Existence results for multi-term fractional differential equations with nonlocal multi-point and multi-strip boundary conditions

Abstract

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point and multi-strip boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We demonstrate the application of the obtained results with the aid of examples. Some new results are also deduced by fixing the parameters involved in the problem at hand.

1 Introduction

Multi-term fractional differential equations involve more than one fractional order differential operators and appear in the mathematical models of many real world problems. Bagley–Torvik [1] and Basset equations [2] are important examples of this class of equations.

Fractional differential equations find useful applications in several disciplines of science and engineering such as blood flow phenomena, virology, bio-engineering, image processing, control theory, etc. For details and examples, see [3–7].

The literature on initial and boundary value problems of differential equations and inclusions containing a single fractional order operator is now much enriched and one can find useful results in a series of articles [8–19] and the references cited therein. However, the topic of boundary value problems of differential equations and inclusions containing more than two fractional order operators needs to be investigated. For some works on differential equations and inclusions involving two fractional order operators (sequential fractional differential equations) can be found in [20–23].

In this paper, we introduce and investigate a new boundary value problem of multi-term fractional differential equations supplemented with nonlocal multi-point and multi-strip boundary conditions given by

$$\begin{aligned}& \bigl(\delta_{2} {}^{\mathrm{c}}D^{\alpha+2}+ \delta_{1} {}^{\mathrm{c}}D^{\alpha+1}+\delta _{0} {}^{\mathrm{c}}D^{\alpha}\bigr)x(t)=f\bigl(t,x(t)\bigr), \quad 0< \alpha< 1, 0< t< 1, \end{aligned}$$

(1.1)

$$\begin{aligned}& x(0)=0,\qquad x(\xi)=\sum_{i=1}^{n} j_{i} x(\eta_{i}),\qquad x(1)= \sum _{i=1}^{k}\lambda_{i} \int_{\upsilon_{i}}^{\sigma_{i}} x(s) \,ds, \end{aligned}$$

(1.2)

where ${}^{\mathrm{c}}D^{\alpha}$ denotes the Caputo fractional derivative of order α, $f : [0, 1] \times{\mathbb {R}} \to{\mathbb {R}}$ is a given continuous function, $0<\xi<\eta_{1}<\eta_{2}<\cdots<\eta_{n}<\upsilon _{1}<\sigma_{1}<\upsilon_{2}<\sigma_{2}<\cdots<\upsilon_{k}<\sigma_{k}<1$, $j_{i} \in{\mathbb {R}}$, $i=1,\ldots,n$, $\lambda_{i} \in\mathbb{R}$, $i=1,\ldots, k$, $\delta_{i}$ are real numbers $\{i=0,1,2\}$, with $\delta_{2}\neq0$.

The rest of the paper is organized as follows. In Sect. 2, we recall some preliminary ideas of fractional calculus and prove some important lemmas. Section 3 contains existence and uniqueness results for the problem (1.1)–(1.2) with $\delta_{1}^{2}-4\delta_{0}\delta _{2}>0$, which are obtained by applying some well-known theorems of the fixed point theory. Though the tools of the fixed point theory are standard, their exposition helps to develop the existence theory for the given problem. In Sects. 4 and 5, we outline the idea for dealing with the problem (1.1)–(1.2) involving the cases $\delta _{1}^{2}-4\delta_{0}\delta_{2}=0$ and $\delta_{1}^{2}-4\delta_{0}\delta_{2}<0$, respectively. The last section describes the importance and the scope of the obtained work.

2 Basic results

Before presenting some auxiliary results, let us recall some preliminary concepts of fractional calculus [24, 25].

Definition 2.1

Let g be a locally integrable real-valued function on $-\infty\leq a< t< b\leq+\infty$. The Riemann–Liouville fractional integral $I_{a} ^{q}$ of order $q\in\mathbb {R}$ ($q>0$) is defined as

$$I_{a} ^{q} g ( t ) = (g*K_{q} ) (t) = \frac{1}{{\Gamma ( q )}} \int _{a}^{t} { ( {t - s} )^{q - 1} g ( s )} \,ds, $$

where $K_{q}(t)=\frac{t^{q-1}}{\Gamma(q)}$, Γ denotes the Euler gamma function.

Definition 2.2

Let $g\in L^{1}[a,b]$, $-\infty\leq a< t< b\leq+\infty$ and $g*K_{m-q}\in W^{m,1}[a,b]$, $m=[q]+1$, $q>0$, where $W^{m,1}[a,b]$ is the Sobolev space defined as

$$W^{m,1}[a,b]= \biggl\{ g\in L^{1}[a,b]: \frac{d^{m}}{dt^{m}}g\in L^{1}[a,b] \biggr\} . $$

The Riemann–Liouville fractional derivative $D_{a} ^{q}$ of order $q>0$ ($m-1< q< m$, $m\in\mathbb{N}$) is defined as

$$D_{a} ^{q} g ( t ) = \frac{{d^{m} }}{{dt^{m} }}I_{a} ^{1 - q } g ( t )=\frac{1}{{\Gamma ( m-q )}}\frac{d^{m}}{dt^{m}} \int _{a}^{t} { ({t - s} )^{m-1-q} g ( s )} \,ds. $$

Definition 2.3

Let $g\in L^{1}[a,b]$, $-\infty\leq a< t< b\leq+\infty$ and $g*K_{m-q}\in W^{m,1}[a,b]$, $m=[q]$, $q>0$. The Caputo fractional derivative ${{}^{\mathrm{c}}D_{a}^{q}}$ of order $q\in\mathbb {R}$ ($m-1< q< m$, $m\in \mathbb{N}$) is defined as

$$\begin{aligned} {{}^{\mathrm{c}}D_{a}^{q}} g ( t ) = D_{a} ^{q} \biggl[ g ( t ) - g ( a ) -g' ( a ) \frac{(t-a)}{1!}-\cdots- g^{(m-1)} ( a )\frac {(t-a)^{m-1}}{(m-1)!} \biggr]. \end{aligned}$$

If $g\in C^{m}[a,b]$, then the Caputo fractional derivative ${{}^{\mathrm{c}}D_{a}^{q}}$ of order $q\in\mathbb {R}$ ($m-1< q< m$, $m\in\mathbb{N}$) is defined as

$${{}^{\mathrm{c}}D_{a}^{q}} [ g ] ( t ) = I_{a} ^{1 - q } f^{(m)} ( t )=\frac{1}{{\Gamma ( m-q )}} \int _{a}^{t} { ({t - s} )^{m-1-q} g^{(m)} ( s )} \,ds. $$

In the sequel, the Riemann–Liouville fractional integral $I_{a}^{q}$ and the Caputo fractional derivative ${{}^{\mathrm{c}}D_{a}^{q}}$ with $a=0$ are respectively denoted by $I^{q}$ and ${{}^{\mathrm{c}}D^{q}}$.

Property 2.4

([24])

With the given notations, the following equality holds:

$$ I^{q} \bigl({{}^{\mathrm{c}}D^{q} \varphi(t)}\bigr)=\varphi(t)-c_{0}-c_{1}t-\cdots -c_{n-1}t^{n-1}, \quad t>0, n-1< q< n, $$

(2.1)

where $c_{i}$ ($i=1,\ldots, n-1$) are arbitrary constants.

Definition 2.5

A function $x\in C^{3}[0,1]$ satisfying (1.1)–(1.2) is called a solution of this problem on $[0,1]$.

The following lemma associated with the linear variant of problem (1.1)–(1.2) plays an important role in the sequel.

Lemma 2.6

For any $y \in C([0,1],{\mathbb {R}})$ and $\delta_{1}^{2}-4\delta_{0}\delta_{2}>0$, the solution of linear multi-term fractional differential equation

$$ \bigl(\delta_{2} {}^{\mathrm{c}}D^{\alpha+2}+ \delta_{1} {}^{\mathrm{c}}D^{\alpha+1}+\delta _{0} {}^{\mathrm{c}}D^{\alpha}\bigr)x(t)=y(t), \quad 0< \alpha< 1, 0< t< 1, $$

(2.2)

supplemented with the boundary conditions (1.2) is given by

$$\begin{aligned} x(t) =& \frac{1}{\hat{\delta}} \Biggl\{ \int _{0}^{t} \int_{0}^{s} \bigl(e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr)\frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)}y(u)\,du \,ds \\ &{}+\rho_{1}(t) \Biggl[ \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi -s)}-e^{m_{1}(\xi-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )}y(u)\,du \,ds \\ &{}-\sum_{i=1}^{n} j_{i} \int_{0}^{\eta_{i}} \int_{0}^{s} \bigl(e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )}y(u)\,du \,ds \Biggr] \\ &{}+\rho_{2}(t) \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)}y(u)\,du \,ds \\ &{}-\sum_{i=1}^{k}\lambda_{i} \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}} \biggr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )}y(u)\,du \,ds \Biggr] \Biggr\} , \end{aligned}$$

(2.3)

where

$$\begin{aligned} &m_{1}=\frac{-\delta_{1}-\sqrt{\delta_{1}^{2}-4\delta _{0}\delta_{2}}}{2\delta_{2}}, \qquad m_{2}=\frac{-\delta_{1}+\sqrt{\delta _{1}^{2}-4\delta_{0}\delta_{2}}}{2\delta_{2}}, \\ &\delta_{2}(m_{2}-m_{1})= \frac {\sqrt{\delta_{1}^{2}-4\delta_{0}\delta_{2}}}{2}=\hat{\delta}, \\ &\rho_{1}(t)=\frac{\omega_{4}\varrho_{1}(t)-\omega _{3}\varrho_{2}(t)}{\mu_{1}}, \qquad \rho_{2}(t)= \frac{\omega_{1}\varrho _{2}(t)-\omega_{2}\varrho_{1}(t)}{\mu_{1}}, \\ &\varrho_{1}(t)=\frac{\delta _{2}[m_{1}(1-e^{m_{2}t})-m_{2}(1-e^{m_{1}t})]}{\delta_{0}}, \\ &\varrho _{2}(t)= \hat{\delta}\bigl(e^{m_{2}t}-e^{m_{1}t}\bigr), \qquad m_{1} m_{2}=\frac{\delta _{0}}{\delta_{2}}, \qquad \mu_{1}=\omega_{1}\omega_{4}- \omega_{2}\omega_{3}\neq 0, \\ &\omega_{1}=\frac{\delta_{2}}{\delta_{0}} \Biggl[m_{2}\Biggl(1-\sum _{i=1}^{n} j_{i}-e^{m_{1}\xi}+ \sum_{i=1}^{n} j_{i}e^{m_{1}\eta_{i}} \Biggr) \\ &\hphantom{\omega_{1}={}}{}-m_{1}\Biggl(1-\sum_{i=1}^{n} j_{i}-e^{m_{2}\xi}+\sum_{i=1}^{n} j_{i}e^{m_{2}\eta_{i}}\Biggr) \Biggr], \\ &\omega_{2}=\hat{\delta} \Biggl(e^{m_{1}\xi}-e^{m_{2}\xi }-\sum _{i=1}^{n} j_{i}e^{m_{1}\eta_{i}}+ \sum_{i=1}^{n} j_{i}e^{m_{2}\eta_{i}} \Biggr), \\ &\omega_{3}=\frac{\delta_{2}}{\delta_{0}} \Biggl[m_{2} \Biggl(1-e^{m_{1}}-\sum_{i=1}^{k} \lambda_{i} (\sigma_{i}-\upsilon_{i})+\sum _{i=1}^{k}\lambda_{i}/m_{1} \bigl(e^{m_{1}\sigma_{i}}-e^{m_{1}\upsilon_{i}}\bigr) \Biggr) \\ &\hphantom{\omega_{3}={}}{}-m_{1} \Biggl(1-e^{m_{2}}-\sum _{i=1}^{k}\lambda _{i} ( \sigma_{i}-\upsilon_{i})+\sum _{i=1}^{k}\lambda_{i}/m_{2} \bigl(e^{m_{2}\sigma _{i}}-e^{m_{2}\upsilon_{i}}\bigr) \Biggr) \Biggr], \\ &\omega_{4}=\hat{\delta} \Biggl(e^{m_{1}}-e^{m_{2}}- \sum_{i=1}^{k}\lambda_{i}/m_{1} \bigl(e^{m_{1} \sigma_{i}}-e^{m_{1} \upsilon_{i}}\bigr)+\sum_{i=1}^{k} \lambda_{i}/m_{2}\bigl(e^{m_{2} \sigma_{i}}-e^{m_{2} \upsilon_{i}} \bigr) \Biggr). \end{aligned}$$

