• Journal of the Chungcheong Mathematical Society
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• v.24 no.1
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• pp.19-34
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• 2011

A bounded linear operator T on a complex Banach space X is said to have property (I) provided that T has Bishop's property (${\beta}$) and there exists an integer p > 0 such that for a closed subset F of ${\mathbb{C}}$ ${X_T}(F)={E_T}(F)=\bigcap_{{\lambda}{\in}{\mathbb{C}}{\backslash}F}(T-{\lambda})^PX$ for all closed sets $F{\subseteq}{\mathbb{C}}$, where $X_T$(F) denote the analytic spectral subspace and $E_T$(F) denote the algebraic spectral subspace of T. Easy examples are provided by normal operators and hyponormal operators in Hilbert spaces, and more generally, generalized scalar operators and subscalar operators in Banach spaces. In this paper, we prove that if T has property (I), then the quasi-nilpotent part $H_0$(T) of T is given by $$KerT^P=\{x{\in}X:r_T(x)=0\}={\bigcap_{{\lambda}{\neq}0}(T-{\lambda})^PX$$ for all sufficiently large integers p, where ${r_T(x)}=lim\;sup_{n{\rightarrow}{\infty}}{\parallel}T^nx{\parallel}^{\frac{1}{n}}$. We also prove that if T has property (I) and the spectrum ${\sigma}$(T) is finite, then T is algebraic. Finally, we prove that if $T{\in}L$(X) has property (I) and has decomposition property (${\delta}$) then T has a non-trivial invariant closed linear subspace.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.171-183
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• 2005

In this paper, we introduce and study a class of general strongly nonlinear quasivariational inequalities in Hilbert spaces. We prove the existence and uniqueness of solution and convergence of the perturbed the three-step iterative sequences with errors for this kind of general strongly nonlinear quasivariational inquality problems involving relaxed Lipschitz, relaxed monotone, and strongly monotone mappings. Our results extend, improve, and unify many known results due to Liu-Ume-Kang, Kim-Kyung, Zeng and others.

• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.69-82
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• 2010

In this paper, we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an asymptotically k-strict pseudo-contractive mapping in the setting of real Hilbert spaces. We establish some weak and strong convergence theorems of the sequences generated by our proposed scheme. Our results are more general than the known results which are given by many authors. In particular, necessary and sufficient conditions for strong convergence of our iterative scheme are obtained.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1443-1455
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• 2017

Let $M_{z^N}(N{\in}{\mathbb{Z}}^d_+)$ be a bounded multiplication operator on a class of Hilbert spaces with orthogonal basis $\{z^n:n{\in}{\mathbb{Z}}^d_+\}$. In this paper, we prove that each reducing subspace of $M_{z^N}$ is the direct sum of some minimal reducing subspaces. For the case that d = 2, we find all the minimal reducing subspaces of $M_{z^N}$ ($N=(N_1,N_2)$, $N_1{\neq}N_2$) on weighted Bergman space $A^2_{\alpha}({\mathbb{B}}_2)$(${\alpha}$ > -1) and Hardy space $H^2({\mathbb{B}}_2)$, and characterize the structure of ${\mathcal{V}}^{\ast}(z^N)$, the commutant algebra of the von Neumann algebra generated by $M_{z^N}$.

• East Asian mathematical journal
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• v.24 no.2
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• pp.177-186
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• 2008

In this paper, we introduce and study a new class of system of parametric general quasivariational-like inequalities. Using $\eta$-subdifferential and $\eta$-proximal mappings of proper functionals in Hilbert spaces, we prove the equivalence between the system of parametric general quasivariational-like inequalities and a xed point problem and construct two iterative algorithms. A few existence and uniqueness results as well as the sensitivity analysis of solutions are also established for the system of parametric general quasivariational-like inequalities, and some convergence results of iterative sequence generated by the algorithms are proved. Our results extend a few known results in the literature.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.669-680
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• 2011

In this paper, some strong convergence theorems are obtained for hybrid method for modified Ishikawa iteration process of asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups in Hilbert spaces. The results presented in this article generalize and improve results of Tae-Hwa Kim and Hong-Kun Xu and others. The convergence rate of the iteration process presented in this article is faster than hybrid method of Tae-Hwa Kim and Hong-Kun Xu and others.

• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.313-321
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• 2017

In this note we establish a new non-convex hybrid iteration algorithm corresponding to Khan iterative process [4] and prove strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in Hilbert spaces. Moreover, the main results are applied to get the common fixed points of finite family of quasi-asymptotically nonexpansive mappings. The results presented in this article are interesting extensions of some current results.

• Korean Journal of Mathematics
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• v.24 no.2
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• pp.199-213
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• 2016

The resolvent operator approach of [2] is applied to solve a set-valued variational inclusion problem in ordered Hilbert spaces. The resolvent operator under consideration is called relaxed resolvent operator and we demonstrate some of its properties. To obtain the solution of a set-valued variational inclusion problem, an iterative algorithm is developed and weak-RRD set-valued mapping is used. The problem as well as main result of this paper are more general than many previous problems and results available in the literature.

• Communications of the Korean Mathematical Society
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• v.22 no.2
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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.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.791-815
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• 2014

We show amongst other that if $f,g:[a,b]{\rightarrow}\mathbb{C}$ are two functions of bounded variation and such that the Riemann-Stieltjes integral $\int_a^bf(t)dg(t)$ exists, then for any continuous functions $h:[a,b]{\rightarrow}\mathbb{C}$, the Riemann-Stieltjes integral $\int_{a}^{b}h(t)d(f(t)g(t))$ exists and $${\int}_a^bh(t)d(f(t)g(t))={\int}_a^bh(t)f(t)d(g(t))+{\int}_a^bh(t)g(t)d(f(t))$$. Using this identity we then provide sharp upper bounds for the quantity $$\|\int_a^bh(t)d(f(t)g(t))\|$$ and apply them for trapezoid and Ostrowski type inequalities. Some applications for continuous functions of selfadjoint operators on complex Hilbert spaces are given as well.