On the Existence and Stability of Variable Order Caputo Type Fractional Differential Equations

Abstract

In the theory of differential equations, the study of existence and the uniqueness of the solutions are important. In the last few decades, many researchers have had a keen interest in finding the existence–uniqueness solution of constant fractional differential equations, but literature focusing on variable order is limited. In this article, we consider a Caputo type variable order fractional differential equation. First, we present the existence–uniqueness of a solution of the considered problem. Secondly, By borrowing the idea from the theory of ordinary differential equations, we extend the continuation theorem for the variable order fractional differential equation. Further, we prove the global existence results. Finally, we present different types of Ulam–Hyers stability results, which have never been studied before for the Caputo type variable order fractional differential equation.

1. Introduction

Over the past few years, the study of fractional calculus [1,2,3,4,5] has broadened, because of its applications in nearly all disciplines of science and engineering. Compared with derivatives of integer orders, non derivatives of integer orders are weakly singular and non-local. As the next stage of the dynamical system not only depends on its present stage, but also on all of its ongoing stages, the theoretical research is more complicated. In 1993, reference [6] first introduced the concept of variable order fractional calculus. Recently, some researchers have extended fractional calculus from a constant fractional order to variable fractional order. As far as we know, variable-order fractional operators rely on their non-stationary power-law kernels, which more accurately describe the memory and genetic properties of many complex physical phenomena and processes [7]. For this reason, variable-order fractional differential equations have received more attention due to their suitability for modeling and a large number of phenomena covering many fields of science and engineering [8,9,10,11].

The existence–uniqueness of the solution of fractional differential equations (FDEs) is a interesting research area [12,13,14,15]. There is some literature on the existence of a solution of variable order FDEs, and the results are interesting. In [16], the author studied the existence of generalized FDE solutions with non-autonomous variable orders. Yufeng et al. [17] discussed the existence and uniqueness of variable order FDEs by considering the iterative series with the contraction mapping principle. Jiang et al. [18] provide the existence of the solution of a variable order fractional differential equation with two point boundary values. A tempered variable order FDE was studied in [19] for the Mittag–Leffler stability. We can find more interesting results on variable order FDEs in [20,21,22].

Researchers also have a keen interest in investigating the stability analysis of fractional-order problems of science and engineering. In the literature, we can find many ways to approach the stability analysis. Some researchers [23,24] have studied the local stability and Mittag–Leffler stability for constant order FDEs; to the best of our knowledge, there are limited works on the Ulam stability for constant order FDEs, but there is no work for variable order FDEs. The Ulam–Hyers (UH) stability is simple and easy way to investigate fractional differential systems. The history of UH stability goes back to the middle of the 19th century. In 1940, Ulam [25,26] raised a question in a seminar held at Wisconsin University—“Under what conditions does there exist an additive mapping near an approximately additive mapping?” In 1941, Hyers [27] obtained an interesting solution to Ulam’s question, by considering the Banach spaces. Therefore, this type of stability is called the Ulam–Hyers stability. In 1978, Rassias further explored the UH stability for linear and nonlinear mappings. Many researchers then generalized these findings in a variety of areas.

Inspired by the above work, we first established a result about the uniqueness of local existence, and then we extended the continuity theorem of ordinary differential equations (ODE) to the continuity theorem of the Caputo type variable order fractional differential equation (VOFDE). In addition, we propose the global existence of (1) solutions. To the best of our knowledge, the continuation theorem, global existence, and the Ulam–Hyers type stability of (1) have not previously been studied.

Now, consider the Caputo type variable order fractional initial value problem as

$$\left\{\begin{array}{ccc}{}_C{\mathrm{D}}_{0,t}^{\alpha \left(t\right)}u\left(t\right)\hfill & =f(t,u),\hfill & 0<\alpha \left(t\right)<1,\hfill \\ {u\left(t\right)|}_{t=0}\hfill & =u_0,\hfill & u\in \mathbb{R},t\in (0,+\infty ),\hfill \end{array}\right.$$

(1)

where ${}_C{\mathrm{D}}^{\alpha \left(t\right)}(·)$ is the Caputo derivative with the variable order $\alpha \left(t\right)$ defined in (3).

The rest of this work is presented as follows: Section 2 provides some definitions and lemmas for the variable fractional calculus. Section 3 addresses the existence–uniqueness and the continuation theorem of variable-order FDE. The global solutions of variable-order FDEs are investigated in Section 4. Ulam–Hyers stability is discussed in Section 5. The conclusioniis presented at the end.

2. Preliminaries

In this paper, our main focus is on the variable order Caputo derivative. We obtained the fractional derivative and integral with the variable-order by extending the fractional derivative and integral of the constant order [1,4,28,29].

Definition 1.

Reference [6] The variable order Riemann–Liouvilleiintegral of function $f\left(u\right)$ is

$${}_{RL}{\mathrm{D}}_{0,t}^{-\alpha \left(t\right)}f\left(u\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(u-\lambda )}^{\alpha \left(t\right)-1}f\left(\lambda \right)\mathrm{d}\lambda ,t>0,\alpha \left(t\right)>0.$$

(2)

Definition 2.

Reference [6] The variable order Riemann–Liouville derivative of function $f\left(u\right)$ is defined as

$${}_{RL}{\mathrm{D}}_{0,t}^{\alpha \left(t\right)}f\left(u\right)=\frac{1}{\mathrm{\Gamma }(n-\alpha (t\left)\right)}\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}{\int }_0^t{(u-\lambda )}^{n-\alpha \left(t\right)-1}f\left(\lambda \right)\mathrm{d}\lambda ,$$

(3)

where $t,\alpha \left(t\right)>0$.

