Let us introduce the following assumption:
- (H1):
-
Let \(n\in N\) 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
$$\begin{aligned} I_{\ell }(t)= \textstyle\begin{cases} 1 & \text{for } t\in J_{\ell }, \\ 0 & \text{for } \text{elsewhere}. \end{cases}\displaystyle \end{aligned}$$
For each \(\ell \in \{1, 2,\ldots,n \}\), the symbol \(E_{\ell }= C(J_{\ell },\Re )\), indicates the Banach space of continuous functions \(x:J_{\ell } \to \Re \) 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 left Caputo fractional derivative of variable order \(u(t)\) for the function \(x(t) \in C(J,\Re )\), 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), the BVP (1) can be written for any \(t \in J_{\ell }, \ell = 1, 2, \ldots, n\) in the form
$$\begin{aligned} \int _{0}^{T_{1}}\frac{(t-s)^{1-u_{1}}}{\Gamma (2-u_{1})}x^{(2)}(s) \,ds +\cdots+ \int _{T_{\ell -1}}^{t}\frac{(t-s)^{1-u_{\ell }}}{\Gamma (2-u_{\ell })}x^{(2)}(s) \,ds = f_{1}\bigl(t, x(t), {}^{c}D^{u(t)}_{0^{+}}x(t) \bigr). \end{aligned}$$
(6)
In what follows we shall introduce the solution to the BVP (1).
Definition 3.1
The BVP (1) has a solution, if there are functions \(x_{\ell }, \ell =1, 2,\ldots, n\), so that \(x_{\ell } \in C([0, T_{\ell }], \Re )\), fulfilling Eq. (6), and \(x_{\ell }(0) = 0 = x_{\ell }(T_{\ell })\).
Let the function \(x \in C(J, \Re )\) be such that \(x(t) \equiv 0\) on \(t \in [0, T_{\ell -1}]\) and such that it solves the integral equation (6). Then (6) is reduced to
$$\begin{aligned} {}^{c}D^{u_{\ell }}_{T_{\ell -1}^{+}} x(t)= f_{1}\bigl(t, x(t), {}^{c}D^{u_{ \ell }}_{T_{\ell -1}^{+}}x(t)\bigr),\quad t \in J_{\ell }. \end{aligned}$$
We shall deal with the following BVP:
$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{u_{\ell }}_{T_{\ell -1}^{+}} x(t)= f_{1}(t, x(t), {}^{c}D^{u_{ \ell }}_{T_{\ell -1}^{+}}x(t)),\quad t \in J_{\ell } \\ x(T_{{\ell -1}})=0, \qquad x(T_{\ell })=0. \end{cases}\displaystyle \end{aligned}$$
(7)
For our purpose, the upcoming lemma will be a corner stone of the solution of the BVP (7).
Lemma 3.1
Let \(\ell \in \{1,2,\ldots,n\}\) be a natural number, \(f_{1}\in C(J_{\ell } \times \Re \times \Re, \Re )\) and there exists a number \(\delta \in (0, 1)\) such that \(t^{\delta } f_{1}\in C(J_{\ell } \times \Re \times \Re, \Re )\).
Then the function \(x \in E_{\ell }\) is a solution of the BVP (7) if and only if x solves the integral equation
$$\begin{aligned} x(t) =-(T_{\ell }-T_{{\ell -1}})^{-1}(t-T_{\ell -1})I^{u_{\ell }}_{T_{ \ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(t), \end{aligned}$$
(8)
where
$$\begin{aligned} y(t) =f_{1} \bigl(t, -(T_{\ell }-T_{{\ell -1}})^{-1}(t-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(t), y(t) \bigr),\quad t \in J_{\ell }. \end{aligned}$$
Proof
We presume that \(x \in E_{\ell }\) is solution of the BVP (7) and we take \({}^{c}D^{u_{\ell }}_{T_{\ell -1}^{+}} x(t)= y(t)\). Employing the operator \(I^{u_{\ell }}_{T_{\ell -1}^{+}}\) to both sides of (7) and regarding Lemma 2.1, we find
$$\begin{aligned} x(t)=\omega _{1} + \omega _{2}(t-T_{{\ell -1}})+I^{u_{\ell }}_{T_{ \ell -1}^{+}}y(t),\quad t \in J_{\ell }. \end{aligned}$$
By \(x(T_{\ell -1}) = 0\), we get \(\omega _{1}=0\).
