# Existence of solutions for fractional Sturm-Liouville boundary value problems with $$p(t)$$-Laplacian operator

## Abstract

This paper is concerned with the solvability for fractional Sturm-Liouville boundary value problems with $$p(t)$$-Laplacian operator at resonance using Mawhin’s continuation theorem. Sufficient conditions for the existence of solutions have been acquired, and they would extend the existing results. Furthermore, an example is provided to illustrate the main result.

## Introduction

The last two decades have witnessed a wide application of fractional differential equations in various fields of natural science and engineering technology (see ). Introduced by Bagley and Torvik , the famous fractional differential model is used to describe radial vibration of a rigid plate connected to a massless spring immersing vertical in the ideal fluid:

$$ay'' ( t ) + bD_{t}^{\frac{3}{2}}y ( t ) + cy ( t ) = f ( t ) ,$$

where $$a,b,c > 0$$, and the fractional derivative represents damping. With some theoretical discussions conducted regarding boundary value problem (BVP for short) of differential equations so far, valuable results have been obtained for BVP of fractional differential equations (see ). For instance, Kosmatov  studied the existence of solution for the following BVP of fractional differential equations using coincidence degree theory:

$$\textstyle\begin{cases} {D^{\alpha}}u ( t ) = f ( {t,u ( t ) ,u' ( t ) } ) , \quad a \cdot e \cdot t \in ( {0,1} ) , \\ D_{0 + }^{\alpha- 2}u ( 0 ) = 0, \quad \quad \eta u ( \xi ) = u ( 1 ) , \end{cases}$$

where $${{D^{\alpha}}}$$ is a Caputo fractional derivative, and $$1 < \alpha\le2$$.

It is generally known that the p-Laplacian equations normally derive from nonlinear elastic mechanics and non-Newtonian fluid theory. However, in view of their significance in theory and practice, more and more attention is being paid to the existence of solutions for fractional p-Laplacian BVP. Consequently, important results have been achieved in this regard by some researchers (see ). Chen and Liu  discussed the solvability of the following anti-periodic BVP:

$$\textstyle\begin{cases} {}_{0}^{C}D_{t}^{\beta}{\phi_{p}} ( {{}_{0}^{C}D_{t}^{\alpha}u} ) = f ( {t,u} ) , \quad t \in[0,1], \\ u ( 0 ) = - u ( 1 ) , \quad\quad {}_{0}^{C}D_{t}^{\alpha}u ( 0 ) = - {}_{0}^{C}D_{t}^{\alpha}u ( 1 ) , \end{cases}$$

where $$0 < \alpha$$, $$\beta\le1$$, $${\phi_{p}} ( \cdot )$$ is a p-Laplacian operator defined by $${\phi_{p}} ( s ) = {\vert s \vert ^{p - 2}}s$$ ($$s \ne0$$, $$p > 1$$), $${\phi _{p}} ( 0 ) = 0$$. With Schaefer’s fixed point theorem, the existence of solutions for BVP was obtained.

Mahmudov and Unul  studied the BVP

$$\textstyle\begin{cases} D_{0 + }^{\beta}{\varphi_{p}} ( {D_{0 + }^{\alpha}x ( t ) } ) = f ( {t,x ( t ) ,D_{0 + }^{\gamma}x ( t ) } ) , \quad t \in [ {0,1} ] , \\ x ( 0 ) + {\mu_{1}}x ( 1 ) = {\sigma_{1}}\int _{0}^{1} {g ( {s,x ( s ) } ) \,\mathrm{d}s} , \\ x' ( 0 ) + {\mu_{2}}x' ( 1 ) = {\sigma_{2}}\int _{0}^{1} {h ( {s,x ( s ) } ) \,\mathrm{d}s} , \\ D_{0 + }^{\alpha}x ( 0 ) = 0, \quad\quad D_{0 + }^{\alpha}x ( 1 ) = \nu D_{0 + }^{\alpha}x ( \eta ) , \end{cases}$$

where $$1 < \alpha\le2$$, $$0 < \beta$$, $$\gamma\le1$$, $$0 < \eta< 1$$, $$\nu,{\mu_{i}},{\sigma_{i}} > 0$$ ($${i = 1,2}$$), $$D_{0 + }^{\alpha}$$ is a Caputo fractional derivative, $${\varphi_{p}} ( \cdot )$$ is a p-Laplacian operator, f, g, h are continuous functions. By constructing the Green’s functions of BVP and by using the fixed point theory, the existence and uniqueness of the solutions were obtained under suitable conditions.

As far as we are concerned, the $$p ( t )$$-Laplacian operator is a non-standard growth operator by nature, and it mainly derives from elasticity theory, nonlinear electrorheological fluids and image restoration. A lot of research regarding BVP of fractional differential equations with $$p ( t )$$-Laplacian operator have been quite limited so far (see ). Specifically, Shen and Liu  studied the existence of solutions for the following BVP with $$p ( t )$$-Laplacian operator at nonresonance and resonance by using Schaefer’s fixed point theorem and Mawhin’s continuation theorem:

$$\textstyle\begin{cases} D_{0 + }^{\beta}{\varphi_{p ( t ) }} ( {D_{0 + }^{ \alpha}x ( t ) } ) + f ( {t,x ( t ) } ) = 0, \quad t \in ( {0,1} ) , \\ x ( 0 ) = 0, \quad\quad D_{0 + }^{\alpha- 1}x ( 1 ) = \gamma I_{0 + }^{\alpha- 1}x ( \eta ) , \quad\quad D_{0 + }^{\alpha}x ( 0 ) = 0, \end{cases}$$

where $$\dim\operatorname{Ker}L=1$$, $$1 < \alpha\le2$$, $$0 < \beta\le1$$, $$\gamma> 1$$, $$0 < \eta< 1$$, $${\varphi_{p ( t ) }} ( \cdot )$$ is a $$p ( t )$$-Laplacian operator, $$p ( t ) > 1$$, $$p ( t ) \in{C^{1}} [ {0,1} ]$$.

Inspired by the above findings, this paper studies the BVP subjected to Sturm-Liouville type integral boundary conditions for fractional differential equations with $$p ( t )$$-Laplacian operator:

$$\textstyle\begin{cases} {}^{C}D_{0 + }^{\beta}{\phi_{p ( t ) }} ( {{}^{C}D_{0 + }^{\alpha}x ( t ) } ) = f ( {t,x ( t ) ,x' ( t ) } ) , \\ x ( 0 ) + bx' ( 0 ) = \gamma\int_{0}^{\xi} {x ( t ) \,\mathrm{d}t} , \\ x ( 1 ) - mx' ( 1 ) = \sigma\int_{0}^{\eta} {x ( t ) \,\mathrm{d}t} , \\ {}^{C}D_{0 + }^{\alpha}x ( 0 ) = 0, \end{cases}$$
(1.1)

where $$0 < \beta\le1$$, $$1 < \alpha\le2$$, $${}^{C}D_{0 + }^{\beta}$$, $${}^{C}D_{0 + }^{\alpha}$$ are Caputo fractional derivatives, $$b, m, \xi, \eta\in ( {0,1} )$$, $$\sigma> 0$$, $$\gamma> 0$$, $$f: [ {0,1} ] \times{\mathbb{R}^{2}} \to{\mathbb{R}}$$ is continuous. $${\phi_{p ( t ) }} ( \cdot )$$ is the $$p ( t )$$-Laplacian operator, $${p ( t ) > 1}$$, $$p ( t ) \in{C^{1}} [ {0,1} ]$$, and

$$\gamma\xi= 1, \quad\quad \sigma\eta= 1, \quad\quad b = \frac{1}{2}\xi, \quad\quad m = 1 - \frac{1}{2}\eta,$$
(1.2)

which leads to BVP (1.1) is resonant. It is also assumed that

$$C = ( {\alpha+ 2} ) \bigl[ {1 - m ( {\alpha+ 1} ) } \bigr] + \xi ( {\alpha+ 1} ) ( {m\alpha- 1} ) + {\eta^{\alpha}} ( {\xi- \eta} ) \ne0.$$
(1.3)

It is worth noting that $$p(t)=p$$ herein, meaning it could be the famous p-Laplacian operator. Since the $$p ( t )$$-Laplacian operator is a nonlinear operator, it is more difficult to construct a projection operator. So our results serve as a further development for the previous findings in this sense. Furthermore, we also observe that few scholars have ever considered fractional Sturm-Liouville BVP with $$p ( t )$$-Laplacian operator before. The kernel space herein is extended to higher dimensions as well. To be specific, it is assumed that $$\dim\operatorname{Ker}L=2$$ in the article. In comparison with the case when $$\dim\operatorname{Ker}L=1$$, the system is more complex.

