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Implicit nonlinear fractional differential equations of variable order

Abstract

In this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

Introduction

The idea of fractional calculus is to replace the natural numbers in the derivative’s order with rational ones. Although it seems an elementary consideration, it has an exciting relevance explaining some physical phenomena. Especially in the last two decades, significant numbers of papers appeared on this topic, some papers deal with the existence of solutions to problems of variable order; see e.g. [3, 4, 9, 10, 12].

In particular, [2] Benchohra et al. studied the existence and uniqueness results for the following nonlinear implicit fractional differential equations:

$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{u}_{0^{+}}x(t)= f(t, x(t), {}^{c}D^{u}_{0^{+}}x(t)),\quad t\in [0, T], 0 < T < +\infty, 1 < u \leq 2, \\ x(0)=x_{0},\qquad x(T)=x_{1}, \end{cases}\displaystyle \end{aligned}$$

where f is a given function, \(x_{0}, x_{1}\in \Re \), and \({}^{c}D^{u}_{0^{+}}\) is the Caputo fractional derivative of order u.

Inspired by [2] and [3, 4, 9, 10, 12], we deal with the boundary value problem (BVP)

$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{u(t)}_{0^{+}}x(t)= f_{1}(t, x(t), {}^{c}D^{u(t)}_{0^{+}}x(t)),\quad t\in J:= [0, T] \\ x(0)=0, \qquad x(T)=0, \end{cases}\displaystyle \end{aligned}$$
(1)

where \(u: J \rightarrow (1, 2]\), \(f_{1}:J\times \Re \times \Re \rightarrow \Re \) is a continuous function and \({}^{c}D^{u(t)}_{0^{+}}\) is the Caputo fractional derivative of variable-order \(u(t)\).

In this paper, we shall look for a solution of (1). Further, we study the stability of the obtained solution of (1) in the sense of Ulam–Hyers (UH).

Preliminaries

This section introduces some important fundamental definitions that will be needed for obtaining our results in the next sections.

The symbol \(C(J, \Re )\) represents the Banach space of continuous functions \(x:J \to \Re \) with the norm

$$\begin{aligned} \Vert x \Vert =\operatorname{Sup} \bigl\{ \bigl\vert x(t) \bigr\vert : t \in J\bigr\} . \end{aligned}$$

For \(- \infty < a_{1} < a_{2} < + \infty \), we consider the mappings \(u(t): [a_{1}, a_{2}]\rightarrow (0, +\infty ) \) and \(v(t): [a_{1}, a_{2}]\rightarrow (n-1, n)\), \(n \in N \). Then the left Caputo fractional integral (CFI) of variable-order \(u(t)\) for the function \(f_{2}(t)\) [7, 8, 11] is

$$\begin{aligned} I^{u(t)}_{a_{1}^{+}}f_{2}(t)= \int _{a_{1}}^{t} \frac{(t-s)^{u(t)-1}}{\Gamma (u(t))}f_{2}(s) \,ds,\quad t> a_{1}, \end{aligned}$$
(2)

and the left Caputo fractional derivative (CFD) of variable-order \(v(t)\) for the function \(f_{2}(t)\) [7, 8, 11] is

$$\begin{aligned} {}^{c}D^{v(t)}_{a_{1}^{+}}f_{2}(t)= \int _{a_{1}}^{t} \frac{(t-s)^{n-v(t)-1}}{\Gamma (n-v(t))}f^{(n)}_{2}(s) \,ds,\quad t> a_{1}. \end{aligned}$$
(3)

As anticipated, in the case of \(u(t)\) and \(v(t)\) being constant, then CFI and CFD coincide with the standard Caputo fractional derivative and integral, respectively; see e.g. [68].

Recall the following pivotal observation.

