Existence of solutions
Let us introduce the following assumption:
-
(H1)
Let \(n\in \mathbb{N}\) be an integer, \({\mathcal{P}} =\{J_{1}:=[0,T_{1}], J_{2}:=(T_{1},T_{2}], J_{3}:=(T_{2},T_{3}],\ldots J_{n}:=(T_{n-1},T] \}\) be a partition of the interval J, and let \(u(t): J \rightarrow (1,2]\) be a piecewise constant function with respect to \({\mathcal{P}}\), i.e.,
$$\begin{aligned} u(t)=\sum_{\ell =1}^{n}u_{\ell}I_{\ell}(t)= \textstyle\begin{cases} u_{1} & \text{if } t\in J_{1}, \\ u_{2} & \text{if } t\in J_{2}, \\ \vdots \\ u_{n}& \text{if } t\in J_{n}, \end{cases}\displaystyle \end{aligned}$$
where \(1< u_{\ell} \leq 2 \) are constants, and \(I_{\ell}\) is the indicator of the interval \(J_{\ell}:=(T_{\ell -1},T_{\ell}], \ell =1,2,\ldots,n\), (with \(T_{0}=0, T_{n}=T\)) such that
$$\begin{aligned} I_{\ell}(t)= \textstyle\begin{cases} 1& \text{for } t\in J_{\ell}, \\ 0& \text{for elsewhere}. \end{cases}\displaystyle \end{aligned}$$
Further, for a given set U of functions \(u: J \to X\), let us denote
$$\begin{aligned} U(t)= \bigl\{ u(t), u \in U \bigr\} , \quad t\in J, \end{aligned}$$
and
$$\begin{aligned} U(J)= \bigl\{ U(t):v\in U, t\in J \bigr\} . \end{aligned}$$
For each \(\ell \in \{1, 2,\ldots,n \}\), the symbol \(E_{\ell}= C(J_{\ell}, \mathbb{R})\) indicates the Banach space of continuous functions \(x:J_{\ell} \to \mathbb{R}\) equipped with the norm
$$\begin{aligned} \Vert x \Vert _{E_{\ell}}=\sup_{t\in J_{\ell}} \bigl\vert x(t) \bigr\vert . \end{aligned}$$
Then, for any \(t \in J_{\ell}, \ell = 1, 2,\ldots, n\), the (CFD) of variable order \(u(t)\) for function \(x(t) \in C(J,\mathbb{R})\), defined by (3), could be presented as a sum of left Caputo fractional derivatives of constant orders \(u_{\ell}, \ell = 1, 2,\ldots, n\):
$$\begin{aligned} {}^{c}D^{u(t)}_{0^{+}}x(t) = \int _{0}^{T_{1}} \frac{(t-s)^{1-u_{1}}}{\Gamma (2-u_{1})}x^{(2)}(s) \,ds +\cdots+ \int _{T_{ \ell -1}}^{t}\frac{(t-s)^{1-u_{\ell}}}{\Gamma (2-u_{\ell})}x^{(2)}(s) \,ds. \end{aligned}$$
(5)
Thus, according to (5), (BVP)(1) can be written, for any \(t \in J_{\ell}, \ell = 1, 2,\ldots, n\), in the form
$$\begin{aligned} \int _{0}^{T_{1}}\frac{(t-s)^{1-u_{1}}}{\Gamma (2-u_{1})}x^{(2)}(s) \,ds +\cdots+ \int _{T_{\ell -1}}^{t}\frac{(t-s)^{1-u_{\ell}}}{\Gamma (2-u_{\ell})}x^{(2)}(s) \,ds = f_{1} \bigl(t, x(t) \bigr),\quad t \in J_{\ell}. \end{aligned}$$
(6)
In what follows we shall introduce the solution to BVP (1).
Definition 3.1
BVP (1) has a solution if there are functions \(x_{\ell}, \ell =1, 2,\ldots, n\), so that \(x_{\ell} \in C([0, T_{\ell}], \mathbb{R})\) fulfilling equation (6) and \(x_{\ell}(0) = 0 = x_{\ell}(T_{\ell})\).
Let the function \(x \in C(J, \mathbb{R})\) be such that \(x(t) \equiv 0\) on \(t \in [0, T_{\ell -1}]\) and it solves integral equation (6). Then (6) is reduced to
$$\begin{aligned} {}^{c}D^{u_{\ell}}_{T_{\ell -1}^{+}} x(t)= f_{1} \bigl(t, x(t) \bigr),\quad t \in J_{ \ell}. \end{aligned}$$
We shall deal with the following BVP:
$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{u_{\ell}}_{T_{\ell -1}^{+}} x(t)= f_{1}(t, x(t)),\quad t \in J_{ \ell}, \\ x(T_{{\ell -1}})=0,\qquad x(T_{\ell})=0. \end{cases}\displaystyle \end{aligned}$$
(7)
For our purpose, the upcoming lemma will be a corner stone of the solution of BVP (7).
Lemma 3.1
Let \(\ell \in \{1,2,\ldots,n\}\) be a natural number, \(f_{1}\in C(J_{\ell} \times \mathbb{R}, \mathbb{R})\), and there exists a number \(\delta \in (0, 1)\) such that \(t^{\delta} f_{1}\in C(J_{\ell} \times \mathbb{R}, \mathbb{R})\).
