Existence Results for Differential Equations with Tempered Ψ–Caputo Fractional Derivatives

Abstract

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered $\Psi$–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution.

1. Introduction

Fractional calculus involves derivatives and integrals of non-integer order. Nowadays, they are defined in various ways as interpolations between classic integer-order derivatives and integrals. For the basic theory on this topic, we refer the reader to [1,2].

The problem of the existence of a unique solution of an initial value problem (IVP) for nonlinear fractional differential equations with the well-known Caputo derivative was studied in [3]. Here, the case of multiple Caputo fractional derivatives of the solution on the right side was also mentioned, along with appropriate references. In [4], this problem was investigated for equations with right sides depending on the integer-order derivatives of the solution, as is typical for higher-order ordinary differential equations in classic calculus.

Fractional calculus was generalized to tempered fractional calculus in [5] (also known as substantial fractional calculus [6]), and to $\Psi$–fractional calculus in [7] (also called a fractional integral and fractional derivative of a function with respect to another function [2,8]). In [9,10], these two approaches were unified, and definitions of a tempered $\Psi$–Hilfer fractional integral of any positive order and of a tempered $\Psi$–Caputo fractional derivative (T$\Psi$CFD) of order $\alpha \in (n-1,n)$, $n\in \mathbb{N}$ of a $C^n$–function x were introduced. Later, in [11], the definition of T$\Psi$CFD was provided for any $C^{n-1}$–function x using a Riemann–Liouville-type fractional derivative. In the same paper, a Henry–Gronwall-type inequality was used to derive results regarding the existence of a unique solution to the IVP of the type

$$\begin{array}{cc}\hfill {{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)& =f(t,x(t\left)\right),t>a,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill x_{\lambda ,\Psi }^{\left[0\right]}\left(a\right)& =x_a^0,\hfill \end{array}$$

(2)

assuming that f fulfills a particular Lipschitz-type condition,

$$t^{1-\gamma }\parallel f(t,x)-f(t,y)\parallel \le L\parallel x-y\parallel$$

(3)

for some $\gamma \in (0,1)$. In the present paper, we remove the need for the term $t^{1-\gamma }$ in the above condition and prove the existence and uniqueness results for IVPs, where the right-hand side may depend not only on the solution itself but also on its T$\Psi$CFDs of lower orders, ordinary integer-order derivatives, or tempered $\Psi$–Hilfer fractional integrals of the solution. The results follow from the Banach fixed point theorem applied to a corresponding operator defined by the right sides of a system of fractional integral equations equivalent to the original IVP.

The structure of the paper is as follows. In the next section, we briefly recall the known definitions and derive the auxiliary results. Section 3 is devoted to the main results of this paper, namely the results on the existence of a unique solution regarding the IVPs involving T$\Psi$CFD with various right-hand sides. Section 4 presents several examples illustrating the theoretical outcomes. Finally, we conclude the paper and outline the possible further research directions in Section 5.

Throughout the paper, the set of all the positive (non-negative) integers is denoted as $\mathbb{N}$ (${\mathbb{N}}_0$). By $\parallel ·\parallel$, we mean any vector norm on ${\mathbb{R}}^N$. For simplicity, we consider the empty sum property, i.e., ${\sum }_{k=a}^b{f}_k=0$, whenever $a>b$ regardless of whether functions $f_k$ are defined or not.

2. Preliminaries

Here, we recall known definitions and prove preliminary results useful for the main section. For the sake of brevity, we introduce a notation for a class of functions,

$${\mathcal{P}}_S^k:=\{\Psi \in C^k\left(S\right)\mid {\Psi }^{\prime }\left(t\right)>0\forall t\in S\},$$

where $k\in \mathbb{N}$.

Next, we recall a definition of the fractional integral (see [9,10]).

Definition 1.

Let $\alpha >0$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,b]}^1$. Then, tempered Ψ–Hilfer fractional integral of function $x\in C[a,b]$ of order $\alpha >0$ is defined by

$$I_a^{\alpha ,\lambda ,\Psi }x\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)x\left(s\right)ds$$

for $t\in [a,b]$, where $\Gamma (·)$ is the Euler gamma function and

$$\kappa (\alpha ;t,s)=\kappa (\alpha ,\lambda ,\Psi ;t,s):={(\Psi \left(t\right)-\Psi \left(s\right))}^{\alpha -1}{\mathrm{e}}^{-\lambda (\Psi (t)-\Psi (s\left)\right)}{\Psi }^{\prime }\left(s\right).$$

The operator $I_a^{\alpha ,\lambda ,\Psi }$ has an important property described in the next result.

Lemma 1.

Let $\alpha >0$, $\lambda \ge 0$, $\Psi \in {\mathcal{P}}_{[a,b]}^1$, and $x\in C[a,b]$. The operator $I_a^{\alpha ,\lambda ,\Psi }$ has the following properties:

1. 

If $0<\alpha <1$, then $I_a^{\alpha ,\lambda ,\Psi }x\in C[a,b]$.

2. 

For any $\alpha \ge 1$, if $\Psi \in C^{\lfloor \alpha \rfloor }[a,b]$, then $I_a^{\alpha ,\lambda ,\Psi }x\in C^{\lfloor \alpha \rfloor }[a,b]$, where $\lfloor ·\rfloor$ is the floor function.

Proof. 

By the change of variable $\sigma =\Psi \left(t\right)-\Psi \left(s\right)$, we write

$$I_a^{\alpha ,\lambda ,\Psi }x\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\Psi \left(t\right)-\Psi \left(a\right)}{\sigma }^{\alpha -1}{\mathrm{e}}^{-\lambda \sigma }x\left({\Psi }^{-1}(\Psi \left(t\right)-\sigma )\right)d\sigma ={\displaystyle \frac{F(\Psi (t)-\Psi (a\left)\right)}{\Gamma \left(\alpha \right)}}$$

for $F\left(t\right)={\int }_0^t{\sigma }^{\alpha -1}K(t,\sigma )d\sigma ,$ where $K(t,s)={\mathrm{e}}^{-\lambda s}x\left({\Psi }^{-1}(t+\Psi \left(a\right)-s)\right)$ is continuous on $M=\{(t,s)\in {\mathbb{R}}^2\mid t\in [0,\Psi \left(b\right)-\Psi \left(a\right)],s\in [0,t]\}$. To prove that $I_a^{\alpha ,\lambda ,\Psi }x\in C[a,b]$, it is sufficient to show that $F\in C[0,\Psi (b)-\Psi (a\left)\right]$.

Let $\epsilon >0$ be fixed, and take $0\le t_2\le t_1\le \Psi \left(b\right)-\Psi \left(a\right)$ such that $t_1-t_2<\delta$. Then,

$$\begin{array}{c}\hfill |F\left(t_1\right)-F\left(t_2\right)|=\left|{\int }_0^{t_1}{\sigma }^{\alpha -1}K(t_1,\sigma )d\sigma -{\int }_0^{t_2}{\sigma }^{\alpha -1}K(t_2,\sigma )d\sigma \right|\le S_1+S_2,\end{array}$$

(4)

where

$$\begin{array}{cc}\hfill S_1& =\left|{\int }_0^{t_1}{\sigma }^{\alpha -1}K(t_1,\sigma )d\sigma -{\int }_0^{t_2}{\sigma }^{\alpha -1}K(t_1,\sigma )d\sigma \right|\le {\int }_{t_2}^{t_1}{\sigma }^{\alpha -1}\left|K(t_1,\sigma )\right|d\sigma \hfill \\ \hfill & \le K{\int }_{t_2}^{t_1}{\sigma }^{\alpha -1}d\sigma \le K{\int }_{t_2}^{t_1}{(\sigma -t_2)}^{\alpha -1}d\sigma ={\displaystyle \frac{K{(t_1-t_2)}^{\alpha }}{\alpha }}<{\displaystyle \frac{K{\delta }^{\alpha }}{\alpha }}\hfill \end{array}$$

for $K={max}_{(t,s)\in M}\left|K(t,s)\right|$, and

$$\begin{array}{cc}\hfill S_2& =\left|{\int }_0^{t_2}{\sigma }^{\alpha -1}K(t_1,\sigma )d\sigma -{\int }_0^{t_2}{\sigma }^{\alpha -1}K(t_2,\sigma )d\sigma \right|\hfill \\ \hfill & \le {\int }_0^{t_2}{\sigma }^{\alpha -1}|K(t_1,\sigma )-K(t_2,\sigma )|d\sigma .\hfill \end{array}$$

The continuity of $K(·,·)$ means that

$$\begin{array}{c}\forall {\epsilon }_K>0\exists {\delta }_K>0\forall (t_1,s_1),(t_2,s_2)\in M:\\ |t_1-t_2|+|s_1-s_2|<{\delta }_K\Rightarrow |K(t_1,s_1)-K(t_2,s_2)|<{\epsilon }_K.\end{array}$$

Let us take ${\epsilon }_K={\displaystyle \frac{\alpha \epsilon }{2{(\Psi \left(b\right)-\Psi \left(a\right))}^{\alpha }}}$. If $\delta >0$ is so small that $\delta \le min\left\{{\delta }_K,{\left({\displaystyle \frac{\alpha \epsilon }{2K}}\right)}^{1/\alpha }\right\}$, then

$$|K(t_1,\sigma )-K(t_2,\sigma )|<{\epsilon }_K\forall \sigma \in [0,t_2].$$

Consequently,

$$|F\left(t_1\right)-F\left(t_2\right)|<{\displaystyle \frac{K{\delta }^{\alpha }}{\alpha }}+{\displaystyle \frac{{\epsilon }_K{t}_2^{\alpha }}{\alpha }}\le {\displaystyle \frac{K{\delta }^{\alpha }}{\alpha }}+{\displaystyle \frac{{\epsilon }_K{(\Psi \left(b\right)-\Psi \left(a\right))}^{\alpha }}{\alpha }}\le \epsilon ,$$

which proves the continuity of F and, consequently, Statement 1.

