Stability of the Banach space valued Chebyshev differential equation (Q712654)

Let \(E\) be a complex Banach space with a norm \(\|\cdot\|\) and \(I=(-1,1)\) the open interval of \(\mathbb R\). Denote by \(C(I,E)\) the linear space of all \(E\)-valued continuous functions on \(I\) and by \(C^2(I,E)\) the linear subspace of all functions \(f\in C(I,E)\) which are strongly twice differentiable and such that \(f''\) is continuous. Then, the authors prove that the \(E\)-valued Chebyshev differential equation \[ (1-x^2)y''(x)-xy'(x)+n^2y(x)=0,\quad x\in I, \] has Hyers-Ulam stability, i.e., for each \(\varepsilon >0\) and \(f\in C^2(I,E)\) satisfying \[ \|(1-x^2)f''(x)-xf'(x)+n^2f(x)\|\leq\varepsilon,\quad\forall x\in I, \] there exists \(h\in C^2(I,E)\) such that \[ (1-x^2)h''(x)-xh'(x)+n^2h(x)=0\quad \text{and}\quad \|f(x)-h(x)\|\leq\frac{\pi^2\varepsilon}8,\quad\forall x\in I. \]