A Revisit to n-Normed Spaces Through Its Quotient Spaces

H. Batkunde; H.  Gunawan

doi:10.30970/ms.53.2.181-191

H. Batkunde Analysis and Geometry Research Group,Faculty of Mathematics and Natural Sciences,Bandung Institute of Technology, Bandung Indonesia
H. Gunawan Analysis and Geometry Research Group, Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Bandung 40132, Indonesia

DOI: https://doi.org/10.30970/ms.53.2.181-191

Keywords: continuous mappings; fixed points; n-normed spaces; quotient spaces

Abstract

In this paper, we define several types of continuous mapping in $n$-normed spaces with respect to the norms of its quotient spaces. Then, we show that all types of the continuity are equivalent. We also study contractive mappings on $n$-normed spaces using these norms. In particular, we prove a fixed point theorem for contractive mappings on a closed and bounded set in the $n$-normed space with respect to the norms of its quotient spaces.
In the last section we prove a fixed point theorem and give some remarks on the $p$-summable sequence space as an $n$-normed space.

Author Biography

H. Batkunde, Analysis and Geometry Research Group,Faculty of Mathematics and Natural Sciences,Bandung Institute of Technology, Bandung Indonesia

Department of Mathematics

References

H. Batkunde, H. Gunawan, "On the topology of n-normed spaces with respect to norms of its quotient spaces Adv. Stud. Contemp. Math., 29 (2019), 89-98.

H. Batkunde, H. Gunawan, O. Neswan, "n-Normed spaces with norms of its quotient spaces J. Phys.: Conf. Series, 1097 (2018), 012079.

S. Ekariani, H. Gunawan, M. Idris, "A contractive mapping theorem on the n-normed space of p-summable sequences J. Math. Anal., 4 (2013), 1-7.

N. Eghbali, J. M. Rassia, M. Taheri, "On the Stability of a k -Cubic Functional Equation in Intuitionistic Fuzzy n-Normed Spaces Results in Mathematics, 70 (2016), 233-248.

S. Gahler, "Lineare 2-normierte raume Math. Nachr., 28 (1964), 1-13.

S. Gahler, "Untersuchungen uber verallgemeinerte m-metrische raume. I Math. Nachr., 40 (1969), 16-189.

S. Gahler, "Untersuchungen uber verallgemeinerte m-metrische raume. II Math. Nachr., 40 (1969), 229-264.

S. Gahler, "Untersuchungen uber verallgemeinerte m-metrische raume. III Math. Nachr., 41 (1969), 23-36.

H. Gunawan, Mashadi: "On n-normed spaces Int. J. Math. Math. Sci., 27 (2001), 631-639.

H. Gunawan, E. Sukaesih, O. Neswan, "Fixed point theorems on bounded sets in an n-normed space J. Math. Anal., 3 (2015), 51-58.

H. Gunawan, "The space of p-summable sequences and its natural n-norm Bull. Austral. Math. Soc., 64 (2001), 137-147.

X. Huang X, Tan A, "Mappings of preserving n-distance one in n-normed spaces Aequationes Math., 92 (2018), 401-413.

S. Meitei, Singh, "On bounded n-linear operators J. Math. Comput. Sci., 8 (2018), 196-215.

K. Raj, A. K. Sharma, S K. Sharma, S. Singh, "Some double sequence spaces defined by a sequence of Orlicz functions over n-normed spaces Lobacevskii J. Math., 8(2) (2012), 183-190.

J. Srivastava, P. Singh, "On some Cauchy sequences defined in `p considered as n-normed space J. Rajasthan Acad. Phys. Sci., 15 (2016), 107-116.