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.


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