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Some existence results on boundary value problems for fractional p-Laplacian equation at resonance

Boundary Value Problems20162016:51

https://doi.org/10.1186/s13661-016-0566-y

  • Received: 9 November 2015
  • Accepted: 17 February 2016
  • Published:

Abstract

Two boundary value problems of the fractional p-Laplacian equation at resonance are considered in this paper. By using the continuation theorem due to Ge, we obtain some existence results for such boundary value problems.

Keywords

  • fractional differential equation
  • p-Laplacian operator
  • boundary value problem
  • continuation theorem
  • resonance

MSC

  • 34A08
  • 34B15

1 Introduction

Consider the following fractional p-Laplacian equation:
$$\begin{aligned} D_{0^{+}}^{\beta}\phi_{p} \bigl(D_{0^{+}}^{\alpha}x(t)\bigr)=f\bigl(t,x(t),D_{0^{+}}^{\alpha}x(t)\bigr), \quad t\in[0,1], \end{aligned}$$
(1.1)
with the boundary value conditions either
$$\begin{aligned} x(0)=x(1), \qquad D_{0^{+}}^{\alpha}x(0)=0 , \end{aligned}$$
(1.2)
or
$$\begin{aligned} x(0)=x(1), \qquad D_{0^{+}}^{\alpha}x(1)=0, \end{aligned}$$
(1.3)
where \(0<\alpha\), \(\beta\leq1\), \(\phi_{p}(s)=\vert s\vert ^{p-2}s\) (\(p>1\)), \(D_{0^{+}}^{\alpha}\) is a Caputo fractional derivative, and \(f:[0,1]\times \mathbb{R}^{2}\rightarrow\mathbb{R}\) is a continuous function.

In the last two decades, the theory of fractional calculus has gained popularity due to its wide applications in various fields of engineering and the sciences [18]. Moreover, the p-Laplacian equations often exist in non-Newtonian fluid theory, nonlinear elastic mechanics, and so on.

Recently, many important results on the p-Laplacian equations or the fractional differential equations have been given. We refer the reader to [931]. However, as far as we know, there is little work about boundary value problems (BVPs for short) for the fractional differential equations with p-Laplacian operator at resonance.

Note that BVP (1.1)-(1.2) (or BVP (1.1)-(1.3)) happens to be at resonance because its associated homogeneous BVP
$$\begin{aligned} \textstyle\begin{cases} D_{0^{+}}^{\beta}\phi_{p}(D_{0^{+}}^{\alpha}x(t))=0, \quad t\in[0,1],\\ x(0)=x(1),\qquad D_{0^{+}}^{\alpha}x(0)=0 \quad (\mbox{or }x(0)=x(1), D_{0^{+}}^{\alpha}x(1)=0), \end{cases}\displaystyle \end{aligned}$$
has a solution \(x(t)=c\), \(\forall c\in\mathbb{R}\).

The rest of this paper is organized as follows. Section 2 contains some definitions, lemmas and notations. In Section 3, some related lemmas are stated and proved which are useful in the proof of our main results. In Section 4 and Section 5, in view of the continuation theorem due to Ge, we establish two theorems about the existence of solutions for BVP (1.1)-(1.2) (Theorem 4.1) and BVP (1.1)-(1.3) (Theorem 5.1).

2 Preliminaries

We give here some definitions and lemmas about the fractional calculus.

Definition 2.1

[32]

The Riemann-Liouville fractional integral operator of order \(\alpha>0\) of a function \(x:(0,+\infty )\rightarrow\mathbb{R}\) is given by
$$\begin{aligned} I_{0^{+}}^{\alpha}x(t)=\frac{1}{\Gamma(\alpha)} \int_{0}^{t} (t-s)^{\alpha-1}x(s)\,ds, \end{aligned}$$
provided that the right side integral is pointwise defined on \((0,+\infty)\).

Definition 2.2

[32]

The Caputo fractional derivative of order \(\alpha>0\) of a continuous function \(x:(0,+\infty)\rightarrow \mathbb{R}\) is given by
$$\begin{aligned} D_{0^{+}}^{\alpha}x(t) =&I_{0^{+}}^{n-\alpha} \frac{d^{n}x(t)}{d t^{n}} \\ =&\frac{1}{\Gamma(n-\alpha)} \int_{0}^{t}(t-s)^{n-\alpha-1}x^{(n)}(s)\,ds, \end{aligned}$$
where n is the smallest integer greater than or equal to α, provided that the right side integral is pointwise defined on \((0,+\infty)\).

Lemma 2.1

[8]

Let \(\alpha>0\). Assume that \(x,D_{0^{+}}^{\alpha}x\in L([0,1],\mathbb{R})\). Then the following equality holds:
$$\begin{aligned} I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}x(t)=x(t)+c_{0}+c_{1}t+ \cdots+c_{n-1}t^{n-1}, \end{aligned}$$
where \(c_{i}\in{\mathbb{R}}\), \(i=0,1,\ldots,n-1\), and n is the smallest integer greater than or equal to α.

Lemma 2.2

[33]

For any \(u,v\geq0\),
$$\begin{aligned}& \phi_{p}(u+v)\leq\phi_{p}(u)+\phi_{p}(v), \quad \textit{if } p< 2; \\& \phi_{p}(u+v)\leq2^{p-2}\bigl(\phi_{p}(u)+ \phi_{p}(v)\bigr), \quad \textit{if } p\geq2. \end{aligned}$$

Next we introduce an extension of Mawhin’s continuation theorem [34, 35] which allows us to deal with the more general abstract operator equations, such as BVPs of p-Laplacian equations.

Let X and Z be Banach spaces with norms \(\Vert \cdot \Vert _{X}\) and \(\Vert \cdot \Vert _{Z}\), respectively.

Definition 2.3

[35]

A continuous operator \(M:\operatorname {dom}M\cap X\rightarrow Z\) is said to be a quasi-linear operator if
  1. (1)

    \(\operatorname {Im}M=M(\operatorname {dom}M\cap X)\) is a closed subset of Z,

     
  2. (2)

    \(\operatorname {Ker}M=\{x\in \operatorname {dom}M\cap X|Mx=0\}\) is linearly homeomorphic to \(\mathbb{R}^{n}\) with \(n<\infty\).