Definition 3.

Reference [6] The variable order Caputo derivative of $f\left(u\right)$ with order is

$${}_C{\mathrm{D}}_{0,t}^{\alpha \left(t\right)}f\left(u\right)=\frac{1}{\mathrm{\Gamma }(n-\alpha (t\left)\right)}{\int }_0^t{(u-\lambda )}^{\alpha \left(t\right)-1}{f}^{\left(n\right)}\left(\lambda \right)\mathrm{d}\lambda ,t>0,\alpha \left(t\right)>0.$$

(4)

Definition 4.

Reference [6] The definitions of variable order derivatives 2 and 3 are not often equivalent; however, they can be linked by the following relationship [7]

$${}_{RL}{\mathrm{D}}_{0,t}^{\alpha \left(t\right)}f\left(t\right)=\sum _{k=0}^{n-1}\frac{f^{\left(k\right)}\left(0\right)t^{k-\alpha \left(t\right)}}{\mathrm{\Gamma }(1+k-\alpha (t\left)\right)}={}_C{\mathrm{D}}_{0,t}^{\alpha \left(t\right)}f\left(t\right).$$

(5)

When $0<\alpha \left(t\right)<1$, then the relation between variable order derivatives 2 and 3 can be defined as

$${}_C{\mathrm{D}}_{0,t}^{\alpha \left(t\right)}f\left(t\right)={}_{RL}{\mathrm{D}}_{0,t}^{\alpha \left(t\right)}\left[f\left(t\right)-f\left(0\right)\right].$$

(6)

Lemma 1.

References [2,3,6]. We assume that $f(x,t)$ is a continuous function. Then the second kind of nonlinear Volterra integral equation is equivalent to variable order initial problem (1) as

$$u\left(t\right)=u_0+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda .$$

(7)

However, every solution of Equation (7) is also the solution of problem (1) and vice versa.

Proof. 

By applying the operator ${}_{RL}{\mathrm{D}}_{0,t}^{-\alpha \left(t\right)}$ to both sides of (1), and using the relation (6) and the initial conditions ${u\left(t\right)|}_{t=0}=u_0$, we can reduce the problem (1) into the equivalent Volterra nonlinear integral Equation (7). The proof is complete. □

Lemma 2.

References [2,3,6]. Suppose that $S\subseteq C[0,T]$. Then S is called pre-compact if $\left\{u\right(t):u\in M\}$ is uniformly bounded and equicontinuous on $[0,T]$.

Lemma 3.

References [2,3,6]. Suppose that X is a Banach space, and $S\subset X$, where S is the closed bounded convex set, and assume that $T:S\to S$ is the continuous completely. Then there exists a fixed point of T in S.

Lemma 4.

References [2,3,6]. Suppose that a non-empty closed set S is a subset of a Banach space X, and assume that $a_n\ge 0,$ then ${\displaystyle \sum _{n=0}^{\infty }}{a}_n$ converges, $\forall n\in \mathbb{N}$. Furthermore assume, $A:S⟶S$ satisfies the

$$∥P^n{u}_1-P^n{u}_2∥\le a_n∥u_1-u_2∥,u_1,u_2\in S.$$

Then, for any $u_1^{*}\in S$ is defined the unique fixed point of P.

3. Existence, Uniqueness, and Continuation Theorems

First, we prove the local existence uniqueness of the solution of problem (1). For this, we make the following hypothesis.

Hypothesis 1 (H1).

Suppose that$f(t,u):[0,+\infty )\times \mathbb{R}\to \mathbb{R}$in (1) be a continuous function. The function f fulfills the Lipschitz condition, i.e., $\left|f(t,u_1)-f(t,u_2)\right|\le L\left|u_1-u_2\right|$, where $L>0$.

Hypothesis 2 (H2).

Assume that$f(t,u)$have weak singularity, with respect totthen ∃ a constant $\eta \in \left(0.1\right]$ such that $(\Im u)\left(t\right)=t^{\eta }f(t,u)$ is a continuous bounded map defined on $[0,T]\times [0,T],andT>0$.

Theorem 1.

We assume that conditions (H1) and (H2) are holds. Then the problem (1) has at least one solution and $u\in C[0,h^{*}],forsome{h}^{*}\in (0,T]$.

Proof. 

Let

$$\Omega =\{u\in C[0,T]:\parallel u-u_0{\parallel }_{C[0,T]}=\underset{t\in [0,T]}{sup}|u-u_0|\le \varphi \},$$

where $\varphi >0$. Because ℑ is bounded, so a constant $N>0$ exits, such that

$$sup\left\{\right|(\Im u)\left(t\right)|:t\in [0,T],u\in \mathrm{\Omega }\}\le N.$$

Again, we let

$${\mathrm{\Lambda }}_{h^{*}}=\left\{u:u\in C[0,h^{*}],\underset{t\in [0,T]}{sup}|u-u_0|\le \varphi \right\},$$

where $h^{*}=min\{{\left(\frac{\varphi \mathrm{\Gamma }\left(\alpha \right(t)+1-\eta )}{N\mathrm{\Gamma }(1-\eta )}\right)}^{\frac{1}{\alpha \left(t\right)-\eta }},T\},$$\alpha \left(t\right)>\eta$.

