• 대한수학회지
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• 제49권1호
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• pp.187-200
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• 2012

In this paper, we introduced a new extended extragradient iteration algorithm for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of equilibrium problems for a monotone and Lipschitz-type continuous mapping. And we show that the iterative sequences generated by this algorithm converge strongly to the common element in a real Hilbert space.

• Nonlinear Functional Analysis and Applications
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• 제27권3호
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• pp.553-568
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• 2022

In this paper, we introduce and study two different iterative hybrid projection algorithms for solving a fixed point problem of nonexpansive mappings. The first algorithm is generated by the combination of the inertial method and the hybrid projection method. On the other hand, the second algorithm is constructed by the convex combination of three updated vectors and the hybrid projection method. The strong convergence of the two proposed algorithms are proved under very mild assumptions on the scalar control. For illustrating the advantages of these two newly invented algorithms, we created some numerical results to compare various numerical performances of our algorithms with the algorithm proposed by Dong and Lu [11].

• Nonlinear Functional Analysis and Applications
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• 제27권1호
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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.

• Kyungpook Mathematical Journal
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• 제59권4호
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• pp.703-723
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• 2019

In this paper, we consider a common solution of three problems in real Hilbert spaces: the Ky Fan inequality problem, the variational inequality problem and the fixed point problem for demicontractive single-valued and quasi-nonexpansive multi-valued mappings. To find the solution we present a new iterative algorithm and prove a strong convergence theorem under mild conditions. Moreover, we provide a numerical example to illustrate the convergence behavior of the proposed iterative method.

• 대한수학회논문집
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• 제19권1호
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• pp.63-74
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• 2004

Suppose that X is a real Banach space, K is a nonempty closed convex subset of X and T : $K\;\rightarrow\;K$ is a uniformly continuous ${\phi}$-hemicontractive operator or a Lipschitz ${\phi}-hemicontractive$ operator. In this paper we prove that under certain conditions the three-step iteration methods with errors converge strongly to the unique fixed point of T. Our results extend the corresponding results of Chang [1], Chang et a1. [2], Chidume [3]-[7], Chidume and Osilike [9], Deng [10], Liu and Kang [13], [14], Osilike [15], [16] and Tan and Xu [17].

• Nonlinear Functional Analysis and Applications
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• 제27권1호
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• pp.203-221
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• 2022

In this paper, we introduce new hybrid inertial contraction projection algorithms for solving variational inequality problems over the intersection of the fixed point sets of demicontractive mappings in a real Hilbert space. The proposed algorithms are based on the hybrid steepest-descent method for variational inequality problems and the inertial techniques for finding fixed points of nonexpansive mappings. Strong convergence of the iterative algorithms is proved. Several fundamental experiments are provided to illustrate computational efficiency of the given algorithm and comparison with other known algorithms

• Journal of applied mathematics & informatics
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• 제30권3_4호
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• pp.395-411
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• 2012

Several methods with order higher than that of Newton methods which are of order 2 have been reported in literature for solving nonlinear equations. The focus of most of these methods was to economize on/minimize the number of function evaluations per iterations. We have demonstrated here that there are several computational pit-falls, such as the violation of fixed-point theorem, that one could encounter while using these methods. Further it was also shown that the overall computational complexity could be more in these high-order methods than that in the second-order Newton method.

• 대한수학회보
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• 제45권4호
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• pp.717-728
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• 2008

Let H be a real Hilbert space and S = {T(s) : $0\;{\leq}\;s\;<\;{\infty}$} be a nonexpansive semigroup on H such that $F(S)\;{\neq}\;{\emptyset}$ For a contraction f with coefficient 0 < $\alpha$ < 1, a strongly positive bounded linear operator A with coefficient $\bar{\gamma}$ > 0. Let 0 < $\gamma$ < $\frac{\bar{\gamma}}{\alpha}$. It is proved that the sequences {$x_t$} and {$x_n$} generated by the iterative method $$x_t\;=\;t{\gamma}f(x_t)\;+\;(I\;-\;tA){\frac{1}{{\lambda}_t}}\;{\int_0}^{{\lambda}_t}\;T(s){x_t}ds,$$ and $$x_{n+1}\;=\;{\alpha}_n{\gamma}f(x_n)\;+\;(I\;-\;{\alpha}_nA)\frac{1}{t_n}\;{\int_0}^{t_n}\;T(s){x_n}ds,$$ where {t}, {${\alpha}_n$} $\subset$ (0, 1) and {${\lambda}_t$}, {$t_n$} are positive real divergent sequences, converges strongly to a common fixed point $\tilde{x}\;{\in}\;F(S)$ which solves the variational inequality $\langle({\gamma}f\;-\;A)\tilde{x},\;x\;-\;\tilde{x}{\rangle}\;{\leq}\;0$ for $x\;{\in}\;F(S)$.

• 대한수학회논문집
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• 제22권2호
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• pp.247-258
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• 2007

In this paper a new class of system of generalized nonlinear variational inequalities involving strongly monotone, relaxed co coercive and relaxed generalized monotone mappings in Hilbert spaces is introduced and studied. Based on the projection method, an equivalence between the system of generalized nonlinear variational inequalities and the fixed point problem is established, which is used to suggest some new iterative algorithms for computing approximate solutions of the system of generalized nonlinear variational inequalities. A few sufficient conditions which ensure the existence and uniqueness of solution of the system of generalized nonlinear variational inequalities are given, and the convergence analysis of iterative sequences generated by the algorithms are also discussed.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2008년도 춘계학술대회논문집
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• pp.389-392
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• 2008

This paper predicts the changes of natural frequencies due to the changes of mass at different point mass stations by using iterative calculation Transfer Matrices Method for different boundary conditions of a single beam structure (fixed-free and fixed-fixed beam). Firstly, the first three natural frequencies of an original beam are obtained using Transfer Matrices Method to verify the accuracy of the obtained results. The results are then compared with the exact solutions before purposely changing the parameter of mass. Both beams are modeled as discrete continuous systems with six-lumped-mass system. A single beam is broken down into a point mass and a massless beam which represent a single station and expressed in matrix form. The assembled matrices are used to determine the value of natural frequencies using numerical interpolation method corresponding to their mode number by manipulating some elements in the assembled matrix.