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Solvability of mixed Hilfer fractional functional boundary value problems with p-Laplacian at resonance

Abstract

This article investigates the existence of solutions of mixed Hilfer fractional differential equations with p-Laplacian under the functional boundary conditions at resonance. By defining Banach spaces with appropriate norms, constructing suitable operators, and using the extension of the continuity theorem, some of the current results are extended to the nonlinear situation, and some new existence results of the problem are obtained. Finally, an example is given to verify our main results.

1 Introduction

The fractional differential equations have become an important research field because of the in-depth development of fractional calculus theory and its wide applications in many sciences such as physics, engineering, biology and so on [15].

There are various definitions of fractional derivatives, such as Riemann–Liouville and Caputo fractional derivatives [6, 7]. On this basis, a more generalized fractional derivative “Hilfer” derivative has been studied [8]. The Hilfer fractional derivative is an extension of the Riemann–Liouville and Caputo fractional derivatives. Hilfer fractional differential equations are very suitable for describing processes with memory and hereditary properties. They have the advantages of simple modeling and accurate description of complex systems, and have become one of the important tools for mathematical modeling of mechanical and physical processes. Therefore, fractional differential equations with Hilfer derivative have gradually become a research hotspot [911].

Ri et al. [11] considered the following multi-point boundary value problems of the Hilfer fractional differential equations at resonance:

$$ \textstyle\begin{cases} D^{\alpha ,\beta}_{0+}x(t)=f (t,x(t) ),\quad 0< t\leq T, \\ I^{1-\gamma}_{0+} u(0)=\sum_{i=1}^{m}c_{i}x{(\tau _{i})}, \end{cases} $$

where \(0<\alpha <1\), \(0\leq \beta \leq 1\), \(\tau _{i} \in (0,T]\), \(D^{\alpha ,\beta}_{0+}\) is Hilfer fractional derivative of order α and type β.

In the past, the boundary conditions of boundary value problems were generally specific. In recent years, some scholars have changed the boundary value conditions into abstract conditions, which contains many specific boundary conditions. And many achievements have been made in the study of functional boundary value problems [1217].

Zhao and Liang [15] first used Mawhin’s coincidence degree theory to discuss the solvability of functional boundary value problems:

$$ \textstyle\begin{cases} x''(t)=f(t,x(t),x'(t)),\quad 0< t< 1, \\ \Gamma _{1}(x)=0,\qquad \Gamma _{2}(x)=0, \end{cases} $$

where \(\Gamma _{1},\Gamma _{2}:C^{1}[0,1]\rightarrow R\) are continuous linear functionals. It was discussed according to the six situations of non-resonance and resonance, and some existence results of the solution of the functional boundary value problems were obtained.

However, the existence of solutions under the condition of \(\Gamma _{1}(t)\Gamma _{2}(1)=\Gamma _{1}(1)\Gamma _{2}(t)\) was not discussed in [15]. Furthermore, Kosmatov and Jiang [16] considered the solvability of functional boundary value problems under the condition \(\Gamma _{1}(t)\Gamma _{2}(1)=\Gamma _{1}(1)\Gamma _{2}(t)\):

$$ \textstyle\begin{cases} x''(t)=f(t,x(t),x'(t)),\quad t\in (0,1), \\ \Gamma _{1}(x)=0,\qquad \Gamma _{2}(x)=0, \end{cases} $$

where \(\Gamma _{1}\), \(\Gamma _{2}\) are linear functionals. The conditions in [15] were supplemented here, and the solvability of functional boundary value problems was analyzed more comprehensively.

The p-Laplacian operator originated from the research of turbulence in porous media. Leibenson [18] first considered the following p-Laplacian equation:

$$ \bigl(\phi _{p}\bigl(x'(t)\bigr) \bigr)'=f\bigl(t,x(t),x'(t)\bigr). $$

Later, many scholars conducted more in-depth research on the p-Laplacian operator and obtained some excellent results [1921].

Jiang [22] considered the solvability of fractional differential equations with p-Laplacian by the extended continuous theorem:

$$ \textstyle\begin{cases} D^{\beta}_{0+}(\varphi _{p}(D^{\alpha}_{0+}u))(t)+f(t,u(t),D^{ \alpha -1}_{0+}u(t),D^{\alpha}_{0+}u(t))=0, \\ u(0)=D^{\alpha}_{0+}u(0)=0,\qquad u(1)=\int _{0}^{1}h(t)u(t)\,dt , \end{cases} $$

where \(0<\beta \leq 1\), \(1<\alpha \leq 2\), \(\varphi _{p}(s)={|s|^{p-2}}s\), \(p>1\), \(\int _{0}^{1}h(t)t^{\alpha -1}\,dt =1\), \(D^{\alpha}_{0+}\) is the Riemann–Liouville fractional derivative.

Based on the above literature, this paper studies the solvability of mixed Hilfer fractional functional boundary value problems with p-Laplacian operator at resonance:

$$ \textstyle\begin{cases} D^{\alpha _{1},\beta _{1}}_{1-}\varphi _{p}{(D^{\alpha _{2},\beta _{2}}_{0+}u(t))}=f(t, u(t), D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t),D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t), D^{\alpha _{2},\beta _{2}}_{0+}u(t)), \\ u(0)=0, \qquad D^{\alpha _{2},\beta _{2}}_{0+}u(1)=0, \qquad T_{1}(u)=T_{2}(u)=0, \quad t\in [0,1], \end{cases} $$
(1.1)

where \(0<\alpha _{1}<1\), \(2<\alpha _{2}<3\), \(0\leq \beta _{1},\beta _{2}\leq 1\), \(\gamma _{1}=\alpha _{1}+\beta _{1}-\alpha _{1}\beta _{1}\), \(\gamma _{2}=\alpha _{2}+3\beta _{2}-\alpha _{2}\beta _{2}\), \(\varphi _{p}(s)={|s|^{p-2}}s\), \(p>1\), \(\varphi _{p}(0)=0\), \(D^{\alpha ,\beta}_{a\pm}\) is Hilfer right-/left-sided fractional derivative of order α and type β, \(f\in C([0,1]\times \mathbb{R}^{4}, \mathbb{R})\) and \(T_{1}, T_{2}:C[0,1]\rightarrow \mathbb{R}\) are linear bounded functionals.

2 Preliminaries

Definition 2.1

([23])

Let X and Y be two Banach spaces with norms \(\| \cdot \| _{X}\), \(\| \cdot \| _{Y}\), respectively. A continuous operator L: \(X\cap \operatorname{dom}L\rightarrow Y\) is said to be quasilinear if

  1. (i)

    \(\operatorname{Im}L:=L(X\cap \operatorname{dom}L)\) is a closed subset of Y,

  2. (ii)

    \(\operatorname{Ker}L:=\{x\in X\cap \operatorname{dom}L: Lx=0\}\) is linearly homeomorphic to \(\mathbb{R}^{n}\), \(n<\infty \),

where domL denotes the domain of the operator L.

Let \({X_{1}=\operatorname{Ker}L}\) and \(X_{2}\) be the complement space of \(X_{1}\) in X, then \({X=X_{1}\oplus X_{2}}\). Let \(P: X\rightarrow X_{1}\) be the projector and \(\Omega \subset X\) be an open and bounded set with the origin \(\theta \in \Omega \).

Definition 2.2

([22])

Suppose that \(N_{\lambda}:\overline{\Omega}\rightarrow Y\), \(\lambda \in [0,1]\) is a continuous and bounded operator. Denote \(N_{1}\) by N. Let \(\Sigma _{\lambda}=\{x\in \overline{\Omega}:Lx=N_{\lambda}x\}\). \(N_{\lambda}\) is said to be L-quasicompact in Ω̅ if there exists a vector subspace \({Y_{1}}\) of Y satisfying \({\operatorname{dim}Y_{1}=\operatorname{dim}X_{1}}\) and two operators Q and R such that for \(\lambda \in [0,1]\),

  1. (a)

    \(\operatorname{Ker}Q=\operatorname{Im}L\),

  2. (b)

    \(QN_{\lambda}x=\theta \), \(\lambda \in (0,1)\Leftrightarrow QNx= \theta \),

  3. (c)

    \(R(\cdot ,0)\) is the zero operator and \(R(\cdot ,\lambda )\mid _{\Sigma _{\lambda}}=(I-P)\mid _{\Sigma _{ \lambda}}\),

  4. (d)

    \(L[P+R(\cdot ,\lambda )]=(I-Q)N_{\lambda}\),

where \(Q:Y\rightarrow Y_{1}\), \({QY=Y_{1}}\) is continuous, bounded and satisfies \(Q(I-Q)=0\) and \(R:\overline{\Omega}\times [0,1]\rightarrow X_{2}\) is continuous and compact.

Lemma 2.3

([22])

Let X and Y be two Banach spaces with the norms \(\| \cdot \| _{X}\), \(\| \cdot \| _{Y}\), respectively, and let \(\Omega \subset X\) be an open and bounded nonempty set. Suppose that \(L:\operatorname{dom}L\cap X\rightarrow Y\) is a quasilinear operator and that \(N_{\lambda}:\overline{\Omega}\rightarrow Y\), \(\lambda \in [0,1]\) is L-quasicompact. In addition, if the following conditions hold:

  1. (a)

    \(Lx\neq N_{\lambda}x\), \(\forall x\in \partial \Omega \cap \operatorname{dom}L\), \(\lambda \in (0,1)\),

  2. (b)

    \(\deg\{JQN,\Omega \cap \operatorname{Ker}L,0\}\neq 0\),

then the abstract equation \({Lx=Nx}\) has at least one solution in \(\operatorname{dom}L\cap \overline{\Omega}\), where \({N=N_{1}}\), \(J:\operatorname{Im}Q\rightarrow \operatorname{Ker}L\) is a homeomorphism with \({J(\theta )=\theta}\).

Definition 2.4

([6])

The left-sided and right-sided Riemann–Liouville fractional integrals of order \(\alpha >0\) of a function \(y:(0,+\infty )\rightarrow R\) are given by

$$ I _{0+}^{\alpha }y(t)=\frac{1}{\Gamma (\alpha )} \int _{0}^{t}(t-s)^{ \alpha -1}y(s)\,ds ,\qquad I^{\alpha}_{1^{-}}y(t)=\frac{1}{\Gamma (\alpha )} \int _{t}^{1}(s-t)^{\alpha -1}y(s)\,ds . $$

Definition 2.5

([6])

The left-sided and right-sided Riemann–Liouville fractional derivatives of order \(\alpha >0\) of a function \(y:(0,+\infty )\rightarrow R\) are given by

$$ D _{0^{+}}^{\alpha }y(t)=\frac{d^{n}}{dt^{n}} \bigl(I_{0^{+}}^{n-\alpha}y\bigr) (t),\qquad D _{1^{-}}^{\alpha }y(t)=(-1)^{n} \frac{d^{n}}{dt^{n}}\bigl(I_{1^{-}}^{n- \alpha}y\bigr) (t), $$

where \(n=[\alpha ]+1\).

