• 대한수학회지
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• 제61권4호
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• pp.743-760
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• 2024

In this paper, we are devoted to study a forth order curve flow for a smooth closed curve in centro-affine geometry. Firstly, a new evolutionary equation about this curve flow is proposed. Then the related geometric quantities and some meaningful conclusions are obtained through the equation. Next, we obtain finite order differential inequalities for energy by applying interpolation inequalities, Cauchy-Schwartz inequalities, etc. After using a completely new symbolic expression, the n-order differential inequality for energy is considered. Finally, by the means of energy estimation, we prove that the forth order curve flow has a smooth solution all the time for any closed smooth initial curve.

• 대한수학회지
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• 제45권2호
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• pp.331-353
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• 2008

Some new nonlinear retarded integral inequalities of Gronwall-like type are established, which mainly generalized some results given by Cho, Dragomir and Kim (J. Korean Math. Soc. 43 (2006), No.3, pp. 563-578) and can be used in the analysis of various problems in the theory of certain classes of differential equations and integral equations. Applications examples are also indicated.

• Kyungpook Mathematical Journal
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• 제51권2호
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• pp.165-175
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• 2011

The purpose of this note is to give Opial type inequalities with nonlocal conditions. Also, a reverse of the original inequality with y(a) = y(b) = 0 is derived. We apply these inequalities to second-order differential equations with nonlocal conditions to derive several necessary conditions for the existence of solutions.

• Journal of applied mathematics & informatics
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• 제9권2호
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• pp.635-643
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• 2002

The asymptotic behaviour of the solutions of initial - boundary value problem for a class of nonlinear parabolic differential - functional equations is studied via the method of differential inequalities in order to obtain oscillation criterion for the solutions.

• 충청수학회지
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• 제24권4호
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• pp.935-943
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• 2011

We study the h-stability for linear impulsive differential equations and their perturbations by using the impulsive integral inequalities.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제15권1호
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• pp.43-55
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• 2011

We here investigate the Liouville type theorems of slow diffusion differential inequality and its coupled system with variable coefficients in cone. First, we give the definition of global weak solution, and then we establish the universal estimate (does not depend on the initial value) of solution by constructing test function. At last, we obtain the nonexistence of non-negative non-trivial global weak solution within the appropriate critical exponent. The main feature of this method is that we need not use comparison theorem or the maximum principle.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제21권1호
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• pp.11-21
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• 2014

The present paper is concerned with the notions of Lipschitz and asymptotic stability for perturbed nonlinear differential system knowing the corresponding stability of nonlinear differential system. We investigate Lipschitz and asymtotic stability for perturbed nonlinear differential systems. The main tool used is integral inequalities of the Bihari-type, in special some consequences of an extension of Bihari's result to Pinto and Pachpatte, and all that sort of things.

• 충청수학회지
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• 제28권4호
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• pp.583-590
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• 2015

This paper deals with linear impulsive fractional differential equations involving the Caputo derivative with non-integer order q. We provide exact solutions of linear impulsive fractional differential equations with constant coefficient by mean of the Mittag-Leffler functions. Then we apply the exact solutions to improve impulsive integral inequalities with singularity.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제22권1호
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• pp.1-11
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• 2015

The present paper is concerned with the notions of Lipschitz and asymptotic for perturbed functional differential system knowing the corresponding stability of functional differential system. We investigate Lipschitz and asymptotic stability for perturbed functional differential systems. The main tool used is integral inequalities of the Bihari-type, and all that sort of things.

• 대한수학회지
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• 제56권1호
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• pp.127-147
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• 2019

In this paper we establish some new explicit solutions for impulsive linear fractional differential equations with impulses at fixed times, which provides a handy tool in deriving singular integral-sum inequalities and an impulsive fractional comparison principle. Thus we study the Mittag-Leffler stability of impulsive differential equations with the Caputo fractional derivative by using the impulsive fractional comparison principle and piecewise continuous functions of Lyapunov's method. Also, we give some examples to illustrate our results.