It is obvious, ${\mathrm{\Lambda }}_{h^{*}}\subseteq C[0,h^{*}]$ is bounded closed, nonempty, and a convex subset. We can see that $h^{*}\le T$, now let us define ℵ as

$$(\aleph u)\left(t\right)=u_0+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda ,t\in [0,h^{*}].$$

(8)

By using (8), we have

$$|(\aleph u)\left(t\right)-u_0|\le \frac{N}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}{\lambda }^{-\eta }\mathrm{d}\lambda \le \frac{N\mathrm{\Gamma }(1-\eta )}{\mathrm{\Gamma }\left(\alpha \right(t)+1-\eta )}{h}^{\alpha \left(t\right)-\eta }\le \varphi ,$$

for any $u\in C[0,h^{*}]$, which shows that $\aleph {\mathrm{\Lambda }}_h\subset {\mathrm{\Lambda }}_h$.

Next, we prove the continuity of operator ℵ. Let $u_n,u\in {\mathrm{\Lambda }}_{h^{*}}$, such that $\parallel u_n{-u\parallel }_{C[0,h^{*}]}$ approaches to 0 as n approaches to ∞. Because the operator ℑ is continuous so $\parallel \Im u_n{-\Im u\parallel }_{[0,h^{*}]}$ approaches 0 as n approaches to ∞. Now

$$\begin{array}{cc}\hfill |(\aleph u_n)\left(t\right)-(\aleph u)\left(t\right)|& =|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u_n\left(\lambda \right))\mathrm{d}\lambda \hfill \\ \hfill & -\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda |\hfill \\ \hfill & \le \frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}|f(\lambda ,u_n\left(\lambda \right))-f(\lambda ,u\left(\lambda \right))|\mathrm{d}\lambda \hfill \\ \hfill & \le \frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}{\lambda }^{-\eta }|(\Im u_n)\left(\lambda \right)-(\Im u)\left(\lambda \right)|\mathrm{d}\lambda \hfill \\ \hfill & \le \frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}{\lambda }^{-\eta }\mathrm{d}\lambda {\parallel (\Im u_n)\left(\lambda \right)-(\Im u)\left(\lambda \right)\parallel }_{[0,h^{*}]}.\hfill \end{array}$$

We have

$$\parallel (\aleph u_n)\left(\lambda \right)-(\aleph u)\left(\lambda \right){\parallel }_{[0,h^{*}]}\le \frac{\mathrm{\Gamma }(1-\alpha (t\left)\right)}{\mathrm{\Gamma }\left(\alpha \right(t)+1-\eta )}{h}^{\alpha \left(t\right)-\eta }{\parallel (\Im u_n)\left(\lambda \right)-(\Im u)\left(\lambda \right)\parallel }_{[0,h^{*}]}.$$

Then $\parallel (\aleph u_n)\left(\lambda \right)-(\aleph u)\left(\lambda \right){\parallel }_{[0,h^{*}]}$ approaches to 0 as n approaches to ∞. Thus, ℵ is continuous.

Next, we prove the continuity of the $\aleph {\mathrm{\Lambda }}_{h^{*}}$. For this, we let $u\in {\mathrm{\Lambda }}_{h^{*}}$ and $t_1,t_2\in [0,h^{*}]$, and $t_1\le t_2$. Note that, for any $ϵ>0$,

$$\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}{\lambda }^{-\eta }\mathrm{d}\lambda =\frac{\mathrm{\Gamma }(1-\eta )}{\mathrm{\Gamma }\left(\alpha \right(t)+1-\lambda )}{t}^{\alpha \left(t\right)-\eta }\to 0,\mathrm{as}t\to 0^{+},$$

where $\eta \in [0,1)$. Then there exists a $\tilde{\eta }>0$,

$$\frac{2N}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}{\lambda }^{-\eta }\mathrm{d}\lambda <ϵ,t\in [0,h^{*}],$$

holds. In this case, for $t_1,t_2\in [0,\tilde{\eta }]$, one has

$$\begin{array}{cc}\hfill & |\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_1}{(t_1-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda -\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda |\hfill \\ \hfill & \le \frac{N}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_1}{(t_1-\lambda )}^{\alpha \left(t\right)-1}{\lambda }^{-\eta }\mathrm{d}\lambda +\frac{N}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}{\lambda }^{-\eta }\mathrm{d}\lambda \hfill \\ \hfill & <ϵ.\hfill \end{array}$$

(9)

One can get for $\frac{\tilde{\eta }}{2}\le t_1\le t_2\le h^{*}$

$$\begin{array}{cc}\hfill |(\aleph u)\left(t_1\right)-(\aleph u)\left(t_2\right)|& =|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_1}{(t_1-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda \hfill \\ \hfill & -\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda |\hfill \\ \hfill & \le \left|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_1}[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}]f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda \right|\hfill \\ \hfill & +\left|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{t_1}^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda \right|.\hfill \end{array}$$

(10)

In (10), on the right hand side, the first term can be written as

$$\begin{array}{cc}\hfill & \left|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_1}[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}]f(\lambda ,x\left(\lambda \right))\mathrm{d}\lambda \right|\hfill \\ \hfill & \le \frac{N}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_1}\left|[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}]{\lambda }^{-\eta }\right|\mathrm{d}\lambda \hfill \\ \hfill & \le \frac{N}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{\tilde{\eta }/2}\left|[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}]{\lambda }^{-\eta }\right|\mathrm{d}\lambda \hfill \\ \hfill & +\frac{N{\left(\frac{\tilde{\eta }}{2}\right)}^{-\eta }}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{\frac{\tilde{\eta }}{2}}^{t_1}\left|[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}]\right|\mathrm{d}\lambda \hfill \\ \hfill & \le \frac{2N}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{\frac{{\eta }_1}{2}}{\left(\frac{\tilde{\delta }}{2}-\lambda \right)}^{\alpha \left(t\right)-1}{\lambda }^{-\eta }\mathrm{d}\lambda +\frac{N{\left(\frac{\tilde{\eta }}{2}\right)}^{-\eta }}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}\left[{(t_2-t_1)}^{\alpha \left(t\right)}\right.\hfill \\ \hfill & \left.+{\left(t_1-\frac{\tilde{\eta }}{2}\right)}^{\alpha \left(t\right)}-{\left(t_2-\frac{\tilde{\eta }}{2}\right)}^{\alpha \left(t\right)}\right]\hfill \\ \hfill & \le ϵ+\frac{N{\left(\frac{\tilde{\eta }}{2}\right)}^{-\eta }}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}\left[{(t_2-t_1)}^{\alpha \left(t\right)}+{\left(t_1-\frac{\tilde{\eta }}{2}\right)}^{\alpha \left(t\right)}-{\left(t_2-\frac{\tilde{\eta }}{2}\right)}^{\alpha \left(t\right)}\right].\hfill \end{array}$$