## Preliminaries

To facilitate understanding, we would firstly make a brief introduction about the concepts and lemmas regarding fractional derivatives and integrals in the article. For more details, please refer to the references hereunder (see ).

### Definition 2.1

()

Let X, Y be real Banach spaces and $$L:\operatorname{dom}L\subset X\rightarrow Y$$ be a linear map. If $$\dim \operatorname{Ker}L = \operatorname{codim} \operatorname{Im} L < + \infty$$ and ImL is a closed subset in Y, then the map L is a Fredholm operator with index zero. If there exist such continuous projections as $$P:X\rightarrow X$$ and $$Q:Y\rightarrow Y$$, which meet the conditions that $$\operatorname{Im}P = \operatorname{Ker}L$$ and $$\operatorname{Ker}Q=\operatorname{Im}L$$, then $$L |_{\operatorname{dom}L\cap\operatorname{Ker}P}:\operatorname {dom}L\cap\operatorname{Ker}P \rightarrow\operatorname{Im}L$$ is reversible. We denote the inverse map by $$K_{P}$$, set $${K_{P}} = L_{P}^{ - 1}$$ and $${K_{P,Q}} = {K_{P}} ( {I - Q} )$$. If Ω is an open bounded subset of X and $$\operatorname{dom} L \cap\Omega\ne\varnothing$$, the map N is L-compact on Ω̅ when $$QN:\overline{\Omega}\to Y$$ is bounded and $${K_{P}} ( {I - Q} ) N:\overline{\Omega}\to X$$ is compact.

### Theorem 2.1

()

Let L be a Fredholm operator of index zero and N be L-compact on Ω̅. Assume that the following conditions are satisfied:

$${(a_{1})}$$ :

$$Lx\neq\lambda Nx$$ for every $$(x,\lambda)\in[( \operatorname{dom}L \setminus\operatorname{Ker}L ) \cap\partial \Omega]\times(0,1)$$.

$${(a_{2})}$$ :

$$Nx\notin\operatorname{Im}L$$ for every $$x\in\operatorname {Ker}L\cap \partial\Omega$$.

$${(a_{3})}$$ :

$$\operatorname{deg}(QN|_{\operatorname{Ker}L},\Omega\cap \operatorname{Ker}L,0) \neq0$$.

Then the equation $$Lx=Nx$$ has at least one solution in $$\operatorname{dom}L \cap\Omega$$.

### Definition 2.2

()

The Riemann-Liouville fractional integral of order α ($$\alpha> 0$$) for the function $$x:(0, + \infty) \to\mathbb{R}$$: is defined as

$$I_{0 + }^{\alpha}x(t) = \frac{1}{{\Gamma(\alpha)}} \int_{0}^{t} {{(t - s)}^{\alpha- 1}}x(s) \, \mathrm{d}s,$$

assume that the right-hand side integral is defined on $$(0, + \infty )$$.

### Definition 2.3

()

The Caputo fractional integral of order α ($$\alpha> 0$$) for the function $$x:(0, + \infty) \to \mathbb{R}$$: is defined as

$${}^{C}D_{0 + }^{\alpha}x(t) =I_{0 + }^{n - \alpha} \frac{{{d^{n}}x(t)}}{ {d{t^{n}}}} = \frac{1}{{\Gamma(n - \alpha)}} \int_{0}^{t} {{(t - s)} ^{n - \alpha- 1}}x^{(n)}(s) \,\mathrm{d}s,$$

where $$n = [ \alpha]+1$$, provided that the right-hand side integral is defined on $$(0, + \infty)$$.

### Lemma 2.1

()

Let $$n-1<\alpha\leq n$$, if $${}^{C}D_{0+}^{ \alpha}x(t)\in C[0,1]$$, then

$$I_{0+}^{\alpha} {}^{C}D_{0+}^{\alpha}x(t)=x(t)+c_{0}+c_{1}t+c_{2}t ^{2}+\cdots+c_{n-1}t^{n-1},$$

where $$c_{i} \in\mathbb{R}$$, $$i = 0,1, \ldots,n-1$$, $$n = [ \alpha]+1$$.

### Lemma 2.2

()

Let $$n-1<\alpha\leq n$$, then the fractional differential $${}^{C}D_{0+}^{\alpha}x(t)=0$$ has the following form:

$$x(t)=c_{0}+c_{1}t+c_{2}t^{2}+ \cdots+c_{n-1}t^{n-1},$$

where $$c_{i} \in\mathbb{R}$$, $$i = 0,1, \ldots,n-1$$, $$n = [ \alpha]+1$$.

### Lemma 2.3

()

For any $$( {t,x} ) \in [ {0,1} ] \times\mathbb{R}$$, $${\varphi_{p ( t ) }} ( x ) = {\vert x \vert ^{p ( t ) - 2}} x$$ is a homeomorphism from $$\mathbb{R}$$ to $$\mathbb{R}$$ and strictly monotone increasing for any fixed t. Moreover, its inverse operator $$\varphi_{p ( t ) }^{ - 1} ( \cdot )$$ is defined by

$$\textstyle\begin{cases} \varphi_{p ( t ) }^{-1} ( x ) = {\vert x \vert ^{\frac{{2 - p ( t ) }}{{p ( t ) - 1}}}}x,\quad x \in\mathbb{R}\backslash \{ 0 \} , \\ \varphi_{p ( t ) }^{-1} ( 0 ) = 0, \quad x = 0 , \end{cases}$$

which is continuous and sends bounded sets into bounded sets.

Since Mawhin’s continuation theorem is applicable to linear operators, the following lemma needs to be introduced in this paper.

### Lemma 2.4

BVP (1.1) is equivalent to the following problem:

$$\textstyle\begin{cases} D_{0 + }^{\alpha}x ( t ) = \phi_{p ( t ) }^{ - 1} ( {I_{0 + }^{\beta}f ( {t,x ( t ) ,x' ( t ) } ) } ) , \quad t \in ( {0,1} ) , \\ x ( 0 ) + bx' ( 0 ) = \gamma\int_{0}^{\xi} {x ( t ) \,\mathrm{d}t} , \quad\quad x ( 1 ) - mx' ( 1 ) = \sigma\int_{0}^{\eta}{x ( t ) \,\mathrm{d}t} . \end{cases}$$
(2.1)

### Proof

On the one hand, by Definition 2.2, we have

$${\phi_{p ( t ) }} \bigl( {D_{0 + }^{\alpha}x ( t ) } \bigr) = I_{0 + }^{\beta}f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) + c, \quad c \in\mathbb{R}.$$

Based on the boundary condition $${}^{C}D_{0 + }^{\alpha}x ( 0 ) = 0$$, we get $$c=0$$. Thus,

\begin{aligned}& {\phi_{p ( t ) }} \bigl( {D_{0 + }^{\alpha}x ( t ) } \bigr) = I_{0 + }^{\beta}f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) , \\& {D_{0 + }^{\alpha}x ( t ) = \phi_{p ( t ) }^{ - 1} \bigl( {I_{0 + }^{\beta}f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) } \bigr) }. \end{aligned}

On the other hand, if $${D_{0 + }^{\alpha}x ( t ) = \phi_{p ( t ) }^{ - 1} ( {I_{0 + }^{\beta}f ( {t,x ( t ) ,x' ( t ) } ) } ) }$$, for $$t=0$$, we have $$D_{0 + }^{\alpha}x ( 0 ) = 0$$. Multiplying both sides of the equation by the operator $${\phi_{p ( t ) }}$$ and $${D_{0 + }^{\beta}}$$, we get

$$D_{0 + }^{\beta}{\phi_{p ( t ) }} \bigl( {D_{0 + }^{\alpha }x ( t ) } \bigr) = f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) .$$