     

Definition 2.4

[35]

Let \(Z_{1}\) be a subspace of Z. An operator \(Q:Z\rightarrow Z_{1}\) is said to be a semi-projector provided that
  1. (1)

    \(Q^{2}z=Qz\), \(\forall z\in Z\),

     
  2. (2)

    \(Q(\lambda z)=\lambda Qz\), \(\forall z\in Z\), \(\lambda\in\mathbb{R}\).

     

Set \(X_{1}=\operatorname {Ker}M\) and let \(X_{2}\) be the complement space of \(X_{1}\) in X, then \(X=X_{1}\oplus X_{2}\). Suppose \(Z_{1}\) is a subspace of Z and \(Z_{2}\) is the complement space of \(Z_{1}\) in Z such that \(Z=Z_{1}\oplus Z_{2}\). Let \(P:X\rightarrow X_{1}\) be a projector and \(Q:Z\rightarrow Z_{1}\) a semi-projector, and \(\Omega\subset X\) an open bounded set with the origin \(\theta\in\Omega\).

Definition 2.5

[35]

A continuous operator \(N_{\lambda}:\overline{\Omega}\rightarrow Z\), \(\lambda\in[0,1]\) is said to be M-compact in Ω̅ if there is a vector subspace \(Z_{1}\) of Z with \(\dim Z_{1}=\dim X_{1}\), and an operator \(R:\overline {\Omega}\times[0,1]\rightarrow X_{2}\) being continuous and compact such that
$$\begin{aligned}& (I-Q)N_{\lambda}(\overline{\Omega})\subset \operatorname {Im}M \subset(I-Q)Z, \end{aligned}$$
(2.1)
$$\begin{aligned}& QN_{\lambda}x=\theta, \quad \lambda\in(0,1) \quad \Leftrightarrow\quad QNx=\theta, \end{aligned}$$
(2.2)
$$\begin{aligned}& R(\cdot,0) \mbox{ is the zero operator}\quad \mbox{and}\quad R(\cdot, \lambda)|_{\sum _{\lambda}}=(I-P)|_{\sum_{\lambda}}, \end{aligned}$$
(2.3)
$$\begin{aligned}& M\bigl(P+R(\cdot,\lambda)\bigr)=(I-Q)N_{\lambda}, \end{aligned}$$
(2.4)
where \(\lambda\in[0,1]\), \(N=N_{1}\), and \(\sum_{\lambda}=\{x\in\overline{\Omega }|Mx=N_{\lambda}x\}\).

Lemma 2.3

[35]

Suppose \(M:\operatorname {dom}M\cap X\rightarrow Z\) is a quasi-linear operator and \(N_{\lambda}:\overline{\Omega}\rightarrow Z\), \(\lambda\in[0,1]\) is M-compact in Ω̅. In addition, if
(C1): 

\(Mx\neq N_{\lambda}x\) for every \((x,\lambda)\in[(\operatorname {dom}M\setminus \operatorname {Ker}M)\cap\partial\Omega]\times(0,1)\);

(C2): 

\(QNx\neq0\) for every \(x\in \operatorname {Ker}M\cap\partial\Omega\);

(C3): 

\(\deg \{JQN,\Omega\cap \operatorname {Ker}M,0\}\neq0\),

where \(N=N_{1}\) and \(J:Z_{1}\rightarrow X_{1}\) is a homeomorphism with \(J(\theta)=\theta\), then the abstract equation \(Mx=Nx\) has at least one solution in \(\operatorname {dom}M\cap\overline{\Omega}\).

We set \(Z=C([0,1],\mathbb{R})\) with the norm \(\Vert z\Vert _{0}=\max_{t\in [0,1]}\vert z(t)\vert \), and \(X=\{x\in Z|D_{0^{+}}^{\alpha}{x\in Z}, x(0)=x(1),D_{0^{+}}^{\alpha}x(0)=0\}\), \(X^{1}=\{x\in Z|D_{0^{+}}^{\alpha}x\in Z,x(0)=x(1),D_{0^{+}}^{\alpha}x(1)=0\}\) with the norm \(\Vert x\Vert _{X}=\max\{\Vert x\Vert _{0},\Vert D_{0^{+}}^{\alpha}x \Vert _{0}\}\). By using linear functional analysis theory, we can prove X, \(X^{1}\) are Banach spaces.

3 Related lemmas

We will give some lemmas that are useful in the proof of our main results.

Define the operator \(M:\operatorname {dom}M\cap X\rightarrow Z\) by
$$\begin{aligned} Mx=D_{0^{+}}^{\beta}\phi_{p} \bigl(D_{0^{+}}^{\alpha}x\bigr), \end{aligned}$$
(3.1)
where \(\operatorname {dom}M=\{x\in X|D_{0^{+}}^{\beta}\phi_{p}(D_{0^{+}}^{\alpha}x)\in Z\} \). For \(\lambda\in[0,1]\), we define \(N_{\lambda}:X\rightarrow Z\) by
$$\begin{aligned} N_{\lambda}x(t)=\lambda f\bigl(t,x(t),D_{0^{+}}^{\alpha}x(t)\bigr), \quad \forall t\in[0,1]. \end{aligned}$$
(3.2)
Then BVP (1.1)-(1.2) is equivalent to the equation
$$\begin{aligned} Mx=Nx,\quad x\in \operatorname {dom}M, \end{aligned}$$
where \(N=N_{1}\).

Lemma 3.1

The operator M, defined by (3.1), is a quasi-linear operator.

Proof

The proof will be given in the following two steps.

Step 1. KerM is linearly homeomorphic to \(\mathbb{R}\).

