Open Access

Existence of solutions of fractional boundary value problems with p-Laplacian operator

Boundary Value Problems20152015:99

https://doi.org/10.1186/s13661-015-0358-9

Received: 19 March 2015

Accepted: 25 May 2015

Published: 14 June 2015

Abstract

In this paper, the existence of the solutions of the fractional differential equation with p-Laplacian operator and integral conditions is discussed. By Green’s functions and the fixed point theorems, we state and prove the existence and uniqueness results of the problem. Two examples are given to illustrate the results.

Keywords

existence and uniquenessfractional calculus p-Laplacian

1 Introduction

Differential equations are useful in modern physics, engineering, and in various fields of science. In these days, the theory on existence and uniqueness of boundary value problems of linear and/or nonlinear fractional equations has attracted the attention of many authors. There are comprehensive studies in this area. At the same time, it is known that the p-Laplacian operator is also used in analyzing mechanics, physics and dynamic systems, and the related fields of mathematical modeling. However, there are few studies of the existence and uniqueness of boundary conditions of fractional differential equations with the p-Laplacian operator, see [127] and the references therein.

Zhang et al. [4] studied the eigenvalue problem for a class of singular p-Laplacian fractional differential equations involving a Riemann-Stieltjes integral boundary condition:
$$\begin{aligned} &{-}D_{t}^{\beta} \bigl( \phi_{p} \bigl( D_{t}^{\alpha}x \bigr) \bigr) ( t ) =\lambda f \bigl( t,x ( t ) \bigr) ,\quad t\in ( 0,1 ),\\ &x ( 0 ) =0,\qquad D_{t}^{\alpha}x(0)=0,\\ &x ( 1 ) =\int_{0}^{1}x ( s )\,dA ( s ) , \end{aligned}$$
where \(D_{t}^{\beta}\) and \(D_{t}^{\alpha}\) are standard Riemann-Liouville derivatives with \(1<\alpha\leq2\), \(0<\beta\leq1\), A is a function of the bounded variation, and \(\int_{0}^{1}x ( s )\,dA ( s ) \) is the standard Riemann-Stieltjes integral. In their study, the results are based on upper and lower solution methods and the Schauder fixed point theorem.
In [5], Su et al. studied the existence criteria of non-negative solutions of nonlinear p-Laplacian fractional differential equations with first order derivative,
$$\left \{ \textstyle\begin{array}{@{}l} \varphi_{p} ( {}^{c}D^{\alpha}u(t) ) =\varphi_{p} ( \lambda ) f ( t,u(t),u^{{\prime}}(t) ) , \quad \mbox{for }t\in ( 0,1 ), \\ k_{0}u(0)-k_{1}u(1)=0,\\ m_{0}u(0)-m_{1}u(1)=0,\\ x^{ ( r ) } ( 0 ) =0,\quad r=2,3,\ldots, [\alpha ] , \end{array}\displaystyle \right . $$
where \(\varphi_{p}\) is p-Laplacian operator, i.e. \(\varphi _{p}(s)=\vert s\vert ^{p-2}s\), \(p>1\), and \(\varphi_{p}^{-1}=\varphi_{q}\), \(\frac{1}{p}+\frac{1}{q}=1\), \({}^{c}D^{\alpha}\) is the Caputo derivative and we have the function \(f(t,u,u^{{\prime}}): [ 0,1 ] \times[ 0,\infty ) \times ( -\infty,+\infty ) \rightarrow [ 0,\infty ) \) which satisfies the Carathéodory type conditions. Moreover, the nonlinear alternative of Leray-Schauder type and Banach fixed point theorems are used.
Han et al. [6] studied nonlinear fractional differential equations with p-Laplacian operator and boundary value conditions,
$$\begin{aligned} &D_{0+}^{\alpha} \bigl( \varphi_{p} \bigl( D_{0+}^{\alpha}u(t) \bigr) \bigr) +a(t)f(u) =0, \quad\mbox{for }0< t< 1,\\ &u(0) =\gamma u(\xi)+\lambda,\\ &\varphi_{p} \bigl( D_{0+}^{\alpha}u(0) \bigr) = \bigl( \varphi_{p} \bigl( D_{0+}^{\alpha}u(1) \bigr) \bigr) ^{{\prime}}= \bigl( \varphi_{p} \bigl( D_{0+}^{\alpha}u(0) \bigr) \bigr) , \end{aligned}$$
where \(0<\alpha\leq1\), \(2<\beta\leq3\), and \(D_{0+}^{\alpha}\), \(D_{0+}^{\beta}\) are Caputo fractional derivatives, \(\varphi_{p}(s)=\vert s\vert ^{p-2}s\), \(p>1\), and \(\varphi_{p}^{-1}=\varphi_{q}\), \(\frac{1}{p}+\frac {1}{q}=1\), and the parameters are \(0\leq\gamma<1\), \(0\leq\xi\leq1\), \(\lambda>0\). The continuous functions \(a: ( 0,1 ) \rightarrow [ 0,\infty ) \) and \(f: [ 0,\infty ) \rightarrow [ 0,\infty ) \) are given. The Green’s function properties and the Schauder fixed point theorem are used.
In [2], Liu et al. studied the solvability of the Caputo fractional differential equation with boundary value conditions involving the p-Laplacian operator. The existence and uniqueness of the problem is found by the Banach fixed point theorem. The problem is given in the following:
$$\bigl( \varphi_{p} \bigl( D_{0+}^{\alpha}x(t) \bigr) \bigr) ^{{\prime}}=f\bigl(t,x(t)\bigr),\quad \mbox{for }t\in ( 0,1 ), $$
with boundary value conditions
$$\begin{aligned} &x(0) =r_{0}x(1),\\ &x^{{\prime}}(0) =r_{1}x^{{\prime}}(1),\\ &x^{ ( j ) }(0) =0, \end{aligned}$$
where \(i=2,3,\ldots, [ \alpha ] -1\). Here, \(\varphi_{p}\) is the p-Laplacian operator and \(D_{0+}^{\alpha}\) is the Caputo fractional derivative, \(1<\alpha\in R\), and the nonlinear function \(f\in C ( [ 0,1 ] \times R,R ) \) is given.
In [7], Lu et al. studied the existence of nonnegative solutions of a nonlinear fractional boundary value problem with the p-Laplacian operator:
$$\begin{aligned} &D_{0+}^{\beta} \bigl( \varphi_{p} \bigl( D_{0+}^{\alpha}u(t) \bigr) \bigr) =f\bigl(t,u(t)\bigr), \quad\mbox{for }0< t< 1,\\ &u(0) =u^{{\prime}}(0)=u^{{\prime}}(1)=0,\\ &D_{0+}^{\alpha}u(0) =D_{0+}^{\alpha}u(1)=0, \end{aligned}$$
where \(2<\alpha\leq3\), \(1<\beta\leq2\), and \(D_{0+}^{\alpha}\), \(D_{0+}^{\beta}\) are the standard Riemann-Liouville fractional derivatives. Green’s functions, the Guo-Krasnoselskii theorem, and the Leggett-Williams fixed point theorems are used.
In [1], Wang and Xiang used upper and lower solutions method to find the existence results of at least one non-negative solution of the p-Laplacian fractional boundary value problem, which is given in the following:
$$\begin{aligned} &D_{0+}^{\gamma} \bigl( \phi_{p} \bigl( D_{0+}^{\alpha}u(t) \bigr) \bigr) =f\bigl(t,u(t)\bigr), \quad\mbox{for }0< t< 1,\\ &u(0) =0,\qquad u^{{\prime}}(1)=au(\xi),\\ &D_{0+}^{\alpha}u(0) =0,\qquad D_{0+}^{\alpha}u(1)=bD_{0+}^{\alpha}u( \eta), \end{aligned}$$
where \(1<\alpha,\gamma\leq2\), \(0\leq a,b\leq1\), \(0<\xi,\eta<1\), and also \(D_{0+}^{\alpha}\), \(D_{0+}^{\gamma}\) are Riemann-Liouville fractional operators.
In this paper, we focus on the existence of solutions of the fractional differential equation
$$ D_{0+}^{\beta}\phi_{p} \bigl( D_{0+}^{\alpha}u(t) \bigr) =f\bigl(t,u(t),D_{0+} ^{\gamma}u(t)\bigr), $$
(1)
with the p-Laplacian operator and integral boundary conditions,
$$\begin{aligned} &u(0)+\mu_{1}u(1) =\sigma_{1}\int_{0}^{1}g \bigl(s,u(s)\bigr)\,ds,\\ &u^{{\prime}}(0)+\mu_{2}u^{{\prime}}(1) =\sigma_{2} \int_{0}^{1}h\bigl(s,u(s)\bigr)\,ds, \\ &D_{0+}^{\alpha}u(0) =0, \\ &D_{0+}^{\alpha}u(1) =\upsilon D_{0+}^{\alpha}u( \eta), \end{aligned}$$
(2)
where \(D_{0+}^{\alpha}\), \(D_{0+}^{\beta}\) are for the Caputo fractional differential equation with \(1<\alpha\leq2\), \(1<\beta\leq2\), ν, \(\mu_{i}\), \(\sigma _{i}\) (\(i=1,2\)) are non-negative parameters. f, g, h are continuous functions. By the Green’s functions and fixed point theorems, we state and prove the existence and uniqueness results of the solutions. Two examples are given to illustrate the results.