The proof is complete. □

## Main result

Let $$X = {C^{1}} [ {0,1} ]$$, $$Y = C [ {0,1} ]$$ with the norm $${\Vert x \Vert _{X}} = {\max_{t \in [ {0,1} ] }} \{ {{{\Vert x \Vert }_{\infty}},{{ \Vert {x'} \Vert } _{\infty}}} \}$$, $${\Vert y \Vert _{Y}} = {\Vert y \Vert _{\infty}}$$, where $${\Vert x \Vert _{\infty}} = \max_{t \in [ {0,1} ] } \vert {x ( t ) } \vert$$. By Lemma 2.4, BVP (1.1) is equivalent to the following problems:

$$\textstyle\begin{cases} D_{0 + }^{\alpha}x ( t ) = \phi_{p ( t ) }^{ - 1} ( {I_{0 + }^{\beta}f ( {t,x ( t ) ,x' ( t ) } ) } ) , \quad t \in ( {0,1} ) , \\ x ( 0 ) + bx' ( 0 ) = \gamma\int_{0}^{\xi} {x ( t ) \,\mathrm{d}t} , \quad\quad x ( 1 ) - mx' ( 1 ) = \sigma\int_{0}^{\eta}{x ( t ) \,\mathrm{d}t} . \end{cases}$$
(3.1)

Define the operator $$L:\operatorname{dom}L \subset X \to Y$$ by

$$Lx = D_{0 + }^{\alpha}x ( t ) ,$$
(3.2)

where

\begin{aligned} \operatorname{dom}L &= \biggl\{ x \in X\Big| D_{0 + }^{\alpha}x ( t ) \in Y, x ( 0 ) + bx' ( 0 ) = \gamma \int_{0}^{\xi}{x ( t ) \,\mathrm{d}t} , \\ &\quad x ( 1 ) - mx' ( 1 ) = \sigma \int_{0}^{\eta}{x ( t ) \,\mathrm{d}t} \biggr\} . \end{aligned}

Let $$N:X \to Y$$ as the Nemytskii operator

$$Nx ( t ) = \phi_{p ( t ) }^{ - 1} \bigl( {I_{0 + } ^{\beta}f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) } \bigr) ,\quad \forall t \in [ {0,1} ] .$$

Then BVP (1.1) is equivalent to the following operator equation:

$$Lx = Nx, \quad x \in\operatorname{dom}L.$$

For convenience, define the operators $${T_{1}}, {T_{2}}, {Q_{1}}, {Q_{2}}:Y \to Y$$:

\begin{aligned} &{T_{1}}y = \int_{0}^{\xi}{{{ ( {\xi- s} ) }^{\alpha}}y ( s ) } \,\mathrm{d}s, \\ &{T_{2}}y = \alpha \int_{0}^{1} {{{ ( {1 - s} ) }^{\alpha- 1}}y ( s ) } \,\mathrm{d}s - m\alpha ( {\alpha- 1} ) \int_{0}^{1} {{{ ( {1 - s} ) } ^{\alpha- 2}}y ( s ) } \,\mathrm{d}s - \sigma \int _{0}^{\eta} {{{ ( {\eta- s} ) }^{\alpha}}y ( s ) } \,\mathrm{d}s, \\ &{Q_{1}}y = \frac{1}{\Lambda} \bigl( {{\Lambda_{4}} {T_{1}}y ( t ) - {\Lambda_{3}} {T_{2}}y ( t ) } \bigr) , \\ &{Q_{2}}y = \frac{1}{\Lambda} \bigl( { - {\Lambda_{2}} {T_{1}}y ( t ) + {\Lambda_{1}} {T_{2}}y ( t ) } \bigr) , \end{aligned}

where

\begin{aligned}& {\Lambda_{1}} = \frac{{{\xi^{\alpha+ 1}}}}{{\alpha+ 1}}, \quad\quad { \Lambda _{2}} = 1 - m\alpha- \frac{{{\eta^{\alpha}}}}{{\alpha+ 1}}, \quad\quad { \Lambda_{3}} = \frac{{{\xi^{\alpha+ 2}}}}{{ ( {\alpha+ 1} ) ( {\alpha+ 2} ) }}, \\& {\Lambda_{4}} = \frac{1}{{\alpha+ 1}} - m - \frac{{{\eta^{\alpha+ 1}}}}{{ ( {\alpha+ 1} ) ( {\alpha+ 2} ) }}, \\& \begin{aligned} \Lambda&= \left \vert \begin{matrix} {{\Lambda_{1}}}&{{\Lambda_{2}}}\\ {{\Lambda_{3}}}&{{\Lambda_{4}}} \end{matrix} \right \vert \\ & = \frac{{{\xi^{\alpha+ 1}}}}{{{{ ( {\alpha+ 1} ) }^{2}} ( {\alpha+ 2} ) }} \bigl\{ { ( {\alpha+ 2} ) \bigl[ {1 - m ( {\alpha+ 1} ) } \bigr] + \xi ( {\alpha+ 1} ) ( {m\alpha- 1} ) + {\eta^{\alpha}} ( {\xi- \eta} ) } \bigr\} \\ & = \frac{{{\xi^{\alpha+ 1}}}}{{{{ ( {\alpha+ 1} ) }^{2}} ( {\alpha+ 2} ) }}C. \end{aligned} \end{aligned}

The following theorem is the main result of this paper.

### Theorem 3.1

Assume that the following conditions hold.

$$(\mathrm{H}_{1})$$ :

If the function $$f: [ {0,1} ] \times {{\mathbb{R}} ^{2}} \to{\mathbb{R}}$$ is continuous, and there exist nonnegative functions $$a,b,c\in C[0,1]$$ such that

$$\bigl\vert {f ( {t,u,v} ) } \bigr\vert \le a ( t ) + b ( t ) {\vert u \vert ^{\theta- 1}} + c ( t ) {\vert v \vert ^{\theta- 1}}, \quad \forall t \in [ {0,1} ] , ( {u,v} ) \in{\mathbb{R}^{2}}, 1 < \theta \le{P_{L}},$$

where $${a_{1}} = {\Vert {a} \Vert _{\infty}}$$, $${b_{1}} = {\Vert {b} \Vert _{\infty}}$$, $${c_{1}} = {\Vert {c} \Vert _{\infty}}$$, $${P_{L}} = \min_{t \in [ {0,1} ] } p ( t )$$.

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

There exists a constant $$\mathrm{B}>0$$ such that for $$u \in{\mathbb{R}}$$, if $$\vert u \vert > \mathrm{B}$$, one has either

$$u \cdot{T_{1}}Nu > 0 \quad \textit{or}\quad u \cdot{T_{1}}Nu < 0.$$
$$(\mathrm{H}_{3})$$ :

There exists a constant $$\mathrm{D}>0$$ such that for $$v \in{\mathbb{R}}$$, if $$\vert v \vert > D$$, one has either

$$v \cdot{T_{2}}Nu > 0 \quad \textit{or} \quad v \cdot{T_{2}}Nu < 0.$$

Then BVP (1.1) has at least one solution provided that

$$\frac{{{4^{\theta- 1}} ( {{b_{1}}{{ ( {\alpha+ 1} ) } ^{\theta- 1}} + {c_{1}}{\alpha^{\theta- 1}}} ) }}{{\Gamma ( {\beta+ 1} ) {{ ( {\Gamma ( {\alpha+ 1} ) } ) }^{\theta- 1}}}} < \frac{1}{2}.$$
(3.3)

In order to prove the above theorem, it is necessary to introduce more relevant lemmas, as shown hereunder.

### Lemma 3.1

Let L be defined by (3.2), then

\begin{aligned}& \operatorname{Ker}L = \bigl\{ x \in X| x ( t ) = {c_{0}} + {c_{1}}t, {c_{0}},{c_{1}} \in{\mathbb{R}}, \forall t \in [ {0,1} ] \bigr\} , \end{aligned}
(3.4)
\begin{aligned}& \operatorname{Im} L = \{ y \in Y| {{T_{1}}y = {T_{2}}y = 0} \} . \end{aligned}
(3.5)

### Proof

By Lemma 2.2, $$D_{0 + }^{\alpha}x ( t ) = 0$$ has a solution, i.e.,

$$x ( t ) = {c_{0}} + {c_{1}}t, \quad {c_{0}},{c_{1}} \in{\mathbb{R}}.$$

From (3.1), we can obtain (3.4).