Lemma 2.1

([6])

Let \(\alpha _{1}, \alpha _{2} >0\), \(a_{1} >0\), \(f_{2} \in L(a_{1}, a_{2})\), \({}^{c}D_{a_{1}^{+}}^{\alpha _{1}}f_{2}\in L(a_{1}, a_{2})\). Then the differential equation

$$\begin{aligned} {}^{c}D_{a_{1}^{+}}^{\alpha _{1}}f_{2}=0 \end{aligned}$$

has the unique solution

$$\begin{aligned} f_{2}(t)=\omega _{0}+\omega _{1}(t-a_{1})+ \omega _{2}(t-a_{1})^{2}+\cdots+ \omega _{n-1}(t-a_{1})^{n-1} \end{aligned}$$

and

$$\begin{aligned} I_{a_{1}^{+}}^{\alpha _{1}} {}^{c}D_{a_{1}^{+}}^{\alpha _{1}}f_{2}(t)=f_{2}(t)+ \omega _{0}+\omega _{1}(t-a_{1})+\omega _{2}(t-a_{1})^{2}+\cdots+\omega _{n-1}(t-a_{1})^{n-1} \end{aligned}$$

with \(n-1 < \alpha _{1} \leq n\), \(\omega _{\ell }\in \Re \), \(\ell =0,1,\ldots,n-1\).

Furthermore,

$$\begin{aligned} {}^{c}D_{a_{1}^{+}}^{\alpha _{1}}I_{a_{1}^{+}}^{\alpha _{1}}f_{2}(t)=f_{2}(t) \end{aligned}$$

and

$$\begin{aligned} I_{a_{1}^{+}}^{\alpha _{1}}I_{a_{1}^{+}}^{\alpha _{2}}f_{2}(t)=I_{a_{1}^{+}}^{ \alpha _{2}}I_{a_{1}^{+}}^{\alpha _{1}}f_{2}(t)=I_{a_{1}^{+}}^{ \alpha _{1}+\alpha _{2}}f_{2}(t). \end{aligned}$$

Remark 2.1

([13, 15, 16])

Note that the semigroup property is not fulfilled for general functions \(u(t)\), \(v(t)\), i.e.,

$$\begin{aligned} I_{a_{1}^{+}}^{u(t)}I_{a_{1}^{+}}^{v(t)}f_{2}(t) \neq I_{a_{1}^{+}}^{u(t)+v(t)}f_{2}(t). \end{aligned}$$

Example 2.1

Let

$$\begin{aligned} &u(t)=t,\quad t \in [0, 4],\qquad v(t)=\textstyle\begin{cases} 2, & t \in [0, 1], \\ 3, & t \in ]1, 4], \end{cases}\displaystyle \qquad f_{2}(t)=2, \quad t \in [0, 4], \\ &I_{0^{+}}^{u(t)}I_{0^{+}}^{v(t)}f_{2}(t)= \int _{0}^{t} \frac{(t-s)^{u(t)-1}}{\Gamma (u(t))} \int _{0}^{s} \frac{(s-\tau )^{v(s)-1}}{\Gamma (v(s))}f_{2}( \tau )\,d\tau\, ds \\ &\phantom{I_{0^{+}}^{u(t)}I_{0^{+}}^{v(t)}f_{2}(t)}= \int _{0}^{t}\frac{(t-s)^{t-1}}{\Gamma (t)}\biggl[ \int _{0}^{1} \frac{(s-\tau )}{\Gamma (2)}2\,d\tau + \int _{1}^{s} \frac{(s-\tau )^{2}}{\Gamma (3)}2\,d\tau \biggr] \,ds \\ &\phantom{I_{0^{+}}^{u(t)}I_{0^{+}}^{v(t)}f_{2}(t)}= \int _{0}^{t}\frac{(t-s)^{t-1}}{\Gamma (t)} \biggl[ 2s-1 + \frac{(s-1)^{3}}{3} \biggr] \,ds, \end{aligned}$$

and

$$\begin{aligned} I_{0^{+}}^{u(t)+v(t)}f_{2}(t)= \int _{0}^{t} \frac{(t-s)^{u(t)+v(t)-1}}{\Gamma (u(t)+v(t))}f_{2}(s) \,ds. \end{aligned}$$

