1 Introduction

Consider the Sturm–Liouville fractional differential problem

\begin{aligned}& ^{c}D^{\alpha }\bigl(p(t)u'(t)\bigr)+q(t) u(t)=h(t)f\bigl(u(t)\bigr), \quad t \in (0,T), \end{aligned}
(1.1)
\begin{aligned}& u'(0)=0, \quad \sum_{k=1}^{m} \xi _{k} u(a_{k})=\nu \sum_{j=1}^{n} \eta _{j}u(b_{j}), \end{aligned}
(1.2)

where $$\alpha \in (0,1]$$, $$^{c}D^{\alpha }$$ denotes the Caputo fractional derivative, $$p(t) \in C^{1}({\mathcal{J}}, {\mathcal{R}})$$ and $$q(t)$$ and $$h(t)$$ are absolute continuous functions on $${\mathcal{J}}=[0,T]$$, $$T<\infty$$ with $$p(t)\neq 0$$ for all $$t \in {\mathcal{J}}$$, $$f(u(t)):{\mathcal{R}}\rightarrow {\mathcal{R}}$$ is defined and differentiable on the interval $${\mathcal{J}}$$, $$0\leq a_{1}< a_{2}<\cdots <a_{m}<c$$, $$d\leq b_{1}< b_{2}<\cdots <b_{n} \leq T$$, $$c \leq d$$ and $$\xi _{k}$$, $$\eta _{j}$$ and $$\nu \in {\mathcal{R}}$$. In this work, we discuss the existence and uniqueness of the solution $$u(t) \in C^{1}({\mathcal{J}}, {\mathcal{R}})$$ of the Sturm–Liouville fractional differential equation (1.1) with the multi-point boundary condition (1.2). We also study the continuous dependence of the solution on the coefficients $$\xi _{k}$$ and $$\eta _{j}$$ of the multi-point boundary condition.

As a consequence of our results, we find a unique solution for the ordinary Sturm–Liouville differential equation with the multi-point boundary condition

\begin{aligned}& \begin{gathered} \bigl(p(t)u'(t)\bigr)'+q(t) u(t)=h(t)f\bigl(u(t) \bigr), \quad t \in (0,T), \\ u'(0)=0, \quad \sum_{k=1}^{m} \xi _{k} u(a_{k})=\nu \sum_{j=1}^{n} \eta _{j} u(b_{j}), \end{gathered} \end{aligned}
(1.3)

As an extension of our problem, we deduce the presence of a unique solution for Eqs. (1.1) and (1.3) under the integral conditions

$$u'(0)=0, \quad \int _{a}^{c}u(\theta )\,d\varpi (\theta )=\nu \int _{d}^{e}u(\theta )\,d \vartheta (\theta ),$$

where $$\varpi (\theta )$$ and $$\vartheta (\theta )$$ are an increasing functions and the integrals are meant in the Riemann–Stieltjes sense for $$0 \leq a< c \leq d< e\leq T$$.

Sturm–Liouville operator is an important operator in physics, applied mathematics and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics and wave phenomena; see Joannopoulos [10] and Teschl [25] and the references therein. The existence of solutions and other properties for Sturm–Liouville boundary value problems have received considerable attention from many researchers during the last two decades; see for example Al-Mdallal [1], Bensidhoum and Dib [2], Erturk [7], Hassana [9], Klimek and Argawal [12], Li et al. [13], Lian and Ge [14], Liu et al. [17], Muensawat et al. [20], Xu and Abernathy [26], Yang [27] and the references therein.

Nonlocal and multi-point conditions can be more useful than the standard initial condition to describe some physical phenomena and have widely been studied by several researchers; see Cui and Zou [3], EL-Sayed and Bin-Taher [4, 5], El-Shahed and Nieto [6], Guo et al. [8], Karaaslan [11], Liu et al. [16], Liang et al. [15], Ma [18], Nyamoradi [21, 22], Nyamoradi et al. [23], Zhao and Ge [30], Zhang and Zhong [28], Zhanga and Liu [29], Zhong and Lin [31] and the references therein.

2 Preliminaries and main results

First of all, we introduce some notations and basic facts which are used throughout the paper.

Let $$\Vert u\Vert =\max \{|u(t)|: t\in [0,T]\}$$ is the norm in the space $$C[0,T]$$ and $$\Vert u\Vert _{L_{1}}=\int _{0}^{T}|u(t)| \,dt$$ is the norm in $$L_{1}[0,T]$$.

The Riemann–Liouville fractional integral of order $$\alpha >0$$ for the function $$u(t)\in L_{1}[0,T]$$ is known (see [19], [24]):

$$I^{\alpha } u(t)= \int _{0}^{t} \frac{(t-s)^{\alpha -1}}{\varGamma (\alpha )} u(s) \,ds,$$

and the Caputo fractional derivative of order $$n-1<\alpha \leq n$$ for the function $$u(t)$$ is known:

$$D^{\alpha } u(t)= I^{n-\alpha } \frac{d^{n}}{dt^{n}} u(t)= \int _{0} ^{t} \frac{(t-s)^{n-\alpha -1}}{\varGamma (n-\alpha )} \frac{d^{n} u(s)}{ds ^{n}} \,ds .$$

and we have for α, $$\beta \in {\mathcal{R^{+}}}$$:

$$(r_{1})$$ :

$$I^{\alpha }: L_{1} \rightarrow L_{1}$$ and $$\lim_{\alpha \rightarrow 1} I^{\alpha }f(t)=I^{1}f(t)=\int _{0} ^{t}f(s)\,ds$$.

$$(r_{2})$$ :

$$I^{\alpha } I^{\beta } f(t)= I^{\alpha +\beta } f(t)$$.

$$(r_{3})$$ :

If $$f(t)$$ is absolutely continuous on $${\mathcal{J}}$$, then $$\lim_{\alpha \rightarrow 1} ^{c}D^{\alpha } f(t)= D f(t)$$ and

$$D I^{\alpha } f(t)=\frac{t^{\alpha -1}}{\varGamma (\alpha )} f(0) + I ^{\alpha } D f(t).$$
$$(r_{4})$$ :

$$I^{\alpha } t^{\gamma }=\frac{\varGamma (\gamma +1) t ^{\alpha +\gamma }}{\varGamma (\alpha +\gamma +1)}$$, $$\gamma +1>0$$.

Here we investigate the Sturm–Liouville fractional differential equation (1.1) with the multi-point boundary condition (1.2) under the following assumptions.

$$(D_{1})$$ :

The function $$f: {\mathcal{R}}\rightarrow {\mathcal{R}}$$ is defined and differentiable on the interval $$[0, T]$$ and $$\frac{ \partial f}{\partial u}$$ is bounded on $${\mathcal{J}}$$ with $$|\frac{\partial f}{\partial u}| \leq {\mathcal{K}}$$.

$$(D_{2})$$ :

The function $$p(t) \in C^{1}({\mathcal{J}}, {\mathcal{R}})$$ with $$p(t)\neq 0$$ for all $$t \in {\mathcal{J}}$$, $$\inf_{[0,T]}|p(t)|=p$$ and $$q(t)$$ and $$h(t)$$ are absolute continuous functions on $${\mathcal{J}}$$.

Lemma 2.1

Let the assumptions $$(D_{1})$$$$(D_{2})$$ be satisfied. Then problem (1.1)(1.2) is equivalent to the integral equation

\begin{aligned} u(t) &=E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{\alpha }q(s) u(s)\,ds-\nu E \sum_{j=1}^{n}\eta _{j} \int _{0}^{b_{j}} \frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds \\ &\quad {} -E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds +\nu E\sum_{j=1}^{n}\eta _{j} \int _{0}^{b_{j}}\frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds \\ & \quad {}- \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds+ \int _{0}^{t} \frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds, \end{aligned}
(2.1)

with $$u\in C^{1}({\mathcal{J}},{\mathcal{R}})$$ and $$u{''}(t)\in L_{1}( {\mathcal{J}}, {\mathcal{R}})$$, where $$E = \frac{1}{\sum_{k=1} ^{m} \xi _{k} - \nu \sum_{j=1}^{n} \eta _{j} }$$.