(2.4)

Proof

Applying the operator $I^{\alpha}$ on (2.2) and using (2.1), we get

$$ \bigl(\delta_{2}D^{2}+\delta_{1}D+ \delta_{0}\bigr)x(t)= \int_{0}^{t} \frac{(t-s)^{\alpha -1}}{\Gamma(\alpha)}y(s) \,ds+c_{1}, $$

(2.5)

where $c_{1}$ is an arbitrary constant. By the method of variation of parameters, the solution of (2.5) can be written as

$$\begin{aligned} x(t) =& c_{2} e^{m_{1}t}+c_{3} e^{m_{2}t}- \frac{1}{\hat{\delta }} \biggl[ \int_{0}^{t} e^{m_{1}(t-s)} \biggl( \int_{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)}y(u) \,du+c_{1} \biggr)\,ds \\ &{}+ \int_{0}^{t} e^{m_{2}(t-s)} \biggl( \int_{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)}y(u) \,du+c_{1} \biggr)\,ds \biggr], \end{aligned}$$

(2.6)

where $m_{1}$ and $m_{2}$ are given by (2.4). Using $x(0)=0$ in (2.6) and simplifying the coefficient of $c_{1}$, we get

$$\begin{aligned} x(t) =& c_{1} \biggl(\frac{\delta _{2}[m_{2}(1-e^{m_{1}t})-m_{1}(1-e^{m_{2}t})]}{\delta_{0}\hat{\delta}} \biggr)+c_{2} \bigl(e^{m_{1}t}-e^{m_{2}t} \bigr) \\ &{}+ \frac{1}{\hat{\delta}} \biggl[ \int_{0}^{t} \bigl(e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr) \biggl( \int_{0}^{s}\frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)}y(u)\,du \biggr)\,ds \biggr], \end{aligned}$$

(2.7)

which, together with the conditions $x(\xi)=\sum_{i=1}^{n} j_{i} x(\eta _{i})$ and $x(1)=\sum_{i=1}^{k} \lambda_{i} \int_{\upsilon_{i}}^{\sigma_{i}} x(s) \,ds$, yields the following system of equations in the unknown constants $c_{1}$ and $c_{2}$:

$$\begin{aligned}& c_{1}\omega_{1}+c_{2}\omega_{2} = V_{1}, \end{aligned}$$

(2.8)

$$\begin{aligned}& c_{1}\omega_{3}+c_{2}\omega_{4} = V_{2} , \end{aligned}$$

(2.9)

where δ̂ and $\omega_{i}$ ($i=1, 2, 3, 4$) are given by (2.4), and

$$\begin{aligned}& \begin{aligned} V_{1} &=- \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi-s)}-e^{m_{1}(\xi-s)} \bigr) \frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}y(u)\,du \,ds \\ &\quad {}+\sum_{i=1}^{n} j_{i} \int_{0}^{\eta_{i}} \int_{0}^{s} \bigl(e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )}y(u)\,du \,ds, \end{aligned} \\& \begin{aligned} V_{2} &= - \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr) \frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}y(u)\,du \,ds \\ &\quad {}+\sum_{i=1}^{k} \lambda_{i} \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl[\frac{(e^{m_{1}(\sigma-s)}-1)}{m_{1}}- \frac{(e^{m_{2}(\sigma -s)}-1)}{m_{2}} \biggr]\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}y(u)\,du \,ds. \end{aligned} \end{aligned}$$

Solving the system (2.8)–(2.9), we find that

$$ c_{1} = \frac{V_{1}\omega_{4}-V_{2}\omega_{2}}{\mu_{1}}, \qquad c_{2} = \frac {V_{2}\omega_{1}-V_{1}\omega_{3}}{\mu_{1}}. $$

Substituting the value of $c_{1}$ and $c_{2}$ in (2.7), we obtain the solution (2.3). This completes the proof. □

Lemma 2.7

For any $y \in C([0,1],{\mathbb {R}})$ and $\delta_{1}^{2}-4\delta_{0}\delta_{2}=0$, the solution of linear multi-term fractional differential equation

$$ \bigl(\delta_{2} {}^{\mathrm{c}}D^{\alpha+2}+ \delta_{1} {}^{\mathrm{c}}D^{\alpha+1}+\delta _{0} {}^{\mathrm{c}}D^{\alpha}\bigr)x(t)=y(t),\quad 0< \alpha< 1, 0< t< 1, $$

(2.10)

supplemented with the boundary conditions (1.2) is given by

$$\begin{aligned} x(t) =& \frac{1}{\delta_{2}} \Biggl\{ \int_{0}^{t} \int _{0}^{s} (t-s)e^{m(t-s)} \frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}y(u)\,du \,ds \\ &{}+ \chi_{1}(t) \Biggl[ \int_{0}^{\xi}\int_{0}^{s} (\xi -s)e^{m(\xi-s)} \frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}y(u)\,du \,ds \\ &{}-\sum_{i=1}^{n} j_{i} \int_{0}^{\eta_{i}} \int_{0}^{s} (\eta_{i}-s)e^{m(\eta _{i}-s)} \frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}y(u)\,du \,ds \Biggr] \\ &{}+ \chi_{2}(t) \Biggl[ \int_{0}^{1} \int_{0}^{s} (1-s)e^{m(1-s)} \frac{(s-u)^{\alpha-1}}{\Gamma(q)}y(u)\,du \,ds \\ &{}-\sum_{i=1}^{k}\lambda_{i} \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{m(\sigma_{i}-s)e^{m(\sigma_{i}-s)}-e^{m(\sigma _{i}-s)}+1}{m^{2}} \biggr) \frac{(s-u)^{q-1}}{\Gamma(q)}y(u)\,du \,ds \Biggr] \Biggr\} , \end{aligned}$$

(2.11)

where

$$\begin{aligned} &m=\frac{-\delta_{1}}{2\delta_{2}}, \qquad \chi_{1}(t)=\frac {\varpi_{3}z_{2}(t)-\varpi_{4}z_{1}(t)}{\mu_{2}}, \qquad \chi_{2}(t)= \frac {\varpi_{2}z_{1}(t)-\varpi_{1}z_{2}(t)}{\mu_{2}}, \\ &z_{1}(t)=\frac{mt e^{mt}-e^{mt}+1}{m^{2}},\qquad z_{2}(t)= \delta_{2} t e^{mt}, \\ &\varpi_{1}=\frac{m\xi e^{m\xi}-e^{m\xi}+1-\sum_{i=1}^{n} j_{i}(m \eta_{i} e^{m\eta_{i}}- e^{m\eta_{i}}+1)}{m^{2}}, \\ &\varpi_{2}=\delta_{2} \Biggl(\xi e^{m\xi}-\sum _{i=1}^{n} j_{i} \eta_{i} e^{m\eta_{i}}\Biggr), \\ &\varpi_{3}=\frac{1}{m^{3}}\Biggl[m^{2} e^{m}-m e^{m}+m-m\sum_{i=1}^{k} \lambda_{i}\bigl(\sigma_{i} e^{m\sigma_{i}}- \upsilon_{i} e^{m\upsilon _{i}}\bigr) \\ &\hphantom{\varpi_{3}={}}{}+2\sum_{i=1}^{k} \lambda_{i} \bigl(e^{m\sigma_{i}}-e^{m\upsilon _{i}}\bigr)-m \sum _{i=1}^{k}\lambda_{i}( \sigma_{i}-\upsilon_{i})\Biggr], \\ &\varpi_{4}=\delta_{2} \biggl(\frac{m^{2} e^{m}-m\sum_{i=1}^{k}\lambda_{i}(\sigma_{i} e^{m\sigma_{i}}-\upsilon_{i} e^{m\upsilon _{i}})+\sum_{i=1}^{k}\lambda_{i}( e^{m\sigma_{i}}-e^{m\upsilon _{i}})}{m^{2}} \biggr), \\ &\mu_{2}=\varpi_{1}\varpi_{4}- \varpi_{2}\varpi_{3}\neq 0. \end{aligned}$$

(2.12)

Proof

Since the proof is similar to that of Lemma 2.6, we omit it. □

Lemma 2.8

For any $y \in C([0,1],{\mathbb {R}})$ and $\delta_{1}^{2}-4\delta_{0}\delta_{2}<0$, the solution of linear multi-term fractional differential equation

$$ \bigl(\delta_{2} {}^{\mathrm{c}}D^{\alpha+2}+ \delta_{1} {}^{\mathrm{c}}D^{\alpha+1}+\delta _{0} {}^{\mathrm{c}}D^{\alpha}\bigr)x(t)=y(t), \quad 0< \alpha< 1, 0< t< 1, $$

(2.13)

supplemented with the boundary conditions (1.2) is given by

$$\begin{aligned} x(t) =& \frac{1}{\delta_{2} b} \Biggl\{ \int_{0}^{t} \int_{0}^{s} e^{-a(t-s)}\sin{b (t-s)} \frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)}y(u)\,du \,ds \\ &{}+ \tau_{1}(t) \Biggl[ \int_{0}^{\xi}\int_{0}^{s} e^{-a(\xi -s)}\sin{b (\xi-s)} \frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}y(u)\,du \,ds \\ &{}-\sum_{i=1}^{n} j_{i} \int_{0}^{\eta_{i}} \int_{0}^{s} e^{-a(\eta_{i}-s)}\sin {b ( \eta_{i}-s)}\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}y(u)\,du \,ds \Biggr] \\ &{}+ \tau_{2}(t) \biggl[ \int_{0}^{1} \int_{0}^{s} e^{-a(1-s)}\sin {b (1-s)} \frac{(s-u)^{q-1}}{\Gamma(q)}y(u)\,du \,ds \\ &{}-\frac{\sum_{i=1}^{k}\lambda_{i}}{a^{2}+b^{2}} \int_{\upsilon_{i}}^{\sigma _{i}} \int_{0}^{s} \bigl(b-be^{-a(\sigma_{i}-s)}\cos{b( \sigma_{i}-s)} \\ &{}-a e^{-a(\sigma_{i}-s)}\sin{b(\sigma_{i}-s)} \bigr) \frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)}y(u)\,du \,ds \biggr] \Biggr\} , \end{aligned}$$

