Existence results for sequential generalized Hilfer fractional differential inclusions with multi-point boundary conditions (Q2107704)

In this paper, the authors obtain existence results for the following sequential Hilfer-type fractional differential inclusions subject to multi-point boundary conditions of the form \[ \begin{cases} \displaystyle \Big({}^{H}D_{a+}^{r_1,r_2; \xi}-k~{}^{H}D_{a+}^{r_1-1,r_2; \xi}\Big)\varpi(t)\in Q(t,\varpi(t)), ~~ t\in (a,b), \\ \displaystyle\varpi(a)=0,~~ \varpi(b)=\sum_{i=1}^{m}\lambda_i \varpi(\theta_i), \end{cases}\tag{1} \] where \({}^{H}D_{a+}^{r_1,r_2; \xi}\) is the \(\xi\)-Hilfer fractional derivative of order \(r_1\in (1,2)\) and type \(r_1\in [0,1],\) \(k, \lambda_i\in\mathbb{R},\) \(0\le a\le \theta_1<\theta_2<\ldots<\theta_m\le b\) and \(Q: [a,b]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})\) is a multi-valued map (\(\mathcal{P}(\mathbb{R}\) is the family of all nonempty subsets of \(\mathbb{R}\)). \vskip 0.5cm \textbf{Reviewer's remark}: The problem (1) is a special case, for \(\mu_i=0, i=1,2,\ldots, n,\) of the problem \[ \begin{cases} \displaystyle \Big({}^HD^{\alpha,\beta;\psi}+k{}~^HD^{\alpha-1,\beta;\psi}\Big)x(t)\in F(t,x(t)),~~~t\in [a,b], \\ \displaystyle x(a)=0,~~~x(b)=\sum_{i=1}^{n}\mu_i\int_a^{\eta_i} \psi'(s)x(s)ds+\sum_{j=1}^{m}\theta_j x(\xi_j), \end{cases}\tag{2} \] studied by \textit{S. Sitho} et al. [``Boundary value problems for \(\psi\)-Hilfer type sequential fractional differential equations and inclusions with integral multi-point boundary conditions'', Mathematics 9, No. 9, Article ID 1001, 18 p. (2021; \url{doi:10.3390/ math9091001})].