Let $\alpha \ge 1$ and $\Psi \in C^1[a,b]$. Then, as before, we derive (4) with $S_2<{\displaystyle \frac{{\epsilon }_K{(\Psi \left(b\right)-\Psi \left(a\right))}^{\alpha }}{\alpha }}$, but this time

$$S_1\le K{\int }_{t_2}^{t_1}{\sigma }^{\alpha -1}d\sigma ={\displaystyle \frac{K(t_1^{\alpha }-t_2^{\alpha })}{\alpha }}=K{\tau }^{\alpha -1}(t_1-t_2)\le K{(\Psi \left(b\right)-\Psi \left(a\right))}^{\alpha -1}\delta$$

for some $\tau \in [t_2,t_1]$. So, it is sufficient to take

$${\epsilon }_K={\displaystyle \frac{\alpha \epsilon }{2{(\Psi \left(b\right)-\Psi \left(a\right))}^{\alpha }}}\mathrm{and}0<\delta \le min\left\{{\delta }_K,{\displaystyle \frac{\epsilon }{2K{(\Psi \left(b\right)-\Psi \left(a\right))}^{\alpha -1}}}\right\}$$

to see that $I_a^{\alpha ,\lambda \Psi }x\in C[a,b]$.

If $\alpha =1$, then for the derivative we have

$${\displaystyle \frac{d}{dt}}\left[I_a^{1,\lambda ,\Psi }x\left(t\right)\right]={\displaystyle \frac{d}{dt}}\left[{\int }_a^t{\mathrm{e}}^{-\lambda (\Psi (t)-\Psi (s\left)\right)}{\Psi }^{\prime }\left(s\right)x\left(s\right)ds\right]={\Psi }^{\prime }\left(t\right)\left(x\left(t\right)-\lambda I_a^{1,\lambda ,\Psi }x\left(t\right)\right).$$

Note that all terms on the right side are continuous. So, $I_a^{1,\lambda ,\Psi }x\in C^1[a,b]$.

If $\alpha >1$, then

$${\displaystyle \frac{d}{dt}}\left[I_a^{\alpha ,\lambda ,\Psi }x\left(t\right)\right]={\Psi }^{\prime }\left(t\right)\left(I_a^{\alpha -1,\lambda ,\Psi }x\left(t\right)-\lambda I_a^{\alpha ,\lambda ,\Psi }x\left(t\right)\right).$$

(5)

From the previous arguments, we know that both terms in the bracket are continuous (we employ Statement 1 if $1<\alpha <2$), i.e., $I_a^{\alpha ,\lambda ,\Psi }x\in C^1[a,b]$.

Let $\alpha \ge 2$ be fixed and let $\mathbb{N}\ni k<\lfloor \alpha \rfloor$ be such that $I_a^{\tilde{\alpha },\lambda ,\Psi }x\in C^k[a,b]$ for all $\tilde{\alpha }\ge k$. Then, in (5), we have $\alpha -1\ge \lfloor \alpha \rfloor -1\ge k$ and $\alpha >k$. Hence, both terms in the bracket are $C^k$–smooth by the induction hypothesis. So is ${\Psi }^{\prime }$, since $\Psi \in C^{\lfloor \alpha \rfloor }[a,b]$ by the assumption. As a consequence, $I_a^{\alpha ,\lambda ,\Psi }x\in C^{k+1}[a,b]$. By induction, it follows that $I_a^{\alpha ,\lambda ,\Psi }x\in C^{\lfloor \alpha \rfloor }[a,b]$. This completes the proof. □

Next, we recall a definition of T$\Psi$CFD that is provided by the use of tempered $\Psi$–Riemann–Liouville fractional derivative [11] (see also [9]).

Definition 2.

Let $n\in \mathbb{N}$, $n-1<\alpha <n$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,b]}^n$. Tempered Ψ–Riemann–Liouville fractional derivative of function $x\in C^{n-1}[a,b]$ of order α is defined by

$$\begin{array}{cc}\hfill {{}^{RL}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}{\left(I_a^{n-\alpha ,\lambda ,\Psi }x\left(t\right)\right)}_{\lambda ,\Psi }^{\left[n\right]}\hfill \\ \hfill & ={\displaystyle \frac{{\mathrm{e}}^{-\lambda \Psi \left(t\right)}}{\Gamma (n-\alpha )}}{\left({\displaystyle \frac{1}{{\Psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{\int }_a^t{(\Psi \left(t\right)-\Psi \left(s\right))}^{n-\alpha -1}{\Psi }^{\prime }\left(s\right){\mathrm{e}}^{\lambda \Psi \left(s\right)}x\left(s\right)ds,\hfill \end{array}$$

where

$$x_{\lambda ,\Psi }^{\left[n\right]}\left(t\right)={\left({\displaystyle \frac{1}{{\Psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n\left({\mathrm{e}}^{\lambda \Psi \left(t\right)}x\left(t\right)\right).$$

Definition 3.

Let $n\in \mathbb{N}$, $n-1<\alpha <n$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,b]}^n$. Tempered Ψ–Caputo fractional derivative of function $x\in C^{n-1}[a,b]$ of order α is defined by

$${{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)={{}^{RL}D}_a^{\alpha ,\lambda ,\Psi }\left[x\left(t\right)-{\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{x_{\lambda ,\Psi }^{\left[k\right]}\left(a\right)}{k!}}{(\Psi \left(t\right)-\Psi \left(a\right))}^k\right].$$

(6)

We refer the reader to [11] ([Lemmas 1 and 2]) for basic properties of $I_a^{\alpha ,\lambda ,\Psi }$ and ${{}^{C}D}_a^{\alpha ,\lambda ,\Psi }$. Further properties, required for this paper, are proved below. The first one extends ([12] [Lemma 1]).

Lemma 2.

Let $n\in \mathbb{N}$, $n-1<\alpha <n$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,b]}^n$. Then, for $x\in C^{n-1}[a,b]$, $m-1<\beta <m\in \mathbb{N}$, $0<\beta <\alpha$,

$$I_a^{\alpha -\beta ,\lambda ,\Psi }{{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)={{}^{C}D}_a^{\beta ,\lambda ,\Psi }x\left(t\right)-{\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=m}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{k-\beta }}{\Gamma (k+1-\beta )}}{x}_{\lambda ,\Psi }^{\left[k\right]}\left(a\right).$$

In particular, if $m=n$, then $I_a^{\alpha -\beta ,\lambda ,\Psi }{{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)={{}^{C}D}_a^{\beta ,\lambda ,\Psi }x\left(t\right)$.

Proof. 

Using [11] [Lemma 2],

$$\begin{array}{cc}\hfill I_a^{\alpha -\beta ,\lambda ,\Psi }{{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)& ={{}^{C}D}_a^{\beta ,\lambda ,\Psi }{I}_a^{\alpha ,\lambda ,\Psi }{{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)\hfill \\ \hfill & ={{}^{C}D}_a^{\beta ,\lambda ,\Psi }\left[x\left(t\right)-{\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}{x}_{\lambda ,\Psi }^{\left[k\right]}\left(a\right)\right]\hfill \\ \hfill & ={{}^{C}D}_a^{\beta ,\lambda ,\Psi }x\left(t\right)-{\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=m}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{k-\beta }}{\Gamma (k+1-\beta )}}{x}_{\lambda ,\Psi }^{\left[k\right]}\left(a\right).\hfill \end{array}$$

The particular case is due to the empty sum property. □

Lemma 3.

Let $\alpha >0$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,b]}^1$. Then,

$$I_a^{\alpha ,\lambda ,\Psi }\left[{\mathrm{e}}^{-\lambda \Psi \left(t\right)}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}\right]={\mathrm{e}}^{-\lambda \Psi \left(t\right)}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{k+\alpha }}{\Gamma (k+\alpha +1)}}$$

for each $k\in {\mathbb{N}}_0$.

Proof. 