In (10), on the right hand side, the second term can be written as

$$\begin{array}{cc}\hfill \left|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{t_1}^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda \right|& \le \frac{N{\left(\frac{{\eta }_1}{2}\right)}^{-\eta }}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{t_1}^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}\mathrm{d}\lambda \hfill \\ \hfill & \le \frac{N{\left(\frac{{\eta }_1}{2}\right)}^{-\eta }}{\mathrm{\Gamma }\left(\alpha \right(t)+1)}{(t_2-t_1)}^{\alpha \left(t\right)}.\hfill \end{array}$$

(11)

From the above discussion, there exists a $(\frac{\tilde{\eta }}{2}>)\widetilde{{\eta }_1}>0$ for $\frac{\tilde{\eta }}{2}\le t_1\le t_2\le h^{*}$ and $|t_1-t_2|<\widetilde{{\eta }_1}$,

$$|(\aleph u)\left(t_1\right)-(\aleph u)\left(t_2\right)|<2ϵ.$$

(12)

By using (9) and (12), it is clear that $\{(\aleph u)\left(t\right):u\in {\mathrm{\Lambda }}_{h^{*}}\}$ is equicontinuous. Someone can easily find that $\{(\aleph u)\left(t\right):u\in {\mathrm{\Lambda }}_{h^{*}}\}$ is uniformly bounded because of $\aleph {\mathrm{\Lambda }}_{h^{*}}\subset {\mathrm{\Lambda }}_{h^{*}}$. So, $\aleph {\mathrm{\Lambda }}_{h^{*}}$ is pre-compact, and the operator ℵ is continuous completely. By using Lemma 3 and Lemma 2, it proves the local existence of the problem (1). □

Theorem 2.

Let us assume that conditions (H1) and (H2) are fulfilled. Then there exists the unique solution of IVP (1) for $u\in C[0,h^{*}]$, where $h^{*}\in [0,T]$.

Proof. 

By using Lemma 1, the problem (1) and Equation (7) are equivalent. So we have to prove only that the problem (7) has one solution only. First, we have a non-empty and closed subset of the Banach space in the form

$${\mathrm{\Lambda }}_{{}^{*}h}=\left\{u:u\in C[0,h^{*}],\underset{t\in [0,T]}{sup}|u-u_0|\le \varphi \right\},$$

Again, we introduce the operator ℵ as

$$(\aleph u)\left(t\right)=u_0+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda ,t\in [0,h^{*}].$$

Now, we obtain the fixed point problem from the uniqueness of the solution to the integral Equation (7), i.e., $u\left(t\right)=(\aleph u)\left(t\right)$. So, we only prove that ℵ has a unique fixed point.

We have

$$\begin{array}{cc}\hfill |(\aleph u)\left(t\right)-u_0|& \le \frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}\left|f(\lambda ,u(\lambda \left)\right)\right|\mathrm{d}\lambda \hfill \\ \hfill & \le \frac{{∥f∥}_{C[0,h^{*}]}\mathrm{\Gamma }(1-\alpha \left(t\right))}{\mathrm{\Gamma }(1-\eta )}\frac{\varphi \mathrm{\Gamma }\left(\alpha \right(t)+1-\eta )}{{∥f∥}_{C[0,h^{*}]}\mathrm{\Gamma }(1-\alpha \left(t\right))}=\varphi ,\mathrm{for}\mathrm{any}u\in {\mathrm{\Lambda }}_{h^{*}}.\hfill \end{array}$$

Hence, $\aleph u\in {\mathrm{\Lambda }}_{h^{*}},$ if $u\in {\mathrm{\Lambda }}_{h^{*}}.$

Next, for any $0\le t_1\le t_2\le h^{*}$,

$$\begin{array}{cc}\hfill & |(\aleph u)\left(t_1\right)-(\aleph u)\left(t_2\right)|\hfill \\ \hfill & =|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_1}{(t_1-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda -\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda |\hfill \\ \hfill & \le \frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_1}[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}]\left|f(\lambda ,u\left(\lambda \right))\right|\mathrm{d}\lambda \hfill \\ \hfill & +\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{t_1}^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}\left|f(\lambda ,u\left(\lambda \right))\right|\mathrm{d}\lambda \hfill \\ \hfill & \le \frac{{∥f∥}_{C[0,h^{*}]}}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_1}[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}]\mathrm{d}\lambda +\frac{{∥f∥}_{C[0,h^{*}]}}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{t_1}^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}\mathrm{d}\lambda \hfill \\ \hfill & =\frac{{∥f∥}_{C[0,h^{*}]}}{\mathrm{\Gamma }(1+\alpha (t\left)\right)}[t_1{}^{\alpha \left(t\right)}-t_2{}^{\alpha \left(t\right)}+{(t_2-t_1)}^{\alpha \left(t\right)}].\hfill \end{array}$$

This shows that $\aleph u$ is continuous.