Let \(x(t)\) satisfy \(x(T_{\ell })=0\). So, we observe that
$$\begin{aligned} \omega _{2} = -(T_{\ell }-T_{{\ell -1}})^{-1} I^{u_{\ell }}_{T_{\ell -1}^{+}} y(T_{\ell }). \end{aligned}$$
Then we find
$$\begin{aligned} x(t) =-(T_{\ell }-T_{{\ell -1}})^{-1}(t-T_{\ell -1})I^{u_{\ell }}_{T_{ \ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(t), \end{aligned}$$
where
$$\begin{aligned} y(t) =f_{1} \bigl(t, -(T_{\ell }-T_{{\ell -1}})^{-1}(t-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(t), y(t) \bigr),\quad t \in J_{\ell }. \end{aligned}$$
Conversely, let \(x \in E_{\ell }\) be a solution of the integral equation (8). Regarding the continuity of the function \(t^{\delta } f_{1}\) and Lemma 2.1, we deduce that x is the solution of the BVP (7).
We will prove the existence result for the BVP (7). This result is based on Theorem 2.1. □
Theorem 3.1
Let the conditions of Lemma 3.1be satisfied and there exist constants \(K, L >0\), such that \(t^{\delta }|f_{1}(t,y_{1}, z_{1})- f_{1}(t,y_{2}, z_{2})|\leq K|y_{1}-y_{2}|+ L|z_{1}-z_{2}|\), for any \(y_{i}, z_{i} \in \Re \), \(i = 1, 2\), \(t\in J_{\ell }\). and the inequality
$$\begin{aligned} \frac{2(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}(T_{\ell }^{1-\delta }-T_{\ell -1}^{1-\delta })}{(1-\delta )\Gamma (u_{\ell })} \biggl( 2K\frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}+L\biggr)< 1 , \end{aligned}$$
(9)
holds.
Then the BVP (7) possesses at least one solution in \(E_{\ell }\).
Proof
We construct the operators
$$\begin{aligned} W_{1}, W_{2}: E_{\ell } \rightarrow E_{\ell } \end{aligned}$$
as follows:
$$\begin{aligned} W_{1}y(t) =-(T_{\ell }-T_{{\ell -1}})^{-1}(t-T_{\ell -1})I^{u_{\ell }}_{T_{ \ell -1}^{+}} y(T_{\ell }),\qquad W_{2}y(t) =I^{u_{\ell }}_{T_{\ell -1}^{+}}y(t), \end{aligned}$$
(10)
where
$$\begin{aligned} y(t) =f_{1} \bigl(t, -(T_{\ell }-T_{{\ell -1}})^{-1}(t-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(t), y(t) \bigr),\quad t \in J_{\ell }. \end{aligned}$$
It follows from the properties of fractional integrals and from the continuity of the function \(t^{\delta }f_{1}\) that the operators \(W_{1}, W_{2}: E_{\ell }\) → \(E_{\ell }\) defined in (10) are well defined.
Let
$$\begin{aligned} R_{\ell } \geq \frac{\frac{2f^{\star }(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}}{1-\frac{2(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}(T_{\ell }^{1-\delta } -T_{\ell -1}^{1-\delta })}{(1-\delta )\Gamma (u_{\ell })} ( 2K\frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}+L)}, \end{aligned}$$
where
$$\begin{aligned} f^{\star }= \sup_{t\in J_{\ell }} \bigl\vert f_{1}(t, 0, 0) \bigr\vert . \end{aligned}$$
We consider the set
$$\begin{aligned} B_{R_{\ell }}=\bigl\{ y \in E_{\ell }, \Vert y \Vert _{E_{\ell }}\leq R_{\ell }\bigr\} . \end{aligned}$$
Clearly \(B_{R_{\ell }}\) is nonempty, closed, convex and bounded.