Next, we prove $$\operatorname{Im} L = \{ y \in Y| {{T_{1}}y = {T_{2}}y = 0} \}$$.

If $$y \in\operatorname{Im} L$$, there exists $$x \in\operatorname {dom}L$$ such that $$y = Lx \in Y$$. By (2.1) ,we get

$$x ( t ) = \frac{1}{{\Gamma ( \alpha ) }} \int_{0} ^{t} {{{ ( {t - s} ) }^{\alpha- 1}}y ( s ) } \,\mathrm{d}s + c_{0}+ {c_{1}}t.$$

In view of the conditions of (3.1), we have

\begin{aligned}& \int_{0}^{\xi}{{{ ( {\xi- s} ) }^{\alpha}}y ( s ) } \,\mathrm{d}s=0, \\& \alpha \int_{0}^{1} {{{ ( {1 - s} ) }^{\alpha- 1}}y ( s ) } \,\mathrm{d}s - m\alpha ( {\alpha- 1} ) \int_{0}^{1} {{{ ( {1 - s} ) }^{\alpha- 2}}y ( s ) } \,\mathrm{d}s - \sigma \int_{0}^{\eta}{{{ ( {\eta- s} ) }^{\alpha}}y ( s ) } \,\mathrm{d}s=0, \end{aligned}

i.e., $${T_{1}}y = {T_{2}}y = 0$$. On the other hand, if $${T_{1}}y = {T_{2}}y = 0$$ for $$y \in Y$$, let $$x ( t ) = I_{0 + }^{ \alpha}y ( t )$$, then $$x \in\operatorname{dom}L$$ and $$D_{0 + } ^{\alpha}x ( t ) = y ( t )$$. Thus, $$y \in \operatorname{Im} L$$. □

### Lemma 3.2

Let L be defined by (3.2), then L is a Fredholm operator of index zero. The linear projection operators $$P:X \to X$$ and $$Q:Y \to Y$$ can be defined as follows:

\begin{aligned}& Px ( t ) = x ( 0 ) + x' ( 0 ) t, \quad \forall t \in [ {0,1} ] , \\& Qy ( t ) = {Q_{1}}y + {Q_{2}}y \cdot t, \quad \forall t \in [ {0,1} ] . \end{aligned}

In addition, $${K_{P}}:\operatorname{Im} L \to\operatorname{dom}L \cap\operatorname{Ker}P$$ is defined as

$${K_{P}}y ( t ) = \frac{1}{{\Gamma ( \alpha ) }} \int_{0}^{t} {{{ ( {t - s} ) }^{\alpha- 1}}y ( s ) } \,\mathrm{d}s, \quad \forall t \in [ {0,1} ] .$$

### Proof

Clearly, $$\operatorname{Im} P = \operatorname{Ker}L$$ and $${P^{2}}x = Px$$. By $$x = ( {x - Px} ) + Px$$, we obtain $$x = \operatorname{Ker}P + \operatorname{Ker}L$$. After a simple calculation, we get $$\operatorname{Ker}L \cap\operatorname{Ker}P = \{ 0 \}$$. Thus, we have

$$x = \operatorname{Ker}L \oplus\operatorname{Ker}P.$$

The next step is to prove $$\operatorname{Ker} Q = \operatorname{Im} L$$. It is clear that $$\operatorname{Im} L \subset\operatorname{Ker} Q$$. On the other hand, if $$y \in\operatorname{Ker} Q \subset Y$$, then $${Q_{1}}y = {Q_{2}}y = 0$$, i.e.,

$$\textstyle\begin{cases} \frac{1}{\Lambda} ( {{\Lambda_{4}}{T_{1}}y - {\Lambda_{3}}{T _{2}}y} ) = 0, \\ \frac{1}{\Lambda} ( { - {\Lambda_{2}}{T_{1}}y + {\Lambda_{1}} {T_{2}}y} ) = 0. \end{cases}$$

By (1.3), we have $$\Lambda\ne0$$. Hence, $${T_{1}}y = {T_{2}}y = 0$$. Thus we get $$y \in\operatorname{Im} L$$ and $$\operatorname{Ker} Q \subset\operatorname{Im} L$$. For $$y \in Y$$, we get

\begin{aligned}& Q_{1}^{2}y = \frac{1}{\Lambda} \bigl( {{\Lambda_{4}} {T_{1}} ( {{Q_{1}}y} ) - { \Lambda_{3}} {T_{2}} ( {{Q_{1}}y} ) } \bigr) = \frac{1}{\Lambda} ( {{\Lambda_{4}} {\Lambda_{1}} - { \Lambda_{3}} {\Lambda_{2}}} ) {Q_{1}}y = {Q_{1}}y, \\& {Q_{2}} ( {{Q_{1}}y} ) = \frac{1}{\Lambda} \bigl( { - { \Lambda_{2}} {T_{1}} ( {{Q_{1}}y} ) + { \Lambda_{1}} {T_{2}} ( {{Q_{1}}y} ) } \bigr) = \frac{1}{\Lambda} ( { - {\Lambda_{2}} {\Lambda _{1}} + { \Lambda_{1}} {\Lambda_{2}}} ) {Q_{1}}y = 0, \\& {Q_{1}} ( {{Q_{2}}y \cdot t} ) = \frac{1}{\Lambda} \bigl( {{\Lambda _{4}} {T_{1}} ( {{Q_{2}}y \cdot t} ) - {\Lambda_{3}} {T_{2}} ( {{Q_{2}}y \cdot t} ) } \bigr) = \frac{1}{\Lambda} ( {{\Lambda_{4}} {\Lambda_{3}} - {\Lambda_{3}} {\Lambda_{4}}} ) {Q_{2}}y = 0, \\& {Q_{2}} ( {{Q_{2}}y \cdot t} ) = \frac{1}{ \Lambda} \bigl( { - {\Lambda_{2}} {T_{1}} ( {{Q_{2}}y \cdot t} ) + {\Lambda_{1}} {T_{2}} ( {{Q_{2}}y \cdot t} ) } \bigr) = \frac{1}{\Lambda} ( { - {\Lambda_{2}} { \Lambda_{3}} + {\Lambda _{1}} {\Lambda_{4}}} ) {Q_{2}}y = {Q_{2}}y. \end{aligned}

Therefore, we have

$${Q^{2}}y = {Q_{1}} ( {{Q_{1}}y + {Q_{2}}y \cdot t} ) + {Q_{2}} ( {{Q_{1}}y + {Q_{2}}y \cdot t} ) t = {Q_{1}}y + {Q_{2}}y \cdot t = Qy.$$

If $$y \in Y$$, let $$y = ( {y - Qy} ) + Qy$$, where $$y - Qy \in\operatorname{Ker} Q = \operatorname{Im} L$$, $$Qy \in \operatorname{Im} Q$$. It follows from $$\operatorname{Ker} Q = \operatorname{Im} L$$ and $${Q^{2}}y = Qy$$ that $$\operatorname{Im} Q \cap \operatorname{Im} L = \{ 0 \}$$. Then we get $$Y = \operatorname{Im} L \oplus\operatorname{Im} Q$$. Thus,

$$\dim\operatorname{Ker} L = \dim\operatorname{Im} Q = \operatorname{codim} \operatorname{Im} L = 2.$$

It implies that L is a Fredholm operator of index zero.

The last step is to prove that $${K_{P}}$$ is the inverse operator of $$L| {_{\operatorname{dom}L \cap\operatorname{Ker} P}}$$. In fact, for $$y \in\operatorname{Im} L$$, we have

$$L{K_{P}}y = D_{0 + }^{\alpha}I_{0 + }^{\alpha}y = y.$$
(3.6)

Additionally, for $$x \in\operatorname{dom}L \cap\operatorname{Ker} P$$, we have $$x ( 0 ) = x' ( 0 ) = 0$$ and $${K_{P}}Lx ( t ) = I_{0 + }^{\alpha}D_{0 + }^{\alpha}x ( t ) = x ( t ) + {c_{0}} + {c_{1}}t$$. With the boundary condition $$x ( 0 ) = x' ( 0 ) = 0$$, we get

$${K_{P}}Lx = x.$$
(3.7)

Combining (3.6) with (3.7), we obtain $${K_{P}} = { ( L| {_{\operatorname{dom}L \cap\operatorname{Ker} P}} ) ^{ - 1}}$$. The proof is complete. □

Theorem 3.1 is proved by the following three steps.