However, we have

$${∥{\aleph }^{n}u-{\aleph }^n\tilde{u}∥}_{C[0,t]}\le \frac{{\left(L{\delta }^{\alpha \left(t\right)}\right)}^n}{\mathrm{\Gamma }(1+n\alpha (t\left)\right)}{∥u-\tilde{u}∥}_{C[0,t]},\mathrm{for}\mathrm{every}n\in \mathbb{N}\mathrm{and}t\in [0,h^{*}].$$

(13)

For $n=0$, the Equation (13) is true. By the fundamental concept of induction, the case $n-1$ is also true, one can get

$$\begin{array}{cc}\hfill {∥{\aleph }^{n}u-{\aleph }^n\tilde{u}∥}_{C[0,t]}& ={∥\aleph \left({\aleph }^{n-1}u\right)-\aleph \left({\aleph }^{n-1}\tilde{u}\right)∥}_{C[0,t]}\hfill \\ \hfill & \frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}\underset{0\in [\delta ,t]}{max}\left|{\int }_0^{\delta }{(\delta -\lambda )}^{\alpha \left(t\right)-1}[f(\lambda ,{\aleph }^{n-1}u\left(\lambda \right))-f(\lambda ,{\aleph }^{n-1}\tilde{u}\left(\lambda \right))]\mathrm{d}\lambda \right|.\hfill \end{array}$$

By the Lipschitz condition and the induction hypothesis, the result (13) is obvious.

Now,

$${\displaystyle \sum _{n=0}^{\infty }}\frac{{\left(L{\delta }^{\alpha \left(t\right)}\right)}^n}{\mathrm{\Gamma }(1+n\alpha (t\left)\right)}=E_{\alpha \left(t\right)}\left(L{\delta }^{\alpha \left(t\right)}\right),$$

where $E_{\alpha }(·)$ is the Mittag–Leffler function, defined as $E_{\alpha }\left(z\right)={\displaystyle \sum _{k=0}^{\infty }}\frac{z^k}{1+\alpha k}$. Thus, we can apply the Lemma 4 and deduce the uniqueness of IVP (1). □

Theorem 3.

If condition (H1) and (H2) are holds, then $u=u\left(t\right),$$t\in (0,\varsigma )$ is non-continuable if only for some $\xi \in (0,\frac{\varsigma }{2})$ and any bounded closed subset $X\subset [\xi ,+\infty )\times \mathbb{R}$ then $\exists t^{*}\in [\xi ,\varsigma )$, such that $(t^{*},u\left(t^{*}\right))\notin X$.

Proof. 

We provide the proof in two steps. Let, ∃ $X\subset [\xi ,+\infty )\times \mathbb{R}$ and $\left\{\right(t,u\left(t\right)):t\in [\xi ,\varsigma \left)\right\}\subset X$. The compactness of X⇒ $\varsigma <+\infty$. A positive K exists by (H1) so that

$${sup}_{(t,u)\in X}\left|f(t,u)\right|\le K.$$

Step 1. For ${lim}_{t\to {\varsigma }^{-}}u\left(t\right)$ exists. Let

$$G\left(t\right)={\int }_0^{\xi }{(t-\lambda )}^{\alpha \left(t\right)-1}{\lambda }^{-\eta }\mathrm{d}\lambda ,2\xi \le t\le \varsigma .$$

One can see that $G\left(t\right)$ is continuous uniformly on $[2\xi ,\varsigma ]$. For all $2\xi \le t_1<t_2\le \varsigma ,$ we have

$$\begin{array}{cc}\hfill & |u\left(t_1\right)-u\left(t_2\right)|\hfill \\ \hfill & =|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_1}{(t_1-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda -\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda |\hfill \\ \hfill & \le \left|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{\xi }[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}]{\lambda }^{-\eta }(\Im u)\left(\lambda \right)\mathrm{d}\lambda \right|\hfill \\ \hfill & +\left|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{\xi }^{t_1}[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}]f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda \right|\hfill \\ \hfill & +|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{t_1}^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda |\hfill \\ \hfill & \le \frac{{\parallel \Im u\parallel }_{[0,\xi ]}}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{\xi }\left[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}\right]{\lambda }^{-\eta }\mathrm{d}\lambda \hfill \\ \hfill & +\frac{K}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{\xi }^{t_1}[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}]\mathrm{d}\lambda +\frac{K}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{t_1}^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}\mathrm{d}\lambda \hfill \\ \hfill & \le |G\left(t_1\right)-G\left(t_2\right)|\frac{{\parallel \Im u\parallel }_{[0,\xi ]}}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}+\frac{K}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}[2{(t_2-t_1)}^{\alpha \left(t\right)}+{(t_1-\xi )}^{\alpha \left(t\right)}-{(t_2-\xi )}^{\alpha \left(t\right)}].\hfill \end{array}$$

Since $G\left(t\right)$ is continuous and from th Cauchy convergence criterion, it follows that ${lim}_{t\to {\varsigma }^{-}}u\left(t\right)=u^{*}$ exists.