From Definition 1, we directly obtain

$$\begin{array}{cc}\hfill I_a^{\alpha ,\lambda ,\Psi }\left[{\mathrm{e}}^{-\lambda \Psi \left(t\right)}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}\right]& ={\displaystyle \frac{{\mathrm{e}}^{-\lambda \Psi \left(t\right)}}{\Gamma \left(\alpha \right)k!}}{\int }_a^t{(\Psi \left(t\right)-\Psi \left(s\right))}^{\alpha -1}{(\Psi \left(s\right)-\Psi \left(a\right))}^k{\Psi }^{\prime }\left(s\right)ds\hfill \\ \hfill & ={\displaystyle \frac{{\mathrm{e}}^{-\lambda \Psi \left(t\right)}}{\Gamma \left(\alpha \right)k!}}{(\Psi \left(t\right)-\Psi \left(a\right))}^{\alpha +k}{\int }_0^1{\sigma }^{\alpha -1}{(1-\sigma )}^{k}d\sigma \hfill \end{array}$$

by the change of variable $\Psi \left(t\right)-\Psi \left(s\right)=\sigma (\Psi (t)-\Psi (a\left)\right)$. Using the Euler beta function $B(·,·)$, we continue with the computations as follows:

$$\begin{array}{cc}\hfill I_a^{\alpha ,\lambda ,\Psi }\left[{\mathrm{e}}^{-\lambda \Psi \left(t\right)}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}\right]& ={\displaystyle \frac{{\mathrm{e}}^{-\lambda \Psi \left(t\right)}}{\Gamma \left(\alpha \right)k!}}{(\Psi \left(t\right)-\Psi \left(a\right))}^{\alpha +k}B(\alpha ,k+1)\hfill \\ \hfill & ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{k+\alpha }}{\Gamma (k+\alpha +1)}}.\hfill \end{array}$$

□

Finally, we show how to rewrite $x_{\lambda ,\Psi }^{\left[k\right]}$ using the ordinary derivatives of x and vice versa.

Lemma 4.

Let $n\in \mathbb{N}$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,b]}^n$. Then, for each $k\in \{0,1,\dots ,n\}$, there are functions ${\alpha }_i^k,{\beta }_i^k\in C[a,b]$, $i=0,1,\dots ,k$ such that

$$x_{\lambda ,\Psi }^{\left[k\right]}\left(t\right)=\sum _{i=0}^k{\alpha }_i^k\left(t\right)x^{\left(i\right)}\left(t\right),x^{\left(k\right)}\left(t\right)=\sum _{i=0}^k{\beta }_i^k\left(t\right)x_{\lambda ,\Psi }^{\left[i\right]}\left(t\right)$$

(7)

for any $t\in [a,b]$ and $x\in C^n[a,b]$.

Proof. 

Let $x\in C^n[a,b]$. Denote $y\left(t\right):=x\left({\Psi }^{-1}\left(t\right)\right)$, $A\left(t\right):={\mathrm{e}}^{\lambda t}y\left(t\right)$. Let $\lambda >0$. Then, for each $k=1,2,\dots ,n$,

$$\begin{array}{cc}\hfill x_{\lambda ,\Psi }^{\left[k\right]}\left(t\right)={\left({\displaystyle \frac{1}{{\Psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^k\left({\mathrm{e}}^{\lambda \Psi \left(t\right)}x\left(t\right)\right)& ={\left({\displaystyle \frac{1}{{\Psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{k}A(\Psi \left(t\right))\hfill \\ \hfill & ={\left({\displaystyle \frac{1}{{\Psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{k-1}{A}^{\prime }(\Psi \left(t\right))=\cdots =A^{\left(k\right)}(\Psi \left(t\right))\hfill \\ \hfill & ={\left.A^{\left(k\right)}\left(t\right)\right|}_{t=\Psi \left(t\right)}={\left.{\left({\mathrm{e}}^{\lambda t}y\left(t\right)\right)}^{\left(k\right)}\right|}_{t=\Psi \left(t\right)}.\hfill \end{array}$$

By the application of the Leibnitz rule, we obtain

$$x_{\lambda ,\Psi }^{\left[k\right]}\left(t\right)=\sum _{i=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{i}}\right){\left({\mathrm{e}}^{\lambda t}\right)}^{(k-i)}{y}^{\left(i\right)}\left(t\right){|}_{t=\Psi \left(t\right)}=\sum _{i=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{i}}\right){\lambda }^{k-i}{\mathrm{e}}^{\lambda t}{y}^{\left(i\right)}\left(t\right){|}_{t=\Psi \left(t\right)}.$$

Using Faà di Bruno formula with Bell polynomials (see [13,14]) for the derivative of function y yields

$$\begin{array}{cc}\hfill x_{\lambda ,\Psi }^{\left[k\right]}\left(t\right)& ={\mathrm{e}}^{\lambda \Psi \left(t\right)}\sum _{i=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{i}}\right){\lambda }^{k-i}{\left[\sum _{j=0}^i{x}^{\left(j\right)}\left({\Psi }^{-1}\left(t\right)\right)B_{i,j}\left({\left({\Psi }^{-1}\left(t\right)\right)}^{\prime },\dots ,{\left({\Psi }^{-1}\left(t\right)\right)}^{(i-j+1)}\right)\right]}_{t=\Psi \left(t\right)}\hfill \\ \hfill & ={\mathrm{e}}^{\lambda \Psi \left(t\right)}\sum _{j=0}^k{x}^{\left(j\right)}\left(t\right){\chi }_{j,k}\left(t\right),\hfill \end{array}$$

(8)

where

$${\chi }_{j,k}\left(t\right)=\sum _{i=j}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{i}}\right){\lambda }^{k-i}{B}_{i,j}\left({\left({\Psi }^{-1}\left(t\right)\right)}^{\prime },\dots ,{\left({\Psi }^{-1}\left(t\right)\right)}^{(i-j+1)}\right){|}_{t=\Psi \left(t\right)}$$

and $B_{i,j}$ is the incomplete exponential Bell polynomial provided by

$$B_{i,j}(x_1,\dots ,x_{i-j+1})=\sum _{\begin{array}{c}{q}_1,\dots ,q_{i-j+1}\in {\mathbb{N}}_0\\ q_1+\cdots +q_{i-j+1}=j\\ q_1+2q_2+\cdots +(i-j+1)q_{i-j+1}=i\end{array}}{\displaystyle \frac{i!}{q_1!\cdots q_{i-j+1}!}}\prod _{r=1}^{i-j+1}{\left({\displaystyle \frac{x_r}{r!}}\right)}^{q_r}$$

(see, e.g., [15]). Note that $B_{i,0}=0$ whenever $i>0$, and $B_{0,0}=1$. Hence, $x_{\lambda ,\Psi }^{\left[k\right]}\left(t\right)$ depends on the derivatives ${\left({\Psi }^{-1}\left(t\right)\right)}^{\prime }$, …, ${\left({\Psi }^{-1}\left(t\right)\right)}^{\left(k\right)}$ only. In particular, $\Psi \in C^n[a,b]$ is sufficient for $x_{\lambda ,\Psi }^{\left[n\right]}\left(t\right)$. Setting ${\chi }_{0,0}\left(t\right)\equiv 1$, one can extend (8) to $k=0$.

If $\lambda =0$, then $x_{0,\Psi }^{\left[k\right]}\left(t\right)={\left.y^{\left(k\right)}\left(t\right)\right|}_{t=\Psi \left(t\right)}$, Leibnitz rule is not needed, and one obtains

$$x_{0,\Psi }^{\left[k\right]}\left(t\right)=\sum _{j=0}^k{x}^{\left(j\right)}\left(t\right){\left[B_{k,j}\left({\left({\Psi }^{-1}\left(t\right)\right)}^{\prime },\dots ,{\left({\Psi }^{-1}\left(t\right)\right)}^{(k-j+1)}\right)\right]}_{t=\Psi \left(t\right)}$$

which is exactly (8) with $\lambda =0$. This proves the first one of identities (7) with ${\alpha }_j^k\left(t\right)={\mathrm{e}}^{\lambda \Psi \left(t\right)}{\chi }_{j,k}\left(t\right)$.

The following matrix identity concludes the previous results:

$$\left(\begin{array}{c}{x}_{\lambda ,\Psi }^{\left[0\right]}\left(t\right)\\ x_{\lambda ,\Psi }^{\left[1\right]}\left(t\right)\\ ⋮\\ x_{\lambda ,\Psi }^{\left[n\right]}\left(t\right)\end{array}\right)={\mathrm{e}}^{\lambda \Psi \left(t\right)}\left(\begin{array}{ccccc}{\chi }_{0,0}\left(t\right)& 0& \dots & \dots & 0\\ {\chi }_{0,1}\left(t\right)& {\chi }_{1,1}\left(t\right)& \ddots & & ⋮\\ ⋮& ⋮& \ddots & \ddots & ⋮\\ ⋮& ⋮& & \ddots & 0\\ {\chi }_{0,n}\left(t\right)& {\chi }_{1,n}\left(t\right)& \dots & \dots & {\chi }_{n,n}\left(t\right)\end{array}\right)\left(\begin{array}{c}x\left(t\right)\\ x^{\prime }\left(t\right)\\ ⋮\\ x^{\left(n\right)}\left(t\right)\end{array}\right).$$

Let $\mathbb{X}$ denote the above lower-triangular matrix. Since

$$\begin{array}{cc}\hfill det\mathbb{X}& =\prod _{j=0}^n{\chi }_{j,j}\left(t\right)=\prod _{j=0}^n{B}_{j,j}\left({\left({\Psi }^{-1}\left(t\right)\right)}^{\prime }\right){|}_{t=\Psi \left(t\right)}\hfill \\ \hfill & =\prod _{j=0}^n{B}_{j,j}\left({\left({\Psi }^{\prime }\left(t\right)\right)}^{-1}\right)=\prod _{j=0}^n{\left({\Psi }^{\prime }\left(t\right)\right)}^{-j}={\left({\Psi }^{\prime }\left(t\right)\right)}^{-{\displaystyle \frac{n(n+1)}{2}}}\ne 0,\hfill \end{array}$$

we obtain

$$\left(\begin{array}{c}x\left(t\right)\\ x^{\prime }\left(t\right)\\ ⋮\\ x^{\left(n\right)}\left(t\right)\end{array}\right)={\mathrm{e}}^{-\lambda \Psi \left(t\right)}{\mathbb{X}}^{-1}\left(\begin{array}{c}{x}_{\lambda ,\Psi }^{\left[0\right]}\left(t\right)\\ x_{\lambda ,\Psi }^{\left[1\right]}\left(t\right)\\ ⋮\\ x_{\lambda ,\Psi }^{\left[n\right]}\left(t\right)\end{array}\right),$$

where ${\mathbb{X}}^{-1}={\left({\tilde{\chi }}_{i,j}\right)}_{0\le i,j\le n}$ is a nonsingular, lower-triangular matrix (see, e.g., ([16], [Proposition 3.7])). This completes the proof of the second identity of (7) with ${\beta }_j^k\left(t\right)={\mathrm{e}}^{-\lambda \Psi \left(t\right)}{\tilde{\chi }}_{k,j}\left(t\right)$. □

3. Existence Results

In this section, we prove the results on the existence of a unique solution of an IVP for various classes of differential equations involving T$\Psi$CFDs.