So, we get

$$\begin{aligned} &I_{0^{+}}^{u(t)}I_{0^{+}}^{v(t)}f_{2}(t)|_{t=3}= \int _{0}^{3} \frac{(3-s)^{2}}{\Gamma (3)} \biggl[ 2s-1 + \frac{(s-1)^{3}}{3} \biggr] \,ds \\ &\phantom{I_{0^{+}}^{u(t)}I_{0^{+}}^{v(t)}f_{2}(t)|_{t=3}}=\frac{21}{10}, \\ &I_{0^{+}}^{u(t)+v(t)}f_{2}(t)|_{t=3}= \int _{0}^{3} \frac{(3-s)^{u(t)+v(t)-1}}{\Gamma (u(t)+v(t))}f_{2}(s) \,ds \\ &\phantom{I_{0^{+}}^{u(t)+v(t)}f_{2}(t)|_{t=3}}= \int _{0}^{1}\frac{(3-s)^{4}}{\Gamma (5)}2\,ds + \int _{1}^{3} \frac{(3-s)^{5}}{\Gamma (6)}2\,ds \\ &\phantom{I_{0^{+}}^{u(t)+v(t)}f_{2}(t)|_{t=3}}=\frac{1}{12} \int _{0}^{1}\bigl(s^{4}-12s^{3}+54s^{2}-108s+81 \bigr)\,ds \\ &\phantom{I_{0^{+}}^{u(t)+v(t)}f_{2}(t)|_{t=3}=}{}+\frac{1}{60} \int _{1}^{3}\bigl(-s^{5}+15s^{4}-90s^{3}+270s^{2}-405s+243 \bigr)\,ds \\ &\phantom{I_{0^{+}}^{u(t)+v(t)}f_{2}(t)|_{t=3}}=\frac{665}{180}. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} I_{0^{+}}^{u(t)}I_{0^{+}}^{v(t)}f_{2}(t)|_{t=3} \neq I_{0^{+}}^{u(t)+v(t)}f_{2}(t)|_{t=3}. \end{aligned}$$

Lemma 2.2

([18])

Let \(u: J \rightarrow (1, 2]\) be a continuous function, then, for \(f_{2} \in C_{\delta }(J, {\Re })=\{ f_{2}(t)\in C(J, {\Re }), t^{ \delta }f_{2}(t) \in C(J, {\Re }), 0 \leq \delta \leq 1 \}\), the variable order fractional integral \(I^{u(t)}_{0^{+}}f_{2}(t)\) exists for any points on J.

Lemma 2.3

([18])

Let \(u: J \rightarrow (1, 2]\) be a continuous function, then \(I^{u(t)}_{0^{+}} f_{2}(t)\in C(J, \Re )\) for \(f_{2} \in C(J, \Re )\).

Definition 2.1

([5, 14, 17])

Let \(I \subset \Re \), I is called a generalized interval if it is either an interval, or \(\{a_{1}\}\) or .

A finite set \({\mathcal{P}}\) is called a partition of I if each x in I lies in exactly one of the generalized intervals E in \({\mathcal{P}}\).

A function \(g: I \rightarrow \Re \) is called piecewise constant with respect to partition \({\mathcal{P}}\) of I if for any \(E \in {\mathcal{P}}\), g is constant on E.

Theorem 2.1

(Krasnoselskii fixed point theorem [6])

Let S be a closed, bounded and convex subset of a real Banach space E and let \(W_{1}\) and \(W_{2}\) be operators on S satisfying the following conditions:

(i) \(W_{1}(S)+W_{2}(S)\subset S\),

(ii) \(W_{1}\) is continuous on S and \(W_{1}(S)\) is a relatively compact subset of E,

(iii) \(W_{2}\) is a strict contraction on S, i.e., there exists \(k\in [0,1)\), such that

$$\begin{aligned} \bigl\Vert W_{2}(x)-W_{2}(y) \bigr\Vert \leq k \Vert x-y \Vert \end{aligned}$$

for every \(x, y \in S\).

