On stability of non-surjective $ (\varepsilon, s) $-isometries of uniformly convex Banach spaces

Yuqi Sun; Xiaoyu Wang; Jing Dong; Jiahong Lv; Yuqi Sun; Xiaoyu Wang; Jing Dong; Jiahong Lv

doi:10.3934/math.20241094

AIMS Mathematics

2024, Volume 9, Issue 8: 22500-22512. doi: 10.3934/math.20241094

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Research article

On stability of non-surjective $ (\varepsilon, s) $-isometries of uniformly convex Banach spaces

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

Received: 23 May 2024 Revised: 05 July 2024 Accepted: 11 July 2024 Published: 19 July 2024
MSC : 46B04, 46B20

In this paper, we established two results concerning non-surjective $ (\varepsilon, s) $-isometries of uniformly convex Banach spaces, which extended some known results of Dolinar and Jung.
- stability,
- $ (\varepsilon, s) $-isometry,
- linear isometry,
- uniformly convex space
Citation: Yuqi Sun, Xiaoyu Wang, Jing Dong, Jiahong Lv. On stability of non-surjective $ (\varepsilon, s) $-isometries of uniformly convex Banach spaces[J]. AIMS Mathematics, 2024, 9(8): 22500-22512. doi: 10.3934/math.20241094

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Abstract

In this paper, we established two results concerning non-surjective $ (\varepsilon, s) $-isometries of uniformly convex Banach spaces, which extended some known results of Dolinar and Jung.

