Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition

1. Fractional Differential Equations, Inclusions and Systems

Paper [1] is focused on the fractional differential equation which contains a variable coefficient, subject to Dirichlet boundary conditions

$$\left\{\begin{array}{c}-D_{0+}^{\alpha }\mathfrak{u}(\mathfrak{t})+\mathfrak{a}(\mathfrak{t})\mathfrak{u}(\mathfrak{t})=\mathfrak{f}(\mathfrak{t}),\mathfrak{t}\in (0,1),\hfill \\ \mathfrak{u}(0)=\mathfrak{u}(1)=0,\hfill \end{array}\right.$$

(1)

where $\alpha \in (1,2)$, the function $\mathfrak{a}\in L^1[0,1]$ may have singularities, $\mathfrak{f}\in L^1[0,1]$, and $D_{0+}^{\alpha }\mathfrak{u}$ denotes the Riemann–Liouville fractional derivative of function $\mathfrak{u}$ of order $\alpha$. The author presents some properties of the Green function associated with problem (1), including positivity properties, and proves the existence and uniqueness of a solution for (1). Then he shows that there exists exactly one positive solution for the singular fractional boundary value problem

$$\left\{\begin{array}{c}-D_{0+}^{\alpha }\mathfrak{u}(\mathfrak{t})+\mathfrak{a}(\mathfrak{t})\mathfrak{u}(\mathfrak{t})=\mathfrak{g}(\mathfrak{t},\mathfrak{u}(\mathfrak{t}),\mathfrak{u}(\mathfrak{t})),\mathfrak{t}\in (0,1),\hfill \\ \mathfrak{u}(0)=\mathfrak{u}(1)=0,\hfill \end{array}\right.$$

with $\alpha \in (1,2)$, under some assumptions on the nonlinear nonnegative singular function $\mathfrak{g}:(0,1)\times (0,+\infty )\times (0,+\infty )\to [0,+\infty )$. The author also investigates the semipositone problem

$$\left\{\begin{array}{c}-D_{0+}^{\alpha }\mathfrak{u}(\mathfrak{t})=\mu \mathfrak{F}(\mathfrak{t},\mathfrak{u}(\mathfrak{t}),\mathfrak{t}\in (0,1),\hfill \\ \mathfrak{u}(0)=\mathfrak{u}(1)=0,\hfill \end{array}\right.$$

(2)

where $\alpha \in (1,2)$, $\mu >0$ is a parameter, and $\mathfrak{F}$ may be singular at $\mathfrak{t}=0$ or $\mathfrak{t}=1$, and may change its sign. By applying the Guo–Krasnosel’skii fixed point theorem, he gives an interval for $\mu$ such that problem (2) has at least one positive solution.

Paper [2] is devoted to the fractional boundary value problems

$$\left\{\begin{array}{c}{}^{c}D^{{\rho }_2}\mathfrak{u}(\mathfrak{t})+{\mathfrak{w}}_1(\mathfrak{t})I^{{\rho }_1}\mathfrak{u}(\mathfrak{t})=\mathfrak{f}(\mathfrak{t},\mathfrak{u}(\mathfrak{t})),\mathfrak{t}\in [0,1],\hfill \\ {\displaystyle \mathfrak{u}(0)=\omega {\int }_0^1\left|\mathfrak{u}(\mathfrak{s})\right|d\mathfrak{s},}\hfill \end{array}\right.$$

(3)

and

$$\left\{\begin{array}{c}{\displaystyle {}^{c}D^{{\rho }_2}\mathfrak{v}(\mathfrak{t})+{\mathfrak{w}}_1(\mathfrak{t})I^{{\varrho }_1}\mathfrak{v}(\mathfrak{t})+{\displaystyle \sum _{i=1}^l}{I}^{{\gamma }_i}{\mathfrak{f}}_i(\mathfrak{t},\mathfrak{v}(\mathfrak{t}))=\mathfrak{F}(\mathfrak{t},\mathfrak{v}(\mathfrak{t})),\mathfrak{t}\in [0,1],}\hfill \\ {\displaystyle \mathfrak{v}(0)={\int }_0^1{\mathfrak{w}}_2(\mathfrak{s})\left|\mathfrak{v}(\mathfrak{s})\right|d\mathfrak{s},\mathfrak{v}(1)={\mathfrak{w}}_3(\mathfrak{v}).}\hfill \end{array}\right.$$

(4)