Then there exists \(x \in S\) such that \(W_{1}(x)+W_{2}(x)=x\).

Definition 2.2

([1])

Equation (1) is (UH) stable if there exists \(c_{f_{1}}>0\), such that, for any \(\epsilon >0\) and for every solution \(z \in C(J, \Re )\) of the following inequality:

$$\begin{aligned} \bigl\vert {}^{c}D^{u(t)}_{0^{+}}z(t)- f_{1}\bigl(t, z(t), {}^{c}D^{u(t)}_{0^{+}}z(t) \bigr) \bigr\vert \leq \epsilon,\quad t\in J, \end{aligned}$$
(4)

there exists a solution \(x \in C(J, \Re )\) of Eq. (1) with

$$\begin{aligned} \bigl\vert z(t)- x(t) \bigr\vert \leq c_{f_{1}} \epsilon,\quad t\in J. \end{aligned}$$

Existence of solutions

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 \(C(J, \Re )\). □

Ulam–Hyers stability

Theorem 4.1

Let the conditions (H1), (H2) and inequality (9) be satisfied. Then BVP (1) is (UH) stable.

Proof

Let \(\epsilon >0\) an arbitrary number and the function \(z(t)\) from \(z \in C(J_{\ell }, \Re )\) satisfy inequality (4).

For any \(\ell \in \{1,2,\ldots,n\}\) we define the functions \(z_{1}(t)\equiv z(t), t \in [0, T_{1}]\) 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 (CFI) \(I^{u_{\ell }}_{T_{\ell -1}^{+}}\) of both sides of the inequality (4), we obtain

$$\begin{aligned} &\biggl\vert z_{\ell }(t)+\frac{(T_{\ell }-T_{{\ell -1}})^{-1}(t-T_{\ell -1})}{{\Gamma (u_{\ell })}} \\ &\qquad{}\times \int _{T_{{\ell -1}}}^{T_{\ell }}(T_{\ell }-s)^{u_{\ell -1}} f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{ \ell -1}^{+}} z_{\ell }(T_{\ell })+ I^{u_{\ell }}_{T_{\ell -1}^{+}}z_{ \ell }(s), z_{\ell }(s) \bigr)\,ds \\ &\qquad{}-\frac{1}{\Gamma (u_{\ell })} \int _{T_{{\ell -1}}}^{t}(t-s)^{u_{ \ell -1}} f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} z_{\ell }(T_{\ell })\\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}}z_{ \ell }(s), z_{\ell }(s) \bigr)\,ds \biggr\vert \\ &\quad\leq \epsilon \int _{T_{\ell -1}}^{t} \frac{(t-s)^{u_{\ell }-1}}{\Gamma (u_{\ell })}\,ds \\ &\quad\leq \epsilon \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}. \end{aligned}$$

According to Theorem 3.2, BVP (1) has a solution \(x \in C(J, \Re )\) defined by \(x(t) = x_{\ell }(t)\) for \(t \in J_{\ell }, \ell = 1, 2,\ldots, n\), where

$$\begin{aligned} {x}_{\ell }= \textstyle\begin{cases} 0, & t \in [0, T_{\ell -1}], \\ \widetilde{x}_{\ell },& t \in J_{\ell }, \end{cases}\displaystyle \end{aligned}$$
(12)

and \(\widetilde{x}_{\ell } \in E_{\ell }\) is a solution of (7). According to Lemma 3.1 the integral equation