Definition 2.6

([8])

The right-/left-sided Hilfer fractional derivative of order α and type β for a function \(y:(0,+\infty )\rightarrow R\) is given by

$$ D^{\alpha ,\beta}_{a\pm}y(t)=(\pm )^{n}I^{\beta (n-\alpha )}_{a\pm} \frac{d^{n}}{dt^{n}}\bigl(I^{(1-\beta )(n-\alpha )}_{a\pm}y\bigr) (t),\quad {n-1}< \alpha < n, 0\leq \beta \leq 1. $$

Remark

  1. (1)

    The operator \(D^{\alpha ,\beta}_{a\pm}\) can also be written as \(D^{\alpha ,\beta}_{a\pm}=I^{\beta (n-\alpha )}_{a\pm}D^{\gamma}_{a \pm}\), \(\gamma =\alpha +n\beta -\alpha \beta \).

  2. (2)

    If \(\beta =0\), then the Riemann–Liouville fractional derivative can be presented as \(D^{\alpha}_{a\pm}=D^{\alpha ,0}_{a\pm}\).

  3. (3)

    If \(\beta =1\), then the Caputo fractional derivative can be presented as \({}^{C} D^{\alpha}_{a\pm}=D^{\alpha ,1}_{a\pm}\).

Lemma 2.7

([6])

For \({n-1}< \alpha \leq n\), \(n \in N\), the general solution of the fractional differential equation \(D _{1-}^{\alpha }u(t)=0\) is given by

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

where \(c_{i}\in \mathbb{R}\), \(i=1,2,\ldots ,n\), \(n=[\alpha ]+1\).

Lemma 2.8

([6])

Let \(\alpha >0\), \(n=[\alpha ]+1\), if \(y\in L_{1}(0,1)\) and \(I^{n-\alpha}_{0+}y \in AC^{n}[0,1]\), then the following holds:

$$ I^{\alpha}_{0+}D^{\alpha}_{0+}y(t)=y(t)- \sum_{j=1}^{n} \frac{(I^{n-\alpha}_{0+}y(t))^{(n-j)}|_{t=0}}{\Gamma (\alpha -j+1)}t^{ \alpha -j}. $$

Lemma 2.9

([6])

For \({n-1}< \alpha \leq n\), \(n \in N\), the general solution of the fractional differential equation \(D _{0+}^{\alpha }u(t)=0\) is given by

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

where \(c_{i}\in \mathbb{R}\), \(i=1,2,\ldots ,n\), \(n=[\alpha ]+1\).

Lemma 2.10

([6])

If \(\alpha >0\), \(\beta >-1\), and \(\beta \neq \alpha -i\), \(i=1,2,\ldots ,[\alpha ]+1\), then

$$ D^{\alpha}_{0+}t^{\beta}= \frac{\Gamma (\beta +1)}{\Gamma (\beta -\alpha +1)}t^{\beta -\alpha},\qquad D^{\alpha}_{0+}t^{\alpha -i}=0. $$

Lemma 2.11

([6])

If \({\alpha}>{\beta}>0\), and \(y\in L_{1}(\mathbb{R}^{+})\), then

$$ D^{\beta}_{0+}I^{\alpha}_{0+}y(t)=I^{\alpha -\beta}_{0+}y(t),\qquad D^{ \alpha}_{0+}I^{\beta}_{0+}y(t)=D^{\alpha -\beta}_{0+}y(t). $$

In particular, when \(\beta =k\in \mathbb{N}\) and \(\alpha >k\), then

$$ \frac{d^{k}}{dt^{k}}I^{\alpha}_{0+}y(t)=I^{\alpha -k}_{0+}y(t). $$

Lemma 2.12

([24])

For any \(u,v\geq 0\), then

  1. (1)

    \(\varphi _{p}(u+v)\leq \varphi _{p}(u)+\varphi _{p}(v)\), \(1< p \leq 2\),

  2. (2)

    \(\varphi _{p}(u+v)\leq 2^{p-2}(\varphi _{p}(u)+\varphi _{p}(v))\), \(p \geq 2\),

where \(\varphi _{p}(s)=|s|^{p-2}s=s^{p-1}\), \(s\geq 0\).

3 Main results

Take

$$ X= \bigl\{ u\mid u(t), D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t), D^{\alpha _{2}-1, \beta _{2}}_{0+}u(t), D^{\alpha _{2},\beta _{2}}_{0+}u(t) \in C[0,1] \bigr\} ,\qquad {Y=C[0,1]}, $$

with norms

$$ \Vert u \Vert _{X}=\max_{t\in [0,1]} \bigl\{ \Vert u \Vert _{\infty}, \bigl\Vert D^{ \alpha _{2}-2,\beta _{2}}_{0+}u \bigr\Vert _{\infty}, \bigl\Vert D^{\alpha _{2}-1,\beta _{2}}_{0+}u \bigr\Vert _{\infty}, \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty} \bigr\} ,\qquad \Vert y \Vert _{Y}= \Vert y \Vert _{\infty}, $$

where \(\|y\|_{\infty}=\max_{t\in [0,1]} |y(t)|\).

Lemma 3.1

\((X,\|\cdot \|)\), \((Y,\|\cdot \|)\) are Banach spaces.

Proof

It is easy to see that \((Y,\|\cdot \|)\) is a Banach space. Next, we prove that \((X,{\|\cdot \|})\) is also a Banach space. Suppose that \(\{u_{n}\}\) is a Cauchy sequence of X, then \(\{u_{n}\}\), \(\{D^{\alpha _{2}-2,\beta _{2}}_{0+}u_{n}\}\), \(\{D^{\alpha _{2}-1,\beta _{2}}_{0+}u_{n}\}\), \(\{D^{\alpha _{2},\beta _{2}}_{0+}u_{n}\}\) are Cauchy sequences of \(C[0,1]\). So, there exist functions \(u,v, w,g\in C[0,1]\) such that \(u_{n}\), \(D^{\alpha _{2}-2,\beta _{2}}_{0+}u_{n}\), \(D^{\alpha _{2}-1,\beta _{2}}_{0+}u_{n}\), \(D^{\alpha _{2},\beta _{2}}_{0+}u_{n}\) converge uniformly to u, v, w, g on \([0,1]\), respectively. We need to prove that \(D^{\alpha _{2}-2,\beta _{2}}_{0+}u=v\), \(D^{\alpha _{2}-1,\beta _{2}}_{0+}u=w\), \(D^{\alpha _{2},\beta _{2}}_{0+}u=g\). By Lemma 2.8, we get

$$ I^{\alpha _{2}-2}_{0+}D^{\alpha _{2}-2,\beta _{2}}_{0+}u_{n}=I^{ \alpha _{2}-2}_{0+}I^{\beta _{2}(3-\alpha _{2})}_{0+}D^{\gamma _{2}-2}_{0+}u_{n}=I^{ \gamma _{2}-2}_{0+}D^{\gamma _{2}-2}_{0+}u_{n}=u_{n}+ct^{\gamma _{2}-3}. $$

So, we have

$$ \frac{1}{\Gamma (\alpha _{2}-2)} \int _{0}^{t}(t-s)^{\alpha _{2}-3}D^{ \alpha _{2}-2,\beta _{2}}_{0+}u_{n}(s) \,ds =u_{n}+ct^{\gamma _{2}-3}. $$

Let \(n\rightarrow \infty \), we get

$$ \frac{1}{\Gamma (\alpha _{2}-2)} \int _{0}^{t}(t-s)^{\alpha _{2}-3}v(s)\,ds =u+ct^{ \gamma _{2}-3}. $$
(3.1)

Applying \(D^{\gamma _{2}-2}_{0+}\) and \(I^{\beta _{2}(3-\alpha _{2})}_{0+}\) to the both sides of (3.1), we obtain

$$ I^{\beta _{2}(3-\alpha _{2})}_{0+}D^{\gamma _{2}-2}_{0+}I^{\alpha _{2}-2}_{0+}v(t)=I^{ \beta _{2}(3-\alpha _{2})}_{0+}D^{\gamma _{2}-2}_{0+}u=D^{\alpha _{2}-2, \beta _{2}}_{0+}u. $$

Therefore, from Lemma 2.11 and Lemma 2.8, we get \(v=D^{\alpha _{2}-2,\beta _{2}}_{0+}u\).

Since \(I^{\alpha _{2}-1}_{0+}D^{\alpha _{2}-1,\beta _{2}}_{0+}u_{n}=u_{n}+c_{1}t^{ \gamma _{2}-2}+c_{2}t^{\gamma _{2}-3}\) and \(I^{\alpha _{2}}_{0+}D^{\alpha _{2},\beta _{2}}_{0+}u_{n}=u_{n}+c_{1}t^{ \gamma _{2}-1}+c_{2}t^{\gamma _{2}-2}+c_{3}t^{\gamma _{2}-3}\), similar to the above proof we can get \(w=D^{\alpha _{2}-1,\beta _{2}}_{0+}u\) and \(g=D^{\alpha _{2},\beta _{2}}_{0+}u\). So, \((X,\|\cdot \|)\) is a Banach space. The proof is completed. □

In order to obtain our main results, we always suppose that the following conditions hold:

\((H_{1})\):

\(T_{1}(t^{\gamma _{2}-1})T_{2}(t^{\gamma _{2}-2})=T_{1}(t^{\gamma _{2}-2})T_{2}(t^{ \gamma _{2}-1})\).

\((H_{2})\):

Functionals \(T_{i}:X\rightarrow \mathbb{R}\) are linear bounded with the respective norms \(\|T_{i}\|\), \(i=1,2\). And the functionals \(T_{1}\), \(T_{2}\) satisfy the relations \(T_{1}(t^{\gamma _{2}-1})=\delta _{2}\), \(T_{1}(t^{\gamma _{2}-2})=\delta _{1}\), \(T_{2}(t^{\gamma _{2}-1})=k\delta _{2}\), \(T_{2}(t^{\gamma _{2}-2})=k\delta _{1}\), where \(\delta _{1}, \delta _{2}, k\in \mathbb{R}\), \(\delta ^{2}_{1}+\delta ^{2}_{2}\neq 0\).

\((H_{3})\):

Functional \(G(y)=(T_{2}-kT_{1})(I^{\alpha _{2}}_{0+}\varphi _{q}(I^{\alpha _{1}}_{1-}y))\), \(\frac{1}{p}+\frac{1}{q}=1\) is increasing.