Here ${\rho }_2\in (0,1]$, ${\rho }_1\ge 0$, ${\varrho }_2\in (1,2]$, ${\varrho }_1\ge 0$, $\omega$ is a constant, ${\gamma }_i\ge 0$ for all $i=1,\dots ,l$, ($l\in \mathbb{N}$); ${\mathfrak{w}}_1,{\mathfrak{w}}_2\in C[0,1]$, and ${\mathfrak{w}}_3:C[0,1]\to \mathbb{R}$ is a functional satisfying some conditions; ${}^{c}D^{\kappa }$ is the Caputo fractional derivative of order $\kappa$, for $\kappa \in \{{\rho }_2,{\varrho }_2\}$, $I^{\kappa }$ denotes the Riemann–Liouville fractional integral of order $\kappa$, for $\kappa \in \{{\rho }_1,{\varrho }_1,{\gamma }_i,i=1,\dots ,l\}$, and the functions $\mathfrak{f},\mathfrak{F},{\mathfrak{f}}_i:[0,1]\times \mathbb{R}\to \mathbb{R}$, $i=1,\dots ,l$ satisfy certain assumptions. By employing the Banach contraction principle, a functional inverse operator approach, and the Mittag-Leffler function, the authors prove the existence and uniqueness of solutions to problem (3), as well as, the existence, uniqueness and stability of solutions for problem (4). They also show the existence of solutions to (4) by using the Leray–Schauder fixed point theorem. An application of the inverse operator method to solve a Caputo time-fractional convection partial differential equation in ${\mathbb{R}}^n$ is finally addressed.

In paper [3] the authors examine the fractional differential equation with sequential derivatives and an integral term supplemented with multipoint boundary conditions

$$\left\{\begin{array}{c}{\displaystyle {}^{ABC}D_{a+}^{\varsigma }\mathfrak{u}(\mathfrak{t})=\mathfrak{f}(\mathfrak{t},\mathfrak{u}(\mathfrak{t}){}^{ABC}D_{a+}^{\vartheta }\mathfrak{u}(\mathfrak{t}),{}^{ABC}D_{a+}^{\varrho }{}^{ABC}D_{a+}^{\kappa }\mathfrak{u}(\mathfrak{t}))))+{\int }_a^{\mathfrak{t}}\mathfrak{F}(\mathfrak{t},\mathfrak{u}(\mathfrak{t}))d\mathfrak{t},\mathfrak{t}\in [a,T],}\hfill \\ {\displaystyle \mathfrak{u}(a)={\displaystyle \sum _{i=1}^m}{c}_i\mathfrak{u}({\mathfrak{t}}_i),}\hfill \end{array}\right.$$

(5)

where $\varsigma ,\vartheta ,\varrho ,\kappa \in [0,1]$, ${\mathfrak{t}}_i\in (a,T)$ and $c_i\in \mathbb{R}$ for all $i=1,\dots ,m$, with ${\sum }_{i=1}^m{c}_i\ne 1$; $\mathfrak{f}:[a,T]\times {\mathbb{R}}^3\to \mathbb{R}$ and $\mathfrak{F}:[a,T]\times \mathbb{R}\to \mathbb{R}$ are continuous functions which satisfy some additional conditions, and ${}^{ABC}D_{a+}^k\mathfrak{u}$ is the Atangana–Baleanu–Caputo fractional derivative of function $\mathfrak{u}$ of order k, for $k\in \{\varsigma ,\vartheta ,\varrho ,\kappa \}$. Based on the Schauder fixed point theorem and the Banach contraction principle, they study the existence and uniqueness of solutions for (5). Besides, the authors establish the Hyers–Ulam stability of problem (5).

Paper [4] deals with the fractional differential equation with sequential derivatives subject to nonlocal boundary conditions

$$\left\{\begin{array}{c}{}^{k,H}D_{0+}^{\vartheta ,\beta ;\psi }\left({}^{k,C}D_{0+}^{\gamma ;\psi }\mathfrak{u}(\mathfrak{t})\right)=\mathfrak{f}(\mathfrak{t},\mathfrak{u}(\mathfrak{t})),\mathfrak{t}\in [0,b],\hfill \\ {\lambda }_1\mathfrak{u}(0)+{\lambda }_2\mathfrak{u}(b)=0,\hfill \\ {\lambda }_3{}^{k,C}D_{0+}^{{\zeta }_k+\gamma -k;\psi }\mathfrak{u}(0)+{\lambda }_4{}^{k,C}D_{0+}^{{\zeta }_k+\gamma -k;\psi }\mathfrak{u}(b)=0,\hfill \end{array}\right.$$