From Lemma 2.1, the homogeneous equation \(D_{0^{+}}^{\beta}\phi _{p}(D_{0^{+}}^{\alpha}x(t))=0\) has the following solutions:
$$\begin{aligned} x(t)=d_{2}+\frac{\phi_{q}(d_{1})}{\Gamma(\alpha+1)}t^{\alpha}, \quad d_{1},d_{2}\in\mathbb{R}. \end{aligned}$$
Thus, by the boundary value condition \(D_{0^{+}}^{\alpha}x(0)=0\), one has
$$\begin{aligned} \operatorname {Ker}M=\bigl\{ x\in X|x(t)=d, \forall t\in[0,1],d\in \mathbb{R}\bigr\} . \end{aligned}$$
Obviously, \(\operatorname {Ker}M\simeq\mathbb{R}\).

Step 2. ImM is a closed subset of Z.

Take \(x\in \operatorname {dom}M\) and consider the equation \(D_{0^{+}}^{\beta}\phi _{p}(D_{0^{+}}^{\alpha}x(t))=z(t)\). Then we have \(z\in Z\) and
$$\begin{aligned} \phi_{p}\bigl(D_{0^{+}}^{\alpha}x(t)\bigr) =d_{1}+I_{0^{+}}^{\beta}z(t), \quad d_{1}\in \mathbb{R}. \end{aligned}$$
By the condition \(D_{0^{+}}^{\alpha}x(0)=0\), one has \(d_{1}=0\). Thus we get
$$\begin{aligned} x(t) =d_{2}+I_{0^{+}}^{\alpha}\phi_{q} \bigl(I_{0^{+}}^{\beta}z\bigr) (t),\quad d_{2}\in \mathbb{R}, \end{aligned}$$
where \(\phi_{q}\) is understood as the operator \(\phi_{q}:Z\rightarrow Z\) defined by \(\phi_{q}(x)(t)=\phi_{q}(x(t))\). Hence, from the condition \(x(0)=x(1)\), we obtain
$$\begin{aligned} I_{0^{+}}^{\alpha}\phi_{q} \bigl(I_{0^{+}}^{\beta}z\bigr) (1)=0. \end{aligned}$$
(3.3)
Suppose \(z\in Z\) and satisfies (3.3). Let \(x(t)=I_{0^{+}}^{\alpha}\phi_{q}(I_{0^{+}}^{\beta}z)(t)\), then we have \(x\in \operatorname {dom}M\) and
$$\begin{aligned} Mx(t)=D_{0^{+}}^{\beta}\phi_{p}\bigl[D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}\phi _{q}\bigl(I_{0^{+}}^{\beta}z\bigr)\bigr](t)=z(t). \end{aligned}$$
Hence we obtain
$$\begin{aligned} \operatorname {Im}M= \biggl\{ z\in Z\Big| \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl( \int_{0}^{s}(s-\tau)^{\beta-1}z(\tau)\,d\tau \biggr)\,ds=0 \biggr\} . \end{aligned}$$
Obviously, \(\operatorname {Im}M\subset Z\) is closed.

Therefore, by Definition 2.3, M is a quasi-linear operator. □

Let \(X_{1}=\operatorname {Ker}M\) and define the continuous operators \(P:X\rightarrow X\), \(Q:Z\rightarrow Z\) by
$$\begin{aligned}& Px(t)=x(0), \quad \forall t\in[0,1], \\& Qz(t)=\phi_{p} \biggl[\frac{1}{\rho} \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl( \int_{0}^{s}(s-\tau)^{\beta-1}z(\tau)\,d\tau \biggr)\,ds \biggr], \quad \forall t\in[0,1], \end{aligned}$$
where \(\rho=\frac{1}{\beta^{q-1}}\int_{0}^{1}(1-s)^{\alpha-1}s^{\beta (q-1)}\,ds>0\). It is easy to see that P is a projector and \(Q^{2}z=Qz\), \(Q(\lambda z)=\lambda Qz\), \(\forall z\in Z\), \(\lambda\in\mathbb{R}\), that is, Q is a semi-projector. Moreover, \(X_{1}=\operatorname {Im}P\) and \(\operatorname {Im}M=\operatorname {Ker}Q\).

Lemma 3.2

Let \(\Omega\subset X\) be an open bounded set. Then the operator \(N_{\lambda}\), defined by (3.2), is M-compact in Ω̅.

Proof

Choose \(X_{2}=\operatorname {Ker}P\), \(Z_{1}=\operatorname {Im}Q\) and define the operator \(R:\overline{\Omega}\times[0,1]\rightarrow X_{2}\) by
$$\begin{aligned} R(x,\lambda) (t) =&I_{0^{+}}^{\alpha}\phi_{q} \bigl[I_{0^{+}}^{\beta}(I-Q)N_{\lambda}x\bigr](t) \\ =&\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}\phi_{q} \biggl[\frac{1}{\Gamma(\beta)} \\ &{}\cdot \int_{0}^{s}(s-\tau)^{\beta-1} \bigl(\lambda f \bigl(\tau,x(\tau),D_{0^{+}}^{\alpha}x(\tau)\bigr)-QN_{\lambda}x(\tau)\bigr)\,d\tau \biggr]\,ds. \end{aligned}$$
Obviously, \(\dim Z_{1}=\dim X_{1}=1\). The remainder of the proof will be given in the following two steps.

Step 1. \(R:\overline{\Omega}\times[0,1]\rightarrow X_{2}\) is continuous and compact.

