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

Bashir Ahmad; Ymnah Alruwaily; Ahmed Alsaedi; Sotiris K. Ntouyas

doi:10.3390/math7121220

,

and

¹

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²

Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece

^*

Author to whom correspondence should be addressed.

Mathematics2019, 7(12), 1220;https://doi.org/10.3390/math7121220

Version Notes

Order Reprints

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

$1 .$: Throughout the original paper, the co-efficient of $g_{3} (t)$ (see (5), (21), (28), proof of Theorem 6) is

$\int_{a}^{b} (\int_{a}^{s} \frac{{(s - u)}^{q - 1}}{Γ (q)} \hat{f} (u) d u) d A (s) .$

It should be

$[\sum_{i = 1}^{n - 2} α_{i} \int_{a}^{η_{i}} \frac{{(η_{i} - s)}^{q - 1}}{Γ (q)} \hat{f} (s) d s + \int_{a}^{b} (\int_{a}^{s} \frac{{(s - u)}^{q - 1}}{Γ (q)} \hat{f} (u) d u) d A (s)] .$
2.: In (15) of the original paper, it was

$I_{3} = \int_{a}^{b} (\int_{a}^{s} \frac{{(s - u)}^{q - 1}}{Γ (q)} \hat{f} (u) d u) d A (s) .$

It should be

$I_{3} = \sum_{i = 1}^{n - 2} α_{i} \int_{a}^{η_{i}} \frac{{(η_{i} - s)}^{q - 1}}{Γ (q)} \hat{f} (s) d s + \int_{a}^{b} (\int_{a}^{s} \frac{{(s - u)}^{q - 1}}{Γ (q)} \hat{f} (u) d u) d A (s) .$
$3 .$: Throughout the original paper, the coefficient of $\bar{g_{3}}$ (for instance, see (19), proofs of Theorems 2, 4, 6, and 7) is $\int_{a}^{b} \frac{{(s - a)}^{q}}{Γ (q + 1)} d A (s) .$ It should be

$[\sum_{i = 1}^{n - 2} | α_{i} | \frac{{(η_{i} - a)}^{q}}{Γ (q + 1)} + \int_{a}^{b} \frac{{(s - a)}^{q}}{Γ (q + 1)} d A (s)] .$

The coefficient of $\frac{| (g_{3} (t_{2}) - g_{3} (t_{1})) |}{Γ (q + 1)}$ in the proof of Theorem 2 is $\int_{a}^{b} (\int_{a}^{s} {(s - a)}^{q} d A (s)) .$ It should be

$[\sum_{i = 1}^{n - 2} | α_{i} | {(η_{i} - a)}^{q} + \int_{a}^{b} {(s - a)}^{q} d A (s)] .$
$4 .$: Throughout the original paper, the co-efficient of $| g_{3} (t) |$ (see proofs of Theorems 2, 4, 6, and 7) and $| g_{3} (t_{2}) - g_{3} (t_{1}) |$ in the proof Theorem 2 is

$\int_{a}^{b} (\int_{a}^{s} \frac{{(s - u)}^{q - 1}}{Γ (q)} | f (u, x (u)) | d u) d A (s) .$

It should be

$[\sum_{i = 1}^{n - 2} | α_{i} | \int_{a}^{η_{i}} \frac{{(η_{i} - s)}^{q - 1}}{Γ (q)} | f (s, x (s)) | d s + \int_{a}^{b} (\int_{a}^{s} \frac{{(s - u)}^{q - 1}}{Γ (q)} | f (u, x (u)) | d u) d A (s)] .$
$5 .$: In the proofs of Theorems 6 and 7, the coefficient of $| g_{3} (t) |$ is

$\int_{a}^{b} (\int_{a}^{s} \frac{{(s - u)}^{q - 1}}{Γ (q)} | f (u, x (u)) - f (u, y (u)) | d u) d A (s) .$

It should be

$\begin{matrix} [\sum_{i = 1}^{n - 2} | α_{i} | \int_{a}^{η_{i}} \frac{{(η_{i} - s)}^{q - 1}}{Γ (q)} | f (s, x (s)) - f (s, y (s)) | d s \\ + \int_{a}^{b} (\int_{a}^{s} \frac{{(s - u)}^{q - 1}}{Γ (q)} | f (u, x (u)) - f (u, y (u)) | d u) d A (s)] . \end{matrix}$
$6 .$: In Example 2, $Λ \approx 0.243646, ξ Λ \approx 0.097458 < 1$ in the original paper. These values should be $Λ \approx 0.261226, ξ Λ \approx 0.104490 < 1 .$
$7 .$: In Example 3 of the original paper, $Λ \approx 0.272140, Λ - \frac{{(b - a)}^{q}}{Γ (q + 1)} \approx 0.204166$ and $δ < 14.693960 .$ The corrected values of these parameters are $Λ \approx 0.326742, Λ - \frac{{(b - a)}^{q}}{Γ (q + 1)} \approx 0.258768$ and $δ < 11.5 .$
$8 .$: In Example 4, $δ < 11.023738$ in the original paper. It should be $δ < 9.1 .$
$9 .$: In Example 5, $L Λ \approx 0.102053 .$ It should be $L Λ \approx 0.097959 .$
$10 .$: In the Conclusions, the coefficient of $\bar{g_{3}}$ is $\frac{{(b - a)}^{q + 1}}{Γ (q + 2)} .$ It should be

$\sum_{i = 1}^{n - 2} | α_{i} | \frac{{(η_{i} - a)}^{q}}{Γ (q + 1)} + \frac{{(b - a)}^{q + 1}}{Γ (q + 2)} .$

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

References

Ahmad, B.; Alruwaily, Y.; Alsaedi, A.; Ntouyas, S.K. Existence and Stability Results for a Fractional Order Differential Equation with Non-Conjugate Riemann-Stieltjes Integro-Multipoint Boundary Conditions. Mathematics 2019, 7, 249. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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

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