(6)

where $\vartheta \in (0,1)$, $\beta \in [0,1]$, ${}^{k,H}D_{0+}^{\vartheta ,\beta ;\psi }\mathfrak{u}$ is the $(k,\psi )$-Hilfer fractional derivative of function $\mathfrak{u}$ of order $\vartheta$ with the parameter $\beta$, ${}^{k,C}D_{0+}^{\gamma ;\psi }\mathfrak{u}$ and ${}^{k,C}D_{0+}^{{\zeta }_k+\gamma -k;\psi }\mathfrak{u}$ are the $(k,\psi )$-Caputo fractional derivatives of $\mathfrak{u}$ of orders $\gamma \in (0,1)$ and ${\zeta }_k+\gamma -k>0$, respectively, with ${\zeta }_k=\vartheta +\beta (k-\vartheta )$. In addition, $k>0$, ${\lambda }_i\in \mathbb{R}$, $i=1,\dots ,4$, and $\mathfrak{f}:[0,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous function. The authors prove the existence and uniqueness of solutions to problem (6) through the use of the fixed point theorems of Banach, Schaefer and Krasnosel’skii, and the Leray–Schauder nonlinear alternative. In addition, they investigate the existence of solutions for the multi-valued version of problem (6), namely the following fractional inclusion with boundary conditions

$$\left\{\begin{array}{c}{}^{k,H}D_{0+}^{\vartheta ,\beta ;\psi }\left({}^{k,C}D_{0+}^{\gamma ;\psi }\mathfrak{u}(\mathfrak{t})\right)\in \mathfrak{F}(\mathfrak{t},\mathfrak{u}(\mathfrak{t})),\mathfrak{t}\in [0,b],\hfill \\ {\lambda }_1\mathfrak{u}(0)+{\lambda }_2\mathfrak{u}(b)=0,\hfill \\ {\lambda }_3{}^{k,C}D_{0+}^{{\zeta }_k+\gamma -k;\psi }\mathfrak{u}(0)+{\lambda }_4{}^{k,C}D_{0+}^{{\zeta }_k+\gamma -k;\psi }\mathfrak{u}(b)=0,\hfill \end{array}\right.$$

(7)

where $\mathfrak{F}:[0,b]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is a multi-valued map, and $\mathcal{P}(\mathbb{R})$ represents the family of all nonempty subsets of $\mathbb{R}$. The authors consider two cases for $\mathfrak{F}$, specifically: the convex-valued case and the nonconvex-valued case. In the proof of the main results for problem (7), they employ the Leray–Schauder nonlinear alternative for Kakutani maps combined with the closed graph operator theorem in the first case, and the Covitz–Nadler fixed point theorem for multi-valued contractions in the second case.

Paper [5] is concerned with the system of fractional differential equations

$$\left\{\begin{array}{c}{}^{H}D_{1+}^{\alpha }\mathfrak{u}(\mathfrak{t})+\mathfrak{f}(\mathfrak{t},\mathfrak{u}(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}){}^{H}I_{1+}^{{\gamma }_1}\mathfrak{u}(\mathfrak{t}){}^{H}I_{1+}^{{\delta }_1}\mathfrak{v}(\mathfrak{t}))=0,\mathfrak{t}\in [1,e],\hfill \\ {}^{H}D_{1+}^{\beta }\mathfrak{v}(\mathfrak{t})+\mathfrak{g}(\mathfrak{t},\mathfrak{u}(\mathfrak{t}),\mathfrak{v}(\mathfrak{t}){}^{H}I_{1+}^{{\gamma }_2}\mathfrak{u}(\mathfrak{t}){}^{H}I_{1+}^{{\delta }_2}\mathfrak{v}(\mathfrak{t}))=0,\mathfrak{t}\in [1,e],\hfill \end{array}\right.$$

(8)

supplemented with the coupled nonlocal boundary conditions

$$\left\{\begin{array}{c}{\mathfrak{u}}^{(i)}(1)=0,i=0,\dots ,n-2,{\mathfrak{v}}^{(j)}(1)=0,j=0,\dots ,m-2,\hfill \\ {\displaystyle {}^{H}D_{1+}^{\varsigma }\mathfrak{u}(e)={\displaystyle \sum _{i=1}^{\mathfrak{a}}}{\int }_1^e{}^{H}D_{1+}^{{\varrho }_i}\mathfrak{u}(\mathfrak{s})d{\mathfrak{H}}_i(\mathfrak{s})+{\displaystyle \sum _{i=1}^{\mathfrak{b}}}{\int }_1^e{}^{H}D_{1+}^{{\sigma }_i}\mathfrak{v}(\mathfrak{s})d{\mathfrak{K}}_i(\mathfrak{s}),}\hfill \\ {\displaystyle {\int }_1^e{}^{H}D_{1+}^{\vartheta }\mathfrak{v}(e)={\displaystyle \sum _{i=1}^{\mathfrak{c}}}{\int }_1^e{}^{H}D_{1+}^{{\eta }_i}\mathfrak{u}(\mathfrak{s})d{\mathfrak{P}}_i(\mathfrak{s})+{\displaystyle \sum _{i=1}^{\mathfrak{d}}}{\int }_1^e{}^{H}D_{1+}^{{\theta }_i}\mathfrak{v}(\mathfrak{s})d{\mathfrak{Q}}_i(\mathfrak{s}).}\hfill \end{array}\right.$$