We start with a short remark about condition (3).

Remark 1.

If a continuous function f satisfies

$$\parallel f(t,x)-f(t,y)\parallel \le F\left(t\right)\parallel x-y\parallel$$

for all $t\in [a,b]$ and some function F bounded (or constant) on $[a,b]$, then f satisfies condition (3) with $L={sup}_{t\in [a,b]}F\left(t\right)t^{1-\gamma }$ for arbitrary fixed $\gamma \in (0,1)$, assuming that $a>0$. Conversely, if (3) holds with $a>0$, then

$$\parallel f(t,x)-f(t,y)\parallel \le L{t}^{\gamma -1}\parallel x-y\parallel \le a^{\gamma -1}L\parallel x-y\parallel .$$

Therefore, [11] [Theorem 3] holds with classic Lipschitz condition

$$\parallel f(t,x)-f(t,y)\parallel \le L\parallel x-y\parallel$$

instead of (3). Nevertheless, unlike [11], in this paper, we do not require $a>0$ or $\Psi \left(t\right)\le t$.

In this section, we prove the existence of an interval $I_{\theta }=[a,a+\theta ),$ on which a solution of a particular IVP exists. Using a better estimation in a corresponding Lipschitz condition (i.e., smaller L) results in a larger interval $I_{\theta }$.

3.1. Right Side Depending on ${{}^{C}D}_a^{\beta ,\lambda ,\Psi }x\left(t\right)$

First, let us consider the following IVP

$$\begin{array}{cc}\hfill {{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)& =f(t,x\left(t\right),{{}^{C}D}_a^{\beta ,\lambda ,\Psi }x\left(t\right)),t>a,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill x_{\lambda ,\Psi }^{\left[0\right]}\left(a\right)& =x_a^0\hfill \end{array}$$

(10)

for some $x_a^0\in {\mathbb{R}}^N$, where $0<\beta <\alpha <1$, $\lambda \ge 0$, $\Psi \in {\mathcal{P}}_{[a,\infty )}^1$, and $f\in C([a,\infty )\times {\mathbb{R}}^N\times {\mathbb{R}}^N,{\mathbb{R}}^N)$. Here, the differential operator is to be understood component-wise.

Definition 4.

We say that $x\in C[a,a+\theta )$, $0<\theta \le \infty$ is a solution of (or that it solves) IVP (9), (10) if ${{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)$, ${{}^{C}D}_a^{\beta ,\lambda ,\Psi }x\left(t\right)$ exist and are continuous on $(a,a+\theta )$, and x fulfills Equation (9) for all $t\in (a,a+\theta )$ and initial condition (10).

By [11] [Theorem 1], x is a solution of (9), (10) if and only if it satisfies the integral equation

$$x\left(t\right)={\mathrm{e}}^{-\lambda \Psi \left(t\right)}{x}_a^0+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f(s,x\left(s\right),{{}^{C}D}_a^{\beta ,\lambda ,\Psi }x\left(s\right))ds.$$

(11)

Let us denote $y\left(t\right)={{}^{C}D}_a^{\beta ,\lambda ,\Psi }x\left(t\right)$. Applying the operator $I_a^{\alpha -\beta ,\lambda ,\Psi }$ to Equation (9), provides, by Lemma 2,

$$y\left(t\right)=I_a^{\alpha -\beta ,\lambda ,\Psi }f(t,x\left(t\right),y\left(t\right))={\displaystyle \frac{1}{\Gamma (\alpha -\beta )}}{\int }_a^t\kappa (\alpha -\beta ;t,s)f(s,x\left(s\right),y\left(s\right))ds.$$

(12)

Thus, if x is a solution of (9), (10), then it solves the equation

$$x\left(t\right)={\mathrm{e}}^{-\lambda \Psi \left(t\right)}{x}_a^0+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f(s,x\left(s\right),y\left(s\right))ds,$$

(13)

where y is provided by (12).

Now, suppose that $(x,y)$ is a solution of (13), (12). Applying the operator ${{}^{C}D}_a^{\beta ,\lambda ,\Psi }$ on Equation (13), we obtain, by [11] [Lemma 2],

$${{}^{C}D}_a^{\beta ,\lambda ,\Psi }x\left(t\right)={{}^{C}D}_a^{\beta ,\lambda ,\Psi }{I}_a^{\alpha ,\lambda ,\Psi }f(t,x\left(t\right),y\left(t\right))=I_a^{\alpha -\beta ,\lambda ,\Psi }f(t,x\left(t\right),y\left(t\right))=y\left(t\right).$$

Hence, (11) holds, implying that x solves differential equation (9) as well as initial condition (10). We have proved the next statement.

Proposition 1.

Function x is a solution of (9), (10) if and only if Equations (13), (12) are fulfilled.

Next, we present our result on the existence of a unique solution of (9), (10).

Theorem 1.

Let $0<\beta <\alpha <1$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,\infty )}^1$. If there exists $L>0$ such that

$$\parallel f(t,x,y)-f(t,u,v)\parallel \le L(\parallel x-u\parallel +\parallel y-v\parallel )$$

(14)

for all $t\ge a$, $x,y,u,v\in {\mathbb{R}}^N$, then there exists $\theta >0$ such that there is a unique solution x of IVP (9), (10) on the interval $I_{\theta }=[a,a+\theta )$.

Proof. 

Let us denote $C_{\theta }$ the Banach space of continuous functions defined on $I_{\theta }$, equipped with the norm ${\parallel x\parallel }_{C_{\theta }}={sup}_{t\in I_{\theta }}\parallel x\left(t\right)\parallel$. Let us define an operator $\mathcal{F}$ on $C_{\theta }\times C_{\theta }$ by

$$\begin{array}{cc}\hfill \mathcal{F}(x,y)\left(t\right)& =\left({\mathrm{e}}^{-\lambda \Psi \left(t\right)}{x}_a^0+I_a^{\alpha ,\lambda ,\Psi }f(t,x\left(t\right),y\left(t\right)),I_a^{\alpha -\beta ,\lambda ,\Psi }f(t,x\left(t\right),y\left(t\right))\right)\hfill \\ \hfill & =\left({\mathrm{e}}^{-\lambda \Psi \left(t\right)}{x}_a^0+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f(s,x\left(s\right),y\left(s\right))ds,\right.\hfill \\ \hfill & \left.{\displaystyle \frac{1}{\Gamma (\alpha -\beta )}}{\int }_a^t\kappa (\alpha -\beta ;t,s)f(s,x\left(s\right),y\left(s\right))ds\right).\hfill \end{array}$$

Clearly, $\mathcal{F}:C_{\theta }\times C_{\theta }\to C_{\theta }\times C_{\theta }$. On $C_{\theta }\times C_{\theta }$, we consider the norm ${\parallel (x,y)\parallel }_{C_{\theta }\times C_{\theta }}={\parallel x\parallel }_{C_{\theta }}+{\parallel y\parallel }_{C_{\theta }}$. Then, we have

$$\begin{array}{cc}\hfill {\parallel \mathcal{F}(x,y)-\mathcal{F}(u,v)\parallel }_{C_{\theta }\times C_{\theta }}& =\underset{t\in I_{\theta }}{sup}\parallel I_a^{\alpha ,\lambda ,\Psi }(f(t,x\left(t\right),y\left(t\right))-f(t,u\left(t\right),v\left(t\right)))\parallel \hfill \\ \hfill & +\underset{t\in I_{\theta }}{sup}\parallel I_a^{\alpha -\beta ,\lambda ,\Psi }(f(t,x\left(t\right),y\left(t\right))-f(t,u\left(t\right),v\left(t\right)))\parallel .\hfill \end{array}$$

For short, we denote

$${\phi }_{\alpha }\left(t\right):={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)ds$$

(15)

for any $\alpha >0$. By the change of variable $\sigma =\Psi \left(t\right)-\Psi \left(s\right)$, one obtains

$${\phi }_{\alpha }\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\Psi \left(t\right)-\Psi \left(a\right)}{\sigma }^{\alpha -1}{\mathrm{e}}^{-\lambda \sigma }d\sigma =\left\{\begin{array}{cc}{\displaystyle \frac{\gamma (\alpha ,\lambda (\Psi \left(t\right)-\Psi \left(a\right)\left)\right)}{{\lambda }^{\alpha }\Gamma \left(\alpha \right)}},\hfill & \lambda >0,\hfill \\ {\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{\alpha }}{\Gamma (\alpha +1)}},\hfill & \lambda =0,\hfill \end{array}\right.$$

where $\gamma (·,·)$ is the lower incomplete gamma function [17] defined as

$$\gamma (\alpha ,z)={\int }_0^z{\sigma }^{\alpha -1}{\mathrm{e}}^{-\sigma }d\sigma .$$

