Skip to content

Advertisement

  • Research
  • Open Access

Solutions for a class of fractional Langevin equations with integral and anti-periodic boundary conditions

Boundary Value Problems20182018:152

https://doi.org/10.1186/s13661-018-1070-3

  • Received: 11 March 2018
  • Accepted: 18 September 2018
  • Published:

Abstract

In this paper, we consider a class of fractional Langevin equations with integral and anti-periodic boundary conditions. By using some fixed point theorems and the Leray–Schauder degree theory, several new existence results of solutions are obtained.

Keywords

  • Fractional Langevin equation
  • Mixed boundary condition
  • Leray–Schauder degree theory
  • Fixed point theorem

MSC

  • 34A08
  • 34B15
  • 34B18

1 Introduction

In this paper, we consider the existence of solutions for the following fractional Langevin equation with integral and anti-periodic conditions:
$$\begin{aligned} \textstyle\begin{cases} ^{c}D^{\beta }_{0^{+}}(^{c}D^{\alpha }_{0^{+}}+\lambda)x(t)=f(t,x(t)),\quad 0< t< 1, \\ x(0)=0,\qquad x(1)=\mu \int_{0}^{1}x(s)\,ds,\qquad {}^{c}D^{\alpha }_{0^{+}}x(0)+^{c}D ^{\alpha }_{0^{+}}x(1)=0, \end{cases}\displaystyle \end{aligned}$$
(1)
where \(0<\alpha <1\), \(1<\beta \leq 2\), \(\lambda >0\), and \(\mu >0\) are real numbers, \({}^{c}D^{\alpha }_{0^{+}}x(t)\) and \({}^{c}D^{\beta }_{0^{+}}x(t)\) are the Caputo fractional derivatives, and \(f\in C([0,1]\times R,R)\).

Theory of fractional differential equations has important application in many areas. It has become a new research field in differential equations [13]. There are a lot of good research results on boundary value problems of fractional differential equations [424]. Recently fractional Langevin equations have been studied by some scholars (see, for example, [2527]).

In [28], via fixed point theorems, Ahmad et al. discussed the existence of solutions for fractional Langevin equations with three-point nonlocal boundary value conditions.

In [29], Li et al. investigated the infinite-point boundary value problem of fractional Langevin equations. By means of the nonlinear alternative and Leray–Schauder degree theory, they got some existence results for the boundary value problem.

To our knowledge, for fractional Langevin equation, the boundary value problem with integral and anti-periodic boundary conditions is rarely studied, so the research of this paper is new.

The rest of this paper is organized as follows. In Sect. 2, we present some preliminaries and lemmas. In Sect. 3, we adopt some fixed point theorems and the Leray–Schauder degree theory to obtain the existence of solutions for boundary value problem (1).

2 Relevant lemmas

Some necessary definitions from fractional calculus theory can be found in [2, 3]. We omit them here.

Lemma 2.1

Let \(h\in C(0,1)\cap L(0,1)\), \(0<\alpha <1\), \(1<\beta \leq 2\), then the following problem
$$\begin{aligned} \textstyle\begin{cases} ^{c}D^{\beta }_{0^{+}}(^{c}D^{\alpha }_{0^{+}}+\lambda)x(t)=h(t),\quad t \in (0,1) \\ x(0)=0,\qquad x(1)=\mu \int_{0}^{1}x(s)\,ds,\qquad {}^{c}D^{\alpha }_{0^{+}}x(0)+^{c}D ^{\alpha }_{0^{+}}x(1)=0 \end{cases}\displaystyle \end{aligned}$$
(2)
has a solution \(x(t)\) satisfying
$$\begin{aligned} x(t) =&\frac{1}{\Gamma (\alpha +\beta)} \int_{0}^{t}(t-s)^{\alpha + \beta -1}h(s)\,ds - \frac{t^{\alpha }(2t-\alpha -1)}{(1-\alpha)\Gamma ( \alpha +\beta)} \int_{0}^{1}(1-s)^{\alpha +\beta -1}h(s)\,ds \\ &{}+ \frac{\lambda t^{\alpha }(2t-\alpha -1)}{(1-\alpha)\Gamma (\alpha)} \int_{0}^{1}(1-s)^{\alpha -1}x(s)\,ds \\ &{} -\frac{t^{\alpha }(1-t)}{(1- \alpha)\Gamma (\alpha +1)\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1}h(s)\,ds \\ &{} +\frac{\lambda \mu t^{\alpha }(1-t)}{(1-\alpha)\Gamma (\alpha +1)} \int_{0}^{1}x(s)\,ds \\ &{} +\frac{\mu t^{\alpha }(2t-\alpha -1)}{1-\alpha } \int_{0}^{1}x(s)\,ds -\frac{\lambda }{\Gamma (\alpha)} \int_{0}^{t}(t-s)^{ \alpha -1}x(s)\,ds. \end{aligned}$$