Define operators \(L:\operatorname{dom}L\cap X\rightarrow Y\) and \(N_{\lambda}:X\rightarrow Y\) as follows

$$\begin{aligned} &Lu(t)= D^{\alpha _{1},\beta _{1}}_{1-}\varphi _{p}{ \bigl(D^{\alpha _{2}, \beta _{2}}_{0+}u(t)\bigr)}, \\ &N_{\lambda}u(t)=\lambda f\bigl(t, u(t), D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t),D^{ \alpha _{2}-1,\beta _{2}}_{0+}u(t), D^{\alpha _{2},\beta _{2}}_{0+}u(t)\bigr),\quad t \in [0,1], \lambda \in [0,1], \end{aligned}$$

where

$$\begin{aligned} \operatorname{dom}L={}& \bigl\{ u(t)\mid u(t)\in X, D^{\alpha _{1},\beta _{1}}_{1-} \varphi _{p}{\bigl(D^{ \alpha _{2},\beta _{2}}_{0+}u(t)\bigr)}\in Y, u(0)=0, \\ &D^{\alpha _{2}, \beta _{2}}_{0+}u(1)=0, T_{1}(u)=T_{2}(u)=0 \bigr\} . \end{aligned}$$

Lemma 3.2

Suppose that (\(H_{1}\)) holds, then L is a quasilinear operator.

Proof

It is easy to get that \(\operatorname{Ker}L=\{u\in \operatorname{dom}L\mid u(t)=c(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{ \gamma _{2}-1}), c\in \mathbb{R}\}\).

For \({y\in \operatorname{Im}L}\), there exists \({u\in \operatorname{dom}L}\) such that \({D^{\alpha _{1},\beta _{1}}_{1-}\varphi _{p}{(D^{\alpha _{2},\beta _{2}}_{0+}u(t))} =y(t)}\). According to Remark, we get

$$ I^{\beta _{1}(1-\alpha _{1})}_{1-}D^{\gamma _{1}}_{1-}\varphi _{p}{\bigl(D^{ \alpha _{2},\beta _{2}}_{0+}u(t)\bigr)}=y(t). $$
(3.2)

Thus, applying \(D^{\beta _{1}(1-\alpha _{1})}_{1-}\) to the both sides of (3.2), and by Lemma 2.7, we have

$$ D^{\alpha _{2},\beta _{2}}_{0+}u(t)=\varphi _{q}{ \bigl(I^{\alpha _{1}}_{1-}y(t)+c_{1}(1-t)^{ \gamma _{1}-1} \bigr)}. $$

Since \(D^{\alpha _{2},\beta _{2}}_{0+}u(1)=0\), we can get

$$ D^{\alpha _{2},\beta _{2}}_{0+}u(t)=\varphi _{q}{ \bigl(I^{\alpha _{1}}_{1-}y(t)\bigr)} . $$
(3.3)

Applying \(D^{\beta _{2}(3-\alpha _{2})}_{0+}\) to the both sides of (3.3), and because of \(u(0)=0\), we obtain

$$ u(t)=I^{\alpha _{2}}_{0+}\varphi _{q}{ \bigl(I^{\alpha _{1}}_{1-}y(t)\bigr)}+c_{2}t^{ \gamma _{2}-1}+c_{3}t^{\gamma _{2}-2}. $$

The functional boundary condition \(T_{1}(u)=T_{2}(u)=0\) implies that

$$\begin{aligned}& T_{1}(u)=T_{1}\bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-}y(t)\bigr)} \bigr)+c_{2} \delta _{2}+c_{3}\delta _{1}=0, \\& T_{2}(u)=T_{2}\bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-}y(t)\bigr)} \bigr)+kc_{2} \delta _{2}+kc_{3}\delta _{1}=0. \end{aligned}$$

Obviously,

$$ (T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-}y(t)\bigr)} \bigr)=0. $$
(3.4)

Hence, \(\operatorname{Im}L\subseteq \{y\in Y|(T_{2}-kT_{1})(I^{\alpha _{2}}_{0+} \varphi _{q}{(I^{\alpha _{1}}_{1-}y(t))})=0 \}\).

Conversely, if \(y\in Y\) and satisfies (3.4), let

$$ u(t)=I^{\alpha _{2}}_{0+}\varphi _{q}{ \bigl(I^{\alpha _{1}}_{1-}y(t)\bigr)}+ \frac{T_{1}(I^{\alpha _{2}}_{0+}\varphi _{q}{(I^{\alpha _{1}}_{1-}y(t))})}{\delta ^{2}_{1}+\delta ^{2}_{2}}\bigl( \delta _{2}t^{\gamma _{2}-1}+\delta _{1}t^{\gamma _{2}-2} \bigr).$$

It is easy to prove that \(u(t)\) satisfies the boundary conditions of problem (1.1), and we have

$$\begin{aligned} Lu(t)&=I^{\beta _{1}(1-\alpha _{1})}_{1-}D^{\gamma _{1}}_{1-} \varphi _{p}\bigl(I^{ \beta _{2}(3-\alpha _{2})}_{0+}D^{\gamma _{2}}_{0+}I^{\alpha _{2}}_{0+} \varphi _{q}\bigl(I^{\alpha _{1}}_{1-}y(t)\bigr) \bigr) \\ &=I^{\beta _{1}(1-\alpha _{1})}_{1-}D^{\beta _{1}(1-\alpha _{1})}_{1-}y(t)=y(t). \end{aligned}$$

Therefore,

$$ \operatorname{Im}L\supseteq \bigl\{ y\in Y\mid (T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+}\varphi _{q}{ \bigl(I^{ \alpha _{1}}_{1-}y(t)\bigr)}\bigr)=0 \bigr\} . $$

In summary, we get

$$ \operatorname{Im}L= \bigl\{ y\in Y\mid (T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+}\varphi _{q}{ \bigl(I^{ \alpha _{1}}_{1-}y(t)\bigr)}\bigr)=0 \bigr\} . $$

Clearly, \(\operatorname{Im}L\subset Y\) is closed. So, L is a quasilinear operator. □

Define the operator \(P:X\rightarrow \operatorname{Ker}L\) by

$$ Pu(t)= \frac{\delta _{2}D^{\gamma _{2}-2}_{0+}u(0)-\delta _{1}D^{\gamma _{2}-1}_{0+}u(0)}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})}\bigl( \delta _{2}t^{\gamma _{2}-2}- \delta _{1}t^{\gamma _{2}-1}\bigr). $$

It is clear that \(P^{2}u=Pu\) and \(\operatorname{Im}P=\operatorname{Ker}L\), \(X=\operatorname{Ker}L\oplus \operatorname{Ker}P\). So, \(P:X\rightarrow \operatorname{Ker}L\) is a projector.

Define the operator \(Q:Y\rightarrow R\) by

$$ Qy(t)=c, $$

where c satisfies

$$ (T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-}\bigl(y(t)-c \bigr)\bigr)}\bigr)=0. $$
(3.5)

Next, we will prove that c is the unique constant satisfying (3.5). For \(y\in Y\), let

$$\begin{aligned} F(c)=(T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-}\bigl(y(t)-c \bigr)\bigr)}\bigr). \end{aligned}$$

Obviously, \(F(c)\) is continuous and strictly decreasing in \(\mathbb{R}\). We make \(c_{1}=\min_{t\in [0,1]}y(t)\), \(c_{2}=\max_{t\in [0,1]}y(t)\). It is easy to see that \(F(c_{1})\geq 0\), \(F(c_{2})\leq 0\), then, there exists a unique constant \(c\in [c_{1},c_{2}]\) such that \(F(c)=0\).

Lemma 3.3

\(Q:Y\rightarrow Y_{1}\) is continuous, bounded and \(Q(I-Q)y=Q(y-Qy)=0\), \(y\in Y\), \(QY=Y_{1}\), where \(Y_{1}=\mathbb{R}\).

Proof

For \(y_{1}, y_{2}\in Y\), assume \({Qy_{1}=c_{1}}\), \({Qy_{2}=c_{2}}\). Since \(\varphi _{q}\) is strictly increasing, if \(c_{2}-c_{1}> \max_{t\in [0,1]}(y_{2}(t)-y_{1}(t))\), then

$$\begin{aligned} \begin{aligned} 0&= (T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-} \bigl(y_{2}(t)-c_{2}\bigr)\bigr)}\bigr) \\ &= (T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-} \bigl(y_{1}(t)-c_{1}+y_{2}(t)-y_{1}(t)-(c_{2}-c_{1}) \bigr)\bigr)}\bigr) \\ &< (T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-} \bigl(y_{1}(t)-c_{1}\bigr)\bigr)}\bigr)=0. \end{aligned} \end{aligned}$$

A contradiction. On the other hand, if \(c_{2}-c_{1}< \min_{t\in [0,1]}(y_{2}(t)-y_{1}(t))\), then

$$\begin{aligned} 0&= (T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-} \bigl(y_{2}(t)-c_{2}\bigr)\bigr)}\bigr) \\ &= (T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-} \bigl(y_{1}(t)-c_{1}+y_{2}(t)-y_{1}(t)-(c_{2}-c_{1}) \bigr)\bigr)}\bigr) \\ &>(T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+} \varphi _{q}{\bigl(I^{\alpha _{1}}_{1-} \bigl(y_{1}(t)-c_{1}\bigr)\bigr)}\bigr)=0. \end{aligned}$$

A contradiction, too. So, we can get

$$ \min_{t\in [0,1]}\bigl(y_{2}(t)-y_{1}(t) \bigr)\leq c_{2}-c_{1}\leq \max_{t\in [0,1]} \bigl(y_{2}(t)-y_{1}(t)\bigr), \quad \text{i.e.}\quad \vert c_{2}-c_{1} \vert \leq \| y_{2}-y_{1} \| _{\infty}. $$

Therefore, Q is continuous. In addition, if \(\Omega \subset Y\) is bounded, then \(Q(\Omega )\) is bounded, \(i.e\)., Q is bounded. According to the definition of Q, we can easily know that Q is not a projector but satisfies \(Q(I-Q)Y=Q(Y-QY)=0\), \(y\in Y\) and \(QY=Y_{1}\). □

Lemma 3.4

Define an operator \(R: X\times [0,1]\rightarrow X_{2}\) as

$$\begin{aligned} R(u,\lambda ) (t)&= I^{\alpha _{2}}_{0+}\varphi _{q}\bigl(I^{\alpha _{1}}_{1-}(I-Q)N_{ \lambda}u(t) \bigr) \\ &\quad{}- \frac{T_{1}(I^{\alpha _{2}}_{0+}\varphi _{q}(I^{\alpha _{1}}_{1-}(I-Q)N_{\lambda}u(t)))}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})}\bigl( \delta _{2}\Gamma (\gamma _{2}-1)t^{\gamma _{2}-1}+\delta _{1}\Gamma ( \gamma _{2})t^{\gamma _{2}-2}\bigr), \end{aligned}$$

where \(\operatorname{Ker}L\oplus X_{2}=X\).