Step 1. Let

$${\Omega_{1}} = \bigl\{ {x \in\operatorname{dom}L \setminus \operatorname{Ker} L | {Lx = \lambda Nx}, \lambda\in ( {0,1} ) } \bigr\} .$$

For any $$x \in{\Omega_{1}}$$, $$x \notin\operatorname{Ker} L$$, we have $$Nx \in\operatorname{Im} L = \operatorname{Ker} Q$$, then $$QNx = 0$$. By (3.5), we get

\begin{aligned}& {T_{1}}Nx = \int_{0}^{\xi}{{{ ( {\xi- s} ) }^{\alpha}}} \phi_{p ( s ) }^{ - 1} \bigl( {I_{0 + }^{\beta}f \bigl( {s,x ( s ) ,x' ( s ) } \bigr) } \bigr) \,\mathrm{d}s = 0, \\& \begin{aligned} {T_{2}}Nx &= \alpha \int_{0}^{1} {{{ ( {1 - s} ) }^{ \alpha- 1}} \phi_{p ( s ) }^{ - 1} \bigl( {I_{0 + }^{\beta}f \bigl( {s,x ( s ) ,x' ( s ) } \bigr) } \bigr) } \,\mathrm{d}s \\ & \quad {} - m\alpha ( {\alpha- 1} ) \int_{0}^{1} {{{ ( {1 - s} ) } ^{\alpha- 2}} \phi_{p ( s ) }^{ - 1} \bigl( {I_{0 + }^{\beta }f \bigl( {s,x ( s ) ,x' ( s ) } \bigr) } \bigr) } \,\mathrm{d}s \\ & \quad {}- \sigma \int_{0}^{\eta}{{{ ( {\eta- s} ) }^{\alpha}} \phi_{p ( s ) }^{ - 1} \bigl( {I_{0 + }^{\beta}f \bigl( {s,x ( s ) ,x' ( s ) } \bigr) } \bigr) } \,\mathrm{d}s = 0. \end{aligned} \end{aligned}

From $$(\mathrm{H}_{2})$$ and $$(\mathrm{H}_{3})$$, there exist two constants $${\varepsilon_{1}},{\varepsilon_{2}} \in ( {0,1} )$$ such that $$\vert {x ( {{\varepsilon_{1}}} ) } \vert \le {\mathrm{B}}$$ and $$\vert {x' ( {{\varepsilon_{2}}} ) } \vert \le {\mathrm{D}}$$. Furthermore, by $$x ( t ) = I_{0 + }^{\alpha}D_{0 + }^{ \alpha}x ( t ) + {c_{0}} + {c_{1}}t$$, we get

$$x' ( t ) = I_{0 + }^{\alpha- 1}D_{0 + }^{\alpha}x ( t ) + {c_{1}} = \frac{1}{{\Gamma ( {\alpha- 1} ) }} \int _{0}^{t} {{{ ( {t - s} ) }^{\alpha- 2}}D_{0 + }^{\alpha}x ( s ) \,\mathrm{d}s} + {c_{1}}.$$

Let $$t = {\varepsilon_{2}}$$, then

$$x' ( \varepsilon_{2} ) = \frac{1}{{\Gamma ( {\alpha- 1} ) }} \int_{0}^{{\varepsilon_{2}}} {{{ ( {{\varepsilon_{2}} - s} ) }^{\alpha- 2}}D_{0 + }^{\alpha}x ( s ) \,\mathrm{d}s} + {c_{1}}.$$

Since $$\vert {x' ( {{\varepsilon_{2}}} ) } \vert \le {\mathrm{D}}$$, we have

\begin{aligned} \vert {{c_{1}}} \vert &\le \bigl\vert {x' ( {{\varepsilon_{2}}} ) } \bigr\vert + \frac{1}{{\Gamma ( {\alpha- 1} ) }} \int _{0}^{{\varepsilon_{2}}} {{{ ( {{\varepsilon_{2}} - s} ) } ^{\alpha- 2}} \bigl\vert {D_{0 + }^{\alpha}x ( s ) } \bigr\vert \,\mathrm{d}s} \\ &\le{\mathrm{D}} + \frac{{\varepsilon_{2}^{\alpha- 1}}}{ {\Gamma ( \alpha ) }} { \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{\infty}} \\ & \le{\mathrm{D}} + \frac{1}{{\Gamma ( \alpha ) }} { \bigl\Vert {D_{0 + }^{\alpha }x} \bigr\Vert _{\infty}}. \end{aligned}

Then

\begin{aligned} { \bigl\Vert {x'} \bigr\Vert _{\infty}} & \le\frac{1}{{\Gamma ( {\alpha- 1} ) }} \int_{0}^{t} {{{ ( {t - s} ) }^{\alpha - 2}} \bigl\vert {D_{0 + }^{\alpha}x ( s ) } \bigr\vert \,\mathrm{d}s} + \vert {{c_{1}}} \vert \\ &\le\frac{{{t^{\alpha- 1}}}}{{\Gamma ( \alpha ) }} { \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{ \infty}} + \mathrm{D} + \frac{1}{{\Gamma ( \alpha ) }} \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{\infty} \\ &\le\frac{2}{ {\Gamma ( \alpha ) }} { \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{\infty}} + \mathrm{D}. \end{aligned}

Let $$t = {\varepsilon_{1}}$$, then

$$x ( {{\varepsilon_{1}}} ) = \frac{1}{{\Gamma ( \alpha ) }} \int_{0}^{{\varepsilon_{1}}} {{{ ( {{\varepsilon_{1}} - s} ) }^{\alpha- 1}}D_{0 + }^{\alpha}x ( s ) \,\mathrm{d}s} + {c_{0}} + {c_{1}} {\varepsilon_{1}}.$$

From $$\vert {x ( {{\varepsilon_{1}}} ) } \vert \le{\mathrm{B}}$$, we have

\begin{aligned} \vert {{c_{0}}} \vert & \le \bigl\vert {x ( {{\varepsilon_{1}}} ) } \bigr\vert + \frac{1}{{\Gamma ( \alpha ) }} \int_{0}^{{\varepsilon_{1}}} {{{ ( {{\varepsilon_{1}} - s} ) } ^{\alpha- 1}} \bigl\vert {D_{0 + }^{\alpha}x ( s ) } \bigr\vert \,\mathrm{d}s} + \vert {{c_{1}}} \vert \\ &\le{\mathrm{B}} + \frac{ {\varepsilon_{1}^{\alpha}}}{{\Gamma ( {\alpha+ 1} ) }} { \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{\infty}} + \mathrm{D} + \frac{1}{ {\Gamma ( \alpha ) }} { \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{\infty}} \\ &\le{\mathrm{B}} + \mathrm{D} + \frac{{1 + \alpha}}{ {\Gamma ( {\alpha+ 1} ) }} { \bigl\Vert {D_{0 + }^{\alpha }x} \bigr\Vert _{\infty}}. \end{aligned}

Then

\begin{aligned} {\Vert {x} \Vert _{\infty}} & \le \frac{1}{{\Gamma ( \alpha ) }} \int_{0}^{t} {{{ ( {t - s} ) }^{\alpha- 1}} \bigl\vert {D_{0 + }^{\alpha}x ( s ) } \bigr\vert \,\mathrm{d}s} + \vert {{c_{0}}} \vert + \vert {{c_{1}}} \vert \\ &\le\frac{{{t^{\alpha }}}}{{\Gamma ( {\alpha+ 1} ) }} { \bigl\Vert {D_{0 + }^{\alpha }x} \bigr\Vert _{\infty}}+ {\mathrm{B}} + 2{\mathrm{D}} + \frac {1}{{\Gamma ( \alpha ) }} { \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{ \infty}} + \frac{{1 + \alpha}}{{\Gamma ( {\alpha+ 1} ) }} { \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{\infty}} \\ &\le{\mathrm{B}} + 2{\mathrm{D }}+\frac{{2 ( {\alpha+ 1} ) }}{{\Gamma ( {\alpha+ 1} ) }} { \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{\infty}}. \end{aligned}