By the definition of R, we obtain
$$\begin{aligned} D_{0^{+}}^{\alpha}Rx(t)=\phi_{q}\bigl[I_{0^{+}}^{\beta}(I-Q)N_{\lambda}x\bigr](t). \end{aligned}$$
Clearly, the operators R, \(D_{0^{+}}^{\alpha}R\) are compositions of the continuous operators. So R, \(D_{0^{+}}^{\alpha}R\) are continuous in Z. Hence R is a continuous operator, and \(R(\overline{\Omega})\), \(D_{0^{+}}^{\alpha}R(\overline{\Omega})\) are bounded in Z. Furthermore, there exists a constant \(T>0\) such that \(|I_{0^{+}}^{\beta}(I-Q)N_{\lambda}x(t)|\leq T\), \(\forall x\in\overline{\Omega}\), \(t\in[0,1]\). Thus, based on the Arzelà-Ascoli theorem, we need only to show \(R(\overline {\Omega})\subset X\) is equicontinuous.
For \(0\leq t_{1}< t_{2}\leq1\), \(x\in\overline{\Omega}\), we have
$$\begin{aligned} & \bigl\vert Rx(t_{2})-Rx(t_{1})\bigr\vert \\ &\quad = \frac{1}{\Gamma(\alpha)} \biggl\vert \int_{0}^{t_{2}}(t_{2}-s)^{\alpha-1} \phi_{q}\bigl[I_{0^{+}}^{\beta}(I-Q)N_{\lambda}x(s) \bigr]\,ds \\ &\qquad{} - \int_{0}^{t_{1}}(t_{1}-s)^{\alpha-1} \phi_{q}\bigl[I_{0^{+}}^{\beta}(I-Q)N_{\lambda}x(s) \bigr]\,ds\biggr\vert \\ &\quad \leq \frac{T^{q-1}}{\Gamma(\alpha)} \biggl\{ \int_{0}^{t_{1}}\bigl[(t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha-1} \bigr]\,ds + \int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha-1}\,ds \biggr\} \\ &\quad = \frac{T^{q-1}}{\Gamma(\alpha+1)}\bigl[t_{1}^{\alpha}-t_{2}^{\alpha} +2(t_{2}-t_{1})^{\alpha}\bigr]. \end{aligned}$$
As \(t^{\alpha}\) is uniformly continuous in \([0,1]\), we obtain \(R(\overline{\Omega})\subset Z\) is equicontinuous. A similar proof can show that \(I_{0^{+}}^{\beta}(I-Q)N_{\lambda}(\overline{\Omega})\subset Z\) is equicontinuous. This, together with the uniformly continuity of \(\phi _{q}(s)\) on \([-T,T]\), shows that \(D_{0^{+}}^{\alpha}R(\overline{\Omega })\subset Z\) is equicontinuous. Thus we find R is compact.

Step 2. Equations (2.1)-(2.4) are satisfied.

For \(x\in\overline{\Omega}\), it is easy to show that \(Q(I-Q)N_{\lambda}x=QN_{\lambda}x-Q^{2}N_{\lambda}x=0\). So \((I-Q)N_{\lambda}x\in \operatorname {Ker}Q=\operatorname {Im}M\). Moreover, for \(z\in \operatorname {Im}M\subset Z\), one has \(Qz=0\). Thus \(z=z-Qz=(I-Q)z\in(I-Q)Z\). Hence (2.1) holds. Since \(QN_{\lambda}x=\lambda QNx\), (2.2) holds too.

For \(x\in\sum_{\lambda}\), we have \(Mx=N_{\lambda}x\in \operatorname {Im}M=\operatorname {Ker}Q\). So \(QN_{\lambda}x=0\). From the condition \(D_{0^{+}}^{\alpha }x(0)=0\), one has \(I_{0^{+}}^{\beta}D_{0^{+}}^{\beta}\phi_{p}(D_{0^{+}}^{\alpha}x)=\phi_{p}(D_{0^{+}}^{\alpha}x)\). Thus we obtain
$$\begin{aligned} R(x,\lambda) (t) =&I_{0^{+}}^{\alpha}\phi_{q} \bigl(I_{0^{+}}^{\beta}N_{\lambda}x\bigr) (t) \\ =&I_{0^{+}}^{\alpha}\phi_{q} \bigl[I_{0^{+}}^{\beta}D_{0^{+}}^{\beta}\phi _{p}\bigl(D_{0^{+}}^{\alpha}x\bigr)\bigr](t) \\ =&x(t)-x(0) \\ =&(I-P)x(t). \end{aligned}$$
Furthermore, when \(\lambda=0\), we have \(N_{\lambda}x(t)\equiv0\), which yields \(R(x,0)(t)\equiv0\), \(\forall x\in\overline{\Omega}\). Hence (2.3) holds.
For \(x\in\overline{\Omega}\), one has
$$\begin{aligned} M\bigl(Px+R(x,\lambda)\bigr) (t) =&D_{0^{+}}^{\beta}\phi_{p}\bigl[D_{0^{+}}^{\alpha}\bigl(Px+R(x,\lambda) \bigr)\bigr](t) \\ =&D_{0^{+}}^{\beta}\phi_{p}\bigl[D_{0^{+}}^{\alpha}I_{0^{+}}^{\alpha}\phi_{q}\bigl(I_{0^{+}}^{\beta}(I-Q)N_{\lambda}x\bigr)\bigr](t) \\ =&(I-Q)N_{\lambda}x(t), \end{aligned}$$
which implies that (2.4) holds.

Therefore, by Definition 2.5, \(N_{\lambda}\) is M-compact in Ω̅. □

4 Solutions of BVP (1.1)-(1.2)

We will give a theorem on the existence of solutions for BVP (1.1)-(1.2).

Theorem 4.1

Let \(f:[0,1]\times\mathbb{R}^{2}\rightarrow\mathbb {R}\) be continuous. Assume that:
(H1): 
there exist nonnegative functions \(a,b,c\in Z\) such that
$$\begin{aligned} \bigl\vert f(t,x,y)\bigr\vert \leq a(t)+b(t)\vert x\vert ^{p-1}+c(t)\vert y\vert ^{p-1}, \quad \forall t\in [0,1],(x,y)\in\mathbb{R}^{2}; \end{aligned}$$
(H2): 
there exists a constant \(A>0\) such that, for \(\forall x\in \operatorname {dom}M\setminus \operatorname {Ker}M\) satisfying \(\vert x(t)\vert >A\) for \(\forall t\in [0,1]\), we have
$$\begin{aligned} \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl( \int_{0}^{s}(s-\tau)^{\beta-1}f\bigl(\tau,x( \tau),D_{0^{+}}^{\alpha}x(\tau)\bigr)\,d\tau \biggr)\,ds\neq0; \end{aligned}$$
(H3): 
there exists a constant \(B>0\) such that, for \(\forall r\in \mathbb{R}\) with \(\vert r\vert >B\), we have either
$$\begin{aligned} \phi_{q}(r) \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl( \int_{0}^{s}(s-\tau)^{\beta-1}f(\tau,r,0)\,d\tau \biggr)\,ds>0 \end{aligned}$$
(4.1)
or
$$\begin{aligned} \phi_{q}(r) \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl( \int_{0}^{s}(s-\tau)^{\beta-1}f(\tau,r,0)\,d\tau \biggr)\,ds< 0. \end{aligned}$$
(4.2)
Then BVP (1.1)-(1.2) has at least one solution, provided that
$$ \begin{aligned} &\gamma_{1}:=\frac{1}{\Gamma(\beta+1)} \biggl[ \frac{2^{p-1}\Vert b\Vert _{0}}{(\Gamma(\alpha+1))^{p-1}}+\Vert c\Vert _{0} \biggr]< 1,\quad \textit{if } p< 2; \\ &\gamma_{2}:=\frac{1}{\Gamma(\beta+1)} \biggl[\frac{2^{2p-3}\Vert b\Vert _{0}}{(\Gamma(\alpha+1))^{p-1}}+\Vert c \Vert _{0} \biggr]< 1, \quad \textit{if } p\geq2. \end{aligned} $$
(4.3)