(9)

Here $\alpha \in (n-1,n]$, $\beta \in (m-1,m]$, $n,m\in \mathbb{N}$, $n,m\ge 3$; $\mathfrak{a},\mathfrak{b},\mathfrak{c},\mathfrak{d}\in \mathbb{N}$; $\varsigma \ge 1$, $0\le {\varrho }_i\le \varsigma <\alpha -1$, $i=1,\dots ,\mathfrak{a}$, $0\le {\eta }_k\le \varsigma$, $k=1,\dots ,\mathfrak{c}$, $\vartheta \ge 1$, $0\le {\sigma }_j\le \vartheta <\beta -1$, $j=1,\dots ,\mathfrak{b}$, $0\le {\theta }_{\iota }\le \vartheta$, $\iota =1,\dots ,\mathfrak{d}$; ${\gamma }_i,{\delta }_i>0$, $i=1,2$; ${}^{H}D_{1+}^{\kappa }$ is the Hadamard fractional derivative of order $\kappa$, for $\kappa \in \{\alpha ,\varsigma ,\beta ,\vartheta ,{\varrho }_i,{\sigma }_j,{\eta }_k,{\theta }_{\iota },$$i=1,\dots ,\mathfrak{a}$, $j=1,\dots ,\mathfrak{b}$, $k=1,\dots ,\mathfrak{c}$, $\iota =1,\dots ,\mathfrak{d}\}$, and ${}^{H}I_{1+}^{\kappa }$ is the Hadamard fractional integral of order $\kappa$, for $\kappa \in \{{\gamma }_i,{\delta }_i,i=1,2\}$. The integrals appearing in the boundary conditions (9) are Riemann–Stieltjes integrals where ${\mathfrak{H}}_i,{\mathfrak{K}}_j,{\mathfrak{P}}_k,{\mathfrak{Q}}_{\iota }$, $i=1,\dots ,\mathfrak{a}$, $j=1,\dots ,\mathfrak{b}$, $k=1,\dots ,\mathfrak{c}$, and $\iota =1,\dots ,\mathfrak{d}$ are bounded variation functions. Moreover, the functions $\mathfrak{f},\mathfrak{g}:[1,e]\times {\mathbb{R}}_{+}^4\to {\mathbb{R}}_{+}$ are given (${\mathbb{R}}_{+}=[0,\infty )$). The authors explore the existence, uniqueness and multiplicity of positive solutions for problem (8) and (9). In the main results, they utilize various fixed point theorems, such as the Guo–Krasnosel’skii fixed point theorem, the Schauder fixed point theorem, the Banach contraction mapping principle, and the Leggett–Williams fixed point theorem.

Paper [6] analyzes the system of fractional differential equations with homomorphisms and delayed or advanced arguments, subject to nonlocal boundary conditions

$$\left\{\begin{array}{c}{\left[{\mathfrak{H}}_1(D_{a+}^{\alpha ,\psi }\mathfrak{u}(\mathfrak{t}))\right]}^{\prime }={\mathfrak{h}}_1(\mathfrak{t}){\mathfrak{f}}_1(\mathfrak{t},\mathfrak{u}({\tau }_1(\mathfrak{t})),\mathfrak{v}({\tau }_2(\mathfrak{t}))),\mathfrak{t}\in (a,b),\hfill \\ {\left[{\mathfrak{H}}_2(D_{a+}^{\beta ,\psi }\mathfrak{v}(\mathfrak{t}))\right]}^{\prime }={\mathfrak{h}}_2(\mathfrak{t}){\mathfrak{f}}_2(\mathfrak{t},\mathfrak{u}({\tau }_1(\mathfrak{t})),\mathfrak{v}({\tau }_2(\mathfrak{t}))),\mathfrak{t}\in (a,b),\hfill \\ \mathfrak{u}(a)=0,D_{a+}^{{\alpha }_i,\psi }\mathfrak{u}(a)=0,\mathfrak{u}(b)=I_{a+}^{\mu ,\psi }\mathfrak{u}(b),i=1,\dots ,n-2,\hfill \\ \mathfrak{v}(a)=0,D_{a+}^{{\beta }_j,\psi }\mathfrak{v}(a)=0,\mathfrak{v}(b)=I_{a+}^{\nu ,\psi }\mathfrak{v}(b),j=1,\dots ,m-2.\hfill \end{array}\right.$$