(2.14)

where

$$\begin{aligned} &m_{1,2}=-a\pm b i, \qquad a= \frac{\delta_{1}}{2\delta _{2}}, \qquad b=\frac{\sqrt{4\delta_{0}\delta_{2}-{\delta_{1}}^{2}}}{2\delta _{2}}, \\ &\tau_{1}(t)=\frac{q_{3}\nu_{2}(t)-q_{4}\nu_{1}(t)}{\mu_{3}},\qquad \tau_{2}(t)= \frac{q_{2}\nu_{1}(t)-q_{1}\nu_{2}(t)}{\mu_{3}}, \\ &\nu_{1}(t)=\frac{b+be^{-at}\cos{bt}-a e^{-at}\sin {bt}}{a^{2}+b^{2}}, \qquad \nu_{2}(t)= \delta_{2}b e^{-at}\sin{bt}, \\ &q_{1}=\frac{b-be^{-a\xi}\cos{b\xi}-a e^{-a\xi}\sin {b\xi}-\sum_{i=1}^{n} j_{i}(b-be^{-a\eta_{i}}\cos{b\eta_{i}}-a e^{-a\eta _{i}}\sin{b\eta_{i}})}{a^{2}+b^{2}}, \\ &q_{2}=\delta_{2}b \Biggl(e^{-a\xi}\sin{b\xi}-\sum _{i=1}^{n} j_{i} e^{-a\eta_{i}}\sin{b\eta_{i}} \Biggr), \\ &q_{3}=\frac{1}{a^{2}+b^{2}} \Biggl[b-be^{-a}\cos{b}-a e^{-a}\sin{b}-b\sum_{i=1}^{k} \lambda_{i}(\sigma_{i}-\upsilon_{i}) \\ &\hphantom{q_{3}={}}{}-\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\sum_{i=1}^{k} \lambda _{i}\bigl(e^{-a\sigma_{i}}\sin{b\sigma_{i}}-e^{-a\upsilon_{i}} \sin{b\upsilon _{i}}\bigr) \\ &\hphantom{q_{3}={}}{}-\frac{2ab}{a^{2}+b^{2}}\sum_{i=1}^{k} \lambda _{i}\bigl(e^{-a\sigma_{i}}\cos{b\sigma_{i}}-e^{-a\upsilon_{i}} \cos{b\upsilon _{i}}\bigr) \Biggr], \\ &q_{4}=\delta_{2} b \biggl[e^{-a}\sin{b}+ \frac{\sum_{i=1}^{k}\lambda_{i}}{a^{2}+b^{2}}\bigl(be^{-a\sigma_{i}}\cos{b\sigma _{i}}-be^{-a\upsilon_{i}} \cos{b\upsilon_{i}} \\ &\hphantom{q_{4}={}}{}+a e^{-a\sigma_{i}}\sin{b\sigma_{i}}-a e^{-a\upsilon_{i}}\sin {b\upsilon_{i}}\bigr) \biggr], \\ &\mu_{3}=q_{1}q_{4}-q_{2}q_{3} \neq0. \end{aligned}$$

(2.15)

Proof

We do not provide the proof as it is similar to that of Lemma 2.6. □

3 Existence and uniqueness results for the case ${\delta _{1}}^{2}-4\delta_{0}\delta_{2}>0$

Denote by $\mathcal{C}=C([0,1],\mathbb{R})$ the Banach space of all continuous functions from $[0,1]$ to $\mathbb{R}$ endowed with the norm defined by $\|x\|=\sup{\{|x(t)|:t \in[0,1]\}}$.

By Lemma 2.6, we transform the problem (1.1)-(1.2) with ${\delta_{1}}^{2}-4\delta_{0}\delta_{2}>0$ into a fixed point problem as

$$ x=\mathcal{J}x, $$

(3.1)

where the operator $\mathcal{J}: \mathcal{C} \rightarrow\mathcal {C}$ is defined by

$$\begin{aligned} (\mathcal{J}x) (t) =& \frac{1}{\hat{\delta }} \Biggl\{ \int_{0}^{t} \int_{0}^{s} \bigl(e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr)\frac {(s-u)^{\alpha-1}}{\Gamma(\alpha)}f\bigl(u,x(u)\bigr)\,du \,ds \\ &{}+\rho_{1}(t) \Biggl[ \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi -s)}-e^{m_{1}(\xi-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )}f\bigl(u,x(u)\bigr)\,du \,ds \\ &{}-\sum_{i=1}^{n} j_{i} \int_{0}^{\eta_{i}} \int_{0}^{s} \bigl(e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )}f\bigl(u,x(u)\bigr)\,du \,ds \Biggr] \\ &{}+\rho_{2}(t) \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)}f\bigl(u,x(u)\bigr)\,du \,ds \\ &{}-\sum_{i=1}^{k}\lambda_{i} \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{(e^{m_{2}(\sigma-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma -s)}-1)}{m_{1}} \biggr) \\ &{}\times\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )}f\bigl(u,x(u)\bigr)\,du \,ds \Biggr] \Biggr\} , \end{aligned}$$

with $\rho_{1}(t)$ and $\rho_{2}(t)$ given by (2.4).

In the sequel, for the sake of computational convenience, we set

$$\begin{aligned} &\widehat{\rho}_{1}=\max _{t\in[0,1]}{ \bigl\vert \rho _{1}(t) \bigr\vert }, \qquad \widehat{\rho}_{2}=\max_{t\in[0,1]}{ \bigl\vert \rho _{2}(t) \bigr\vert }, \\ &\varepsilon= \max_{t\in[0,1]} \bigl\vert m_{2} \bigl(1-e^{m_{1}t}\bigr)-m_{1}\bigl(1-e^{m_{2}t}\bigr) \bigr\vert , \\ &\phi =\frac{ \vert \delta_{2} \vert }{ \vert \delta_{0}\hat{\delta } \vert \Gamma(\alpha+1)} \Biggl\{ \varepsilon+ \widehat{\rho}_{1} \Biggl[\xi ^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\xi} \bigr)-m_{1}\bigl(1-e^{m_{2}\xi}\bigr) \bigr\vert \\ &\hphantom{\phi ={}}{}+\sum_{i=1}^{n} \vert j_{i} \vert \eta_{i}^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\eta_{i}}\bigr)-m_{1} \bigl(1-e^{m_{2}\eta_{i}}\bigr) \bigr\vert \Biggr] \\ &\hphantom{\phi ={}}{}+\widehat{\rho}_{2} \Biggl[ \bigl\vert m_{2}\bigl(1-e^{m_{1}}\bigr)-m_{1} \bigl(1-e^{m_{2}}\bigr) \bigr\vert \\ &\hphantom{\phi ={}}{}+\frac{ \vert \delta_{2} \vert }{ \vert \delta_{0} \vert }\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma_{i}^{\alpha}\bigl\vert m^{2}_{2}\bigl(m_{1}( \sigma_{i}-\upsilon _{i})-e^{m_{1}(\sigma_{i}-\upsilon_{i})}+1\bigr) \\ &\hphantom{\phi ={}}{}-m^{2}_{1}\bigl(m_{2}( \sigma_{i}-\upsilon_{i})-e^{m_{2}(\sigma _{i}-\upsilon_{i})}+1\bigr) \bigr\vert \Biggr]\Biggr\} , \\ &\phi_{1}=\phi-\frac{ \vert \delta_{2} \vert \varepsilon}{ \vert \delta _{0}\hat{\delta} \vert \Gamma(\alpha+1)}. \end{aligned}$$

(3.2)

Now the stage is set to present our main results. In the first result, we use Krasnoselskii’s fixed point theorem to prove the existence of solutions for the problem (1.1)–(1.2) with $\delta _{1}^{2}-4\delta_{0}\delta_{2}>0$.

Theorem 3.1

(Krasnoselskii’s fixed point theorem [26])

Let Y be a bounded, closed, convex, and nonempty subset of a Banach space X. Let $F_{1}$ and $F_{2}$ be operators satisfying the conditions: (i) $F_{1}y_{1}+F_{2}y_{2} \in Y$ whenever $y_{1},y_{2} \in Y$; (ii) $F_{1}$ is compact and continuous; (iii) $F_{2}$ is a contraction mapping. Then there exists a $y\in Y$ such that $y=F_{1}y+F_{2}y$.

In the forthcoming analysis, we need the following assumptions:

(A₁):

$|f(t,x)-f(t,y)|\le\ell\|x-y\|$, for all $t\in[0,1]$, $x, y \in\mathbb{R}$, $\ell>0$.

(A₂):

$|f(t,x)|\leq\vartheta(t)$, for all $(t,x)\in [0,1]\times\mathbb{R}$ and $\vartheta\in C([0,1],\mathbb{R}^{+})$.

Theorem 3.2

Let $f:[0,1]\times\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function satisfying conditions (A₁) and (A₂). Then the problem (1.1)–(1.2), with $\delta_{1}^{2}-4\delta _{0}\delta_{2}>0$, has at least one solution on $[0,1]$ if

$$ \ell\phi_{1} < 1, $$

(3.3)

where $\phi_{1}$ is given by (3.2).

Proof

Setting $\sup_{t\in[0,1]}|\vartheta(t)|=\|\vartheta\|$, we can fix

$$\begin{aligned} r \geq& \frac{|\delta_{2}|\|\vartheta\|}{|\delta_{0}\hat {\delta}|\Gamma(\alpha+1)} \Biggl\{ \varepsilon+ \widehat{\rho }_{1} \Biggl[\xi^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\xi} \bigr)-m_{1}\bigl(1-e^{m_{2}\xi}\bigr) \bigr\vert \\ &{}+\sum_{i=1}^{n}| j_{i}| \eta_{i}^{\alpha}\bigl\vert m_{2} \bigl(1-e^{m_{1}\eta _{i}}\bigr)-m_{1}\bigl(1-e^{m_{2}\eta_{i}}\bigr) \bigr\vert \Biggr] +\widehat{\rho}_{2} \Biggl[ \bigl\vert m_{2} \bigl(1-e^{m_{1}}\bigr)-m_{1}\bigl(1-e^{m_{2}}\bigr) \bigr\vert \\ &{}+\frac{\delta_{2}}{\delta_{0}}\sum_{i=1}^{k} | \lambda _{i}|\bigl\vert m^{2}_{2} \bigl(m_{1}(\sigma_{i}-\upsilon_{i})-e^{m_{1}(\sigma_{i}-\upsilon_{i})}+1 \bigr) -m^{2}_{1}\bigl(m_{2}(\sigma_{i}- \upsilon_{i})-e^{m_{2}(\sigma_{i}-\upsilon _{i})}+1\bigr) \bigr\vert \Biggr] \Biggr\} , \end{aligned}$$

and define $B_{r}=\{ x \in\mathcal{C} : \|x\|\leq r\}$. Introduce the operators $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ on $B_{r}$ as follows:

$$ (\mathcal{J}_{1} x) (t)=\frac{1}{\hat{\delta}} \int_{0}^{t} \int_{0}^{s} \bigl(e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)}f\bigl(u, x(u)\bigr)\,du \,ds $$

(3.4)

and

$$\begin{aligned} (\mathcal{J}_{2} x) (t) =& \frac{1}{\hat{\delta}} \Biggl\{ \rho _{1}(t) \Biggl[ \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi-s)}-e^{m_{1}(\xi -s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}f\bigl(u, x(u)\bigr)\,du \,ds \\ &{}-\sum_{i=1}^{n} j_{i} \int_{0}^{\eta_{i}} \int_{0}^{s} \bigl(e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )}f\bigl(u, x(u)\bigr)\,du \,ds \Biggr] \\ &{}+\rho_{2}(t) \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)}f\bigl(u, x(u)\bigr)\,du \,ds \\ &{}-\sum_{i=1}^{k}\lambda_{i} \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}} \biggr) \\ &{}\times\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}f\bigl(u, x(u)\bigr)\,du \,ds \Biggr] \Biggr\} . \end{aligned}$$