Proof

The proof will be given in the following four steps.

Step 1. \(\Omega_{1}=\{x\in \operatorname {dom}M\setminus \operatorname {Ker}M|Mx=N_{\lambda}x,\lambda\in(0,1)\}\) is bounded.

For \(x\in\Omega_{1}\), one has \(Nx\in \operatorname {Im}M=\operatorname {Ker}Q\). Thus we have
$$\begin{aligned} \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl( \int_{0}^{s}(s-\tau)^{\beta-1}f\bigl(\tau,x( \tau),D_{0^{+}}^{\alpha}x(\tau)\bigr)\,d\tau \biggr)\,ds=0. \end{aligned}$$
From (H2), there exists a constant \(\xi\in[0,1]\) such that \(|x(\xi )|\leq A\). By Lemma 2.1, one has
$$\begin{aligned} x(t)=x(\xi)-I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}x( \xi)+I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}x(t), \end{aligned}$$
which together with
$$\begin{aligned} \bigl\vert I_{0^{+}}^{\alpha}D_{0^{+}}^{\alpha}x(t)\bigr\vert =&\frac{1}{\Gamma(\alpha)}\biggl\vert \int_{0}^{t}(t-s)^{\alpha-1}D_{0^{+}}^{\alpha}x(s)\,ds\biggr\vert \\ \leq&\frac{1}{\Gamma(\alpha)}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}\cdot\frac{1}{\alpha }t^{\alpha} \\ \leq&\frac{1}{\Gamma(\alpha+1)}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0},\quad \forall t\in[0,1] , \end{aligned}$$
(4.4)
and \(\vert x(\xi)\vert \leq A\) yields
$$\begin{aligned} \Vert x\Vert _{0}\leq A+\frac{2}{\Gamma(\alpha+1)}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}. \end{aligned}$$
(4.5)
Then, from (H1), we have
$$\begin{aligned} \bigl\vert I_{0^{+}}^{\beta}Nx(t)\bigr\vert =& \frac{1}{\Gamma(\beta)}\biggl\vert \int_{0}^{t}(t-s)^{\beta -1}f \bigl(s,x(s),D_{0^{+}}^{\alpha}x(s)\bigr)\,ds\biggr\vert \\ \leq&\frac{1}{\Gamma(\beta)} \int_{0}^{t}(t-s)^{\beta -1}\bigl(a(s)+b(s)\bigl\vert x(s)\bigr\vert ^{p-1} \\ &{}+c(s)\bigl\vert D_{0^{+}}^{\alpha}x(s)\bigr\vert ^{p-1}\bigr)\,ds \\ \leq&\frac{1}{\Gamma(\beta)}\bigl(\Vert a\Vert _{0}+\Vert b\Vert _{0}\Vert x\Vert _{0}^{p-1} +\Vert c\Vert _{0}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}^{p-1}\bigr)\cdot\frac{1}{\beta}t^{\beta} \\ \leq&\frac{1}{\Gamma(\beta+1)} \biggl[\Vert a\Vert _{0}+\Vert c\Vert _{0}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}^{p-1} \\ &{}+\Vert b\Vert _{0} \biggl(A+ \frac{2}{\Gamma(\alpha+1)}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0} \biggr)^{p-1} \biggr], \quad \forall t\in[0,1]. \end{aligned}$$
(4.6)
By \(Mx=N_{\lambda}x\), \(D_{0^{+}}^{\alpha}x(0)=0\), and Lemma 2.1, one has
$$\begin{aligned} \phi_{p}\bigl(D_{0^{+}}^{\alpha}x(t)\bigr)=\lambda I_{0^{+}}^{\beta}Nx(t), \end{aligned}$$
which, together with \(\vert \phi_{p}(D_{0^{+}}^{\alpha}x(t))\vert =\vert D_{0^{+}}^{\alpha}x(t)\vert ^{p-1}\) and (4.6), implies
$$\begin{aligned} \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}^{p-1} \leq& \frac{1}{\Gamma(\beta+1)} \biggl[\Vert a\Vert _{0}+\Vert c\Vert _{0}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}^{p-1} \\ &{}+\Vert b\Vert _{0} \biggl(A+ \frac{2}{\Gamma(\alpha+1)}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0} \biggr)^{p-1} \biggr]. \end{aligned}$$
(4.7)
If \(p<2\), from (4.7) and Lemma 2.2, we have
$$\begin{aligned} \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}^{p-1} \leq& \frac{1}{\Gamma(\beta+1)} \biggl[\Vert a\Vert _{0}+A^{p-1} \Vert b\Vert _{0} \\ &{}+ \biggl(\frac{2^{p-1}\Vert b\Vert _{0}}{(\Gamma(\alpha+1))^{p-1}} +\Vert c\Vert _{0} \biggr) \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}^{p-1} \biggr]. \end{aligned}$$
Then, based on (4.3), one has
$$\begin{aligned} \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0} \leq \biggl[\frac{\Vert a\Vert _{0}+A^{p-1}\Vert b\Vert _{0}}{ (1-\gamma_{1})\Gamma(\beta+1)} \biggr]^{q-1}:=K_{1}. \end{aligned}$$
(4.8)
Thus, from (4.5), we have
$$\begin{aligned} \Vert x\Vert _{0}\leq A+\frac{2K_{1}}{\Gamma(\alpha+1)}. \end{aligned}$$
(4.9)
Similarly, if \(p\geq2\), we obtain
$$\begin{aligned}& \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0} \leq \biggl[\frac{\Vert a\Vert _{0}+2^{p-2}A^{p-1}\Vert b\Vert _{0}}{ (1-\gamma_{2})\Gamma(\beta+1)} \biggr]^{q-1}:=K_{2}, \end{aligned}$$
(4.10)
$$\begin{aligned}& \Vert x\Vert _{0}\leq A+\frac{2K_{2}}{\Gamma(\alpha+1)}. \end{aligned}$$
(4.11)
Therefore, combining (4.8), (4.10) with (4.9), (4.11), we have
$$\begin{aligned} \Vert x\Vert _{X} =&\max\bigl\{ \Vert x\Vert _{0}, \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}\bigr\} \\ \leq&\max \biggl\{ K_{1},K_{2},A+\frac{2K_{1}}{\Gamma(\alpha+1)},A+ \frac{2 K_{2}}{\Gamma(\alpha+1)} \biggr\} :=K. \end{aligned}$$
That is, \(\Omega_{1}\) is bounded.