Then the function \(x \in E_{\ell}\) is a solution of BVP (7) if and only if x solves the integral equation
$$\begin{aligned} x(t) =-(T_{\ell}-T_{{\ell -1}})^{-1}(t-T_{\ell -1}) I^{u_{\ell}}_{T_{ \ell -1}^{+}} f_{1} \bigl(T_{\ell}, x(T_{\ell}) \bigr) +I^{u_{\ell}}_{T_{\ell -1}^{+}}f_{1} \bigl(t, x(t) \bigr). \end{aligned}$$
(8)
Proof
We presume that \(x \in E_{\ell}\) is a solution of BVP (7). Employing the operator \(I^{u_{\ell}}_{T_{\ell -1}^{+}}\) to both sides of (7) and regarding Lemma 2.1, we find
$$\begin{aligned} x(t)=\omega _{1} + \omega _{2}(t-T_{{\ell -1}})+ \frac{1}{\Gamma (u_{\ell})} \int _{T_{{\ell -1}}}^{t}(t-s)^{u_{\ell}-1} f_{1} \bigl(s, x(s) \bigr)\,ds,\quad t \in J_{\ell}. \end{aligned}$$
By \(x(T_{\ell -1}) = 0\), we get \(\omega _{1}=0\).
Let \(x(t)\) satisfy \(x(T_{\ell})=0\). So, we observe that
$$\begin{aligned} \omega _{2} = -(T_{\ell}-T_{{\ell -1}})^{-1} I^{u_{\ell}}_{T_{\ell -1}^{+}} f_{1} \bigl(T_{\ell}, x(T_{\ell}) \bigr). \end{aligned}$$
Then we find
$$\begin{aligned} x(t) =-(T_{\ell}-T_{{\ell -1}})^{-1}(t-T_{\ell -1})I^{u_{\ell}}_{T_{ \ell -1}^{+}} f_{1} \bigl(T_{\ell}, x(T_{\ell}) \bigr)+ I^{u_{\ell}}_{T_{\ell -1}^{+}}f_{1} \bigl(t, x(t) \bigr),\quad t \in J_{\ell}. \end{aligned}$$
Conversely, let \(x \in E_{\ell}\) be a solution of integral equation (8). Regarding the continuity of function \(t^{\delta} f_{1}\) and Lemma 2.1, we deduce that x is the solution of BVP (7). □
We are now in a position to prove the existence of solution for (BVP) (7) based on the concept of (MNCK) and (DFPT).
Theorem 3.1
Let the conditions of Lemma 3.1be satisfied and there exist a constant \(K >0\) such that
$$\begin{aligned} t^{\delta} \bigl\vert f_{1}(t,y_{1})- f_{1}(t,y_{2}) \bigr\vert \leq K \vert y_{1}-y_{2} \vert \end{aligned}$$
(9)
for any \(y_{1}, y_{2} \in \mathbb{R}\), \(t\in J_{\ell}\), and the inequality
$$\begin{aligned} \frac{2K(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}(T_{\ell}^{1-\delta}-T_{\ell -1}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})} < 1 \end{aligned}$$
(10)
holds. Then BVP (7) possesses at least one solution in \(E_{\ell}\).
Proof
We construct the operator
$$\begin{aligned} W: E_{\ell} \rightarrow E_{\ell} \end{aligned}$$
as follows:
$$\begin{aligned} Wx(t) ={}&{-}(T_{\ell}-T_{{\ell -1}})^{-1}(t-T_{\ell -1}) I^{u_{\ell}}_{T_{ \ell -1}^{+}} f_{1} \bigl(T_{\ell}, x(T_{\ell}) \bigr) \\ &{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{{\ell -1}}}^{t}(t-s)^{u_{\ell -1}} f_{1} \bigl(s, x(s) \bigr)\,ds,\quad t \in J_{\ell}. \end{aligned}$$
(11)
It follows from the properties of fractional integrals and from the continuity of function \(t^{\delta}f_{1}\) that the operator \(W: E_{\ell}\) → \(E_{\ell}\) defined in (11) is well defined.
Let
$$\begin{aligned} R_{\ell} \geq \frac{\frac{2f^{\star}(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell})}}{1-\frac{2(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}(T_{\ell}^{1-\delta}-T_{\ell -1}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})} (K+ L\frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell}+1)})} \end{aligned}$$
with
$$\begin{aligned} f^{\star}= \sup_{t\in J_{\ell}} \bigl\vert f_{1}(t, 0) \bigr\vert . \end{aligned}$$
We consider the set
$$\begin{aligned} B_{R_{\ell}}= \bigl\{ x \in E_{\ell}, \Vert x \Vert _{E_{\ell}}\leq R_{\ell} \bigr\} . \end{aligned}$$
Clearly, \(B_{R_{\ell}}\) is nonempty, closed, convex, and bounded. □
Now, we demonstrate that W satisfies the assumption of Theorem 2.1. We shall prove it in four phases.
Step 1: Claim: \(W(B_{R_{\ell}})\subseteq (B_{R_{\ell}})\).
For \(x \in B_{R_{\ell}}\) and by (H2), we get
$$\begin{aligned} \bigl\vert Wx(t) \bigr\vert \leq{}& \frac{(T_{\ell}-T_{\ell -1})^{-1}(t-T_{\ell -1})}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, x(s) \bigr) \bigr\vert \,ds\\ &{}+ \frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t}(t-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, x(s) \bigr) \bigr\vert \,ds \\ \leq{}& \frac{2}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{ \ell}-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, x(s) \bigr) \bigr\vert \,ds \\ \leq{}& \frac{2}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{ \ell}-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, x(s) \bigr)-f_{1}(s, 0) \bigr\vert \,ds\\ &{}+ \frac{2}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{\ell}-s)^{u_{ \ell}-1} \bigl\vert f_{1}(s, 0) \bigr\vert \,ds \\ \leq{}& \frac{2}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{ \ell}-s)^{u_{\ell}-1} s^{-\delta} \bigl(K \bigl\vert x(s) \bigr\vert \bigr)\,ds + \frac{2f^{\star}(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell})} \\ \leq{}& \frac{2(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}} s^{-\delta} \bigl(K \bigl\vert x(s) \bigr\vert \bigr)\,ds + \frac{2f^{\star}(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell})} \\ \leq{}& \frac{2K(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}(T_{\ell}^{1-\delta}-T_{\ell -1}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})} R_{\ell} + \frac{2f^{\star}(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell})} \\ \leq{}& R_{\ell}, \end{aligned}$$
which means that \(W(B_{R_{\ell}}) \subseteq B_{R_{\ell}} \).