2 Preliminaries

The basic definitions are given in the following.

Definition 1

The Riemann-Liouville fractional integral of order \(\alpha>0\) for a function \(f:(0,+\infty)\rightarrow R\) is defined as
$$I_{0+}^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha -1}f(s)\,ds, $$
provided that the right hand side of the integral is pointwise defined on \((0,+\infty)\) and Γ is the gamma function.

Definition 2

The Caputo derivative of order \(\alpha>0\) for a function \(f:(0,+\infty)\rightarrow R\) is written as
$$D_{0+}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}(t-s)^{n-\alpha-1}f^{ ( n ) }(s)\,ds, $$
where \(n= [ \alpha ] +1\), \([ \alpha ] \) is the integral part of α.

Lemma 3

Let \(u\in C(0,1)\cap L^{1}(0,1)\) with the fractional derivative of order \(\alpha>0\) that belongs to \(C(0,1)\cap L^{1}(0,1)\). Then
$$I_{0+}^{\alpha}D_{0+}^{\alpha}u(t)=u(t)+c_{1}t^{\alpha-1}+c_{2}t^{\alpha -2}+ \cdots+c_{n}t^{\alpha-n}, $$
for \(c_{i}\in R\) (\(i=1,2,\ldots,n\)), where n is the smallest integer greater than or equal to α.

Lemma 4

Let \(\alpha>0\). Then the differential equation \(D_{0+}^{\alpha}f(t)=0\) has solutions
$$f(t)=k_{0}+k_{1}t+k_{2}t^{2}+ \cdots+k_{n-1}t^{n-1} $$
and
$$I_{0+}^{\alpha}D_{0+}^{\alpha }f(t)=f(t)+k_{0}+k_{1}t+k_{2}t^{2}+ \cdots+k_{n-1}t^{n-1}, $$
where \(k_{i}\in R\) and \(i=1,2,\ldots,n= [ \alpha ] +1\).
The Caputo fractional derivative of order \(n-1<\alpha<n\) for \(t^{\gamma }\) is given by
$$ D_{0+}^{\alpha}t^{\gamma}= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{\Gamma(\gamma+1)}{\Gamma(\gamma-\alpha+1)}t^{\gamma-\alpha}, & \gamma\in N\mbox{ and }\gamma\geq n\mbox{ or }\gamma\notin N\mbox{ and }\gamma>n-1,\\ 0, &\gamma\in \{0,1,\ldots,n-1 \}. \end{array}\displaystyle \right . $$
(3)
Also, for brevity, we set
$$\begin{aligned} &\omega_{1} =\frac{\sigma_{1}}{1+\mu_{1}}-\frac{\sigma_{2}\mu_{1}}{(1+\mu_{1})(1+\mu_{2})},\qquad \omega_{2}=\frac{\sigma_{2}}{1+\mu _{2}},\\ &c_{1}(\eta) =\frac{\upsilon^{p-1}\eta^{\beta-1}}{(1-\upsilon^{p-1}\eta)\Gamma(\beta+1)},\qquad L=ct^{\beta-1}c_{1}( \eta). \end{aligned}$$
We use the following properties of the p-Laplacian operator: \(\phi_{p} ( u ) =\vert u\vert ^{p-2}u\), \(p>1\), and \(\phi_{p}^{-1}=\phi_{q}\), \(\frac{1}{p}+\frac{1}{q}=1\).
  1. (L1)
    If \(1< p<2\), \(uv>0\), \(\vert u\vert ,\vert v\vert \geq r>0\), then
    $$\bigl\vert \phi_{p}(u)-\phi_{p}(v)\bigr\vert \leq(p-1)r^{p-2}\vert u-v\vert . $$
     
  2. (L2)
    If \(p>2\), \(\vert u\vert ,\vert v\vert \leq \mathbb{R} \) then
    $$\bigl\vert \phi_{p}(u)-\phi_{p}(v)\bigr\vert \leq(p-1)R^{p-2}\vert u-v\vert . $$
     
We define two Green’s functions \(G(t,s)\) and \(H(t,s)\),
$$G(t,s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{(t-\tau)^{\alpha-1}}{\Gamma(\alpha)}- ( \frac{\mu_{1} ( 1+\mu_{2} ) +t\mu_{2} ( 1+\mu_{1} ) }{ ( 1+\mu _{1} ) ( 1+\mu_{2} ) } ) \frac{(1-\tau)^{\alpha-1}}{\Gamma(\alpha)}\\ \quad{}+\frac{\mathcal{\mu}_{1}\mathcal{\mu}_{2}(1-\tau)^{\alpha -2}}{(1+\mathcal{\mu }_{1})(1+\mathcal{\mu}_{2})\Gamma(\alpha-1)},& t\geq\tau,\\ - ( \frac{\mu_{1} ( 1+\mu_{2} ) +t\mu_{2} ( 1+\mu _{1} ) }{ ( 1+\mu_{1} ) ( 1+\mu_{2} ) } ) \frac{(1-\tau)^{\alpha-1}}{\Gamma(\alpha)}\\ \quad{}+\frac{\mathcal{\mu}_{1}\mathcal{\mu}_{2}(1-\tau)^{\alpha -2}}{(1+\mathcal{\mu }_{1})(1+\mathcal{\mu}_{2})\Gamma(\alpha-1)},& t\leq\tau, \end{array}\displaystyle \right . $$
and
$$H(t,s)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{ [ (t-s) ] ^{\beta-1}}{\Gamma(\beta)}-\frac{t(1-s)^{\beta -1}}{ ( 1-\upsilon^{p-1}\eta ) \Gamma(\beta)} ,& 0\leq s\leq t\leq1, \eta\leq s,\\ \frac{ [ (t-s) ] ^{\beta-1}}{\Gamma(\beta)}-\frac{t(1-s)^{\beta -1}}{ ( 1-\upsilon^{p-1}\eta ) \Gamma(\beta)}+\frac{t\upsilon ^{p-1}(\eta-s)^{\beta-1}}{ ( 1-\upsilon^{p-1}\eta ) \Gamma (\beta)} ,& 0\leq s\leq t\leq1, \eta\geq s,\\ \frac{-t(1-s)^{\beta-1}}{ ( 1-\upsilon^{p-1}\eta ) \Gamma(\beta)} ,& 0\leq t\leq s\leq1, \eta\leq s,\\ \frac{-t(1-s)^{\beta-1}}{ ( 1-\upsilon^{p-1}\eta ) \Gamma(\beta )}+\frac{t\upsilon^{p-1}(\eta-s)^{\beta-1}}{ ( 1-\upsilon^{p-1}\eta ) \Gamma(\beta)} ,&0\leq t\leq s\leq1, \eta\geq s. \end{array}\displaystyle \right . $$

Lemma 5

Let \(f, g, h\in C(0,1)\), and with \(1<\alpha\leq2\) we have the following fractional boundary value problem:
$$\begin{aligned}& D_{0+}^{\beta}\phi_{p} \bigl( D_{0+}^{\alpha}u(t) \bigr) =f(t), \end{aligned}$$
(4)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{@{}l} u(0)+\mu_{1}u(1)=\sigma_{1}\int_{0}^{1}g(s)\,ds,\\ u^{{\prime}}(0)+\mu_{2}u^{{\prime}}(1)=\sigma_{2}\int_{0}^{1}h(s)\,ds, \end{array}\displaystyle \right . \end{aligned}$$
(5)
$$\begin{aligned}& \begin{aligned} &D_{0+}^{\alpha}u(0) =0,\\ &D_{0+}^{\alpha}u(1) =\nu D_{0+}^{\alpha}u(\eta), \end{aligned} \end{aligned}$$
(6)
it has a unique solution which is given by
$$(\mathcal{T}u) (t)=\int_{0}^{t}G(t,s) \phi_{q} \biggl( \int_{0}^{1}H(t, \tau)f(\tau)\,d\tau \biggr)\,ds+\omega_{1}+\omega_{2}t, $$
with
$$\omega_{1}=\frac{\sigma_{1}}{1+\mu_{1}}-\frac{\sigma_{2}\mu_{1}}{ ( 1+\mu_{1} ) ( 1+\mu_{2} ) }\quad\textit{and}\quad \omega_{2}=\frac{\sigma_{2}}{1+\mu_{2}}. $$