Furthermore, by $$Lx = \lambda Nx$$, we have

\begin{aligned}& D_{0 + }^{\alpha}x ( t ) = \lambda \phi_{p ( t ) } ^{ - 1} \bigl( {I_{0 + }^{\beta}f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) } \bigr), \\& \begin{aligned} {\phi_{p ( t ) }} \bigl( {D_{0 + }^{\alpha}x ( t ) } \bigr) &= {\phi_{p ( t ) }} \bigl( {\lambda\phi_{p ( t ) }^{ - 1} \bigl( {I_{0 + }^{ \beta}f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) } \bigr) } \bigr) \\ &= {\phi_{p ( t ) }} ( \lambda ) I_{0 + }^{\beta}f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) \\ &= {\lambda^{p ( t ) - 1}} I_{0 + }^{\beta}f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) . \end{aligned} \end{aligned}

Combining $$(\mathrm{H}_{1})$$ and $$\lambda\in ( {0,1} )$$, we have

\begin{aligned} { \bigl\vert {D_{0 + }^{\alpha}x ( t ) } \bigr\vert ^{p ( t ) - 1}} & \le\frac{1}{{\Gamma ( \beta ) }} \int_{0}^{t} {{{ ( {t - s} ) }^{\beta- 1}} \bigl\vert {f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) } \bigr\vert \,\mathrm{d}s} \\ & \le\frac{1}{{\Gamma ( {\beta+ 1} ) }} \bigl( {{a_{1}} + {b_{1}}\Vert x \Vert _{\infty}^{\theta- 1} + {c_{1}} \bigl\Vert {x'} \bigr\Vert _{\infty}^{ \theta- 1}} \bigr) \\ &\le \frac{1}{{\Gamma ( {\beta+ 1} ) }} \biggl[ {{a_{1}} + {b_{1}} {{ \biggl( {\mathrm{B} + 2{\mathrm{D }}+\frac{{2 ( {\alpha+ 1} ) }}{{\Gamma ( {\alpha+ 1} ) }}{{ \bigl\Vert {D_{0 + } ^{\alpha}x} \bigr\Vert }_{\infty}}} \biggr) }^{\theta- 1}}} \\ & \quad {} + {c_{1}} {{ \biggl( {\frac{2}{{\Gamma ( \alpha ) }} {{ \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert }_{\infty}} + \mathrm {D}} \biggr) } ^{\theta- 1}} \biggr] . \end{aligned}

According to $${ ( {\vert a \vert + \vert b \vert } ) ^{p}} \le{2^{p}} ( {{{\vert a \vert }^{p}} + {{\vert b \vert } ^{p}}} )$$, $$p > 0$$, we get

$${ \bigl\vert {D_{0 + }^{\alpha}x ( t ) } \bigr\vert ^{p ( t ) - 1}} \le{A_{1}} + {A_{2}} \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{ \infty}^{\theta- 1},$$

where

$${A_{1}} = \frac{{{a_{1}} + {b_{1}}{{ ( {2\mathrm{B} + 4{\mathrm {D}}} ) } ^{\theta- 1}} + {c_{1}}{{ ( {2{\mathrm{D}}} ) }^{\theta- 1}}}}{ {\Gamma ( {\beta+ 1} ) }}, \quad\quad {A_{2}} = \frac{{{b_{1}}{{ ( {\frac{{4 ( {\alpha+ 1} ) }}{{\Gamma ( {\alpha+ 1} ) }}} ) }^{\theta- 1}} + {c_{1}}{{ ( {\frac{4}{{\Gamma ( \alpha ) }}} ) }^{\theta- 1}}}}{{\Gamma ( {\beta+ 1} ) }}.$$

Hence, we have

$${ \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{\infty}} \le {2^{\frac{1}{ {p ( t ) - 1}}}} \bigl( {A_{1}^{\frac{1}{{p ( t ) - 1}}} + A_{2}^{\frac{1}{{p ( t ) - 1}}} \bigl\Vert {D_{0 + }^{ \alpha}x} \bigr\Vert _{\infty}^{\frac{{\theta- 1}}{{p ( t ) - 1}}}} \bigr) .$$

It follows from $$\frac{{\theta- 1}}{{p ( t ) - 1}} \in ( {0,1} ]$$ and $${x^{k}} \le x + 1$$, $$x > 0$$, $$k \in ( {0,1} ]$$ that

$${ \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert _{\infty}} \le{ ( {2 {A_{1}}} ) ^{\frac{1}{{p ( t ) - 1}}}} + { ( {2 {A_{2}}} ) ^{\frac{1}{{p ( t ) - 1}}}} \bigl( {{{ \bigl\Vert {D_{0 + }^{\alpha}x} \bigr\Vert }_{\infty}} + 1} \bigr) .$$

By (3.3), there exists a constant $${\mathrm{M}_{1}} > 0$$ such that $${\Vert {D_{0 + }^{\alpha}x} \Vert _{\infty}} \le {\mathrm{M} _{1}}$$. Thus,

$${\Vert x \Vert _{\infty}} \le{\mathrm{B}} + 2{\mathrm{D }}+ \frac{ {2 ( {\alpha+ 1} ) }}{{\Gamma ( {\alpha+ 1} ) }} {\mathrm{M}_{1}}: = {\mathrm{M}_{2}}, \quad\quad { \bigl\Vert {x'} \bigr\Vert _{\infty}} \le\mathrm{D} + \frac{2}{{\Gamma ( \alpha ) }}{\mathrm {M}_{1}}: = \mathrm{M}_{3},$$

this proves that $${\Omega_{1}}$$ is bounded.

Step 2. Let

$${\Omega_{2}} = \{ x| x \in\operatorname{Ker} L, Nx \in \operatorname{Im} L \} .$$

If $$x ( t ) \in{\Omega_{2}}$$, then $$x ( t ) = {c_{0}} + {c_{1}}t$$, $${c_{0}},{c_{1}} \in{\mathbb{R}}$$ and $${Nx \in\operatorname{Im} L}$$. Thus, we have $${T_{1}}Nx = {T _{2}}Nx = 0$$. When it is combined with $$(\mathrm{H}_{3})$$, we get $$\vert {x' ( t ) } \vert = \vert {c_{1}} \vert \le {\mathrm{D}}$$. According to $$(\mathrm{H}_{2})$$, there exists $${\varepsilon _{1}} \in ( {0,1} )$$ such that $$\vert {x ( {{\varepsilon_{1}}} ) } \vert = \vert {{c_{0}} + {c_{1}}{\varepsilon _{1}}} \vert \le{\mathrm{B}}$$. It is clear that $$\vert {{c_{0}}} \vert \le{\mathrm{B} + \mathrm{D}}$$. So $${\Vert x \Vert _{\infty}} \le \mathrm{B} + 2\mathrm{D}: = {\mathrm{M}_{3}}$$. Thus, $${\Omega_{2}}$$ is bounded.

Step 3. Let

$${\Omega_{3}} = \bigl\{ {x \in\operatorname{Ker} L, \lambda Jx + ( {1 - \lambda} ) QNx = 0, \lambda\in [ {0,1} ] } \bigr\} ,$$

where $$J:\operatorname{Ker} L \to\operatorname{Im} Q$$ is a homeomorphism mapping:

$$J ( {{c_{0}} + {c_{1}}t} ) = \frac{1}{\Lambda} ( {{\Lambda _{4}} {c_{0}} - {\Lambda_{3}} {c_{1}}} ) + \frac{1}{\Lambda} ( { - {\Lambda_{2}} {c_{0}} + { \Lambda_{1}} {c_{1}}} ) t, \quad {c_{0}},{c_{1}} \in{\mathbb{R}}.$$

Let $$x \in{\Omega_{3}}$$, then $$x ( t ) = {c_{0}} + {c_{1}}t$$, $${c_{0}},{c_{1}} \in{\mathbb{R}}$$ and $$\lambda J ( {{c_{0}} + {c_{1}}t} ) + ( {1 - \lambda} ) QN ( {{c_{0}} + {c_{1}}t} ) = 0$$, i.e.,

$$\lambda J ( {{c_{0}} + {c_{1}}t} ) + ( {1 - \lambda} ) \bigl[ {{Q_{1}}N ( {{c_{0}} + {c_{1}}t} ) + {Q_{2}}N ( {{c_{0}} + {c_{1}}t} ) t} \bigr] = 0.$$

