December 2024
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1 Read
Communications in Algebra
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December 2024
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1 Read
Communications in Algebra
January 2024
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31 Reads
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3 Citations
Communications in Algebra
December 2022
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129 Reads
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3 Citations
Journal of Fixed Point Theory and Applications
We use methods from nonstandard analysis to unify and extend results from the theory of asymptotic fixed points on metric spaces. Among the topics we consider are Kirk’s asymptotic fixed point theorem with extensions, Boyd and Wong’s fixed point theorem, and Suzuki’s fixed point theorem for asymptotic contractions of the Meier–Keeler type. We also show how Suzuki’s asymptotic contractions of the final type can be modified to yield uniform convergence on compact and bounded sets.
June 2022
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60 Reads
The roots of a complex polynomial depend continuously on the coefficients; that is, an infinitesimal perturbation of the coefficients results in an infinitesimal perturbation of the roots. A short, straightforward proof of this is possible using infinitesimals.
June 2022
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69 Reads
Let K be an algebraically closed field with an absolute value. This note gives an elementary proof of the classical result that the roots of a polynomial with coefficients in K are continuous functions of the coefficients of the polynomial.
... Continuity of roots of a polynomial,[30,). Let f (z) be a polynomial of degree K with only simple roots. ...
January 2024
Communications in Algebra
... The beauty of asymptotic contractions is that they are not even non-expansive and, hence, they are not iteratively equivalent to any class of contractions. However, at the same time, the most arduous part of research in this topic is to give constructive proofs of metrical fixed point theorems for this kind of mappings (see, for example, [12,17]). ...
December 2022
Journal of Fixed Point Theory and Applications