Then \(R:\overline{\Omega}\times [0,1]\rightarrow X_{2}\) is continuous and compact, where \(\Omega \subset X\) is an open bounded set.

Proof

Obviously, R is continuous. Let A be any bounded set in X, for \(\forall u\in A\), \(D^{\alpha _{2}-1,\beta _{2}}_{0+}u\in A\), \(D^{\alpha _{2}-2,\beta _{2}}_{0+}u\in A\), \(\lambda \in [0,1]\). By the continuity of f and the boundedness of Q, we can get that there exist constants \(k_{1}>0\), \(k_{2}>0\) such that \(|f(t, u(t), D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t), D^{\alpha _{2}-1, \beta _{2}}_{0+}u(t), D^{\alpha _{2},\beta _{2}}_{0+}u(t))|\leq k_{1}\), \(|Qf|\leq k_{2}\) for \(u\in \overline{\Omega}\). Note that

$$\begin{aligned}& \bigl\vert I^{\alpha _{2}}_{0+}\varphi _{q} \bigl(I^{\alpha _{1}}_{1-}(I-Q)N_{ \lambda}u(t)\bigr) \bigr\vert \\& \quad \leq \frac{1}{\Gamma (\alpha _{2})} \int _{0}^{t}(t-s)^{ \alpha _{2}-1}\varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{ \alpha _{1}-1} \bigl\vert (I-Q)N_{\lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{2})} \int _{0}^{t}(t-s)^{\alpha _{2}-1} \varphi _{q} \biggl(\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr)\,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{2}+1)}\varphi _{q} \biggl( \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr), \\& \bigl\vert D^{\alpha _{2}-2,\beta _{2}}_{0+}I^{\alpha _{2}}_{0+} \varphi _{q}\bigl(I^{ \alpha _{1}}_{1-}(I-Q)N_{\lambda}u(t) \bigr) \bigr\vert \\& \quad \leq \int _{0}^{t}(t-s) \varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{ \alpha _{1}-1} \bigl\vert (I-Q)N_{\lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\& \quad \leq \frac{1}{2}\varphi _{q} \biggl( \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr), \\& \bigl\vert D^{\alpha _{2}-1,\beta _{2}}_{0+}I^{\alpha _{2}}_{0+} \varphi _{q}\bigl(I^{ \alpha _{1}}_{1-}(I-Q)N_{\lambda}u(t) \bigr) \bigr\vert \\& \quad \leq \int _{0}^{t} \varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{ \alpha _{1}-1} \bigl\vert (I-Q)N_{\lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\& \quad \leq \varphi _{q} \biggl(\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr), \end{aligned}$$

and

$$\begin{aligned} \bigl\vert D^{\alpha _{2},\beta _{2}}_{0+}I^{\alpha _{2}}_{0+} \varphi _{q}\bigl(I^{ \alpha _{1}}_{1-}(I-Q)N_{\lambda}u(t) \bigr) \bigr\vert \leq \varphi _{q} \biggl( \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr). \end{aligned}$$

Therefore,

$$\begin{aligned} &\bigl\Vert I^{\alpha _{2}}_{0+}\varphi _{q} \bigl(I^{\alpha _{1}}_{1-}(I-Q)N_{ \lambda}u\bigr) \bigr\Vert _{X} \\ &\quad \leq \max \biggl\{ \frac{1}{\Gamma (\alpha _{2}+1)} \varphi _{q} \biggl(\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr), \frac{1}{2} \varphi _{q} \biggl( \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr), \varphi _{q} \biggl( \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr) \biggr\} \\ &\quad = \varphi _{q} \biggl(\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr), \end{aligned}$$

then we have

$$\begin{aligned}& \begin{aligned} \bigl\vert R(u,\lambda ) (t) \bigr\vert &\leq \bigl\Vert I^{\alpha _{2}}_{0+}\varphi _{q} \bigl(I^{ \alpha _{1}}_{1-}(I-Q)N_{\lambda}u\bigr) \bigr\Vert _{X} \\ &\quad{}+ \frac{ \Vert T_{1} \Vert _{\infty}( \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)+ \vert \delta _{1} \vert \Gamma (\gamma _{2}))}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})} \bigl\Vert I^{\alpha _{2}}_{0+} \varphi _{q}\bigl(I^{\alpha _{1}}_{1-}(I-Q)N_{ \lambda}u \bigr) \bigr\Vert _{X} \\ &\leq \biggl[1+ \frac{ \Vert T_{1} \Vert _{\infty}( \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)+ \vert \delta _{1} \vert \Gamma (\gamma _{2}))}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})} \biggr] \bigl\Vert I^{\alpha _{2}}_{0+} \varphi _{q}\bigl(I^{\alpha _{1}}_{1-}(I-Q)N_{ \lambda}u \bigr) \bigr\Vert _{X} \\ &\leq \biggl[1+ \frac{ \Vert T_{1} \Vert _{\infty}( \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)+ \vert \delta _{1} \vert \Gamma (\gamma _{2}))}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})} \biggr]\varphi _{q} \biggl( \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr), \end{aligned} \\& \bigl\vert D^{\alpha _{2}-2,\beta _{2}}_{0+}R(u,\lambda ) (t) \bigr\vert \\& \quad \leq \int _{0}^{t}(t-s)\varphi _{q} \biggl( \frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{\alpha _{1}-1} \bigl\vert (I-Q)N_{ \lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\& \quad \quad{}+ \frac{ \Vert T_{1} \Vert _{\infty} \Vert I^{\alpha _{2}}_{0+}\varphi _{q}(I^{\alpha _{1}}_{1-}(I-Q)N_{\lambda}u) \Vert _{X}}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})} \\& \quad \quad {}\times\bigl( \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)I^{\beta _{2}(3-\alpha _{2})}_{0+} \Gamma ( \gamma _{2})t+ \vert \delta _{1} \vert \Gamma (\gamma _{2})I^{\beta _{2}(3- \alpha _{2})}_{0+}\Gamma (\gamma _{2}-1) \bigr) \\& \quad \leq \int _{0}^{t}(t-s)\varphi _{q} \biggl( \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr)\,ds \\& \quad \quad {}+ \frac{ \Vert T_{1} \Vert _{\infty}\varphi _{q} (\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} )}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})} \biggl( \frac{ \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)\Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)}+ \frac{ \vert \delta _{1} \vert \Gamma (\gamma _{2}-1)\Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+1)} \biggr) \\& \quad \leq \biggl[\frac{1}{2}+ \frac{ \Vert T_{1} \Vert _{\infty} ( \vert \delta _{2} \vert \Gamma (\gamma _{2})+(\beta _{2}(3-\alpha _{2})+1) \vert \delta _{1} \vert \Gamma (\gamma _{2}) )}{(\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1))\Gamma (\beta _{2}(3-\alpha _{2})+2)} \biggr]\varphi _{q} \biggl(\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr), \\& \bigl\vert D^{\alpha _{2}-1,\beta _{2}}_{0+}R(u,\lambda ) (t) \bigr\vert \\& \quad \leq \int _{0}^{t}\varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{ \alpha _{1}-1} \bigl\vert (I-Q)N_{\lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\& \quad \quad{}+ \frac{ \Vert T_{1} \Vert _{\infty} \Vert I^{\alpha _{2}}_{0+}\varphi _{q}(I^{\alpha _{1}}_{1-}(I-Q)N_{\lambda}u) \Vert _{X}}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})} \times \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)I^{\beta _{2}(3-\alpha _{2})}_{0+} \Gamma ( \gamma _{2}) \\& \quad \leq \int _{0}^{t}\varphi _{q} \biggl( \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr)\,ds + \frac{ \Vert T_{1} \Vert _{\infty}\varphi _{q} (\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} )}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})} \times \frac{ \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)\Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+1)} \\& \quad \leq \biggl[1+ \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{2} \vert \Gamma (\gamma _{2})}{(\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1))\Gamma (\beta _{2}(3-\alpha _{2})+1)} \biggr]\varphi _{q} \biggl( \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr), \\& \begin{aligned} \bigl\vert D^{\alpha _{2},\beta _{2}}_{0+}R(u,\lambda ) (t) \bigr\vert & \leq \varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1})} \int _{t}^{1}(s-t)^{ \alpha _{1}-1} \bigl\vert (I-Q)N_{\lambda}u(s) \bigr\vert \,ds \biggr) \\ &\leq \varphi _{q} \biggl(\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr). \end{aligned} \end{aligned}$$

So, R is bounded in \(\overline{\Omega}\times [0,1]\).

For \((u,\lambda )\in \overline{\Omega}\times [0,1]\), \(0\leq t_{1}< t_{2}\leq 1\), we have

