• 한국수학교육학회지시리즈B:순수및응용수학
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• 제30권2호
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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.

• 대한수학회보
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• 제51권2호
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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.

• 대한수학회논문집
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• 제11권3호
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• pp.695-706
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• 1996

In the control system $ \dot{x} = f(t,x(t)) + g(t,x(t))\dot{u}, x(0) = \bar{x}, t \in [0,T], $ this paper shows that the map from u with $L^1(m)$-topology to $x_u$ with $L^1(\mu)$-topology is Lipschitz continuous where f is $C^1$, $\mu$ is the Stieltjes measure derived from the function g which is not smooth in the variable t and $x_u$ is the solution of the above system corresponding to u under the assumption that $\dot{u}$ is bounded.

• 대한수학회보
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• 제54권1호
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• pp.17-28
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• 2017

In this paper, we deal with a class of mean-field backward stochastic differential equations (BSDEs) related to finite state, continuous time Markov chains. We obtain the existence and uniqueness theorem and a comparison theorem for solutions of one-dimensional mean-field BSDEs under Lipschitz condition.

• Journal of applied mathematics & informatics
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• 제38권5_6호
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• pp.469-481
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• 2020

In this paper, we consider a control system for semilinear differential equations in Hilbert spaces with Lipschitz continuous nonlinear term. Our method is to find the equivalence of approximate controllability for the given semilinear system and the linear system excluded the nonlinear term, which is based on results on regularity for the mild solution and estimates of the fundamental solution.

• 대한수학회지
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• 제36권4호
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• pp.737-746
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• 1999

We consider a class of discrete parameter processes on a locally compact Banach space S arising from successive compositions of strictly stationary random maps with state space C(S,S), where C(S,S) is the collection of continuous functions on S into itself. Sufficient conditions for stationary solutions are found. Existence of pth moments and convergence of empirical distributions for trajectories are proved.

• 충청수학회지
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• 제27권3호
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• pp.393-401
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• 2014

In this work, we discuss the existence of mild solutions in the ${\alpha}$-norm for some partial functional integrodifferential equations with infinite delay. We assume that the linear part generates an analytic semigroup on a Banach space X and the nonlinear part is a Lipschitz continuous function with respect to the fractional power norm of the linear part.

• 대한수학회보
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• 제37권1호
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• pp.37-52
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• 2000

This paper is concerned with the impulsive Cauchy problem (equation omitted) where u is a possibly discontinuous vector-valued function and f, $g_{i}$ : $IR^{n}$ -> $IR^{n}$ are suitably smooth functions. We show that the input-output map is Lipschitz continuous and investigate compactness of reachable sets.

• Kyungpook Mathematical Journal
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• 제59권1호
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• pp.101-123
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• 2019

In this paper, we consider a system of nonlinear variational type inclusions involving ($H,{\varphi},{\eta}$)-monotone operators in real Banach spaces. Further, we define a proximal operator associated with an ($H,{\varphi},{\eta}$)-monotone operator and show that it is single valued and Lipschitz continuous. Using proximal point operator techniques, we prove the existence and uniqueness of a solution and suggest an iterative algorithm for the system of nonlinear variational type inclusions. Furthermore, we discuss the convergence of the iterative sequences generated by the algorithms.

• 대한수학회보
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• 제52권2호
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• pp.649-659
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• 2015

The space $BV^2_{\alpha}(I)$ of all the real functions defined on interval $I=[a,b]{\subset}\mathbb{R}$, which are of bounded second ${\alpha}$-variation (in the sense De la Vall$\acute{e}$ Poussin) on I forms a Banach space. In this space we define an operator of substitution H generated by a function $h:I{\times}\mathbb{R}{\rightarrow}\mathbb{R}$, and prove, in particular, that if H maps $BV^2_{\alpha}(I)$ into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.