Three-steps iterative algorithms for mixed variational inequalities

Bnouhachem, Abdellah; Noor, Muhammad Aslam; Rassias, Th. M.

doi:10.1016/j.amc.2006.05.086

Cited by 34 publications

(22 citation statements)

References 19 publications

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“…This alternative equivalent form plays a crucial role in suggesting three-step iterative schemes for solving variational inequalities. It is now well-known [1,2,31] that the three-step iterative methods for solving the variational inequalities perform better numerically than two-step and one-step method. This fact motivates us to consider the three-step iterative schemes.…”

Section: Lemma 31 the Function U ∈ H Is A Solution Of The Mixed Varimentioning

confidence: 98%

“…It is well known that a large class of problems arising in industry, ecology, finance, economics, transportation, network analysis and optimization can be formulated and studied in the framework of (2.1) and (2.7), see [1][2][3] and the references therein.…”

Section: Lemma 21 Let F Be a Differentiable G-convex Function And ϕ mentioning

confidence: 99%

“…Clearly Noor iterations include Mann (onestep) and Ishikawa (two-step) iterations as special cases. It has been shown in [1,2,31] that three-step iterative methods are numerically more efficient than one-step and two-step iterative methods for solving variational inequalities and the related optimization problems.…”

Section: Introductionmentioning

confidence: 99%

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Some iterative schemes for general mixed variational inequalities

Noor

2009

J. Appl. Math. Comput.

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In this paper, we consider a class of variational inequalities which is called the general mixed variational inequality. It is known that the general mixed variational inequalities are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some three-step iterative schemes for finding the common element of the set of fixed points of a nonexpansive mappings and the set of solutions of the mixed variational inequalities. We also study the convergence criteria of threestep iterative method under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results.

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Section: Lemma 31 the Function U ∈ H Is A Solution Of The Mixed Varimentioning

confidence: 98%

Section: Lemma 21 Let F Be a Differentiable G-convex Function And ϕ mentioning

confidence: 99%

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Some iterative schemes for general mixed variational inequalities

Noor

2009

J. Appl. Math. Comput.

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“…In recent years, variational inequality theory has been extended and generalized in several directions, using new and powerful methods, to study a wide class of unrelated problems in a unified and general framework. It turned out that odd-order and nonsymmetric obstacle, free, nonlinear equilibrium, dynamical network, optimal design, bifurcation and chaos, and moving boundary problems arising in various branches of pure and applied sciences can be studied via variational inequalities; see [1,2,[5][6][7][8][12][13]17]. One of the most important and difficult problems in this theory is the development of an efficient and implementable iterative algorithm for solving variational inequalities.…”

Section: Introductionmentioning

confidence: 99%

“…Variational inequalities introduced by Stampacchia [1] in the early sixties have witnessed an explosive growth in theoretical advances, algorithmic development, and applications across all the discipline of pure and applied sciences; see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], and the references therein. In recent years, variational inequality theory has been extended and generalized in several directions, using new and powerful methods, to study a wide class of unrelated problems in a unified and general framework.…”

Section: Introductionmentioning

confidence: 99%

Three-step iterations for nonexpansive mappings and inverse-strongly monotone mappings

Shang

Qin

2009

J Syst Sci Complex

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This paper introduces a three-step iteration for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inversestrongly monotone mapping by viscosity approximation methods in a Hilbert space. The authors show that the iterative sequence converges strongly to a common element of the two sets, which solves some variational inequality. Subsequently, the authors consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudo-contractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inversestrongly monotone mapping. The results obtained in this paper extend and improve the corresponding results announced by Nakajo, Takahashi, and Toyoda.

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A novel iterative scheme for solving delay differential equations and nonlinear integral equations in Banach spaces

Okeke

Ofem

2022

Math Methods in App Sciences

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We propose a three-step iteration process for finding the common fixed points of nonexpansive mapping and strongly pseudocontractive mapping in a real Banach space. We weaken the necessity of condition (C) imposed by a previous study on the mappings by using a quite simple and different method to obtain strong convergence of our proposed iterative scheme to the common fixed point of nonexpansive mapping and strongly pseudocontractive mapping. Numerically, we also show that our proposed iterative scheme converges faster than some existing iterative schemes. Furthermore, we apply our proposed iterative process in solving mixed type Volterra-Fredholm functional nonlinear integral equations and delay differential equations.

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Three-steps iterative algorithms for mixed variational inequalities

Cited by 34 publications

References 19 publications

Some iterative schemes for general mixed variational inequalities

Some iterative schemes for general mixed variational inequalities

Three-step iterations for nonexpansive mappings and inverse-strongly monotone mappings

A novel iterative scheme for solving delay differential equations and nonlinear integral equations in Banach spaces

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