Proof

For $$t>0$$, Eq. (1.1) can be written as

$$I^{1-\alpha } \frac{d}{dt} p(t) u'(t) = -q(t) u(t)+ h(t)f \bigl(u(t)\bigr).$$

Operating by $$I^{\alpha }$$ on both sides, we obtain

\begin{aligned} &I^{1} \frac{d}{dt} p(t) u'(t)= -I^{\alpha } q(t) u(t)+ I^{\alpha } h(t)f\bigl(u(t)\bigr), \\ &p(t) u'(t)-p(0)u'(0)= -I^{\alpha } q(t) u(t)+ I^{\alpha } h(t)f\bigl(u(t)\bigr). \end{aligned}

From (1.2) we have

\begin{aligned} \begin{aligned} &u'(t)= -\frac{1}{p(t)}I^{\alpha }q(t) u(t)+ \frac{1}{p(t)}I^{\alpha }h(t)f\bigl(u(t)\bigr), \\ &u(t)- u(0)= - \int _{0}^{t}\frac{1}{p(s)}I^{\alpha }q(s) u(s)\,ds+ \int _{0}^{t}\frac{1}{p(s)}I^{\alpha }h(s)f \bigl(u(s)\bigr)\,ds. \end{aligned} \end{aligned}
(2.2)

For convenience, put $$A(t)=\int _{0}^{t}\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds$$ and $$B(t)=\int _{0}^{t}\frac{1}{p(s)} I^{\alpha } h(s)f(u(s))\,ds$$, we get

$$\sum_{k=1}^{m} \xi _{k} u(a_{k}) -\sum_{k=1}^{m} \xi _{k} u(0)=- \sum_{k=1}^{m} \xi _{k} A(a_{k})+\sum_{k=1}^{m} \xi _{k} B(a_{k})$$
(2.3)

and

$$\nu \sum_{j=1}^{n}\eta _{j} u(b_{j})-\nu \sum_{j=1}^{n} \eta _{j} u(0)=- \nu \sum_{j=1}^{n} \eta _{j} A(b_{j})+\nu \sum_{j=1}^{n} \eta _{j} B(b _{j}).$$
(2.4)

On subtracting (2.3) from (2.4) and using $$\sum_{k=1}^{m} \xi _{k} u(a_{k})=\nu \sum_{j=1}^{n}\eta _{j} u(b_{j})$$, we obtain

\begin{aligned} u(0) &= E \Biggl[\sum_{k=1}^{m} \xi _{k} A(a_{k})-\nu \sum_{j=1}^{n} \eta _{j} A(b_{j})-\sum_{k=1}^{m} \xi _{k} B(a_{k})+\nu \sum_{j=1}^{n} \eta _{j} B(b_{j}) \Biggr], \end{aligned}

with $$E = \frac{1}{\sum_{k=1}^{m} \xi _{k} - \nu \sum_{j=1}^{n} \eta _{j} }$$, substituting in (2.2) we get (2.1).

Conversely, to complete the equivalence between integral equation (2.1) and problem (1.1)–(1.2), we have from (2.1)

\begin{aligned}& \begin{gathered} u'(t)= -\frac{1}{p(t)}I^{\alpha } q(t) u(t)+ \frac{1}{p(t)}I^{\alpha }h(t) f\bigl(u(t)\bigr) \in C({\mathcal{J}}, { \mathcal{R}}), \\ \frac{d}{dt}\bigl(p(t) u'(t)\bigr)= -\frac{d}{dt} I^{\alpha } q(t) u(t) + \frac{d}{dt} I^{\alpha } h(t) f\bigl(u(t) \bigr), \\ I^{1-\alpha }\frac{d}{dt}\bigl(p(t) u'(t)\bigr)= -I^{1-\alpha }\frac{d}{dt} I ^{\alpha } q(t) u(t) + I^{1-\alpha } \frac{d}{dt} I^{\alpha } h(t) f\bigl(u(t)\bigr). \end{gathered} \end{aligned}
(2.5)

From the definition of Caputo derivative and applying $$(r_{3})$$, we have

\begin{aligned} ^{c}D^{\alpha }\bigl(p(t) u'(t)\bigr)&= -I^{1-\alpha } I^{\alpha } \frac{d}{dt}\bigl(q(t) u(t)\bigr) + I^{1-\alpha } I^{\alpha } \frac{d}{dt}\bigl(h(t) f\bigl(u(t)\bigr) \bigr) \\ &\quad {}- I^{1-\alpha } \frac{t^{\alpha -1}}{\varGamma (\alpha )}q(0)u(0)+I ^{1-\alpha } \frac{t^{\alpha -1}}{\varGamma (\alpha )} h(0) f\bigl(u(0)\bigr). \end{aligned}

Hence, from $$(r_{2})$$ and $$(r_{4})$$ we get

\begin{aligned} ^{c}D^{\alpha }\bigl(p(t) u'(t)\bigr)&= -I^{1} \frac{d}{dt}\bigl(q(t) u(t)\bigr) + I^{1} \frac{d}{dt}\bigl(h(t) f\bigl(u(t)\bigr)\bigr)- q(0)u(0)+h(0)f\bigl(u(0)\bigr) \\ &= - q(t) u(t)+h(t)f\bigl(u(t)\bigr), \end{aligned}

and then we get (1.1).

From (2.5) we have $$u'(0)=0$$. Also by simple computation we can deduce from (2.1) that $$\sum_{k=1}^{m} \xi _{k} u(a_{k})=\nu \sum_{j=1}^{n}\eta _{j} u(b_{j})$$. Then problem (1.1)–(1.2) and Eq. (2.1) are equivalent.

Now from (2.5) using $$(r_{3})$$, we have

\begin{aligned} u{''}(t)&= \frac{d}{dt} \biggl[ \frac{1}{p(t)}I^{\alpha } \bigl( -q(t) u(t)\bigr) + h(t) f\bigl(u(t)\bigr) ) \biggr], \\ &= -\frac{p'(t)}{p^{2}(t)}I^{\alpha } \bigl(-q(t) u(t) + h(t) f\bigl(u(t)\bigr) \bigr) + \frac{1}{p(t)}I^{\alpha } \frac{d}{dt}\bigl(-q(t) u(t) \\ &\quad {} + h(t) f\bigl(u(t)\bigr)\bigr) + \frac{1}{p(t)}\frac{t^{\alpha -1}}{\varGamma ( \alpha )} \bigl(-q(0) u(0) + h(0) f\bigl(u(0)\bigr)\bigr) \end{aligned}

and

\begin{aligned} \bigl\vert u{''}(t) \bigr\vert &\leq \frac{ \vert p'(t) \vert }{ \vert p^{2}(t) \vert } \int _{0}^{t}\frac{(t-s)^{ \alpha -1}}{\varGamma (\alpha )} \bigl( \bigl\vert q(s) \bigr\vert \bigl\vert u(s) \bigr\vert + \bigl\vert h(s) \bigr\vert \bigl\vert f\bigl(u(s)\bigr) \bigr\vert \bigr)\,ds \\ &\quad {}+ \frac{1}{ \vert p(t) \vert } \int _{0}^{t}\frac{(t-s)^{\alpha -1}}{\varGamma ( \alpha )} \biggl( \frac{}{} \bigl\vert q(s) \bigr\vert \frac{}{} \bigl\vert u'(s) \bigr\vert + \bigl\vert q'(s) \bigr\vert \bigl\vert u(s) \bigr\vert \\ &\quad {} + \bigl\vert h(s) \bigr\vert \frac{}{} \biggl\vert \frac{\partial f(u(s))}{\partial u} \biggr\vert \biggl\vert \frac{du}{ds} \biggr\vert + \bigl\vert h'(s) \bigr\vert \bigl\vert f\bigl(u(s)\bigr) \bigr\vert \biggr)\,ds \\ &\quad {}+ \frac{1}{ \vert p(t) \vert }\frac{t^{\alpha -1}}{\varGamma (\alpha )} \bigl( \bigl\vert q(0) \bigr\vert \bigl\vert u(0) \bigr\vert + \bigl\vert h(0) \bigr\vert \bigl\vert f\bigl(u(0)\bigr) \bigr\vert \bigr). \end{aligned}