Step 2: Claim: W is continuous.
We presume that the sequence \((x_{n})\) converges to x in \(E_{\ell}\) and \(t \in J_{\ell}\). Then
$$\begin{aligned} &\bigl\vert (Wx_{n}) (t)-(Wx) (t) \bigr\vert \\ &\quad \leq \frac{(T_{\ell}-T_{\ell -1})^{-1}(t-T_{\ell -1})}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, x_{n}(s) \bigr)-f_{1} \bigl(s, x(s) \bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t}(t-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, x_{n}(s) \bigr)-f_{1} \bigl(s, x(s) \bigr) \bigr\vert \,ds \\ &\quad\leq \frac{2}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{ \ell}-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, x_{n}(s) \bigr)-f_{1} \bigl(s, x(s) \bigr) \bigr\vert \,ds \\ &\quad \leq \frac{2}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}s^{- \delta}(T_{\ell}-s)^{u_{\ell}-1} \bigl(K \bigl\vert x_{n}(s)- x(s) \bigr\vert \bigr)\,ds \\ &\quad\leq \frac{2K(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}}{\Gamma (u_{\ell})} \Vert x_{n}- x \Vert _{E_{\ell}} \int _{T_{\ell -1}}^{T_{\ell}}s^{-\delta}\,ds \\ &\quad\leq \frac{2K(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}({T_{\ell}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})} \Vert x_{n}-x \Vert _{E_{\ell}}, \end{aligned}$$
i.e., we obtain
$$\begin{aligned} \bigl\Vert (Wx_{n})-(Wx) \bigr\Vert _{E_{\ell}}\rightarrow 0\quad \text{as }n \rightarrow \infty. \end{aligned}$$
Ergo, the operator W is continuous on \(E_{\ell}\).
Step 3: Claim: W is bounded and equicontinuous.
By Step 1, we have \(W(B_{R_{\ell}})= \{W(x): x \in B_{R_{\ell}} \} \subset B_{R_{\ell}}\), thus for each \(x \in B_{R_{\ell}}\) we have \(\|W(x)\|_{E_{\ell}} \leq R_{\ell}\), which means that \(W(B_{R_{\ell}})\) is bounded. It remains to indicate that \(W(B_{R_{\ell}})\) is equicontinuous.
For \(t_{1},t_{2}\in J_{\ell}, t_{1} < t_{2}\) and \(x \in B_{R_{\ell}}\), we have
$$\begin{aligned} & \bigl\vert (Wx) (t_{2})-(Wx) (t_{1}) \bigr\vert \\ &\quad= \biggl\vert - \frac{(T_{\ell}-T_{\ell -1})^{-1}(t_{2}-T_{\ell -1})}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell}-1} f_{1} \bigl(s, x(s) \bigr)\,ds \\ &\qquad{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t_{2}}(t_{2}-s)^{u_{ \ell}-1} f_{1} \bigl(s, x(s) \bigr)\,ds+ \frac{(T_{\ell}-T_{\ell -1})^{-1}(t_{1}-T_{\ell -1})}{\Gamma (u_{\ell})} \\ &\qquad{}\times \int _{T_{\ell -1}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell}-1} f_{1} \bigl(s, x(s) \bigr)\,ds- \frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t_{1}}(t_{1}-s)^{u_{ \ell}-1} f_{1} \bigl(s, x(s) \bigr)\,ds \biggr\vert \\ &\quad\leq \frac{(T_{\ell}-T_{\ell -1})^{-1}}{\Gamma (u_{\ell})} \bigl((t_{2}-T_{ \ell -1})-(t_{1}-T_{\ell -1}) \bigr) \int _{T_{\ell -1}}^{T_{\ell}}(T_{ \ell}-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, x(s) \bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t_{1}} \bigl((t_{2}-s)^{u_{ \ell}-1}-(t_{1}-s)^{u_{\ell}-1} \bigr) \bigl\vert f_{1} \bigl(s, x(s) \bigr) \bigr\vert \,ds\\ &\qquad{}+ \frac{1}{\Gamma (u_{\ell})} \int _{t_{1}}^{t_{2}}(t_{2}-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, x(s) \bigr) \bigr\vert \,ds \\ &\quad\leq \frac{(T_{\ell}-T_{\ell -1})^{-1}}{\Gamma (u_{\ell})} \bigl((t_{2}-T_{ \ell -1})-(t_{1}-T_{\ell -1}) \bigr) \int _{T_{\ell -1}}^{T_{\ell}}(T_{ \ell}-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, x(s) \bigr)-f_{1}(s, 0) \bigr\vert \,ds \\ &\qquad{}+\frac{(T_{\ell}-T_{\ell -1})^{-1}}{\Gamma (u_{\ell})} \bigl((t_{2}-T_{ \ell -1})-(t_{1}-T_{\ell -1}) \bigr) \int _{T_{\ell -1}}^{T_{\ell}}(T_{ \ell}-s)^{u_{\ell}-1} \bigl\vert f_{1}(s, 0) \bigr\vert \,ds \\ &\qquad{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t_{1}} \bigl((t_{2}-s)^{u_{ \ell}-1}-(t_{1}-s)^{u_{\ell}-1} \bigr) \bigl\vert f_{1} \bigl(s, x(s) \bigr)-f_{1}(s, 0) \bigr\vert \,ds \\ &\qquad{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t_{1}} \bigl((t_{2}-s)^{u_{ \ell}-1}-(t_{1}-s)^{u_{\ell}-1} \bigr) \bigl\vert f_{1}(s, 0) \bigr\vert \,ds \\ &\qquad{}+\frac{1}{\Gamma (u_{\ell})} \int _{t_{1}}^{t_{2}}(t_{2}-s)^{u_{ \ell}-1} \bigl\vert f_{1} \bigl(s, x(s) \bigr)-f_{1}(s, 0) \bigr\vert \,ds+\frac{1}{\Gamma (u_{\ell})} \int _{t_{1}}^{t_{2}}(t_{2}-s)^{u_{\ell}-1} \bigl\vert f_{1}(s, 0) \bigr\vert \,ds \\ &\quad\leq \frac{(T_{\ell}-T_{\ell -1})^{-1}}{\Gamma (u_{\ell})} \bigl((t_{2}-T_{ \ell -1})-(t_{1}-T_{\ell -1}) \bigr) \int _{T_{\ell -1}}^{T_{\ell}}(T_{ \ell}-s)^{u_{\ell}-1}s^{-\delta} \bigl(K \bigl\vert x(s) \bigr\vert \bigr)\,ds \\ &\qquad{}+\frac{f^{\star}(T_{\ell}-T_{\ell -1})^{-1}}{\Gamma (u_{\ell})} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \int _{T_{\ell -1}}^{T_{ \ell}}(T_{\ell}-s)^{u_{\ell}-1} \,ds \\ &\qquad{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t_{1}}s^{-\delta} \bigl((t_{2}-s)^{u_{\ell}-1}-(t_{1}-s)^{u_{\ell}-1} \bigr) \bigl(K \bigl\vert x(s) \bigr\vert \bigr)\,ds \\ &\qquad{}+\frac{f^{\star}}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t_{1}} \bigl((t_{2}-s)^{u_{\ell}-1}-(t_{1}-s)^{u_{\ell}-1} \bigr)\,ds \\ &\qquad{}+\frac{1}{\Gamma (u_{\ell})} \int _{t_{1}}^{t_{2}}s^{-\delta}(t_{2}-s)^{u_{ \ell}-1} \bigl(K \bigl\vert x(s) \bigr\vert \bigr)\,ds+\frac{f^{\star}}{\Gamma (u_{\ell})} \int _{t_{1}}^{t_{2}}(t_{2}-s)^{u_{ \ell}-1} \,ds \\ &\quad\leq \frac{K(T_{\ell}-T_{\ell -1})^{u_{\ell}-2}}{\Gamma (u_{\ell})} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \Vert x \Vert _{E_{\ell}} \int _{T_{\ell -1}}^{T_{\ell}}s^{-\delta}\,ds \\ &\qquad{}+ \frac{f^{\star}(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}}{\Gamma (u_{\ell}+1)} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr)\\ &\qquad{}+ \frac{K}{\Gamma (u_{\ell})} \Vert x \Vert _{E_{\ell}} \int _{T_{\ell -1}}^{t_{1}}s^{- \delta} \bigl((t_{2}-t_{1})^{u_{\ell}-1} \bigr)\,ds \\ &\qquad{}+ \frac{f^{\star}}{\Gamma (u_{\ell})} \biggl( \frac{(t_{2}-T_{\ell -1})^{u_{\ell}}}{u_{\ell}}- \frac{(t_{2}-t_{1})^{u_{\ell}}}{{u_{\ell}}}- \frac{(t_{1}-T_{\ell -1})^{u_{\ell}}}{{u_{\ell}}} \biggr) \\ &\qquad{}+\frac{K(t_{2}-t_{1})^{u_{\ell}-1}}{\Gamma (u_{\ell})} \Vert x \Vert _{E_{ \ell}} \int _{t_{1}}^{t_{2}}s^{-\delta}\,ds+ \frac{f^{\star}}{\Gamma (u_{\ell})} \frac{(t_{2}-t_{1})^{u_{\ell}}}{u_{\ell}} \\ &\quad\leq \frac{K(T_{\ell}-T_{\ell -1})^{u_{\ell}-2}({T_{\ell}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \Vert x \Vert _{E_{\ell}} \\ &\qquad{}+ \frac{f^{\star}(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}}{\Gamma (u_{\ell}+1)} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \\ &\qquad{}+ \biggl( \frac{K({t_{1}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})(t_{2}-t_{1})^{u_{\ell}-1}}{(1-\delta )\Gamma (u_{\ell})} \biggr) \Vert x \Vert _{E_{\ell}} \\ &\qquad{}+ \frac{f^{\star}}{\Gamma (u_{\ell}+1)} \bigl((t_{2}-T_{\ell -1})^{u_{ \ell}}-(t_{2}-t_{1})^{u_{\ell}}-(t_{1}-T_{\ell -1})^{u_{\ell}} \bigr) \\ &\qquad{}+ \frac{K({t_{2}}^{1-\delta}-{t_{1}}^{1-\delta})(t_{2}-t_{1})^{u_{\ell}-1}}{(1-\delta )\Gamma (u_{\ell})} \Vert x \Vert _{E_{\ell}} \\ &\qquad{}+\frac{f^{\star}(t_{2}-t_{1})^{u_{\ell}}}{\Gamma (u_{\ell}+1)} \\ &\quad\leq \biggl( \frac{K(T_{\ell}-T_{\ell -1})^{u_{\ell}-2}({T_{\ell}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})} \Vert x \Vert _{E_{\ell}}+ \frac{f^{\star}(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}}{\Gamma (u_{\ell}+1)} \biggr)\\ &\qquad{}\times \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \\ &\qquad{}+ \biggl( \frac{K({t_{2}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})} \Vert x \Vert _{E_{\ell}} \biggr) (t_{2}-t_{1})^{u_{\ell}-1}\\ &\qquad{}+ \frac{f^{\star}}{\Gamma (u_{\ell}+1)} \bigl((t_{2}-T_{\ell -1})^{u_{ \ell}}-(t_{1}-T_{\ell -1})^{u_{\ell}} \bigr). \end{aligned}$$
Hence \(\|(Wx)(t_{2})-(Wx)(t_{1})\|_{E_{\ell}}\rightarrow 0\) as \(|t_{2}-t_{1}|\rightarrow 0\). It implies that \(T(B_{R_{\ell}})\) is equicontinuous.
