Approximately additive Schwartz distributions. (English) Zbl 1104.39022
A distributional analogue of the Rassias inequality \(\| f(x+y)-f(x)-f(y)\| \leq \varepsilon(\| x\| ^p+\| y\| ^p)\) was investigated for \(p=4, 6, 8, \dots\) by J. Chung in [J. Math. Anal. Appl. 300, No. 2, 343–350 (2004; Zbl 1066.39028)].
In this paper, the author generalizes his previous stability result to all \(p \geq 0, p \neq 1\), and in fact for a certain general control function \(\psi(x, y)\) instead of \(\varepsilon(\| x\| ^p+\| y\| ^p)\), in the space of Schwartz (tempered) distributions.
In this paper, the author generalizes his previous stability result to all \(p \geq 0, p \neq 1\), and in fact for a certain general control function \(\psi(x, y)\) instead of \(\varepsilon(\| x\| ^p+\| y\| ^p)\), in the space of Schwartz (tempered) distributions.
Reviewer: Mohammad Sal Moslehian (Mashhad)
MSC:
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 35K05 | Heat equation |
| 46F10 | Operations with distributions and generalized functions |
Keywords:
Cauchy functional equation; stability; heat kernel; Gauss transform; Rassias inequality; Schwartz (tempered) distributionsCitations:
Zbl 1066.39028References:
| [1] | Baker, J. A., Distributional methods for functional equations, Aequationes Math., 62, 136-142 (2001) · Zbl 0989.39009 |
| [2] | Chung, J., Hyers-Ulam-Rassias stability of Cauchy equation in the space of Schwartz distributions, J. Math. Anal. Appl., 300, 343-350 (2004) · Zbl 1066.39028 |
| [3] | Chung, J.; Chung, S.-Y.; Kim, D., The stability of Cauchy equations in the space of Schwartz distributions, J. Math. Anal. Appl., 295, 107-114 (2004) · Zbl 1053.39043 |
| [4] | Chung, J.; Chung, S.-Y.; Kim, D., Une caractérisation de l’espace de Schwartz, C. R. Acad. Sci. Paris Sér. I Math., 316, 23-25 (1993) · Zbl 0820.46033 |
| [5] | Chung, J.; Chung, S.-Y.; Kim, D., A characterization for Fourier hyperfunctions, Publ. Res. Inst. Math. Sci., 30, 203-208 (1994) · Zbl 0808.46056 |
| [6] | Chung, J.; Lee, S. Y., Some functional equations in the spaces of generalized functions, Aequationes Math., 65, 267-279 (2003) · Zbl 1035.39015 |
| [7] | Chung, S.-Y., Reformulation of some functional equations in the space of Gevrey distributions and regularity of solutions, Aequationes Math., 59, 108-123 (2000) · Zbl 0945.39013 |
| [8] | Gajda, Z., On stability of additive mappings, Internat. J. Math. Math. Sci., 14, 431-434 (1991) · Zbl 0739.39013 |
| [9] | Găvruta, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184, 431-436 (1994) · Zbl 0818.46043 |
| [10] | Hörmander, L., The Analysis of Linear Partial Differential Operators I (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0521.35002 |
| [11] | Hyers, D. H., On the stability of the linear functional equations, Proc. Natl. Acad. Sci. USA, 27, 222-224 (1941) · Zbl 0061.26403 |
| [12] | Hyers, D. H.; Isac, G.; Rassias, Th. M., Stability of Functional Equations in Several Variables (1998), Birkhäuser · Zbl 0894.39012 |
| [13] | Isac, G.; Rassias, Th. M., On the Hyers-Ulam stability of \(ψ\)-additive mappings, J. Approx. Theory, 72, 131-137 (1993) · Zbl 0770.41018 |
| [14] | Matsuzawa, T., A calculus approach to hyperfunctions III, Nagoya Math. J., 118, 133-153 (1990) · Zbl 0692.46042 |
| [15] | Rassias, Th. M., On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251, 264-284 (2000) · Zbl 0964.39026 |
| [16] | Rassias, Th. M., On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 (1978) · Zbl 0398.47040 |
| [17] | Schwartz, L., Théorie des distributions (1966), Hermann: Hermann Paris · Zbl 0149.09501 |
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