Step 2. \(\Omega_{2}=\{x\in \operatorname {Ker}M|QNx=0\}\) is bounded.

For \(x\in\Omega_{2}\), one has \(x(t)=d\), \(\forall d\in\mathbb{R}\). Then we have
$$\begin{aligned} \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl( \int_{0}^{s}(s-\tau)^{\beta-1}f(\tau,d,0)\,d\tau \biggr)\,ds=0, \end{aligned}$$
which together with (H3) implies \(\vert d\vert \leq B\). Thus we obtain
$$\begin{aligned} \Vert x\Vert _{X}\leq\max\{B,0\}=B. \end{aligned}$$
Hence \(\Omega_{2}\) is bounded.
Step 3. If (4.1) holds, then
$$\begin{aligned} \Omega_{3}=\bigl\{ x\in \operatorname {Ker}M|\lambda Ix+(1-\lambda)JQNx=0, \lambda\in [0,1]\bigr\} \end{aligned}$$
is bounded, where \(J:\operatorname {Im}Q\rightarrow \operatorname {Ker}M\) is a homeomorphism such that \(J(d)=d\), \(\forall d\in\mathbb{R}\). If (4.2) holds, then
$$\begin{aligned} \Omega'_{3}=\bigl\{ x\in \operatorname {Ker}M|{-}\lambda Ix+(1- \lambda)JQNx=0,\lambda\in [0,1]\bigr\} \end{aligned}$$
is bounded.
For \(x\in\Omega_{3}\), we have \(x(t)=d\), \(\forall d\in\mathbb{R}\), and
$$\lambda d =-(1-\lambda)\phi_{p} \biggl[\frac{1}{\rho} \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl( \int_{0}^{s}(s-\tau)^{\beta-1}f(\tau,d,0)\,d\tau \biggr)\,ds \biggr]. $$
If \(\lambda=1\), then \(d=0\). If \(\lambda\in[0,1)\), we can show \(\vert d\vert \leq B\). Otherwise, if \(\vert d\vert >B\), in view of (4.1), one has
$$\begin{aligned} 0\leq\lambda d^{2} =&-(1-\lambda)\phi_{p} \biggl[ \frac{\phi_{q}(d)}{\rho} \int_{0}^{1}(1-s)^{\alpha -1} \\ &{}\cdot\phi_{q} \biggl( \int_{0}^{s}(s-\tau)^{\beta-1}f(\tau,d,0)\,d\tau \biggr)\,ds \biggr]< 0, \end{aligned}$$
which is a contradiction. Hence \(\Omega_{3}\) is bounded.

Similar to the above argument, we can show \(\Omega'_{3}\) is also bounded.

Step 4. All conditions of Lemma 2.3 are satisfied.

Define
$$\begin{aligned} \Omega=\bigl\{ x\in X|\Vert x\Vert _{X}< \max\{K,B\}+1\bigr\} . \end{aligned}$$
Clearly, \((\Omega_{1}\cup\Omega_{2}\cup\Omega_{3})\subset\Omega\) (or \((\Omega _{1}\cup\Omega_{2}\cup\Omega'_{3})\subset\Omega\)). From Lemma 3.1 and Lemma 3.2, M is a quasi-linear operator and \(N_{\lambda}\) is M-compact in Ω̅. Moreover, by the above arguments, we see that the following two conditions are satisfied:
(C1): 

\(Mx\neq N_{\lambda}x\) for every \((x,\lambda)\in[(\operatorname {dom}M\setminus \operatorname {Ker}M)\cap\partial\Omega]\times(0,1)\);

(C2): 

\(QNx\neq0\) for every \(x\in \operatorname {Ker}M\cap\partial\Omega\).

Now we verify the condition (C3) of Lemma 2.3. Let us define the homotopy
$$\begin{aligned} H(x,\lambda)=\pm\lambda Ix+(1-\lambda)JQNx. \end{aligned}$$
According to the above argument, we know
$$\begin{aligned} H(x,\lambda)\neq0,\quad \forall x\in\partial\Omega\cap \operatorname {Ker}M. \end{aligned}$$
Thus we have
$$\begin{aligned} \deg \{JQN,\Omega\cap \operatorname {Ker}M,\theta\} =&\deg \bigl\{ H(\cdot,0), \Omega\cap \operatorname {Ker}M,\theta\bigr\} \\ =&\deg \bigl\{ H(\cdot,1),\Omega\cap \operatorname {Ker}M,\theta\bigr\} \\ =&\deg \{\pm I,\Omega\cap \operatorname {Ker}M,\theta\}\neq0. \end{aligned}$$
So the condition (C3) of Lemma 2.3 is satisfied.

