zbMATH — the first resource for mathematics

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
Full Text:
References:
  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  Gajda, Z., On stability of additive mappings, Internat. J. math. math. sci., 14, 431-434, (1991) · Zbl 0739.39013  Hyers, D.H., On the stability of the linear functional equation, Proc. nat. acad. sci. USA, 27, 222-224, (1941) · Zbl 0061.26403  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  Miura, T., On the hyers – ulam stability of a differentiable map, Sci. math. Japan, 55, 17-24, (2002) · Zbl 1025.47041  T. Miura, S.-E. Takahasi, S. Miyajima, Hyers-Ulam stability of linear differential operator with constant coefficients, Math. Nachr., in press · Zbl 1039.34054  Rassias, T.M., On the stability of the linear mapping in Banach spaces, Proc. amer. math. soc., 72, 297-300, (1978) · Zbl 0398.47040  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  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  Ulam, S.M., Problems in modern mathematics, (1964), Wiley New York, Chapter VI, Science Editions · Zbl 0137.24201  Ulam, S.M., Sets, numbers, and universes. selected works, part III, (1974), MIT Press Cambridge, MA
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.