Now, we demonstrate that \(W_{1}, W_{2}\) satisfy the assumption of Theorem 2.1. We shall prove it in four phases.
STEP 1: Claim: \(W_{1}(B_{R_{\ell }})+ W_{2}(B_{R_{\ell }})\subseteq (B_{R_{\ell }})\).
For \(y \in B_{R_{\ell }}\), we have
$$\begin{aligned} & \bigl\vert (W_{1}y) (t)+(W_{2}y) (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, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{ \ell -1}^{+}} y(T_{\ell })\\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s), y(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, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} y(T_{\ell })\\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s), y(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, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{ \ell -1})I^{u_{\ell }}_{T_{\ell -1}^{+}} y(T_{\ell })\\ &\qquad{}+ I^{u_{\ell }}_{T_{ \ell -1}^{+}}y(s), y(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, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{ \ell -1})I^{u_{\ell }}_{T_{\ell -1}^{+}} y(T_{\ell })\\ &\qquad{}+ I^{u_{\ell }}_{T_{ \ell -1}^{+}}y(s), y(s) \bigr)-f_{1}(s, 0, 0) \bigr\vert \,ds \\ &\qquad{}+\frac{2}{\Gamma (u_{\ell })} \int _{T_{\ell -1}}^{T_{\ell }}(T_{\ell }-s)^{u_{ \ell }-1} \bigl\vert f_{1}(s, 0, 0) \bigr\vert \,ds \\ &\quad\leq \frac{2}{\Gamma (u_{\ell })} \int _{T_{\ell -1}}^{T_{\ell }}(T_{ \ell }-s)^{u_{\ell }-1} s^{-\delta }\bigl(K \bigl\vert -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{ \ell -1})I^{u_{\ell }}_{T_{\ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{ \ell -1}^{+}}y(s) \bigr\vert \\ &\qquad{} + L \bigl\vert y(s) \bigr\vert \bigr)\,ds + \frac{2f^{\star }(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)} \\ &\quad\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 I^{u_{\ell }}_{T_{\ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s) \bigr\vert + L \bigl\vert y(s) \bigr\vert \bigr)\,ds \\ &\qquad{}+ \frac{2f^{\star }(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)} \\ &\quad\leq \frac{2(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \bigl(2K \bigl\Vert I^{u_{\ell }}_{T_{\ell -1}^{+}} y \bigr\Vert _{E_{\ell }}+ L \Vert y \Vert _{E_{\ell }}\bigr) \int _{T_{\ell -1}}^{T_{\ell }} s^{-\delta }\,ds + \frac{2f^{\star }(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)} \\ &\quad\leq \frac{2(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}(T_{\ell }^{1-\delta }-T_{\ell -1}^{1-\delta })}{(1-\delta )\Gamma (u_{\ell })} \biggl( 2K\frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}+L\biggr)R_{ \ell } + \frac{2f^{\star }(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)} \\ &\quad\leq R_{\ell }, \end{aligned}$$
which means that \(W_{1}(B_{R_{\ell }})+ W_{2}(B_{R_{\ell }}) \subseteq B_{R_{\ell }} \).
STEP 2: Claim: \(W_{1}\) is continuous.