Therefore, the operator equation \(Mx=Nx\) has at least one solution in \(\operatorname {dom}M\cap\overline{\Omega}\). That is, BVP (1.1)-(1.2) has at least one solution in X. □

5 Solutions of BVP (1.1)-(1.3)

We will give a theorem on the existence of solutions for BVP (1.1)-(1.3).

Define the operator \(M_{1}:\operatorname {dom}M_{1}\cap X^{1}\rightarrow Z\) by
$$\begin{aligned} M_{1}x=D_{0^{+}}^{\beta}\phi_{p}\bigl(D_{0^{+}}^{\alpha}x\bigr), \end{aligned}$$
(5.1)
where \(\operatorname {dom}M_{1}=\{x\in X^{1}|D_{0^{+}}^{\beta}\phi_{p}(D_{0^{+}}^{\alpha}x)\in Z\}\). Then BVP (1.1)-(1.3) is equivalent to the operator equation
$$\begin{aligned} M_{1}x=Nx,\quad x\in \operatorname {dom}M_{1}, \end{aligned}$$
where \(N=N_{1}\) and \(N_{\lambda}:X^{1}\rightarrow Z\), \(\lambda\in[0,1]\) is defined by (3.2).
By similar arguments to Section 3, we obtain
$$\begin{aligned}& \operatorname {Ker}M_{1}=\bigl\{ x\in X^{1}|x(t)=d, \forall t \in[0,1],d\in\mathbb{R}\bigr\} , \\& \operatorname {Im}M_{1}= \biggl\{ z\in Z\Big| \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl(- \int_{0}^{1}(1-\tau)^{\beta-1}z(\tau)\,d\tau \\& \hphantom{\operatorname {Im}M_{1}=} {} + \int_{0}^{s}(s-\tau)^{\beta-1}z(\tau)\,d\tau \biggr)\,ds=0 \biggr\} . \end{aligned}$$

Lemma 5.1

The operator \(M_{1}\), defined by (5.1), is a quasi-linear operator.

Let \(X_{1}^{1}=\operatorname {Ker}M_{1}\), define the projector \(P_{1}:X^{1}\rightarrow X^{1}\) and the semi-projector \(Q_{1}:Z\rightarrow Z\) by
$$\begin{aligned}& P_{1}x(t)=x(0), \quad \forall t\in[0,1], \\& Q_{1}z(t)=\phi_{p} \biggl[\frac{1}{\rho_{1}} \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl(- \int_{0}^{1}(1-\tau)^{\beta-1}z(\tau)\,d\tau \\& \hphantom{ Q_{1}z(t)=} {}+ \int_{0}^{s}(s-\tau)^{\beta-1}z(\tau)\,d\tau \biggr)\,ds \biggr], \quad \forall t\in[0,1], \end{aligned}$$
where \(\rho_{1}=\frac{1}{\beta^{q-1}}\int_{0}^{1}(1-s)^{\alpha-1}\phi _{q}(-1+s^{\beta})\,ds<0\). Furthermore, let \(\Omega^{1}\subset X^{1}\) be an open bounded set, choose \(X_{2}^{1}=\operatorname {Ker}P_{1}\), \(Z_{1}^{1}=\operatorname {Im}Q_{1}\) and define the operator \(R_{1}:\overline{\Omega^{1}}\times[0,1]\rightarrow X_{2}^{1}\) by
$$\begin{aligned} R_{1}(x,\lambda) (t) =&I_{0^{+}}^{\alpha}\phi_{q}\bigl[I_{0^{+}}^{\beta}(I-Q)N_{\lambda}x+ \tilde {d}\bigl((I-Q)N_{\lambda}x\bigr)\bigr](t) \\ =&\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}\phi_{q} \biggl[\frac{1}{\Gamma(\beta)} \\ &{}\cdot \int_{0}^{s}(s-\tau)^{\beta-1} \bigl(\lambda f \bigl(\tau,x(\tau),D_{0^{+}}^{\alpha}x(\tau)\bigr)-QN_{\lambda}x(\tau)\bigr)\,d\tau \\ &{}-\frac{1}{\Gamma(\beta)} \int_{0}^{1}(1-\tau)^{\beta-1} \bigl((I-Q)N_{\lambda}x(\tau)\bigr)\,d\tau \biggr]\,ds, \end{aligned}$$
where \(\tilde{d}:Z\rightarrow\mathbb{R}\) is defined by
$$\begin{aligned} \tilde{d}(z) =&-I_{0^{+}}^{\beta}z(1) \\ =&-\frac{1}{\Gamma(\beta)} \int_{0}^{1}(1-s)^{\beta-1}z(s)\,ds. \end{aligned}$$

Lemma 5.2

The operator \(N_{\lambda}:X^{1}\rightarrow Z\), \(\lambda\in [0,1]\), defined by (3.2), is M-compact in \(\overline{\Omega^{1}}\).

Our second result, based on Lemma 5.1 and Lemma 5.2, is stated as follows.

