Correction: Alsaedi, A., et al. Existence and Stability Results for a Fractional Order Differential Equation with Non-Conjugate Riemann–Stieltjes Integro-Multipoint Boundary Conditions. Mathematics 2019, 7, 249

In [1], the authors wish to make the following corrections.

$\mathbf{1}.$

Throughout the original paper, the co-efficient of $g_3\left(t\right)$ (see (5), (21), (28), proof of Theorem 6) is

$${\int }_a^b\left({\int }_a^s\frac{{(s-u)}^{q-1}}{\Gamma \left(q\right)}\widehat{f}\left(u\right)du\right)dA\left(s\right).$$

It should be

$$\left[\sum _{i=1}^{n-2}{\alpha }_i{\int }_a^{{\eta }_i}\frac{{({\eta }_i-s)}^{q-1}}{\Gamma \left(q\right)}\widehat{f}\left(s\right)ds+{\int }_a^b\left({\int }_a^s\frac{{(s-u)}^{q-1}}{\Gamma \left(q\right)}\widehat{f}\left(u\right)du\right)dA\left(s\right)\right].$$

2. 

In (15) of the original paper, it was

$$I_3={\int }_a^b\left({\int }_a^s\frac{{(s-u)}^{q-1}}{\Gamma \left(q\right)}\widehat{f}\left(u\right)du\right)dA\left(s\right).$$

It should be

$$I_3=\sum _{i=1}^{n-2}{\alpha }_i{\int }_a^{{\eta }_i}\frac{{({\eta }_i-s)}^{q-1}}{\Gamma \left(q\right)}\widehat{f}\left(s\right)ds+{\int }_a^b\left({\int }_a^s\frac{{(s-u)}^{q-1}}{\Gamma \left(q\right)}\widehat{f}\left(u\right)du\right)dA\left(s\right).$$

$\mathbf{3}.$

Throughout the original paper, the coefficient of $\overline{g_3}$ (for instance, see (19), proofs of Theorems 2, 4, 6, and 7) is ${\displaystyle {\int }_a^b\frac{{(s-a)}^q}{\Gamma (q+1)}dA\left(s\right).}$ It should be

$$\left[\sum _{i=1}^{n-2}|{\alpha }_i|\frac{{({\eta }_i-a)}^q}{\Gamma (q+1)}+{\int }_a^b\frac{{(s-a)}^q}{\Gamma (q+1)}dA\left(s\right)\right].$$

The coefficient of ${\displaystyle \frac{\left|\right(g_3\left(t_2\right)-g_3\left(t_1\right)\left)\right|}{\Gamma (q+1)}}$ in the proof of Theorem 2 is ${\int }_a^b\left({\int }_a^s{(s-a)}^{q}dA\left(s\right)\right).$ It should be

$$\left[\sum _{i=1}^{n-2}|{\alpha }_i|{({\eta }_i-a)}^q+{\int }_a^b{(s-a)}^{q}dA\left(s\right)\right].$$

$\mathbf{4}.$

Throughout the original paper, the co-efficient of $|g_3\left(t\right)|$ (see proofs of Theorems 2, 4, 6, and 7) and $|g_3\left(t_2\right)-g_3\left(t_1\right)|$ in the proof Theorem 2 is

$${\int }_a^b\left({\int }_a^s\frac{{(s-u)}^{q-1}}{\Gamma \left(q\right)}\left|f(u,x\left(u\right))\right|du\right)dA\left(s\right).$$

It should be

$$\left[\sum _{i=1}^{n-2}|{\alpha }_i|{\int }_a^{{\eta }_i}\frac{{({\eta }_i-s)}^{q-1}}{\Gamma \left(q\right)}\left|f(s,x\left(s\right))\right|ds+{\int }_a^b\left({\int }_a^s\frac{{(s-u)}^{q-1}}{\Gamma \left(q\right)}\left|f(u,x\left(u\right))\right|du\right)dA\left(s\right)\right].$$

$\mathbf{5}.$

In the proofs of Theorems 6 and 7, the coefficient of $|g_3\left(t\right)|$ is

$${\int }_a^b\left({\int }_a^s\frac{{(s-u)}^{q-1}}{\Gamma \left(q\right)}|f(u,x\left(u\right))-f(u,y\left(u\right))|du\right)dA\left(s\right).$$

It should be

$$\begin{array}{c}\hfill \left[\sum _{i=1}^{n-2}|{\alpha }_i|{\int }_a^{{\eta }_i}\frac{{({\eta }_i-s)}^{q-1}}{\Gamma \left(q\right)}\right|f(s,x\left(s\right))-f(s,y\left(s\right))|ds\\ \hfill +{\int }_a^b\left({\int }_a^s\frac{{(s-u)}^{q-1}}{\Gamma \left(q\right)}|f(u,x\left(u\right))-f(u,y\left(u\right))|du\right)dA\left(s\right)].\end{array}$$

$\mathbf{6}.$

In Example 2, $\Lambda \approx 0.243646,\xi \Lambda \approx 0.097458<1$ in the original paper. These values should be $\Lambda \approx 0.261226,\xi \Lambda \approx 0.104490<1.$

$\mathbf{7}.$

In Example 3 of the original paper, $\Lambda \approx 0.272140,\Lambda -\frac{{(b-a)}^q}{\Gamma (q+1)}\approx 0.204166$ and $\delta <14.693960.$ The corrected values of these parameters are $\Lambda \approx 0.326742,\Lambda -\frac{{(b-a)}^q}{\Gamma (q+1)}\approx 0.258768$ and $\delta <11.5.$

$\mathbf{8}.$

In Example 4, $\delta <11.023738$ in the original paper. It should be $\delta <9.1.$

$\mathbf{9}.$

In Example 5, $L\Lambda \approx 0.102053.$ It should be $L\Lambda \approx 0.097959.$

$\mathbf{10}.$

In the Conclusions, the coefficient of $\overline{g_3}$ is ${\displaystyle \frac{{(b-a)}^{q+1}}{\Gamma (q+2)}.}$ It should be

$$\sum _{i=1}^{n-2}\left|{\alpha }_i\right|\frac{{({\eta }_i-a)}^q}{\Gamma (q+1)}+\frac{{(b-a)}^{q+1}}{\Gamma (q+2)}.$$

The authors would like to apologize for any inconvenience caused to the readers by these changes.