(3.5)

Observe that $\mathcal{J}=\mathcal{J}_{1}+\mathcal{J}_{2}$. For $x,y \in B_{r}$, we have

$$\begin{aligned}& \|\mathcal{J}_{1}x+\mathcal{J}_{2}y\| \\& \quad = \sup_{t\in[0,1]} \bigl\vert (\mathcal {J}_{1}x) (t)+(\mathcal{J}_{2}y) (t) \bigr\vert \\& \quad \leq \frac{1}{ \vert \hat{\delta} \vert }\sup_{t\in[0,1]} \Biggl\{ \int _{0}^{t} \int_{0}^{s} \bigl(e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr)\frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\& \qquad {}+ \bigl\vert \rho_{1}(t) \bigr\vert \Biggl[ \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi -s)}-e^{m_{1}(\xi-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, y(u)\bigr) \bigr\vert \,du \,ds \\& \qquad {}+\sum_{i=1}^{n} \vert j_{i} \vert \int_{0}^{\eta_{i}} \int_{0}^{s} \bigl(e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, y(u)\bigr) \bigr\vert \,du \,ds \Biggr] \\& \qquad {}+ \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)} \bigl\vert f\bigl(u, y(u)\bigr) \bigr\vert \,du \,ds \\& \qquad {}+\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}}\biggr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, y(u)\bigr) \bigr\vert \,du \,ds\Biggr] \Biggr\} \\& \quad \leq \frac{\|\vartheta\|}{ \vert \hat{\delta} \vert \Gamma{(\alpha +1)}}\sup_{t\in[0,1]} \Biggl\{ t^{\alpha}\int_{0}^{t} \bigl\vert e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr\vert \,ds \\& \qquad {}+ \bigl\vert \rho_{1}(t) \bigr\vert \Biggl[ \xi^{\alpha} \int_{0}^{\xi}\bigl\vert e^{m_{2}(\xi -s)}-e^{m_{1}(\xi-s)} \bigr\vert \,ds+\sum_{i=1}^{n} \vert j_{i} \vert \eta^{\alpha}_{i} \int _{0}^{\eta_{i}} \bigl\vert e^{m_{2}(\eta_{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr\vert \,ds\Biggr] \\& \qquad {}+ \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \bigl\vert e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr\vert \,ds \\& \qquad {}+\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma_{i}^{\alpha}\int_{\upsilon _{i}}^{\sigma_{i}} \biggl\vert \frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac {(e^{m_{1}(\sigma_{i}-s)}-1)}{m_{1}} \biggr\vert \,ds \Biggr] \Biggr\} \\& \quad \leq \frac{ \vert \delta_{2} \vert \|\vartheta\|}{ \vert \delta_{0}\hat{\delta } \vert \Gamma(\alpha+1)}\Biggl\{ \varepsilon+\widehat{\rho}_{1} \Biggl[\xi ^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\xi} \bigr)-m_{1}\bigl(1-e^{m_{2}\xi}\bigr) \bigr\vert \\& \qquad {}+\sum_{i=1}^{n} \vert j_{i} \vert \eta_{i}^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\eta _{i}}\bigr)-m_{1} \bigl(1-e^{m_{2}\eta_{i}}\bigr) \bigr\vert \Biggr] +\widehat{\rho }_{2} \Biggl[ \bigl\vert m_{2}\bigl(1-e^{m_{1}}\bigr)-m_{1} \bigl(1-e^{m_{2}}\bigr) \bigr\vert \\& \qquad {}+\frac{ \vert \delta_{2} \vert }{ \vert \delta_{0} \vert }\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma _{i}^{\alpha}\bigl\vert m^{2}_{2}\bigl(m_{1}( \sigma_{i}-\upsilon_{i})-e^{m_{1}(\sigma_{i}-\upsilon_{i})}+1\bigr) \\& \qquad {}-m^{2}_{1}\bigl(m_{2}(\sigma_{i}- \upsilon_{i})-e^{m_{2}(\sigma_{i}-\upsilon _{i})}+1\bigr) \bigr\vert \Biggr]\Biggr\} \\& \quad \leq r. \end{aligned}$$

Thus $\mathcal{J}_{1}x+\mathcal{J}_{2}y\in B_{r}$. Using assumption (A₁) together with (3.3), we show that $\mathcal{J}_{2}$ is a contraction as follows:

$$\begin{aligned}& \|\mathcal{J}_{2}x-\mathcal{J}_{2}y\| \\& \quad = \sup_{t\in[0,1]} \bigl\vert (\mathcal{J}_{2}x) (t)-(\mathcal{J}_{2}y) (t) \bigr\vert \\& \quad \leq \frac{1}{ \vert \hat{\delta} \vert }\sup_{t\in[0,1]} \Biggl\{ \bigl\vert \rho _{1}(t) \bigr\vert \Biggl[ \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi-s)}-e^{m_{1}(\xi -s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)} \\& \qquad {}\times \bigl\vert f\bigl(u, x(u)\bigr)-f\bigl(u, y(u)\bigr) \bigr\vert \,du \,ds \\& \qquad {} +\sum_{i=1}^{n} \vert j_{i} \vert \int_{0}^{\eta_{i}} \int_{0}^{s} \bigl(e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, x(u)\bigr)-f\bigl(u, y(u)\bigr) \bigr\vert \,du \,ds\Biggr] \\& \qquad {} + \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)} \bigl\vert f\bigl(u, x(u)\bigr)-f\bigl(u, y(u)\bigr) \bigr\vert \,du \,ds \\& \qquad {} +\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}}\biggr) \\& \qquad {} \times\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr)-f\bigl(u, y(u)\bigr) \bigr\vert \,du \,ds\Biggr] \Biggr\} \\& \quad \leq \frac{\ell}{ \vert \hat{\delta} \vert }\sup_{t\in[0,1]} \Biggl\{ \bigl\vert \rho _{1}(t) \bigr\vert \Biggl[\xi^{\alpha}\int_{0}^{\xi}\bigl\vert e^{m_{2}(\xi-s)}-e^{m_{1}(\xi -s)} \bigr\vert \,ds \\& \qquad {} +\sum_{i=1}^{n} \vert j_{i} \vert \eta^{\alpha}_{i} \int_{0}^{\eta_{i}} \bigl\vert e^{m_{2}(\eta_{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr\vert \,ds\Biggr] \\& \qquad {} + \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl\vert e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr\vert \,du \,ds \\& \qquad {} +\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma_{i}^{\alpha}\int_{\upsilon _{i}}^{\sigma_{i}} \biggl\vert \frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac {(e^{m_{1}(\sigma_{i}-s)}-1)}{m_{1}} \biggr\vert \,ds \Biggr] \Biggr\} \|x-y\| \\& \quad \leq \frac{ \vert \delta_{2} \vert \ell}{ \vert \delta_{0}\hat{\delta} \vert \Gamma(\alpha+1)} \Biggl\{ \widehat{\rho}_{1}\Biggl[ \xi^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\xi } \bigr)-m_{1}\bigl(1-e^{m_{2}\xi}\bigr) \bigr\vert \\& \qquad {} +\sum_{i=1}^{n} \vert j_{i} \vert \eta^{\alpha}_{i} \bigl\vert m_{2}\bigl(1-e^{m_{1}\eta _{i}}\bigr)-m_{1} \bigl(1-e^{m_{2}\eta_{i}}\bigr) \bigr\vert \Biggr]+\widehat{\rho }_{2} \Biggl[ \bigl\vert m_{2}\bigl(1-e^{m_{1}}\bigr)-m_{1} \bigl(1-e^{m_{2}}\bigr) \bigr\vert \\& \qquad {} +\frac{ \vert \delta_{2} \vert }{ \vert \delta_{0} \vert }\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma _{i}^{\alpha}\bigl\vert m^{2}_{2}\bigl(m_{1}( \sigma_{i}-\upsilon_{i})-e^{m_{1}(\sigma_{i}-\upsilon _{i})}+1 \bigr) \\& \qquad {} -m^{2}_{1}\bigl(m_{2}(\sigma_{i}- \upsilon_{i})-e^{m_{2}(\sigma_{i}-\upsilon_{i})}+1\bigr) \bigr\vert \Biggr]\Biggr\} \|x-y\| \\& \quad = \ell\phi_{1} \|x-y\|. \end{aligned}$$

Note that continuity of f implies that operator $\mathcal{J}_{1}$ is continuous. Also, $\mathcal{J}_{1}$ is uniformly bounded on $B_{r}$ as

$$ \|\mathcal{J}_{1}x\|= \sup_{t\in[0,1]} \bigl\vert ( \mathcal{J}_{1}x) (t) \bigr\vert \leq \frac{|\delta_{2}|\|\vartheta\|\varepsilon}{|\delta_{0}\hat{\delta }|\Gamma(\alpha+1)}. $$

Now we prove the compactness of operator $\mathcal{J}_{1}$. We define $\sup_{(t,x)\in{[0,1]\times B_{r}}}|f(t,x)|=\overline{f}$. Thus, for $0< t_{1}< t_{2}<1$, we have

$$\begin{aligned}& \bigl\vert (\mathcal{J}_{1}x) (t_{2})-( \mathcal{J}_{1}x) (t_{1}) \bigr\vert \\& \quad =\frac{1}{ \vert \hat {\delta} \vert } \biggl\vert \int_{0}^{t_{1}} \int_{0}^{s} \bigl[ \bigl(e^{m_{2}(t_{2}-s)}-e^{m_{1}(t_{2}-s)} \bigr)- \bigl(e^{m_{2}(t_{1}-s)}-e^{m_{1}(t_{1}-s)}\bigr)\bigr] \\& \qquad {} \times\frac {(s-u)^{\alpha-1}}{\Gamma(\alpha)}f\bigl(u, x(u)\bigr)\,du \,ds \\& \qquad {}+ \int_{t_{1}}^{t_{2}} \int_{0}^{s} \bigl(e^{m_{2}(t_{2}-s)}-e^{m_{1}(t_{2}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}f\bigl(u, x(u)\bigr)\,du \,ds \biggr\vert \\& \quad \leq \frac{ \vert \delta_{2} \vert \overline{f}}{ \vert \delta_{0}\hat{\delta } \vert \Gamma(\alpha+1)}\bigl\{ \bigl(t_{1}^{\alpha}-t_{2}^{\alpha}\bigr) \bigl\vert m_{1}\bigl(1-e^{m_{2}(t_{2}-t_{1})}\bigr)-m_{2} \bigl(1-e^{m_{1}(t_{2}-t_{1})} \bigr) \bigr\vert \\& \qquad {}+ t_{1}^{\alpha}\bigl\vert m_{1} \bigl(e^{m_{2}t_{2}}-e^{m_{2}t_{1}}\bigr)-m_{2} \bigl(e^{m_{1}t_{2}}-e^{m_{1}t_{1}}\bigr) \bigr\vert \bigr\} \rightarrow0, \quad \mbox{as } t_{1}\rightarrow t_{2}, \end{aligned}$$

independent of $x\in B_{r}$. Thus, $\mathcal{J}_{1}$ is relatively compact on $B_{r}$. Hence, by the Arzelá–Ascoli Theorem, $\mathcal {J}_{1}$ is compact on $B_{r}$. Thus all the assumptions of Theorem 3.1 are satisfied. So, by the conclusion of Theorem 3.1, the problem (1.1)–(1.2) with ${\delta_{1}}^{2}-4\delta_{0}\delta _{2}>0$ has at least one solution on $[0,1]$. The proof is completed. □

Remark 3.3

In the above theorem we can interchange the roles of operators $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ to obtain a second result by replacing (3.3) with the following condition:

$$\frac{|\delta_{2}|\ell\varepsilon}{|\delta_{0}\hat{\delta}|\Gamma (\alpha+1)}< 1. $$

In the next result, we prove the uniqueness of solutions for the problem (1.1)–(1.2) with ${\delta_{1}}^{2}-4\delta_{0}\delta _{2}>0$ by applying Banach contraction mapping principle.