Remark 3.1
According to the remark of [12] page 20, we can easily show that inequality (9) and the following inequality
$$\begin{aligned} \zeta \bigl(t^{\delta} \bigl\vert f_{1}(t,B) \bigr\vert \bigr)\leq K\zeta (B) \end{aligned}$$
are equivalent for any bounded sets \(B \subset X\) and for each \(t\in J_{\ell}\).
Step 4: Claim: W is k-set contractions.
For \(U \in B_{R_{\ell}}\), \(t \in J_{\ell}\), we get
$$\begin{aligned} \zeta \bigl(W(U) (t) \bigr)={}&\zeta \bigl({(Wx) (t), x\in U} \bigr) \\ \leq{} & \biggl\{ \frac{(T_{\ell}-T_{\ell -1})^{-1}(t-T_{\ell -1})}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell}-1} \zeta f_{1} \bigl(s, x(s) \bigr)\,ds \\ &{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t}(t-s)^{u_{\ell}-1} \zeta f_{1} \bigl(s, x(s) \bigr)\,ds, x \in U \biggr\} . \end{aligned}$$
Then Remark 3.1 implies that, for each \(s\in J_{i}\),
$$\begin{aligned} \zeta \bigl(W(U) (t) \bigr)\leq{} & \biggl\{ \frac{(T_{\ell}-T_{\ell -1})^{-1}(t-T_{\ell -1})}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell}-1} \biggl[K \widehat{\zeta}(U) \int _{T_{{\ell -1}}}^{T_{\ell}}s^{-\delta}\,ds \biggr] \\ &{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t}(t-s)^{u_{\ell}-1} \biggl[K\widehat{ \zeta}(U) \int _{T_{{\ell -1}}}^{t}s^{-\delta}\,ds \biggr], x \in U \biggr\} \\ \leq{} & \biggl\{ \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}-2}(t-T_{\ell -1})}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}} \biggl[K\widehat{\zeta}(U) \int _{T_{{ \ell -1}}}^{T_{\ell}}s^{-\delta}\,ds \biggr] \\ &{}+\frac{(t-T_{\ell -1})^{u_{\ell}-1}}{\Gamma (u_{\ell})} \int _{T_{ \ell -1}}^{t} \biggl[K\widehat{\zeta}(U) \int _{T_{{\ell -1}}}^{t}s^{- \delta}\,ds \biggr], x \in U \biggr\} \\ \leq {}& \frac{K({T_{\ell}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})(T_{\ell}-T_{\ell -1})^{u_{\ell}-2}(t-T_{\ell -1})}{(1-\delta )\Gamma (u_{\ell})} \widehat{\zeta}(U)\\ &{}+ \frac{K({t}^{1-\delta}-{T_{\ell -1}}^{1-\delta})(t-T_{\ell -1})^{u_{\ell}-1}}{(1-\delta )\Gamma (u_{\ell})} \widehat{ \zeta}(U) \\ \leq {}& \frac{2K({T_{\ell}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}}{(1-\delta )\Gamma (u_{\ell})} \widehat{\zeta}(U). \end{aligned}$$
Thus,
$$\begin{aligned} \widehat{\zeta}(WU)\leq \frac{2K({T_{\ell}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}}{(1-\delta )\Gamma (u_{\ell})} \widehat{\zeta}(U). \end{aligned}$$
Therefore, all conditions of Theorem 2.1 are fulfilled, and thus BVP (7) has at least solution \(\widetilde{x_{\ell}}\in B_{R_{\ell}}\). Since \(B_{R_{\ell}} \subset E_{\ell}\), the claim of Theorem 3.1 is proved.
Now, we will prove the existence result for BVP (1).
Introduce the following assumption:
-
(H2)
Let \(f_{1}\in C(J \times \mathbb{R}, \mathbb{R})\), and there exists a number \(\delta \in (0, 1)\) such that \(t^{\delta} f_{1}\in C(J \times \mathbb{R}, \mathbb{R})\) and there exists a constant \(K >0\) such that \(t^{\delta}|f_{1}(t,y_{1})- f_{1}(t,y_{2})|\leq K|y_{1}-y_{2}|\) for any \(y_{1}, y_{2} \in \mathbb{R}\) and \(t\in J\).
Theorem 3.2
Let conditions (H1), (H2) and inequality (10) be satisfied for all \(\ell \in \{1,2,\ldots,n\}\). Then problem (1) possesses at least one solution in \(C(J, \mathbb{R})\).
Proof
For any \(\ell \in \{1,2,\ldots,n\}\), according to Theorem 3.1, BVP (7) possesses at least one solution \(\widetilde{x_{\ell}}\in E_{\ell}\).
For any \(\ell \in \{1,2,\ldots,n\}\), we define the function
$$\begin{aligned} {x}_{\ell}= \textstyle\begin{cases} 0, \quad t \in [0, T_{\ell -1}], \\ \widetilde{x}_{\ell}, \quad t \in J_{\ell}. \end{cases}\displaystyle \end{aligned}$$
Thus, the function \(x_{\ell} \in C([0, T_{\ell}], \mathbb{R})\) solves the integral equation (6) for \(t \in J_{\ell}\) with \(x_{\ell}(0) =0, x_{\ell}(T_{\ell}) = \widetilde{x}_{\ell}(T_{\ell}) = 0\).
Then the function
$$\begin{aligned} x(t)= \textstyle\begin{cases} x_{1}(t), \quad t \in J_{1}, \\ x_{2}(t)= \textstyle\begin{cases} 0, & t \in J_{1}, \\ \widetilde{x}_{2}, & t \in J_{2}, \end{cases}\displaystyle \\ \vdots \\ x_{n}(t)= \textstyle\begin{cases} 0, & t \in [0, T_{\ell -1}], \\ \widetilde{x}_{\ell},& t \in J_{\ell}, \end{cases}\displaystyle \end{cases}\displaystyle \end{aligned}$$
(12)
is a solution of BVP (1) in \(C(J, \mathbb{R})\). □
Ulam–Hyers stability
Theorem 3.3
Let conditions (H1), (H2) and inequality (10) be satisfied. Then BVP (1) is (UH) stable.