$$\begin{aligned} \widetilde{x}_{\ell }(t)={}& {-} \frac{(T_{\ell }-T_{{\ell -1}})^{-1}(t-T_{\ell -1})}{{\Gamma (u_{\ell })}} \\ &{}\times\int _{T_{{\ell -1}}}^{T_{\ell }}(T_{\ell }-s)^{u_{\ell -1}} f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{\ell -1}^{+}} \widetilde{x}_{\ell }(T_{\ell }) + I^{u_{\ell }}_{T_{\ell -1}^{+}} \widetilde{x}_{\ell }(s), \widetilde{x}_{\ell }(s) \bigr)\,ds \\ &{}+ \frac{1}{\Gamma (u_{\ell })} \int _{T_{{\ell -1}}}^{t}(t-s)^{u_{ \ell -1}} f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} \widetilde{x}_{\ell }(T_{\ell }) \\ &{}+ I^{u_{\ell }}_{T_{ \ell -1}^{+}} \widetilde{x}_{\ell }(s), \widetilde{x}_{\ell }(s) \bigr)\,ds \end{aligned}$$
(13)

holds.

Let \(t \in J_{\ell }, \ell = 1, 2,\ldots, n\). Then by Eqs. (12) and (13) we get

$$\begin{aligned} & \bigl\vert z(t)-x(t) \bigr\vert \\ &\quad= \bigl\vert z(t)-x_{\ell }(t) \bigr\vert \\ &\quad = \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 })}} \\ &\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}^{+}} \widetilde{x}_{\ell }(T_{\ell }) \\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}}\widetilde{x}_{\ell }(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, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} \widetilde{x}_{\ell }(T_{\ell }) \\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}}\widetilde{x}_{\ell }(s), \widetilde{x}_{ \ell }(s) \bigr)\,ds \biggr\vert \\ &\qquad{}+ \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}^{+}} z_{\ell }(T_{\ell }) \\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}}z_{\ell }(s), z_{\ell }(s) \bigr)\,ds -f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{\ell }}_{T_{ \ell -1}^{+}} \widetilde{x}_{\ell }(T_{\ell })\\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}} \widetilde{x}_{\ell }(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, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} z_{\ell }(T_{\ell })\\ &\qquad{}+ I^{u_{\ell }}_{T_{\ell -1}^{+}}z_{ \ell }(s), z_{\ell }(s) \bigr)\,ds \\ &\qquad{}-f_{1} \bigl(s, -(T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1})I^{u_{ \ell }}_{T_{\ell -1}^{+}} \widetilde{x}_{\ell }(T_{\ell })+ I^{u_{\ell }}_{T_{ \ell -1}^{+}} \widetilde{x}_{\ell }(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 })} \\ &\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}) \bigl( I^{u_{\ell }}_{T_{ \ell -1}^{+}} \bigl(z_{\ell }(T_{\ell })- \widetilde{x}_{\ell }(T_{\ell })\bigr)\bigr) \\ &\qquad{}+\bigl(I^{u_{\ell }}_{T_{\ell -1}^{+}}\bigl(z_{\ell }(s)- \widetilde{x}_{\ell }(s)\bigr)\bigr) \bigr\vert +L \bigl\vert \bigl(z_{\ell }(s)-\widetilde{x}_{\ell }(s)\bigr) \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 (T_{\ell }-T_{{\ell -1}})^{-1}(s-T_{\ell -1}) \bigl( I^{u_{ \ell }}_{T_{\ell -1}^{+}} \bigl(z_{\ell }(T_{\ell })- \widetilde{x}_{\ell }(T_{ \ell })\bigr)\bigr) \\ &\qquad{}+\bigl(I^{u_{\ell }}_{T_{\ell -1}^{+}}\bigl(z_{\ell }(s)- \widetilde{x}_{\ell }(s)\bigr)\bigr) \bigr\vert +L \bigl\vert \bigl(z_{\ell }(s)-\widetilde{x}_{\ell }(s)\bigr) \bigr\vert \bigr)\,ds \\ &\quad\leq \epsilon \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}+ \frac{2(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \\ &\qquad{}\times\int _{T_{ \ell -1}}^{T_{\ell }} s^{-\delta } \bigl(K \bigl\vert \bigl( I^{u_{\ell }}_{T_{\ell -1}^{+}} \bigl(z_{\ell }(T_{\ell })- \widetilde{x}_{\ell }(T_{\ell })\bigr)\bigr)+\bigl(I^{u_{\ell }}_{T_{ \ell -1}^{+}} \bigl(z_{\ell }(s)-\widetilde{x}_{\ell }(s)\bigr)\bigr) \bigr\vert \\ &\qquad{}+ L \bigl\vert \bigl(z_{\ell }(s)-\widetilde{x}_{\ell }(s) \bigr) \bigr\vert \bigr)\,ds \\ &\quad\leq \epsilon \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}\\ &\qquad{}+ \frac{2(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}}{\Gamma (u_{\ell })} \bigl(2K \bigl\Vert I^{u_{ \ell }}_{T_{\ell -1}^{+}}(z_{\ell }-\widetilde{x}_{\ell }) \bigr\Vert _{E_{\ell }}+L \Vert z_{\ell }-\widetilde{x}_{\ell } \Vert _{E_{\ell }}\bigr) \int _{T_{\ell -1}}^{T_{ \ell }} s^{-\delta }\,ds \\ &\quad\leq \epsilon \frac{(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 })}\\ &\qquad{}\times \biggl(2K \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)} \Vert z_{ \ell }-\widetilde{x}_{\ell } \Vert _{E_{\ell }}+L \Vert z_{\ell }-\widetilde{x}_{ \ell } \Vert _{E_{\ell }}\biggr) \\ &\quad\leq \epsilon \frac{(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 })} \\ &\qquad{}\times\biggl(2K \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}+L\biggr) \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