Step 2. In this step, we prove that $u\left(t\right)$ is continuable. Because X is a closed subset, then $(\varsigma ,u^{*})\in X$. We have $u\left(\varsigma \right)=u^{*}$ and $u\left(t\right)\in C[0,\varsigma ]$, we define operator Q as follows

$$\left(Qx\right)\left(t\right)=u_1+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{\varsigma }^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,x\left(\lambda \right))\mathrm{d}\lambda ,$$

where

$$u_1=u_0+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{\varsigma }{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,x\left(\lambda \right))\mathrm{d}\lambda ,x\in C[\varsigma ,\varsigma +1],\varsigma \le t\le \varsigma +1.$$

Let

$$W_a=\{(t,x):t\in [\varsigma ,\varsigma +1],|x|\le \underset{t\in [\varsigma ,\varsigma +1]}{max}|u_1\left(t\right)|+a\}.$$

Since f is continuous on $W_a$, we can have $M={max}_{(t,x)\in W_a}\left|f(t,u)\right|$. Let

$$W_h=\{x\in C[\varsigma ,\varsigma +1]:\underset{t\in [\varsigma ,\varsigma +h]}{max}|x\left(t\right)-u_1\left(t\right)|\le a,x\left(\varsigma \right)=u_1\left(\varsigma \right)\},$$

where $h^{*}=min\left\{1,{\left(\frac{M}{\mathrm{\Gamma }\left(\alpha \right(t)+1)a}\right)}^{\alpha \left(t\right)}\right\}$. Thus Q is completely continuous on $W_a$. Set $\left\{x_n\right\}\subseteq C[\varsigma ,\varsigma +h^{*}]$, $\parallel x_n{-x\parallel }_{[\varsigma ,\varsigma +h^{*}]}$ approaches to 0 as n approaches to ∞. Then we have

$$\begin{array}{cc}\hfill |\left(Q{x}_n\right)\left(t\right)-\left(Qx\right)\left(t\right)|& =\left|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{\varsigma }^t{(t-\lambda )}^{\alpha \left(t\right)-1}[f(\lambda ,x_n\left(\lambda \right))-f(\lambda ,y\left(\lambda \right))]\mathrm{d}\lambda \right|\hfill \\ \hfill & \le \frac{h^{\alpha \left(t\right)}}{\mathrm{\Gamma }\left(\alpha \right(t)+1)}{\parallel f(\lambda ,y_n\left(\lambda \right))-f(\lambda ,y\left(\lambda \right))\parallel }_{[\varsigma ,\varsigma +h^{*}]}.\hfill \end{array}$$

Since f is continuous, we can have $\parallel f(\lambda ,x_n\left(\lambda \right))-f(\lambda ,x\left(\lambda \right)){\parallel }_{[\varsigma ,\varsigma +h^{*}]}$ approaches to 0 as n approaches to ∞. Moreover, $\parallel \left(Q{x}_n\right)\left(t\right)-\left(Qx\right)\left(t\right){\parallel }_{{}_{[\varsigma ,\varsigma +h^{*}]}}$ approaches to 0 as n approaches to ∞, which shows that Q is continuous.

Next, we show that $Q{W}_h$ is equicontinuous. For any $x\in W_h$ we have $\left(Qx\right)\left(\varsigma \right)=u_1\left(\varsigma \right)$ and

$$\begin{array}{cc}\hfill |\left(Qx\right)\left(t\right)-u_1|& =\left|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{\varsigma }^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,x\left(\lambda \right))\mathrm{d}\lambda \right|\hfill \\ \hfill & \le \frac{M{(t-\varsigma )}^{\alpha \left(t\right)}}{\mathrm{\Gamma }\left(\alpha \right(t)+1)}\le \frac{Mh{*}^{\alpha \left(t\right)}}{\mathrm{\Gamma }\left(\alpha \right(t)+1)}\le a.\hfill \end{array}$$

Thus $Q{W}_{h^{*}}\subset W_{h^{*}}$. Set $J\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{\varsigma }{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,x\left(\lambda \right))\mathrm{d}\lambda$. We know that $J\left(t\right)$ is continuous on $[\varsigma ,\varsigma +1]$. For all $x\in W_{h^{*}},t_1,t_2\in [\varsigma ,\varsigma +h^{*}]$, we have

$$\begin{array}{c}\hfill \left|\left(Qx\right)\left(t_1\right)-\left(Qx\right)\left(t_2\right)\right|\le \left|\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^{\varsigma }\left[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}\right]f(\lambda ,x\left(\lambda \right))\mathrm{d}\lambda \right|\\ \hfill +\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}\left|{\int }_{\varsigma }^{t_1}\left[{(t_1-\lambda )}^{\alpha \left(t\right)-1}-{(t_2-\lambda )}^{\alpha \left(t\right)-1}\right]f(\lambda ,x\left(\lambda \right))\mathrm{d}\lambda \right|\\ \hfill +\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}\left|{\int }_{t_1}^{t_2}{(t_2-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,x\left(\lambda \right))\mathrm{d}\lambda \right|\\ \hfill \le |J\left(t_1\right)-J\left(t_2\right)|+\frac{M}{\mathrm{\Gamma }\left(\alpha \right(t)+1)}[2{(t_2-t_1)}^{\alpha \left(t\right)}+{(t_1-\varsigma )}^{\alpha \left(t\right)}-{(t_2-\varsigma )}^{\alpha \left(t\right)}].\end{array}$$

Since $J\left(t\right)$ is uniform continuity on $[\varsigma ,\varsigma +h^{*}]$ and (13), we conclude that $\{\left(Qx\right)\left(t\right):x\in W_{h^{*}}\}$ is equicontinuous. Thus Q is continuous completely. By Lemma 3, operator Q has a fixed point $\tilde{u}\left(t\right)\in W_{h^{*}}$, i.e.,

$$\begin{array}{cc}\hfill \tilde{u}\left(t\right)& =u_1+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_{\varsigma }^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,\tilde{u}\left(\lambda \right))\mathrm{d}\lambda ,t\in [\varsigma ,\varsigma +h],\hfill \\ \hfill & =u_0+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,\tilde{u}\left(\lambda \right))\mathrm{d}\lambda ,\hfill \end{array}$$

