Correction: Comparison principles for the time-fractional diffusion equations with the Robin boundary conditions. Part I: Linear equations

Correction to: Fractional Calculus and Applied Analysis (2023) 26:1504–1544 https://doi.org/10.1007/s13540-023-00182-z

Much to our regret, the definition of the space $H_{\alpha }(0,T),\ 0<\alpha <1$ provided in our recently published paper (Fract. Calc. Appl. Anal. 26(4) (2023), 1504–1544, https://doi.org/10.1007/s13540-023-00182-z) on the bottom of page 1507 is not correct for $\alpha = 1/2$. Fortunately, this definition was not directly used in our paper and thus it had no influence on other results and derivations. However, we decided to submit a corrigendum with a correct definition of the space $H_{\alpha }(0,T)$ that is valid for all $\alpha \in (0,\, 1)$.

In our paper, we defined $H_{\alpha }(0,T),\ 0< \alpha <1$ as the closure of the space ${_{0}C^1[0,T]}:= \{ u \in C^1[0,T]; u(0) = 0\}$ in the Sobolev-Slobodeckij space $H^{\alpha }(0,T)$, i.e., as $H_{\alpha }(0,T) = \overline{{_{0}C^1[0,T]}}^{H^{\alpha }(0,T)}$. This definition is correct for $0< \alpha <1$ and $\alpha \not = 1/2$ because in this case the relation $\overline{{_{0}C^1[0,T]}}^{H^{\alpha }(0,T)} = \overline{{_{0}C^1[0,T]}}^{H_{\alpha }(0,T)}$ holds valid. However, it is not the case for $\alpha = 1/2$ and thus the definition has to be reformulated as follows:

$$\begin{aligned} H_{\alpha }(0,T) = \left\{ \begin{array}{rl} &{}H^{\alpha }(0,T), \quad 0<\alpha<\frac{1}{2}, \\ &{}\left\{ v \in H^{\frac{1}{2}}(0,T);\, \int ^T_0 \frac{\vert v(t)\vert ^2}{t} dt< \infty \right\} , \quad \alpha =\frac{1}{2}, \\ &{} \{ v \in H^{\alpha }(0,T);\, v(0) = 0\}, \quad \frac{1}{2}< \alpha < 1 \end{array}\right. \end{aligned}$$

with the norm defined by

$$\begin{aligned} \Vert v\Vert _{H_{\alpha }(0,T)} = \left\{ \begin{array}{rl} &{}\Vert v\Vert _{H^{\alpha }(0,T)}, \quad \alpha \ne \frac{1}{2}, \\ &{}\left( \Vert v\Vert _{H^{\frac{1}{2}}(0,T)}^2 + \int ^T_0 \frac{\vert v(t)\vert ^2}{t}dt\right) ^{\frac{1}{2}}, \quad \alpha =\frac{1}{2}. \end{array}\right. \end{aligned}$$

For the space $H_{\alpha }(0,T)$, the property $H_{\alpha }(0,T) = \overline{{_{0}C^1[0,T]}}^{H_{\alpha }(0,T)}$ holds valid for $0<\alpha <1$ and thus any function from $H_{\alpha }(0,T)$ can be approximated by the functions from ${_{0}C^1[0,T]}$. In our paper, we employed only this approximation property and did not directly use the incorrect definition of the space $H_{\alpha }(0,T)$ for $\alpha = 1/2$.