Therefore, we get

\begin{aligned} \int _{0}^{T} \bigl\vert u{''}(t) \bigr\vert \,dt&\leq \int _{0}^{T}\frac{ \vert p'(t) \vert }{ \vert p^{2}(t) \vert } \int _{0}^{t}\frac{(t-s)^{\alpha -1}}{\varGamma (\alpha )} \bigl( \bigl\vert q(s) \bigr\vert \bigl\vert u(s) \bigr\vert + \bigl\vert h(s) \bigr\vert \bigl\vert f\bigl(u(s)\bigr) \bigr\vert \bigr)\,ds \,dt \\ &\quad {}+ \int _{0}^{T}\frac{1}{ \vert p(t) \vert } \int _{0}^{t}\frac{(t-s)^{\alpha -1}}{ \varGamma (\alpha )} \biggl( \bigl\vert q(s) \bigr\vert \frac{}{} \bigl\vert u'(s) \bigr\vert + \bigl\vert q'(s) \bigr\vert \bigl\vert u(s) \bigr\vert \quad \\ &\quad{} + \bigl\vert h(s) \bigr\vert \frac{}{} \biggl\vert \frac{\partial f(u(s))}{\partial u} \biggr\vert \bigl\vert u'(s) \bigr\vert + \bigl\vert h'(s) \bigr\vert \bigl\vert f\bigl(u(s)\bigr) \bigr\vert \biggr)\,ds \,dt \\ &\quad {}+\bigl( \bigl\vert q(0) \bigr\vert \bigl\vert u(0) \bigr\vert + \bigl\vert h(0) \bigr\vert \bigl\vert f\bigl(u(0)\bigr) \bigr\vert \bigr) \int _{0}^{T} \frac{1}{ \vert p(t) \vert }\frac{t ^{\alpha -1}}{\varGamma (\alpha )}\,dt \\ &= \int _{0}^{T}\bigl( \bigl\vert q(s) \bigr\vert \bigl\vert u(s) \bigr\vert + \bigl\vert h(s) \bigr\vert \bigl\vert f \bigl(u(s)\bigr) \bigr\vert \bigr) \,ds \int _{s}^{T}\frac{ \vert p'(t) \vert }{ \vert p ^{2}(t) \vert }\frac{(t-s)^{\alpha -1}}{\varGamma (\alpha )} \,dt \\ &\quad {}+ \int _{0}^{T} \biggl( \bigl\vert q(s) \bigr\vert \bigl\vert u'(s) \bigr\vert + \bigl\vert q'(s) \bigr\vert \bigl\vert u(s) \bigr\vert + \bigl\vert h(s) \bigr\vert \biggl\vert \frac{ \partial f(u(s))}{\partial u} \biggr\vert \bigl\vert u'(s) \bigr\vert \\ &\quad{} +\frac{}{} \bigl\vert h'(s) \bigr\vert \bigl\vert f(u) \bigr\vert \biggr)\,ds \int _{s}^{T} \frac{1}{ \vert p(t) \vert }\frac{(t-s)^{\alpha -1}}{\varGamma (\alpha )} \,dt \\ &\quad {}+ \bigl( \bigl\vert q(0) \bigr\vert \bigl\vert u(0) \bigr\vert + \bigl\vert h(0) \bigr\vert \bigl\vert f\bigl(u(0)\bigr) \bigr\vert \bigr) \int _{0}^{T} \frac{1}{ \vert p(t) \vert }\frac{t^{\alpha -1}}{\varGamma (\alpha )}\,dt \\ &\leq \bigl( \Vert q \Vert \Vert u \Vert + \Vert h \Vert \Vert f \Vert \bigr)\frac{ \Vert p' \Vert T^{\alpha +1}}{p ^{2} \varGamma (\alpha +1)}+ \bigl( \Vert q \Vert \bigl\Vert u' \bigr\Vert T+ \bigl\Vert q' \bigr\Vert _{L_{1}} \Vert u \Vert \\ &\quad {}+{\mathcal{K}} \Vert h \Vert \bigl\Vert u' \bigr\Vert T+ \bigl\Vert h' \bigr\Vert _{L_{1}} \Vert f \Vert \bigr)\frac{T^{\alpha }}{p \varGamma (\alpha +1)} \\ &\quad {}+\frac{T^{\alpha }}{p \varGamma (\alpha +1)} \bigl( \bigl\vert q(0) \bigr\vert \bigl\vert u(0) \bigr\vert + \bigl\vert h(0) \bigr\vert \bigl\vert f\bigl(u(0)\bigr) \bigr\vert \bigr). \end{aligned}

Then $$u{''}(t)\in L_{1}({\mathcal{J}}, {\mathcal{R}})$$. □

Define an operator $$\mathcal{H}$$ associated with the integral equation (2.1) as follows:

\begin{aligned} {\mathcal{H}}u(t)&= E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}} \frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds-\nu E \sum_{j=1}^{n}\eta _{j} \int _{0}^{b_{j}}\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds \\ &\quad {}-E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds - \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds \\ &\quad {}+ \nu E \sum_{j=1}^{n}\eta _{j} \int _{0}^{b_{j}}\frac{1}{p(s)}I^{ \alpha } h(s) f\bigl(u(s)\bigr)\,ds + \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } h(s) f \bigl(u(s)\bigr)\,ds. \end{aligned}

Theorem 2.2

Assume that the hypotheses $$(D_{1})$$$$(D_{2})$$ hold, and the coefficients $$\xi _{k}$$, $$\eta _{j}$$ and $$\nu \in {\mathcal{R}}$$ with $$\sum_{k=1}^{m} \xi _{k}-\nu \sum_{j=1}^{n}\eta _{j} \neq 0$$. If

$${\mathcal{G}}=\frac{T^{\alpha +1}}{p\varGamma (\alpha +2)} \Biggl[ \Biggl( \vert E \vert \Biggl(\sum_{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr) \bigl( \Vert q \Vert + {\mathcal{K}} \Vert h \Vert \bigr) \Biggr]< 1,$$
(2.6)

then Sturm–Liouville fractional differential equation (1.1) with the multi-point boundary condition (1.2) has a unique solution $$u\in C^{1}({\mathcal{J}},{\mathcal{R}})$$.

Proof

Let $$\sup_{t \in {\mathcal{J}}}|f(0)|={\mathcal{M}}$$; it follows from the Lipschitz condition that

$$\bigl\vert f\bigl(u(t)\bigr) \bigr\vert = \bigl\vert f(u)-f(0)+f(0) \bigr\vert \leq {\mathcal{K}} \vert u \vert + \bigl\vert f(0) \bigr\vert \leq {\mathcal{K}} \Vert u \Vert +{\mathcal{M}}.$$
(2.7)

Firstly, we show that the operator $${\mathcal{H}}$$ satisfies the relation $${\mathcal{H}} B_{r}\subset B_{r}$$, where $$B_{r}=\{u \in C( {\mathcal{J}}, {\mathcal{R}}): \Vert u\Vert \leq r\}$$ and

$$r>\frac{\frac{{\mathcal{M}} \Vert h \Vert T^{\alpha +1}}{p\varGamma (\alpha +2)} [ \vert E \vert (\sum_{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert )+1 ]}{1 - {\mathcal{G}} }.$$