Proof

By applying \(I_{0+}^{\beta}\) to both sides of (4), we get
$$\begin{aligned}& \phi_{p} \bigl( D_{0+}^{\alpha}u(t) \bigr) =\int _{0}^{t}\frac{(t-s)^{\beta-1}}{\Gamma ( \beta ) }f(s)\,ds-b_{1}-b_{2}t,\quad b_{1},b_{2}\in R,\\& D_{0+}^{\alpha}u(t)=\phi_{q} \biggl( \int _{0}^{t}\frac{(t-s)^{\beta-1}}{\Gamma ( \beta ) }f(s)\,ds-b_{1}-b_{2}t \biggr) . \end{aligned}$$
Using the boundary conditions \(D_{0+}^{\alpha}u(0)=0\) and \(D_{0+}^{\alpha }u(1)=\nu D_{0+}^{\alpha}u(\eta)\), we have
$$\phi_{q} ( -b_{1} ) =0\quad\Longrightarrow\quad b_{1}=0, $$
and secondly,
$$\begin{aligned} \phi_{q} \biggl( \int_{0}^{1} \frac{(1-s)^{\beta-1}}{\Gamma ( \beta ) }f(s)\,ds-b_{2} \biggr) & =\nu\phi_{q} \biggl( \int_{0}^{\eta}\frac{(\eta-s)^{\beta-1}}{\Gamma ( \beta ) }f(s)\,ds-b_{2} \eta \biggr) \\ & =\phi_{q} \biggl( \nu^{\frac{1}{q-1}} \biggl( \int _{0}^{\eta}\frac{(\eta-s)^{\beta-1}}{\Gamma ( \beta ) }f(s)\,ds-b_{2}\eta \biggr) \biggr) . \end{aligned}$$
Moreover, since \(\phi_{p}\) is one-to-one,
$$\begin{aligned}& I_{0+}^{\beta}f ( 1 ) -b_{2} =\nu^{p-1} \bigl( I_{0+}^{\beta }f ( \eta ) -b_{2}\eta \bigr) =\nu^{p-1}I_{0+}^{\beta}f ( \eta ) - \nu^{p-1}b_{2}\eta,\\& I_{0+}^{\beta}f ( 1 ) -\nu^{p-1}I_{0+}^{\beta}f ( \eta ) = \bigl( 1-\nu^{p-1}\eta \bigr) b_{2}. \end{aligned}$$
Then
$$\begin{aligned} b_{2} & =\frac{1}{ ( 1-\nu^{p-1}\eta ) }I_{0+}^{\beta}f ( 1 ) - \frac{\nu^{p-1}}{ ( 1-\nu^{p-1}\eta ) }I_{0+}^{\beta }f ( \eta )\\ & =\frac{1}{ ( 1-\nu^{p-1}\eta ) }\int_{0}^{1}\frac{(1-s)^{\beta-1}}{\Gamma ( \beta ) }f(s)\,ds-\frac{\nu^{p-1} }{ ( 1-\nu^{p-1}\eta ) }\int_{0}^{\eta} \frac{(\eta -s)^{\beta-1}}{\Gamma ( \beta ) }f(s)\,ds. \end{aligned}$$
Since \(\phi_{p} ( D_{0+}^{\alpha}u(t) ) =I_{0+}^{\beta}f ( t ) -b_{1}-b_{2}t\),
$$\begin{aligned}& \begin{aligned}[b] \phi_{p} \bigl( D_{0+}^{\alpha}u(t) \bigr) ={}&\int _{0}^{t}\frac{(t-s)^{\beta-1}}{\Gamma ( \beta ) }f\bigl(s,u(s) \bigr)\,ds-\frac {t}{ ( 1-\nu^{p-1}\eta ) }\int_{0}^{1} \frac{(1-s)^{\beta -1}}{\Gamma ( \beta ) }f\bigl(s,u(s)\bigr)\,ds \\ &{} +\frac{t\nu^{p-1}}{ ( 1-\nu^{p-1}\eta ) }\int_{0}^{\eta} \frac{(\eta-s)^{\beta-1}}{\Gamma ( \beta ) }f\bigl(s,u(s)\bigr)\,ds \\ ={}&\int_{0}^{1}H(t,s)f(s)\,ds, \end{aligned}\\& D_{0+}^{\alpha}u(t)=\phi_{q} \biggl( \int _{0}^{1}H(t,s)f(s)\,ds \biggr) , \\& u(t)=\int_{0}^{t}\frac{(t-\tau)^{\alpha-1}}{\Gamma ( \alpha ) } \phi_{q} \biggl( \int_{0}^{1}H(t,s)f(s)\,ds \biggr)\,d\tau-c_{1}-c_{2}t. \end{aligned}$$
(7)
By the boundary conditions (5), we get
$$\begin{aligned}& -c_{1}+\mu_{1} \biggl( \int_{0}^{1} \frac{(1-\tau)^{\alpha-1}}{\Gamma ( \alpha ) }\phi_{p} \biggl( \int_{0}^{1}H( \tau,s)f(s)\,ds \biggr)\,d\tau-c_{1}-c_{2} \biggr) = \sigma_{1}\int_{0}^{1}g(s)\,ds, \\& \mu_{1}\int_{0}^{1}\frac{(1-\tau)^{\alpha-1}}{\Gamma ( \alpha ) } \phi_{p} \biggl( \int_{0}^{1}H( \tau,s)f(s)\,ds \biggr)\,d\tau-c_{2}\mu_{1}-\sigma_{1} \int_{0}^{1}g(s)\,ds =c_{1} ( 1+ \mu_{1} ) , \\& \begin{aligned}[b] c_{1} ={}&\frac{\mu_{1}}{ ( 1+\mu_{1} ) }\int_{0}^{1}\frac{(1-\tau)^{\alpha-1}}{\Gamma ( \alpha ) }\phi_{p} \biggl( \int_{0}^{1}H( \tau,s)f(s)\,ds \biggr)\,d\tau-c_{2}\frac{\mu_{1}}{ ( 1+\mu_{1} ) }\\ &{} -\frac{\sigma_{1}}{ ( 1+\mu_{1} ) }\int_{0}^{1}g(s)\,ds, \end{aligned}\\& c_{2}=\frac{\mu_{2}}{ ( 1+\mu_{2} ) }\int_{0}^{1}\frac{(1-\tau)^{\alpha-2}}{\Gamma ( \alpha-1 ) }\phi_{p} \biggl( \int_{0}^{1}H( \tau,s)f(s)\,ds \biggr)\,d\tau-\frac{\sigma_{2}}{ ( 1+\mu_{2} ) }\int_{0}^{1}h(s)\,ds. \end{aligned}$$
(8)
Inserting \(c_{2}\) into (8), we get the values of \(c_{1}\), and inserting \(c_{1}\) and \(c_{2}\) into (7), we have
$$\begin{aligned} u(t) ={}&\int_{0}^{t}\frac{(t-\tau)^{\alpha-1}}{\Gamma ( \alpha ) } \phi_{p} \biggl( \int_{0}^{1}H(t,s)f \bigl(s,u(s)\bigr)\,ds \biggr)\,d\tau \\ &{} - \biggl( \frac{\mu_{1} ( 1+\mu_{2} ) +t\mu_{2} ( 1+\mu _{1} ) }{ ( 1+\mu_{1} ) ( 1+\mu_{2} ) } \biggr) \int_{0}^{1} \frac{(1-\tau)^{\alpha-1}}{\Gamma ( \alpha ) }\phi_{p} \biggl( \int_{0}^{1}H( \tau,s)f\bigl(s,u(s)\bigr)\,ds \biggr)\,d\tau \\ &{} +\frac{\mu_{1}\mu_{2}}{ ( 1+\mu_{1} ) ( 1+\mu_{2} ) }\int_{0}^{1} \frac{(1-\tau)^{\alpha-2}}{\Gamma ( \alpha-1 ) }\phi_{p} \biggl( \int_{0}^{1}H( \tau,s)f\bigl(s,u(s)\bigr)\,ds \biggr)\,d\tau \\ &{} +\frac{\sigma_{1}}{1+\mu_{1}}\int_{0}^{1}g\bigl(s,u ( s ) \bigr)\,ds- \biggl( \frac{\sigma_{2}\mu_{1}-t\sigma_{2} ( 1+\mu_{1} ) }{ ( 1+\mu_{1} ) ( 1+\mu_{2} ) } \biggr) \int_{0}^{1}h \bigl(s,u ( s ) \bigr)\,ds. \end{aligned}$$
 □

Lemma 6

The functions \(G(t,s)\) and \(H(t,s)\) are continuous on \([ 0,1 ] \times [ 0,1 ] \) and \(H(t,s)\) satisfies the following properties:
  1. (1)

    \(H(t,s)\leq0\), for \(t,s\in [ 0,1 ] \),

     
  2. (2)

    \(H(t,s)\geq H(s,s)\), for \(t,s\in [ 0,1 ] \),

     
  3. (3)
    the Green’s function \(H(t,s)\) satisfies the following condition:
    $$0\leq\int_{0}^{1}\bigl\vert H(t,s)\bigr\vert \,ds\leq\frac{B(\beta ,\beta)}{ ( 1-\upsilon^{p-1}\eta ) \Gamma(\beta)}, $$
    where B is the Beta function.
     

