Correction: Pavlačková, M.; Taddei, V. Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces. Mathematics 2022, 10, 672

The authors would like to correct parts of Step (b) and Step (d) in Theorem 1 in [1] since both Steps contain a mistake. In particular:

Step (b) Proving that Σ_m has a closed graph.

On page 11, the part of Step (b) from line 18 has to be corrected as follows:

Let us now prove that $f\in S_{G_m,q}^1.$ Due to Mazur’s convexity theorem, for each k ∈ N, there exists $p_k$ ∈ N and positive numbers ${\beta }_{k,i}, i=0, \dots , p_k$, such that ${\sum }_{i=0}^{p_k}{\beta }_{k,i}=1$ and $r_k:={\sum }_{i=0}^{p_k}{\beta }_{k,i} f_{k+i}\to f$ in $L^1(\left[a,b\right],E)$. From the sequence ${\left\{r_k\right\}}_k$, we extract a subsequence, denoted as the sequence as usual, such that $r_k\left(t\right)\to f\left(t\right)$, for all $t\in [a,b]\backslash N_1$ with $\mathsf{\lambda }\left(N_1\right)=0.$ Moreover, for all $t\in [a,b]\backslash N_2$ with $\mathsf{\lambda }\left(N_2\right)=0$, $G_m(t, ·, ·)$ is weakly u.s.c.

Put $N=N_1{\cup N}_2$ and consider ${\mathrm{t}}_0\in [a,b]\backslash N.$ Then, for every weak neighborhood V of $G_m(t_0, q(t_0), \dot{q}(t_0\left)\right)$, there exists a weak neighborhood W of $\left(q\right(t_0), \dot{q}(t_0\left)\right)$ such that $G_m(t, x,y)\subset V$ when $(x,y)\in W.$ Since the uniform convergence implies the weak pointwise convergence, it follows that $q_k\left(t_0\right)$ $\rightharpoonup$ $q\left(t_0\right)$ and ${\dot{q}}_k\left(t_0\right)$ $\rightharpoonup$ $\dot{q}\left(t_0\right)$. Thus, there exists $\overline{k}$ such that, for all $k\ge \overline{k}$, ${(q}_k\left(t_0\right), {\dot{q}}_k\left(t_0\right))\in W,$ yielding that $f_k\left(t_0\right) \in G_m(t_0, q_k(t_0),{\dot{q}}_k(t_0\left)\right) \subset V,$ i.e., that $r_k\left(t_0\right)\in V,$ because $G_m$ is convex valued. Since $r_k\left(t_0\right)\to f\left(t_0\right),$ it follows that $f\left(t_0\right)\in \overline{V},$ for every weak neighborhood V of $G_m(t_0, q(t_0), \dot{q}(t_0\left)\right).$ Since $G_m$ is closed valued, the proof is complete.

Given $\Phi \in E^{*}$ and t ∈ [$a, b$], consider the operator Φ: $L^1\left(\left[a,t\right],E\right)\to \mathrm{R}$ defined by

$$\mathsf{\Phi }\left(p\right):=\Phi \left({\int }_a^{t}S\left(t-s\right)p\left(s\right) ds\right).$$

Since $S\left(t-s\right)$ is bounded and linear, for every $t, s$, Φ is clearly linear and bounded. Moreover, $f_k\rightharpoonup f$ also in $L^1\left(\left[a,t\right],E\right),$ and hence, we have that

$$\Phi \left({\int }_a^{t}S\left(t-s\right)f_k\left(s\right) ds\right)=\mathsf{\Phi }\left(f_k\right)\to \mathsf{\Phi }\left(f\right)=\Phi \left({\int }_a^{t}S\left(t-s\right)f\left(s\right) ds\right).$$

By the arbitrariness of $\Phi ,$ we conclude that

$${\int }_a^{t}S\left(t-s\right)f_k\left(s\right) ds\rightharpoonup {\int }_a^{t}S\left(t-s\right)f\left(s\right) ds.$$

Hence, since $P_m$ is a linear and bounded operator taking values in the finite-dimensional space $E_m,$ it holds that

$${\int }_a^t{P}_{m }S\left(t-s\right)f_k\left(s\right) ds\to {\int }_a^t{P}_{m }S\left(t-s\right)f\left(s\right) ds.$$

The conclusion, then, can be completed like in last two lines of Part (b) [1].

Step (d) showing that Σ_m maps bounded sets into relatively compact sets.

The part of Step (d) starting twelve lines below formula (25) has to be corrected as follows:

According to Lemma 3, $\left(t,x\right)\to C\left(t\right)x$ is continuous in [$a, b$] × E and, hence, uniformly continuous in the pre-compact set [$a, b$] × $I_1$.

According to Lemma 2 (h), since S(s) is also linear and bounded, for every $s$ ∈ [$a, b$], we obtain by the same reasoning that

$$\left\{S\left(s\right)f\left(s\right):q\in n{B}_{m, }, f\in S_{G_m,q}^1\right\}\subset E_m\cap X \subset X$$

is relatively compact, hence

$$I_2=\left\{{\int }_a^{t_0}S\left(s\right)f\left(s\right) ds:q\in n{B}_{m, }, f\in S_{G_m,q}^1\right\}$$

is relatively compact in X.

According to (11), $\left(t,x\right)\to C’\left(t\right)x$ is continuous in [$a, b$] × X and, hence, uniformly continuous in the pre-compact set [$a, b$] × $I_2$.

The conclusion, then, can be completed like in last five lines of Part (d) [1].

With these corrections, the order of some equations have been adjusted accordingly.

A correction has been made to Author Contributions, Funding and Acknowledgments. The correct information appears below as follows:

Author Contributions: Writing—original draft preparation, M.P. and V.T.; writing—review and editing, M.P. and V.T.; funding acquisition, M.P. and V.T. All authors have read and agreed to the published version of the manuscript.

Funding: This research was funded by the European Structural and Investment Funds (Operational Programme Research, Development and Education) and by the Ministry of Education, Youth and Sports of the Czech Republic under the Grant No. CZ.02.2.69/0.0/0.0/18_054/0014592 The Advancement of Capacities for Research and Development at Moravian Business College Olomouc. This research was performed within the framework of the grant MIUR-PRIN 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)”.

Acknowledgments: Taddei is the member of Gruppo Nazionale di Analisi Matematica, Probabilitá e le loro Applicazioni of Istituto Nazionale di Alta Matematica.

The authors state that the scientific conclusions are unaffected, and that this Correction does not influence the results contained in [1]. This correction was approved by the Academic Editor. The original publication has also been updated.