For $$u\in B_{r}$$, we have

\begin{aligned}& \vert {\mathcal{H}}u \vert \\& \quad \leq \vert E \vert \sum _{k=1}^{m} \vert \xi _{k} \vert \int _{0}^{a_{k}} \frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert q(s) \bigr\vert \bigl\vert u(s) \bigr\vert \,ds+ \vert \nu \vert \vert E \vert \sum _{j=1}^{n} \vert \eta _{j} \vert \int _{0}^{b_{j}} \frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert q(s) \bigr\vert \bigl\vert u(s) \bigr\vert \,ds \\& \quad \quad {}+ \vert E \vert \sum_{k=1}^{m} \vert \xi _{k} \vert \int _{0}^{a_{k}}\frac{1}{ \vert p(s) \vert }I ^{\alpha } \bigl\vert h(s) \bigr\vert \bigl\vert f\bigl(u(s)\bigr) \bigr\vert \,ds \\& \quad \quad {}+ \vert \nu \vert \vert E \vert \sum _{j=1}^{n} \vert \eta _{j} \vert \int _{0}^{b_{j}} \frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert h(s) \bigr\vert \bigl\vert f\bigl(u(s)\bigr) \bigr\vert \,ds \\& \quad \quad {}+ \int _{0}^{t}\frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert q(s) \bigr\vert \bigl\vert u(s) \bigr\vert \,ds+ \int _{0} ^{t}\frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert h(s) \bigr\vert \bigl\vert f\bigl(u(s)\bigr) \bigr\vert \,ds \\& \quad \leq \frac{ \vert E \vert \Vert q \Vert \Vert u \Vert }{p} \Biggl[ \sum_{k=1}^{m} \vert \xi _{k} \vert \int _{0}^{a_{k}} \int _{0}^{s}\frac{(s-\zeta )^{\alpha -1}}{ \varGamma (\alpha )}\,d\zeta \,ds \\& \quad \quad {}+ \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \int _{0}^{b_{j}} \int _{0}^{s}\frac{(s- \zeta )^{\alpha -1}}{\varGamma (\alpha )}\,d\zeta \,ds \Biggr] \\& \quad \quad{} +\frac{ \vert E \vert \Vert h \Vert ({\mathcal{K}} \Vert u \Vert +{\mathcal{M}})}{p} \Biggl[\sum_{k=1}^{m} \vert \xi _{k} \vert \int _{0}^{a_{k}} \int _{0}^{s}\frac{(s- \zeta )^{\alpha -1}}{\varGamma (\alpha )}\,d\zeta \,ds \\& \quad \quad {}+ \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \int _{0}^{b_{j}} \int _{0}^{s}\frac{(s- \zeta )^{\alpha -1}}{\varGamma (\alpha )}\,d\zeta \,dt \Biggr] + \frac{ \Vert q \Vert \Vert u \Vert }{p} \int _{0}^{t} \int _{0}^{s}\frac{(s-\zeta )^{\alpha -1}}{\varGamma (\alpha )}\,d\zeta \,ds \\& \quad \quad {} +\frac{ \Vert h \Vert ({\mathcal{K}} \Vert u \Vert +{\mathcal{M}})}{p} \int _{0} ^{t} \int _{0}^{s}\frac{(s-\zeta )^{\alpha -1}}{\varGamma (\alpha )}\,d \zeta \,ds \\& \quad \leq \frac{ \vert E \vert \Vert q \Vert \Vert u \Vert }{p} \Biggl[ \sum_{k=1}^{m} \vert \xi _{k} \vert \frac{a_{k}^{\alpha +1}}{\varGamma (\alpha +2)}+ \vert \nu \vert \sum _{j=1} ^{n} \vert \eta _{j} \vert \frac{b_{j}^{\alpha +1}}{\varGamma (\alpha +2)} \Biggr] \\& \quad \quad {}+\frac{ \vert E \vert \Vert h \Vert ({\mathcal{K}} \Vert u \Vert +{\mathcal{M}})}{p} \Biggl[ \sum_{k=1}^{m} \vert \xi _{k} \vert \frac{a_{k}^{\alpha +1}}{\varGamma (\alpha +2)}+ \vert \nu \vert \sum _{j=1}^{n} \vert \eta _{j} \vert \frac{b_{j}^{\alpha +1}}{\varGamma ( \alpha +2)} \Biggr] \\& \quad \quad {}+\frac{ \Vert q \Vert \Vert u \Vert }{p} \frac{T^{\alpha +1}}{\varGamma (\alpha +2)}+\frac{ \Vert h \Vert ({\mathcal{K}} \Vert u \Vert + {\mathcal{M}})}{p} \frac{T^{\alpha +1}}{\varGamma (\alpha +2)} \\& \quad \leq \frac{T^{\alpha +1}}{p\varGamma (\alpha +2)} \Biggl[ \Biggl( \vert E \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr) \bigl( \Vert q \Vert + { \mathcal{K}} \Vert h \Vert \bigr) \Biggr] \Vert u \Vert \\& \quad \quad {}+\frac{{\mathcal{M}} \Vert h \Vert T^{\alpha +1}}{p\varGamma (\alpha +2)} \Biggl[ \vert E \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr] \\& \quad \leq \frac{T^{\alpha +1}}{p\varGamma (\alpha +2)} \Biggl[ \Biggl( \vert E \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr) \bigl( \Vert q \Vert + { \mathcal{K}} \Vert h \Vert \bigr) \Biggr]r \\& \quad \quad {}+\frac{{\mathcal{M}} \Vert h \Vert T^{\alpha +1}}{p\varGamma (\alpha +2)} \Biggl[ \vert E \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr] \leq r, \end{aligned}

which proves that $${\mathcal{H}}B_{r} \subset B_{r}$$.

Now for $$u,v\in C({\mathcal{J}},{\mathcal{R}})$$ and $$t\in {\mathcal{J}}$$, we have

\begin{aligned}& \bigl\vert ({\mathcal{H}}u) (t) -({\mathcal{H}}v) (t) \bigr\vert \\& \quad \leq \vert E \vert \sum_{k=1}^{m} \vert \xi _{k} \vert \int _{0}^{a_{k}}\frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert q(s) \bigr\vert \bigl\vert u(s)-v(s) \bigr\vert \,ds \\& \quad \quad{} + \vert \nu \vert \vert E \vert \sum _{j=1}^{n} \vert \eta _{j} \vert \int _{0}^{b_{j}} \frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert q(s) \bigr\vert \bigl\vert u(s)-v(s) \bigr\vert \,ds \\& \quad \quad{} + \vert E \vert \sum_{k=1}^{m} \vert \xi _{k} \vert \int _{0}^{a_{k}} \frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert h(s) \bigr\vert \bigl\vert f\bigl(u(s)\bigr)-f\bigl(v(s)\bigr) \bigr\vert \,ds \\& \quad \quad{} + \vert \nu \vert \vert E \vert \sum _{j=1}^{n} \vert \eta _{j} \vert \int _{0}^{b_{j}} \frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert h(s) \bigr\vert \bigl\vert f\bigl(u(s)\bigr)-f\bigl(v(s)\bigr) \bigr\vert \,ds \\& \quad \quad {}+ \int _{0}^{t}\frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert q(s) \bigr\vert \bigl\vert u(s)-v(s) \bigr\vert \,ds + \int _{0}^{t}\frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert h(s) \bigr\vert \bigl\vert f\bigl(u(s)\bigr)-f\bigl(v(s)\bigr) \bigr\vert \,ds \\& \quad \leq \frac{ \vert E \vert \Vert q \Vert \Vert u-v \Vert }{p} \Biggl[\sum_{k=1}^{m} \vert \xi _{k} \vert \int _{0}^{a_{k}} \int _{0}^{s}\frac{(s-\zeta )^{\alpha -1}}{\varGamma (\alpha )}\,d\zeta \,ds \\& \quad \quad{}+ \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \int _{0}^{b_{j}} \int _{0}^{s}\frac{(s- \zeta )^{\alpha -1}}{\varGamma (\alpha )}\,d\zeta \,ds \Biggr] \\& \quad \quad {}+\frac{{\mathcal{K}} \vert E \vert \Vert h \Vert \Vert u-v \Vert }{p} \Biggl[\sum_{k=1} ^{m} \vert \xi _{k} \vert \int _{0}^{a_{k}} \int _{0}^{s}\frac{(s-\zeta )^{\alpha -1}}{ \varGamma (\alpha )}\,d\zeta \,ds \\& \quad \quad{}+ \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \int _{0}^{b_{j}} \int _{0}^{s}\frac{(s- \zeta )^{\alpha -1}}{\varGamma (\alpha )}\,d\zeta \,ds \Biggr] \\& \quad \quad{} +\frac{ \Vert q \Vert \Vert u-v \Vert }{p} \int _{0}^{t} \int _{0}^{s}\frac{(s- \zeta )^{\alpha -1}}{\varGamma (\alpha )}\,d\zeta \,ds+\frac{{\mathcal{K}} \Vert h \Vert \Vert u-v \Vert }{p} \int _{0}^{t} \int _{0} ^{s}\frac{(s-\zeta )^{\alpha -1}}{\varGamma (\alpha )}\,d\zeta \,ds. \end{aligned}

By a similar calculation to the one above, we obtain

\begin{aligned} & \bigl\vert ({\mathcal{H}}u) (t) -({\mathcal{H}}v) (t) \bigr\vert \\ &\quad \leq \frac{T^{\alpha +1}}{p\varGamma (\alpha +2)} \Biggl[ \Biggl( \vert E \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr) \bigl( \Vert q \Vert + {\mathcal{K}} \Vert h \Vert \bigr) \Biggr] \Vert u-v \Vert . \end{aligned}

Therefore, we get

\begin{aligned} & \Vert {\mathcal{H}}u -{\mathcal{H}}v \Vert \\ & \quad \leq \frac{T^{\alpha +1}}{p\varGamma (\alpha +2)} \Biggl[ \Biggl( \vert E \vert \Biggl( \sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr) \bigl( \Vert q \Vert + {\mathcal{K}} \Vert h \Vert \bigr) \Biggr] \Vert u-v \Vert . \end{aligned}

It follows from (2.6) that $$\mathcal{H}$$ is a contraction mapping and by applying the Banach fixed point theorem problem (1.1)–(1.2) has a unique solution $$u\in C^{1}({\mathcal{J}}, {\mathcal{R}})$$. □