(10)

Here $0\le a<b$, $\alpha \in (n-1,n)$, $\beta \in (m-1,m)$; ${\alpha }_i\in (i,i+1)$ for $i=1,\dots ,n-2$; ${\beta }_j\in (j,j+1)$ for $j=1,\dots ,m-2$; $\mu ,\nu \ge 0$; ${\mathfrak{H}}_1,{\mathfrak{H}}_2:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ denote two positive increasing homomorphisms, ${\mathfrak{h}}_1,{\mathfrak{h}}_2\in C([a,b],{\mathbb{R}}^{+})$, ${\mathfrak{f}}_1,{\mathfrak{f}}_2\in C([a,b]\times {\mathbb{R}}_0^{+}\times {\mathbb{R}}_0^{+},{\mathbb{R}}^{+})$, (${\mathbb{R}}^{+}=(0,\infty )$, ${\mathbb{R}}_0^{+}=[0,\infty )$), and ${\tau }_1,{\tau }_2\in C([a,b],[a,b])$; $D_{a+}^{\kappa ,\psi }$ is the $\psi$-Caputo fractional derivative of order $\kappa$, for $\kappa \in \{\alpha ,\beta ;{\alpha }_i,i=1,\dots ,n-2;{\beta }_j,j=1,\dots ,m-2\}$; and $I_{a+}^{\kappa ,\psi }$ denotes the $\psi$-Riemann–Liouville fractional integral of order $\kappa$, for $\kappa \in \{\mu ,\nu \}$. Making use of the generalized Avery–Henderson fixed point theorem, the authors prove the existence of at least three positive solutions for problem (10). As applications, they obtain multiplicity results for the positive solutions of two fractional boundary value problems with Caputo fractional derivatives and Caputo–Hadamard fractional derivatives, respectively, and Laplacian operators.

2. Resonant Problems for Fractional Differential Equations

Paper [7] is devoted to the fractional differential equation on an infinite interval at resonance

$${}^{H}D_{1+}^{\alpha }\mathfrak{x}(\mathfrak{t})=\mathfrak{f}\left(\mathfrak{t},\mathfrak{x}(\mathfrak{t}),{}^{H}D_{1+}^{\alpha -n+1}\mathfrak{x}(\mathfrak{t}),{}^{H}D_{1+}^{\alpha -n+2}\mathfrak{x}(\mathfrak{t}),\dots ,{}^{H}D_{1+}^{\alpha -1}\mathfrak{x}(\mathfrak{t})\right),\mathfrak{t}\in (1,\infty ),$$

(11)

supplemented with the nonlocal boundary conditions

$$\left\{\begin{array}{c}{\displaystyle {\mathfrak{x}}^{(k)}(1)=0,k=0,\dots ,n-3;{}^{H}D_{1+}^{\alpha -2}\mathfrak{x}(1)={\int }_1^A{}^{H}D_{1+}^{\alpha -2}\mathfrak{x}(\mathfrak{s})d\mathfrak{P}(\mathfrak{s}),}\hfill \\ {\displaystyle {}^{H}D_{1+}^{\alpha -1}\mathfrak{x}(\infty )={\int }_1^B{}^{H}D_{1+}^{\alpha -1}\mathfrak{x}(\mathfrak{s})d\mathfrak{Q}(\mathfrak{s}),}\hfill \end{array}\right.$$

(12)

where $\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n\ge 3$; $A,B>1$, and $\mathfrak{f}:(1,\infty )\times {\mathbb{R}}^n\to \mathbb{R}$ is a Carathéodory function. The integrals from conditions (12) are Riemann–Stieltjes integrals with $\mathfrak{P}:[1,A]\to \mathbb{R}$ and $\mathfrak{Q}:[1,B]\to \mathbb{R}$ functions of bounded variation satisfying some resonant conditions, and ${}^{H}D_{1+}^{\kappa }$ is the Hadamard fractional derivative of order $\kappa$, for $\kappa \in \{\alpha ,\alpha -1,\dots ,\alpha -n+1\}$. Since the associated homogeneous boundary value problem admits nontrivial solutions, the problem (11) and (12) is a resonant one. By way of the Mawhin continuation theorem, the authors examine the existence of solutions to (11) and (12).