Note that ${\phi }_{\alpha }$ is increasing and ${\phi }_{\alpha }\left(t\right)\to 0^{+}$ as $t\to a^{+}$ for any $\lambda \ge 0$. Using property (14), we obtain

$$\begin{array}{cc}\hfill \parallel I_a^{\xi ,\lambda ,\Psi }(f(t,x\left(t\right),y\left(t\right))-f(t,u\left(t\right),v\left(t\right)))\parallel & \le L{\phi }_{\xi }{\left(t\right)(\parallel x-u\parallel }_{C_{\theta }}{+\parallel y-v\parallel }_{C_{\theta }})\hfill \\ \hfill & \le L{\phi }_{\xi }(a+\theta ){\parallel (x,y)-(u,v)\parallel }_{C_{\theta }\times C_{\theta }}\hfill \end{array}$$

(16)

for any $\xi >0$ and $t\in I_{\theta }$. Therefore,

$${\parallel \mathcal{F}(x,y)-\mathcal{F}(u,v)\parallel }_{C_{\theta }\times C_{\theta }}\le L({\phi }_{\alpha }(a+\theta )+{\phi }_{\alpha -\beta }(a+\theta )){\parallel (x,y)-(u,v)\parallel }_{C_{\theta }\times C_{\theta }}.$$

Taking $\theta >0$ as sufficiently small yields

$$L({\phi }_{\alpha }(a+\theta )+{\phi }_{\alpha -\beta }(a+\theta ))<1.$$

The Banach fixed point theorem implies that there exists a unique point $(\widehat{x},\widehat{y})\in C_{\theta }\times C_{\theta }$ such that $(\widehat{x},\widehat{y})=\mathcal{F}(\widehat{x},\widehat{y})$. This means that Equations (13), (12) are solved by $(\widehat{x},\widehat{y})$, and, by Proposition 1, $\widehat{x}$ is a unique solution of (9), (10) on $I_{\theta }$. □

In the rest of Section 3, for short, we denote $C_{\theta }^{\zeta }=\underset{\underset{\zeta }{⏟}}{C_{\theta }\times \cdots \times C_{\theta }}$ for $\zeta \in \mathbb{N}$ with the norm ${\parallel x\parallel }_{C_{\theta }^{\zeta }}={\sum }_{i=1}^{\zeta }{\parallel x_i\parallel }_{C_{\theta }}$ for $x=(x_1,x_2,\dots ,x_{\zeta })\in C_{\theta }^{\zeta }$.

Next, we consider the IVP of a higher-order fractional derivative on the left side and the right side depending on multiple fractional derivatives of lower orders of the solution,

$$\begin{array}{cc}\hfill {{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)& =f(t,x\left(t\right),{{}^{C}D}_a^{{\beta }_1,\lambda ,\Psi }x\left(t\right),\dots ,{{}^{C}D}_a^{{\beta }_{\mathfrak{B}},\lambda ,\Psi }x\left(t\right)),t>a,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill x_{\lambda ,\Psi }^{\left[k\right]}\left(a\right)& =x_a^k,k=0,1,\dots ,n-1\hfill \end{array}$$

(18)

for some $x_a^k\in {\mathbb{R}}^N$, $k=0,1,\dots ,n-1$, where $n-1<\alpha <n\in \mathbb{N}$, $m_i-1<{\beta }_i<m_i\in \mathbb{N}$, ${\beta }_i<\alpha$ for each $i=1,2,\dots ,\mathfrak{B}$, $\lambda \ge 0$, $\Psi \in {\mathcal{P}}_{[a,\infty )}^n$, and $f\in C([a,\infty )\times {\mathbb{R}}^{(1+\mathfrak{B})N},{\mathbb{R}}^N)$. Function $x\in C^{n-1}[a,a+\theta )$ is a solution of IVP (17), (18) if it satisfies conditions analogous to those in Definition 4. So will it be in other IVPs considered below whenever $n-1<\alpha <n$. The next proposition can be proved exactly as Proposition 1.

Proposition 2.

Function x is a solution of (17), (18) if and only if there is $y=(y_1,y_2,\dots ,y_{\mathfrak{B}})$ such that $(x,y)$ fulfills the following system of integral equations

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill x\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}{x}_a^k\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f(s,x\left(s\right),y_1\left(s\right),\dots ,y_{\mathfrak{B}}\left(s\right))ds,\hfill \end{array}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill y_i\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=m_i}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{k-{\beta }_i}}{\Gamma (k+1-{\beta }_i)}}{x}_a^k\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\alpha -{\beta }_i)}}{\int }_a^t\kappa (\alpha -{\beta }_i;t,s)f(s,x\left(s\right),y_1\left(s\right),\dots ,y_{\mathfrak{B}}\left(s\right))ds,i=1,2,\dots ,\mathfrak{B}.\hfill \end{array}\hfill \end{array}$$

(20)

The existence result for a solution of (17), (18) follows.

Theorem 2.

Let $n\in \mathbb{N}$, $n-1<\alpha <n$, $m_i-1<{\beta }_i<m_i\in \mathbb{N}$, ${\beta }_i<\alpha$ for each $i=1,2,\dots ,\mathfrak{B}$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,\infty )}^n$. If there exists $L>0$ such that

$$\parallel f(t,x,y_1,\dots ,y_{\mathfrak{B}})-f(t,u,v_1,\dots ,v_{\mathfrak{B}})\parallel \le L\left(\parallel x-u\parallel +\sum _{i=1}^{\mathfrak{B}}\parallel y_i-v_i\parallel \right)$$

(21)

for all $t\ge a$, $x,y_i,u,v_i\in {\mathbb{R}}^N$ for each $i=1,2,\dots ,\mathfrak{B}$, then there exists $\theta >0$ such that there is a unique solution x of IVP (17), (18) on the interval $I_{\theta }=[a,a+\theta )$.

Proof. 

Let $C_{\theta }$ denote the Banach space of continuous functions defined on $I_{\theta }$, endowed with the supremum norm as in the proof of Theorem 1.

For short, denote $y=(y_1,y_2,\dots ,y_{\mathfrak{B}})$. Let us define an operator $\mathcal{F}=({\mathcal{F}}_0,{\mathcal{F}}_1,\dots ,{\mathcal{F}}_{\mathfrak{B}})$ on $C_{\theta }^{1+\mathfrak{B}}$ by

$${\mathcal{F}}_0(x,y)\left(t\right)={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}{x}_a^k+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f(s,x\left(s\right),y\left(s\right))ds$$

and

$$\begin{array}{cc}\hfill {\mathcal{F}}_i(x,y)\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=m_i}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{k-{\beta }_i}}{\Gamma (k+1-{\beta }_i)}}{x}_a^k\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\alpha -{\beta }_i)}}{\int }_a^t\kappa (\alpha -{\beta }_i;t,s)f(s,x\left(s\right),y\left(s\right))ds\hfill \end{array}$$

for $i=1,2,\dots ,\mathfrak{B}$. Clearly, $\mathcal{F}:C_{\theta }^{1+\mathfrak{B}}\to C_{\theta }^{1+\mathfrak{B}}$. From Lemma 1, we know even more, namely that $\mathcal{F}:C_{\theta }^{1+\mathfrak{B}}\to C^{n-1}\left(I_{\theta }\right)\times \tilde{Y}$ with $\tilde{Y}=C^{\lfloor \alpha -{\beta }_1\rfloor }\left(I_{\theta }\right)\times \cdots \times C^{\lfloor \alpha -{\beta }_{\mathfrak{B}}\rfloor }\left(I_{\theta }\right)$. Hence, if we show that $\mathcal{F}$ has a fixed point in $C_{\theta }^{1+\mathfrak{B}}$, then x is smooth enough, and, due to Proposition 2, it is the unique solution for the IVP.

