On a new nonlinear convex structure

Andreea Bejenaru; Mihai Postolache; Andreea Bejenaru; Mihai Postolache

doi:10.3934/math.2024103

AIMS Mathematics

2024, Volume 9, Issue 1: 2063-2083. doi: 10.3934/math.2024103

Previous Article Next Article

Research article

On a new nonlinear convex structure

Andreea Bejenaru ^{1
,
,},
Mihai Postolache ^2,3,4

1.
Department of Mathematics and Informatics, National University of Science and Technology Politehnica Bucharest, 060042 Bucharest, Romania
2.
The Key Laboratory of Intelligent Information and Big Data Processing of NingXia Province, and Health Big Data Research Institute, North Minzu University, 750021 Yinchuan, China
3.
Department of Mathematics and Informatics, National University of Science and Technology Politehnica Bucharest, 060042 Bucharest, Romania
4.
Gh. Mihoc-C. Iacob Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, 050711 Bucharest, Romania

Received: 01 October 2023 Revised: 08 November 2023 Accepted: 09 November 2023 Published: 20 December 2023
MSC : 46B20, 47H09, 52A01

In this work we start from near vector spaces, which we endow with some additional properties that allow convex analysis. The seminormed structure used here will also be improved by adding properties such as the null condition and null equality, thus resulting in a new type of space, which is still weaker than the conventional Banach structures: pre-convex regular near-Banach space. On the newly defined structure, we introduce the concept of uniform convexity and analyze several resulting properties. The major outcomes prove a remarkable resemblance to the classical properties resulting from uniform convexity on hyperbolic metric spaces or modular function spaces, including the famous Browder-Göhde fixed point theorem.
- near-Banach spaces,
- uniform convexity,
- type function,
- asymptotic center,
- nonexpansive mapping
Citation: Andreea Bejenaru, Mihai Postolache. On a new nonlinear convex structure[J]. AIMS Mathematics, 2024, 9(1): 2063-2083. doi: 10.3934/math.2024103

Related Papers:

Abstract

In this work we start from near vector spaces, which we endow with some additional properties that allow convex analysis. The seminormed structure used here will also be improved by adding properties such as the null condition and null equality, thus resulting in a new type of space, which is still weaker than the conventional Banach structures: pre-convex regular near-Banach space. On the newly defined structure, we introduce the concept of uniform convexity and analyze several resulting properties. The major outcomes prove a remarkable resemblance to the classical properties resulting from uniform convexity on hyperbolic metric spaces or modular function spaces, including the famous Browder-Göhde fixed point theorem.

References

[1]	J. A. Clarkson, Uniformly convex spaces, Trans. Am. Math. Soc., 40 (1936), 396–414. https://doi.org/10.2307/1989630
[2]	F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci., 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041 doi: 10.1073/pnas.54.4.1041
[3]	D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr., 30 (1965), 251–258. https://doi.org/10.1002/mana.19650300312 doi: 10.1002/mana.19650300312
[4]	K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.
[5]	M. Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc., 78 (1972), 206–208. https://doi.org/10.1090/S0002-9904-1972-12918-5 doi: 10.1090/S0002-9904-1972-12918-5
[6]	M. Edelstein, Fixed point theorems in uniformly convex Banach spaces, Proc. Amer. Math. Soc., 44 (1974), 369–374. https://doi.org/10.2307/2040439 doi: 10.2307/2040439
[7]	K. Goebel, T. Sekowski, A. Stachura, Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball, Nonlinear Anal.-Theor., 4 (1980), 1011–1021. https://doi.org/10.1016/0362-546X(80)90012-7 doi: 10.1016/0362-546X(80)90012-7
[8]	S. Reich, I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal., 15 (1990), 537–558. https://doi.org/10.1016/0362-546X(90)90058-O doi: 10.1016/0362-546X(90)90058-O
[9]	M. A. Khamsi, A. R. Khan, Inequalities in metric spaces with applications, Nonlinear Anal.-Theor., 74 (2011), 4036–4045. https://doi.org/10.1016/j.na.2011.03.034 doi: 10.1016/j.na.2011.03.034
[10]	M. A. Khamsi, W. M. Kozlowski, S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal., 14 (1990), 935–953. https://doi.org/10.1016/0362-546X(90)90111-S doi: 10.1016/0362-546X(90)90111-S
[11]	H. C. Wu, Near fixed point theorems in near-Banach spaces, AIMS Mathematics, 8 (2023), 1269–1303. https://doi.org/10.3934/math.2023064 doi: 10.3934/math.2023064

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)