References

[1]	S. Mazur, S. Ulam, Sur les transformations isométriques d'espaces vectoriels normés, CR Acad. Sci. Paris, 194 (1932), 946–948.
[2]	T. Figiel, On non linear isometric embeddings of normed linear spaces, Bull. Acad. Polon. Sci. Math. Astro. Phys., 16 (1968), 185–188.
[3]	G. Godefroy, N. J. Kalton, Lipschitz-free Banach spaces, Stud. Math., 159 (2003), 121–141. http://dx.doi.org/10.4064/sm159-1-6 doi: 10.4064/sm159-1-6
[4]	Y. Dutrieux, G. Lancien, Isometric embeddings of compact spaces into Banach spaces, J. Funct. Anal., 255 (2008), 494–501. http://dx.doi.org/10.1016/j.jfa.2008.04.002 doi: 10.1016/j.jfa.2008.04.002
[5]	L. X. Cheng, Y. Zhou, On perturbed metric-preserved mappings and their stability characterizations, J. Funct. Aanl., 266 (2014), 4995–5015. http://dx.doi.org/10.1016/j.jfa.2014.02.019 doi: 10.1016/j.jfa.2014.02.019
[6]	D. H. Hyers, S. M. Ulam, On approximate isometries, Bull. Amer. Math. Soc., 51 (1945), 288–292. http://dx.doi.org/10.1090/S0002-9904-1945-08337-2
[7]	P. M. Gruber, Stability of isometries, Trans. Amer. Math. Soc., 245 (1978), 263–277. http://dx.doi.org/10.2307/1998866
[8]	J. Gevirtz, Stability of isometries on Banach spaces, Proc. Amer. Math. Soc., 89 (1983), 633–636. http://dx.doi.org/10.2307/2044596 doi: 10.2307/2044596
[9]	D. H. Hyers, S. M. Ulam, Approximate isometries of the space of continuous functions, Ann. Math., 48 (1947), 285–289. http://dx.doi.org/10.2307/1969171 doi: 10.2307/1969171
[10]	M. Omladič, P. Šemrl, On non linear perturbations of isometries, Math. Ann., 303 (1995), 617–628. http://dx.doi.org/10.1007/bf01461008 doi: 10.1007/bf01461008
[11]	S. W. Qian, $\varepsilon$-Isometric embeddings, Proc. Amer. Math. Soc., 123 (1995), 1797–1803. http://dx.doi.org/10.1090/s0002-9939-1995-1260178-5 doi: 10.1090/s0002-9939-1995-1260178-5
[12]	P. Šemrl, J. Väisälä, Nonsurjective nearisometries of Banach spaces, J. Funct. Anal., 198 (2003), 268–278. http://dx.doi.org/10.1016/s0022-1236(02)00049-6 doi: 10.1016/s0022-1236(02)00049-6
[13]	L. X. Cheng, Y. B. Dong, W. Zhang, On stability of nonsurjective $\varepsilon$-isometries of Banach spaces, J. Funct. Anal., 264 (2013), 713–734. http://dx.doi.org/10.1016/j.jfa.2012.11.008 doi: 10.1016/j.jfa.2012.11.008
[14]	L. X. Cheng, Q. J. Cheng, K. Tu, J. C. Zhang, A universal theorem for stability of $\varepsilon$-isometries on Banach spaces, J. Funct. Anal., 269 (2015), 199–214. http://dx.doi.org/10.1016/j.jfa.2015.04.015 doi: 10.1016/j.jfa.2015.04.015
[15]	L. X. Cheng, D. X. Dai, Y. B. Dong, Y. Zhou, Universal stability of Banach spaces of $\varepsilon$-isometries, Stud. Math., 221 (2014), 141–149. http://dx.doi.org/10.4064/sm221-2-3 doi: 10.4064/sm221-2-3
[16]	D. X. Dai, Y. B. Dong, Stablility of Banach spaces via nonlinear $\varepsilon$-isometries, J. Math. Anal. Appl., 414 (2014), 996–1005. http://dx.doi.org/10.1016/j.jmaa.2014.01.028 doi: 10.1016/j.jmaa.2014.01.028
[17]	I. A. Vestfrid, Stability of almost surjective $\varepsilon$-isometries of Banach spaces, J. Funct. Anal., 269 (2015), 2165–2170. http://dx.doi.org/10.1016/j.jfa.2015.04.009 doi: 10.1016/j.jfa.2015.04.009
[18]	Y. Zhou, Z. H. Zhang, C. Y. Liu, Stability of $\varepsilon$-isometric embeddings into Banach spaces of continuous functions, J. Math. Anal. Appl., 453 (2017), 805–820. http://dx.doi.org/10.1016/j.jmaa.2017.04.039 doi: 10.1016/j.jmaa.2017.04.039
[19]	G. Dolinar, Generalized stability of isometries, J. Math. Anal. Appl., 242 (2000), 39–56. https://doi.org/10.1006/jmaa.1999.6649 doi: 10.1006/jmaa.1999.6649
[20]	L. Cǎdariu, V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math., 4 (2003).
[21]	V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91–96.
[22]	L. Cǎdariu, V. Radu, On the stability of the Cauchy functional equation: A fixed point approach, Grazer Math. Ber., 346 (2004), 43–52.
[23]	S. M. Jung, A fixed point approach to the stability of isometries, J. Math. Anal. Appl., 329 (2007), 879–890. http://dx.doi.org/10.1016/j.jmaa.2006.06.098 doi: 10.1016/j.jmaa.2006.06.098
[24]	G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math., 20 (1975), 326–350. http://dx.doi.org/10.1007/bf02760337 doi: 10.1007/bf02760337
[25]	B. Margolis, J. B. Diaz, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305–309. http://dx.doi.org/10.1090/s0002-9904-1968-11933-0 doi: 10.1090/s0002-9904-1968-11933-0
[26]	J. A. Baker, Isometries in normed spaces, Amer. Math. Monthly, 78 (1971), 655–658. http://dx.doi.org/10.1080/00029890.1971.11992823 doi: 10.1080/00029890.1971.11992823
[27]	J. Diestel, Geometry of Banach spaces-selected topics, Berlin: Springer-Verlag, 1975.
[28]	L. X. Cheng, Q. Q. Fang, S. J. Luo, L. F. Sun, On non-surjective coarse isometries between Banach spaces, Quaest. Math., 42 (2019), 347–362. http://dx.doi.org/10.2989/16073606.2018.1448900 doi: 10.2989/16073606.2018.1448900
[29]	Y. Q. Sun, W. Zhang, Non-surjective coarse isoemtries between $L_p$ spaces, J. Math. Anal. Appl., 489 (2020), 124165. http://dx.doi.org/10.1016/j.jmaa.2020.124165 doi: 10.1016/j.jmaa.2020.124165
[30]	Y. Q. Sun, W. Zhang, Coarse isometries between finite dimensional Banach spaces, Acta Math. Sci., 41 (2021), 1493–1502. http://dx.doi.org/10.1007/s10473-021-0506-5 doi: 10.1007/s10473-021-0506-5

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