Proof

The proofs of properties (1)-(2) are given in [1]. Thus we will prove property (3) for any \(t,s\in [ 0,1 ] \). The Green’s function \(H(t,s)\) is negative. Therefore,
$$0\leq\int_{0}^{1}\bigl\vert H(t,s)\bigr\vert \,ds\leq\int_{0}^{1}\bigl\vert H(s,s) \bigr\vert \,ds\leq\frac{B(\beta,\beta)}{ ( 1-\upsilon^{p-1}\eta ) \Gamma(\beta)}. $$
 □

3 Existence and uniqueness results

In this section, we state and prove existence and uniqueness results of the fractional BVP (1)-(2) by using the Banach fixed point theorem. Our study concerns the space
$$C_{\gamma} \bigl( [ 0,1 ] ,R \bigr) = \bigl\{ u\in C \bigl( [ 0,1 ] ,R \bigr) ,D_{0+}^{\gamma}u\in C \bigl( [ 0,1 ] ,R \bigr) \bigr\} , $$
which is shown in the form
$$\Vert u\Vert _{\gamma}=\Vert u\Vert _{c}+\bigl\Vert D_{0+}^{\gamma}u\bigr\Vert _{c}, $$
where \(\Vert \cdot \Vert _{c}\) is the sup norm in \(C ( [ 0,1 ] ,R ) \).
The following notations will be used throughout this paper:
$$\begin{aligned}& \Delta_{1}=\frac{1}{\Gamma ( \alpha+1 ) } \biggl[ 1+\frac {\vert \mu_{1}\vert \vert 1+\mu_{2}\vert +\vert \mu_{2}\vert \vert 1+\mu_{1}\vert }{\vert 1+\mu _{1}\vert \vert 1+\mu_{2}\vert } \biggr] +\frac{1}{\Gamma ( \alpha ) } \biggl[ \frac{\vert \mu_{1}\vert \vert \mu_{2}\vert }{\vert 1+\mu_{1}\vert \vert 1+\mu_{2}\vert } \biggr] ,\\& \Delta_{2}=\frac{1}{\Gamma ( \alpha-\gamma+1 ) } \biggl[ 1+\frac{\vert \mu_{2}\vert }{\Gamma ( 2-\gamma ) \vert 1+\mu_{2}\vert } \biggr] ,\\& \Delta_{g}=\frac{\vert \sigma_{1}\vert }{\vert 1+\mu _{1}\vert },\\& \Delta_{h_{1}}=\frac{\vert \sigma_{2}\vert \vert \mu _{1}+\vert 1+\mu_{1}\vert \vert }{\vert 1+\mu _{1}\vert \vert 1+\mu_{2}\vert }, \qquad\Delta_{h_{2}}=\frac{\vert \sigma_{2}\vert }{\Gamma(2-\gamma)\vert 1+\mu_{2}\vert }. \end{aligned}$$
To state and prove our first result, we pose the following conditions:

(A1) The function \(f: [ 0,1 ] \times R\times R\rightarrow R\) is jointly continuous.

(A2) There exists a function \(l_{f}\in L^{\frac{1}{\tau}} ( [ 0,1 ] ,R^{+} ) \) such that
$$\bigl\vert f(t,u_{1},u_{2})-f(t,v_{1},v_{2}) \bigr\vert \leq l_{f}(t) \bigl( \vert u_{1}-v_{1} \vert +\vert u_{2}-v_{2}\vert \bigr) , $$
for all \((t,u_{1},u_{2}), ( t,v_{1},v_{2} ) \in [ 0,1 ] \times R\times R\).
(A3) The functions g and h are jointly continuous and there exists \(l_{g},l_{h}\in L^{1} ( [ 0,1 ] ,R^{+} ) \) such that
$$\bigl\vert g(t,u)-g(t,v)\bigr\vert \leq l_{g}(t)\vert u-v \vert $$
and
$$\bigl\vert h(t,u)-h(t,v)\bigr\vert \leq l_{h}(t)\vert u-v \vert , $$
for each \(( t,u ) , ( t,v ) \in [ 0,1 ] \times R\).
Next, we define an operator, \(\mathcal{T}_{0}\) which is \(\mathcal{T}_{0}:C [ 0,1 ] \rightarrow C [ 0,1 ] \) as follows:
$$\mathcal{T}_{0}x(t)=\phi_{q} \biggl( \int _{0}^{1}H(t,s)f\bigl(s,x(s),D_{0+}^{\gamma}x(s) \bigr)\,ds \biggr) . $$

Lemma 7

Assume (A1)-(A3) hold and \(q>2\). There exists a constant \(l_{T_{0}}>0\) such that
$$\bigl\vert \mathcal{T}_{0}u(t)-\mathcal{T}_{0}v(t)\bigr\vert \leq l_{\mathcal{T}_{0}}\Vert u-v\Vert _{\gamma}, $$
for all \(u,v\in B_{r}\). We have
$$l_{\mathcal{T}_{0}}= ( q-1 ) L_{H}^{q-2}\Vert l_{f}\Vert _{\infty}\int_{0}^{1} \bigl\vert H(s,s)\bigr\vert \,ds\leq ( q-1 ) L_{H}^{q-2} \Vert l_{f}\Vert _{\infty}\frac {B(\beta,\beta)}{ ( 1-\upsilon^{p-1}\eta ) \Gamma(\beta)}. $$

Proof

If \(p>2\) and \(t>0\) we have the following estimation:
$$\begin{aligned} \biggl\vert \int_{0}^{1}H(t,s)f \bigl(s,u(s),D_{0+}^{\gamma}u(s)\bigr)\,ds\biggr\vert & \leq\int_{0}^{1}\bigl\vert H(t,s)\bigr\vert \bigl\vert f\bigl(s,u(s),D_{0+}^{\gamma}u(s)\bigr)\bigr\vert \,ds \\ & \leq\int_{0}^{1}\bigl\vert H(t,s)\bigr\vert l_{f} ( s ) \bigl( \bigl\vert u(s)\bigr\vert +\bigl\vert D_{0+}^{\gamma}u(s)\bigr\vert +\bigl\vert f ( s,0,0 ) \bigr\vert \bigr)\,ds \\ & \leq \bigl( \Vert l_{f}\Vert _{\infty} \Vert u\Vert _{\gamma}+M \bigr) \int_{0}^{1}\bigl\vert H(s,s)\bigr\vert \,ds \\ & \leq \bigl( \Vert l_{f}\Vert _{\infty}r+M \bigr) \int _{0}^{1}\bigl\vert H(s,s)\bigr\vert \,ds \\ & =L_{H}, \end{aligned}$$
where \(M=\max_{s\in [ 0,1 ] }\vert f ( s,0,0 ) \vert \). Now using the property (L2), we get the desired inequality,
$$\begin{aligned} & \bigl\vert ( \mathcal{T}_{0}u ) ( t ) - ( \mathcal{T}_{0}v ) ( t ) \bigr\vert \\ &\quad =\biggl\vert \phi_{q} \biggl( \int_{0}^{1}H(t,s)f \bigl(s,u(s),D_{0+}^{\gamma}u(s)\bigr)\,ds \biggr) - \phi_{q} \biggl( \int_{0}^{1}H(t,s)f \bigl(s,v(s),D_{0+}^{\gamma}v(s)\bigr)\,ds \biggr) \biggr\vert \\ &\quad \leq ( q-1 ) L_{H}^{q-2}\biggl\vert \int _{0}^{1}H(t,s) \bigl( f \bigl(s,u(s),D_{0+}^{\gamma}u(s)\bigr)-f\bigl(s,v(s),D_{0+}^{\gamma }v(s) \bigr) \bigr)\,ds\biggr\vert \\ &\quad \leq ( q-1 ) L_{H}^{q-2}\Vert l_{f}\Vert _{\infty }\Vert u-v\Vert _{\gamma}\int_{0}^{1} \bigl\vert H(s,s)\bigr\vert \,ds \\ & \quad\leq ( q-1 ) L_{H}^{q-2}\Vert l_{f}\Vert _{\infty }\frac{B(\beta,\beta)}{ ( 1-\upsilon^{p-1}\eta ) \Gamma(\beta )}\Vert u-v\Vert _{\gamma} \\ &\quad =l_{\mathcal{T}_{0}}\Vert u-v\Vert _{\gamma}. \end{aligned}$$
 □

Theorem 8

Assume (A1)-(A3) hold. If
$$ \Biggl\{ l_{\mathcal{T}_{0}} \Biggl( \sum_{i=1}^{2} \Delta_{i} \Biggr) +\Delta_{g}\Vert l_{g} \Vert _{1}+ \biggl( \sum_{i=1}^{2} \Delta h_{i} \biggr) \Vert l_{h}\Vert _{1} \Biggr\} < 1, $$
(9)
then our BVP (1)-(2) has a unique solution on \([ 0,1 ] \).