Proof

From the relevant lemma in [23], it follows that
$$\begin{aligned}& \bigl(^{c}D^{\alpha }_{0^{+}}+\lambda\bigr)x(t)=I_{0^{+}}^{\beta }h(t)+c_{0}+c _{1}t, \\& {}^{c}D^{\alpha }_{0^{+}}x(t)=I_{0^{+}}^{\beta }h(t)+c_{0}+c_{1}t- \lambda x(t)=\frac{1}{\Gamma (\beta)} \int_{0}^{t}(t-s)^{\beta -1}h(s)\,ds +c_{0}+c_{1}t-\lambda x(t), \\& x(t)=I_{0^{+}}^{\alpha +\beta }h(t)+I_{0^{+}}^{\alpha }c_{0}+I_{0^{+}} ^{\alpha }c_{1}t-I_{0^{+}}^{\alpha }\lambda x(t)+c_{2} \\& \hphantom{x(t)}=\frac{1}{\Gamma (\alpha +\beta)} \int_{0}^{t}(t-s)^{\alpha +\beta -1}h(s)\,ds + \frac{c_{0}}{\Gamma (\alpha +1)}t^{\alpha } +\frac{c_{1}}{\Gamma ( \alpha +2)}t^{\alpha +1} \\& \hphantom{x(t)=} -\frac{\lambda }{\Gamma (\alpha)} \int_{0} ^{t}(t-s)^{\alpha -1}x(s) \,ds+c_{2}. \end{aligned}$$
By the boundary value conditions \(x(0)=0\), \(x(1)=\mu \int_{0}^{1}x(s)\,ds\), and \({}^{c}D^{\alpha }_{0^{+}}x(0)+\) \({}^{c}D^{\alpha }_{0^{+}}x(1)=0\), we can get
$$\begin{aligned}& c_{2}=0, \\& \frac{1}{\Gamma (\alpha +\beta)} \int_{0}^{1}(1-s)^{\alpha +\beta -1}h(s)\,ds + \frac{c_{0}}{\Gamma (\alpha +1)} +\frac{c_{1}}{\Gamma (\alpha +2)} -\frac{ \lambda }{\Gamma (\alpha)} \int_{0}^{1}(1-s)^{\alpha -1}x(s)\,ds \\& \quad =\mu \int_{0}^{1}x(s)\,ds, \end{aligned}$$
and
$$ c_{0}+\frac{1}{\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1}h(s)\,ds +c _{0}+c_{1}-\lambda \mu \int_{0}^{1}x(s)\,ds=0. $$
By direct computation, we have
$$\begin{aligned} c_{0} =&\frac{\Gamma (\alpha +2)}{(1-\alpha)\Gamma (\alpha +\beta)} \int_{0}^{1}(1-s)^{\alpha +\beta -1}h(s)\,ds - \frac{\lambda \alpha (1+ \alpha)}{1-\alpha } \int_{0}^{1}(1-s)^{\alpha -1}x(s)\,ds \\ &{} +\frac{\lambda \mu }{1-\alpha } \int_{0}^{1}x(s)\,ds -\frac{\mu \Gamma (\alpha +2)}{1-\alpha } \int_{0}^{1}x(s)\,ds -\frac{1}{(1- \alpha)\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1}h(s)\,ds, \end{aligned}$$
and
$$\begin{aligned} c_{1} =&\frac{2\lambda \alpha (1+\alpha)}{1-\alpha } \int_{0}^{1}(1-s)^{ \alpha -1}x(s)\,ds + \frac{1+\alpha }{(1-\alpha)\Gamma (\beta)} \int _{0}^{1}(1-s)^{\beta -1}h(s)\,ds \\ &{} -\frac{\lambda \mu (1+\alpha)}{1- \alpha } \int_{0}^{1}x(s)\,ds \\ &{} +\frac{2\mu \Gamma (\alpha +2)}{1-\alpha } \int_{0}^{1}x(s)\,ds -\frac{2 \Gamma (\alpha +2)}{(1-\alpha)\Gamma (\alpha +\beta)} \int_{0}^{1}(1-s)^{ \alpha +\beta -1}h(s)\,ds. \end{aligned}$$
Thus, the solution of (2) satisfies
$$\begin{aligned} x(t) =&\frac{1}{\Gamma (\alpha +\beta)} \int_{0}^{t}(t-s)^{\alpha + \beta -1}h(s)\,ds - \frac{t^{\alpha }(2t-\alpha -1)}{(1-\alpha)\Gamma ( \alpha +\beta)} \int_{0}^{1}(1-s)^{\alpha +\beta -1}h(s)\,ds \\ &{} + \frac{\lambda t^{\alpha }(2t-\alpha -1)}{(1-\alpha)\Gamma (\alpha)} \int_{0}^{1}(1-s)^{\alpha -1}x(s)\,ds \\ &{} -\frac{t^{\alpha }(1-t)}{(1- \alpha)\Gamma (\alpha +1)\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1}h(s)\,ds \\ &{} +\frac{\lambda \mu t^{\alpha }(1-t)}{(1-\alpha)\Gamma (\alpha +1)} \int_{0}^{1}x(s)\,ds +\frac{\mu t^{\alpha }(2t-\alpha -1)}{1-\alpha } \int_{0}^{1}x(s)\,ds \\ &{} -\frac{\lambda }{\Gamma (\alpha)} \int_{0}^{t}(t-s)^{ \alpha -1}x(s)\,ds. \end{aligned}$$

This completes the proof. □

3 Main results

Let \(E=C[0,1]\). Obviously, the space E is a Banach space if it is endowed with the norm as follows:
$$ \Vert x \Vert =\max_{t\in [0,1]} \bigl\vert x(t) \bigr\vert . $$
Define an operator \(T:E\rightarrow E\) as
$$\begin{aligned} (Tx) (t) =&\frac{1}{\Gamma (\alpha +\beta)} \int_{0}^{t}(t-s)^{\alpha + \beta -1}f\bigl(s,x(s)\bigr) \,ds -\frac{t^{\alpha }(2t-\alpha -1)}{(1-\alpha) \Gamma (\alpha +\beta)} \\ &{}\times \int_{0}^{1}(1-s)^{\alpha +\beta -1}f\bigl(s,x(s)\bigr) \,ds +\frac{\lambda t^{ \alpha }(2t-\alpha -1)}{(1-\alpha)\Gamma (\alpha)} \int_{0}^{1}(1-s)^{ \alpha -1}x(s)\,ds \\ &{} - \frac{t^{\alpha }(1-t)}{(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1}f\bigl(s,x(s)\bigr) \,ds+\frac{\lambda \mu t^{\alpha }(1-t)}{(1- \alpha)\Gamma (\alpha +1)} \int_{0}^{1}x(s)\,ds \\ &{} +\frac{\mu t^{\alpha }(2t-\alpha -1)}{1-\alpha } \int_{0}^{1}x(s)\,ds -\frac{ \lambda }{\Gamma (\alpha)} \int_{0}^{t}(t-s)^{\alpha -1}x(s)\,ds,\quad t \in [0,1]. \end{aligned}$$
(3)

It is easy to see that the solution for (1) is equivalent to the fixed point of T.

