• East Asian mathematical journal
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• v.28 no.5
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• pp.569-578
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• 2012

In this paper, the necessary and sufficient conditions for the strong convergence of a modified Mann iteration process to a fixed point of an asymptotically demicontractive mapping in real Banach spaces are considered. Presented results improve and extend the results of Igbokwe [3], Liu [4], Moore and Nnoli [6] and Osilike [7].

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.45-58
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• 2022

In this paper, iterative algorithms for approximating a common fixed point of a countable family of multi-valued demicontractive maps in the setting of Hadamard spaces are presented. Under different mild conditions, the sequences generated are shown to strongly convergent and ∆-convergent to a common fixed point of the considered family, accordingly. Our theorems complement many results in the literature.

• Kyungpook Mathematical Journal
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• v.59 no.4
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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.

• Bulletin of the Korean Mathematical Society
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• v.51 no.5
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• pp.1527-1538
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• 2014

The purpose of this paper is to address the multiple split common fixed point problem. We present two different methods to approximate a solution of the problem. One is cyclic iteration method; the other is simultaneous iteration method. Under appropriate assumptions on the operators and iterative parameters, we prove both the proposed algorithms converge to the solution of the multiple split common fixed point problem. Our results generalize and improve some known results in the literatures.

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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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

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.149-167
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• 2022

In this paper, we propose a viscosity iterative algorithm for approximating a common solution of finite family of variational inequality problem and fixed point problem for finite family of multi-valued type-one demicontractive mappings in real Hilbert spaces. A strong convergence result of the aforementioned problems were proved and some consequences of our result was also displayed. In addition, we discuss an application of our main result to convex minimization problem. The result presented in this article complements and extends many recent results in literature.

• Nonlinear Functional Analysis and Applications
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• v.29 no.2
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• pp.307-332
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• 2024

Two inertial extragradient-type algorithms are introduced for solving convex pseudomonotone variational inequalities with fixed point problems, where the associated mapping for the fixed point is a 𝜌-demicontractive mapping. The algorithm employs variable step sizes that are updated at each iteration, based on certain previous iterates. One notable advantage of these algorithms is their ability to operate without prior knowledge of Lipschitz-type constants and without necessitating any line search procedures. The iterative sequence constructed demonstrates strong convergence to the common solution of the variational inequality and fixed point problem under standard assumptions. In-depth numerical applications are conducted to illustrate theoretical findings and to compare the proposed algorithms with existing approaches.

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.269-276
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• 2002

Let E be a real Banach space. Let T : E longrightarrow E be a map with F(T) : = { x $\in$ E : Tx = x} $\neq$ 0 and satisfying the accretive-type condition $\lambda\$\mid$x-Tx\$\mid$^2$, for all $x\inE,\;x^*\inf(T)\;and\;\lambda >0$. We prove some necessary and sufficient conditions for the convergence of the Mann iterative sequence to a fixed point of T.

• Bulletin of the Korean Mathematical Society
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• v.37 no.1
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• pp.153-169
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• 2000

Let E be a real q-uniformly smooth Banach space which admits a weakly sequentially continuous duality map, and K a nonempty closed convex subset of E. Let T : K -> K be a mapping such that $F(T)\;=\;{x\;{\in}\;K\;:\;Tx\;=\;x}\;{\neq}\;0$ and (I - T) satisfies the accretive-type condition: $\;{\geq}\;{\lambda}$\mid$$\mid$x-Tx$\mid$$\mid$^2$, for all $x\;{\in}\;K,\;x^*\;{\in}\;F(T)$ and for some ${\lambda}\;>\;0$. The weak and strong convergence of the Ishikawa iteration method to a fixed point of T are investigated. An application of our results to the approximation of a solution of a certain linear operator equation is also given. Our results extend several important known results from the Mann iteration method to the Ishikawa iteration method. In particular, our results resolve in the affirmative an open problem posed by Naimpally and Singh (J. Math. Anal. Appl. 96 (1983), 437-446).