Proof

Let us define the operator \(\mathcal{T}:C_{\gamma} ( [ 0,1 ] ,R ) \rightarrow C_{\gamma} ( [ 0,1 ] ,R ) \) to transform our BVP (1)-(2) into a fixed point problem,
$$\begin{aligned} & (\mathcal{T}u) (t) \\ &\quad =\int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} \mathcal{T}_{0}\bigl(f\bigl(s,u(s),D_{0+}^{\gamma}u(s) \bigr)\bigr)\,ds \\ &\qquad{} -\frac{\mu_{1}}{ ( 1+\mu_{1} ) }\int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}\mathcal{T}_{0}\bigl(f\bigl(s,u(s),D_{0+}^{\gamma}u(s) \bigr)\bigr)\,ds \\ &\qquad{} +\frac{\mu_{1}\mu_{2}}{(1+\mu_{1})(1+\mu_{2})}\int_{0}^{1}\frac{(1-s)^{\alpha-2}}{\Gamma(\alpha-1)}\mathcal{T}_{0}\bigl(f\bigl(s,u(s),D_{0+} ^{\gamma}u(s)\bigr)\bigr)\,ds \\ &\qquad{} -\frac{\mu_{2}t}{(1+\mu_{2})}\int_{0}^{1} \frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}\mathcal{T}_{0}\bigl(f\bigl(s,u(s),D_{0+}^{\gamma }u(s) \bigr)\bigr)\,ds \\ &\qquad{} +\frac{\sigma_{1}}{(1+\mu_{1})}\int_{0}^{1}g\bigl(s,u(s) \bigr)\,ds-\frac {\sigma_{2}(\mu_{1}-(1+\mu_{1})t)}{ ( 1+\mu_{2} ) (1+\mu_{1})}\int_{0}^{1}h \bigl(s,u(s)\bigr)\,ds. \end{aligned}$$
(10)
Taking the γth fractional derivative, we get
$$\begin{aligned} & D_{0+}^{\gamma}(\mathcal{T}u) (t) \\ &\quad =\int_{0}^{t}\frac{(t-s)^{\alpha-\gamma-1}}{\Gamma(\alpha -\gamma)} \mathcal{T}_{0}\bigl(f\bigl(s,u(s),D_{0+}^{\gamma}u(s) \bigr)\bigr)\,ds \\ &\qquad{} -\frac{\mu_{2}}{(1+\mu_{2})}\frac{t^{1-\gamma}}{\Gamma(2-\gamma)} \int_{0}^{1} \frac{(1-s)^{\alpha-\gamma-1}}{\Gamma(\alpha-\gamma)}\mathcal{T}_{0}\bigl(f\bigl(s,u(s),D_{0+}^{\gamma}u(s) \bigr)\bigr)\,ds \\ &\qquad{} +\frac{\sigma_{2}}{(1+\mu_{2})}\frac{t^{1-\gamma}}{\Gamma (2-\gamma)}\int_{0}^{1}h \bigl(s,u(s)\bigr)\,ds \end{aligned}$$
(11)
for \(t\in [ 0,1 ] \). Since f, g, h are continuous, the expression (10) and (11) are well defined. Clearly, the fixed point of the operator \(\mathcal{T}\) is the solution of the problem (1)-(2). To show the existence and uniqueness of the solution, the Banach fixed point theorem is used and then we show \(\mathcal{T}\) is contraction. We have
$$\begin{aligned} & \bigl\vert (\mathcal{T}u) (t)-(\mathcal{T}v) (t)\bigr\vert \\ &\quad \leq\int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}l_{\mathcal{T}_{0}} \Vert u-v\Vert _{\gamma}\,ds \\ & \qquad{}+\frac{\vert \mu_{1}\vert }{\vert 1+\mu_{1}\vert }\int_{0}^{1} \frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}l_{\mathcal{T}_{0}}\Vert u-v\Vert _{\gamma}\,ds \\ & \qquad{}+\frac{\vert \mu_{1}\vert \vert \mu_{2}\vert }{\vert 1+\mu_{1}\vert \vert 1+\mu_{2}\vert }\int_{0}^{1} \frac{(1-s)^{\alpha-2}}{\Gamma(\alpha-1)}l_{\mathcal{T}_{0}}\Vert u-v\Vert _{\gamma}\,ds \\ & \qquad{}+\frac{\vert \mu_{2}\vert }{\vert 1+\mu_{2}\vert }\int_{0}^{1} \frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}l_{\mathcal{T}_{0}}\Vert u-v\Vert _{\gamma}\,ds \\ &\qquad{} +\frac{\vert \sigma_{1}\vert }{\vert 1+\mu _{1}\vert }\int_{0}^{1}l_{g}(s) \bigl( \bigl\vert u ( s ) -v ( s ) \bigr\vert +\bigl\vert D_{0+}^{\gamma}u ( s ) -D_{0+}^{\gamma}v ( s ) \bigr\vert \bigr)\,ds \\ & \qquad{}+\frac{\vert \sigma_{2}\vert \vert \mu_{1}+\vert 1+\mu_{1}\vert \vert }{\vert 1+\mu_{2}\vert \vert 1+\mu_{1}\vert }\int_{0}^{1}l_{h}(s) \bigl( \bigl\vert u ( s ) -v ( s ) \bigr\vert +\bigl\vert D_{0+}^{\gamma}u ( s ) -D_{0+}^{\gamma}v ( s ) \bigr\vert \bigr)\,ds \\ &\quad\leq \biggl\{ l_{\mathcal{T}_{0}} \biggl( \int_{0}^{t} \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\,ds+\frac{\vert \mu_{1}\vert \vert 1+\mu _{2}\vert +\vert \mu_{2}\vert \vert 1+\mu _{1}\vert }{\vert 1+\mu_{1}\vert \vert 1+\mu _{2}\vert }\int_{0}^{1} \frac{(1-s)^{\alpha-1}}{\Gamma(\alpha )}\,ds \\ &\qquad{} +\frac{\vert \mu_{1}\vert \vert \mu _{2}\vert }{\vert 1+\mu_{1}\vert \vert 1+\mu _{2}\vert }\int_{0}^{1} \frac{(1-s)^{\alpha-2}}{\Gamma(\alpha -1)}\,ds \biggr) \\ &\qquad{} +\frac{\vert \sigma_{1}\vert }{\vert 1+\mu _{1}\vert }\int_{0}^{1}l_{g}(s)\,ds+ \frac{\vert \sigma _{2}\vert \vert \mu_{1}+\vert 1+\mu_{1}\vert \vert }{\vert 1+\mu_{2}\vert \vert 1+\mu _{1}\vert }\int_{0}^{1}l_{h}(s)\,ds \biggr\} \Vert u-v\Vert _{\gamma} \\ &\quad \leq \biggl\{ l_{\mathcal{T}_{0}} \biggl( \frac{1}{\Gamma(\alpha+1)}+\frac{\vert \mu_{1}\vert \vert 1+\mu_{2}\vert +\vert \mu _{2}\vert \vert 1+\mu_{1}\vert }{\Gamma(\alpha +1)\vert 1+\mu_{1}\vert \vert 1+\mu_{2}\vert } +\frac{\vert \mu_{1}\vert \vert \mu_{2}\vert }{\Gamma(\alpha)\vert 1+\mu_{1}\vert \vert 1+\mu _{2}\vert } \biggr) \\ & \qquad{} +\frac{\vert \sigma_{1}\vert }{\vert 1+\mu _{1}\vert }\Vert l_{g}\Vert _{1}+ \frac{\vert \sigma _{2}\vert \vert \mu_{1}+\vert 1+\mu_{1}\vert \vert }{\vert 1+\mu_{2}\vert \vert 1+\mu _{1}\vert }\Vert l_{h}\Vert _{1} \biggr\} \Vert u-v\Vert _{\gamma}. \end{aligned}$$
(12)
By using the Hölder inequality, we get
$$\begin{aligned}& \bigl\vert \mathcal{T}u ( t ) -\mathcal{T}v ( t ) \bigr\vert \leq \bigl\{ l_{\mathcal{T}_{0}}\Delta_{1}+\Delta _{g}\Vert l_{g}\Vert _{1}+\Delta_{h_{1}}\Vert l_{h}\Vert _{1} \bigr\} \Vert u-v\Vert _{\gamma}, \end{aligned}$$
(13)
$$\begin{aligned}& \bigl\vert D_{0+}^{\gamma}(\mathcal{T}u) (t)-D_{0+}^{\gamma}(\mathcal{T}v) (t)\bigr\vert \\& \quad \leq\int_{0}^{t}\frac{(t-s)^{\alpha-\gamma-1}}{\Gamma(\alpha -\gamma)}l_{\mathcal{T}_{0}} \Vert u-v\Vert _{\gamma}\,ds \\& \qquad{} +\frac{\vert \mu_{2}\vert }{\vert 1+\mu_{2}\vert }\frac{t^{1-\gamma}}{\Gamma(2-\gamma)}\int_{0}^{1} \frac{(1-s)^{\alpha-\gamma-1}}{\Gamma(\alpha-\gamma)}l_{\mathcal{T}_{0}}\Vert u-v\Vert _{\gamma}\,ds \\& \qquad{} +\frac{\vert \sigma_{2}\vert }{\vert 1+\mu _{2}\vert }\frac{t^{1-\gamma}}{\Gamma(2-\gamma)}\int _{0}^{1}l_{h}(s) \bigl( \bigl\vert u ( s ) -v ( s ) \bigr\vert +\bigl\vert D_{0+}^{\gamma}u ( s ) -D_{0+}^{\gamma }v ( s ) \bigr\vert \bigr)\,ds \\& \quad \leq \biggl\{ \frac{l_{\mathcal{T}_{0}}}{\Gamma(\alpha-\gamma)}\int_{0}^{t}(t-s)^{\alpha-\gamma-1}\,ds \\& \qquad{} +\frac{l_{\mathcal{T}_{0}}t^{1-\gamma} \vert \mu _{2}\vert }{\Gamma(\alpha-\gamma)\Gamma ( 2-\gamma ) \vert 1+\mu_{2}\vert }\int_{0}^{1}(1-s)^{\alpha-\gamma -1}\,ds \\& \qquad{} +\frac{\vert \sigma_{2}\vert t^{1-\gamma}}{ \vert 1+\mu_{2}\vert \Gamma ( 2-\gamma ) }\int _{0}^{1}l_{h}(s)\,ds \biggr\} \Vert u-v\Vert _{\gamma} \\& \quad \leq \biggl\{ l_{\mathcal{T}_{0}} \biggl( \frac{1}{\Gamma(\alpha -\gamma+1)}+\frac{\vert \mu_{2}\vert }{\Gamma(\alpha-\gamma+1)\Gamma ( 2-\gamma ) \vert 1+\mu_{2}\vert } \biggr) \\& \qquad{} +\frac{\vert \sigma_{2}\vert }{\vert 1+\mu _{2}\vert \Gamma ( 2-\gamma ) }\int _{0}^{1}l_{h}(s)\,ds \biggr\} \Vert u-v\Vert _{\gamma} \\& \quad \leq \biggl\{ \frac{l_{\mathcal{T}_{0}}}{\Gamma(\alpha-\gamma +1)} \biggl( 1+\frac{\vert \mu_{2}\vert }{\Gamma ( 2-\gamma ) \vert 1+\mu_{2}\vert } \biggr) \\& \qquad{} +\frac{\vert \sigma_{2}\vert }{\vert 1+\mu _{2}\vert \Gamma ( 2-\gamma ) }\Vert l_{h}\Vert _{1} \biggr\} \Vert u-v\Vert _{\gamma}. \end{aligned}$$
(14)
Similarly,
$$ \bigl\vert D_{0+}^{\gamma} \bigl( \mathcal{T}u ( t ) \bigr) -D_{0+}^{\gamma} \bigl( \mathcal{T}v ( t ) \bigr) \bigr\vert \leq \bigl\{ l_{\mathcal{T}_{0}}\Delta_{2}+\Delta_{h_{2}}\Vert l_{h}\Vert _{1} \bigr\} \Vert u-v\Vert _{\gamma}. $$
(15)
With the help of (13)-(15), we find that
$$\begin{aligned} & \Vert Tu-Tv\Vert _{\gamma} \\ &\quad \leq \bigl\{ l_{\mathcal{T}_{0}} ( \Delta_{1}+\Delta_{2} ) + \Delta_{g}\Vert l_{g}\Vert _{1}+ ( \Delta_{h_{1}}+\Delta _{h_{2}} ) \Vert l_{h}\Vert _{1} \bigr\} \Vert u-v\Vert _{\gamma} \\ &\quad = \Biggl\{ l_{\mathcal{T}_{0}} \Biggl( \sum_{i=1}^{2} \Delta_{i} \Biggr) +\Delta_{g}\Vert l_{g} \Vert _{1}+ \Biggl( \sum_{i=1}^{2} \Delta_{h_{i}} \Biggr) \Vert l_{h}\Vert _{1} \Biggr\} \Vert u-v\Vert _{\gamma}. \end{aligned}$$
Thus \(\mathcal{T}\) is a contraction mapping by condition (9). By the Banach fixed point theorem, \(\mathcal{T}\) has a fixed point which is the solution of the BVP. □