Lemma 3.1

  1. (i)

    \(\max_{t\in [0,1]}|t^{\alpha }(2t-\alpha -1)|=\max \{(\frac{ \alpha }{2})^{\alpha },1-\alpha \}\);

     
  2. (ii)

    \(\max_{t\in [0,1]}t^{\alpha }(1-t)=\frac{\alpha^{\alpha }}{(1+ \alpha)^{1+\alpha }}\).

     

Proof

(i) Let \(g(t)=t^{\alpha }(2t-\alpha -1)\), \(0\leq t\leq 1\), then \(g'(t)=(\alpha +1)t^{\alpha -1}(2t-\alpha)\), \(g(0)=0\), \(g(1)=1-\alpha >0\).

When \(0\leq t<\frac{\alpha }{2}\), \(g'(t)\leq 0\); when \(\frac{\alpha }{2}< t\leq 1\), \(g'(t)\geq 0\). In conclusion, \(|g(t)|_{ \max }=\max \{|g(\frac{\alpha }{2})|,g(1))\}=\max \{( \frac{\alpha }{2})^{\alpha },1-\alpha \}\).

(ii) Let \(g(t)=t^{\alpha }(1-t)\), \(0\leq t\leq 1\), then \(g'(t)=t^{ \alpha -1}[\alpha -(\alpha +1)t]\).

When \(0\leq t<\frac{\alpha }{1+\alpha }\), \(g'(t)\geq 0\); when \(\frac{\alpha }{1+\alpha }< t\leq 1\), \(g'(t)\leq 0\). So \(g(t)_{\max }=g(\frac{ \alpha }{1+\alpha })=\frac{\alpha^{\alpha }}{(1+\alpha)^{1+\alpha }}\).

The proof is completed. □

Let \(\eta =\frac{1}{1-\alpha }\max \{(\frac{\alpha }{2})^{\alpha },1- \alpha \}\).

Lemma 3.2

\(T:E\rightarrow E\) is completely continuous.