Corollary 2.3

Let the assumptions $$(D_{1})$$$$(D_{2})$$ be satisfied, and the coefficients $$\xi _{k}$$, $$\eta _{j}$$ and $$\nu \in {\mathcal{R}}$$ with $$\sum_{k=1}^{m} \xi _{k}-\nu \sum_{j=1}^{n}\eta _{j} \neq 0$$. If

$$\frac{T^{2}}{2p} \Biggl[ \Biggl( \vert E \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr) \bigl( \Vert q \Vert + {\mathcal{K}} \Vert h \Vert \bigr) \Biggr]< 1,$$
(2.8)

then the ordinary Sturm–Liouville differential equation

$$\bigl(p(t)u'(t)\bigr)'+q(t) u(t)=h(t)f\bigl(u(t) \bigr), \quad t \in (0,T) ,$$
(2.9)

under the conditions

$$u'(0)=0,\quad \sum_{k=1}^{m} \xi _{k} u(a_{k})=\nu \sum_{j=1}^{n} \eta _{j} u(b _{j})$$

has a unique solution $$u\in C^{1}({\mathcal{J}},{\mathcal{R}})$$ if and only if u solves the integral equation

\begin{aligned} u(t) &=E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)} \int _{0} ^{s} q(\zeta ) u(\zeta )\,d\zeta \,ds -\nu E \sum_{j=1}^{n}\eta _{j} \int _{0}^{b_{j}}\frac{1}{p(s)} \int _{0}^{s} q(\zeta ) u(\zeta ) \,d\zeta \,ds \\ & \quad {}-E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)} \int _{0}^{s} h(\zeta ) f\bigl(u(\zeta )\bigr) \,d \zeta \,ds \\ &\quad {}+ \nu E\sum_{j=1}^{n}\eta _{j} \int _{0}^{b_{j}}\frac{1}{p(s)} \int _{0}^{s} h(\zeta ) f\bigl(u(\zeta )\bigr) \,d \zeta \,ds \\ &\quad {}- \int _{0}^{t}\frac{1}{p(s)} \int _{0}^{s} q(\zeta ) u(\zeta )\,d \zeta \,ds+ \int _{0}^{t}\frac{1}{p(s)} \int _{0}^{s} h(\zeta ) f\bigl(u(\zeta )\bigr)\,d \zeta \,ds. \end{aligned}
(2.10)

Proof

Taking the limit as $$\alpha \rightarrow 1$$ in (1.1) and applying $$(r_{3})$$ we get the ordinary Sturm–Liouville differential equation (2.9). Also from the continuity of the solution of (1.1)–(1.2) (cf. (2.1) and (2.6)) and applying $$(r_{1})$$ we obtain (2.10) and (2.8), respectively. □

Example

Consider the Sturm–Liouville fractional differential equation

$$^{c}D^{1/2}\bigl(e^{t} u'(t)\bigr)+\frac{\sqrt{t}}{100} u(t)= \frac{t}{20(t+1)} \tanh u(t), \quad t \in [0,1],$$
(2.11)

under the conditions

$$u'(0)= 0, \quad 4 u\biggl(\frac{1}{4} \biggr)-3 u\biggl(\frac{1}{3}\biggr)=\frac{1}{10} \biggl(7 u\biggl( \frac{2}{3}\biggr) - 6 u\biggl(\frac{3}{4}\biggr) \biggr).$$
(2.12)

Here, $$\alpha =\frac{1}{2}$$, $$p(t)=e^{t}$$, $$q(t)=\frac{ \sqrt{t}}{100}$$, $$h(t)=\frac{t}{20 (t+1)}$$, $$f(u(t))=\tanh u(t)$$, $$\xi _{1}=4$$, $$\xi _{2}=-3$$, $$\nu =\frac{1}{10}$$, $$\eta _{1}=7$$, $$\eta _{2}=-6$$, $$T=1$$, $$p=1$$, $$\Vert q\Vert =\frac{1}{100}$$, $$\Vert h\Vert =\frac{1}{40}$$, $$|\frac{\partial f(u)}{\partial u}|\leq 1={\mathcal{K}}$$, $$|E|= \frac{10}{9}$$, then $${\mathcal{G}}=0.2691383936 <1$$.

Therefore by Theorem 2.2, the Sturm–Liouville fractional differential equation (2.11) under the conditions (2.12) has a unique continuous solution.

3 Continuous dependence

In this section we study the continuous dependence (on the coefficient $$\xi _{k}$$ and $$\eta _{j}$$ of the multi-point condition) of the solution of the Sturm–Liouville fractional differential equation (1.1) with the multi-point boundary condition (1.2).

Definition 3.1

The solution of the fractional Sturm–Liouville differential equation (1.1) is continuously dependent on the data $$\xi _{k}$$ and $$\eta _{j}$$ if for any $$\epsilon >0$$, there exist $$\delta _{1}(\epsilon )$$ and $$\delta _{2}(\epsilon )$$ such that, for any two solutions $$u(t)$$ and $$\tilde{u}(t)$$ of (1.1) with the initial data (1.2) and

$$u'(0)=0, \quad \sum_{k=1}^{m} \tilde{\xi }_{k} u(a_{k})=\nu \sum _{j=1} ^{n}\tilde{\eta }_{j} u(b_{j}),$$
(3.1)

respectively, one has $$\sum_{k=1}^{m}|\xi _{k}-\tilde{\xi }_{k}|< \delta _{1}$$ and $$\sum_{j=1}^{n}|\eta _{j}-\tilde{\eta }_{j}|< \delta _{2}$$, then $$\Vert u-\tilde{u}\Vert < \epsilon$$ for all $$t \in {\mathcal{J}}$$.

Theorem 3.2

Let the assumptions of Theorem 2.2 be satisfied. Then the solution of Sturm–Liouville problem (1.1)(1.2) is continuously dependent on the coefficients $$\xi _{k}$$ and $$\eta _{j}$$ of the multi-point boundary condition.

Proof

Let $$u(t)$$ as defined in Eq. (2.1) be the solution of the multi-point problem (1.1)–(1.2) and

\begin{aligned} \tilde{u}(t) &=\tilde{E} \sum_{k=1}^{m} \tilde{\xi }_{k} \int _{0}^{a _{k}}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds -\nu \tilde{E} \sum_{j=1}^{n}\tilde{ \eta }_{j} \int _{0}^{b _{j}}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds \\ &\quad {}-\tilde{E} \sum_{k=1}^{m} \tilde{\xi }_{k} \int _{0}^{a_{k}} \frac{1}{p(s)}I^{\alpha } h(s) f\bigl(\tilde{u}(s)\bigr)\,ds + \nu \tilde{E}\sum_{j=1}^{n}\tilde{ \eta }_{j} \int _{0}^{b _{j}}\frac{1}{p(s)}I^{\alpha } h(s) f\bigl(\tilde{u}(s)\bigr)\,ds \\ &\quad {}- \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds+ \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } h(s) f \bigl(\tilde{u}(s)\bigr)\,ds \end{aligned}

be the solution of the nonlocal problem (1.1) and (3.1). Then

\begin{aligned} \bigl\vert u(t)-\tilde{u}(t) \bigr\vert & = \Biggl\vert E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I ^{\alpha } q(s) u(s)\,ds-\tilde{E} \sum_{k=1}^{m} \tilde{\xi }_{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds \\ & \quad {}-\nu E \sum_{j=1}^{n}\eta _{j} \int _{0}^{b_{j}}\frac{1}{p(s)}I ^{\alpha } q(s) u(s)\,ds +\nu \tilde{E} \sum_{j=1}^{n}\tilde{ \eta }_{j} \int _{0}^{b_{j}}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds \\ &\quad {}-E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{ \alpha } h(s) f\bigl(u(s)\bigr)\,ds+\tilde{E} \sum_{k=1}^{m} \tilde{\xi }_{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{\alpha } h(s) f\bigl(\tilde{u}(s)\bigr)\,ds \\ &\quad {}+ \nu E\sum_{j=1}^{n}\eta _{j} \int _{0}^{b_{j}}\frac{1}{p(s)}I ^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds -\nu \tilde{E}\sum_{j=1}^{n} \tilde{\eta }_{j} \int _{0}^{b_{j}} \frac{1}{p(s)}I^{\alpha } h(s) f\bigl(\tilde{u}(s)\bigr)\,ds \\ &\quad {}- \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds+ \int _{0} ^{t}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds \\ &\quad {}+ \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } h(s) f \bigl(u(s)\bigr)\,ds- \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } h(s) f \bigl(\tilde{u}(s)\bigr)\,ds \Biggr\vert . \end{aligned}
(3.2)