We presume that the sequence \((y_{n})\) converges to y in \(E_{\ell }\) and \(t \in J_{\ell }\). Then
$$\begin{aligned} & \bigl\vert (W_{1}y_{n}) (t)-(W_{1}y) (t) \bigr\vert \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{-1}(t-T_{\ell -1})}{\Gamma (u_{\ell })} \\ &\qquad{}\times\int _{T_{\ell -1}}^{T_{\ell }}(T_{\ell }-s)^{u_{\ell }-1} \bigl\vert f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{ \ell -1}^{+}} y_{n}(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y_{n}(s), y_{n}(s) \bigr) \\ &\qquad{}-f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s), y(s) \bigr) \bigr\vert \,ds \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{-1}(t-T_{\ell -1})}{\Gamma (u_{\ell })} \\ &\qquad{}\times\int _{T_{\ell -1}}^{T_{\ell }}(T_{\ell }-s)^{u_{\ell }-1} s^{-\delta } \bigl(K \bigl\vert -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{ \ell -1}^{+}} \bigl(y_{n}(T_{\ell })-y(T_{\ell })\bigr) \\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}}\bigl(y_{n}(s)-y(s)\bigr) \bigr\vert +L \bigl\vert \bigl(y_{n}(s)-y(s)\bigr) \bigr\vert \bigr) \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \int _{T_{\ell -1}}^{T_{\ell }} s^{-\delta } \bigl(K \bigl\vert I^{u_{\ell }}_{T_{ \ell -1}^{+}} \bigl(y_{n}(T_{\ell })-y(T_{\ell }) \bigr)\\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}}\bigl(y_{n}(s)-y(s)\bigr) \bigr\vert +L \bigl\vert \bigl(y_{n}(s)-y(s)\bigr) \bigr\vert \bigr) \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \bigl(2K \bigl\Vert I^{u_{\ell }}_{T_{\ell -1}^{+}} (y_{n}-y) \bigr\Vert _{E_{\ell }}+ L \Vert y_{n}-y \Vert _{E_{ \ell }}\bigr) \int _{T_{\ell -1}}^{T_{\ell }} s^{-\delta }\,ds \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}(T_{\ell }^{1-\delta }-T_{\ell -1}^{1-\delta })}{(1-\delta )\Gamma (u_{\ell })} \biggl( 2K\frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}+L\biggr) \Vert y_{n}-y \Vert _{E_{\ell }}, \end{aligned}$$
i.e., we obtain
$$\begin{aligned} \bigl\Vert (W_{1}y_{n})-(W_{1}y) \bigr\Vert _{E_{\ell }}\rightarrow 0 \quad\text{as } n \rightarrow \infty. \end{aligned}$$
Ergo, the operator \(W_{1}\) is a continuous on \(E_{\ell }\).
STEP 3: \(W_{1}\) is compact
Now, we will show that \(W_{1}(B_{R_{\ell }})\) is relatively compact, meaning that \(W_{1}\) is compact. Clearly \(W_{1}(B_{R_{\ell }})\) is uniformly bounded because by Step 1, we have \(W_{1}(B_{R_{\ell }})= \{W_{1}(y): y \in B_{R_{\ell }} \}\subset W_{1}(B_{R_{ \ell }})+ W_{2}(B_{R_{\ell }})\subseteq (B_{R_{\ell }})\) thus for each \(y \in B_{R_{\ell }}\) we have \(\|W_{1}(y)\|_{E_{\ell }} \leq R_{\ell }\), which means that \(W_{1}(B_{R_{\ell }})\) is bounded. It remains to show that \(W_{1}(B_{R_{\ell }})\) is equicontinuous.