Proof

Since the continuity of f, \(T:E\rightarrow E\) is continuous. For any bounded set \(D\subset E\), there exists \(K>0\) such that \(\forall x\in D\), \(\|x\|\leq K\). There exists a constant \(L_{1}>0\) such that \(|f(t,x)|\leq L_{1}\) for any \(t\in [0,1]\) and \(x\in [-K,K]\). Then \(\forall x\in D\), it follows that
$$\begin{aligned} \bigl\vert (Tx) (t) \bigr\vert \leq& \frac{1}{\Gamma (\alpha +\beta)} \int_{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert \,ds \\ &{} +\frac{\eta }{\Gamma (\alpha +\beta)} \int_{0}^{1}(1-s)^{\alpha +\beta -1} \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert \,ds \\ &{} +\frac{\lambda \eta }{\Gamma (\alpha)} \int_{0}^{1}(1-s)^{\alpha -1} \bigl\vert x(s) \bigr\vert \,ds \\ &{} +\frac{\alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma ( \alpha +1)\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1} \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert \,ds \\ &{} +\frac{\lambda \mu \alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)} \int_{0}^{1} \bigl\vert x(s) \bigr\vert \,ds + \mu \eta \int_{0}^{1} \bigl\vert x(s) \bigr\vert \,ds \\ &{} +\frac{\lambda }{\Gamma (\alpha)} \int_{0}^{t}(t-s)^{\alpha -1} \bigl\vert x(s) \bigr\vert \,ds \\ \leq& \frac{L_{1}}{\Gamma (\alpha +\beta)} \int_{0}^{t}(t-s)^{ \alpha +\beta -1}\,ds + \frac{L_{1}\eta }{\Gamma (\alpha +\beta)} \int _{0}^{1}(1-s)^{\alpha +\beta -1}\,ds \\ &{} +\frac{\eta \lambda \Vert x \Vert }{\Gamma (\alpha)} \int_{0}^{1}(1-s)^{\alpha -1}\,ds \\ &{} +\frac{\alpha^{\alpha }L_{1}}{(1+\alpha)^{1+\alpha }(1-\alpha) \Gamma (\alpha +1)\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1}\,ds \\ &{} +\frac{ \lambda \mu \alpha^{\alpha } \Vert x \Vert }{(1+\alpha)^{1+\alpha }(1-\alpha) \Gamma (\alpha +1)} \\ &{} +\eta \mu \Vert x \Vert +\frac{\lambda \Vert x \Vert }{\Gamma (\alpha)} \int_{0}^{t}(t-s)^{ \alpha -1}\,ds \\ =&\frac{L_{1}t^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)} +\frac{ \eta L_{1}}{\eta \Gamma (\alpha +\beta +1)} +\frac{\lambda \Vert x \Vert }{ \Gamma (\alpha +1)} \\ &{} +\frac{\alpha^{\alpha }L_{1}}{(1+\alpha)^{1+ \alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} \\ &{} +\frac{\lambda \mu \alpha^{\alpha } \Vert x \Vert }{(1+\alpha)^{1+\alpha }(1- \alpha)\Gamma (\alpha +1)} +\eta \mu \Vert x \Vert +\frac{\lambda \Vert x \Vert t^{ \alpha }}{\Gamma (\alpha +1)} \\ \leq& \frac{(1+\eta)L_{1}}{\Gamma (\alpha +\beta +1)} +\frac{(1+ \eta)\lambda \Vert x \Vert }{\Gamma (\alpha +1)} +\frac{\alpha^{\alpha }L_{1}}{(1+ \alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} \\ &{} +\frac{\lambda \mu \alpha^{\alpha } \Vert x \Vert }{(1+\alpha)^{1+\alpha }(1- \alpha)\Gamma (\alpha +1)} +\eta \mu \Vert x \Vert \\ \leq& \frac{(1+\eta)L_{1}}{\Gamma (\alpha +\beta +1)} +\frac{(1+ \eta)\lambda K}{\Gamma (\alpha +1)} +\frac{\alpha^{\alpha }L_{1}}{(1+ \alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} \\ &{} +\frac{\lambda \mu \alpha^{\alpha }K}{(1+\alpha)^{1+\alpha }(1- \alpha)\Gamma (\alpha +1)} +\eta \mu K, \end{aligned}$$
which implies that TD is uniformly bounded.
In addition, for \(x\in D\), \(0\leq t_{1}< t_{2}\leq 1\), we have
$$\begin{aligned}& \bigl\vert (Tx) (t_{2})-(Tx) (t_{1}) \bigr\vert \\& \quad =|\frac{1}{\Gamma (\alpha +\beta)} \int_{0} ^{t_{2}}(t_{2}-s)^{\alpha +\beta -1}f \bigl(s,x(s)\bigr)\,ds -\frac{t_{2}^{\alpha }(2t_{2}-\alpha -1)}{(1-\alpha)\Gamma (\alpha +\beta)} \\& \qquad {}\times \int_{0}^{1}(1-s)^{\alpha +\beta -1}f\bigl(s,x(s)\bigr) \,ds +\frac{\lambda t_{2} ^{\alpha }(2t_{2}-\alpha -1)}{(1-\alpha)\Gamma (\alpha)} \int_{0} ^{1}(1-s)^{\alpha -1}x(s)\,ds \\& \qquad {} -\frac{t_{2}^{\alpha }(1-t_{2})}{(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1}f\bigl(s,x(s)\bigr) \,ds +\frac{\lambda \mu t_{2}^{\alpha }(1-t_{2})}{(1-\alpha)\Gamma (\alpha +1)} \int_{0} ^{1}x(s)\,ds \\& \qquad {} +\frac{\mu t_{2}^{\alpha }(2t_{2}-\alpha -1)}{1-\alpha } \int_{0}^{1}x(s)\,ds -\frac{\lambda }{\Gamma (\alpha)} \int_{0}^{t_{2}}(t_{2}-s)^{\alpha -1}x(s) \,ds \\& \qquad {} -\frac{1}{\Gamma (\alpha +\beta)} \int_{0}^{t_{1}}(t_{1}-s)^{\alpha + \beta -1}f \bigl(s,x(s)\bigr)\,ds \\& \qquad {} +\frac{t_{1}^{\alpha }(2t_{1}-\alpha -1)}{(1- \alpha)\Gamma (\alpha +\beta)} \int_{0}^{1}(1-s)^{\alpha +\beta -1}f\bigl(s,x(s)\bigr) \,ds \\& \qquad {} -\frac{\lambda t_{1}^{\alpha }(2t_{1}-\alpha -1)}{(1-\alpha)\Gamma ( \alpha)} \int_{0}^{1}(1-s)^{\alpha -1}x(s)\,ds \\& \qquad {} +\frac{t_{1}^{\alpha }(1-t _{1})}{(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta)} \int_{0}^{1}(1-s)^{ \beta -1}f\bigl(s,x(s)\bigr) \,ds \\& \qquad {} -\frac{\lambda \mu t_{1}^{\alpha }(1-t_{1})}{(1-\alpha)\Gamma ( \alpha +1)} \int_{0}^{1}x(s)\,ds \\& \qquad {} -\frac{\mu t_{1}^{\alpha }(2t_{1}- \alpha -1)}{1-\alpha } \int_{0}^{1}x(s)\,ds +\frac{\lambda }{\Gamma ( \alpha)} \int_{0}^{t_{1}}(t_{1}-s)^{\alpha -1}x(s) \,ds \\& \quad = \biggl\vert \int_{0}^{t_{1}}\frac{f(s,x(s))}{\Gamma (\alpha +\beta)} \bigl[(t_{2}-s)^{ \alpha +\beta -1}-(t_{1}-s)^{\alpha +\beta -1}\bigr] \,ds + \int_{t_{1}}^{t _{2}}\frac{f(s,x(s))}{\Gamma (\alpha +\beta)}(t_{2}-s)^{\alpha + \beta -1} \,ds \\& \qquad {} +\frac{(\alpha +1)(t_{2}^{\alpha }-t_{1}^{\alpha })+2(t_{1}^{\alpha +1}-t _{2}^{\alpha +1})}{(1-\alpha)\Gamma (\alpha +\beta)} \int_{0}^{1}(1-s)^{ \alpha +\beta -1}f\bigl(s,x(s)\bigr) \,ds \\& \qquad {} +\frac{\lambda (\alpha +1)(t_{1}^{\alpha }-t_{2}^{\alpha })+2\lambda (t_{2}^{\alpha +1} -t_{1}^{\alpha +1})}{(1-\alpha)\Gamma (\alpha)} \int_{0}^{1}(1-s)^{\alpha -1}x(s)\,ds \\& \qquad {} +\frac{(t_{2}^{\alpha +1}-t_{1}^{\alpha +1})+(t_{1}^{\alpha }-t_{2} ^{\alpha })}{(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta)} \int_{0} ^{1}(1-s)^{\beta -1}f\bigl(s,x(s)\bigr) \,ds \\& \qquad {} +\frac{\lambda \mu (t_{1}^{\alpha +1}-t_{2}^{\alpha +1})+\lambda \mu (t_{2}^{\alpha }-t_{1}^{\alpha })}{(1-\alpha)\Gamma (\alpha +1)} \int_{0}^{1}x(s)\,ds \\& \qquad {} +\frac{2\mu (t_{2}^{\alpha +1}-t_{1}^{\alpha +1})+\mu (\alpha +1)(t _{1}^{\alpha }-t_{2}^{\alpha })}{1-\alpha } \int_{0}^{1}x(s)\,ds \\& \qquad {} + \int_{0}^{t_{1}}\frac{\lambda x(s)}{\Gamma (\alpha)}\bigl[(t_{1}-s)^{ \alpha -1}-(t_{2}-s)^{\alpha -1} \bigr]\,ds - \int_{t_{1}}^{t_{2}}\frac{\lambda x(s)}{\Gamma (\alpha)}(t_{2}-s)^{\alpha -1} \,ds \biggr\vert \\& \quad \leq \frac{L_{1} \vert t_{2}^{\alpha +\beta }-t_{1}^{\alpha +\beta } \vert }{ \Gamma (\alpha +\beta +1)} +\frac{L_{1} \vert (\alpha +1)(t_{2}^{\alpha }-t _{1}^{\alpha })+2(t_{1}^{\alpha +1}-t_{2}^{\alpha +1}) \vert }{(1-\alpha) \Gamma (\alpha +\beta +1)} \\& \qquad {} +\frac{ \vert \lambda (\alpha +1)(t_{1}^{\alpha }-t_{2}^{\alpha }) +2\lambda (t_{2}^{\alpha +1}-t_{1}^{\alpha +1}) \vert \Vert x \Vert }{(1-\alpha)\Gamma ( \alpha +1)} \\& \qquad {} +\frac{L_{1} \vert (t_{2}^{\alpha +1}-t_{1}^{\alpha +1})+(t_{1}^{\alpha }-t _{2}^{\alpha }) \vert }{(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} +\frac{ \vert \lambda \mu (t_{1}^{\alpha +1}-t_{2}^{\alpha +1})+\lambda \mu (t_{2} ^{\alpha }-t_{1}^{\alpha }) \vert \Vert x \Vert }{(1-\alpha)\Gamma (\alpha +1)} \\& \qquad {} +\frac{ \vert 2\mu (t_{2}^{\alpha +1}-t_{1}^{\alpha +1})+\mu (\alpha +1)(t _{1}^{\alpha }-t_{2}^{\alpha }) \vert \Vert x \Vert }{1-\alpha } +\frac{\lambda \Vert x \Vert \vert t_{1}^{\alpha }-t_{2}^{\alpha } \vert }{\Gamma (\alpha +1)} \\& \quad \leq \frac{L_{1}}{\Gamma (\alpha +\beta +1)}\bigl(t_{2}^{\alpha +\beta }-t _{1}^{\alpha +\beta }\bigr) +\biggl[\frac{2L_{1}}{(1-\alpha)\Gamma (\alpha + \beta +1)} + \frac{2\lambda K}{(1-\alpha)\Gamma (\alpha +1)} \\& \qquad {} +\frac{L_{1}}{(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} +\frac{ \lambda \mu K}{(1-\alpha)\Gamma (\alpha +1)} +\frac{2\mu K}{1-\alpha }\biggr] \bigl(t_{2}^{\alpha +1}-t_{1}^{\alpha +1}\bigr) \\& \qquad {} +\biggl[\frac{L_{1}(\alpha +1)}{(1-\alpha)\Gamma (\alpha +\beta +1)} +\frac{ \lambda K(\alpha +1)}{(1-\alpha)\Gamma (\alpha +1)} +\frac{L_{1}}{(1- \alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} \\& \qquad {} +\frac{\lambda \mu K}{(1-\alpha)\Gamma (\alpha +1)}+\frac{\mu K( \alpha +1)}{1-\alpha } +\frac{\lambda K}{\Gamma (\alpha +1)}\biggr] \bigl(t_{2} ^{\alpha }-t_{1}^{\alpha }\bigr), \end{aligned}$$
which implies that TD is equicontinuous. Thus, by the Arzel àAscoli theorem, \(T:E\rightarrow E\) is completely continuous.