Now

\begin{aligned} & \Biggl\vert E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{ \alpha } q(s) u(s)\,ds-\tilde{E} \sum_{k=1}^{m} \tilde{\xi }_{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds \Biggr\vert \\ &\quad = \Biggl\vert E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}} \frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds-\tilde{E} \sum_{k=1}^{m} \tilde{\xi }_{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds \\ &\qquad {}-E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds+E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}} \frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds \\ &\qquad {} -\tilde{E} \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{ \alpha } q(s) \tilde{u}(s)\,ds+\tilde{E} \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds \Biggr\vert \\ &\quad \leq \vert E \vert \sum_{k=1}^{m} \vert \xi _{k} \vert \int _{0}^{a_{k}} \frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert q(s) \bigr\vert \bigl\vert u(s)-\tilde{u}(s) \bigr\vert \,ds \\ &\qquad{} + \vert E-\tilde{E} \vert \sum_{k=1}^{m} \vert \xi _{k} \vert \int _{0}^{a_{k}} \frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert q(s) \bigr\vert \bigl\vert \tilde{u}(s) \bigr\vert \,ds \\ &\qquad{}+ \vert \tilde{E} \vert \sum_{k=1}^{m} \vert \xi _{k}-\tilde{\xi }_{k} \vert \int _{0}^{a_{k}} \frac{1}{ \vert p(s) \vert }I^{\alpha } \bigl\vert q(s) \bigr\vert \bigl\vert \tilde{u}(s) \bigr\vert \,ds \\ &\quad \leq \frac{ \vert E \vert \Vert q \Vert \Vert u-\tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)} \sum_{k=1}^{m} \vert \xi _{k} \vert + \vert \tilde{E} \vert \sum _{k=1} ^{m} \vert \xi _{k}-\tilde{\xi }_{k} \vert \frac{ \Vert q \Vert \Vert \tilde{u} \Vert T^{ \alpha +1}}{p \varGamma (\alpha +2)} \\ &\qquad{} + \vert E \vert \vert \tilde{E} \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k}-\tilde{\xi }_{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j}-\tilde{\eta }_{j} \vert \Biggr) \frac{ \Vert q \Vert \Vert \tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)} \sum_{k=1} ^{m} \vert \xi _{k} \vert \\ &\quad \leq \frac{ \vert E \vert \Vert q \Vert \Vert u-\tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)} \sum_{k=1}^{m} \vert \xi _{k} \vert +\delta _{1} \vert \tilde{E} \vert \frac{ \Vert q \Vert \Vert \tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)} \\ &\qquad {} + \bigl(\delta _{1} + \vert \nu \vert \delta _{2} \bigr) \vert E \vert \vert \tilde{E} \vert \frac{ \Vert q \Vert \Vert \tilde{u} \Vert T^{\alpha +1}}{p \varGamma ( \alpha +2)} \sum_{k=1}^{m} \vert \xi _{k} \vert . \end{aligned}
(3.3)

Similar to (3.3) we can obtain

\begin{aligned} & \Biggl\vert -\nu E \sum_{j=1}^{n}\eta _{j} \int _{0}^{b_{j}}\frac{1}{p(s)}I ^{\alpha } q(s) u(s)\,ds +\nu \tilde{E} \sum_{j=1}^{n}\tilde{ \eta }_{j} \int _{0}^{b_{j}}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds \Biggr\vert \\ &\quad \leq \frac{ \vert \nu \vert \vert E \vert \Vert q \Vert \Vert u-\tilde{u} \Vert T^{ \alpha +1}}{p \varGamma (\alpha +2)} \sum_{j=1}^{n} \eta _{j} + \vert \nu \vert \delta _{2} \vert \tilde{E} \vert \frac{ \Vert q \Vert \Vert \tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)} \\ &\qquad{}+ \bigl(\delta _{1} + \vert \nu \vert \delta _{2} \bigr) \vert \nu \vert \vert E \vert \vert \tilde{E} \vert \frac{ \Vert q \Vert \Vert \tilde{u} \Vert T^{\alpha +1}}{p \varGamma ( \alpha +2)} \sum_{j=1}^{n}\eta _{j}. \end{aligned}

Also by a similar calculation to the one of (3.3) and by using (2.7), we obtain the following inequalities:

\begin{aligned} & \Biggl\vert -E \sum_{k=1}^{m} \xi _{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{ \alpha } h(s) f\bigl(u(s)\bigr)\,ds+\tilde{E} \sum_{k=1}^{m} \tilde{\xi }_{k} \int _{0}^{a_{k}}\frac{1}{p(s)}I^{\alpha } h(s) f\bigl(\tilde{u}(s)\bigr)\,ds \Biggr\vert \\ &\quad \leq \frac{{\mathcal{K}} \vert E \vert \Vert h \Vert \Vert u-\tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)} \sum_{k=1}^{m} \vert \xi _{k} \vert + \delta _{1} \vert \tilde{E} \vert \frac{ \Vert h \Vert ({\mathcal{K}} \Vert \tilde{u} \Vert +{\mathcal{M}}) T^{\alpha +1}}{p \varGamma (\alpha +2)} \\ &\qquad {}+ \bigl(\delta _{1} + \vert \nu \vert \delta _{2} \bigr) \vert E \vert \vert \tilde{E} \vert \frac{ \Vert h \Vert ({\mathcal{K}} \Vert \tilde{u} \Vert +{\mathcal{M}}) T^{\alpha +1}}{p \varGamma (\alpha +2)} \sum_{k=1}^{m} \vert \xi _{k} \vert , \\ & \Biggl\vert \nu E \sum_{j=1}^{n} \eta _{j} \int _{0}^{b_{j}}\frac{1}{p(s)}I ^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds-\nu \tilde{E} \sum_{j=1}^{n} \tilde{\eta }_{j} \int _{0}^{b_{j}}\frac{1}{p(s)}I^{\alpha } h(s) f\bigl( \tilde{u}(s)\bigr)\,ds \Biggr\vert \\ &\quad \leq \frac{ {\mathcal{K}} \vert \nu \vert \vert E \vert \Vert h \Vert \Vert u- \tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)} \sum_{j=1}^{n} \vert \eta _{j} \vert + \delta _{2} \vert \nu \vert \vert \tilde{E} \vert \frac{ \Vert h \Vert ({\mathcal{K}} \Vert \tilde{u} \Vert +{\mathcal{M}}) T^{\alpha +1}}{p \varGamma (\alpha +2)} \\ &\qquad{}+ \bigl(\delta _{1} + \vert \nu \vert \delta _{2} \bigr) \vert \nu \vert \vert E \vert \vert \tilde{E} \vert \frac{ \Vert h \Vert ({\mathcal{K}} \Vert \tilde{u} \Vert +{\mathcal{M}}) T^{\alpha +1}}{p \varGamma (\alpha +2)} \sum_{j=1}^{n} \vert \eta _{j} \vert . \end{aligned}

Also it is easy to obtain

\begin{aligned} & \biggl\vert - \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds+ \int _{0} ^{t}\frac{1}{p(s)}I^{\alpha } q(s) \tilde{u}(s)\,ds \\ &\qquad{} + \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } h(s) f \bigl(u(s)\bigr)\,ds- \int _{0} ^{t}\frac{1}{p(s)}I^{\alpha } h(s) f\bigl(\tilde{u}(s)\bigr)\,ds \biggr\vert \\ &\quad \leq \frac{ \Vert q \Vert \Vert u-\tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)}+\frac{{\mathcal{K}} \Vert h \Vert \Vert u-\tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)}. \end{aligned}

With the final five inequalities, we have after reducing similar terms in Eq. (3.2):

\begin{aligned} & \bigl\vert u(t)-\tilde{u}(t) \bigr\vert \\ &\quad \leq \frac{ \vert E \vert \Vert q \Vert \Vert u-\tilde{u} \Vert T^{\alpha +1}}{p \varGamma ( \alpha +2)} \Biggl(\sum_{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum _{j=1}^{n} \vert \eta _{j} \vert \Biggr) + \bigl(\delta _{1} + \vert \nu \vert \delta _{2} \bigr) \vert \tilde{E} \vert \frac{ \Vert q \Vert \Vert \tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)} \\ &\qquad {}+ \bigl(\delta _{1} + \vert \nu \vert \delta _{2} \bigr) \vert E \vert \vert \tilde{E} \vert \frac{ \Vert q \Vert \Vert \tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)} \Biggl(\sum_{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum _{j=1}^{n} \vert \eta _{j} \vert \Biggr) \\ &\qquad {}+\frac{{\mathcal{K}} \vert E \vert \Vert h \Vert \Vert u-\tilde{u} \Vert T^{ \alpha +1}}{p \varGamma (\alpha +2)} \Biggl(\sum_{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum _{j=1}^{n} \vert \eta _{j} \vert \Biggr) \\ &\qquad {}+ \bigl(\delta _{1} + \vert \nu \vert \delta _{2} \bigr) \vert \tilde{E} \vert \frac{ \Vert h \Vert ({\mathcal{K}} \Vert \tilde{u} \Vert +{\mathcal{M}} ) T^{\alpha +1}}{p \varGamma (\alpha +2)} \\ &\qquad {}+ \bigl(\delta _{1} + \vert \nu \vert \delta _{2} \bigr) \vert E \vert \vert \tilde{E} \vert \frac{ \Vert h \Vert ({\mathcal{K}} \Vert \tilde{u} \Vert +{\mathcal{M}}) T^{\alpha +1}}{p \varGamma (\alpha +2)} \sum_{k=1}^{m} \Biggl( \vert \xi _{k} \vert + \vert \nu \vert \sum _{j=1}^{n} \vert \eta _{j} \vert \Biggr) \\ &\qquad {}+ \frac{ \Vert q \Vert \Vert u-\tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)}+\frac{{\mathcal{K}} \Vert h \Vert \Vert u-\tilde{u} \Vert T^{\alpha +1}}{p \varGamma (\alpha +2)}, \end{aligned}