Then

\begin{aligned}& \lambda ( {{\Lambda_{4}} {c_{0}} - {\Lambda_{3}} {c_{1}}} ) + ( {1 - \lambda} ) \bigl[ {{\Lambda_{4}} {T_{1}}N ( {{c_{0}} + {c_{1}}t} ) - { \Lambda_{3}} {T_{2}}N ( {{c_{0}} + {c_{1}}t} ) } \bigr] = 0, \\& \lambda ( { - {\Lambda_{2}} {c_{0}} + { \Lambda_{1}} {c_{1}}} ) + ( {1 - \lambda} ) \bigl[ { - { \Lambda_{2}} {T_{1}}N ( {{c_{0}} + {c_{1}}t} ) + {\Lambda_{1}} {T_{2}}N ( {{c_{0}} + {c_{1}}t} ) } \bigr] = 0. \end{aligned}

Thus,

\begin{aligned}& \lambda{c_{0}} + ( {1 - \lambda} ) {T_{1}}N ( {{c_{0}} + {c_{1}}t} ) = 0, \end{aligned}
(3.8)
\begin{aligned}& \lambda{c_{1}} + ( {1 - \lambda} ) {T_{2}}N ( {{c_{0}} + {c_{1}}t} ) = 0. \end{aligned}
(3.9)

According to (3.8) and the first part of $$(\mathrm{H}_{2})$$, we have $$\vert {{c_{0}}} \vert \le{\mathrm{B}}$$. Otherwise, if $$\vert {{c _{0}}} \vert > \mathrm{B}$$, by the first part of $$(\mathrm {H}_{2})$$, we have

$$\lambda c_{0}^{2} + ( {1 - \lambda} ) {c_{0}} {T_{1}}N ( {{c_{0}} + {c_{1}}t} ) > 0,$$

which is contradictory to (3.8). Similarly, by (3.9) and the first part of $$(\mathrm{H}_{3})$$, we have $${\vert {{c_{1}}} \vert \le {\mathrm{D}}}$$. Otherwise, if $$\vert {{c_{1}}} \vert > \mathrm{D}$$, by the first part of $$(\mathrm{H}_{3})$$, we have

$$\lambda c_{1}^{2} + ( {1 - \lambda} ) {c_{1}} {T_{2}}N ( {{c_{0}} + {c_{1}}t} ) > 0,$$

which is contradictory to (3.9). Hence, $${\Omega_{3}}$$ is bounded.

Let

$$\Omega= \bigl\{ {x \in X, {{\Vert x \Vert }_{\infty}} < \max \{ {{ \mathrm{M}_{2}} ,{\mathrm{M}_{3}}} \} + 1} \bigr\} .$$

As indicated by Lemma 3.2, L is a Fredholm operator of index zero. Based on the Arzela-Ascoli theorem, we obtain that N is L-compact on Ω̅. Then, by Step 1 and Step 2, we get

$$( {a_{1}} )$$ :

$$Lx \ne\lambda Nx$$, $$( {x,\lambda} ) \in [ { ( {\operatorname{dom}L\backslash\operatorname {Ker} L} ) \cap\partial\Omega} ] \times ( {0,1} )$$.

$$( {a_{2}} )$$ :

$$Nx \notin\operatorname{Im} L$$, $$x \in\operatorname{Ker} L \cap\partial\Omega$$.

Let

$$H ( {x,\lambda} ) = \lambda J ( x ) + ( {1 - \lambda} ) QNx.$$

According to Step 3, we have $$H ( {x,\lambda} ) \ne0$$ for $$x \in\operatorname{Ker} L \cap\partial\Omega$$, then

\begin{aligned} \deg ( QN| {_{\operatorname{Ker} L}, \Omega\cap\operatorname {Ker} L, 0} ) &= \deg \bigl( {H ( { \cdot,0} ) , \Omega\cap\operatorname{Ker} L, 0} \bigr) \\ &= \deg \bigl( {H ( { \cdot,1} ) , \Omega\cap\operatorname{Ker} L, 0} \bigr) \\ & = \deg ( {I, \Omega\cap\operatorname{Ker} L, 0} ) \ne0. \end{aligned}

Condition $$(a_{3})$$ of Theorem 2.1 is thus met. Through Theorem 2.1, we get that $$Lx=Nx$$ has at least one fixed point in $$\operatorname{dom}L \cap \overline{ \Omega}$$. Hence, BVP (1.1) has at least one solution.

### Remark 3.1

The proof process would be similar to that of Step 3 if the second inequality of both $$(\mathrm{H}_{2})$$ and $$(\mathrm{H}_{3})$$, or the first of $$(\mathrm{H}_{2})$$ and the second of $$(\mathrm {H}_{3})$$, or the first of $$(\mathrm{H}_{3})$$ and the second of $$(\mathrm{H}_{2})$$ hold. It is hence omitted herein.

### Corollary 3.1

Assume that the conditions of Theorem  3.1 hold. If $$\eta\le\xi$$, BVP (1.1) has at least one solution.

### Proof

Here we just need to verify the condition $$C \neq0$$.

By (1.2), we have

\begin{aligned} C & = ( {\alpha+ 2} ) \bigl[ {1 - m ( {\alpha+ 1} ) } \bigr] + \xi ( {\alpha+ 1} ) ( {m\alpha- 1} ) + {\eta^{\alpha}} ( {\xi- \eta} ) \\ &= ( {\alpha+ 2} ) + ( {\alpha+ 1} ) \biggl( {\xi\alpha- \alpha- 2 - \frac{1}{2}\xi\alpha\eta+ \frac {1}{2}\alpha\eta+ \eta- \xi} \biggr) + {\eta^{\alpha}} ( {\xi - \eta} ) \\ &= 2 ( {\alpha+ 1} ) - \alpha+ ( {\alpha+ 1} ) \biggl( {\xi\alpha- \alpha- 2 - \frac{1}{2}\xi\alpha\eta+ \frac {1}{2}\alpha\eta+ \eta- \xi} \biggr) + {\eta^{\alpha}} ( {\xi - \eta} ) \\ & = ( {\alpha+ 1} ) \biggl[ {\alpha ( {\xi- 1} ) \biggl( {1 - \frac{1}{2} \eta} \biggr) + ( {\eta- \xi} ) } \biggr] + \bigl[ {{\eta^{\alpha}} ( {\xi- \eta} ) - \alpha} \bigr] . \end{aligned}

Obviously, by $$1 < \alpha\le2$$ and $$0 < \eta\le\xi< 1$$, we obtain $$C < 0$$. The proof is complete. □

## Example

### Example 4.1

Consider the following BVP:

$$\textstyle\begin{cases} {}^{C}D_{0 + }^{\frac{2}{3}}{\phi_{ ( {{t^{2}} + 2} ) }} ( {{}^{C}D_{0 + }^{\frac{3}{2}}x ( t ) } ) = \frac{1}{ {10}} + \frac{1}{{30}}\sin ( {x ( t ) } ) + \frac{1}{ {90}}\sin ( {x' ( t ) } ) , \quad t \in ( {0,1} ) , \\ x ( 0 ) + \frac{1}{4}x' ( 0 ) = 2\int_{0}^{ \frac{1}{2}} {x ( t ) \,\mathrm{d}t} , \\ x ( 1 ) - \frac{3}{4}x' ( 1 ) = 2\int_{0}^{ \frac{1}{2}} {x ( t ) \,\mathrm{d}t} , \\ {}^{C}D_{0 + }^{\alpha}x ( 0 ) = 0, \end{cases}$$

where $$p ( t ) = {t^{2}} + 2$$, $$\alpha= \frac{3}{2}$$, $$\beta= \frac{2}{3}$$, $$\theta= 2$$, $$f ( {t,x ( t ) ,x' ( t ) } ) = \frac{1}{{10}} + \frac{1}{{30}}\sin ( {x ( t ) } ) + \frac{1}{{90}}\sin ( {x' ( t ) } )$$, $${a_{1}} = \frac{1}{{10}}$$, $${b_{1}} = \frac{1}{ {30}}$$, $${c_{1}} = \frac{1}{{90}}$$, $$b = \frac{1}{4}$$, $$\xi= \frac{1}{2}$$, $$\eta= \frac{1}{2}$$, $$m = \frac{3}{4}$$, $$\gamma= 2$$, $$\sigma= 2$$, $${{P_{L}} = 2}$$. It is easy to verify that (1.2) and (1.3) hold. Let $$\mathrm{B} = 30$$, $$\mathrm{D} = 90$$, $$\mathrm{C} = - \frac{93}{32} <0$$, if $$x ( t ) > 30$$, $$x' ( t ) > 90$$, then $$f ( {t,x ( t ) ,x' ( t ) } ) > 0$$. Clearly, $$(\mathrm{H}_{1})$$ of Theorem 3.1 holds. By (3.3), we get