The proof is completed. □

Theorem 3.1

Suppose that f satisfies the Lipschitz condition
$$ \bigl\vert f(t,x)-f(t,y) \bigr\vert \leq L \vert x-y \vert ,\quad \forall t\in [0,1],x,y\in R, $$
and
$$\begin{aligned} A =&\frac{(1+\eta)L}{\Gamma (\alpha +\beta +1)} +\frac{(1+\eta)\lambda }{\Gamma (\alpha +1)} +\frac{\alpha^{\alpha }L}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} \\ &{}+\frac{\lambda \mu \alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)} +\eta \mu < 1. \end{aligned}$$
Then (1) has a unique solution.

Proof

Define \(Q=\max_{t\in [0,1]}|f(t,0)|\) and select \(r\geq \frac{\frac{2Q}{\Gamma (\alpha +\beta +1)} +\frac{\alpha^{ \alpha }Q}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta +1)}}{1-A}\), define a closed ball as \(B_{r}=\{x\in E:\|x\| \leq r\}\), then for \(x\in B_{r}\), we have
$$\begin{aligned} \bigl\vert (Tx) (t) \bigr\vert \leq& \frac{1}{\Gamma (\alpha +\beta)} \int_{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert \,ds \\ &{} +\frac{\eta }{\Gamma (\alpha +\beta)} \int_{0}^{1}(1-s)^{\alpha +\beta -1} \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert \,ds \\ &{} +\frac{\lambda \eta }{\Gamma (\alpha)} \int_{0}^{1}(1-s)^{\alpha -1} \bigl\vert x(s) \bigr\vert \,ds \\ &{} +\frac{\alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma ( \alpha +1)\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1} \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert \,ds \\ &{} +\frac{\lambda \mu \alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)} \int_{0}^{1} \bigl\vert x(s) \bigr\vert \,ds + \mu \eta \int_{0}^{1} \bigl\vert x(s) \bigr\vert \,ds \\ &{} +\frac{\lambda }{\Gamma (\alpha)} \int_{0}^{t}(t-s)^{\alpha -1} \bigl\vert x(s) \bigr\vert \,ds \\ \leq& \frac{1}{\Gamma (\alpha +\beta)} \int_{0}^{t}(t-s)^{\alpha + \beta -1}\bigl( \bigl\vert f \bigl(s,x(s)\bigr)-f(s,0) \bigr\vert + \bigl\vert f(s,0) \bigr\vert \bigr) \,ds \\ &{} +\frac{\eta }{\Gamma (\alpha +\beta)} \int_{0}^{1}(1-s)^{\alpha + \beta -1}\bigl( \bigl\vert f \bigl(s,x(s)\bigr)-f(s,0) \bigr\vert + \bigl\vert f(s,0) \bigr\vert \bigr) \,ds \\ &{} +\frac{\lambda \eta }{\Gamma (\alpha)} \int_{0}^{1}(1-s)^{\alpha -1} \bigl\vert x(s) \bigr\vert \,ds +\frac{\alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma ( \alpha +1)\Gamma (\beta)} \\ &{}\times \int_{0}^{1}(1-s)^{\beta -1}\bigl( \bigl\vert f\bigl(s,x(s)\bigr)-f(s,0) \bigr\vert + \bigl\vert f(s,0) \bigr\vert \bigr) \,ds \\ &{} +\frac{\lambda \mu \alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)} \int_{0}^{1} \bigl\vert x(s) \bigr\vert \,ds + \mu \eta \int_{0}^{1} \bigl\vert x(s) \bigr\vert \,ds \\ &{} +\frac{\lambda }{\Gamma (\alpha)} \int_{0}^{t}(t-s)^{\alpha -1} \bigl\vert x(s) \bigr\vert \,ds \\ \leq& \frac{(Lr+Q)t^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)} +\frac{ \eta (Lr+Q)}{\Gamma (\alpha +\beta +1)} +\frac{\lambda r\eta }{\Gamma (\alpha +1)} \\ &{} +\frac{\alpha^{\alpha }(Lr+Q)}{(1+\alpha)^{1+\alpha }(1- \alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} \\ &{} +\frac{\lambda \mu \alpha^{\alpha }r}{(1+\alpha)^{1+\alpha }(1- \alpha)\Gamma (\alpha +1)} +\eta \mu r +\frac{\lambda rt^{\alpha }}{ \Gamma (\alpha +1)} \\ \leq& \frac{(1+\eta)(Lr+Q)}{\Gamma (\alpha +\beta +1)} +\frac{(1+ \eta)\lambda r}{\Gamma (\alpha +1)} +\frac{\alpha^{\alpha }(Lr+Q)}{(1+ \alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} \\ &{} +\frac{\lambda \mu \alpha^{\alpha }r}{(1+\alpha)^{1+\alpha }(1- \alpha)\Gamma (\alpha +1)} +\eta \mu r\leq r, \end{aligned}$$
which implies that \(\|Tx\|\leq r\), that is, \(T(B_{r})\subset B_{r}\). In what follows, for \(x,y\in E\), for each \(t\in [0,1]\), we can get that
$$\begin{aligned}& \bigl\vert (Tx) (t)-(Ty) (t) \bigr\vert \\& \quad \leq \frac{1}{\Gamma (\alpha +\beta)} \int_{0}^{t}(t-s)^{ \alpha +\beta -1} \bigl\vert f \bigl(s,x(s)\bigr)-f\bigl(s,y(s)\bigr) \bigr\vert \,ds \\& \qquad {} +\frac{\eta }{\Gamma (\alpha +\beta)} \int_{0}^{1}(1-s)^{\alpha + \beta -1} \bigl\vert f \bigl(s,x(s)\bigr)-f\bigl(s,y(s)\bigr) \bigr\vert \,ds \\& \qquad {} +\frac{\lambda \eta }{\Gamma (\alpha)} \int_{0}^{1}(1-s)^{\alpha -1} \bigl\vert x(s)-y(s) \bigr\vert \,ds \\& \qquad {} +\frac{\alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma ( \alpha +1)\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1} \bigl\vert f \bigl(s,x(s)\bigr)-f\bigl(s,y(s)\bigr) \bigr\vert \,ds \\& \qquad {} +\frac{\lambda \mu \alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)} \int_{0}^{1} \bigl\vert x(s)-y(s) \bigr\vert \,ds +\mu \eta \int_{0}^{1} \bigl\vert x(s)-y(s) \bigr\vert \,ds \\& \qquad {} +\frac{\lambda }{\Gamma (\alpha)} \int_{0}^{t}(t-s)^{\alpha -1} \bigl\vert x(s)-y(s) \bigr\vert \,ds \\& \quad \leq \frac{(1+\eta)L \Vert x-y \Vert }{\Gamma (\alpha +\beta +1)} +\frac{(1+ \eta)\lambda \Vert x-y \Vert }{\Gamma (\alpha +1)} +\frac{\alpha^{\alpha }L \Vert x-y \Vert }{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma ( \beta +1)} \\& \qquad {} +\frac{\lambda \mu \alpha^{\alpha } \Vert x-y \Vert }{(1+\alpha)^{1+\alpha }(1- \alpha)\Gamma (\alpha +1)} +\eta \mu \Vert x-y \Vert =A \Vert x-y \Vert , \end{aligned}$$
which implies T is a contraction. Thus, by Banach fixed point theorem [30], T has a unique fixed point, that is, (1) has a unique solution.

The proof is completed. □

Theorem 3.2

Suppose there exist \(M>0\) and \(c\geq 0\) with \(\frac{(1+\eta)c}{ \Gamma (\alpha +\beta +1)} + \frac{(1+\eta)\lambda }{\Gamma (\alpha +1)} + \frac{\alpha^{\alpha }c}{(1+ \alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} +\frac{ \lambda \mu \alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha) \Gamma (\alpha +1)} +\eta \mu <1\) such that \(|f(t,x)|\leq c|x|+M\) for \(t\in [0,1]\) and \(x\in R\), then (1) has at least one solution.