and we get

\begin{aligned} \Vert u-\tilde{u} \Vert &\leq \frac{T^{\alpha +1}}{p\varGamma (\alpha +2)} \Biggl[ \Biggl( \vert E \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr) \bigl( \Vert q \Vert + {\mathcal{K}} \Vert h \Vert \bigr) \Biggr] \Vert u-\tilde{u} \Vert \\ & \quad {}+ \bigl(\delta _{1}+ \vert \nu \vert \delta _{2} \bigr) \vert \tilde{E} \vert \frac{ [ \Vert q \Vert \Vert \tilde{u} \Vert + \Vert h \Vert ({\mathcal{K}} \Vert \tilde{u} \Vert + {\mathcal{M}} ) ] T^{\alpha +1}}{p \varGamma (\alpha +2)} \\ &\quad {} \times \Biggl( \vert E \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr), \end{aligned}

then we have

\begin{aligned} (1-{\mathcal{G}}) \Vert u-\tilde{u} \Vert &\leq \bigl(\delta _{1}+ \vert \nu \vert \delta _{2} \bigr) \vert \tilde{E} \vert \frac{ [ \Vert q \Vert \Vert \tilde{u} \Vert + \Vert h \Vert ({\mathcal{K}} \Vert \tilde{u} \Vert + {\mathcal{M}}) ] T ^{\alpha +1}}{p \varGamma (\alpha +2)} \\ & \quad{} \times \Biggl( \vert E \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr). \end{aligned}

Therefore for $$\sum_{k=1}^{m}|\xi _{k}-\tilde{\xi }_{k}|< \delta _{1}$$ and $$\sum_{j=1}^{n}|\eta _{j}-\tilde{\eta }_{j}|< \delta _{2}$$, we can find

\begin{aligned} \Vert u-\tilde{u} \Vert < \epsilon &=(1-{\mathcal{G}})^{-1} \bigl(\delta _{1}+ \vert \nu \vert \delta _{2} \bigr) \vert \tilde{E} \vert \frac{ [ \Vert q \Vert \Vert \tilde{u} \Vert + \Vert h \Vert ({\mathcal{K}} \Vert \tilde{u} \Vert +{\mathcal{M}}) ] T^{\alpha +1}}{p \varGamma (\alpha +2)} \\ &\quad {} \times \Biggl( \vert E \vert \Biggl(\sum _{k=1}^{m} \vert \xi _{k} \vert + \vert \nu \vert \sum_{j=1}^{n} \vert \eta _{j} \vert \Biggr)+1 \Biggr), \end{aligned}

i.e. for every $$\epsilon >0$$, there exist $$\delta _{1}(\epsilon ), \delta _{2}(\epsilon ) >0$$ such that $$\sum_{k=1}^{m}|\xi _{k}- \tilde{\xi }_{k}|< \delta _{1}$$ and $$\sum_{j=1}^{n}|\eta _{j}- \tilde{\eta }_{j}|< \delta _{2}$$, then $$\Vert u-\tilde{u}\Vert <\epsilon$$.

This proves the continuous dependence of the solution of the Sturm–Liouville fractional differential equation (1.1) with the multi-point boundary condition (1.2) on the coefficient $$\xi _{k}$$ and $$\eta _{j}$$ of the multi-point condition. □

4 Integral boundary conditions

Let $$u\in C^{1}({\mathcal{J}}, {\mathcal{R}})$$ be a solution of the problem (1.1) with the multi-point BCs in (1.2). Then we have the following theorem.

Theorem 4.1

Assume that the hypotheses $$(D_{1})$$$$(D_{2})$$ hold. If

$$\frac{T^{\alpha +1}}{p\varGamma (\alpha +2)} \biggl[ \biggl(\frac{\varpi (c)- \varpi (a) + \vert \nu \vert (\vartheta (e)-\vartheta (d))}{ \vert \varpi (c)-\varpi (a) - \nu (\vartheta (e)-\vartheta (d)) \vert }+1 \biggr) \bigl( \Vert q \Vert + {\mathcal{K}} \Vert h \Vert \bigr) \biggr]< 1,$$
(4.1)

then there exists a unique solution $$u \in C^{1}({\mathcal{J}}, {\mathcal{R}})$$ of the fractional differential Sturm–Liouville problem:

$$\begin{gathered} ^{c}D^{\alpha } \bigl(p(t)u'(t)\bigr)+q(t) u(t)=h(t)f\bigl(u(t)\bigr), \quad t \in [0,T], \\ u'(0)=0, \quad \int _{a}^{c}u(\theta )\,d\varpi (\theta )=\nu \int _{d}^{e}u(\theta )\,d \vartheta (\theta ), \quad 0\leq a< c\leq d< e \leq T, \end{gathered}$$
(4.2)

and u solves (4.2) if and only if u solves the integral equation

\begin{aligned} u(t)& = \frac{1}{\varpi (c)-\varpi (a) - \nu (\vartheta (e)-\vartheta (d))} \biggl[ \int _{a}^{c} \int _{0}^{\theta }\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds \,d \varpi (\theta ) \\ &\quad {}- \nu \int _{d}^{e} \int _{0}^{\theta }\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds \,d \vartheta (\theta ) - \int _{a}^{c} \int _{0}^{\theta }\frac{1}{p(s)}I ^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds \,d\varpi (\theta ) \\ &\quad {}+ \nu \int _{d}^{e} \int _{0}^{\theta }\frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds \,d \vartheta (\theta ) \biggr] \\ &\quad {}- \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds+ \int _{0}^{t} \frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds, \end{aligned}
(4.3)

provided that $$\varpi (c)-\varpi (a) \neq \nu (\vartheta (e)-\vartheta (d))$$, $$\varpi (\theta )$$ and $$\vartheta (\theta )$$ are increasing functions and the integrals are meant in the Riemann–Stieltjes sense for $$0 \leq a< c \leq d< e\leq T$$.

Proof

Let $$u \in C[0, T]$$ be a solution of problem (1.1)–(1.2). Let $$\xi _{k}=\varpi (t_{k})-\varpi (t_{k-1})>0$$, $$a_{k} \in (t_{k-1}, t_{k})$$, $$0\leq a=t_{0}< t_{1}< t_{2}<\cdots <t_{m}=c$$, $$\eta _{j}=\vartheta (\tau _{j})-\vartheta (\tau _{j-1})$$, $$b_{j} \in (\tau _{j-1},\tau _{j})$$ and $$d=\tau _{0}<\tau _{1}<\cdots<\tau _{n}=e \leq T$$. Then the multi-point boundary condition in (1.2) will be

$$\sum_{k=1}^{m}\bigl(\varpi (t_{k})-\varpi (t_{k-1})\bigr) u(a_{k})=\nu \sum _{j=1} ^{n}\bigl(\vartheta (\tau _{j})-\vartheta (\tau _{j-1})\bigr) u(b_{j}),$$

from the continuity of solution u of (1.1)–(1.2), we can obtain

$$\lim_{m\rightarrow \infty }\sum_{k=1}^{m} \bigl(\varpi (t_{k})-\varpi (t _{k-1})\bigr) u(a_{k})=\nu \lim_{n\rightarrow \infty }\sum _{j=1}^{n}\bigl( \vartheta (\tau _{j})- \vartheta (\tau _{j-1})\bigr) u(b_{j}),$$

that is, the multi-point boundary condition (1.2) is transformed to the integral condition

$$\int _{a}^{c} u(\theta )\,d\varpi (\theta )=\nu \int _{d}^{e}u(\theta ) \,d \vartheta (\theta ).$$

Next, from the continuity of the solution u (cf. (2.1)), we can get

\begin{aligned} u(t)&= \frac{1}{\sum _{k=1}^{\infty }\xi _{k} - \nu \sum _{j=1}^{\infty }\eta _{j} } \lim_{m\rightarrow \infty }\sum _{k=1}^{m} \bigl( \varpi (t_{k})-\varpi (t_{k-1})\bigr) \int _{0}^{a_{k}}\frac{1}{p(s)}I^{ \alpha } q(s) u(s)\,ds \\ &\quad {}- \frac{\nu }{\sum _{k=1}^{\infty }\xi _{k} - \nu \sum _{j=1}^{\infty }\eta _{j} }\lim_{n\rightarrow \infty }\sum _{j=1}^{n}\bigl( \vartheta (\tau _{j})- \vartheta (\tau _{j-1})\bigr) \int _{0}^{b_{j}} \frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds \\ &\quad {}- \frac{1}{\sum _{k=1}^{\infty }\xi _{k} - \nu \sum _{j=1} ^{\infty }\eta _{j} } \lim_{m\rightarrow \infty } \sum _{k=1}^{m} \bigl( \varpi (t_{k})-\varpi (t_{k-1})\bigr) \int _{0}^{a_{k}}\frac{1}{p(s)}I^{ \alpha } h(s) f\bigl(u(s)\bigr)\,ds \\ &\quad {}+ \frac{\nu }{\sum _{k=1}^{\infty }\xi _{k} - \nu \sum _{j=1}^{\infty }\eta _{j} }\lim_{n\rightarrow \infty }\sum _{j=1}^{n}\bigl( \vartheta (\tau _{j})- \vartheta (\tau _{j-1})\bigr) \int _{0}^{b_{j}} \frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds \\ &\quad {}- \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds+ \int _{0}^{t} \frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds, \end{aligned}

and we see that $$u\in C^{1}({\mathcal{J}}, {\mathcal{R}})$$ solves problem (4.2) if and only if u solves (4.3).