4 Existence results

Theorem 9

Assume:

(iv) There exist non-decreasing functions \(\varphi:[0,\infty )\times{}[0,\infty)\rightarrow{}[0,\infty)\) and \(\psi _{i}:[0,\infty)\rightarrow{}[0,\infty)\), \(i=1,2\), and the functions \(l_{f}\in L^{\frac{1}{\tau}}( [ 0,1 ] ,R^{+})\) and \(l_{g},l_{h}\in L^{1}( [ 0,1 ] ,R^{+})\) such that
$$\begin{aligned} &\bigl\vert f(t,u,v)\bigr\vert \leq l_{f}(t)\varphi\bigl(\vert u\vert +\vert v\vert \bigr),\\ &\bigl\vert g(t,u)\bigr\vert \leq l_{g}(t)\psi_{1} \bigl(\vert u\vert \bigr),\\ &\bigl\vert h(t,u)\bigr\vert \leq l_{h}(t)\psi_{2} \bigl(\vert u\vert \bigr), \end{aligned}$$
for all \(t\in [ 0,1 ] \) and \(u,v\in R\).
(v) There exists a constant \(\mathcal{N}>0\) such that
$$ \biggl[ \frac{\mathcal{N}}{\varphi(\Vert u\Vert _{\gamma })l_{\mathcal{T}_{0}}\sum_{i=1}^{2}\Delta_{i}+\psi_{1}(\Vert u\Vert _{\gamma})\Vert l_{g}\Vert _{1}\Delta_{g}+\psi_{2}(\Vert u\Vert _{\gamma})\Vert l_{h}\Vert _{1}\sum_{i=1}^{2}\Delta_{h_{i}}} \biggr] >1. $$
(16)
Thus problem (1)-(2) has at least one solution on \([ 0,1 ] \).

Proof

Let \(B_{r}= \{ u\in C_{\gamma}([0,1],R):\Vert u\Vert _{\gamma }\leq r \} \).