Proof

First we analyze the a priori bound of solutions of the equation \(x=\sigma Tx\) for some \(\sigma \in [0,1]\).

If \(x=\sigma Tx\) for some \(\sigma [0,1]\), \(x\in E\), then we get
$$\begin{aligned} \forall t\in [0,1],\quad \bigl\vert x(t) \bigr\vert =& \bigl\vert \sigma Tx(t) \bigr\vert \leq \bigl\vert Tx(t) \bigr\vert \leq \frac{1}{ \Gamma (\alpha +\beta)} \int_{0}^{t}(t-s)^{\alpha +\beta -1} \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert \,ds \\ &{} +\frac{\eta }{\Gamma (\alpha +\beta)} \int_{0}^{1}(1-s)^{\alpha + \beta -1} \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert \,ds \\ &{} +\frac{\lambda \eta }{\Gamma (\alpha)} \int _{0}^{1}(1-s)^{\alpha -1} \bigl\vert x(s) \bigr\vert \,ds \\ &{} +\frac{\alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma ( \alpha +1)\Gamma (\beta)} \int_{0}^{1}(1-s)^{\beta -1} \bigl\vert f \bigl(s,x(s)\bigr) \bigr\vert \,ds \\ &{} +\frac{\lambda \mu \alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)} \int_{0}^{1} \bigl\vert x(s) \bigr\vert \,ds + \mu \eta \int_{0}^{1} \bigl\vert x(s) \bigr\vert \,ds \\ &{} +\frac{\lambda }{\Gamma (\alpha)} \int_{0}^{t}(t-s)^{\alpha -1} \bigl\vert x(s) \bigr\vert \,ds \\ \leq& \frac{(1+\eta)2(c \Vert x \Vert +M)}{\Gamma (\alpha +\beta +1)} +\frac{(1+ \eta)\lambda \Vert x \Vert }{\Gamma (\alpha +1)} \\ &{}+\frac{\alpha^{\alpha }(c \Vert x \Vert +M)}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma ( \beta +1)} \\ &{} +\frac{\lambda \mu \alpha^{\alpha } \Vert x \Vert }{(1+\alpha)^{1+\alpha }(1- \alpha)\Gamma (\alpha +1)} +\eta \mu \Vert x \Vert . \end{aligned}$$
So
$$\begin{aligned} \Vert x \Vert \leq& \frac{(1+\eta)(c \Vert x \Vert +M)}{\Gamma (\alpha +\beta +1)} +\frac{(1+ \eta)\lambda \Vert x \Vert }{\Gamma (\alpha +1)} + \frac{\alpha^{\alpha }(c \Vert x \Vert +M)}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma ( \beta +1)} \\ &{} +\frac{\lambda \mu \alpha^{\alpha } \Vert x \Vert }{(1+\alpha)^{1+\alpha }(1- \alpha)\Gamma (\alpha +1)} +\eta \mu \Vert x \Vert . \end{aligned}$$
This implies
$$\begin{aligned} \Vert x \Vert \leq& \frac{\frac{2M}{\Gamma (\alpha +\beta +1)} +\frac{ \alpha^{\alpha }M}{(1+\alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1) \Gamma (\beta +1)}}{1-(\frac{(1+\eta)c}{\Gamma (\alpha +\beta +1)} +\frac{(1+ \eta)\lambda }{\Gamma (\alpha +1)} +\frac{\alpha^{\alpha }c}{(1+ \alpha)^{1+\alpha }(1-\alpha)\Gamma (\alpha +1)\Gamma (\beta +1)} +\frac{ \lambda \mu \alpha^{\alpha }}{(1+\alpha)^{1+\alpha }(1-\alpha) \Gamma (\alpha +1)} +\eta \mu)} \\ :=& B. \end{aligned}$$

Let \(\gamma =B+1\). Set \(P_{\gamma }=\{x\in E: \parallel x\parallel <\gamma \}\), then \(\forall x\in \partial P_{\gamma }\), \(\parallel x\parallel =\gamma >B\). Now, we consider \(T: \overline{P_{\gamma }}\rightarrow E\). By the above analysis, it follows that \(x\neq \sigma Tx\) for \(\forall x\in \partial P_{\gamma }, \forall \sigma \in [0,1]\).

Define operators \(H_{\sigma }: E\rightarrow E\) (\(\forall \sigma \in [0,1]\)), as \(H_{\sigma }(x)=x-\sigma Tx\). It is easy to see that
$$ \forall \sigma \in [0,1], \forall x\in \partial P_{\gamma },\quad H_{\sigma }(x)=x-\sigma Tx\neq 0. $$
According to Lemma 3.2, T is completely continuous. This yields that \(\forall \sigma \in [0,1]\), \(H_{\sigma }\) is a completely continuous field.
Hence by the homotopy invariance of Leray–Schauder degree, we know that
$$ \operatorname{deg}(H_{\sigma }, P_{\gamma }, 0)=\operatorname{deg}(H_{1}, P_{\gamma }, 0)=\operatorname{deg}(H_{0}, P_{\gamma }, 0)=\operatorname{deg}(I, P_{\gamma }, 0)=1\neq 0. $$
By the nonzero property of Leray–Schauder degree, the equation \(H_{1}(x)=x-Tx=0\) has at least one solution in \(P_{\gamma }\), that is, problem (1) has at least one solution. The proof is completed. □

Declarations

Acknowledgements

The authors would like to thank the referees for their careful reading and very constructive comments that have led to the present improved version of the original manuscript.

Availability of data and materials

Not applicable.

Funding

The research was supported by the National Natural Science Foundation of China (Grant No. 11371027), Anhui Provincial Natural Science Foundation (1608085MA12), University Science Research Key Project of Anhui Province (KJ2018A0565), and Research Fund Project of Hefei University (17ZR05ZDA).

