• 호남수학학술지
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• 제31권2호
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• pp.233-237
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• 2009

In this paper, we show that: (1) every strongly ${\omega}$-monolithic space X with countable fan-tightness is Fr$\'{e}$chet-Urysohn; (2) a direct proof of that X is Lindel$\"{o}$f when $C_p$(X) is Fr$\'{e}$chet-Urysohn; and (3) X is Lindel$\"{o}$f when X is paraLindel$\"{o}$f and $C_p$(X) is AP. (3) is a generalization of the result of [8]. And we give two questions related to Fr$\'{e}$chet-Urysohn and AP properties on $C_p$(X).

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제18권2호
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• pp.97-111
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• 2011

We provide a semilocal convergence result for approximating a solution of a singular system with constant rank derivatives, using Newton's method in an Euclidean space setting. Our approach uses more precise estimates and a combination of two Lipschitz-type conditions leading to the following advantages over earlier works [13], [16], [17], [29]: tighter bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples are also provided in this study.

• Journal of applied mathematics & informatics
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• 제16권1_2호
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• pp.1-17
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• 2004

The famous Newton-Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton method to a solution of an equation in connection with the Lipschitz continuity of the Frechet-derivative of the operator involved. Using Lipschitz and center-Lipschitz conditions we show that the Newton-Kantorovich hypothesis is weakened. The error bounds obtained under our semilocal convergence result are finer and the information on the location of the solution more precise than the corresponding ones given by the dominating Newton-Kantorovich theorem, and under the same hypotheses/computational cost, since the evaluation of the Lipschitz also requires the evaluation of the center-Lipschitz constant. In the case of local convergence we obtain a larger convergence radius than before. This observation is important in computational mathematics and can be used in connection to projection methods and in the construction of optimum mesh independence refinement strategies.