\begin{aligned}& \frac{{{4^{\theta- 1}} ( {{b_{1}}{{ ( {\alpha+ 1} ) } ^{\theta- 1}} + {c_{1}}{\alpha^{\theta- 1}}} ) }}{{\Gamma ( {\beta+ 1} ) {{ ( {\Gamma ( {\alpha+ 1} ) } ) }^{\theta- 1}}}} = \frac{2}{{5\Gamma ( {\frac{5}{3}} ) \Gamma ( {\frac{5}{2}} ) }} < \frac{1}{2}, \\& Nx ( t ) = \phi_{p ( t ) }^{ - 1} \bigl( {I_{0 + } ^{\beta}f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) } \bigr) = \phi_{ ( {{t^{2}} + 2} ) }^{-1} \bigl( {I_{0 + }^{\frac{2}{3}}f \bigl( {t,x ( t ) ,x' ( t ) } \bigr) } \bigr) > 0. \end{aligned}

So, $$(\mathrm{H}_{2})$$ of Theorem 3.1 holds. Furthermore, by the definition of $${T_{2}}y$$, we have

\begin{aligned} {T_{2}}Nu ( t ) & = \alpha \int_{0}^{1} {{{ ( {1 - s} ) } ^{\alpha- 1}}Nu ( s ) } \,\mathrm{d}s - m\alpha ( {\alpha - 1} ) \int_{0}^{1} {{{ ( {1 - s} ) }^{\alpha- 2}}Nu ( s ) } \,\mathrm{d}s \\ &\quad{}- \sigma \int_{0}^{\eta}{{{ ( {\eta- s} ) }^{\alpha}}Nu ( s ) } \,\mathrm{d}s \\ &= Nu ( s ) \biggl( {1 - m\alpha- \frac{{\sigma{\eta^{\alpha+ 1}}}}{ {\alpha+ 1}}} \biggr) \\ &= Nu ( s ) \biggl( { - \frac{1}{8} - \frac{1}{{5\sqrt{2} }}} \biggr) < 0. \end{aligned}

Thus, $$(\mathrm{H}_{3})$$ of Theorem 3.1 holds. Hence, there exists at least one solution.

## Conclusions

The solvability for fractional Sturm-Liouville BVP with $$p ( t )$$-Laplacian operator is discussed in the article by using Mawhin’s continuation theorem, and the existence of solutions has been obtained (see Theorem 3.1). The kernel space is expanded to higher dimensions on condition that $$\dim\operatorname{Ker}L=2$$, and the system is more complex in comparison with the case when $$\dim\operatorname{Ker}L=1$$. Moreover, when $$p(t)=p$$, the $$p ( t )$$-Laplacian operator will evolve into the famous p-Laplacian operator. Therefore, our results would develop previous findings to some extent.

## References

1. Torvik, P, Bagley, RL: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 725-728 (1984)

2. Matignon, D: Stability results for fractional differential equations with applications to control processing. In: Comput. Eng. Syst. Appl, vol. 2, pp. 963-968 (1996)

3. Bai, J, Feng, X: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16, 2492-2502 (2007)

4. Magin, RL: Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59, 1586-1593 (2010)

5. Leszczynski, JS, Blaszczyk, T: Modeling the transition between stable and unstable operation while emptying a silo. Granul. Matter 13, 429-438 (2011)

6. Nazari, D, Shahmorad, S: Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions. J. Comput. Appl. Math. 234, 883-891 (2010)

7. Ahmad, B, Agarwal, RP: Some new versions of fractional boundary value problems with slit-strips conditions. Bound. Value Probl. 2014, 175 (2014)

8. Kosmatov, N, Jiang, W: Second-order functional problems with a resonance of dimension one. Differ. Equ. Appl. 3, 349-365 (2016)

9. Jiang, W, Kosmatov, N: Solvability of a third-order differential equation with functional boundary conditions at resonance. Bound. Value Probl. 2017, 81 (2017)

10. Bai, Z, Liu, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005)

11. Luo, H, Hu, W: Existence and uniqueness of positive solutions for boundary value problem of nonlinear fractional differential equation. J. Yili Norm. Univ. 24, 1-7 (2012)

12. Kosmatov, N: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 2010, 135 (2010)

13. Wei, Z, Dong, W, Chen, J: Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative. Nonlinear Anal. 73, 3232-3238 (2010)

14. Bai, Z, Zhang, Y: Solvability of fractional three-point boundary value problems with nonlinear growth. Appl. Math. Comput. 218, 1719-1725 (2011)

15. Sun, J, Liu, Y: Existence of solutions for fractional differential systems with antiperiodic boundary conditions. Comput. Math. Appl. 64, 1557-1566 (2012)

16. Hu, J, Lu, G, Zhang, S: Lyapunov stability theorem about fractional system without and with delay. Commun. Nonlinear Sci. Numer. Simul. 20, 905-913 (2015)

17. Kosmatov, N, Jiang, W: Resonant functional problems of fractional order. Chaos Solitons Fractals 91, 573-579 (2016)

18. Rakkiyappan, R, Cao, J, Velmurugan, G: Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans. Neural Netw. Learn. Syst. 26, 84-97 (2015)

19. Wang, J, Xiang, H: Upper and lower solutions method for a class of singular fractional boundary value problems with p-Laplacian operator. Abstr. Appl. Anal. 2010, 1 (2010)

20. Chen, T, Liu, W: Anti-periodic boundary value problem for fractional differential equation with p-Laplacian operator. Appl. Math. Lett. 25, 1671-1675 (2012)

21. Chen, T, Liu, W, Hu, Z: A boundary value problem for fractional differential equation with p-Laplacian operator at resonance. Nonlinear Anal. 75, 3210-3217 (2012)

22. Liu, Z, Lu, L: A class of BVPs for nonlinear fractional differential equations with p-Laplacian operator. Electron. J. Qual. Theory Differ. Equ. 2010, 70 (2010)

23. Tang, X, Yan, C, Liu, Q: Existence of solutions of two-point boundary value problems for fractional p-Laplace differential equations at resonance. J. Appl. Math. Comput. 41, 119-131 (2013)

24. Hu, Z, Liu, W, Liu, J: Existence of solutions of fractional differential equation with p-Laplacian operator at resonance. Abstr. Appl. Anal. 2014, 1 (2014)

25. Mahmudov, NI, Unul, S: Existence of solutions of fractional boundary value problems with p-Laplacian operator. Bound. Value Probl. 2015, 99 (2015)

26. Shen, T, Liu, W: Existence of solutions for fractional integral boundary value problems with $$p(t)$$-Laplacian operator. J. Nonlinear Sci. Appl. 9, 5000-5010 (2016)

27. Fan, X, Zhang, Q, Zhao, D: Eigenvalues of $$p(x)$$-Laplacian Dirichlet problem. J. Math. Anal. Appl. 302, 306-317 (2005)

28. Zhang, Q: Existence of solutions for weighted $$p(r)$$-Laplacian system boundary value problems. Nonlinear Anal. 71, 3596-3611 (2007)

29. Zhang, Q, Wang, Y, Qiu, Z: Existence of solutions and boundary asymptotic behavior of $$p(r)$$-Laplacian equation multi-point boundary value problems. Nonlinear Anal. 72, 2950-2973 (2010)

30. Shen, T, Liu, W, Zhao, R: Fractional boundary value problems with $$p(t)$$-Laplacian operator. Adv. Differ. Equ. 2016, 118 (2016)

31. Mawhin, J: Topological degree and boundary value problems for nonlinear differential equations. In: Topological Methods for Ordinary Differential Equations. Springer, Berlin (1993)

32. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

33. Podlubny, I: Fractional Differential Equations. Academic Press, New York (1999)

## Acknowledgements

This work is supported by the National Natural Science Foundation of China (11271364).

## Author information

Authors

### Corresponding author

Correspondence to Wenbin Liu.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests. 