$$\begin{aligned}& \bigl\vert R(u,\lambda ) (t_{2})-R(u,\lambda ) (t_{1}) \bigr\vert \\& \quad \leq \biggl\vert \frac{1}{\Gamma (\alpha _{2})} \int _{0}^{t_{2}}(t_{2}-s)^{ \alpha _{2}-1} \varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{ \alpha _{1}-1}(I-Q)N_{\lambda}u(x) \,dx \biggr)\,ds \\& \quad \quad{}-\frac{1}{\Gamma (\alpha _{2})} \int _{0}^{t_{1}}(t_{1}-s)^{\alpha _{2}-1} \varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{ \alpha _{1}-1}(I-Q)N_{\lambda}u(x) \,dx \biggr)\,ds \biggr\vert \\& \quad \quad{}+ \frac{ \Vert T_{1} \Vert _{\infty} \Vert I^{\alpha _{2}}_{0+}\varphi _{q}(I^{\alpha _{1}}_{1-}(I-Q)N_{\lambda}u) \Vert _{X}}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1} \Gamma (\gamma _{2})} \\& \quad \quad {}\times \bigl( \vert \delta _{2} \vert \Gamma (\gamma _{2}-1) \bigl(t^{\gamma _{2}-1}_{2}-t^{ \gamma _{2}-1}_{1} \bigr)+ \vert \delta _{1} \vert \Gamma (\gamma _{2}) \bigl(t^{\gamma _{2}-2}_{2}-t^{ \gamma _{2}-2}_{1}\bigr) \bigr) \\& \quad \leq \frac{1}{\Gamma (\alpha _{2})} \int _{0}^{t_{1}}\bigl[(t_{2}-s)^{ \alpha _{2}-1}-(t_{1}-s)^{\alpha _{2}-1} \bigr] \\& \quad \quad {}\times\varphi _{q} \biggl( \frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{\alpha _{1}-1} \bigl\vert (I-Q)N_{ \lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\& \quad \quad{}+\frac{1}{\Gamma (\alpha _{2})} \int _{t_{1}}^{t_{2}}(t_{2}-s)^{ \alpha _{2}-1} \varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{ \alpha _{1}-1} \bigl\vert (I-Q)N_{\lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\& \quad \quad{}+ \frac{ \Vert T_{1} \Vert _{\infty}\varphi _{q} (\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} )}{\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1)} \bigl( \vert \delta _{2} \vert \bigl(t^{\gamma _{2}-1}_{2}-t^{\gamma _{2}-1}_{1} \bigr)+ \vert \delta _{1} \vert (\gamma _{2}-1) \bigl(t^{\gamma _{2}-2}_{2}-t^{\gamma _{2}-2}_{1}\bigr) \bigr) \\& \quad \leq \frac{\varphi _{q} (\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} )}{\Gamma (\alpha _{2})} \biggl[ \int _{0}^{t_{1}}\bigl[(t_{2}-s)^{\alpha _{2}-1}-(t_{1}-s)^{\alpha _{2}-1} \bigr]\,ds + \int _{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha _{2}-1} \,ds \biggr] \\& \quad \quad{}+ \frac{ \Vert T_{1} \Vert _{\infty}\varphi _{q} (\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} )}{\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1)} \bigl( \vert \delta _{2} \vert \bigl(t^{\gamma _{2}-1}_{2}-t^{\gamma _{2}-1}_{1} \bigr)+ \vert \delta _{1} \vert (\gamma _{2}-1) \bigl(t^{\gamma _{2}-2}_{2}-t^{\gamma _{2}-2}_{1}\bigr) \bigr) \\& \quad \leq \varphi _{q} \biggl(\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr) \biggl[ \frac{(t^{\alpha _{2}}_{2}-t^{\alpha _{2}}_{1})}{\Gamma (\alpha _{2}+1)}+ \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{2} \vert }{\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1)}\bigl(t^{ \gamma _{2}-1}_{2}-t^{\gamma _{2}-1}_{1} \bigr) \\& \quad \quad {}+ \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{1} \vert (\gamma _{2}-1)}{\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1)}\bigl(t^{ \gamma _{2}-2}_{2}-t^{\gamma _{2}-2}_{1} \bigr) \biggr], \\& \bigl\vert D^{\alpha _{2}-2,\beta _{2}}_{0+}R(u,\lambda ) (t_{2})-D^{ \alpha _{2}-2,\beta _{2}}_{0+}R(u,\lambda ) (t_{1}) \bigr\vert \\& \quad \leq \biggl\vert \int _{0}^{t_{2}}(t_{2}-s)\varphi _{q} \biggl( \frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{\alpha _{1}-1}(I-Q)N_{ \lambda}u(x) \,dx \biggr)\,ds \\& \quad \quad{}- \int _{0}^{t_{1}}(t_{1}-s)\varphi _{q} \biggl( \frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{\alpha _{1}-1}(I-Q)N_{ \lambda}u(x) \,dx \biggr)\,ds \biggr\vert \\& \quad \quad{}+ \frac{ \Vert T_{1} \Vert _{\infty}\varphi _{q} (\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} )}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})} \\& \quad \quad {}\times\biggl( \frac{ \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)\Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)}t^{ \beta _{2}(3-\alpha _{2})+1}_{2}- \frac{ \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)\Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)}t^{ \beta _{2}(3-\alpha _{2})+1}_{1} \biggr) \\& \quad \leq \int _{0}^{t_{1}}\bigl[(t_{2}-s)-(t_{1}-s) \bigr]\varphi _{q} \biggl( \frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{\alpha _{1}-1} \bigl\vert (I-Q)N_{ \lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\& \quad \quad{}+ \int _{t_{1}}^{t_{2}}(t_{2}-s)\varphi _{q} \biggl( \frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{\alpha _{1}-1} \bigl\vert (I-Q)N_{ \lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\& \quad \quad{}+ \frac{ \Vert T_{1} \Vert _{\infty}\varphi _{q} (\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} ) \vert \delta _{2} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)(\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1))} \bigl(t^{\beta _{2}(3-\alpha _{2})+1}_{2}-t^{\beta _{2}(3-\alpha _{2})+1}_{1} \bigr) \\& \quad \leq \varphi _{q} \biggl(\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr) \biggl[ \int _{0}^{t_{1}}\bigl[(t_{2}-s)-(t_{1}-s) \bigr]\,ds + \int _{t_{1}}^{t_{2}}(t_{2}-s)\,ds \biggr] \\& \quad \quad{}+ \frac{ \Vert T_{1} \Vert _{\infty}\varphi _{q} (\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} ) \vert \delta _{2} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)(\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1))} \bigl(t^{\beta _{2}(3-\alpha _{2})+1}_{2}-t^{\beta _{2}(3-\alpha _{2})+1}_{1} \bigr) \\& \quad \leq \varphi _{q} \biggl(\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr) \biggl[ \frac{(t^{2}_{2}-t^{2}_{1})}{2} \\& \quad \quad {}+ \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{2} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)(\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1))} \bigl(t^{\beta _{2}(3-\alpha _{2})+1}_{2}-t^{\beta _{2}(3-\alpha _{2})+1}_{1} \bigr) \biggr], \\& \bigl\vert D^{\alpha _{2}-1,\beta _{2}}_{0+}R(u,\lambda ) (t_{2})-D^{ \alpha _{2}-1,\beta _{2}}_{0+}R(u,\lambda ) (t_{1}) \bigr\vert \\& \quad \leq \biggl\vert \int _{0}^{t_{2}}\varphi _{q} \biggl( \frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{\alpha _{1}-1}(I-Q)N_{ \lambda}u(x) \,dx \biggr)\,ds \\& \quad \quad{}- \int _{0}^{t_{1}}\varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{\alpha _{1}-1}(I-Q)N_{\lambda}u(x) \,dx \biggr)\,ds \biggr\vert \\& \quad \leq \int _{t_{1}}^{t_{2}}\varphi _{q} \biggl( \frac{1}{\Gamma (\alpha _{1})} \int _{s}^{1}(x-s)^{\alpha _{1}-1} \bigl\vert (I-Q)N_{ \lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\& \quad \leq \varphi _{q} \biggl(\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)} \biggr) (t_{2}-t_{1}). \end{aligned}$$

So, \(\{R(u,\lambda )\mid (u,\lambda )\in \overline{\Omega}\times [0,1]\}\), \(\{D^{\alpha _{2}-2,\beta _{2}}_{0+}R(u,\lambda )\mid (u,\lambda ) \in \overline{\Omega}\times [0,1]\}\) and \(\{D^{\alpha _{2}-1,\beta _{2}}_{0+}R(u,\lambda )\mid (u,\lambda ) \in \overline{\Omega}\times [0,1]\}\) are equicontinuous. Next, we prove that \(\{D^{\alpha _{2},\beta _{2}}_{0+}R(u,\lambda )\mid (u,\lambda )\in \overline{\Omega}\times [0,1]\}\) is also equicontinuous.

For \((u,\lambda )\in \overline{\Omega}\times [0,1]\), \(0\leq t_{1}< t_{2}\leq 1\), then

$$\begin{aligned} & \bigl\vert D^{\alpha _{2},\beta _{2}}_{0+}R(u,\lambda ) (t_{2})-D^{\alpha _{2}, \beta _{2}}_{0+}R(u,\lambda ) (t_{1}) \bigr\vert \\ &\quad = \biggl\vert \varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1})} \int _{t_{2}}^{1}(s-t_{2})^{ \alpha _{1}-1}(I-Q)N_{\lambda}u(s) \,ds \biggr) \\ &\quad \quad {}-\varphi _{q} \biggl( \frac{1}{\Gamma (\alpha _{1})} \int _{t_{1}}^{1}(s-t_{1})^{\alpha _{1}-1}(I-Q)N_{ \lambda}u(s) \,ds \biggr) \biggr\vert . \end{aligned}$$

Since

$$\begin{aligned} & \biggl\vert \frac{1}{\Gamma (\alpha _{1})} \int _{t_{2}}^{1}(s-t_{2})^{ \alpha _{1}-1}(I-Q)N_{\lambda}u(s) \,ds -\frac{1}{\Gamma (\alpha _{1})} \int _{t_{1}}^{1}(s-t_{1})^{\alpha _{1}-1}(I-Q)N_{\lambda}u(s) \,ds \biggr\vert \\ &\quad \leq \frac{1}{\Gamma (\alpha _{1})} \biggl( \int _{t_{2}}^{1}\bigl[(s-t_{2})^{ \alpha _{1}-1}-(s-t_{1})^{\alpha _{1}-1} \bigr] \bigl\vert (I-Q)N_{\lambda}u(s) \bigr\vert \,ds \\ &\quad \quad {}+ \int _{t_{1}}^{t_{2}}(s-t_{1})^{\alpha _{1}-1} \bigl\vert (I-Q)N_{\lambda}u(s) \bigr\vert \,ds \biggr) \\ &\quad \leq \frac{2(k_{1}+k_{2})}{\Gamma (\alpha _{1}+1)}(t_{2}-t_{1})^{ \alpha _{1}}, \end{aligned}$$

and

$$\begin{aligned} \biggl\vert \frac{1}{\Gamma (\alpha _{1})} \int _{t}^{1}(s-t)^{\alpha _{1}-1}(I-Q)N_{ \lambda}u(s) \,ds \biggr\vert \leq \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)}, \quad (u, \lambda )\in \overline{\Omega} \times [0,1], \end{aligned}$$

and taking into account that \(\varphi _{q}\) is uniformly continuous in \([-\frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)}, \frac{k_{1}+k_{2}}{\Gamma (\alpha _{1}+1)}]\), we can obtain \(\{D^{\alpha _{2},\beta _{2}}_{0+}R(u,\lambda )\mid (u,\lambda )\in \overline{\Omega}\times [0,1]\}\) is also equicontinuous. By the Arzela–Ascoli theorem, we get that \(R:\Omega \times [0,1]\rightarrow X_{2}\) is compact. □

Lemma 3.5

Assume that \(\Omega \subset X\) is an open and bounded set. Then \(N_{\lambda}\) is L-quasicompact in Ω̅.

Proof

It is obvious that \({\operatorname{Im}P=\operatorname{Ker}L}\), \({\operatorname{dim}\operatorname{Ker}L=\operatorname{dim}\operatorname{Im}Q }\), \({Q(I-Q)=0}\), \({\operatorname{Ker}Q=\operatorname{Im}L}\), \({R(\cdot ,0)=0}\) and that Definition 2.2(b) holds.