Proof
Let \(\epsilon >0\) be an arbitrary number and the function \(z(t)\) from \(z \in C(J_{\ell}, \mathbb{R})\) satisfy inequality (4).
For any \(\ell \in \{1,2,\ldots,n\}\), we define the functions \(z_{1}(t)\equiv z(t), t \in [1, T_{1}]\), and for \(\ell =2,3,\ldots,n\),
$$\begin{aligned} {z}_{\ell}(t)= \textstyle\begin{cases} 0, & t \in [0, T_{\ell -1}], \\ z(t), & t \in J_{\ell}. \end{cases}\displaystyle \end{aligned}$$
For any \(\ell \in \{1,2,\ldots,n\}\), according to equality (5), for \(t \in J\) we get
$$\begin{aligned} {}^{c}D^{u(t)}_{{T_{\ell -1}}^{+}}z_{\ell}(t)= \int _{T_{\ell -1}}^{t} \frac{(t-s)^{1-u_{\ell}}}{\Gamma (2-u_{\ell})}z^{(2)}(s) \,ds. \end{aligned}$$
Taking the (RLFI) \(I^{u_{\ell}}_{T_{\ell -1}^{+}}\) of both sides of inequality (4), we obtain
$$\begin{aligned} & \biggl\vert z_{\ell}(t)+ \frac{(T_{\ell}-T_{{\ell -1}})^{-1}(t-T_{\ell -1})}{{\Gamma (u_{\ell})}} \int _{T_{{\ell -1}}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell -1}} f_{1} \bigl(s, z_{ \ell}(s) \bigr)\,ds \\ &\qquad{}-\frac{1}{\Gamma (u_{\ell})} \int _{T_{{\ell -1}}}^{t}(t-s)^{u_{ \ell -1}} f_{1} \bigl(s, z_{\ell}(s) \bigr)\,ds \biggr\vert \\ &\quad \leq \epsilon \int _{T_{\ell -1}}^{t} \frac{(t-s)^{u_{\ell}-1}}{\Gamma (u_{\ell})}\,ds \\ &\quad \leq \epsilon \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell}+1)}. \end{aligned}$$
According to Theorem 3.2, BVP (1) has a solution \(x \in C(J, \mathbb{R})\) defined by \(x(t) = x_{\ell}(t)\) for \(t \in J_{\ell}, \ell = 1, 2,\ldots, n\), where
$$\begin{aligned} {x}_{\ell}= \textstyle\begin{cases} 0, & t \in [0, T_{\ell -1}], \\ \widetilde{x}_{\ell}, & t \in J_{\ell}, \end{cases}\displaystyle \end{aligned}$$
(13)
and \(\widetilde{x}_{\ell} \in E_{\ell}\) is a solution of BVP (7). According to Lemma 3.1, the integral equation
$$\begin{aligned} \widetilde{x}_{\ell}(t)={}&{ -} \frac{(T_{\ell}-T_{{\ell -1}})^{-1}(t-T_{\ell -1})}{{\Gamma (u_{\ell})}} \int _{T_{{\ell -1}}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell -1}} f_{1} \bigl(s, \widetilde{x}_{\ell}(s) \bigr)\,ds \\ &{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{{\ell -1}}}^{t}(t-s)^{u_{\ell -1}} f_{1} \bigl(s, \widetilde{x}_{\ell}(s) \bigr)\,ds \end{aligned}$$
(14)
holds.
Let \(t \in J_{\ell}, \ell = 1, 2,\ldots, n\). Then by Eqs. (13) and (14) we get
$$\begin{aligned} & \bigl\vert z(t)-x(t) \bigr\vert \\ &\quad = \bigl\vert z(t)-x_{\ell}(t) \bigr\vert = \bigl\vert z_{\ell}(t)-\widetilde{x}_{\ell}(t) \bigr\vert \\ &\quad = \biggl\vert z_{\ell}(t)+ \frac{(T_{\ell}-T_{{\ell -1}})^{-1}(t-T_{\ell -1})}{{\Gamma (u_{\ell})}} \int _{T_{{\ell -1}}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell -1}} f_{1} \bigl(s, \widetilde{x}_{\ell}(s) \bigr)\,ds \\ &\qquad{}-\frac{1}{\Gamma (u_{\ell})} \int _{T_{{\ell -1}}}^{t}(t-s)^{u_{ \ell -1}} f_{1} \bigl(s, \widetilde{x}_{\ell}(s) \bigr)\,ds \biggr\vert \\ &\quad= \biggl\vert z_{\ell}(t)+ \frac{(T_{\ell}-T_{{\ell -1}})^{-1}(t-T_{\ell -1})}{{\Gamma (u_{\ell})}} \int _{T_{{\ell -1}}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell -1}} f_{1} \bigl(s, z_{ \ell}(s) \bigr)\,ds \\ &\qquad{}-\frac{1}{\Gamma (u_{\ell})} \int _{T_{{\ell -1}}}^{t}(t-s)^{u_{ \ell -1}} f_{1} \bigl(s, z_{\ell}(s) \bigr)\,ds \biggr\vert \\ &\qquad{}+ \frac{(T_{\ell}-T_{\ell -1})^{-1}(t-T_{\ell -1})}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, z_{ \ell}(s) \bigr)-f_{1} \bigl(s, \widetilde{x}_{\ell}(s) \bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t}(t-s)^{u_{\ell}-1} \bigl\vert f_{1} \bigl(s, z_{\ell}(s) \bigr)-f_{1} \bigl(s, \widetilde{x}_{\ell}(s) \bigr) \bigr\vert \,ds \\ &\quad\leq \epsilon \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell}+1)}+ \frac{(T_{\ell}-T_{\ell -1})^{-1}(t-T_{\ell -1})}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{T_{\ell}}(T_{\ell}-s)^{u_{\ell}-1} s^{-\delta} \bigl(K \bigl\vert z_{ \ell}(s)-\widetilde{x}_{\ell}(s) \bigr\vert \bigr)\,ds \\ &\qquad{}+\frac{1}{\Gamma (u_{\ell})} \int _{T_{\ell -1}}^{t}(t-s)^{u_{\ell}-1} s^{-\delta} \bigl(K \bigl\vert z_{\ell}(s)-\widetilde{x}_{\ell}(s) \bigr\vert \bigr)\,ds \\ &\quad\leq \epsilon \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell}+1)}+ \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}}{\Gamma (u_{\ell})} \bigl(K \Vert z_{ \ell}-\widetilde{x}_{\ell} \Vert _{E_{\ell}} \bigr) \int _{T_{\ell -1}}^{T_{\ell}} s^{-\delta}\,ds \\ &\qquad{}+\frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}}{\Gamma (u_{\ell})} \bigl(K \Vert z_{ \ell}- \widetilde{x}_{\ell} \Vert _{E_{\ell}} \bigr) \int _{T_{\ell -1}}^{t} s^{- \delta}\,ds \\ &\quad\leq \epsilon \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell}+1)}+ \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}({T_{\ell}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})} \bigl(K \Vert z_{\ell}-\widetilde{x}_{\ell} \Vert _{E_{\ell}} \bigr) \\ &\qquad{}+ \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}({t}^{1-\delta}-{T_{\ell -1}}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})} \bigl(K \Vert z_{\ell}- \widetilde{x}_{\ell} \Vert _{E_{\ell}} \bigr) \\ &\quad\leq \epsilon \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell}+1)}+ \frac{2K(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}({T_{\ell}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})} \Vert z_{\ell}-\widetilde{x}_{\ell} \Vert _{E_{\ell}} \\ &\quad\leq \epsilon \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell}+1)}+\mu \Vert z-x \Vert , \end{aligned}$$
where
$$\begin{aligned} \mu =\max_{\ell = 1, 2,\ldots, n} \frac{2K(T_{\ell}-T_{\ell -1})^{u_{\ell}-1}({T_{\ell}}^{1-\delta}-{T_{\ell -1}}^{1-\delta})}{(1-\delta )\Gamma (u_{\ell})}. \end{aligned}$$
Then
$$\begin{aligned} \Vert z- x \Vert (1-\mu ) \leq \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{\Gamma (u_{\ell}+1)} \epsilon. \end{aligned}$$
We obtain, for each \(t \in J_{\ell}\),
$$\begin{aligned} \bigl\vert z(t)- x(t) \bigr\vert \leq \Vert z- x \Vert \leq \frac{(T_{\ell}-T_{\ell -1})^{u_{\ell}}}{(1-\mu )\Gamma (u_{\ell}+1)} \epsilon:=c_{f_{1}} \epsilon. \end{aligned}$$
Therefore, BVP (1) is (UH) stable. □
Examples
Example 1
Let us consider the following fractional boundary value problem:
$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{u(t)}_{0^{+}}x(t)= \frac{t^{-\frac{1}{3}}e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)(1+ \vert x(t) \vert )}, \quad t\in J:= [0,2], \\ x(0)=0, \qquad x(2)=0. \end{cases}\displaystyle \end{aligned}$$
(15)
Let
$$\begin{aligned} &f_{1}(t, y)= \frac{t^{-\frac{1}{3}}e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)(1+y)},\quad (t,y)\in [0,2]\times [0,+\infty ). \\ & u(t)= \textstyle\begin{cases} \frac{3}{2}, & t \in J_{1}:=[0, 1], \\ \frac{9}{5}, & t \in J_{2}:=]1, 2]. \end{cases}\displaystyle \end{aligned}$$
(16)
Then we have
$$\begin{aligned} t^{\frac{1}{3}} \bigl\vert f_{1}(t,y_{1})-f_{1}(t,y_{2}) \bigr\vert &= \biggl\vert \frac{e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)} \biggl( \frac{1}{1+y_{1}}- \frac{1}{1+y_{2}} \biggr) \biggr\vert \\ &\leq \frac{e^{-t} \vert y_{1}-y_{2} \vert }{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)(1+y_{1})(1+y_{2})} \\ &\leq \frac{e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)} \vert y_{1}-y_{2} \vert \\ &\leq \frac{1}{(e+5)} \vert y_{1}-y_{2} \vert . \end{aligned}$$
Hence, condition (H2) holds with \(\delta =\frac{1}{3}\) and \(K = \frac{1}{e+5}\).
By (16), according to BVP (7), we consider two auxiliary BVPs for Caputo fractional differential equations of constant order:
$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{\frac{3}{2}}_{0^{+}}x(t)= \frac{t^{-\frac{1}{3}}e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)(1+ \vert x(t) \vert )}, \quad t \in J_{1}, \\ x(0)=0,\qquad x(1)=0, \end{cases}\displaystyle \end{aligned}$$
(17)
and
$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{\frac{9}{5}}_{1^{+}}x(t)= \frac{t^{-\frac{1}{3}}e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)(1+ \vert x(t) \vert )}, \quad t \in J_{2}, \\ x(1)=0,\qquad x(2)=0. \end{cases}\displaystyle \end{aligned}$$
(18)
Next, we prove that condition (10) is fulfilled for \(\ell = 1\). Indeed,
$$\begin{aligned} \frac{2K({T_{1}}^{1-\delta}-{T_{0}}^{1-\delta})(T_{1}-T_{0})^{u_{1}-1}}{(1-\delta )\Gamma (u_{1})} =\frac{2}{\frac{2}{3}(e+5)\Gamma (\frac{3}{2})} \simeq 0.4385 < 1. \end{aligned}$$
Accordingly, condition (10) is achieved. By Theorem 3.1, BVP (17) has a solution \(\widetilde{x}_{1} \in E_{1}\).
