There are many opinions about multiplication and division formulas in interval numbers, but there is a common weakness in these formulas: That the division between two equal interval numbers does not produce the identity. Similar to the interval number sequence, many concepts about the convergence of the interval sequence are offered by various authors but they cannot prove that the basic properties of the convergence of the real number sequence can be generalized to the properties of convergence of the interval number sequence. Here, we used the algebra for interval numbers from the author, as contained in Mashadi et al. (2023), that is, the algebra for interval numbers using midpoints, which guarantees the existence of the inverse of any interval number. In this article, we showed that some basic properties of the sequence of real numbers can be generalized to the sequence of interval numbers. In addition to the convergence properties of the interval number sequence, the convergence of the interval number sequence with positive, negative, and fractional powers were also shown. Based on the definition of convergence of interval sequences given along with various basic theorems for convergence given in this paper, it was expected that all theorems related to the convergence of real number sequences can be generalized to interval number sequences; for example, the properties of tail sequences, the Monotone Convergence Theorem, the Existence of Monotone Subsequences, Subsequences and the Bolzano-Weierstrass Theorem, and the Cauchy criterion for the convergence of interval number sequences.
Citation: Mashadi, Rasi Adishamita, Sukono, Igif Gimin Prihanto, Nurnadiah Zamri, Moch Panji Agung Saputra. Convergence of interval fuzzy number sequences[J]. AIMS Mathematics, 2025, 10(10): 24755-24778. doi: 10.3934/math.20251097
There are many opinions about multiplication and division formulas in interval numbers, but there is a common weakness in these formulas: That the division between two equal interval numbers does not produce the identity. Similar to the interval number sequence, many concepts about the convergence of the interval sequence are offered by various authors but they cannot prove that the basic properties of the convergence of the real number sequence can be generalized to the properties of convergence of the interval number sequence. Here, we used the algebra for interval numbers from the author, as contained in Mashadi et al. (2023), that is, the algebra for interval numbers using midpoints, which guarantees the existence of the inverse of any interval number. In this article, we showed that some basic properties of the sequence of real numbers can be generalized to the sequence of interval numbers. In addition to the convergence properties of the interval number sequence, the convergence of the interval number sequence with positive, negative, and fractional powers were also shown. Based on the definition of convergence of interval sequences given along with various basic theorems for convergence given in this paper, it was expected that all theorems related to the convergence of real number sequences can be generalized to interval number sequences; for example, the properties of tail sequences, the Monotone Convergence Theorem, the Existence of Monotone Subsequences, Subsequences and the Bolzano-Weierstrass Theorem, and the Cauchy criterion for the convergence of interval number sequences.
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Mashadi, Rasi Adishamita, Sukono, Igif Gimin Prihanto, Nurnadiah Zamri, Moch Panji Agung Saputra. Convergence of interval fuzzy number sequences[J]. AIMS Mathematics, 2025, 10(10): 24755-24778. doi: 10.3934/math.20251097
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