$$\begin{aligned} \mu =\max_{\ell = 1, 2,\ldots, n} \frac{2(T_{\ell }-T_{\ell -1})^{u_{\ell }-1}({T_{\ell }}^{1-\delta }-{T_{\ell -1}}^{1-\delta })}{(1-\delta )\Gamma (u_{\ell })} \biggl(2K \frac{(T_{\ell }-T_{\ell -1})^{u_{\ell }}}{\Gamma (u_{\ell }+1)}+L\biggr). \end{aligned}$$

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, the BVP (1) is (UH) stable. □

Example

Let us consider the following fractional boundary value problem:

$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{u(t)}_{0^{+}}x(t)= \frac{t^{-\frac{1}{3}}e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)(1+ \vert x(t) \vert + \vert {}^{c}D^{u(t)}_{0^{+}} x(t) \vert )}, \quad t\in J:= [0,2], \\ x(0)=0,\qquad x(2)=0. \end{cases}\displaystyle \end{aligned}$$
(14)

Let

$$\begin{aligned} &f_{1}(t, y, z)= \frac{t^{-\frac{1}{3}}e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)(1+y+z)}, \quad(t,y,z)\in [0,2]\times [0,+\infty )\times [0,+\infty ). \\ &u(t)= \textstyle\begin{cases} \frac{3}{2}, & t \in J_{1}:=[0, 1], \\ \frac{9}{5}, & t \in J_{2}:=\,]1, 2]. \end{cases}\displaystyle \end{aligned}$$
(15)

Then we have

$$\begin{aligned} &t^{\frac{1}{3}} \bigl\vert f_{1}(t,y_{1},z_{1})-f_{1}(t,y_{2},z_{2}) \bigr\vert \\ &\quad=\biggl\vert \frac{e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)} \biggl( \frac{1}{1+y_{1}+z_{1}}- \frac{1}{1+y_{2}+z_{2}} \biggr) \biggr\vert \\ &\quad\leq \frac{e^{-t}( \vert y_{1}-y_{2} \vert + \vert z_{1}-z_{2} \vert )}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)(1+y_{1}+z_{1})(1+y_{2}+z_{2})} \\ &\quad\leq \frac{e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)}\bigl( \vert y_{1}-y_{2} \vert + \vert z_{1}-z_{2} \vert \bigr) \\ &\quad\leq \frac{1}{(e+5)} \vert y_{1}-y_{2} \vert + \frac{1}{(e+5)} \vert z_{1}-z_{2} \vert . \end{aligned}$$

Hence the condition (H2) holds with \(\delta =\frac{1}{3}\) and \(K =L = \frac{1}{e+5}\).