Paper [8] is focused on the fractional functional equation with nonlinear boundary conditions at resonance

$$\left\{\begin{array}{c}{}^{H}D_{0+}^{\alpha ,\beta ;\psi }\mathfrak{u}(\mathfrak{t})=\mathfrak{f}(\mathfrak{t},\mathfrak{u}(\mathfrak{t}),\mathfrak{u}(\lambda \mathfrak{t}),D_{0+}^{\gamma -2;\psi }\mathfrak{u}(\mathfrak{t}),D_{0+}^{\gamma -1;\psi }\mathfrak{u}(\mathfrak{t})),\mathfrak{t}\in (0,1],\hfill \\ I_{0+}^{n-\gamma ;\psi }\mathfrak{u}(0)=0,D_{0+}^{\gamma -i;\psi }\mathfrak{u}(0)=0,i=3,\dots ,n-1,\hfill \\ {\mathcal{B}}_1(\mathfrak{u})=0,{\mathcal{B}}_2(\mathfrak{u})=0,\hfill \end{array}\right.$$

(13)

where $\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n>3$, $\beta \in [0,1]$, $\gamma =\alpha +\beta (n-\alpha )$, and $\lambda \in (0,1)$. Here ${}^{H}D_{0+}^{\alpha ,\beta ;\psi }\mathfrak{u}$ and $D_{0+}^{k;\psi }\mathfrak{u}$ are the $\psi$-Hilfer fractional derivative of function $\mathfrak{u}$ of order $\alpha$ and type $\beta$, and the $\psi$-Riemann–Liouville fractional derivative of $\mathfrak{u}$ of order k, respectively, for $k\in \{\gamma -i,i=,1\dots ,n-1\}$; $I_{0+}^{n-\gamma ;\psi }\mathfrak{u}$ is the $\psi$-Riemann–Liouville fractional integral of $\mathfrak{u}$ of order $n-\gamma$; and the function $\mathfrak{f}\in C([0,1]\times {\mathbb{R}}^4,\mathbb{R})$, where ${\mathcal{B}}_1,{\mathcal{B}}_2$ are continuous linear functionals, satisfies the resonance condition

$$\begin{array}{c}{\mathcal{B}}_1\left({(\psi (\mathfrak{t})-\psi (0))}^{\gamma -1}\right){\mathcal{B}}_2\left({(\psi (\mathfrak{t})-\psi (0))}^{\gamma -2}\right)\hfill \\ ={\mathcal{B}}_1\left({(\psi (\mathfrak{t})-\psi (0))}^{\gamma -2}\right){\mathcal{B}}_2\left({(\psi (\mathfrak{t})-\psi (0))}^{\gamma -1}\right).\hfill \end{array}$$

Applying the Mawhin coincidence theorem, the authors prove the existence of solutions to problem (13).

3. Fractional Differential Equations and Systems with Variable-Order Derivatives

Paper [9] explores the variable-order fractional differential equation with integral boundary conditions

$$\left\{\begin{array}{c}{D}_{0+}^{w(\mathfrak{t})}\mathfrak{v}(\mathfrak{t})+\mathfrak{B}\left(\mathfrak{t},\mathfrak{v}(\mathfrak{t}),D_{0+}^{\gamma (\mathfrak{t})}\mathfrak{v}(\mathfrak{t})\right)=0,\mathfrak{t}\in (0,1),\hfill \\ {\displaystyle \underset{\mathfrak{t}\to 0+}{lim}{\mathfrak{t}}^{i-w(\mathfrak{t})}\mathfrak{v}(\mathfrak{t})=0,i=2,\dots ,n,\mathfrak{v}(1)={\displaystyle \sum _{j=0}^m}{\lambda }_j{I}_{0+}^{\beta (\mathfrak{t})}\mathfrak{v}({\eta }_j).}\hfill \end{array}\right.$$

(14)

Here $w(\mathfrak{t})\in [n-1,n)$ with $n\ge 4$, $\gamma (\mathfrak{t})\in (0,1)$, $\beta (\mathfrak{t})>0$, ${\eta }_j\in (0,1)$ and ${\lambda }_j>0$ for all $j=0,\dots ,m$. The function $\mathfrak{B}:[0,1]\times {\mathbb{R}}^2\to {\mathbb{R}}^{+}$ is a Caratheodory function which satisfies certain assumptions; $D_{0+}^{w(\mathfrak{t})}$ and $D_{0+}^{\gamma (\mathfrak{t})}$ are the Riemann–Liouville fractional derivatives of variable orders $w(\mathfrak{t})$ and $\gamma (\mathfrak{t})$, respectively; and $I_{0+}^{\beta (\mathfrak{t})}$ represents the Riemann–Liouville fractional integral of order $\beta (\mathfrak{t})$. By using an iterative sequential approach and a limit point technique, combined with the method of upper and lower solutions and the Schauder fixed-point theorem, the authors show the existence of positive solutions within subintervals in a suitable Sobolev space for problem (14).