For \(u\in \Sigma _{\lambda}=\{u\in \overline{\Omega}\mid Lu=N_{\lambda}u \}\), we can get \(N_{\lambda}u\in \operatorname{Im}L=\operatorname{Ker}Q\). Thus, we have \(QN_{\lambda}u=0\) and \(N_{\lambda}u=Lu=D^{\alpha _{1},\beta _{1}}_{1-}\varphi _{p} (D^{ \alpha _{2},\beta _{2}}_{0^{+}}u )\), then

$$\begin{aligned} I^{\alpha _{2}}_{0+}\varphi _{q} \bigl(I^{\alpha _{1}}_{1-}N_{\lambda}u(t) \bigr) &= I^{\alpha _{2}}_{0+}\varphi _{q} \bigl(I^{\alpha _{1}}_{1-}I^{ \beta _{1}(1-\alpha _{1})}_{1-}D^{\gamma _{1}}_{1-} \varphi _{p}\bigl(I^{ \beta _{2}(3-\alpha _{2})}_{0+}D^{\gamma _{2}}_{0+}u(t) \bigr) \bigr)=I^{ \gamma _{2}}_{0+}D^{\gamma _{2}}_{0+}u(t) \\ &= u(t)-\frac{D^{\gamma _{2}-1}_{0+}u(0)}{\Gamma (\gamma _{2})}t^{ \gamma _{2}-1}- \frac{D^{\gamma _{2}-2}_{0+}u(0)}{\Gamma (\gamma _{2}-1)}t^{\gamma _{2}-2}- \frac{D^{\gamma _{2}-3}_{0+}u(0)}{\Gamma (\gamma _{2}-2)}t^{\gamma _{2}-3}. \end{aligned}$$

Since \(u(0)=0\), we obtain \(D^{\gamma _{2}-3}_{0+}u(0)=0\). It follows from \(D^{\alpha _{2},\beta _{2}}_{0+}u(1)=u(0)= D^{\alpha _{2},\beta _{2}}_{0+}R(u, \lambda )(1)=R(u,\lambda )(0)=D^{\gamma _{2}-3}_{0+}u(0)=T_{1}(u)=0\) that

$$\begin{aligned} R(u,\lambda )&= I^{\alpha _{2}}_{0+}\varphi _{q} \bigl(I^{\alpha _{1}}_{1-}\bigl(N_{ \lambda}u(t)-QN_{\lambda}u(t) \bigr) \bigr) \\ &\quad{}- \frac{T_{1}(I^{\alpha _{2}}_{0+}\varphi _{q} (I^{\alpha _{1}}_{1-}(N_{\lambda}u(t)-QN_{\lambda}u(t)) ))}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})}\bigl( \delta _{2}\Gamma (\gamma _{2}-1)t^{\gamma _{2}-1}+\delta _{1}\Gamma ( \gamma _{2})t^{\gamma _{2}-2}\bigr) \\ &= u(t)-\frac{D^{\gamma _{2}-1}_{0+}u(0)}{\Gamma (\gamma _{2})}t^{ \gamma _{2}-1}- \frac{D^{\gamma _{2}-2}_{0+}u(0)}{\Gamma (\gamma _{2}-1)}t^{\gamma _{2}-2} \\ &\quad{}+ \frac{T_{1}(\frac{D^{\gamma _{2}-1}_{0+}u(0)}{\Gamma (\gamma _{2})}t^{\gamma _{2}-1})+T_{1}(\frac{D^{\gamma _{2}-2}_{0+}u(0)}{\Gamma (\gamma _{2}-1)}t^{\gamma _{2}-2})}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})}\bigl( \delta _{2}\Gamma (\gamma _{2}-1)t^{\gamma _{2}-1}+\delta _{1}\Gamma ( \gamma _{2})t^{\gamma _{2}-2}\bigr) \\ &= u(t)-\frac{D^{\gamma _{2}-1}_{0+}u(0)}{\Gamma (\gamma _{2})}t^{ \gamma _{2}-1}+ \frac{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)D^{\gamma _{2}-1}_{0+}u(0)+\delta _{1}\delta _{2}\Gamma (\gamma _{2})D^{\gamma _{2}-2}_{0+}u(0)}{\Gamma (\gamma _{2})(\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2}))}t^{ \gamma _{2}-1} \\ &\quad{}-\frac{D^{\gamma _{2}-2}_{0+}u(0)}{\Gamma (\gamma _{2}-1)}t^{\gamma _{2}-2}+ \frac{\delta _{1}\delta _{2}\Gamma (\gamma _{2}-1)D^{\gamma _{2}-1}_{0+}u(0)+\delta ^{2}_{1}\Gamma (\gamma _{2})D^{\gamma _{2}-2}_{0+}u(0)}{\Gamma (\gamma _{2}-1)(\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2}))}t^{ \gamma _{2}-2} \\ &= u(t)- \frac{\delta _{2}D^{\gamma _{2}-2}_{0+}u(0)-\delta _{1}D^{\gamma _{2}-1}_{0+}u(0)}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})}\bigl( \delta _{2}t^{\gamma _{2}-2}- \delta _{1}t^{\gamma _{2}-1}\bigr) \\ &= u(t)-Pu(t)=(I-P)u, \end{aligned}$$

i.e., Definition 2.2(c) holds.

For \(u\in \overline{\Omega}\), we have

$$\begin{aligned} L\bigl[Pu(t)+R(u,\lambda ) (t)\bigr] &= I^{\beta _{1}(1-\alpha _{1})}_{1-}D^{ \gamma _{1}}_{1-} \varphi _{p} \bigl(I^{\beta _{2}(3-\alpha _{2})}_{0+}D^{ \gamma _{2}}_{0+} \bigl(Pu(t)+R(u,\lambda ) (t)\bigr) \bigr) \\ &= I^{\beta _{1}(1-\alpha _{1})}_{1-}D^{\beta _{1}(1-\alpha _{1})}_{1-}(I-Q)N_{ \lambda}u(t) \\ &= (I-Q)N_{\lambda}u(t), \end{aligned}$$

i.e., Definition 2.2(d) holds. Therefore, \(N_{\lambda}\) is L-quasicompact in Ω̅. □

Theorem 3.6

Suppose that \((H_{1})\)\((H_{3})\) and the following conditions hold:

\((H_{4})\):

There exists a constant \(M_{0}>0\) such that if \(|t^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t)|+|t^{- \beta _{2}(3-\alpha _{2})} D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t)|>M_{0}\), then \((T_{2}-kT_{1})(I^{\alpha _{2}}_{0+}\varphi _{q}(I^{\alpha _{1}}_{1-}Nu(t))) \neq 0\).

\((H_{5})\):

There exist nonnegative functions \(a(t), b(t), c(t), d(t), e(t)\in C[0,1]\), such that

$$ \begin{aligned} \bigl\vert f(t,x,y,z,w) \bigr\vert \leq{}& a(t)+b(t)\varphi _{p} \bigl( \vert x \vert \bigr)+c(t)\varphi _{p}\bigl( \vert y \vert \bigr) \\ &{}+d(t) \varphi _{p}\bigl( \vert z \vert \bigr)+e(t) \varphi _{p}\bigl( \vert w \vert \bigr),\quad x,y,z,w\in \mathbb{R}, \end{aligned} $$

where \(\Gamma (\alpha _{1}+1)> A( \frac{2\Gamma (\gamma _{2}-1)+5\Gamma (\beta _{2}(3-\alpha _{2})+1)\Gamma (\alpha _{2}+1)}{2\Gamma (\alpha _{2}+1)\Gamma (\gamma _{2}-1)})^{p-1} \|b\|_{\infty}+A3^{p-1}\|c\|_{\infty}+ A2^{p-1}\|d\|_{\infty}+\|e\|_{ \infty}\), \(A=\max_{p\in (1,+\infty )}\{1,2^{p-2}\}\).

\((H_{6})\):

There exists \(B_{1}>0\) such that one of the following inequalities holds:

$$ (1) \quad cQN\bigl(c\bigl(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1}\bigr)\bigr)>0,\qquad (2)\quad cQN\bigl(c\bigl( \delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1}\bigr)\bigr)< 0. $$

Then problem (1.1) has at least one solution in X.

Lemma 3.7

Suppose that \((H_{4})\) and \((H_{5})\) hold, then \(\Omega _{1}=\{u\mid u\in \operatorname{dom}L \backslash \operatorname{Ker}L,Lu=N_{\lambda}u,\lambda \in (0,1)\}\) is bounded in X.

Proof

For \(u\in \operatorname{dom}L\), according to Lemma 2.8, we obtain

$$ u(t)=I^{\gamma _{2}}_{0+}D^{\gamma _{2}}_{0+}u(t)+c_{1}t^{\gamma _{2}-1}+c_{2}t^{ \gamma _{2}-2}. $$
(3.6)

Applying \(D^{\alpha _{2}-1,\beta _{2}}_{0+}\) and \(D^{\alpha _{2}-2,\beta _{2}}_{0+}\) to both sides of (3.6) respectively, we can get

$$\begin{aligned} &D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t)= \int _{0}^{t}D^{\alpha _{2}, \beta _{2}}_{0+}u(s) \,ds + \frac{c_{1}\Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+1)}t^{ \beta _{2}(3-\alpha _{2})}, \\ &D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t)= \int _{0}^{t}(t-s)D^{\alpha _{2}, \beta _{2}}_{0+}u(s) \,ds + \frac{c_{1}\Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)}t^{ \beta _{2}(3-\alpha _{2})+1} \\ &\hphantom{D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t)={}}{}+ \frac{c_{2}\Gamma (\gamma _{2}-1)}{\Gamma (\beta _{2}(3-\alpha _{2})+1)}t^{ \beta _{2}(3-\alpha _{2})}. \end{aligned}$$

Therefore,

$$\begin{aligned}& c_{1}= \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2})} \biggl(t^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t)-t^{- \beta _{2}(3-\alpha _{2})} \int _{0}^{t}D^{\alpha _{2},\beta _{2}}_{0+}u(s) \,ds \biggr), \end{aligned}$$
(3.7)
$$\begin{aligned}& \begin{aligned}[b] c_{2}&= \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)} \biggl(t^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t)-t^{- \beta _{2}(3-\alpha _{2})} \int _{0}^{t}(t-s)D^{\alpha _{2},\beta _{2}}_{0+}u(s) \,ds \\ &\quad{}- \frac{c_{1}\Gamma (\gamma _{2})t}{\Gamma (\beta _{2}(3-\alpha _{2})+2)} \biggr). \end{aligned} \end{aligned}$$
(3.8)