For $(x,y),(u,v)\in C_{\theta }^{1+\mathfrak{B}}$, $v=(v_1,\dots ,v_{\mathfrak{B}})$, it holds that

$$\begin{array}{cc}\hfill {\parallel \mathcal{F}(x,y)-\mathcal{F}(u,v)\parallel }_{C_{\theta }^{1+\mathfrak{B}}}& =\parallel {\mathcal{F}}_0(x,y)-{\mathcal{F}}_0{(u,v)\parallel }_{C_{\theta }}+\sum _{i=1}^{\mathfrak{B}}{\parallel {\mathcal{F}}_i(x,y)-{\mathcal{F}}_i(u,v)\parallel }_{C_{\theta }}\hfill \\ \hfill & =\sum _{i=0}^{\mathfrak{B}}\underset{t\in I_{\theta }}{sup}\parallel I_a^{\alpha -{\beta }_i,\lambda ,\Psi }(f(t,x\left(t\right),y\left(t\right))-f(t,u\left(t\right),v\left(t\right)))\parallel ,\hfill \end{array}$$

where we set ${\beta }_0:=0$ for simplicity. Using estimation (16), we derive

$${\parallel \mathcal{F}(x,y)-\mathcal{F}(u,v)\parallel }_{C_{\theta }^{1+\mathfrak{B}}}\le \left(L\sum _{i=0}^{\mathfrak{B}}{\phi }_{\alpha -{\beta }_i}(a+\theta )\right){\parallel (x,y)-(u,v)\parallel }_{C_{\theta }^{1+\mathfrak{B}}}.$$

For sufficiently small $\theta >0$, the Banach fixed point theorem yields the existence of the unique fixed point of $\mathcal{F}$, which was to be proven. □

3.2. Right Side Depending on $x_{\lambda ,\Psi }^{\left[k\right]}\left(t\right)$

In this part, we consider the IVP

$$\begin{array}{cc}\hfill {{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)& =f\left(t,x\left(t\right),x_{\lambda ,\Psi }^{\left[k_1\right]}\left(t\right),\dots ,x_{\lambda ,\Psi }^{\left[k_{\mathfrak{K}}\right]}\left(t\right)\right),t>a,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill x_{\lambda ,\Psi }^{\left[k\right]}\left(a\right)& =x_a^k,k=0,1,\dots ,n-1\hfill \end{array}$$

(23)

for some $x_a^k\in {\mathbb{R}}^N$, $k=0,1,\dots ,n-1$, where $n-1<\alpha <n\in \mathbb{N}$, $k_1,k_2,\dots ,k_{\mathfrak{K}}\in {\mathbb{N}}_0$ are such that $k_1<k_2<\cdots <k_{\mathfrak{K}}<n$ for some $\mathfrak{K}\in \mathbb{N}$, $\lambda \ge 0$, $\Psi \in {\mathcal{P}}_{[a,\infty )}^n$, and $f\in C([a,\infty )\times {\mathbb{R}}^{(1+\mathfrak{K})N},{\mathbb{R}}^N)$.

First, we find an equivalent system of integral equations. To achieve this, denote $z_i\left(t\right)=x_{\lambda ,\Psi }^{\left[k_i\right]}\left(t\right)$ for $i=1,2,\dots ,\mathfrak{K}$ in the integral equation

$$\begin{array}{cc}\hfill x\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}{x}_a^k\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f\left(s,x\left(s\right),x_{\lambda ,\Psi }^{\left[k_1\right]}\left(s\right),\dots ,x_{\lambda ,\Psi }^{\left[k_{\mathfrak{K}}\right]}\left(s\right)\right)ds\hfill \end{array}$$

(24)

equivalent to system (22), (23) due to [11] [Theorem 1]. The values of $x_{\lambda ,\Psi }^{\left[k_i\right]}\left(t\right)$ are obtained from this equation by using the identity

$${\left({\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}{x}_a^k\right)}_{\lambda ,\Psi }^{\left[k_i\right]}=\sum _{k=k_i}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{k-k_i}}{(k-k_i)!}}{x}_a^k$$

and [11] [Lemma 1]. So, we obtain the next result.

Proposition 3.

Function x solves (22), (23) if and only if there is $z=(z_1,z_2,\dots ,z_{\mathfrak{K}})$ such that $(x,z)$ fulfills the following system of integral equations

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill x\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}{x}_a^k+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f\left(s,x\left(s\right),z_1\left(s\right),\dots ,z_{\mathfrak{K}}\left(s\right)\right)ds,\hfill \end{array}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill z_i\left(t\right)& =\sum _{k=k_i}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{k-k_i}}{(k-k_i)!}}{x}_a^k\hfill \\ \hfill & +{\displaystyle \frac{{\mathrm{e}}^{\lambda \Psi \left(t\right)}}{\Gamma (\alpha -k_i)}}{\int }_a^t\kappa (\alpha -k_i;t,s)f(s,x\left(s\right),z_1\left(s\right),\dots ,z_{\mathfrak{K}}\left(s\right))ds,i=1,2,\dots ,\mathfrak{K}.\hfill \end{array}\hfill \end{array}$$

(26)

Now, we are able to prove the following statement.

Theorem 3.

Let $n\in \mathbb{N}$, $n-1<\alpha <n$, $k_1,k_2,\dots ,k_{\mathfrak{K}}\in {\mathbb{N}}_0$ be such that $k_1<k_2<\cdots <k_{\mathfrak{K}}<n$ for some $\mathfrak{K}\in \mathbb{N}$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,\infty )}^n$. If there exists $L>0$ such that

$$\parallel f(t,x,z_1,\dots ,z_{\mathfrak{K}})-f(t,u,w_1,\dots ,w_{\mathfrak{K}})\parallel \le L\left(\parallel x-u\parallel +\sum _{i=1}^{\mathfrak{K}}\parallel z_i-w_i\parallel \right)$$

(27)

for all $t\ge a$, $x,z_i,u,w_i\in {\mathbb{R}}^N$ for each $i=1,2,\dots ,\mathfrak{K}$, then there exists $\theta >0$ such that there is a unique solution x for IVP (22), (23) on $I_{\theta }=[a,a+\theta )$.

Proof. 

The proof is similar to the proof of Theorem 2, so we omit some details. As usual, we consider the Banach space $C_{\theta }$. We define the operator $\mathcal{F}=({\mathcal{F}}_0,{\mathcal{F}}_1,\dots ,{\mathcal{F}}_{\mathfrak{K}})$ on $C_{\theta }^{1+\mathfrak{K}}$ so that ${\mathcal{F}}_0(x,z)\left(t\right)$ and ${\mathcal{F}}_i(x,z)\left(t\right)$ are provided by the right sides of (25) and (26), respectively. Using estimations (16) and (27), for $(x,z),(u,w)\in C_{\theta }^{1+\mathfrak{K}}$, we obtain

$$\begin{array}{cc}\hfill & {\parallel \mathcal{F}(x,z)-\mathcal{F}(u,w)\parallel }_{C_{\theta }^{1+\mathfrak{K}}}\hfill \\ \hfill & \le L\left({\phi }_{\alpha }(a+\theta )+{\mathrm{e}}^{\lambda \Psi (a+\theta )}\sum _{i=1}^{\mathfrak{K}}{\phi }_{\alpha -k_i}(a+\theta )\right){\parallel (x,z)-(u,w)\parallel }_{C_{\theta }^{1+\mathfrak{K}}}.\hfill \end{array}$$

The proof is then finished by the Banach fixed point theorem and Proposition 3. □

3.3. Right Side Depending on $x^{\left(k\right)}\left(t\right)$

Here, we consider the problem

$$\begin{array}{cc}\hfill {{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)& =f\left(t,x\left(t\right),x^{\left(k_1\right)}\left(t\right),\dots ,x^{\left(k_{\tilde{\mathfrak{K}}}\right)}\left(t\right)\right),t>a,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill x_{\lambda ,\Psi }^{\left[k\right]}\left(a\right)& =x_a^k,k=0,1,\dots ,n-1\hfill \end{array}$$

(29)

for some $x_a^k\in {\mathbb{R}}^N$, $k=0,1,\dots ,n-1$, where $n\in \mathbb{N}$, $n-1<\alpha <n$, $k_1,k_2,\dots ,k_{\tilde{\mathfrak{K}}}\in {\mathbb{N}}_0$ are such that $k_1<k_2<\cdots <k_{\tilde{\mathfrak{K}}}<n$ for some $\tilde{\mathfrak{K}}\in \mathbb{N}$, $\lambda \ge 0$, $\Psi \in {\mathcal{P}}_{[a,\infty )}^n$, and $f\in C([a,\infty )\times {\mathbb{R}}^{(1+\tilde{\mathfrak{K}})N},{\mathbb{R}}^N)$.

As before, we consider the corresponding integral equation

$$\begin{array}{cc}\hfill x\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}{x}_a^k\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f\left(s,x\left(s\right),x^{\left(k_1\right)}\left(s\right),\dots ,x^{\left(k_{\tilde{\mathfrak{K}}}\right)}\left(s\right)\right)ds\hfill \end{array}$$

(30)

and denote $z_i\left(t\right)=x^{\left(k_i\right)}\left(t\right)$ for $i=1,2,\dots ,\tilde{\mathfrak{K}}$. For the values of $z_i\left(t\right)$, we apply Lemma 4. So, we derive an equivalent system of integral equations:

Proposition 4.

Function x solves (28), (29) if and only if there is $z=(z_1,z_2,\dots ,z_{\tilde{\mathfrak{K}}})$ such that $(x,z)$ fulfills the following system

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill x\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}{x}_a^k\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f\left(s,x\left(s\right),z_1\left(s\right),\dots ,z_{\tilde{\mathfrak{K}}}\left(s\right)\right)ds,\hfill \end{array}\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill z_i\left(t\right)& =\sum _{j=0}^{k_i}{\beta }_j^{k_i}\left(t\right)x_{\lambda ,\Psi }^{\left[j\right]}\left(t\right)=\sum _{j=0}^{k_i}{\beta }_j^{k_i}\left(t\right)\left(\sum _{k=j}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{k-j}}{(k-j)!}}{x}_a^k\right.\hfill \\ \hfill & +\left.{\displaystyle \frac{{\mathrm{e}}^{\lambda \Psi \left(t\right)}}{\Gamma (\alpha -j)}}{\int }_a^t\kappa (\alpha -j;t,s)f(s,x\left(s\right),z_1\left(s\right),\dots ,z_{\tilde{\mathfrak{K}}}\left(s\right))ds\right),i=1,2,\dots ,\tilde{\mathfrak{K}}.\hfill \end{array}\hfill \end{array}$$

(32)

Theorem 4.

