• Korean Journal of Mathematics
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• 제28권4호
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• pp.915-929
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• 2020

We prove a fixed point theorem for generalized asymptotic pointwise nonexpansive mapping in the setting of a hyperbolic space. A one-step iterative scheme approximating common fixed point of two generalized asymptotic pointwise (quasi-) nonexpansive mappings in this setting is provided. We obtain ∆-convergence and strong convergence theorems of the iterative scheme for two generalized asymptotic pointwise nonexpansive mappings in the same setting. Our results generalize and extend some related results in the literature.

• Nonlinear Functional Analysis and Applications
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• 제26권5호
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• pp.961-969
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• 2021

In this paper, we study the ∆-convergence and strong convergence of SP-iteration scheme involving a nonexpansive mapping in partially ordered hyperbolic metric spaces. Also, we give an example to support our main result and compare SP-iteration scheme with the Mann iteration and Ishikawa iteration scheme. Thus, we generalize many previous results.

• Korean Journal of Mathematics
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• 제24권4호
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• pp.647-662
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• 2016

In this paper, we study Lipschitz continuous, the boundedness and compactness of the composition operator $C_{\phi}$ acting between the general hyperbolic Bloch type-classes ${\mathcal{B}}^{\ast}_{p,{\log},{\alpha}}$ and general hyperbolic Besov-type classes $F^{\ast}_{p,{\log}}(p,q,s)$. Moreover, these classes are shown to be complete metric spaces with respect to the corresponding metrics.

• Kyungpook Mathematical Journal
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• 제55권1호
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• pp.13-20
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• 2015

As a vector space provides a fundamental tool for the study of Euclidean geometry, a gyrovector space provides an algebraic tool for the study of hyperbolic geometry. In general, the gyrovector spaces do not satisfy the distributivity with scalar multiplication. In this article, we see under what condition the distributivity with scalar multiplication is satisfied.

• 대한수학회보
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• 제25권2호
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• pp.185-194
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• 1988

In [2], we have shown that the optimal boundary controls for hyperbolic systems in L$^{2}$-spaces can be attained in a feedback form via Riccati operators. A number of authors [1, 5, 7 and 10] have investigated approximations of Riccati operators arising in distributed parameter systems. They assumed bounded controls for parabolic systems. However, we in this paper study Galerkin approximations of Riccati operators and feedback controls for hyperbolic systems with unbounded control actions. Let us briefly introduce some results of [2]. Let .ohm. be an open bounded region in R$^{n}$ with smooth boundary .GAMMA. where n is a fixed positive integer. We consider a strictly hyperbolic differential operator H(x) of order 1 on .ohm. with noncharacteristic boundary on .GAMMA.

• 대한수학회논문집
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• 제31권3호
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• pp.575-583
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• 2016

In this paper, we study the modified S-iteration process for asymptotic pointwise nonexpansive mappings in a uniformly convex hyperbolic metric space. We then prove the convergence of the sequence generated by the modified S-iteration process.

• 충청수학회지
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• 제35권3호
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• pp.269-275
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• 2022

In this note, we verify that the bounded shadowing property and quasi-hyperbolicity of bounded linear operators on Hilbert spaces are preserved under Aluthge transforms.

• 대한수학회보
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• 제50권1호
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• pp.321-341
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• 2013

In this paper, we discuss the a posteriori error estimates of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order $k$ Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order $k(k{\geq}0)$. Using mixed elliptic reconstruction method, a posterior $L^{\infty}(L^2)$-error estimates for both the state and the control approximation are derived. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

• 대한수학회논문집
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• 제30권3호
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• pp.209-219
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• 2015

In this article, we first give metric version of an iteration scheme of Agarwal et al. [1] and approximate fixed points of two finite families of nonexpansive mappings in hyperbolic spaces through this iteration scheme which is independent of but faster than Mann and Ishikawa scheme. Also we consider case of three finite families of nonexpansive mappings. But, we need an extra condition to get convergence. Our convergence theorems generalize and refine many know results in the current literature.

• 대한수학회보
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• 제55권4호
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• pp.1103-1107
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• 2018

Let M be a complete spacelike hypersurface in the (n + 1)-dimensional Minkowski space ${\mathbb{L}}^{n+1}$. Suppose that every unit speed curve X(s) on M satisfies ${\langle}X^{\prime\prime}(s),X^{\prime\prime}s){\rangle}{\geq}-1/r^2$ and there exists a point $p{\in}M$ such that for every unit speed geodesic X(s) of M through the point p, ${\langle}X^{\prime\prime}(s),X^{\prime\prime}s){\rangle}=-1/r^2$ holds. Then, we show that up to isometries of ${\mathbb{L}}^{n+1}$, M is the hyperbolic space $H^n(r)$.