Theorem 3.4

Assume that $f:[0,1]\times\mathbb{R }\rightarrow\mathbb{R}$ is a continuous function such that (A₁) is satisfied. Then there exists a unique solution for the problem (1.1)–(1.2), with ${\delta_{1}}^{2}-4\delta_{0}\delta_{2}>0$, on $[0, 1]$ if $\ell <1/\phi$, where ϕ is given by (3.2).

Proof

Let us define $\sup_{t\in[0,1]}{|f(t,0)|=M}$ and select $\bar{r}\geq\frac{\phi M}{1-\ell\phi}$ to show that $\mathcal{J}B_{\bar{r}} \subset B_{\bar{r}}$, where $B_{\bar{r}}=\{x\in\mathcal{C}:\|x\|\leq\bar{r}\}$ and ${\mathcal {J}}$ is defined by (3.1). Using condition (A₁), we have

$$\begin{aligned} \bigl\vert f(t,x) \bigr\vert =& \bigl\vert f(t,x)-f(t,0)+f(t,0) \bigr\vert \leq \bigl\vert f(t,x)-f(t,0) \bigr\vert + \bigl\vert f(t,0) \bigr\vert \\ \leq& \ell\|x\| + M \leq\ell\bar{r}+M. \end{aligned}$$

(3.6)

Then, for $x\in B_{\bar{r}}$, we obtain

$$\begin{aligned}& \bigl\Vert \mathcal{J}(x) \bigr\Vert \\& \quad = \sup_{t\in[0,1]} \bigl\vert \mathcal{J}(x) (t) \bigr\vert \\& \quad \leq \frac{1}{ \vert \hat{\delta} \vert }\sup_{t\in[0,1]}\Biggl\{ \int _{0}^{t} \int_{0}^{s} \bigl(e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr)\frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\& \qquad {} + \bigl\vert \rho_{1}(t) \bigr\vert \Biggl[ \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi -s)}-e^{m_{1}(\xi-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\& \qquad {} +\sum_{i=1}^{n} \vert j_{i} \vert \int_{0}^{\eta_{i}} \int_{0}^{s} \bigl(e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \Biggr] \\& \qquad {} + \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\& \qquad {} +\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}}\biggr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds\Biggr] \Biggr\} \\& \quad \leq \frac{(\ell\bar{r}+M)}{ \vert \hat{\delta} \vert }\sup_{t\in [0,1]}\Biggl\{ \int_{0}^{t} \bigl\vert e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr\vert \frac {s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \\& \qquad {} + \bigl\vert \rho_{1}(t) \bigr\vert \Biggl[ \int_{0}^{\xi}\bigl\vert e^{m_{2}(\xi-s)}-e^{m_{1}(\xi -s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \\& \qquad {} +\sum_{i=1}^{n} \vert j_{i} \vert \int_{0}^{\eta_{i}} \bigl\vert e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha +1)}}\,ds\Biggr] \\& \qquad {} + \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \bigl\vert e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}} \,ds \\& \qquad {} +\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \biggl\vert \frac {(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}} \biggr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \Biggr] \Biggr\} \\& \quad \leq \frac{ \vert \delta_{2} \vert (\ell\bar{r}+M)}{ \vert \delta_{0}\hat{\delta } \vert \Gamma(\alpha+1)}\Biggl\{ \varepsilon+\widehat{\rho}_{1} \Biggl[\xi ^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\xi} \bigr)-m_{1}\bigl(1-e^{m_{2}\xi}\bigr) \bigr\vert \\& \qquad {} +\sum_{i=1}^{n} \vert j_{i} \vert \eta^{\alpha}_{i} \bigl\vert m_{2}\bigl(1-e^{m_{1}\eta _{i}}\bigr)-m_{1} \bigl(1-e^{m_{2}\eta_{i}}\bigr) \bigr\vert \Biggr] +\widehat{\rho}_{2} \Biggl[ \bigl\vert m_{2}\bigl(1-e^{m_{1}}\bigr)-m_{1} \bigl(1-e^{m_{2}}\bigr) \bigr\vert \\& \qquad {} +\frac{ \vert \delta_{2} \vert }{ \vert \delta_{0} \vert }\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma _{i}^{\alpha}\bigl\vert m^{2}_{2}\bigl(m_{1}( \sigma_{i}-\upsilon_{i})-e^{m_{1}(\sigma_{i}-\upsilon _{i})}+1 \bigr) \\& \qquad {} -m^{2}_{1}\bigl(m_{2}(\sigma_{i}- \upsilon_{i})-e^{m_{2}(\sigma_{i}-\upsilon_{i})}+1\bigr) \bigr\vert \Biggr]\Biggr\} \\& \quad = (\ell\bar{r}+M) \phi\leq\bar{r}, \end{aligned}$$

which clearly shows that $\mathcal{J}x\in B_{\bar{r}}$ for any $x\in B_{\bar{r}}$. Thus $\mathcal{J}B_{\bar{r}}\subset B_{\bar{r}}$. Now, for $x,y\in\mathcal{C}$ and for each $t\in[0,1]$, we have

$$\begin{aligned}& \bigl\Vert (\mathcal{J}x)-(\mathcal{J}y) \bigr\Vert \\ & \quad \leq \frac{1}{ \vert \hat {\delta} \vert }\sup_{t\in[0,1]}\Biggl\{ \int_{0}^{t} \int_{0}^{s} \bigl(e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)} \bigl\vert f\bigl(u, x(u)\bigr)-f\bigl(u,y(u)\bigr) \bigr\vert \,du \,ds \\ & \qquad {} + \bigl\vert \rho_{1}(t) \bigr\vert \Biggl[ \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi -s)}-e^{m_{1}(\xi-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, x(u)\bigr)-f\bigl(u,y(u)\bigr) \bigr\vert \,du \,ds \\ & \qquad {} +\sum_{i=1}^{n} \vert j_{i} \vert \int_{0}^{\eta_{i}} \int_{0}^{s} \bigl(e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, x(u)\bigr)-f\bigl(u,y(u)\bigr) \bigr\vert \,du \,ds\Biggr] \\ & \qquad {} + \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)} \bigl\vert f\bigl(u, x(u)\bigr)-f\bigl(u,y(u)\bigr) \bigr\vert \,du \,ds \\ & \qquad {} +\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}}\biggr) \\ & \qquad {} \times\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr)-f \bigl(u,y(u)\bigr) \bigr\vert \,du \,ds\Biggr] \Biggr\} \\ & \quad \leq \frac{\ell}{ \vert \hat{\delta} \vert }\sup_{t\in[0,1]}\Biggl\{ \int _{0}^{t} \bigl\vert e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr\vert \frac{s^{\alpha}}{\Gamma {(\alpha+1)}}\,ds \\ & \qquad {} + \bigl\vert \rho_{1}(t) \bigr\vert \Biggl[ \int_{0}^{\xi}\bigl\vert e^{m_{2}(\xi-s)}-e^{m_{1}(\xi -s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \\ & \qquad {} +\sum_{i=1}^{n} \vert j_{i} \vert \int_{0}^{\eta_{i}} \bigl\vert e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha +1)}}\,ds\Biggr] \\ & \qquad {} + \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \bigl\vert e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}} \,ds \\ & \qquad {} +\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \biggl\vert \frac {(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}} \biggr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \Biggr] \Biggr\} \|x-y\| \\ & \quad \leq \frac{ \vert \delta_{2} \vert \ell}{ \vert \delta_{0}\hat{\delta} \vert \Gamma (\alpha+1)}\Biggl\{ \varepsilon+\widehat{\rho}_{1} \Biggl[\xi^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\xi} \bigr)-m_{1}\bigl(1-e^{m_{2}\xi}\bigr) \bigr\vert \\ & \qquad {} +\sum_{i=1}^{n} \vert j_{i} \vert \eta^{\alpha}_{i} \bigl\vert m_{2}\bigl(1-e^{m_{1}\eta _{i}}\bigr)-m_{1} \bigl(1-e^{m_{2}\eta_{i}}\bigr) \bigr\vert \Biggr]+\widehat{\rho }_{2} \Biggl[ \bigl\vert m_{2}\bigl(1-e^{m_{1}}\bigr)-m_{1} \bigl(1-e^{m_{2}}\bigr) \bigr\vert \\& \qquad {} +\frac{ \vert \delta_{2} \vert }{ \vert \delta_{0} \vert }\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma _{i}^{\alpha}\bigl\vert m^{2}_{2}\bigl(m_{1}( \sigma_{i}-\upsilon_{i})-e^{m_{1}(\sigma_{i}-\upsilon _{i})}+1\bigr) \\& \qquad {} -m^{2}_{1}\bigl(m_{2}( \sigma_{i}-\upsilon_{i})-e^{m_{2}(\sigma_{i}-\upsilon _{i})}+1\bigr) \bigr\vert \Biggr]\Biggr\} \|x-y\| \\& \quad = \ell\phi \|x-y\|, \end{aligned}$$

where ϕ is given by (3.2) and depends only on the parameters involved in the problem. In view of the condition $\ell <1/\phi$, it follows that $\mathcal{J}$ is a contraction. Thus, by the contraction mapping principle (Banach fixed point theorem), the problem (1.1)–(1.2) with ${\delta_{1}}^{2}-4\delta_{0}\delta _{2}>0$ has a unique solution on $[0,1]$. This completes the proof. □

The next existence result is based on Leray–Schauder nonlinear alternative.

Theorem 3.5

(Nonlinear alternative for single valued maps [27])

Let C be a closed, convex subset of be a Banach space E and U an open subset of C with $0\in U$. Suppose that $F:\overline{U}\to C$ is a continuous, compact (that is, $F(\overline {U})$ is a relatively compact subset of C) map. Then either

• (i) F has a fixed point in U̅, or

• (ii) there is a $u\in\partial U$ (the boundary of U in C) and $\epsilon\in(0,1)$ with $u=\epsilon F(u)$.

We need the following assumptions:

(H₁):

There exist a function $g\in C([0,1],{\mathbb{ R}}^{+})$, and a nondecreasing function $Q :{\mathbb {R}}^{+}\rightarrow{ \mathbb{R}}^{+}$ such that $| f(t,y)| \leq g(t)Q (\| y\| )$, $\forall(t,y)\in[0,1]\times{\mathbb{R}}$.

(H₂):

There exists a constant $K>0$ such that

$$ \frac{K}{\|g\|Q(K)\phi}>1. $$

Theorem 3.6

Let $f: [0,1]\times\mathbb{R} \to\mathbb{R}$ be a continuous function and suppose assumptions (H₁) and (H₂) are satisfied. Then the problem (1.1)–(1.2), with ${\delta_{1}}^{2}-4\delta _{0}\delta_{2}>0$, has at least one solution on $[0,1]$.