Step 1: Let the operator \(\mathcal{T}:C_{\gamma}([0,1],R)\rightarrow C_{\gamma}([0,1],R)\) be given in (10) which defines \(B_{r}\) to be a bounded set. For all \(u\in B_{r}\), we get
$$\begin{aligned} & \bigl\vert (\mathcal{T}u) (t)\bigr\vert \\ &\quad \leq\frac{\varphi(r)}{\Gamma(\alpha)}l_{\mathcal{T}_{0}}\int_{0}^{t}(t-s)^{\alpha-1}\,ds \\ &\qquad{} +\frac{\vert \mu_{1}\vert \vert 1+\mu_{2}\vert +\vert \mu_{2}\vert \vert 1+\mu_{1}\vert }{\vert 1+\mu_{1}\vert \vert 1+\mu_{2}\vert }\frac{\varphi(r)}{\Gamma(\alpha)}l_{\mathcal{T}_{0}}\int_{0}^{1}(1-s)^{\alpha-1}\,ds \\ &\qquad{} +\frac{\vert \mu_{1}\vert \vert \mu_{2}\vert }{\vert 1+\mu_{1}\vert \vert 1+\mu_{2}\vert }\frac{\varphi(r)}{\Gamma(\alpha-1)}l_{\mathcal{T}_{0}}\int _{0}^{1}(1-s)^{\alpha-2}\,ds \\ &\qquad{} +\frac{\vert \sigma_{1}\vert }{\vert 1+\mu_{1}\vert }\psi_{1}(r)\int_{0}^{1} \bigl\vert l_{g}(s)\bigr\vert \,ds +\frac{\vert \sigma_{2}\vert \vert \mu_{1}+\vert 1+\mu_{1}\vert \vert }{\vert 1+\mu_{2}\vert \vert 1+\mu_{1}\vert }\psi_{2}(r)\int_{0}^{1} \bigl\vert l_{h}(s)\bigr\vert \,ds \end{aligned}$$
and
$$\begin{aligned} & \bigl\vert D_{0+}^{\gamma}(\mathcal{T}u) (t)\bigr\vert \\ &\quad \leq\frac{\varphi(r)}{\Gamma(\alpha-\gamma)}l_{\mathcal{T}_{0}}\int_{0}^{t}(t-s)^{\alpha-\gamma-1}\,ds \\ &\qquad{} +\frac{\vert \mu_{2}\vert }{\vert 1+\mu_{2}\vert }\frac{t^{1-\gamma}}{\Gamma(2-\gamma)}\frac{\varphi(r)}{\Gamma(\alpha -\gamma )}l_{\mathcal{T}_{0}}\int _{0}^{1}(1-s)^{\alpha-\gamma-1}\,ds \\ &\qquad{} +\frac{\vert \sigma_{2}\vert }{\vert 1+\mu_{2}\vert }\frac{t^{1-\gamma}}{\Gamma(2-\gamma)}\psi_{2}(r)\int _{0}^{1}\bigl\vert l_{h}(s)\bigr\vert \,ds. \end{aligned}$$
By the Hölder inequality,
$$\begin{aligned} & \bigl\vert (\mathcal{T}u) (t)\bigr\vert \\ & \quad\leq\frac{\varphi(r)l_{\mathcal{T}_{0}}}{\Gamma(\alpha+1)} +\frac{ ( \vert \mu_{1}\vert \vert 1+\mu _{2}\vert +\vert \mu_{2}\vert \vert 1+\mu_{1}\vert ) \varphi(r)l_{\mathcal{T}_{0}}}{\vert 1+\mu_{1}\vert \vert 1+\mu_{2}\vert \Gamma(\alpha+1)} +\frac{\vert \mu_{1}\vert \vert \mu_{2}\vert \varphi(r)l_{\mathcal{T}_{0}}}{\vert 1+\mu_{1}\vert \vert 1+\mu_{2}\vert \Gamma(\alpha)} \\ &\qquad{} +\frac{\vert \sigma_{1}\vert \psi_{1}(r)\Vert l_{g}\Vert _{1}}{\vert 1+\mu_{1}\vert }+\frac{\vert \sigma_{2}\vert \vert \mu_{1}+\vert 1+\mu_{1}\vert \vert \psi_{2}(r)\Vert l_{h}\Vert _{1}}{\vert 1+\mu _{2}\vert \vert 1+\mu_{1}\vert } \\ &\quad \leq\varphi(r)l_{\mathcal{T}_{0}}\Delta_{1}+\Delta_{g} \psi_{1}(r)\Vert l_{g}\Vert _{1}+ \Delta_{h_{1}}\psi_{2}(r)\Vert l_{h}\Vert _{1}, \\ & \bigl\vert D_{0+}^{\gamma}(\mathcal{T}u) (t)\bigr\vert \\ &\quad \leq\frac{\varphi(r)l_{\mathcal{T}_{0}}}{\Gamma(\alpha-\gamma+1)} +\frac{\vert \mu_{2}\vert \varphi(r)l_{\mathcal{T}_{0}}}{\vert 1+\mu_{2}\vert \Gamma(2-\gamma)\Gamma(\alpha-\gamma+1)} +\frac{\vert \sigma_{2}\vert \psi_{2}(r)\Vert l_{h}\Vert _{1}}{\vert 1+\mu_{2}\vert \Gamma(2-\gamma)} \\ &\quad \leq\varphi(r)l_{\mathcal{T}_{0}}\Delta_{2}+\Delta_{h_{2}}\psi _{2}(r)\Vert l_{h}\Vert _{1}. \end{aligned}$$
Therefore,
$$\bigl\Vert ( \mathcal{T}u ) \bigr\Vert _{\gamma}\leq \varphi(r)l_{\mathcal{T}_{0}} ( \Delta_{1}+\Delta_{2} ) +\Delta _{g}\psi_{1}(r)\Vert l_{g}\Vert _{1}+ ( \Delta_{h_{1}}+\Delta_{h_{2}} ) \psi_{2}(r)\Vert l_{h}\Vert _{1}. $$
Step 2: The families \(\{ (\mathcal{T}u):u\in B_{r} \} \) and \(\{ D_{0+}^{\gamma}(\mathcal{T}u):u\in B_{r} \} \) are equicontinuous. For \(t_{1}< t_{2}\), we get
$$\begin{aligned} & \bigl\vert (\mathcal{T}u) (t_{2})-(\mathcal{T}u) (t_{1})\bigr\vert \\ &\quad \leq\frac{\varphi(r)l_{\mathcal{T}_{0}}}{\Gamma(\alpha)} \biggl[ \int_{0}^{t_{1}} \bigl( (t_{1}-s)^{\alpha-1}+(t_{2}-s)^{\alpha -1} \bigr)\,ds-\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha-1}\,ds \biggr] \\ & \qquad{}+\frac{\vert \mu_{2}\vert \vert t_{2}-t_{1}\vert }{\vert 1+\mu_{2}\vert }\frac{\varphi(r)l_{\mathcal{T}_{0}}}{\Gamma(\alpha)}\int_{0}^{t_{1}}(1-s)^{\alpha-1}\,ds \\ &\qquad{} +\frac{\vert \sigma_{2}\vert \vert 1+\mu_{1}\vert \vert t_{2}-t_{1}\vert \psi_{2}(r)}{\vert \mu_{2}\vert \vert 1+\mu_{1}\vert }\int_{0}^{1}\bigl\vert l_{h}(s)\bigr\vert \,ds\rightarrow0 \quad\mbox{as }t_{2} \rightarrow t_{1}. \end{aligned}$$
Similarly,
$$\begin{aligned} & \bigl\vert D_{0+}^{\gamma}(\mathcal{T}u) (t_{2})-D_{0+}^{\gamma}(\mathcal{T}u) (t_{1})\bigr\vert \\ &\quad \leq\frac{\varphi(r)l_{\mathcal{T}_{0}}}{\Gamma(\alpha-\gamma)} \biggl[ \int_{0}^{t_{1}} \bigl( (t_{1}-s)^{\alpha-\gamma-1}+(t_{2}-s)^{\alpha-\gamma-1} \bigr)\,ds-\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha-\gamma-1}\,ds \biggr] \\ &\qquad{} +\frac{\varphi(r)l_{\mathcal{T}_{0}}\vert \mu_{2}\vert \vert t_{2}^{1-\gamma}-t_{1}^{1-\gamma} \vert }{\Gamma(\alpha -\gamma)\vert 1+\mu_{2}\vert \Gamma ( 2-\gamma ) }\int_{0}^{1}(1-s)^{\alpha-\gamma-1}\,ds \\ &\qquad{} +\frac{\vert \sigma_{2}\vert \vert t_{2}^{1-\gamma}-t_{1}^{1-\gamma} \vert \psi_{2}(r)}{\vert 1+\mu_{2}\vert \Gamma ( 2-\gamma ) }\int_{0}^{1}\bigl\vert l_{h}(s)\bigr\vert \,ds\rightarrow0 \quad\mbox{as }t_{2} \rightarrow t_{1}. \end{aligned}$$
Thus \(\{ (\mathcal{T}u):u\in B_{r} \} \) and \(\{ D_{0+}^{\gamma}(\mathcal{T}u):u\in B_{r} \} \) are equicontinuous and relatively compact in \(C([0,1],R)\) by the Arzela-Ascoli theorem. Therefore \(\mathcal{T}(B_{r})\) is a relatively compact subset of \(C_{\gamma }([0,1],R)\) and the operator \(\mathcal{T}\) is compact.
Step 3: Let \(u=\lambda ( \mathcal{T}u ) \) and \(u=\lambda ( D_{0+}^{\gamma}(\mathcal{T}u) ) \) for \(0<\lambda<1\). For each \(t\in [ 0,1 ] \), define \(\overline{\mathcal{M}}= \{ \Vert u\Vert _{\gamma}\in C_{\gamma}([0,1],R),\Vert u\Vert _{\gamma}<\mathcal{N} \} \). Then we get
$$\begin{aligned}& \begin{aligned}[b] \Vert u\Vert _{c} &= \bigl\Vert \lambda ( \mathcal{T}u ) \bigr\Vert _{c} \\ &\leq \varphi\bigl(\Vert u\Vert _{\gamma}\bigr)l_{\mathcal{T}_{0}}\Delta _{1}+\Delta_{g}\psi_{1}\bigl(\Vert u\Vert _{\gamma}\bigr)\Vert l_{g}\Vert _{1} +\Delta_{h_{1}}\psi_{2}\bigl(\Vert u\Vert _{\gamma}\bigr)\Vert l_{h}\Vert _{1}, \end{aligned} \\& \begin{aligned}[b] \Vert u\Vert _{c} & =\bigl\Vert \lambda \bigl( D_{0+}^{\gamma }(\mathcal{T}u) \bigr) \bigr\Vert _{c} \\ & \leq\varphi\bigl(\Vert u\Vert _{\gamma}\bigr)l_{\mathcal{T}_{0}}\Delta _{2}+\Delta_{h_{2}}\psi_{2}\bigl(\Vert u\Vert _{\gamma}\bigr)\Vert l_{h}\Vert _{1}. \end{aligned} \end{aligned}$$
Thus
$$\begin{aligned} \Vert u\Vert _{\gamma} \leq \varphi\bigl(\Vert u\Vert _{\gamma}\bigr)l_{\mathcal{T}_{0}}\sum_{i=1}^{2} \Delta_{i}+\psi_{1}\bigl(\Vert u\Vert _{\gamma} \bigr)\Vert l_{g}\Vert _{1}\Delta_{g} +\psi_{2}\bigl(\Vert u\Vert _{\gamma}\bigr)\Vert l_{h}\Vert _{1}\sum_{i=1}^{2} \Delta_{h_{i}}. \end{aligned}$$
That means
$$\frac{\Vert u\Vert _{\gamma}}{\varphi(\Vert u\Vert _{\gamma})l_{\mathcal{T}_{0}}\sum_{i=1}^{2}\Delta_{i}+\psi_{1}(\Vert u\Vert _{\gamma})\Vert l_{g}\Vert _{1}\Delta_{g}+\psi _{2}(\Vert u\Vert _{\gamma})\Vert l_{h}\Vert _{1}\sum_{i=1}^{2}\Delta_{h_{i}}}\leq1. $$
For a non-negative \(\mathcal{N}\) and \(\Vert u\Vert _{\gamma }<\mathcal{N}\), the operator \(\mathcal{T}\) which is defined in \(\overline{\mathcal{M}}\) to be \(C_{\gamma}([0,1],R)\) is continuous and compact. Therefore \(\mathcal{T}\) has a fixed point in \(\overline {\mathcal{M}}\). □