By (15), according to (7) we consider two auxiliary BVPs for Caputo fractional differential equations of constant order,

$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{\frac{3}{2}}_{0^{+}}x(t)= \frac{t^{-\frac{1}{3}}e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)(1+ \vert x(t) \vert + \vert {}^{c}D^{\frac{3}{2}}x(t) \vert )}, & t \in J_{1}, \\ x(0)=0, \qquad x(1)=0 \end{cases}\displaystyle \end{aligned}$$
(16)

and

$$\begin{aligned} \textstyle\begin{cases} {}^{c}D^{\frac{9}{5}}_{1^{+}}x(t)= \frac{t^{-\frac{1}{3}}e^{-t}}{(e^{e^{\frac{t^{2}}{1+t}}}+4e^{2t}+1)(1+ \vert x(t) \vert + \vert {}^{c}D^{\frac{9}{5}}x(t) \vert )}, \quad t \in J_{2}, \\ x(1)=0, \qquad x(2)=0. \end{cases}\displaystyle \end{aligned}$$
(17)

Next, we prove that the condition (9) is fulfilled for \(\ell = 1\). Indeed,

$$\begin{aligned} \frac{2({T_{1}}^{1-\delta }-{T_{0}}^{1-\delta })(T_{1}-T_{0})^{u_{1}-1}}{(1-\delta )\Gamma (u_{1})} \biggl( \frac{2K(T_{1}-T_{0})^{u_{1}}}{\Gamma (u_{1}+1)} + L \biggr) &= \frac{1}{\frac{2}{3}(e+5)\Gamma (\frac{3}{2})} \biggl( \frac{2}{\Gamma (\frac{5}{2})}+1 \biggr)\\ & \simeq 0.3664< 1. \end{aligned}$$

Accordingly the condition (9) is achieved. By Theorem 3.1, the problem (16) has a solution \(\widetilde{x}_{1} \in E_{1}\).

We prove that the condition (9) is fulfilled for \(\ell = 2\). Indeed,

$$\begin{aligned} \frac{2({T_{2}}^{1-\delta }-{T_{1}}^{1-\delta })(T_{2}-T_{1})^{u_{2}-1}}{(1-\delta )\Gamma (u_{2})} \biggl( \frac{2K(T_{2}-T_{1})^{u_{2}}}{\Gamma (u_{2}+1)} + L \biggr) &= \frac{{2}^{\frac{2}{3}}-1}{\frac{2}{3}\Gamma (\frac{9}{5})} \frac{1}{e+5} \biggl(\frac{2}{\Gamma (\frac{14}{5})}+1 \biggr)\\ &\simeq 0.2682< 1. \end{aligned}$$

Thus, the condition (9) is satisfied.

According to Theorem 3.1, the BVP (17) possesses a solution \(\widetilde{x}_{2} \in E_{2}\).

Then, by Theorem 3.2, the BVP (14) 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 4.1, BVP (14) is (UH) stable.

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Acknowledgements

The third author would like to thank the Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok.

Funding

This research was funded by Thailand Science Research and Innovation Fund, and King Mongkut’s University of Technology North Bangkok with Contract no. KMUTNB-BasicR-64-33-1.

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Correspondence to Kanokwan Sitthithakerngkiet.

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Benkerrouche, A., Souid, M.S., Sitthithakerngkiet, K. et al. Implicit nonlinear fractional differential equations of variable order. Bound Value Probl 2021, 64 (2021). https://doi.org/10.1186/s13661-021-01540-7

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MSC

  • 26A33
  • 34K37

Keywords

  • Fractional differential equations of variable order
  • Boundary value problem
  • Fixed point theorem
  • Ulam–Hyers stability