Proof

Consider the operator $\mathcal{J}: \mathcal{C} \to \mathcal{C}$ defined by (3.1). We show that $\mathcal{J}$ maps bounded sets into bounded sets in $\mathcal{C}= C([0,1], \mathbb{R})$. For a positive number ζ, let ${\mathcal {B}}_{\zeta}= \{x \in\mathcal{C}: \|x\| \le\zeta\}$ be a bounded set in $\mathcal{C}$. Then we have

$$\begin{aligned} \bigl\Vert \mathcal{J}(x) \bigr\Vert =& \sup_{t\in[0,1]} \bigl\vert \mathcal{J}(x) (t) \bigr\vert \\ \leq& \frac{1}{ \vert \hat{\delta} \vert } \sup_{t\in[0,1]}\Biggl\{ \int_{0}^{t} \int_{0}^{s} \bigl(e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr)\frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\ &{}+ \bigl\vert \rho_{1}(t) \bigr\vert \Biggl[ \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi -s)}-e^{m_{1}(\xi-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\ &{}+\sum_{i=1}^{n} \vert j_{i} \vert \int_{0}^{\eta_{i}} \int_{0}^{s} \bigl(e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \Biggr] \\ &{}+ \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\ &{}+\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}}\biggr) \\ &{}\times\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds\Biggr] \Biggr\} \\ \leq&\frac{\|g\|Q(\zeta)}{ \vert \hat{\delta} \vert } \sup_{t\in [0,1]}\Biggl\{ \int_{0}^{t} \bigl\vert e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr\vert \frac {s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \\ &{}+ \bigl\vert \rho_{1}(t) \bigr\vert \Biggl[ \int_{0}^{\xi}\bigl\vert e^{m_{2}(\xi-s)}-e^{m_{1}(\xi -s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \\ &{}+\sum_{i=1}^{n} j_{i} \int_{0}^{\eta_{i}} \bigl\vert e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha +1)}}\,ds\Biggr] \\ &{}+ \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \bigl\vert e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}} \,ds \\ &{}+\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \biggl\vert \frac {(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}} \biggr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \Biggr] \Biggr\} \\ \leq& \frac{ \vert \delta_{2} \vert \|g\|Q(\zeta)}{ \vert \delta_{0}\hat{\delta } \vert \Gamma(\alpha+1)}\Biggl\{ \varepsilon+\widehat{\rho}_{1} \Biggl[\xi^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\xi} \bigr)-m_{1}\bigl(1-e^{m_{2}\xi}\bigr) \bigr\vert \\ &{}+\sum_{i=1}^{n} \vert j_{i} \vert \eta^{\alpha}_{i} \bigl\vert m_{2}\bigl(1-e^{m_{1}\eta _{i}}\bigr)-m_{1} \bigl(1-e^{m_{2}\eta_{i}}\bigr) \bigr\vert \Biggr] \\ &{}+\widehat{\rho }_{2} \Biggl[ \bigl\vert m_{2}\bigl(1-e^{m_{1}}\bigr)-m_{1} \bigl(1-e^{m_{2}}\bigr) \bigr\vert \\ &{}+\frac{ \vert \delta_{2} \vert }{ \vert \delta_{0} \vert }\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma _{i}^{\alpha}\bigl\vert m^{2}_{2}\bigl(m_{1}( \sigma_{i}-\upsilon_{i})-e^{m_{1}(\sigma_{i}-\upsilon _{i})}+1\bigr) \\ &{}-m^{2}_{1}\bigl(m_{2}(\sigma_{i}- \upsilon_{i})-e^{m_{2}(\sigma_{i}-\upsilon _{i})}+1\bigr) \bigr\vert \Biggr]\Biggr\} , \end{aligned}$$

which yields

$$\begin{aligned} \|{\mathcal {J}}x\| \le& \frac{ \vert \delta_{2} \vert \|g\|Q(\zeta)}{ \vert \delta _{0}\hat{\delta} \vert \Gamma(\alpha+1)} \Biggl\{ \varepsilon+\widehat{\rho }_{1}\xi^{\alpha}\bigl\vert m_{2} \bigl(1-e^{m_{1}\xi}\bigr)-m_{1}\bigl(1-e^{m_{2}\xi}\bigr) \bigr\vert \\ &{}+\sum_{i=1}^{n} \vert j_{i} \vert \eta^{\alpha}_{i} \bigl\vert m_{2}\bigl(1-e^{m_{1}\eta _{i}}\bigr)-m_{1} \bigl(1-e^{m_{2}\eta_{i}}\bigr) \bigr\vert \\ &{}+\widehat{\rho }_{2}\Biggl[ \bigl\vert m_{2}\bigl(1-e^{m_{1}}\bigr)-m_{1} \bigl(1-e^{m_{2}}\bigr) \bigr\vert \\ &{}+\frac{ \vert \delta_{2} \vert }{ \vert \delta_{0} \vert }\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma _{i}^{\alpha}\bigl\vert m^{2}_{2}\bigl(m_{1}( \sigma_{i}-\upsilon_{i})-e^{m_{1}(\sigma_{i}-\upsilon_{i})}+1\bigr) \\ &{}-m^{2}_{1}\bigl(m_{2}(\sigma_{i}- \upsilon_{i})-e^{m_{2}(\sigma_{i}-\upsilon _{i})}+1\bigr) \bigr\vert \Biggr] \Biggr\} . \end{aligned}$$

Next we show that ${\mathcal {J}}$ maps bounded sets into equicontinuous sets of $\mathcal{C}$. Let $t_{1}, t_{2} \in[0,1]$ with $t_{1}< t_{2}$ and $y \in{\mathcal {B}}_{\zeta}$, where ${\mathcal {B}}_{\zeta}$ is a bounded set of $\mathcal{C}$. Then we obtain

$$\begin{aligned}& \bigl\vert (\mathcal{J}x) (t_{2})-(\mathcal{J}x) (t_{1}) \bigr\vert \\& \quad \le \frac{1}{ \vert \hat{\delta} \vert }\Biggl\{ \biggl\vert \int_{0}^{t_{1}} \int _{0}^{s} \bigl[\bigl(e^{m_{2}(t_{2}-s)}-e^{m_{1}(t_{2}-s)} \bigr)- \bigl(e^{m_{2}(t_{1}-s)}-e^{m_{1}(t_{1}-s)}\bigr)\bigr] \\& \qquad {}\times\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}f\bigl(u, x(u)\bigr)\,du\,ds \\& \qquad {}+ \int_{t_{1}}^{t_{2}} \int_{0}^{s} \bigl(e^{m_{2}(t_{2}-s)}-e^{m_{1}(t_{2}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)}f\bigl(u, x(u)\bigr)\,du \,ds \biggr\vert \\& \qquad {}+ \bigl\vert \rho_{1}(t_{2})- \rho_{1}(t_{1}) \bigr\vert \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi-s)}-e^{m_{1}(\xi-s)} \bigr) \frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\& \qquad {}+\sum_{i=1}^{n} \vert j_{i} \vert \int_{0}^{\eta_{i}} \int_{0}^{s} \biggl[ \bigl(e^{m_{2}(\eta_{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \biggr] \\& \qquad {} + \bigl\vert \rho_{2}(t_{2})- \rho_{2}(t_{1}) \bigr\vert \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr) \frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\& \qquad {} +\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}}\biggr) \\& \qquad {}\times\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds\Biggr]\Biggr\} \\& \quad \leq \frac{ \vert \delta_{2} \vert \|g\|Q(\zeta)}{ \vert \delta_{0}\hat{\delta } \vert \Gamma(\alpha+1)}\Biggl\{ \bigl(t_{1}^{\alpha}-t_{2}^{\alpha}\bigr) \bigl\vert m_{1}\bigl(1-e^{m_{2}(t_{2}-t_{1})}\bigr)-m_{2} \bigl(1-e^{m_{1}(t_{2}-t_{1})} \bigr\vert \bigr) \\& \qquad {} + t_{1}^{\alpha}\bigl\vert m_{1} \bigl(e^{m_{2}t_{2}}-e^{m_{2}t_{1}}\bigr)-m_{2} \bigl(e^{m_{1}t_{2}}-e^{m_{1}t_{1}} \bigr\vert \bigr) \\& \qquad {} + \bigl\vert \rho_{1}(t_{2})- \rho_{1}(t_{1}) \bigr\vert \Biggl[\xi^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\xi }\bigr)-m_{1} \bigl(1-e^{m_{2}\xi}\bigr) \bigr\vert \\& \qquad {} +\sum_{i=1}^{n} \vert j_{i} \vert \eta^{\alpha}_{i} \bigl\vert m_{2}\bigl(1-e^{m_{1}\eta _{i}}\bigr)-m_{1} \bigl(1-e^{m_{2}\eta_{i}}\bigr) \bigr\vert \Biggr] \\& \qquad {} + \bigl\vert \rho_{2}(t_{2})- \rho_{2}(t_{1}) \bigr\vert \Biggl[ \bigl\vert m_{2}\bigl(1-e^{m_{1}}\bigr)-m_{1} \bigl(1-e^{m_{2}}\bigr) \bigr\vert \\& \qquad {} +\frac{ \vert \delta_{2} \vert }{ \vert \delta_{0} \vert }\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma _{i}^{\alpha}\bigl\vert m^{2}_{1}\bigl(m_{2}( \sigma_{i}-\upsilon_{i})-e^{m_{2}(\sigma_{i}-\upsilon _{i})}+1 \bigr) \\& \qquad {}-m^{2}_{2}\bigl(m_{1}(\sigma_{i}- \upsilon_{i})-e^{m_{1}(\sigma_{i}-\upsilon _{i})}+1\bigr) \bigr\vert \Biggr]\Biggr\} , \end{aligned}$$

which tends to zero independently of $x \in{\mathcal {B}}_{\zeta}$ as $t_{2}- t_{1} \to0$. As ${\mathcal {J}}$ satisfies the above assumptions, it follows by the Arzelá–Ascoli theorem that ${\mathcal {J}}: \mathcal{C} \to \mathcal{C}$ is completely continuous.