5 Examples

Example 10

Consider the following boundary value problem of a fractional differential equation:
$$ \left \{ \textstyle\begin{array}{@{}l} D_{0+}^{\frac{3}{2}} ( \phi_{p}D_{0+}^{\frac{3}{2}}u ) ( t ) =l_{f} ( \frac{\vert u(t)\vert }{\vert u(t)\vert +1}+\frac{\vert D_{0+}^{\frac{3}{2}}u(t)\vert }{\vert D_{0+}^{\frac{3}{2}}u(t)\vert +1} ) ,\\ u(0)+0.1u(1)=0.01\int_{0}^{1}\frac{u(s)}{(1+s)^{2}}\,ds,\\ u^{{\prime}}(0)+0.1u^{{\prime}}(1)=0.01\int_{0}^{1} ( \frac{e^{s}u(s)}{1+2e^{s}}+\frac{1}{2} )\,ds. \end{array}\displaystyle \right . $$
(17)
Here
$$\begin{aligned}& \alpha,\beta =1.5, \qquad\mu_{1},\mu_{2}=0.1,\qquad \sigma _{1},\sigma _{2}=0.01,\\& \upsilon,\eta =0.3, \qquad\tau=0.4,\qquad \gamma=0.01, \end{aligned}$$
and
$$\begin{aligned}& f(t,u,v)=\frac{\vert u\vert }{\vert u\vert +1}+\frac{\vert v\vert }{\vert v\vert +1}, \\& g(t,u)=\frac{u}{(1+s)^{2}},\qquad h(t,u)=\frac{e^{s}u}{ ( 1+2e^{s} ) }+\frac{1}{2}. \end{aligned}$$
Since \(0.88<\Gamma ( 1.5 ) <0.89\), \(\Gamma ( 2 ) =1\), \(\Gamma ( 2.5 ) =1\), we find
$$\begin{aligned} &\Delta_{1} =0.89, \qquad\Delta_{2}=0.82,\qquad \Delta_{g}=0.009, \\ &\Delta_{h_{1}} =0.0099, \qquad\Delta_{h_{2}}=0.009, \qquad l_{g}=l_{h}=1, \end{aligned}$$
with simple calculations. Therefore
$$\begin{aligned} & \bigl\{ l_{\mathcal{T}_{0}} ( \Delta_{1}+\Delta_{2} ) +2 \Delta_{g}\Vert l_{g}\Vert _{1}+ ( \Delta_{h_{1}}+\Delta_{h_{2}} ) \Vert l_{h} \Vert _{1} \bigr\} \\ &\quad < 1.73l_{\mathcal{T}_{0}}+0.04 \\ &\quad < 1. \end{aligned}$$
Then we can choose
$$l_{\mathcal{T}_{0}}< 0.562. $$
Thus all assumptions of Theorem 8 satisfied. Therefore the problem has a unique solution on \([ 0,1 ] \).

Example 11

Consider the following boundary value problem of fractional differential equation:
$$ \left \{ \textstyle\begin{array}{@{}l} D_{0+}^{\frac{3}{2}} ( \phi_{p}D_{0+}^{\frac{3}{2}}u ) ( t ) =\frac{\vert u(t)\vert ^{3}}{9(\vert u(t)\vert ^{3}+3)}+\frac{\vert \sin D_{0+}^{\frac{3}{2}}u(t)\vert }{9(\sin D_{0+}^{\frac{3}{2}}u(t)+1)}+\frac{1}{12},\\ u(0)+0.1u(1)=0,01\int_{0}^{1}\frac{u(s)}{3(1+s)^{2}}\,ds,\\ u^{{\prime}}(0)+0.1u^{{\prime}}(1)=0,01\int_{0}^{1}\frac{e^{s}u(s)}{3(1+e^{s})^{2}}\,ds,\\ D_{0+}^{\frac{3}{2}}u(0)=0,\\ D_{0+}^{\frac{3}{2}}u(1)=0,3D_{0+}^{\frac{3}{2}}u(0,3), \end{array}\displaystyle \right . $$
(18)
where f is given by
$$f(t,u,v)=\frac{\vert u\vert ^{3}}{9(\vert u\vert ^{3}+3)}+\frac{\vert \sin v\vert }{9(\sin v+1)}+\frac{1}{12}. $$
We have
$$\bigl\vert f(t,u,v)\bigr\vert \leq\frac{\vert u\vert ^{3}}{9(\vert u\vert ^{3}+3)}+\frac{\vert \sin v\vert }{9(\sin v+1)}+ \frac{1}{12}, \quad u\in R. $$
Here
$$\begin{aligned}& \alpha,\beta =1.5,\qquad \mu_{1},\mu_{2}=0.1,\qquad \sigma _{1},\sigma _{2}=0.01, \\& \upsilon,\eta =0.3,\qquad \tau=0.4,\qquad \gamma=0.01, \\& \Delta_{1} =0.89,\qquad \Delta_{2}=0.82,\qquad \Delta_{g}=0.009, \\& \Delta_{h_{1}} =0.0099,\qquad \Delta_{h_{2}}=0.009,\qquad l_{g}=l_{h}=0.1, \end{aligned}$$
and \(g(t,u)=\frac{u(s)}{3(1+s)^{2}}\), \(h(t,u)=\frac{e^{s}u(s)}{3(1+e^{s})^{2}}\), \(\varphi(\mathcal{N)=\psi}_{1}\mathcal{ ( \mathcal{N} ) =\psi}_{2}\mathcal{ ( \mathcal{N} ) =N}\). If
$$\begin{aligned}& \frac{\mathcal{N}}{\mathcal{\varphi} ( \mathcal{N} ) ( 0.561 ) ( 0.89+0.82 ) +\psi_{1} ( \mathcal{N} ) ( 0.1 ) ( 0.009 ) +\psi_{2} ( \mathcal{N} ) ( 0.1 ) ( 0.0099+0.009 ) }>1, \\& \frac{\mathcal{N}}{\mathcal{N} ( 0.96 ) +\mathcal{N} ( 0.0009 ) +\mathcal{N} ( 0.0019 ) }>1, \\& \frac{\mathcal{N}}{0.9628\mathcal{N}}\mathcal{>}1, \\& 1.04>1, \end{aligned}$$
then (16) is satisfied. Then there exists at least one solution of the BVP on \([ 0,1 ] \).

Declarations

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

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)
Eastern Mediterranean University

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© Mahmudov and Unul 2015