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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
##### Citations:
Zbl 0398.47040; Zbl 0061.26403; Zbl 0739.39013
Full Text:
##### References:
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