The result will follow from the Leray–Schauder nonlinear alternative once it is shown that the set of all solutions to the equation $x=\theta{\mathcal {J}}x$ is bounded for $\theta\in[ 0,1]$. For that, let x be a solution of $x=\theta{\mathcal {J}}x$ for $\theta\in[ 0,1]$. Then, for $t\in[0,1]$, we have

$$\begin{aligned} \bigl\vert x(t) \bigr\vert =& \bigl\vert \theta{\mathcal {J}}x(t) \bigr\vert \\ \leq& \frac{1}{ \vert \hat{\delta} \vert } \sup_{t\in[0,1]}\Biggl\{ \int_{0}^{t} \int_{0}^{s} \bigl(e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr)\frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\ &{}+ \bigl\vert \rho_{1}(t) \bigr\vert \Biggl[ \int_{0}^{\xi}\int_{0}^{s} \bigl(e^{m_{2}(\xi -s)}-e^{m_{1}(\xi-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\ &{}+\sum_{i=1}^{n} \vert j_{i} \vert \int_{0}^{\eta_{i}} \int_{0}^{s} \bigl(e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \Biggr] \\ &{}+ \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \int_{0}^{s} \bigl(e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr)\frac{(s-u)^{\alpha-1}}{\Gamma (\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds \\ &{}+\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}}\biggr) \\ &{}\times\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)} \bigl\vert f\bigl(u, x(u)\bigr) \bigr\vert \,du \,ds\Biggr] \Biggr\} \\ \leq&\frac{ \Vert g \Vert Q( \Vert x \Vert )}{ \vert \hat{\delta} \vert } \sup_{t\in[0,1]} \Biggl\{ \int_{0}^{t} \bigl\vert e^{m_{2}(t-s)}-e^{m_{1}(t-s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \\ &{}+ \bigl\vert \rho_{1}(t) \bigr\vert \Biggl[ \int_{0}^{\xi}\bigl\vert e^{m_{2}(\xi-s)}-e^{m_{1}(\xi -s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \\ &{}+\sum_{i=1}^{n} \vert j_{i} \vert \int_{0}^{\eta_{i}} \bigl\vert e^{m_{2}(\eta _{i}-s)}-e^{m_{1}(\eta_{i}-s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha +1)}}\,ds\Biggr] \\ &{}+ \bigl\vert \rho_{2}(t) \bigr\vert \Biggl[ \int_{0}^{1} \bigl\vert e^{m_{2}(1-s)}-e^{m_{1}(1-s)} \bigr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}} \,ds \\ &{}+\sum_{i=1}^{k} \vert \lambda_{i} \vert \int_{\upsilon_{i}}^{\sigma_{i}} \biggl\vert \frac {(e^{m_{2}(\sigma_{i}-s)}-1)}{m_{2}}- \frac{(e^{m_{1}(\sigma _{i}-s)}-1)}{m_{1}} \biggr\vert \frac{s^{\alpha}}{\Gamma{(\alpha+1)}}\,ds \Biggr] \Biggr\} \\ \leq& \frac{ \vert \delta_{2} \vert \Vert g \Vert Q( \Vert x \Vert )}{ \vert \delta_{0}\hat{\delta } \vert \Gamma(\alpha+1)}\Biggl\{ \varepsilon+\widehat{\rho}_{1} \Biggl[\xi^{\alpha}\bigl\vert m_{2}\bigl(1-e^{m_{1}\xi} \bigr)-m_{1}\bigl(1-e^{m_{2}\xi}\bigr) \bigr\vert \\ &{}+\sum_{i=1}^{n} \vert j_{i} \vert \eta^{\alpha}_{i} \bigl\vert m_{2}\bigl(1-e^{m_{1}\eta _{i}}\bigr)-m_{1} \bigl(1-e^{m_{2}\eta_{i}}\bigr) \bigr\vert \Biggr] \\ &{}+\widehat{\rho }_{2} \Biggl[ \bigl\vert m_{2}\bigl(1-e^{m_{1}}\bigr)-m_{1} \bigl(1-e^{m_{2}}\bigr) \bigr\vert \\ &{}+\frac{ \vert \delta_{2} \vert }{ \vert \delta_{0} \vert }\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma _{i}^{\alpha}\bigl\vert m^{2}_{1}\bigl(m_{2}( \sigma_{i}-\upsilon_{i})-e^{m_{2}(\sigma_{i}-\upsilon _{i})}+1\bigr) \\ &{}-m^{2}_{2}\bigl(m_{1}(\sigma_{i}- \upsilon_{i})-e^{m_{1}(\sigma_{i}-\upsilon _{i})}+1\bigr) \bigr\vert \Biggr]\Biggr\} \\ =& \Vert g \Vert Q\bigl( \Vert x \Vert \bigr) \phi, \end{aligned}$$

which implies that

$$ \frac{\| x\| }{ \|g\|Q(\|x\|) \phi}\leq1. $$

In view of (H₂), there is no solution x such that $\|x\| \neq K$. Let us set

$$ U=\bigl\{ x\in{\mathcal {C}}: \Vert x \Vert < K\bigr\} . $$

The operator ${\mathcal {J}}:\overline{U}\rightarrow{\mathcal {C}}$ is continuous and completely continuous. From the choice of U, there is no $u\in\partial U$ such that $u=\theta {\mathcal {J}}(u)$ for some $\theta\in(0,1)$. Consequently, by the nonlinear alternative of Leray–Schauder type [27], we deduce that ${\mathcal {J}}$ has a fixed point $u\in\overline{U}$ which is a solution of the problem (1.1)–(1.2) with ${\delta_{1}}^{2}-4\delta_{0}\delta _{2}>0$. The proof is completed. □

Example 3.7

Consider the following multi-term fractional differential equation

$$ \bigl(2 {}^{\mathrm{c}}D^{7/3}+3 {}^{\mathrm{c}}D^{4/3}+ {}^{\mathrm{c}}D^{1/3}\bigr)x(t)=\frac{A}{4(1+t)^{2}}\tan ^{-1}x+\sin{(t+3)}, \quad 0< t< 1, $$

(3.7)

subject to the boundary conditions

$$ x(0)=0, \qquad x(1/6)=2x(1/5)+x(2/5), \qquad x(1)= \int _{1/4}^{3/5}x(s)\,ds+3 \int_{2/3}^{4/5}x(s)\,ds. $$

(3.8)

Here $\alpha=1/3$, $\xi=1/6$, $\eta_{1}=1/5$, $\eta_{2}=2/5$, $\upsilon _{1}=1/4$, $\upsilon_{2}=2/3$, $\sigma_{1}=3/5$, $\sigma_{2}=4/5$ $j_{1}=2$, $j_{2}=1$, $\lambda_{1}=1$, $\lambda_{2}=3$ and

$$ f(t,x)=\frac{A}{4(1+t)^{2}}\tan^{-1}x+\sin{(t+3)}, $$

A is positive number. Clearly, $\delta^{2}_{1}-4\delta_{0}\delta_{2}=1>0$, $|f(t,x)-f(t,y)|\leq\ell|x-y|$ with $\ell=A/4$. Using the given values, we find that $\phi\approx 0.66348$ and $\phi_{1}\approx 0.49011$. Further, we have that $|f(t,x)|\leq\frac{\pi A}{8(1+t)^{2}}+\sin(t+3)=\vartheta(t)$ and $\ell\phi_{1}<1$ when $A<8.16143$. As all the conditions of Theorem 3.2 are satisfied, the conclusion of Theorem 3.2 applies to the problem (3.7)–(3.8). On the other hand, as $\ell\phi<1$ for $A<6.02882$, there exists a unique solution for the problem (3.7)–(3.8) on $[0,1]$ by Theorem 3.4.

Example 3.8

Consider the multi-term fractional differential equation:

$$ \bigl(2 {}^{\mathrm{c}}D^{7/3}+3 {}^{\mathrm{c}}D^{4/3}+ {}^{\mathrm{c}}D^{1/3}\bigr)x(t)=\frac{2}{\sqrt{t^{2}+64}} \biggl(|x| \biggl(\frac{|x|}{|x|+1} \biggr)+\frac{1}{5} \biggr), \quad 0< t< 1, $$

(3.9)

supplemented with the boundary conditions (3.8).

Observe that $\delta^{2}_{1}-4\delta_{2}\delta_{0}=1>0$ and $|f(t,x)|\leq g(t)Q(\|x\|)$ with $g(t)=\frac{2}{\sqrt{t^{2}+64}}$ and $Q(\|x\|)=\|x\|+\frac{1}{5}$. Due to condition (H₂), using $\phi\approx 0.66348$, we find that $K> 0.15908$. Thus, by the conclusion of Theorem 3.6, there exists at least one solution for the equation (3.9) with the boundary conditions (3.8).

4 Existence results for problem (1.1)–(1.2) with $\delta_{1}^{2}-4\delta_{0}\delta_{2}=0$

In view of Lemma 2.7, we can transform problem (1.1)–(1.2) with $\delta_{1}^{2}-4\delta_{0}\delta_{2}=0$ into an equivalent fixed point problem as

$$ x=\mathcal{H}x, $$

(4.1)

where the operator $\mathcal{H}: \mathcal{C} \rightarrow\mathcal{C}$ is defined by

$$\begin{aligned} (\mathcal{H}x) (t) =& \frac{1}{\delta _{2}} \Biggl\{ \int_{0}^{t} \int_{0}^{s} (t-s)e^{m(t-s)} \frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)}f\bigl(u,x(u)\bigr)\,du \,ds \\ &{}+ \chi_{1}(t) \Biggl[ \int_{0}^{\xi}\int_{0}^{s} (\xi -s)e^{m(\xi-s)} \frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}f\bigl(u,x(u)\bigr)\,du \,ds \\ &{}-\sum_{i=1}^{n} j_{i} \int_{0}^{\eta_{i}} \int_{0}^{s} (\eta_{i}-s)e^{m(\eta _{i}-s)} \frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}f\bigl(u,x(u)\bigr)\,du \,ds \Biggr] \\ &{}+ \chi_{2}(t) \Biggl[ \int_{0}^{1} \int_{0}^{s} (1-s)e^{m(1-s)} \frac{(s-u)^{q-1}}{\Gamma(q)}f\bigl(u,x(u)\bigr)\,du \,ds \\ &{}-\sum_{i=1}^{k}\lambda_{i} \int_{\upsilon_{i}}^{\sigma_{i}} \int_{0}^{s} \biggl(\frac{m(\sigma-s)e^{m(\sigma-s)}-e^{m(\sigma-s)}+1}{m^{2}} \biggr) \\ &{}\times\frac{(s-u)^{q-1}}{\Gamma(q)}f\bigl(u,x(u)\bigr)\,du \,ds \Biggr] \Biggr\} , \end{aligned}$$

$\chi_{1}(t)$ and $\chi_{2}(t)$ are defined by (2.12). Moreover, we set

$$ \begin{aligned} &\widehat{\chi}_{1}=\max _{t\in[0,1]}{ \bigl\vert \chi _{1}(t) \bigr\vert }, \qquad \widehat{\chi}_{2}=\max_{t\in[0,1]}{ \bigl\vert \chi _{2}(t) \bigr\vert }, \\ &\beta =\frac{1}{ \vert \delta_{2} \vert m^{2} \Gamma(\alpha +1)}\Biggl\{ (1 +\widehat{\chi}_{2}) \bigl\vert (m-1)e^{m}+1 \bigr\vert \\ &\hphantom{\beta ={}}{} +\widehat{\chi}_{1}\Biggl[\xi^{\alpha}\bigl\vert (m\xi-1) e^{m\xi }+1 \bigr\vert +\sum _{i=1}^{n} \vert j_{i} \vert \eta_{i}^{\alpha}\bigl\vert (m\eta_{i} -1)e^{m\eta _{i}}+1 \bigr\vert \Biggr] \\ &\hphantom{\beta ={}}{} +\widehat{\chi}_{2}\frac{\sum_{i=1}^{k} \vert \lambda_{i} \vert \sigma_{i}^{\alpha}}{ \vert m \vert } \bigl\vert m(\sigma_{i}-\upsilon _{i}) \bigl(e^{m(\sigma_{i}-\upsilon_{i})}+1 \bigr)+2\bigl(1-e^{m(\sigma_{i}-\upsilon _{i})}\bigr) \bigr\vert \Biggr\} , \\ &\beta_{1}=\beta-\frac{ \vert (m-1)e^{m}+1 \vert }{ \vert \delta _{2} \vert m^{2}\Gamma(\alpha+1)}. \end{aligned} $$

(4.2)

Now we present existence results for the problem (1.1)–(1.2) with $\delta_{1}^{2}-4\delta_{0}\delta_{2}=0$ without proof. One can complete the proofs for these results following the arguments used in the previous section.

