• Korean Journal of Mathematics
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• v.24 no.4
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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.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.581-598
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• 1997

This paper is concerned with the impulsive control problem $$ \dot{x}(t) = f(t, x) + g(t, x)\dot{u}(t), t \in [0, T], x(0) = \overline{x}, $$ where u is a possibly discontinuous control function of bounded variation, $f : R \times R^n \mapsto R^n$ is a bounded and Lipschitz continuous function, and $g : R \times R^n \mapsto R^n$ is continuously differentiable w.r.t. the variable x and satisfies $\mid$g(t,\cdot) - g(s,\cdot)$\mid$ \leq \phi(t) - \phi(s)$, for some increasing function $\phi$ and every s < t. We show that the map $u \mapsto x_u$ is Lipschitz continuous when u ranges in the set of step functions whose total variations are uniformly bounded, where $x_u$ is the solution of the impulsive control system corresponding to u. We also define the generalized solution of the impulsive control system corresponding to a measurable control functin of bounded variation.

• Communications of the Korean Mathematical Society
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• v.8 no.2
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• pp.247-254
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• 1993

• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.839-857
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• 2010

We give a set of continuous characterizations for the homogeneous Triebel-Lizorkin spaces and use them to study boundedness properties of Fourier multiplier operators whose symbols satisfy a generalization of H$\ddot{o}$rmander's condition. As an application, we give new direct proofs of the imbedding theorems of the Sobolev type.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.20 no.1
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• pp.239-246
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• 2020

The recently proposed Wasserstein generative adversarial network (WGAN) has improved some of the tricky and unstable training processes that are chronic problems of the generative adversarial network(GAN), but there are still cases where it generates poor samples or fails to converge. In order to solve the problems, this paper proposes algorithms to improve the sampling process so that the discriminator can more accurately estimate the data probability distribution to be modeled and to stably maintain the discriminator should be Lipschitz continuous. Through various experiments, we analyze the characteristics of the proposed techniques and verify their performances.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.369-378
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• 2008

Modeling nonlinear friction effects is a challenging problem. In this paper, a tracking controller is proposed for a system with uncertain nonlinear friction dynamics. Instead of using a specific friction model, we assume that the friction dynamics are represented by a function, which is unknown except its being continuously differentiable and Lipschitz continuous with known Lipschitz constants. It is shown that the scheme results in friction identification and trajectory position and velocity tracking. The analysis is done using Lyapunov-based stability method.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.73-93
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• 2005

In this paper, we attempt to present a new numerical approach to solve non-linear backward stochastic differential equations. First, we present some definitions and theorems to obtain the conditions, from which we can approximate the non-linear term of the backward stochastic differential equation (BSDE) and we get a continuous piecewise linear BSDE correspond with the original BSDE. We use the relationship between backward stochastic differential equations and stochastic controls by interpreting BSDEs as some stochastic optimal control problems, to solve the approximated BSDE and we prove that the approximated solution converges to the exact solution of the original non-linear BSDE in two different cases.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.467-484
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• 2022

In this work, we define bivariate Bernstein-Kantorovich type operators on a triangular domain and obtain some approximation results for these operators. We start off by computing some moment estimates and prove a Korovkin type convergence theorem. Then, we estimate the rate of convergence using the partial and complete modulus of continuity, and derive a Voronovskaya-type asymptotic theorem. Further, we calculate the order of approximation with regard to the Peetre's K-functional and a Lipschitz type class. In addition, we construct the associated GBS type operators and compute the rate of approximation using the mixed modulus of continuity and class of the Lipschitz of Bögel continuous functions for these operators. Finally, we use the two operators to approximate example functions in order to compare their convergence.

• The Pure and Applied Mathematics
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• v.30 no.2
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• pp.191-201
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• 2023

In this paper the control system described by Urysohn type integral equation is studied. It is assumed that control functions are integrally constrained. The trajectory of the system is defined as multivariable continuous function which satisfies the system's equation everywhere. It is shown that the set of trajectories is Lipschitz continuous with respect to the parameter which characterizes the bound of the control resource. An upper estimation for the diameter of the set of trajectories is obtained. The robustness of the trajectories with respect to the fast consumption of the remaining control resource is discussed. It is proved that every trajectory can be approximated by the trajectory obtained by full consumption of the control resource.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.479-499
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• 2014

In this paper, we propose new iteration methods for finding a common point of the solution set of a pseudomonotone equilibrium problem and the solution set of a monotone equilibrium problem. The methods are based on both the extragradient-type method and the viscosity approximation method. We obtain weak convergence theorems for the sequences generated by these methods in a real Hilbert space.