In paper [10], the authors study the system of variable-order fractional differential equations subject to boundary conditions

$$\left\{\begin{array}{c}{}_{0+}{}^{V\hspace{0.33em}\hspace{0.33em}}D_{\mathfrak{t}}^{\zeta (\mathfrak{t})}\mathfrak{u}(\mathfrak{t})=\mathfrak{f}(\mathfrak{t},\mathfrak{u}(\mathfrak{t}),\mathfrak{v}(\mathfrak{t})),\mathfrak{t}\in (0,T),\hfill \\ {}_{0+}{}^{V\hspace{0.33em}\hspace{0.33em}}D_{\mathfrak{t}}^{\xi (\mathfrak{t})}\mathfrak{v}(\mathfrak{t})=\mathfrak{g}(\mathfrak{t},\mathfrak{u}(\mathfrak{t}),\mathfrak{v}(\mathfrak{t})),\mathfrak{t}\in (0,T),\hfill \\ \mathfrak{u}(0)=\mathfrak{u}(T)=\mathfrak{v}(0)=\mathfrak{v}(T)=0,\hfill \end{array}\right.$$

(15)

where $\zeta (\mathfrak{t}),\xi (\mathfrak{t})\in (1,2)$; ${}_{0+}{}^{V\hspace{0.33em}\hspace{0.33em}}D_{\mathfrak{t}}^{\zeta (\mathfrak{t})}$ and ${}_{0+}{}^{V\hspace{0.33em}\hspace{0.33em}}D_{\mathfrak{t}}^{\xi (\mathfrak{t})}$ are variable-order fractional derivatives of Caputo type of orders $\zeta (\mathfrak{t})$ and $\xi (\mathfrak{t})$, respectively, and $\mathfrak{f},\mathfrak{g}:[0,T]\times {\mathbb{R}}^2\to \mathbb{R}$ are Lipschitz continuous functions. Using generalized intervals and piecewise constant functions, together with the Banach fixed point theorem, they establish the existence and uniqueness of solutions to problem (15) in a vector subspace of the Banach space ${(C[0,T])}^2$.

Paper [11] deals with the system of variable-order fractional equations

$$\left\{\begin{array}{c}{}_{0}D_{\mathfrak{t}}^{\alpha (\mathfrak{t})}{\mathfrak{x}}_1(\mathfrak{t})={\mathfrak{x}}_2,\hfill \\ {}_{0}D_{\mathfrak{t}}^{\alpha (\mathfrak{t})}{\mathfrak{x}}_2(\mathfrak{t})={\mathfrak{x}}_2{\mathfrak{x}}_3-{\mathfrak{x}}_1,\hfill \\ {}_{0}D_{\mathfrak{t}}^{\alpha (\mathfrak{t})}{\mathfrak{x}}_3(\mathfrak{t})=1-{\mathfrak{x}}_2^2,\hfill \end{array}\right.$$

(16)

with the initial conditions

$${\mathfrak{x}}_1(0)=0,{\mathfrak{x}}_2(0)=1,{\mathfrak{x}}_3(0)=0,$$

(17)

where ${\mathfrak{x}}_i,i=1,\dots ,3$ represent the state variables, and ${}_{0}D_{\mathfrak{t}}^{\alpha (\mathfrak{t})}$ denotes the Caputo fractional derivative of order $\alpha (\mathfrak{t})$. The authors present a finite difference scheme making use of the two-step Lagrange polynomial interpolation for approximation of the solution to problem (16) and (17). Then they investigate its behavior for various values of $\alpha$ through numerical simulations, at which phase portraits, time series and Lyapunov exponents are used to characterize the transition to chaos.

4. Nabla Fractional Difference Equations

Paper [12] is concerned with the nonlinear nabla fractional difference equation with multipoint summation boundary conditions

$$\left\{\begin{array}{c}-\left({\nabla }_{\varrho (a)}^{\vartheta }\mathfrak{u}\right)(\mathfrak{t})=\mathfrak{f}(\mathfrak{t},\mathfrak{u}(\mathfrak{t})),\mathfrak{t}\in {\mathbb{N}}_{a+2}^b=\{a+2,a+3,\dots ,b\},\hfill \\ {\displaystyle \mathfrak{u}(a)=A,\mathfrak{u}(b)+\mu {\displaystyle \sum _{\mathfrak{s}=a+1}^{b-1}}\mathfrak{u}(\mathfrak{s})=B.}\hfill \end{array}\right.$$

(18)