Let $n\in \mathbb{N}$, $n-1<\alpha <n$, $k_1,k_2,\dots ,k_{\tilde{\mathfrak{K}}}\in {\mathbb{N}}_0$ be such that $k_1<k_2<\cdots <k_{\tilde{\mathfrak{K}}}<n$ for some $\tilde{\mathfrak{K}}\in \mathbb{N}$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,\infty )}^n$. If there exists $L>0$ such that

$$\parallel f(t,x,z_1,\dots ,z_{\tilde{\mathfrak{K}}})-f(t,u,w_1,\dots ,w_{\tilde{\mathfrak{K}}})\parallel \le L\left(\parallel x-u\parallel +\sum _{i=1}^{\tilde{\mathfrak{K}}}\parallel z_i-w_i\parallel \right)$$

(33)

for all $t\ge a$, $x,z_i,u,w_i\in {\mathbb{R}}^N$ for each $i=1,2,\dots ,\tilde{\mathfrak{K}}$, then there exists $\theta >0$ such that there is a unique solution x for IVP (28), (29) on $I_{\theta }=[a,a+\theta )$.

Proof. 

Let us consider an operator $\mathcal{F}=({\mathcal{F}}_0,{\mathcal{F}}_1,\dots ,{\mathcal{F}}_{\tilde{\mathfrak{K}}})$ on $C_{\theta }^{1+\tilde{\mathfrak{K}}}$ such that ${\mathcal{F}}_0(x,z)\left(t\right)$ and ${\mathcal{F}}_i(x,z)\left(t\right)$ are defined by the right sides of (31) and (32), respectively. Then, using (16) and (33), for $(x,z),(u,w)\in C_{\theta }^{1+\tilde{\mathfrak{K}}}$, we derive

$$\begin{array}{cc}\hfill & {\parallel \mathcal{F}(x,z)-\mathcal{F}(u,w)\parallel }_{C_{\theta }^{1+\tilde{\mathfrak{K}}}}\hfill \\ \hfill & \le L\left({\phi }_{\alpha }(a+\theta )+{\mathrm{e}}^{\lambda \Psi (a+\theta )}\sum _{i=1}^{\tilde{\mathfrak{K}}}\sum _{j=0}^{k_i}{\overline{\beta }}_j^{k_i}{\phi }_{\alpha -j}(a+\theta )\right){\parallel (x,z)-(u,w)\parallel }_{C_{\theta }^{1+\tilde{\mathfrak{K}}}},\hfill \end{array}$$

where ${\overline{\beta }}_j^{k_i}={sup}_{t\in I_{\theta }}\left|{\beta }_j^{k_i}\left(t\right)\right|$. If $\theta >0$ is small enough, the statement follows from the Banach fixed point theorem and Proposition 4. □

3.4. Right Side Depending on $I_a^{\gamma ,\lambda ,\Psi }x\left(t\right)$

Finally, consider the IVP

$$\begin{array}{cc}\hfill {{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)& =f\left(t,x\left(t\right),I_a^{{\gamma }_1,\lambda ,\Psi }x\left(t\right),\dots ,I_a^{{\gamma }_{\mathfrak{C}},\lambda ,\Psi }x\left(t\right)\right),t>a,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill x_{\lambda ,\Psi }^{\left[k\right]}\left(a\right)& =x_a^k,k=0,1,\dots ,n-1\hfill \end{array}$$

(35)

for some $x_a^k\in {\mathbb{R}}^N$, $k=0,1,\dots ,n-1$, where $n\in \mathbb{N}$, $n-1<\alpha <n$, $0<{\gamma }_1<{\gamma }_2<\cdots <{\gamma }_{\mathfrak{C}}$ for some $\mathfrak{C}\in \mathbb{N}$ are real, $\lambda \ge 0$, $\Psi \in {\mathcal{P}}_{[a,\infty )}^n$, and $f\in C([a,\infty )\times {\mathbb{R}}^{(1+\mathfrak{C})N},{\mathbb{R}}^N)$.

By [11] [Theorem 1], system (34), (35) is equivalent to the integral equation

$$\begin{array}{cc}\hfill x\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}{x}_a^k\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f\left(s,x\left(s\right),I_a^{{\gamma }_1,\lambda ,\Psi }x\left(s\right),\dots ,I_a^{{\gamma }_{\mathfrak{C}},\lambda ,\Psi }x\left(s\right)\right)ds.\hfill \end{array}$$

(36)

An equivalent system of fractional integral equations is derived in the next statement.

Proposition 5.

Function x solves (34), (35) if and only if there is $p=(p_1,p_2,\dots ,p_{\mathfrak{C}})$ such that $(x,p)$ fulfills the following system of integral equations

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill x\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^k}{k!}}{x}_a^k\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t\kappa (\alpha ;t,s)f(s,x\left(s\right),p_1\left(s\right),\dots ,p_{\mathfrak{C}}\left(s\right))ds,\hfill \end{array}\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill p_i\left(t\right)& ={\mathrm{e}}^{-\lambda \Psi \left(t\right)}\sum _{k=0}^{n-1}{\displaystyle \frac{{(\Psi \left(t\right)-\Psi \left(a\right))}^{k+{\gamma }_i}}{\Gamma (k+{\gamma }_i+1)}}{x}_a^k\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\alpha +{\gamma }_i)}}{\int }_a^t\kappa (\alpha +{\gamma }_i;t,s)f(s,x\left(s\right),p_1\left(s\right),\dots ,p_{\mathfrak{C}}\left(s\right))ds,i=1,2,\dots ,\mathfrak{C}.\hfill \end{array}\hfill \end{array}$$

(38)

Proof. 

Let us denote $p_i\left(t\right)=I_a^{{\gamma }_i,\lambda ,\Psi }x\left(t\right)$, $i=1,2,\dots ,\mathfrak{C}$ in (36). These values are obtained by applying the operator $I_a^{{\gamma }_i,\lambda ,\Psi }$ to this equation and using Lemma 3 and [11] [Lemma 1]. □

An existence result follows.

Theorem 5.

Let $n\in \mathbb{N}$, $n-1<\alpha <n$, $0<{\gamma }_1<{\gamma }_2<\cdots <{\gamma }_{\mathfrak{C}}$ for some $\mathfrak{C}\in \mathbb{N}$ are real, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[a,\infty )}^n$. If there exists $L>0$ such that

$$\parallel f(t,x,p_1,\dots ,p_{\mathfrak{C}})-f(t,u,q_1,\dots ,q_{\mathfrak{C}})\parallel \le L\left(\parallel x-u\parallel +\sum _{i=1}^{\mathfrak{C}}\parallel p_i-q_i\parallel \right)$$

(39)

for all $t\ge a$, $x,p_i,u,q_i\in {\mathbb{R}}^N$ for each $i=1,2,\dots ,\mathfrak{C}$, then there exists $\theta >0$ such that there is a unique solution x for IVP (34), (35) on the interval $I_{\theta }=[a,a+\theta )$.

Proof. 

For the operator $\mathcal{F}=({\mathcal{F}}_0,{\mathcal{F}}_1,\dots ,{\mathcal{F}}_{\mathfrak{C}})$ on $C_{\theta }^{1+\mathfrak{C}}$ with ${\mathcal{F}}_0(x,p)\left(t\right)$ and ${\mathcal{F}}_i(x,p)\left(t\right)$ defined by the right-hand sides of (37) and (38), respectively, using (16), (39), for $(x,p),(u,q)\in C_{\theta }^{1+\mathfrak{C}}$, we derive

$${\parallel \mathcal{F}(x,p)-\mathcal{F}(u,q)\parallel }_{C_{\theta }^{1+\mathfrak{C}}}\le L\left({\phi }_{\alpha }(a+\theta )+\sum _{i=1}^{\mathfrak{C}}{\phi }_{\alpha +{\gamma }_i}(a+\theta )\right){\parallel (x,p)-(u,q)\parallel }_{C_{\theta }^{1+\mathfrak{C}}}.$$

If $\theta >0$ is small enough, the Banach fixed point theorem and Proposition 5 yield the result. □

3.5. General Right Side

Combining Theorems 1–5, we immediately obtain an existence and uniqueness result for a more general IVP. Here, we omit the proof since it can be easily derived from the proofs of the mentioned theorems.

Corollary 1.

Let $n\in \mathbb{N}$, $n-1<\alpha <n$, $\mathfrak{B},\mathfrak{K},\tilde{\mathfrak{K}},\mathfrak{C}\in {\mathbb{N}}_0$, $\lambda \ge 0$, $\Psi \in {\mathcal{P}}_{[a,\infty )}^n$, and

1. 

$m_i-1<{\beta }_i<m_i\in \mathbb{N}$, ${\beta }_i<\alpha$ for each $i=1,2,\dots ,\mathfrak{B}$,

2. 

$k_1,k_2,\dots ,k_{\mathfrak{K}}\in {\mathbb{N}}_0$ be such that $k_1<k_2<\cdots <k_{\mathfrak{K}}<n$,

3. 