For \(u\in \Omega _{1}\), we have \(Lu=N_{\lambda}u\), \(N_{\lambda}u \in \operatorname{Im}L=\operatorname{Ker}Q\), we get \(QN_{\lambda}u(t)=0\). It follows from \((H_{4})\) that there exists \(t_{0}\in (0,1]\), such that \(|t_{0}^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t_{0})|+|t_{0}^{- \beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t_{0})| \leq M_{0}\), then \(|t_{0}^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t_{0})| \leq M_{0}\) and \(|t_{0}^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t_{0})| \leq M_{0}\). Taking \(t=t_{0}\) into equations (3.7) and (3.8), we have

$$\begin{aligned}& \vert c_{1} \vert \leq \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2})} \bigl(M_{0}+ \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty} \bigr), \\& \begin{aligned} \vert c_{2} \vert &\leq \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)} \biggl(M_{0}+\frac{1}{2} \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty}+ \frac{ \vert c_{1} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)} \biggr) \\ &\leq \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)} \biggl(2M_{0}+\frac{3}{2} \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty} \biggr). \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} \bigl\Vert D^{\alpha _{2}-1,\beta _{2}}_{0+}u \bigr\Vert _{\infty}\leq M_{0}+2 \bigl\Vert D^{ \alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty}, \qquad \bigl\Vert D^{\alpha _{2}-2, \beta _{2}}_{0+}u \bigr\Vert _{\infty}\leq 3\bigl(M_{0}+ \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty}\bigr). \end{aligned}$$

Since \(u(t)\) in (3.6) can also be written as \(u(t)=I^{\alpha _{2}}_{0+}D^{\alpha _{2},\beta _{2}}_{0+}u(t)+c_{1}t^{ \gamma _{2}-1}+c_{2}t^{\gamma _{2}-2}\), then

$$\begin{aligned} \Vert u \Vert _{\infty}\leq{}& \frac{3\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)}M_{0} \\ &{}+ \frac{2\Gamma (\gamma _{2}-1)+5\Gamma (\beta _{2}(3-\alpha _{2})+1)\Gamma (\alpha _{2}+1)}{2\Gamma (\alpha _{2}+1)\Gamma (\gamma _{2}-1)} \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty}. \end{aligned}$$

According to \(Lu(t)=N_{\lambda}u(t)\) and boundary conditions, we can get

$$ u(t)=I^{\alpha _{2}}_{0+}\varphi _{q} \bigl(I^{\alpha _{1}}_{1-}N_{\lambda}u(t) \bigr)+c_{3}t^{ \gamma _{2}-1}+c_{4}t^{\gamma _{2}-2}. $$

Therefore,

$$\begin{aligned}& \bigl\vert \varphi _{p}\bigl(D^{\alpha _{2},\beta _{2}}_{0+}u(t) \bigr) \bigr\vert \\& \quad = \bigl\vert \varphi _{p} \bigl(I^{ \beta _{2}(3-\alpha _{2})}_{0+}D^{\gamma _{2}}_{0+}I^{\alpha _{2}}_{0+} \varphi _{q}\bigl(I^{\alpha _{1}}_{1-}N_{\lambda}u(t) \bigr)\bigr) \bigr\vert = \bigl\vert I^{\alpha _{1}}_{1-}N_{ \lambda}u(t) \bigr\vert \\& \quad \leq \frac{\lambda}{\Gamma (\alpha _{1})} \int _{t}^{1}(s-t)^{\alpha _{1}-1} \bigl\vert f\bigl(s,u(s),D^{ \alpha _{2}-2,\beta _{2}}_{0+}u(s),D^{\alpha _{2}-1,\beta _{2}}_{0+}u(s),D^{ \alpha _{2},\beta _{2}}_{0+}u(s) \bigr) \bigr\vert \,ds \\& \quad \leq \frac{\lambda}{\Gamma (\alpha _{1})} \int _{t}^{1}(s-t)^{\alpha _{1}-1} \bigl(a(t)+b(t)\varphi _{p}\bigl( \bigl\vert u(s) \bigr\vert \bigr)+c(t)\varphi _{p}\bigl( \bigl\vert D^{\alpha _{2}-2, \beta _{2}}_{0+}u(t) \bigr\vert \bigr) \\& \quad \quad{}+d(t)\varphi _{p}\bigl( \bigl\vert D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t) \bigr\vert \bigr)+e(t) \varphi _{p}\bigl( \bigl\vert D^{\alpha _{2},\beta _{2}}_{0+}u(t) \bigr\vert \bigr) \bigr)\,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{1}+1)} \bigl( \Vert a \Vert _{\infty}+ \Vert b \Vert _{ \infty} \Vert u \Vert ^{p-1}_{\infty}+ \Vert c \Vert _{\infty} \bigl\Vert D^{\alpha _{2}-2,\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{\infty}+ \Vert d \Vert _{\infty} \bigl\Vert D^{\alpha _{2}-1,\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{\infty} \\& \quad \quad{}+ \Vert e \Vert _{\infty} \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{\infty} \bigr) \\& \quad \leq \frac{1}{\Gamma (\alpha _{1}+1)} \biggl[ \Vert a \Vert _{\infty}+ \Vert b \Vert _{ \infty} \biggl( \frac{2\Gamma (\gamma _{2}-1)+5\Gamma (\beta _{2}(3-\alpha _{2})+1)\Gamma (\alpha _{2}+1)}{2\Gamma (\alpha _{2}+1)\Gamma (\gamma _{2}-1)} \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty} \\& \quad \quad{}+ \frac{3M_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)} \biggr)^{p-1}+ \Vert c \Vert _{\infty} \bigl(3M_{0}+3 \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty} \bigr)^{p-1} \\& \quad \quad{}+ \Vert d \Vert _{\infty} \bigl(M_{0}+2 \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{ \infty} \bigr)^{p-1}+ \Vert e \Vert _{\infty} \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{ \infty} \biggr]. \end{aligned}$$

It is known that \(|\varphi _{p}(D^{\alpha _{2},\beta _{2}}_{0+}u(t))|=|D^{\alpha _{2}, \beta _{2}}_{0+}u(t)|^{p-1}\), then

$$\begin{aligned}& \bigl\vert D^{\alpha _{2},\beta _{2}}_{0+}u(t) \bigr\vert ^{p-1} \\& \quad \leq \frac{1}{\Gamma (\alpha _{1}+1)} \biggl[ \Vert a \Vert _{\infty}+ \Vert b \Vert _{\infty} \biggl( \frac{2\Gamma (\gamma _{2}-1)+5\Gamma (\beta _{2}(3-\alpha _{2})+1)\Gamma (\alpha _{2}+1)}{2\Gamma (\alpha _{2}+1)\Gamma (\gamma _{2}-1)} \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty} \\& \quad \quad{}+ \frac{3M_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)} \biggr)^{p-1}+ \Vert c \Vert _{\infty} \bigl(3M_{0}+3 \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty} \bigr)^{p-1} \\& \quad \quad{}+ \Vert d \Vert _{\infty} \bigl(M_{0}+2 \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{ \infty} \bigr)^{p-1}+ \Vert e \Vert _{\infty} \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{ \infty} \biggr]. \end{aligned}$$

If \(1< p\leq 2\), then

$$\begin{aligned}& \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{\infty} \\& \quad \leq \frac{1}{\Gamma (\alpha _{1}+1)} \biggl[ \Vert a \Vert _{\infty}+ \Vert b \Vert _{\infty} \biggl( \frac{2\Gamma (\gamma _{2}-1)+5\Gamma (\beta _{2}(3-\alpha _{2})+1)\Gamma (\alpha _{2}+1)}{2\Gamma (\alpha _{2}+1)\Gamma (\gamma _{2}-1)} \biggr)^{p-1} \\& \quad \quad {}\times\bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{\infty} \\& \quad \quad{}+ \Vert b \Vert _{\infty} \biggl( \frac{3M_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)} \biggr)^{p-1}+ \Vert c \Vert _{\infty} (3M_{0} )^{p-1}+ \Vert c \Vert _{\infty}3^{p-1} \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{\infty} \\& \quad \quad{}+ \Vert d \Vert _{\infty}M^{p-1}_{0}+ \Vert d \Vert _{\infty}2^{p-1} \bigl\Vert D^{\alpha _{2}, \beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{\infty}+ \Vert e \Vert _{\infty} \bigl\Vert D^{\alpha _{2}, \beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{\infty} \biggr]. \end{aligned}$$

Consequently,

$$\begin{aligned}& \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+} \bigr\Vert _{\infty} \\& \quad \leq \biggl( \frac{ \Vert a \Vert _{\infty}+ \Vert b \Vert _{\infty} (\frac{3M_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)} )^{p-1}+ \Vert c \Vert _{\infty} (3M_{0} )^{p-1}+ \Vert d \Vert _{\infty}M^{p-1}_{0}}{\Gamma (\alpha _{1}+1)- [ \Vert b \Vert _{\infty} (\frac{2\Gamma (\gamma _{2}-1)+5\Gamma (\beta _{2}(3-\alpha _{2})+1)\Gamma (\alpha _{2}+1)}{2\Gamma (\alpha _{2}+1)\Gamma (\gamma _{2}-1)} )^{p-1}+ \Vert c \Vert _{\infty}3^{p-1}+ \Vert d \Vert _{\infty}2^{p-1}+ \Vert e \Vert _{\infty} ]} \biggr)^{p-1}. \end{aligned}$$

If \(p>2\), then

$$\begin{aligned}& \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+} \bigr\Vert _{\infty} \\& \quad \leq \biggl( \frac{ \Vert a \Vert _{\infty}+2^{p-1} \Vert b \Vert _{\infty} (\frac{3M_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)} )^{p-1}+2^{p-1} \Vert c \Vert _{\infty} (3M_{0} )^{p-1}+2^{p-1} \Vert d \Vert _{\infty}M^{p-1}_{0}}{\Gamma (\alpha _{1}+1)-2^{p-1} [ \Vert b \Vert _{\infty} (\frac{2\Gamma (\gamma _{2}-1)+5\Gamma (\beta _{2}(3-\alpha _{2})+1)\Gamma (\alpha _{2}+1)}{2\Gamma (\alpha _{2}+1)\Gamma (\gamma _{2}-1)} )^{p-1}+ \Vert c \Vert _{\infty}3^{p-1}+ \Vert d \Vert _{\infty}2^{p-1}+\frac{ \Vert e \Vert _{\infty}}{2^{p-1}} ]} \biggr)^{p-1}. \end{aligned}$$