Here, $\vartheta \in (1,2)$, $\varrho (a)=a-1$ and $A,B,a,b\in \mathbb{R}$, $\mu >0$. The function $\mathfrak{u}:{\mathbb{N}}_a^b\to \mathbb{R}$ is the unknown function, where ${\mathbb{N}}_a^b=\{a,a+1,\dots ,b\}$ with ($b-a\in \mathbb{N}$). The function $\mathfrak{f}:{\mathbb{N}}_{a+2}^b\times \mathbb{R}\to \mathbb{R}$ is continuous with respect to the second variable, and ${\nabla }_{\varrho (a)}^{\vartheta }\mathfrak{u}$ represents the ${\vartheta }^{\mathrm{th}}$-order Riemann–Liouville nabla fractional difference of $\mathfrak{u}$ based on $\varrho (a)$. By utilizing the Brouwer and the Banach fixed point theorems, the authors prove the existence and uniqueness of solutions to problem (18). The generalized Ulam-Hyers stability and the generalized Ulam–Hyers–Rassias stability for (18) are also obtained.

Paper [13] examines the nabla fractional difference equation supplemented with multipoint boundary conditions

$$\left\{\begin{array}{c}-\left({\nabla }_{k-1}^{\nu }\mathfrak{u}\right)(\mathfrak{t})=\mathfrak{f}(\mathfrak{t},\mathfrak{u}(\mathfrak{t})),\mathfrak{t}\in {\mathbb{N}}_{k+2}^l=\{k+2,\dots ,l\},\hfill \\ {\displaystyle \mathfrak{u}(k)=0,\mathfrak{u}(l)=\mu {\displaystyle \sum _{\mathfrak{s}=k+1}^d}\mathfrak{u}(\mathfrak{s}),}\hfill \end{array}\right.$$

(19)

where $\nu \in (1,2]$, $d\in {\mathbb{N}}_{k+1}^{l-1}=\{k+1,\dots ,l-1\}$, $\mu >0$ is a parameter, and $\mathfrak{f}:{\mathbb{N}}_{k+2}^l\times \mathbb{R}\to \mathbb{R}$ is a continuous function. Through the use of the Leray–Schauder nonlinear alternative, the authors establish the existence of solutions to problem (19). Based on the Krasnosel’skii–Zabreiko fixed-point theorem, they also show the existence of nontrivial solutions and positive solutions to (19) when the parameter $\mu$ belongs to a certain interval. Using the Leray–Schauder nonlinear alternative, the authors establish the existence of solutions to problem (19). Based on the Krasnosel’skii–Zabreiko fixed-point theorem, they also show the existence of nontrivial and positive solutions to (19) when the parameter lies in value problems with fractional Laplacian

In paper [14], the authors analyze the fractional Kirchhoff equation with a mass constraint condition

$$\left\{\begin{array}{c}\left({\displaystyle a+b{\int }_{{\mathbb{R}}^N}{\left|{(-\Delta )}^{\frac{s}{2}}\mathfrak{u}\right|}^{2}dx}\right){(-\Delta )}^s\mathfrak{u}+\lambda \mathfrak{u}=\mathfrak{f}(\mathfrak{u})+{\left|\mathfrak{u}\right|}^{2_s^{\ast }-2}\mathfrak{u},\mathrm{in}{\mathbb{R}}^N,\hfill \\ {\displaystyle {\int }_{{\mathbb{R}}^N}{\left|\mathfrak{u}\right|}^{2}dx=m^2,}\hfill \end{array}\right.$$

(20)

where $N\ge 1$, $s\in (0,1)$, $s>N/4$; $a,b>0$, $2_s^{\ast }={\displaystyle \frac{2N}{N-2s}}$ is the fractional Sobolev critical exponent, $m>0$ is a given constant, and the function $\mathfrak{f}:\mathbb{R}\to \mathbb{R}$ is continuous and odd, satisfying some additional assumptions. Here, ${(-\Delta )}^s$ is the pseudo-differential operator (the fractional Laplacian) defined by

$${\displaystyle {(-\Delta )}^s\mathfrak{u}(x)=C(N,s)\mathrm{P}.\mathrm{V}.{\int }_{{\mathbb{R}}^N}\frac{\mathfrak{u}(x)-\mathfrak{u}(y)}{{|x-y|}^{N+2s}}dy,}$$

where $C(N,s)$ is a positive constant, and P.V. is the Cauchy principal value of the integral. Using a constrained minimization method and characterizations of the mountain pass levels, they prove the existence of normalized solutions $(\mathfrak{u},\lambda )\in H^s({\mathbb{R}}^N)\times \mathbb{R}$ to problem (20).

Finally, I would like to extend my sincere thanks to all the authors for their contributions to this Special Issue. I hope that the results presented herein will prove valuable to researchers in the field of fractional differential equations and stimulate further advances in this rapidly evolving area of research.