(14)

where

$$\tilde{u}\left(t\right)=\left\{\begin{array}{cc}u\left(t\right),\hfill & t\in (0,\varsigma ]\hfill \\ \tilde{u}\left(t\right),\hfill & t\in [\varsigma ,\varsigma +h^{*}]\hfill \end{array}\right.$$

It follows that $\tilde{u}\left(t\right)\in C[0,\varsigma +h^{*}]$ and

$$\tilde{u}\left(t\right)=u_0+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha -1}f(\lambda ,\tilde{u}\left(\lambda \right))\mathrm{d}\lambda .$$

(15)

By Lemma 1, $\tilde{u}\left(t\right)$ of Equation (15), define a solution of problem (1) on $(0,\varsigma +h^{*}]$. This results in a contradiction because $u\left(t\right)$ is non-continuable. The proof is complete. □

4. Global Solution

Theorem 4.

Assume that (H1) is hold. We consider that $u\left(t\right)$ is a solution of problem (1) on $(0,\varsigma )$. For $\epsilon >0$, and if $u\left(t\right)$ is bounded on $[\epsilon ,\varsigma )$, then $\varsigma =+\infty$.

First, we give the following lemma before continuing the next discussion, which will be relevant in our analysis.

Lemma 5

([30,31]). Let r be a real function defined on $[0,\beta ]\times [0,+\infty )$. Assume then $\exists c>0$ and $\alpha \left(t\right)\in (0,1)$, such that

$$r\left(t\right)\le q\left(t\right)+c{\int }_0^t\frac{r\left(\lambda \right)}{{(t-\lambda )}^{\alpha \left(t\right)}}\mathrm{d}\lambda ,$$

where $q(·)>0$ is a locally integrable function on $[0,\beta ]$ Then $\exists h=h\left(\alpha \right(t\left)\right)$, such that for $0\le t\le \beta$, we have

$$r\left(t\right)\le q\left(t\right)+hc{\int }_0^t\frac{q\left(\lambda \right)}{{(t-\lambda )}^{\alpha \left(t\right)}}\mathrm{d}\lambda .$$

Theorem 5.

Assume that (H1) is the hold and three continuous functions $g\left(t\right)>0$, $h\left(t\right)>$, $j\left(t\right)>0$ defined on $[0,+\infty )\times [0,+\infty )$, such that $\left|f\right(t,u\left)\right|\le g\left(t\right)h\left(\right|u\left|\right)+j\left(t\right)$, where $h\left(s\right)\le s$ for $s\in [0,\infty )$. Then there is one solution of (1) in $C[0,+\infty )$.

Proof. 

By using the Theorem 1, one can easily conclude the local existence of the solution of (1). By using Lemma 1, $u\left(t\right)$ satisfies the following equation

$$u\left(t\right)=u_0+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda .$$

Let the maximum interval of $u\left(t\right)$ is $[0,\varsigma )$, where $(\varsigma <+\infty )$. Then

$$\begin{array}{cc}\hfill \left|u\right(t\left)\right|& =|u_0+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda |\hfill \\ \hfill & \le u_0+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}\left(g\left(\lambda \right)h\right(\left|u\right|)+j\left(\lambda \right))\mathrm{d}\lambda \hfill \\ \hfill & \le u_0+\frac{{\parallel g\parallel }_{[0,\varsigma ]}}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}\left(h\right(\left|u\right|)\mathrm{d}\lambda +\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}j\left(\lambda \right)\mathrm{d}\lambda .\hfill \end{array}$$

Taking $r\left(t\right)=\left|u\left(t\right)\right|,q\left(t\right)=u_0+\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}j\left(\lambda \right)\mathrm{d}\lambda$, $c=\frac{{\parallel g\parallel }_{[0,\varsigma ]}}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}$. By Lemma (5), $r\left(t\right)=\left|u\right(t\left)\right|$ is bounded on $[0,\varsigma )$. Thus, for any $\epsilon \in (0,\varsigma ),$$u\left(t\right)$ is bounded on $[\epsilon ,\varsigma )$. By Theorem 4, the solution of problem (1) exists on $(0,+\infty )$. □

The next theorem ensures the existence uniqueness of the global solution of (1) on ${\mathbb{R}}^{+}$.

Theorem 6.

Assume (H1) is hold, and a continuous function $p\left(t\right)>0$ exists and defined on $[0,\infty )$, such that $|f(t,u)-f(t,\tilde{u})|\le p\left(t\right)|u-\tilde{u}|$. Then the unique solution of (1) exists in $C[0,+\infty )$.

For proof that this theorem is simple and straightforward, we leave it for the interested readers.

5. Ulam Stability Results

Now, we consider the Ulam stability for (1). Let $ϵ>0$ and $\phi$ be a continuous function defined on $[0,+\infty )\to {\mathbb{R}}^{+}$. Consider these inequalities:

$$\left|{}_C{D}_{0,t}^{\alpha \left(t\right)}u\left(t\right)-f(t,u\left(t\right))\right|\le ϵ;$$

(16)

$$\left|{}_C{D}_{0,t}^{\alpha \left(t\right)}u\left(t\right)-f(t,u\left(t\right))\right|\le \phi \left(t\right);$$

(17)

$$\left|{}_C{D}_{0,t}^{\alpha \left(t\right)}u\left(t\right)-f(t,u\left(t\right))\right|\le ϵ\phi \left(t\right).$$

(18)

Definition 5.

IVP (1) is Ulam–Hyers stable if there exists a real number $c_f>0$, such that for each $ϵ>0$ and for each solution $u\in C[0,+\infty )$ of inequality (16), there exists a solution $v\in C[0,+\infty )$ of (1) with

$$\left|u\left(t\right)-v\left(t\right)\right|\le ϵc_f.$$

Definition 6.

