• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.729-740
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• 2012

In this paper, we prove common fixed point theorems for occasionally weakly compatible maps in Menger spaces with a continuous t-norm of H-type. As application to our results, we obtain the corresponding fixed point theorems in metric spaces. Our results improve and generalize many known results in Menger spaces as well as in metric spaces.

• 제어로봇시스템학회:학술대회논문집
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• 2004.08a
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• pp.1847-1852
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• 2004

This article provides the connection between feedback stabilization and interpolation conditions for n-D linear systems (n > 1). In addition to internal stability, if one demands performance as a design goal, then there results an n-D matrix Nevanlinna-Pick interpolation problem. Application of recent work on Nevanlinna-Pick interpolation on the polydisk yields a solution of the problem for the 2-D case. The same analysis applies in the n-D case (n > 2), but leads to solutions which are contractive in a norm (the "Schur-Agler norm") somewhat stronger than the $H^{\infty}$ norm. This is an analogous version of the connection between the standard $H^{\infty}$ control problem and an interpolation problem of Nevanlinna-Pick type in the classical 1-D linear time-invariant systems.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.149-167
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• 2019

In this work, we study the existence, uniqueness and stability in the ${\alpha}$-norm of solutions for some stochastic partial functional integrodifferential equations. We suppose that the linear part has an analytic resolvent operator in the sense given in Grimmer [8] and the nonlinear part satisfies a $H{\ddot{o}}lder$ type condition with respect to the ${\alpha}$-norm associated to the linear part. Firstly, we study the existence of the mild solutions. Secondly, we study the exponential stability in pth moment (p > 2). Our results are illustrated by an example. This work extends many previous results on stochastic partial functional differential equations.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.815-828
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• 2021

Under some mild conditions for the weight function K, the boundedness, compactness and essential norm of integral operators T_g and I_g from Morrey type spaces H^p_K to weighted BMOA spaces $BMOAK_{K,{\frac{2}{p}}}$ investigated in this paper.

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.553-569
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• 2021

In this paper, we present and analyze a fully discrete numerical method for solving the time-fractional diffusion wave equation: ∂^β_tu - div(a∇u) = f, 1 < β < 2. We first construct a difference formula to approximate ∂^β_tu by using an interpolation of derivative type. The truncation error of this formula is of O(△t^2+δ-β)-order if function u(t) ∈ C^2,δ[0, T] where 0 ≤ δ ≤ 1 is the Hölder continuity index. This error order can come up to O(△t^3-β) if u(t) ∈ C³ [0, T]. Then, in combinination with the linear finite volume discretization on spatial domain, we give a fully discrete scheme for the fractional wave equation. We prove that the fully discrete scheme is unconditionally stable and the discrete solution admits the optimal error estimates in the H¹-norm and L₂-norm, respectively. Numerical examples are provided to verify the effectiveness of the proposed numerical method.

• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.363-370
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• 2005

In this paper, we obtain a new fixed point theorem in complete probabilistic ${\Delta}$-inner product space. As an example of applications, we utilize the results of this paper to study the existence and uniqueness of solutions for linear Valterra integral equation.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.117-129
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• 1996

In this paper, some existence theorem of solutins for nonlinear operator equations with probabilistic contractor in probabilistic normed spaces are given.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1195-1219
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• 2017

In this paper, the limit behavior of solution for the $Schr{\ddot{o}}dinger$ equation with random dispersion and time-dependent nonlinear loss/gain: $idu+{\frac{1}{{\varepsilon}}}m({\frac{t}{{\varepsilon}^2}}){\partial}_{xx}udt+{\mid}u{\mid}^{2{\sigma}}udt+i{\varepsilon}a(t){\mid}u{\mid}^{2{\sigma}_0}udt=0$ is studied. Combining stochastic Strichartz-type estimates with $L^2$ norm estimates, we first derive the global existence for $L^2$ and $H^1$ solution of the stochastic $Schr{\ddot{o}}dinger$ equation with white noise dispersion and time-dependent loss/gain: $idu+{\Delta}u{\circ}d{\beta}+{\mid}u{\mid}^{2{\sigma}}udt+ia(t){\mid}u{\mid}^{2{\sigma}_0}udt=0$. Secondly, we prove rigorously the global diffusion-approximation limit of the solution for the former as ${\varepsilon}{\rightarrow}0$ in one-dimensional $L^2$ subcritical and critical cases.