Finally, substituting by $$\xi _{k}$$ and $$\eta _{j}$$ in (2.6) we get (4.1). □

Corollary 4.2

Assume that the hypotheses $$(D_{1})$$$$(D_{2})$$ hold. If

$$\frac{T^{\alpha +1}}{p\varGamma (\alpha +2)} \biggl[ \biggl(\frac{(c-a) + \vert \nu \vert (e-d)}{ \vert (c-a) - \nu (e-d) \vert }+1 \biggr) \bigl( \Vert q \Vert + {\mathcal{K}} \Vert h \Vert \bigr) \biggr]< 1,$$

then there exists a unique solution $$u \in C^{1}({\mathcal{J}}, {\mathcal{R}})$$ of the Sturm–Liouville fractional differential problem:

$$\begin{gathered} ^{c}D^{\alpha } \bigl(p(t)u'(t)\bigr)+q(t) u(t)=h(t)f\bigl(u(t)\bigr), \quad t \in [0,T], \\ u'(0)=0, \quad \int _{a}^{c}u(\theta )\,d\theta =\nu \int _{d}^{e}u( \theta )\,d\theta , \quad 0\leq a< c \leq d< e \leq T, \end{gathered}$$
(4.4)

and solving (4.4) is equivalent to finding a solution $$u \in C^{1}({\mathcal{J}}, {\mathcal{R}})$$ to the integral equation

\begin{aligned} u(t)&= \frac{1}{c-a- \nu (e-d)} \biggl[ \int _{a}^{c} \int _{0}^{\theta } \frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds \,d\theta \\ &\quad {}-\nu \int _{d}^{e} \int _{0}^{\theta }\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds \,d \theta - \int _{a}^{c} \int _{0}^{\theta }\frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds \,d\theta \\ &\quad {}+\nu \int _{d}^{e} \int _{0}^{\theta }\frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds \,d \theta \biggr] \\ &\quad {}- \int _{0}^{t}\frac{1}{p(s)}I^{\alpha } q(s) u(s)\,ds+ \int _{0}^{t} \frac{1}{p(s)}I^{\alpha } h(s) f\bigl(u(s)\bigr)\,ds, \end{aligned}

provided that $$c-a \neq \nu (e-d)$$.

Corollary 4.3

Let the assumptions $$(D_{1})$$$$(D_{2})$$ be satisfied. If

$$\frac{T^{2}}{2p} \biggl[ \biggl(\frac{\varpi (c)-\varpi (a) + \vert \nu \vert ( \vartheta (e)-\vartheta (d))}{ \vert \varpi (c)-\varpi (a) - \nu (\vartheta (e)-\vartheta (d)) \vert }+1 \biggr) \bigl( \Vert q \Vert + {\mathcal{K}} \Vert h \Vert \bigr) \biggr]< 1,$$

then the ordinary Sturm–Liouville differential problem

\begin{aligned}& \bigl(p(t)u'(t)\bigr)' +q(t) u(t)=h(t)f \bigl(u(t)\bigr), \quad t \in [0,T], \end{aligned}
(4.5)
\begin{aligned}& u'(0)=0, \quad \int _{a}^{c}u(\theta )\,d\varpi (\theta )=\nu \int _{d}^{e}u(\theta )\,d \vartheta (\theta ), \quad 0\leq a< c\leq d< e \leq T, \end{aligned}
(4.6)

has a unique solution $$u \in C^{1}({\mathcal{J}}, {\mathcal{R}})$$ and solving (4.5), (4.6) is equivalent to finding a solution u to the integral equation

\begin{aligned} u(t)&= \frac{1}{\varpi (c)-\varpi (a) - \nu (\vartheta (e)-\vartheta (d))} \biggl[ \int _{a}^{c} \int _{0}^{\theta }\frac{1}{p(s)} \int _{0}^{s} q( \zeta ) u(\zeta ) \,d\zeta \,ds \,d \varpi (\theta ) \\ &\quad {}-\nu \int _{d}^{e} \int _{0}^{\theta }\frac{1}{p(s)} \int _{0}^{s} q( \zeta ) u(\zeta )\,d\zeta \,ds \,d \vartheta (\theta ) \\ &\quad {}- \int _{a}^{c} \int _{0}^{\theta }\frac{1}{p(s)} \int _{0}^{s} h( \zeta ) f\bigl(u(\zeta )\bigr)\,d \zeta \,ds \,d\varpi (\theta ) \\ &\quad {}+\nu \int _{d}^{e} \int _{0}^{\theta }\frac{1}{p(s)} \int _{0}^{s} h( \zeta ) f\bigl(u(\zeta )\bigr)\,d \zeta \,ds \,d\vartheta (\theta ) \biggr] \\ &\quad {}- \int _{0}^{t}\frac{1}{p(s)} \int _{0}^{s} q(\zeta ) u(\zeta ) \,d \zeta \,ds+ \int _{0}^{t}\frac{1}{p(s)} \int _{0}^{s} h(\zeta ) f\bigl(u( \zeta )\bigr) \,d \zeta \,ds, \end{aligned}

provided that $$\varpi (c)-\varpi (a) \neq \nu (\vartheta (e)-\vartheta (d))$$.

Proof

Taking the limit as $$\alpha \rightarrow 1$$ for (4.1), (4.2) and (4.3) and applying $$(r_{1})$$ and $$(r_{3})$$ we get the result. □

Example

Consider the Sturm–Liouville fractional differential equation

$$^{c}D^{3/4}\bigl(24 \bigl(t^{2}+1\bigr) u'(t)\bigr)+\sqrt[3]{t} u(t)=\sin t \tan ^{-1} u(t), \quad t \in (0,1),$$
(4.7)

under the conditions

$$u'(0)= 0, \quad \int _{0}^{\frac{1}{2}}u(\theta ) \,d\bigl(\theta ^{2}+1\bigr)= \frac{1}{100} \int _{3/4}^{1}u(\theta ) \,d\bigl(\theta ^{3}+2\bigr).$$
(4.8)

Here, $$\alpha =3/4$$, $$p(t)=24(t^{2}+1)$$, $$q(t)=\sqrt[3]{t}$$, $$h(t)= \sin t$$, $$f(u(t))=\tan ^{-1} u(t)$$, $$c=1/2$$, $$\varpi (1/2)=1.25$$, $$a=0$$, $$\varpi (0)=1$$, $$\nu =\frac{1}{100}$$, $$e=1$$, $$\vartheta (1)=3$$, $$d=3/4$$, $$\vartheta (3/4)=2.41875$$, $$T=1$$, $$p=24$$, $$\Vert q\Vert =1$$, $$\Vert h\Vert =1$$, $$|\frac{ \partial f(u)}{\partial u}|\leq 1={\mathcal{K}}$$, $$\varpi (c)-\varpi (a) - \nu (\vartheta (e)-\vartheta (d))=0.24421875\neq 0$$.

Then

\begin{aligned} &\frac{T^{\alpha +1}}{p\varGamma (\alpha +2)} \biggl[ \biggl(\frac{\varpi (c)- \varpi (a) + \vert \nu \vert (\vartheta (e)-\vartheta (d))}{ \vert \varpi (c)-\varpi (a) - \nu (\vartheta (e)-\vartheta (d)) \vert }+1 \biggr) \bigl( \Vert q \Vert + {\mathcal{K}} \Vert h \Vert \bigr) \biggr] \\ & \quad =0.0530467051< 1. \end{aligned}

Therefore, by Theorem 4.1, the Sturm–Liouville fractional differential equation (4.7) under the conditions (4.8) has a unique continuous solution.