We prove that condition (10) is fulfilled for \(\ell = 2\). Indeed,
$$\begin{aligned} \frac{2K({T_{2}}^{1-\delta}-{T_{1}}^{1-\delta})(T_{2}-T_{1})^{u_{2}-1}}{(1-\delta )\Gamma (u_{2})} =\frac{2}{e+5} \frac{{2}^{\frac{2}{3}}-1}{\frac{2}{3}\Gamma (\frac{9}{5})}\simeq 0.2451 < 1. \end{aligned}$$
Thus, condition (10) is satisfied.
According to Theorem 3.1, BVP (18) possesses a solution \(\widetilde{x}_{2} \in E_{2}\).
Then, by Theorem 3.2, BVP (15) has a solution
$$\begin{aligned} x(t)= \textstyle\begin{cases} \widetilde{x}_{1}(t), & t \in J_{1}, \\ x_{2}(t), & t \in J_{2}, \end{cases}\displaystyle \end{aligned}$$
where
$$\begin{aligned} x_{2}(t)= \textstyle\begin{cases} 0, & t \in J_{1}, \\ \widetilde{x}_{2}(t), & t \in J_{2}. \end{cases}\displaystyle \end{aligned}$$
According to Theorem 3.3, BVP (15) is (UH) stable.
Example 2
Let us consider the following fractional boundary value problem:
$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{u(t)}_{0^{+}}x(t)=\frac{t^{-\frac{1}{2}}}{5e^{t}(1+ \vert x(t) \vert )}, \quad t\in J:= [0,3], \\ x(0)=0,\qquad x(3)=0. \end{cases}\displaystyle \end{aligned}$$
(19)
Let
$$\begin{aligned} f_{1}(t, y)=\frac{t^{-\frac{1}{2}}}{5e^{t}(1+y)},\quad (t,y)\in [0,3] \times [0,+\infty ). \end{aligned}$$
$$\begin{aligned} u(t)= \textstyle\begin{cases} \frac{9}{6}, \quad t \in J_{1}:=[0, 1], \\ \frac{6}{5}, \quad t \in J_{2}:=]1, 3]. \end{cases}\displaystyle \end{aligned}$$
(20)
Then we have
$$\begin{aligned} t^{\frac{1}{2}} \bigl\vert f_{1}(t,y_{1})-f_{1}(t,y_{2}) \bigr\vert &= \biggl\vert \frac{1}{5e^{t}} \biggl(\frac{1}{1+y_{1}}- \frac{1}{1+y_{2}} \biggr) \biggr\vert \\ &\leq \frac{ \vert y_{2}-y_{1} \vert }{5e^{t}(1+y_{1})(1+y_{2})} \\ &\leq \frac{1}{5} \vert y_{1}-y_{2} \vert . \end{aligned}$$
Hence condition (H2) holds with \(\delta =\frac{1}{2}\) and \(K =\frac{1}{5}\).
By (20), according to (7), we consider two auxiliary BVPs for Caputo fractional differential equations of constant order:
$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{\frac{9}{6}}_{0^{+}}x(t)= \frac{t^{-\frac{1}{2}}}{5e^{t}(1+ \vert x(t) \vert )}, \quad t \in J_{1}, \\ x(0)=0,\qquad x(1)=0, \end{cases}\displaystyle \end{aligned}$$
(21)
and
$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{\frac{6}{5}}_{1^{+}}x(t)= \frac{t^{-\frac{1}{2}}}{5e^{t}(1+ \vert x(t) \vert )}, \quad t \in J_{2}, \\ x(1)=0, \qquad x(3)=0. \end{cases}\displaystyle \end{aligned}$$
(22)
Next, we prove that condition (10) is fulfilled for \(\ell = 1\). Indeed,
$$\begin{aligned} \frac{2K({T_{1}}^{1-\delta}-{T_{0}}^{1-\delta})(T_{1}-T_{0})^{u_{1}-1}}{(1-\delta )\Gamma (u_{1})} =\frac{2\frac{1}{5}}{\frac{1}{2}\Gamma (\frac{9}{6})} \simeq 0.9027 < 1. \end{aligned}$$
Accordingly, condition (10) is achieved. By Theorem 3.1, BVP (21) has a solution \(\widetilde{x}_{1} \in E_{1}\).
We prove that condition (10) is fulfilled for \(\ell = 2\). Indeed,
$$\begin{aligned} \frac{2K({T_{2}}^{1-\delta}-{T_{1}}^{1-\delta})(T_{2}-T_{1})^{u_{2}-1}}{(1-\delta )\Gamma (u_{2})} = \frac{2\frac{1}{5}(\sqrt{3}-1)(2)^{0.2}}{\frac{1}{2}\Gamma (\frac{6}{5})} \simeq 0.7326 < 1. \end{aligned}$$
Thus, condition (10) is satisfied.
According to Theorem 3.1, BVP (22) possesses a solution \(\widetilde{x}_{2} \in E_{2}\).
Then, by Theorem 3.2, BVP (19) has a solution
$$\begin{aligned} x(t)= \textstyle\begin{cases} \widetilde{x}_{1}(t), & t \in J_{1}, \\ x_{2}(t), & t \in J_{2}, \end{cases}\displaystyle \end{aligned}$$
where
$$\begin{aligned} x_{2}(t)= \textstyle\begin{cases} 0, & t \in J_{1}, \\ \widetilde{x}_{2}(t), & t \in J_{2}. \end{cases}\displaystyle \end{aligned}$$
According to Theorem 3.3, BVP (19) is (UH) stable.