For \(t_{1},t_{2}\in J_{\ell }, t_{1} < t_{2}\) and \(y \in B_{R_{\ell }}\), we have
$$\begin{aligned} &\bigl\vert (W_{1}y) (t_{2})-(W_{1}y) (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, -(T_{ \ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{\ell -1}^{+}} y(T_{\ell }) \\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s), y(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, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{\ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s), y(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) \\ &\qquad{}\times \int _{T_{\ell -1}}^{T_{\ell }}(T_{\ell }-s)^{u_{\ell }-1} \bigl\vert f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{ \ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s), y(s) \bigr) \bigr\vert \,ds \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-2}}{\Gamma (u_{\ell })} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \\ &\qquad{}\times \int _{T_{\ell -1}}^{T_{\ell }} \bigl\vert f_{1} \bigl(s, -(T_{\ell }-T_{{ \ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{\ell -1}^{+}} y(T_{ \ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s), y(s) \bigr)-f_{1}(s, 0, 0) \bigr\vert \,ds \\ &\qquad{}+\frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-2}}{\Gamma (u_{\ell })} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \int _{T_{\ell -1}}^{T_{ \ell }} \bigl\vert f_{1}(s, 0, 0) \bigr\vert \,ds \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-2}}{\Gamma (u_{\ell })} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \\ &\qquad{}\times \int _{T_{\ell -1}}^{T_{\ell }} s^{-\delta } \bigl(K \bigl\vert -(T_{\ell }-T_{{ \ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{\ell -1}^{+}} y(T_{ \ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s) \bigr\vert + L \bigl\vert y(s) \bigr\vert \bigr) )\,ds \\ &\qquad{}+ \frac{f^{\star }(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-2}}{\Gamma (u_{\ell })} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \\ &\qquad{}\times\int _{T_{\ell -1}}^{T_{ \ell }} s^{-\delta } \bigl(K \bigl\vert I^{u_{\ell }}_{T_{\ell -1}^{+}} y(T_{\ell })+ I^{u_{ \ell }}_{T_{\ell -1}^{+}}y(s) \bigr\vert + L \bigl\vert y(s) \bigr\vert \bigr) )\,ds \\ &\qquad{}+ \frac{f^{\star }(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-2}}{\Gamma (u_{\ell })} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \bigl(2K \bigl\Vert I^{u_{\ell }}_{T_{ \ell -1}^{+}} y \bigr\Vert _{E_{\ell }}+ L \Vert y \Vert _{E_{\ell }}\bigr) \int _{T_{\ell -1}}^{T_{ \ell }} s^{-\delta } \,ds \\ &\qquad{}+ \frac{f^{\star }(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \\ &\quad\leq \frac{(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) \\ &\qquad{}\times\biggl( 2K \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}+L\biggr) \Vert y \Vert _{E_{ \ell }} \\ &\qquad{}+ \frac{f^{\star }(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr) \\ &\quad\leq \biggl[ \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-2}(T_{\ell }^{1-\delta }-T_{\ell -1}^{1-\delta })}{(1-\delta )\Gamma (u_{\ell })} \biggl( 2K\frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}+L\biggr) \Vert y \Vert _{E_{\ell }}\\ &\qquad{}+ \frac{f^{\star }(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \biggr] \\ &\qquad{}\times \bigl((t_{2}-T_{\ell -1})-(t_{1}-T_{\ell -1}) \bigr). \end{aligned}$$
Hence \(\|(W_{1}y)(t_{2})-(W_{1}y)(t_{1})\|_{E_{\ell }}\rightarrow 0\) as \(|t_{2}-t_{1}|\rightarrow 0\). It implies that \(W_{1}(B_{R_{\ell }})\) is equicontinuous.
