Methods for Variational Inequality Problem Over the Intersection of Fixed Point Sets of Quasi-Nonexpansive Operators

Cegielski, Andrzej; Zalas, Rafał

doi:10.1080/01630563.2012.716807

Cited by 44 publications

(34 citation statements)

References 26 publications

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“…If A = I in the problem (1.6), then (1.6) reduces to the CFP for demicontractive (quasi-nonexpansive) operators which the results in the papers [1,7,8,16,19,30] can be applied to solve. Furthermore, if S = P C and T = P Q , then the problem (1.6) reduces to the SFP (1.1) which the results in [27] can applied to solve.…”

Section: Remark 42mentioning

confidence: 98%

Another look at the split common fixed point problem for demicontractive operators

Shehu

Cholamjiak

2015

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In this article, we take a look at the recent results of Moudafi (Inverse Probl 26:587-600, 2010), Tang et al. (Math Model Anal 17:457-466, 2012) and Wang and Cui (Math Model Anal 18:537-542, 2013), which weak convergence results were obtained for the split common fixed point problem for demicontractive mappings. We introduce a new algorithm for solving the split common fixed point problem for demicontractive mappings and then prove strong convergence of the sequence in real Hilbert spaces. We also apply our results to the split common null point problem in real Hilbert spaces. Finally, we give numerical results to demonstrate its convergence.

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Section: Remark 42mentioning

confidence: 98%

Another look at the split common fixed point problem for demicontractive operators

Shehu

Cholamjiak

2015

RACSAM

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“…Quasi-nonexpansive operators satisfying implication (2.16) were investigated in [16,17,13,53] under the name approximately shrinking. It is worth mentioning that in a general Hilbert space the demi-closedness of U − Id is only a necessary condition for implication (2.16) and even a firmly nonexpansive mapping may not satisfy it.…”

Section: Opial's Closedness Principlementioning

confidence: 99%

“…See, for example, the papers of Bauschke (with F := Id −a) [2], Deutsch and Yamada (F := ∇f ) [24], Xu and Kim [48], Hirstoaga (Q := Id −µF ) [32], Cegielski and Zalas [16,17], Aoyama and Kohsaka [1], Zalas' PhD thesis [53] and Cegielski [12,13]. As a matter of fact, Yamada and Ogura [50,Theorem 5] showed that (1.12) could be applied to a more general case with weaker monotonicity assumptions, such as paramonotonicity (see, for example, [51]).…”

Section: Introductionmentioning

confidence: 99%

“…Such a condition will be satisfied if, for example, the normalized vectors F x and x form an acute angle bounded away from π/2 outside some bounded set. In order to guarantee the convergence of any trajectory {x k } ∞ k=0 generated by Algorithm 1.1, we impose, following Cegielski and Zalas [16,17,53], a certain regularity condition on the family {T k : R n → R n | k = 0, 1, 2, . .…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 3.16], while weaker forms of it can be found in[16,,[17, Definition 6.1] and [12, Definition 4.1]. Similar regularity conditions were proposed by many authors; see, for example, Bauschke and Borwein [4, Definitions 3.7 and 4.8], Yamada and Ogura [50, Definition 1], Hirstoaga [32, Condition 2.2(iii)], Takahashi, Takeuchi and Kubota [44, NSTcondition(I)], Cegielski [11, Theorem 3.6.2, Definition 5.8.5] and Aoyama and Kohsaka [1, Condition (Z)].…”

mentioning

confidence: 97%

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Iterative methods for solving variational inequalities in Euclidean space

Gibali

Reich

Zalas

2015

J. Fixed Point Theory Appl.

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In this paper, we investigate the convergence properties of an iterative method for solving variational inequalities in Euclidean space. We show that under certain assumptions the method can be applied to variational inequalities defined over the common fixed point set of a given infinite family of cutter operators. The main step of our method consists in the computation of the metric projection onto a certain superhalf-space, which is constructed using the input data defining the problem. Moreover, in the case where the common fixed point set is a finite intersection of the fixed point sets of cutters for which Opial's closedness principle holds, we show that the iterative method can be easily combined with either sequential, composition or convex combination type methods. We also discuss ways of applying our method to more general classes of operators such as quasi-nonexpansive and demi-contractive ones.Mathematics Subject Classification. 47H05, 47H09, 47H10, 47J20, 47J25, 65K15.

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Extrapolated simultaneous block‐iterative cutter methods and applications

Buong

Anh²

2023

Math Methods in App Sciences

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Our goal of this paper is to solve the problem of common fixed point of a quite large family of demiclosed cutters on Hilbert spaces. We introduce two new extrapolated simultaneous block‐iterative methods. The first method converges weakly to a common fixed point of the family and the second one is a specific combination of the first and the steepest descent method and converges in norm. Its strong convergence is proved without additional conditions on the cutters such as the approximately shrinking property of each cutter and bounded regular assumption on their fixed point sets, assumed recently in the literature. Some particular cases of the last method, applications and computational experiments to a convex optimization over the intersection of level sets are given for illustration and comparison.

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Methods for Variational Inequality Problem Over the Intersection of Fixed Point Sets of Quasi-Nonexpansive Operators

Cited by 44 publications

References 26 publications

Another look at the split common fixed point problem for demicontractive operators

Another look at the split common fixed point problem for demicontractive operators

Iterative methods for solving variational inequalities in Euclidean space

Extrapolated simultaneous block‐iterative cutter methods and applications

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