Functions of bounded $ {\bf (2, k)} $-variation in 2-normed spaces

Cure Arenas Jaffeth; Ferrer Sotelo Kandy; Ferrer Villar Osmin; Cure Arenas Jaffeth; Ferrer Sotelo Kandy; Ferrer Villar Osmin

doi:10.3934/math.20241175

AIMS Mathematics

2024, Volume 9, Issue 9: 24166-24183. doi: 10.3934/math.20241175

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Functions of bounded $ {\bf (2, k)} $-variation in 2-normed spaces

1.
Departament of Mathematics, University of Sucre, Cra. 28 # 5-267, Puerta Roja, Sincelejo, Sucre, Colombia
2.
Facultad de Ingenierías y Arquitectura, Universidad Pontificia Bolivariana, Cra 6 # 97A-99 Barrio Mocari, Montería, Colombia

Received: 01 June 2024 Revised: 30 June 2024 Accepted: 05 July 2024 Published: 15 August 2024
MSC : 42C15, 46C05, 46C20

In this work, the notion of function of bounded variation in 2-normed spaces was established (Definition 4.2), the set of functions of bounded $ (2, k) $-variation was endowed with a norm (Theorem 4.5), and it was proved that such a set is a Banach space (Theorem 4.6). In addition, the fundamental properties of the functions of bounded $ (2, k) $-variation in the formalism of 2-Hilbert and 2-normed spaces were studied (see Theorems 4.1, 4.3, 4.4). Also, it was shown how to endow a 2-normed space with a function of bounded $ (2, k) $-variation from a classical Hilbert space (Proposition 4.1). A series of examples and counterexamples are presented that enrich the results obtained in this work (4.1 and 4.2).
- 2-product,
- 2-norm,
- 2-Hilbert,
- 2-Banach,
- $ (2, k) $-variation
Citation: Cure Arenas Jaffeth, Ferrer Sotelo Kandy, Ferrer Villar Osmin. Functions of bounded $ {\bf (2, k)} $-variation in 2-normed spaces[J]. AIMS Mathematics, 2024, 9(9): 24166-24183. doi: 10.3934/math.20241175

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Abstract

In this work, the notion of function of bounded variation in 2-normed spaces was established (Definition 4.2), the set of functions of bounded $ (2, k) $-variation was endowed with a norm (Theorem 4.5), and it was proved that such a set is a Banach space (Theorem 4.6). In addition, the fundamental properties of the functions of bounded $ (2, k) $-variation in the formalism of 2-Hilbert and 2-normed spaces were studied (see Theorems 4.1, 4.3, 4.4). Also, it was shown how to endow a 2-normed space with a function of bounded $ (2, k) $-variation from a classical Hilbert space (Proposition 4.1). A series of examples and counterexamples are presented that enrich the results obtained in this work (4.1 and 4.2).

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