).…”

Section: Applications To Multilevel Split Variational Inclusion Problemsmentioning

confidence: 99%

On the multilevel nonlinear problem and its convergence algorithms

He¹

2016

J. Nonlinear Sci. Appl.

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“…Replacing F by

I - f

with

f = a I + (1 - a) u

, where

a \in (0, 1)

and

u \in E

are fixed, and re‐denoting

k : = k - 1

in , we obtain

\begin{matrix} true \\ right \begin{matrix} x^{2 k + 1} & = J_{β_{k}}^{A} (t_{k} u + (1 - t_{k}) x^{2 k} + e^{k}), k = 0, 1, \dots, \\ x^{2 k} & = J_{γ_{k}}^{B} (t_{k} u + (1 - t_{k}) x^{2 k - 1} + {\overset{e}{true}}^{k}), k = 1, 2, \dots, \end{matrix} \end{matrix}

for a given

x^{0} \in E

. Algorithm is the method of alternating resolvents, whose general form was studied in with additional conditions on resolvent parameters

β_{k}

and

γ_{k}

, when A and B are two maximal monotone mappings in Hilbert spaces. These additional conditions were dropped for algorithm and its equivalent form

\begin{matrix} true \\ right \begin{matrix} z^{2 k + 1} & = t_{k} u + (1 - t_{k}) J_{r_{k}}^{A} z^{2 k} + \end{matrix} \end{matrix}

…”

Section: Resultsmentioning

confidence: 99%

“…Algorithm is the method of alternating resolvents, whose general form was studied in with additional conditions on resolvent parameters

β_{k}

and

γ_{k}

, when A and B are two maximal monotone mappings in Hilbert spaces. These additional conditions were dropped for algorithm and its equivalent form

\begin{matrix} true \\ right \begin{matrix} z^{2 k + 1} & = t_{k} u + (1 - t_{k}) J_{r_{k}}^{A} z^{2 k} + e^{k}, k = 0, 1, \dots, \\ z^{2 k} & = t_{k} u + (1 - t_{k}) J_{r_{k}}^{B} z^{2 k - 1} + {\overset{e}{true}}^{k}, k = 1, 2, \dots, \end{matrix} \end{matrix}

(see, and ). So, and can be considered as extensions of the results in of and from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces.…”

Section: Resultsmentioning

confidence: 99%

“…These additional conditions were dropped for algorithm and its equivalent form

\begin{matrix} true \\ right \begin{matrix} z^{2 k + 1} & = t_{k} u + (1 - t_{k}) J_{r_{k}}^{A} z^{2 k} + e^{k}, k = 0, 1, \dots, \\ z^{2 k} & = t_{k} u + (1 - t_{k}) J_{r_{k}}^{B} z^{2 k - 1} + {\overset{e}{true}}^{k}, k = 1, 2, \dots, \end{matrix} \end{matrix}

(see, and ). So, and can be considered as extensions of the results in of and from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. By the similar argument as the above, we have the modification of alternating resolvent method,

\begin{matrix} true \\ right \begin{matrix} x^{2 k + 1} & = (1 - t_{k}) J_{β_{k}}^{A} x^{2 k} + e^{k}, k = 0, 1, \dots, \\ x^{2 k} & = (1 - t_{k}) J_{γ_{k}}^{B} x^{2 k - 1} + {\overset{e}{true}}^{k}, k = 1, 2, \dots, \end{matrix} \end{matrix}

and the following result.…”

Section: Resultsmentioning

confidence: 99%

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Steepest‐descent proximal point algorithms for a class of variational inequalities in Banach spaces

Buong

2018

Mathematische Nachrichten

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In this paper, we present a new approach to the problem of finding a common zero for a system of m‐accretive mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. We propose an implicit iteration method and two explicit ones, based on compositions of resolvents with the steepest‐descent method. We show that our results contain some iterative methods in literature as special cases. An extension of the Xu's regularization method for the proximal point algorithm from Hilbert spaces onto Banach ones under simple conditions of convergence and a new variant for the method of alternating resolvents are obtained. Numerical experiments are given to affirm efficiency of the methods.

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Product of Resolvents on Hadamard Manifolds

Ahmadi,

Khatibzadeh

2024

Mediterr. J. Math.

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On the method of alternating resolvents

Cited by 10 publications

References 16 publications

On the multilevel nonlinear problem and its convergence algorithms

On the multilevel nonlinear problem and its convergence algorithms

Steepest‐descent proximal point algorithms for a class of variational inequalities in Banach spaces

Product of Resolvents on Hadamard Manifolds

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