STEP 4: \(W_{2}\) is a strict contraction
For \(x(t), y(t) \in E_{\ell }\), we obtain
$$\begin{aligned} & \bigl\vert (W_{2}x) (t)-(W_{2}y) (t) \bigr\vert \\ &\quad= \biggl\vert \frac{1}{\Gamma (u_{\ell })} \int _{T_{\ell -1}}^{t}(t-s)^{u_{ \ell }-1} f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} x(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}x(s), x(s) \bigr)\,ds \\ &\qquad{}-\frac{1}{\Gamma (u_{\ell })} \int _{T_{\ell -1}}^{t}(t-s)^{u_{\ell }-1}f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{ \ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s), y(s) \bigr)\,ds \biggr\vert \\ &\quad\leq \frac{1}{\Gamma (u_{\ell })} \int _{T_{\ell -1}}^{t}(t-s)^{u_{ \ell }-1} \bigl\vert f_{1} \bigl(s,-(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} x(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}x(s), x(s) \bigr) \\ &\qquad{}-f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} y(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}y(s), y(s) \bigr) \bigr\vert \,ds \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \int _{T_{\ell -1}}^{t}s^{-\delta } \bigl(K \bigl\vert (T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{ \ell -1}) \bigl( I^{u_{\ell }}_{T_{\ell -1}^{+}} (x-y) (T_{\ell })\bigr) \\ &\qquad{}+\bigl(I^{u_{\ell }}_{T_{\ell -1}^{+}}(x-y) (s)\bigr) \bigr\vert +L \bigl\vert (x-y) (s) \bigr\vert \bigr)\,ds \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \int _{T_{\ell -1}}^{t}s^{-\delta } (K|\bigl( I^{u_{\ell }}_{T_{\ell -1}^{+}} (x-y) (T_{\ell })+I^{u_{\ell }}_{T_{\ell -1}^{+}}(x-y) (s) \vert +L \bigl\vert (x-y) (s) \bigr\vert \bigr)\,ds \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} (2K\|\bigl( I^{u_{\ell }}_{T_{\ell -1}^{+}} (x-y) \|_{E_{\ell }}+L\|x-y\|_{E_{ \ell }} \bigr) \int _{T_{\ell -1}}^{t}s^{-\delta }\,ds \\ &\quad\leq \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}(T_{\ell }^{1-\delta }-T_{\ell -1}^{1-\delta })}{(1-\delta )\Gamma (u_{\ell })} \biggl(2K\frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}+L \biggr) \Vert x-y \Vert _{E_{\ell }}. \end{aligned}$$
Consequently by (9), the operator \(W_{2}\) is a strict contraction.
Therefore, all conditions of Theorem 2.1 are fulfilled and thus there exists \(\widetilde{x_{\ell }}\in B_{R_{\ell }}\), such that \(W_{1}\widetilde{x_{\ell }}+W_{2}\widetilde{x_{\ell }}=\widetilde{x_{ \ell }}\), which is a solution of the BVP (7). Since \(B_{R_{\ell }} \subset E_{\ell }\), the claim of Theorem 3.1 is proved.
Now, we will prove the existence result for the BVP (1).
Introduce the following assumption:
- (H2):
-
Let \(f_{1}\in C(J \times \Re \times \Re, \Re )\) and there exists a number \(\delta \in (0, 1)\) such that \(t^{\delta } f_{1}\in C(J \times \Re \times \Re, \Re )\) and there exist constants \(K, L >0\), such that \(t^{\delta }|f_{1}(t,y_{1}, z_{1})- f_{1}(t,y_{2}, z_{2})|\leq K|y_{1}-y_{2}|+ L|z_{1}-z_{2}|\), for any \(y_{1}, y_{2}, z_{1}, z_{2} \in \Re \) and \(t\in J\). □
Theorem 3.2
Let the conditions (H1), (H2) and inequality (9) be satisfied for all \(\ell \in \{1,2,\ldots,n\}\).
Then the problem (1) possesses at least one solution in \(C(J, \Re )\).
Proof
For any \(\ell \in \{1,2,\ldots,n\}\) according to Theorem 3.1 the 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
$$\begin{aligned} {x}_{\ell }= \textstyle\begin{cases} 0, & t \in [0, T_{\ell -1}], \\ \widetilde{x}_{\ell }, & t \in J_{\ell }. \end{cases}\displaystyle \end{aligned}$$
Thus, the function \(x_{\ell } \in C([0, T_{\ell }], \Re )\) 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
$$\begin{aligned} x(t)= \textstyle\begin{cases} x_{1}(t),& t \in J_{1}, \\ x_{2}(t)=\textstyle\begin{cases} 0, & t \in J_{1}, \\ \widetilde{x}_{2},& t \in J_{2}, \end{cases}\displaystyle \\ \vdots \\ x_{n}(t)=\textstyle\begin{cases} 0, & t \in [0, T_{\ell -1}], \\ \widetilde{x}_{\ell }, & t \in J_{\ell }, \end{cases}\displaystyle \end{cases}\displaystyle \end{aligned}$$
(11)
is a solution of the BVP (1) in \(C(J, \Re )\). □