Theorem 4.1

Let $f:[0,1]\times\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function satisfying conditions (A₁) and (A₂). Then the problem (1.1)–(1.2), with $\delta_{1}^{2}-4\delta_{0}\delta_{2}=0$, has at least one solution on $[0,1]$ if $\ell\beta_{1} < 1$, where $\beta _{1}$ is given by (4.2).

Theorem 4.2

Assume that $f:[0,1]\times\mathbb{R }\rightarrow\mathbb{R}$ is a continuous function and condition (A₁) is satisfied. Then there exists a unique solution for problem (1.1)–(1.2), with $\delta_{1}^{2}-4\delta_{0}\delta_{2}=0$, on $[0, 1]$ if $\ell<1/\beta$, where β is given by (4.2).

Theorem 4.3

Let $f: [0,1]\times\mathbb{R} \to\mathbb{R}$ be a continuous function. In addition, suppose that (H₁) and the following condition hold:

($\mathrm{H}'_{2}$):

There exists a constant $K_{1}>0$ such that $\frac{K_{1}}{\|g\|Q(K_{1})\beta}>1$, where β is defined by (4.2).

Then the problem (1.1)–(1.2), with $\delta_{1}^{2}-4\delta _{0}\delta_{2}=0$, has at least one solution on $[0,1]$.

Example 4.4

Let us consider the multi-term fractional differential equation

$$ \bigl( {}^{\mathrm{c}}D^{7/3}+2 {}^{\mathrm{c}}D^{4/3}+ {}^{\mathrm{c}}D^{1/3}\bigr)x(t)=\biggl(\frac{|x|}{1+|x|}+t\biggr) \frac {B}{2\sqrt{t^{2}+4}}+\cos t, \quad 0< t< 1, $$

(4.3)

supplemented with the boundary conditions (3.8), where

$$ f(t,x)= \biggl(\frac{|x|}{1+|x|}+t \biggr)\frac{B}{2\sqrt{t^{2}+4}}+\cos t $$

and B is positive number.

Obviously, $\delta^{2}_{1}-4\delta_{0}\delta_{2}=0$, and $|f(t,x)-f(t,y)|\leq\ell|x-y|$ with $\ell=B/4$. Using the given values, we find that $\beta\approx0.39636$ and $\beta_{1}\approx0.10045$. It is easy to check that $|f(t,x)|\leq\frac {B(1+t)}{2\sqrt{t^{2}+4}}+\cos t=\vartheta(t)$ and $\ell\beta_{1}<1$ when $B<39.82081$. As all the condition of Theorem 4.1 are satisfied, equation (4.3) with the boundary data (3.8) has at least one solution on $[0,1]$. On the other hand, $\ell\beta <1$ whenever $B<10.091836$, so there exists a unique solution for equation (4.3) with the boundary data (3.8) on $[0,1]$ by Theorem 4.2.

5 Existence results for problem (1.1)–(1.2) with $\delta_{1}^{2}-4\delta_{0}\delta_{2}<0$

By Lemma 2.8, the fixed point problem equivalent to the problem (1.1)–(1.2) with $\delta_{1}^{2}-4\delta_{0}\delta _{2}<0$ can be written as

$$ x=\mathcal{K}x, $$

(5.1)

where the operator $\mathcal{K}: \mathcal{C} \rightarrow\mathcal{C}$ is defined by

$$\begin{aligned} (\mathcal{K}x) (t) =& \frac{1}{\delta_{2} b} \Biggl\{ \int_{0}^{t} \int_{0}^{s} e^{-a(t-s)}\sin{b (t-s)} \frac {(s-u)^{\alpha-1}}{\Gamma(\alpha)}f\bigl(u,x(u)\bigr)\,du \,ds \\ &{}+ \tau_{1}(t) \Biggl[ \int_{0}^{\xi}\int_{0}^{s} e^{-a(\xi -s)}\sin{b (\xi-s)} \frac{(s-u)^{\alpha-1}}{\Gamma(\alpha )}f\bigl(u,x(u)\bigr)\,du \,ds \\ &{}-\sum_{i=1}^{n} j_{i} \int_{0}^{\eta_{i}} \int_{0}^{s} e^{-a(\eta_{i}-s)}\sin {b ( \eta_{i}-s)}\frac{(s-u)^{\alpha-1}}{\Gamma(\alpha)}f\bigl(u,x(u)\bigr)\,du \,ds \Biggr] \\ &{}+ \tau_{2}(t) \biggl[ \int_{0}^{1} \int_{0}^{s} e^{-a(1-s)}\sin {b (1-s)} \frac{(s-u)^{q-1}}{\Gamma(q)}f\bigl(u,x(u)\bigr)\,du \,ds \\ &{}-\frac{\sum_{i=1}^{k}\lambda_{i}}{a^{2}+b^{2}} \int_{\upsilon_{i}}^{\sigma _{i}} \int_{0}^{s} \bigl(b-be^{-a(\sigma-s)}\cos{b( \sigma-s)} \\ &{} -a e^{-a(\sigma-s)}\sin{b(\sigma-s)} \bigr)\frac{(s-u)^{\alpha -1}}{\Gamma(\alpha)}f\bigl(u,x(u) \bigr)\,du \,ds \biggr] \Biggr\} , \end{aligned}$$

$\tau_{1}(t)$ and $\tau_{2}(t)$ are defined by (2.15).

Further, we set

$$ \begin{aligned} &\widehat{\tau}_{1}=\max _{t\in[0,1]}{ \bigl\vert \tau _{1}(t) \bigr\vert }, \qquad \widehat{\tau}_{2}=\max_{t\in[0,1]}{ \bigl\vert \tau _{2}(t) \bigr\vert } \\ & \gamma =\frac{1}{ \vert \delta_{0} b \vert \Gamma(\alpha +1)}\Biggl\{ (1 +\widehat{\tau}_{2}) \bigl\vert b-be^{-a}\cos b-ae^{-a}\sin b \bigr\vert \\ &\hphantom{\gamma ={}}{}+\widehat{\tau}_{1}\Biggl[\xi^{\alpha}\bigl\vert b-be^{-a \xi }\cos{b\xi}-ae^{-a \xi}\sin{b\xi} \bigr\vert \\ &\hphantom{\gamma ={}}{} +\sum_{i=1}^{n} \vert j_{i} \vert \eta_{i}^{\alpha}\bigl\vert b-be^{-a \eta_{i}}\cos{b\eta _{i}}-ae^{-a \eta_{i}}\sin{b \eta_{i}} \bigr\vert \Biggr] \\ &\hphantom{\gamma ={}}{} +\widehat{\tau}_{2}\sum _{i=1}^{k} \vert \lambda_{i} \vert \sigma _{i}^{\alpha}\bigl\vert b(\sigma_{i}- \upsilon_{i})-e^{-a(\sigma_{i}-\upsilon _{i})}\sin{b(\sigma_{i}- \upsilon_{i})} \bigr\vert \Biggr\} , \\ &\gamma_{1}=\gamma-\frac{ \vert b-be^{-a}\cos b-a e^{-a}\sin b \vert }{ \vert \delta_{0} b \vert \Gamma(\alpha+1)},\qquad a=\frac{\delta_{1}}{2\delta_{2}}, \qquad b=\frac{\sqrt{4\delta_{0}\delta_{2}-{\delta_{1}}^{2}}}{2\delta _{2}}. \end{aligned} $$

(5.2)

As before, we can formulate existence results for the problem (1.1)–(1.2) with $\delta_{1}^{2}-4\delta_{0}\delta_{2}<0$ as follows.

Theorem 5.1

Let $f:[0,1]\times\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function satisfying conditions (A₁) and (A₂). Then the problem (1.1)–(1.2), with $\delta_{1}^{2}-4\delta_{0}\delta_{2}<0$, has at least one solution on $[0,1]$ provided that $\ell\gamma_{1} < 1$, where γ is given by (5.2).

Theorem 5.2

Assume that $f:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function such that (A₁) is satisfied. Then there exists a unique solution for the problem (1.1)–(1.2), with $\delta _{1}^{2}-4\delta_{0}\delta_{2}<0$, on $[0, 1]$ if $\ell<1/\gamma$, where γ is given by (5.2).

Theorem 5.3

Let $f: [0,1]\times\mathbb{R} \to\mathbb{R}$ be a continuous function. Further, suppose that (H₁) and the following condition hold:

($\mathrm{H}''_{2}$):

There exists a constant $K_{2}>0$ such that $\frac{K_{2}}{ \|g\|Q(K_{2})\gamma}>1$, where γ is defined by (5.2).

Then the problem (1.1)–(1.2), with $\delta_{1}^{2}-4\delta _{0}\delta_{2}<0$, has at least one solution on $[0,1]$.

Example 5.4

Consider the following multi-term fractional differential equation

$$ \bigl(2 {}^{\mathrm{c}}D^{7/3}+2 {}^{\mathrm{c}}D^{4/3}+ {}^{\mathrm{c}}D^{1/3}\bigr)x(t)=f(t,x), \quad 0< t< 1, $$

(5.3)

equipped with the boundary conditions (3.8), where

$$ f(t,x)=\frac{L}{3\sqrt{t^{3}+9}}\bigl(\cos x+e^{-2t}\bigr), \quad L>0. $$

Clearly, $\delta^{2}_{1}-4\delta_{0}\delta_{2}=-4<0$ and $|f(t,x)-f(t,y)|\leq\ell|x-y|$ with $\ell=L/9$. Using the given values, it is found that $\gamma\approx 0.57912$ and $\gamma _{1}\approx 0.38098$. Further, it is easy to check that $|f(t,x)|\leq\frac {L(1+e^{-2t})}{3\sqrt{t^{3}+9}}=\vartheta(t)$ and $\ell\gamma_{1}<1$ when $L<23.62329$. As all the conditions of Theorem 5.1 are satisfied, equation (5.3) with the boundary conditions (3.8) has at least one solution on $[0,1]$. On the other hand, since $\ell\gamma<1$ for $L<15.54082$, there exists a unique solution for equation (5.3) with the boundary conditions (3.8) on $[0,1]$ by Theorem 5.2.

6 Conclusions

We have derived existence results for a multi-term fractional differential equation associated with different combinations ($\delta_{1}^{2}-4\delta_{0}\delta_{2}>0$, $\delta_{1}^{2}-4\delta _{0}\delta_{2}=0$, and $\delta_{1}^{2}-4\delta_{0}\delta_{2}<0$) of the constants involved in the equation equipped with nonlocal multi-point and multi-strip boundary conditions. Our results are not only new in the given context, but also yield some interesting new results as special cases of the obtained work. For instance, by taking $\lambda _{i}=0$, $i=i, \ldots, k$ in the results of this paper, we obtain new results for the multi-term fractional differential equation (1.1) associated with the boundary condition of the form: $x(0)=0$, $x(\xi)=\sum_{i=1}^{n} j_{i} x(\eta_{i})$, $x(1)= 0$. Our results correspond to those for (1.1) with the nonlocal multi-strip boundary condition: $x(0)=0$, $x(\xi)=0$, $x(1)= \sum_{i=1}^{k}\lambda_{i} \int_{\upsilon_{i}}^{\sigma_{i}} x(s) \,ds$ if we fix $j_{i}=0$, $i=1,\ldots, n$ in the obtained results.