${\tilde{k}}_1,{\tilde{k}}_2,\dots ,{\tilde{k}}_{\tilde{\mathfrak{K}}}\in {\mathbb{N}}_0$ be such that ${\tilde{k}}_1<{\tilde{k}}_2<\cdots <{\tilde{k}}_{\tilde{\mathfrak{K}}}<n$,

4. 

$0<{\gamma }_1<{\gamma }_2<\cdots <{\gamma }_{\mathfrak{C}}$ for some $\mathfrak{C}\in \mathbb{N}$ are real.

If there exists $L>0$ such that

$$\begin{array}{cc}\hfill & \parallel f(t,x,y,z,\tilde{z},p)-f(t,u,v,w,\tilde{w},q)\parallel \hfill \\ \hfill & \le L\left(\parallel x-u\parallel +\sum _{i=1}^{\mathfrak{B}}\parallel y_i-v_i\parallel +\sum _{i=1}^{\mathfrak{K}}\parallel z_i-w_i\parallel +\sum _{i=1}^{\tilde{\mathfrak{K}}}\parallel {\tilde{z}}_i-{\tilde{w}}_i\parallel +\sum _{i=1}^{\mathfrak{C}}\parallel p_i-q_i\parallel \right)\hfill \end{array}$$

for all $t\ge a$, $x,u\in {\mathbb{R}}^N$, $y=(y_1,\dots ,y_{\mathfrak{B}})\in {\mathbb{R}}^{\mathfrak{B}N}$, $v=(v_1,\dots ,v_{\mathfrak{B}})\in {\mathbb{R}}^{\mathfrak{B}N}$, $z=(z_1,\dots ,z_{\mathfrak{K}})\in {\mathbb{R}}^{\mathfrak{K}N}$, $w=(w_1,\dots ,w_{\mathfrak{K}})\in {\mathbb{R}}^{\mathfrak{K}N}$, $\tilde{z}=({\tilde{z}}_1,\dots ,{\tilde{z}}_{\tilde{\mathfrak{K}}})\in {\mathbb{R}}^{\tilde{\mathfrak{K}}N}$, $\tilde{w}=({\tilde{w}}_1,\dots ,{\tilde{w}}_{\tilde{\mathfrak{K}}})\in {\mathbb{R}}^{\tilde{\mathfrak{K}}N}$, $p=(p_1,\dots ,p_{\mathfrak{C}})\in {\mathbb{R}}^{\mathfrak{C}N}$, $q=(q_1,\dots ,q_{\mathfrak{C}})\in {\mathbb{R}}^{\mathfrak{C}N}$, then there exists $\theta >0$ such that there is a unique solution x for the IVP

$$\begin{array}{cc}\hfill {{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)& =f\left(t,x\left(t\right),{{}^{C}D}_a^{{\beta }_1,\lambda ,\Psi }x\left(t\right),\dots ,{{}^{C}D}_a^{{\beta }_{\mathfrak{B}},\lambda ,\Psi }x\left(t\right),\right.\hfill \\ \hfill & x_{\lambda ,\Psi }^{\left[k_1\right]}\left(t\right),\dots ,x_{\lambda ,\Psi }^{\left[k_{\mathfrak{K}}\right]}\left(t\right),x^{\left({\tilde{k}}_1\right)}\left(t\right),\dots ,x^{\left({\tilde{k}}_{\tilde{\mathfrak{K}}}\right)}\left(t\right),\hfill \\ \hfill & \left.I_a^{{\gamma }_1,\lambda ,\Psi }x\left(t\right),\dots ,I_a^{{\gamma }_{\mathfrak{C}},\lambda ,\Psi }x\left(t\right)\right),t>a,\hfill \\ \hfill x_{\lambda ,\Psi }^{\left[k\right]}\left(a\right)& =x_a^k,k=0,1,\dots ,n-1\hfill \end{array}$$

on the interval $I_{\theta }=[a,a+\theta )$.

4. Examples

Here, we present several examples to illustrate the use of the main results. We start with an easy observation.

Remark 2.

Theorems 1–5 as well as Corollary 1 remain valid if the corresponding Lipschitz conditions are satisfied only for all $t\in [a,a+\Theta ]$ with some $\Theta >0$. In this case, $\theta \le \Theta$ in the statements.

Example 1.

Let us consider the IVP

$$\begin{array}{cc}\hfill {{}^{C}D}_a^{\alpha ,\lambda ,\Psi }x\left(t\right)& =x\left(t\right),t>a\ge 1,\hfill \\ \hfill x_{\lambda ,\Psi }^{\left[0\right]}\left(a\right)& =x_a^0\hfill \end{array}$$

for some $x_a^0\in \mathbb{R}$, $0<\alpha <1$, $\lambda >0$, $\Psi \left(t\right)=\sqrt{t}$.

Clearly, the right-hand side satisfies (14) with $L=1$. Hence, Theorem 1 yields the existence of a unique solution on $I_{\theta }$ for some $\theta >0$. Note that the same result follows by [11] [Theorem 3 and Remark 1] when Remark 1 is applied. However, Theorem 1 can be applied for any $\Psi \in {\mathcal{P}}_{[a,\infty )}^1$, and $\lambda \ge 0$.

Example 2.

Let us consider the IVP for a generalized version of a fractional harmonic vibration equation [18] with viscoelastic damping,

$$\begin{array}{cc}\hfill M{{}^{C}D}_0^{\alpha ,\lambda ,\Psi }x\left(t\right)+C{{}^{C}D}_0^{\beta ,\lambda ,\Psi }x\left(t\right)+Kx\left(t\right)& =1,t>0,\hfill \\ \hfill x\left(0\right)& =c_0,\hfill \\ \hfill x^{\prime }\left(0\right)& =c_1\hfill \end{array}$$

for some $c_0,c_1\in \mathbb{R}$ and positive real constants M, C, K, where $1<\beta <\alpha <2$, $\lambda \ge 0$, and $\Psi \in {\mathcal{P}}_{[0,\infty )}^2$.

First, note that this system can be written in the form of (17), (18) as

$$\begin{array}{cc}\hfill {{}^{C}D}_0^{\alpha ,\lambda ,\Psi }x\left(t\right)& ={\displaystyle \frac{1}{M}}-{\displaystyle \frac{C}{M}}{{}^{C}D}_0^{\beta ,\lambda ,\Psi }x\left(t\right)-{\displaystyle \frac{K}{M}}x\left(t\right),t>0,\hfill \\ \hfill x_{\lambda ,\Psi }^{\left[0\right]}\left(0\right)& ={\mathrm{e}}^{\lambda \Psi \left(0\right)}{c}_0,\hfill \\ \hfill x_{\lambda ,\Psi }^{\left[1\right]}\left(0\right)& ={\mathrm{e}}^{\lambda \Psi \left(0\right)}\left(\lambda c_0+{\displaystyle \frac{c_1}{{\Psi }^{\prime }\left(0\right)}}\right).\hfill \end{array}$$

Then, condition (21) holds with $L=max\{C/M,K/M\}$, and the existence of a unique solution follows from Theorem 2.

Example 3.

Let us consider the following IVP

$$\begin{array}{cc}\hfill {{}^{C}D}_0^{4.2,\lambda ,\Psi }x\left(t\right)& =(1-sint){{}^{C}D}_0^{\pi ,\lambda ,\Psi }x\left(t\right)-t^{3/2}{x}^{\prime }\left(t\right),t>0,\hfill \\ \hfill x_{\lambda ,\Psi }^{\left[k\right]}\left(0\right)& =x_0^k,k=0,1,\dots ,4\hfill \end{array}$$

for some $x_0^k\in \mathbb{R}$, $k=0,1,\dots ,4$, where $\lambda \ge 0$ and $\Psi \in {\mathcal{P}}_{[0,\infty )}^5$.

Denoting $y={{}^{C}D}_0^{\pi ,\lambda ,\Psi }x$, $z=x^{\prime }$, for $f(t,x,y,z)=(1-sint)y-t^{3/2}z$, we have

$$\left|f\right(t,x,y,z)-f(t,u,v,w\left)\right|\le |y-v|+|z-w|,\forall t\in [0,1].$$

The existence of a unique solution on $I_{\theta }$ for some $0<\theta \le 1$ follows from Corollary 1 and Remark 2.

5. Conclusions and Discussion

In the present paper, IVPs for fractional differential equations with T$\Psi$CFDs and right sides depending on time, the solution itself, its integer- or non-integer-order derivatives, or its fractional integrals have been considered. The local existence of a unique solution has been proved for each problem, assuming that the corresponding right side satisfies a Lipschitz condition. Using an equivalent system of integral equations, it was possible to formulate the problem of the existence as a fixed point problem and thus to extend the results of [11] to more general right sides. At once, this is a generalization of other known results, such as [4], to equations with tempered $\Psi$–Caputo derivatives.

Conversely, further research is required to weaken the global Lipschitz continuity assumption to derive results on the existence of solutions of IVPs with more general right sides, e.g., ${\left({{}^{C}D}_a^{\beta ,\lambda ,\Psi }x\left(t\right)\right)}^2$. The study of the existence of global solutions is another possible avenue of research. Moreover, for the purpose of this paper, it was sufficient to apply the fractional derivatives and fractional integrals to continuous or even smoother functions. It would be interesting to consider a more general class of functions, such as $L^1$ functions, or investigate the Hölder continuity of $I_a^{\alpha ,\lambda ,\Psi }$ using Lemma 1 (see, e.g., [19]).