Theorem 5.1

Let \(f:[0,1]\times\mathbb{R}^{2}\rightarrow\mathbb {R}\) be continuous. Assume that:
(H4): 
there exists a constant \(A_{1}>0\) such that, for \(\forall x\in \operatorname {dom}M_{1}\setminus \operatorname {Ker}M_{1}\) satisfying \(|x(t)|>A_{1}\) for \(\forall t\in[0,1]\), we have
$$\begin{aligned}& \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl(- \int_{0}^{1}(1-\tau)^{\beta-1}f\bigl(\tau,x( \tau),D_{0^{+}}^{\alpha}x(\tau)\bigr)\,d\tau \\& \quad {}+ \int_{0}^{s}(s-\tau)^{\beta-1}f\bigl(\tau,x( \tau),D_{0^{+}}^{\alpha}x(\tau )\bigr)\,d\tau \biggr)\,ds\neq0; \end{aligned}$$
(H5): 
there exists a constant \(B_{1}>0\) such that, for \(\forall r_{1}\in \mathbb{R}\) with \(|r_{1}|>B_{1}\), we have either
$$\begin{aligned}& \phi_{q}(r_{1}) \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl(- \int_{0}^{1}(1-\tau)^{\beta-1}f( \tau,r_{1},0)\,d\tau \\& \quad {} + \int_{0}^{s}(s-\tau)^{\beta-1}f( \tau,r_{1},0)\,d\tau \biggr)\,ds>0 \end{aligned}$$
or
$$\begin{aligned}& \phi_{q}(r_{1}) \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl(- \int_{0}^{1}(1-\tau)^{\beta-1}f( \tau,r_{1},0)\,d\tau \\& \quad {}+ \int_{0}^{s}(s-\tau)^{\beta-1}f( \tau,r_{1},0)\,d\tau \biggr)\,ds< 0, \end{aligned}$$
and (H1) is true. Then BVP (1.1)-(1.3) has at least one solution, provided that
$$ \begin{aligned} &\delta_{1}:=\frac{2}{\Gamma(\beta+1)} \biggl[ \frac{2^{p-1}\Vert b\Vert _{0}}{(\Gamma(\alpha+1))^{p-1}}+\Vert c\Vert _{0} \biggr]< 1,\quad \textit{if } p< 2; \\ &\delta_{2}:=\frac{2}{\Gamma(\beta+1)} \biggl[\frac{2^{2p-3}\Vert b\Vert _{0}}{(\Gamma(\alpha+1))^{p-1}}+\Vert c \Vert _{0} \biggr]< 1,\quad \textit{if } p\geq2. \end{aligned} $$
(5.2)

Proof

Let
$$\begin{aligned} \Omega_{1}^{1}=\bigl\{ x\in \operatorname {dom}M_{1} \setminus \operatorname {Ker}M_{1}|M_{1}x=N_{\lambda}x,\lambda \in(0,1)\bigr\} . \end{aligned}$$
Now we prove \(\Omega_{1}^{1}\) is bounded.
For \(x\in\Omega_{1}^{1}\), one has \(Nx\in \operatorname {Im}M_{1}=\operatorname {Ker}Q_{1}\). Thus we have
$$\begin{aligned}& \int_{0}^{1}(1-s)^{\alpha-1} \phi_{q} \biggl(- \int_{0}^{1}(1-\tau)^{\beta-1}f\bigl(\tau,x( \tau),D_{0^{+}}^{\alpha}x(\tau)\bigr)\,d\tau \\& \quad {}+ \int_{0}^{s}(s-\tau)^{\beta-1}f\bigl(\tau,x( \tau),D_{0^{+}}^{\alpha}x(\tau )\bigr)\,d\tau \biggr)\,ds=0. \end{aligned}$$
From (H4), there exists a constant \(\eta\in[0,1]\) such that \(\vert x(\eta )\vert \leq A_{1}\). Hence, by (4.4), one has
$$\begin{aligned} \Vert x\Vert _{0}\leq A_{1}+ \frac{2}{\Gamma(\alpha+1)}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}. \end{aligned}$$
(5.3)
Since \(M_{1}x=N_{\lambda}x\), \(D_{0^{+}}^{\alpha}x(1)=0\), one has
$$\begin{aligned} \phi_{p}\bigl(D_{0^{+}}^{\alpha}x(t)\bigr)=-\lambda I_{0^{+}}^{\beta}Nx(1)+\lambda I_{0^{+}}^{\beta}Nx(t), \end{aligned}$$
which together with (4.6) and (5.3) implies
$$\begin{aligned} \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}^{p-1} \leq& \frac{2}{\Gamma(\beta+1)} \biggl[\Vert a\Vert _{0}+\Vert c\Vert _{0}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}^{p-1} \\ &{}+\Vert b\Vert _{0} \biggl(A_{1}+\frac{2}{\Gamma(\alpha+1)}\bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0} \biggr)^{p-1} \biggr]. \end{aligned}$$
(5.4)
If \(p<2\), from (5.4) and Lemma 2.2, we have
$$\begin{aligned} \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}^{p-1} \leq& \frac{2}{\Gamma(\beta+1)} \biggl[\Vert a\Vert _{0}+A_{1}^{p-1} \Vert b\Vert _{0} \\ &{}+ \biggl(\frac{2^{p-1}\Vert b\Vert _{0}}{(\Gamma(\alpha+1))^{p-1}} +\Vert c\Vert _{0} \biggr) \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0}^{p-1} \biggr]. \end{aligned}$$
Then, in view of (5.2), one has
$$\begin{aligned} \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0} \leq \biggl[\frac{2(\Vert a\Vert _{0}+A_{1}^{p-1}\Vert b\Vert _{0})}{ (1-\delta_{1})\Gamma(\beta+1)} \biggr]^{q-1}:=T_{1}. \end{aligned}$$
(5.5)
Similarly, if \(p\geq2\), we obtain
$$\begin{aligned} \bigl\Vert D_{0^{+}}^{\alpha}x\bigr\Vert _{0} \leq \biggl[\frac{2(\Vert a\Vert _{0}+2^{p-2}A_{1}^{p-1}\Vert b\Vert _{0})}{ (1-\delta_{2})\Gamma(\beta+1)} \biggr]^{q-1}:=T_{2}. \end{aligned}$$
(5.6)
Therefore, from (5.3), (5.5), and (5.6), we have
$$\begin{aligned} \Vert x\Vert _{X} \leq\max \biggl\{ T_{1},T_{2},A_{1}+ \frac{2T_{1}}{\Gamma(\alpha+1)},A_{1}+\frac{2 T_{2}}{\Gamma(\alpha+1)} \biggr\} . \end{aligned}$$
That is, \(\Omega_{1}^{1}\) is bounded.

The remainder of proof are similar to the proof of Theorem 4.1, so we omit the details. □

Declarations

Acknowledgements

The authors would like to thank the anonymous referee for his/her valuable comments, which have improved the presentation and quality of the manuscript. This research was supported by the Fundamental Research Funds for the Central Universities (2015XKMS072).

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)
Department of Mathematics, China University of Mining and Technology, Xuzhou, 221116, P.R. China

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