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A characterization of Hyers-Ulam stability of first order linear differential operators. (English) Zbl 1045.47037

Let \(E_1\), \(E_2\) be two real Banach spaces and \(f: E_1\to E_2\) is a mapping such that \(f(tx)\) is continuous in \(t\in\mathbb{R}\) (the set of real numbers), for each fixed \(x\in E_1\). Th. M. Rassias [Proc. Am. Math. Soc. 72, 297–300 (1978; Zbl 0398.47040)] introduced the following inequality: Assume that there exist \(\theta\geq 0\) and \(p\in [0,1)\) such that \[ \| f(x+ y)- f(x)- f(y)\|\leq \theta(\| x\|^p+\| y\|^p) \] for every \(x,y\in E_1\). Then there exists a unique linear mapping \(T: E_1\to E_2\) such that \(\| f(x)- T(x)\|\leq 2\theta\| x\|^p/(2-2^p)\) for every \(x\in E_1\). D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222–224 (1941; Zbl 0061.26403)] had obtained the result for \(p= 0\).
Rassias’ proof also works for \(p< 0\). In 1990, the reviewer, during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for \(p\geq 1\). In 1991, Z. Gajda [Int. J. Math. Math. Sci. 14, 431–434 (1991; Zbl 0739.39013)], following the reviewer’s approach, gave an affirmative solution to this question for \(p> 1\).
The authors of the present paper consider the following problem: Let \(X\) be a complex Banach space and \(h: \mathbb{R}\to\mathbb{C}\) a continuous function. Assume that \(T_h: C^1(\mathbb{R}, X)\to C(\mathbb{R}, X)\) is the linear differential operator defined by \(T_hu= u'+ hu\). Then a very essential and interesting necessary and sufficient condition is obtained in order for the operator \(T_h\) to be stable in the sense of Hyers-Ulam.

MSC:

47E05 General theory of ordinary differential operators
39B42 Matrix and operator functional equations
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References:

[1] Alsina, C.; Ger, R., On some inequalities and stability results related to the exponential function, J. Inequal. Appl., 2, 373-380 (1998) · Zbl 0918.39009
[2] Gajda, Z., On stability of additive mappings, Internat. J. Math. Math. Sci., 14, 431-434 (1991) · Zbl 0739.39013
[3] Hyers, D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27, 222-224 (1941) · Zbl 0061.26403
[4] Miura, T.; Takahasi, S.-E.; Choda, H., On the Hyers-Ulam stability of real continuous function valued differentiable map, Tokyo J. Math., 24, 467-476 (2001) · Zbl 1002.39039
[5] Miura, T., On the Hyers-Ulam stability of a differentiable map, Sci. Math. Japan, 55, 17-24 (2002) · Zbl 1025.47041
[6] T. Miura, S.-E. Takahasi, S. Miyajima, Hyers-Ulam stability of linear differential operator with constant coefficients, Math. Nachr., in press; T. Miura, S.-E. Takahasi, S. Miyajima, Hyers-Ulam stability of linear differential operator with constant coefficients, Math. Nachr., in press · Zbl 1039.34054
[7] Rassias, T. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 (1978) · Zbl 0398.47040
[8] Rassias, T. M.; Šemrl, P., On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., 114, 989-993 (1992) · Zbl 0761.47004
[9] Takahasi, S.-E.; Miura, T.; Miyajima, S., On the Hyers-Ulam stability of the Banach space-valued differential equation \(y\)′=λy, Bull. Korean Math. Soc., 39, 309-315 (2002) · Zbl 1011.34046
[10] Ulam, S. M., Problems in Modern Mathematics (1964), Wiley: Wiley New York, Chapter VI, Science Editions · Zbl 0137.24201
[11] Ulam, S. M., Sets, Numbers, and Universes. Selected Works, Part III (1974), MIT Press: MIT Press Cambridge, MA
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