If there exists a real number $c_f\in \left({\mathbb{R}}^{+},{\mathbb{R}}^{+}\right)$ with $c_f\left(0\right)=0$, for every $ϵ>0$ and for every solution of $u\in C$ of inequality (17), then ∃ a solution $v\in C$ of (1) with

$$\left|u\left(t\right)-v\left(t\right)\right|\le c_f\left(ϵ\right)$$

then IVP (1) is the generalized Ulam–Hyers stable.

Definition 7.

If there is a number $c_{f,\phi }\in \mathbb{R}$, for every $ϵ>0$ and for every solution $u\in C$ of (18) then ∃ a solution $v\in C$ of (1) with

$$\left|u\left(t\right)-v\left(t\right)\right|\le ϵc_{f,\phi }\phi \left(t\right),$$

then IVP (1) is Ulam–Hyers–Rassias stable, with respect to φ.

Definition 8.

If there is a number $c_{f,\phi }\in \mathbb{R}$, for every solution $u\in C$ of (17), then ∃ a solution $v\in C$ of (1) with

$$\left|u\left(t\right)-v\left(t\right)\right|\le c_{f,\phi }\phi \left(t\right).$$

and then IVP (1) is the generalized Ulam–Hyers–Rassias stable with respect to φ

Remark 1.

Clearly, we can see that: (i) Definition 5 ⟹ Definition 6; (ii) Definition 7 ⟹ Definition 8; (iii) Definition 7 ⟹ Definition 5.

Hypothesis 3 (H3).

Assume that$\phi$is an increasing function and belongs to$C[0,+\infty )$. Then there exists${\chi }_{\phi }>0$, such that

$$\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}\phi \left(\lambda \right)\mathrm{d}\lambda \le {\chi }_{\phi }\phi \left(t\right),t\in [0,+\infty ).$$

Lemma 6

([32]). Let x and y be a continuous function defined on $[0,T]\times [0,+\infty )$ where $T\le \infty .$ If y is increasing and there are constants $\mu \ge 0$ and $p>0$, such that

$$x\left(t\right)\le y\left(t\right)+\mu {\displaystyle \underset{0}{\overset{t}{\int }}}{(t-\lambda )}^{p-1}x\left(\lambda \right)\mathrm{d}\lambda ,t\in [0,T),$$

then

$$x\left(t\right)\le y\left(t\right)+{\displaystyle \underset{0}{\overset{t}{\int }}}\left[{\displaystyle \sum _{k=0}^{\infty }}\frac{{\left(\mu \mathrm{\Gamma }\left(p\right)\right)}^k}{\mathrm{\Gamma }\left(kp\right)}{(t-\lambda )}^{p-1}y\left(\lambda \right)\right]\mathrm{d}\lambda ,t\in [0,T).$$

If $y\left(t\right)=c,$ is a constant on $t\in [0,T)$, then we have

$$x\left(t\right)\le c{E}_p\left(\mu \mathrm{\Gamma }\left(p\right)t^p\right),t\in [0,T),$$

where $E_p($·) is the Mittag–Leffler function.

Theorem 7.

If the condition (H3) is satisfied, then IVP (1) is the generalized Ulam–Hyers–Rassias stable.

Proof. 

Suppose that u is a solution of (17) on $C[0,+\infty )$, and we assume that v is a solution of (1). Thus, we have

$$\begin{array}{cc}\hfill \left|u\left(t\right)-u_0\left(t\right)-\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda \right|& \\ \hfill & \le \frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}\phi \left(\lambda \right)\mathrm{d}\lambda \hfill \\ \hfill & \le {\chi }_{\phi }\phi \left(t\right).\hfill \end{array}$$

From these relations, it follows

$$\begin{array}{cc}\hfill \left|u\right(t)-v(t\left)\right|& \le \left|u\left(t\right)-u_0\left(t\right)-\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}f(\lambda ,u\left(\lambda \right))\mathrm{d}\lambda \right|\hfill \\ \hfill & +\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}\left|f(\lambda ,u(\lambda \left)\right)-f(\lambda ,v(\lambda \left)\right)\right|\mathrm{d}\lambda \hfill \\ \hfill & \le {\chi }_{\phi }\phi \left(t\right)+\frac{L}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t{(t-\lambda )}^{\alpha \left(t\right)-1}|u\left(\lambda \right)-v\left(\lambda \right)|\mathrm{d}\lambda .\hfill \end{array}$$

By Lemma (6), there exists a constant $L^{*}>0$ independent of ${\chi }_{\phi }\phi \left(t\right)$, such that $|u\left(t\right)-v\left(t\right)|\le L^{*}{\chi }_{\phi }\phi \left(t\right):=c_{f,\phi }\phi \left(t\right).$ Thus, IVP (1) is the generalized Ulam–Hyers–Rassias stable. □

Corollary 1.

By using the same arguments of Theorem 7, one can prove that IVP (1) with inequality (18) is Ulam–Hyers–Rassias stable.

Corollary 2.

Under the same steps of Theorem 7, with inequality (16), one can prove that IVP (1) is Ulam–Hyers stable.

6. Conclusions

First, we derived new local existence–uniqueness theorems for the variable order Caputo order FDE. Next, we proved the new continuation theorems to establish the global existence of the variable order FDEs. Finally, we gave the lemmas, which showed that our considered problem is Ulam–Hyers type stable. Readers can find the existence–uniqueness solution of Hadamard, Caputo–Hadamard, and the Hilfer type variable order differential equations by reading this work.