Authors’ contributions

The authors have equal contributions to each part of this paper. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Anhui University, Hefei, China

References

  1. Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, New Tork (1999) MATHGoogle Scholar
  2. Yukunthorn, W., Ntouyas, S.K., Tariboon, J.: Nonlinear fractional Caputo–Langevin equation with nonlocal Riemann–Liouville fractional integral conditions. Adv. Differ. Equ. 2014, 315 (2014) MathSciNetView ArticleGoogle Scholar
  3. Zhao, K.H., Gong, P.: Existence of positive solutions for a class of higher-order Caputo fractional differential equation. Qual. Theory Dyn. Syst. 14(1), 157–171 (2015) MathSciNetView ArticleGoogle Scholar
  4. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014) View ArticleGoogle Scholar
  5. Zhang, X.: Positive solutions for singular higher-order fractional differential equations with nonlocal conditions. J. Appl. Math. Comput. 49, 69–89 (2015) MathSciNetView ArticleGoogle Scholar
  6. Agarwal, R.P., Zhou, Y., He, Y.Y.: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59(3), 1095–1100 (2010) MathSciNetView ArticleGoogle Scholar
  7. Zhang, L., Wang, G., Ahmad, B., Agarwal, R.P.: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 249, 51–56 (2013) MathSciNetView ArticleGoogle Scholar
  8. Langevin, P.: On the theory of Brownian motion. C. R. Acad. Bulgare Sci. 146, 530 (1908) Google Scholar
  9. Zhao, K.H.: Triple positive solutions for two classes of delayed nonlinear fractional FDEs with nonlinear integral boundary value conditions. Bound. Value Probl. 2015, 181 (2015) MathSciNetView ArticleGoogle Scholar
  10. Zhang, S.: Existence of positive solution for some class of nonlinear fractional differential equations. J. Math. Anal. Appl. 278(1), 136–148 (2003) MathSciNetView ArticleGoogle Scholar
  11. Cui, Y., Zou, Y.: Existence results and the monotone iterative technique for nonlinear fractional differential systems with coupled four-point boundary value problems. Abstr. Appl. Anal. 2014, Article ID 242591 (2014) MathSciNetGoogle Scholar
  12. Ahmad, B., Nieto, J.J., Alsaedi, A., El-Shahed, M.: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal. 13, 599–606 (2012) MathSciNetView ArticleGoogle Scholar
  13. Li, X., Sun, S., Sun, Y.: Existence of solutions for fractional Langevin equation with infinite-point boundary conditions. J. Appl. Math. Comput. 53(1), 1–10 (2016) MathSciNetView ArticleGoogle Scholar
  14. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
  15. Xu, Y.F.: Fractional boundary value problems with integral and anti-periodic boundary conditions. Bull. Malays. Math. Sci. Soc. 39, 571–587 (2016) MathSciNetView ArticleGoogle Scholar
  16. Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980) MATHGoogle Scholar
  17. Lutz, E.: Fractional Langevin equation. Phys. Rev. E 64(5), 051106 (2011) MathSciNetView ArticleGoogle Scholar
  18. Henderson, J., Luca, R.: Existence and multiplicity of positive solutions for a system of fractional boundary value problems. Bound. Value Probl. 2014, 60 (2014) MathSciNetView ArticleGoogle Scholar
  19. Zhang, K.: On sign-changing solution for some fractional differential equations. Bound. Value Probl. 2017, 59 (2017) MathSciNetView ArticleGoogle Scholar
  20. Guo, L., Liu, L., Wu, Y.: Iterative unique positive solutions for singular p-Laplacian fractional differential equation system with several parameters. Nonlinear Anal., Model. Control 23(2), 182–203 (2018) MathSciNetView ArticleGoogle Scholar
  21. Zhang, X., Liu, L., Wu, Y., Wiwatanapataphee, B.: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252–263 (2015) MathSciNetMATHGoogle Scholar
  22. Jiang, J., Liu, W., Wang, H.: Positive solutions to singular Dirichlet-type boundary value problems of nonlinear fractional differential equations. Adv. Differ. Equ. 2018, 169 (2018) MathSciNetView ArticleGoogle Scholar
  23. Jiang, J., Liu, L.: Existence of solutions for a sequential fractional differential system with coupled boundary conditions. Bound. Value Probl. 2016, 159 (2016) MathSciNetView ArticleGoogle Scholar
  24. Jiang, J., Liu, L., Wu, Y.: Positive solutions to singular fractional differential system with coupled boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 18, 3061–3074 (2013) MathSciNetView ArticleGoogle Scholar
  25. Giacomoni, J., Mukherjee, T., Sreenadh, K.: Positive solutions of fractional elliptic equation with critical and singular nonlinearity. Adv. Nonlinear Anal. 6(3), 327–354 (2017) MathSciNetMATHGoogle Scholar
  26. Denton, Z., Ramírez, J.D.: Existence of minimal and maximal solutions to RL fractional integro-differential initial value problems. Opusc. Math. 37(5), 705–724 (2017) MathSciNetView ArticleGoogle Scholar
  27. Guan, Y., Zhao, Z., Lin, X.: On the existence of solutions for impulsive fractional differential equations. Adv. Math. Phys. 2017, Article ID 1207456 (2017) MathSciNetView ArticleGoogle Scholar
  28. Wang, Y., Jiang, J.: Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian. Adv. Differ. Equ. 2017, 337 (2017) MathSciNetView ArticleGoogle Scholar
  29. Xiang, M., Zhang, B., Radulescu, V.D.: Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian. Nonlinearity 29(10), 3186–3205 (2016) MathSciNetView ArticleGoogle Scholar
  30. Lyons, J.W., Neugebauer, J.T.: Positive solutions of a singular fractional boundary value problem with a fractional boundary condition. Opusc. Math. 37(3), 421–434 (2017) MathSciNetView ArticleGoogle Scholar

Copyright

© The Author(s) 2018

Advertisement