Set \(A=\max_{p\in (1,+\infty )}\{1,2^{p-2}\}\), then the above inequality is equivalent to

$$\begin{aligned}& \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+} \bigr\Vert _{\infty} \\& \quad \leq \biggl( \frac{ \Vert a \Vert _{\infty}+A \Vert b \Vert _{\infty} (\frac{3M_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)} )^{p-1}+A \Vert c \Vert _{\infty} (3M_{0} )^{p-1}+A \Vert d \Vert _{\infty}M^{p-1}_{0}}{\Gamma (\alpha _{1}+1)- [A \Vert b \Vert _{\infty} (\frac{2\Gamma (\gamma _{2}-1)+5\Gamma (\beta _{2}(3-\alpha _{2})+1)\Gamma (\alpha _{2}+1)}{2\Gamma (\alpha _{2}+1)\Gamma (\gamma _{2}-1)} )^{p-1}+A \Vert c \Vert _{\infty}3^{p-1}+A \Vert d \Vert _{\infty}2^{p-1}+ \Vert e \Vert _{\infty} ]} \biggr)^{p-1} \\& \quad :=M_{1}. \end{aligned}$$

Therefore,

$$\begin{aligned} & \bigl\Vert D^{\alpha _{2}, \beta _{2}}_{0+} \bigr\Vert _{\infty}\leq M_{1},\qquad \bigl\Vert D^{ \alpha _{2}-1,\beta _{2}}_{0+} \bigr\Vert _{\infty}\leq M_{0}+2M_{1},\qquad \bigl\Vert D^{ \alpha _{2}-2,\beta _{2}}_{0+} \bigr\Vert _{\infty}\leq 3(M_{0}+M_{1}), \\ & \Vert u \Vert _{\infty}\leq \frac{3\Gamma (\beta _{2}(3-\alpha _{2})+1)}{\Gamma (\gamma _{2}-1)}M_{0}+ \frac{2\Gamma (\gamma _{2}-1)+5\Gamma (\beta _{2}(3-\alpha _{2})+1)\Gamma (\alpha _{2}+1)}{2\Gamma (\alpha _{2}+1)\Gamma (\gamma _{2}-1)}M_{1}:=M_{2}, \end{aligned}$$

we can get

$$\begin{aligned} \Vert u \Vert _{X}&=\max \bigl\{ \Vert u \Vert _{\infty}, \bigl\Vert D^{\alpha _{2}-2,\beta _{2}}_{0+}u \bigr\Vert _{\infty}, \bigl\Vert D^{\alpha _{2}-1,\beta _{2}}_{0+}u \bigr\Vert _{\infty}, \bigl\Vert D^{ \alpha _{2},\beta _{2}}_{0+}u \bigr\Vert _{\infty}\bigr\} \\ &\leq \max \bigl\{ M_{2},3(M_{0}+M_{1}),M_{0}+2M_{1},M_{1} \bigr\} :=M_{3}. \end{aligned}$$

Hence, we can conclude that \(\Omega _{1}\) is bounded in X. □

Lemma 3.8

Suppose that \((H_{1})\)\((H_{3})\) and \((H_{6})\) hold, then \(\Omega _{2}=\{u|u\in \operatorname{Ker}L, QNu=0\}\) is bounded in X.

Proof

Let \(u\in \Omega _{2}\), we have \(u(t)=c(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1})\), \(c \in \mathbb{R}\).

Since \(QNu(t)=0\), according to \((H_{6})\), there exists a constant \(B_{1}>0\) such that \(|c|\leq B_{1}\), then

$$\begin{aligned}& \Vert u \Vert _{\infty}\leq B_{1}\bigl( \vert \delta _{2} \vert + \vert \delta _{1} \vert \bigr), \\& \bigl\Vert D^{\alpha _{2}-2,\beta _{2}}_{0+}u \bigr\Vert _{\infty}\leq B_{1} \biggl( \frac{ \vert \delta _{2} \vert (\beta _{2}(3-\alpha _{2})+1)\Gamma (\gamma _{2}-1)+ \vert \delta _{1} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)} \biggr), \\& \bigl\Vert D^{\alpha _{2}-1,\beta _{2}}_{0+}u \bigr\Vert _{\infty}\leq \frac{B_{1} \vert \delta _{1} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+1)}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert u \Vert _{X}\leq{}& \max \biggl\{ B_{1}\bigl( \vert \delta _{2} \vert + \vert \delta _{1} \vert \bigr),B_{1} \biggl( \frac{ \vert \delta _{2} \vert (\beta _{2}(3-\alpha _{2})+1)\Gamma (\gamma _{2}-1)+ \vert \delta _{1} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)} \biggr), \\ & \frac{B_{1} \vert \delta _{1} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+1)} \biggr\} \\ :={}&M_{4}, \end{aligned}$$

we can conclude that \(\Omega _{2}\) is bounded in X. □

Proof of Theorem 3.6

Let \(\Omega \supset \overline{\Omega _{1}}\cup \overline{\Omega _{2}} \cup \{u\mid u\in X,\|u\|_{X}\leq \max \{M_{3},M_{4}\}+1\}\) be an open and bounded set of X. By Lemma 3.7 and Lemma 3.8, we can get \(Lu\neq N_{\lambda}u\), \(u\in \operatorname{dom}L\cap \partial \Omega \) and \(QNu\neq 0\), \(u\in \operatorname{Ker}L\cap \partial \Omega \).

Let \(H(u,\xi )=\rho \xi u+(1-\xi )JQNu\), \(\xi \in [0,1]\), \(u\in \operatorname{Ker}L\cap \overline{\Omega}\), where \(J:\operatorname{Im}Q\rightarrow \operatorname{Ker}L\) is a homeomorphism with \(Jc=c(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1})\),

$$ \rho =\textstyle\begin{cases} 1, &\mbox{if }(H_{6}) \mbox{ (1) holds}, \\ -1, &\mbox{if }(H_{6}) \mbox{ (2) holds} . \end{cases} $$

For \(u\in \operatorname{Ker}L\cap \partial \Omega \), we have \(u(t)=c(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1})\). Therefore

$$ H(u,\xi )=\rho \xi c\bigl(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{ \gamma _{2}-1}\bigr)+(1-\xi )QN\bigl(c\bigl(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{ \gamma _{2}-1} \bigr)\bigr) \bigl(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1}\bigr). $$

If \(\xi =1\), then \(H(u,1)=\rho c(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1}) \neq 0\). If \(\xi =0\), then \(H(u,0)=QN(c(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1}))( \delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1})\neq 0\). If \(0<\xi <1\), suppose \(H(u,\xi )=0\), then \(\rho \xi c(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1})=-(1- \xi )QN(c(\delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1}))( \delta _{2}t^{\gamma _{2}-2}-\delta _{1}t^{\gamma _{2}-1})\). So, \(c=- (\frac{1-\xi}{\rho \xi} )QN(c(\delta _{2}t^{\gamma _{2}-2}- \delta _{1}t^{\gamma _{2}-1}))\). By \((H_{6})\), we get

$$ c^{2}=- \biggl(\frac{1-\xi}{\rho \xi} \biggr)cQN\bigl(c\bigl(\delta _{2}t^{\gamma _{2}-2}- \delta _{1}t^{\gamma _{2}-1} \bigr)\bigr)< 0. $$

A contradiction. That is, \(H(u,\xi )\neq 0\), \(u\in \operatorname{Ker}L\cap \partial \Omega \), \(\xi \in [0,1]\).

Therefore, via the homotopy property of degree, we obtain

$$\begin{aligned} \deg(JQN,\Omega \cap \operatorname{Ker}L,0)&=\deg\bigl( H(\cdot ,0),\Omega \cap \operatorname{Ker}L,0\bigr) \\ &=\deg\bigl(H(\cdot ,1),\Omega \cap \operatorname{Ker}L,0\bigr) \\ &= \deg(\rho I,\Omega \cap \operatorname{Ker}L,0)\neq 0. \end{aligned}$$

Applying Lemma 2.3, we conclude that boundary value problem (1.1) has at least one solution in X. □

For another result of problem (1.1), suppose that the inequality \(|t^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t)|+|t^{- \beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t)|>M_{0}\) in condition \((H_{4})\) is replaced by \(|t^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t)|>M'_{0}\) or \(|t^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t)|>M''_{0}\), which will cause the proof of Lemma 3.7 to change, but the result of Theorem 3.6 can still be obtained, as shown below.

Theorem 3.9

Suppose that \((H_{1})\)\((H_{3})\), \((H_{6})\) and the following conditions hold:

\((H_{7})\):

There exists a constant \(M'_{0}>0\) such that if \(|t^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t)|>M'_{0}\), then \((T_{2}-kT_{1})(I^{\alpha _{2}}_{0+}\varphi _{q}(I^{\alpha _{1}}_{1-}Nu(t))) \neq 0\).

\((H_{8})\):

There exist nonnegative functions \(a(t), b(t), c(t), d(t), e(t)\in C[0,1]\), such that

$$ \begin{aligned} \bigl\vert f(t,x,y,z,w) \bigr\vert \leq{}& a(t)+b(t)\varphi _{p} \bigl( \vert x \vert \bigr)+c(t)\varphi _{p}\bigl( \vert y \vert \bigr) \\ &{}+d(t) \varphi _{p}\bigl( \vert z \vert \bigr)+e(t) \varphi _{p}\bigl( \vert w \vert \bigr), \quad x,y,z,w\in R, \end{aligned} $$

where \(L(C_{1}M+C_{2}) (\|b\|_{\infty}+\|c\|_{\infty}+\|d\|_{\infty}+\|e \|_{\infty} )^{q-1}<1\), \(L=\max_{q\in (1,+\infty )}\{1,2^{q-2} \}\),

$$\begin{aligned}& \begin{aligned} C_{1}={}&\max \biggl\{ \vert \delta _{2} \vert + \vert \delta _{1} \vert , \frac{ \vert \delta _{2} \vert (\beta _{2}(3-\alpha _{2})+1)\Gamma (\gamma _{2}-1)+ \vert \delta _{1} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)}, \\ &\frac{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+1)} \biggr\} , \end{aligned} \\& \begin{aligned} C_{2}={}&\max \biggl\{ 1+ \frac{ \Vert T_{1} \Vert _{\infty}( \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)+ \vert \delta _{1} \vert \Gamma (\gamma _{2}))}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})}, \\ & \frac{1}{2}+ \frac{ \Vert T_{1} \Vert _{\infty} ( \vert \delta _{2} \vert \Gamma (\gamma _{2})+(\beta _{2}(3-\alpha _{2})+1) \vert \delta _{1} \vert \Gamma (\gamma _{2}) )}{(\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1))\Gamma (\beta _{2}(3-\alpha _{2})+2)}, \\ & 1+ \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{2} \vert \Gamma (\gamma _{2})}{(\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1))\Gamma (\beta _{2}(3-\alpha _{2})+1)} \biggr\} \times \varphi _{q} \biggl( \frac{1}{\Gamma (\alpha _{1}+1)} \biggr), \end{aligned} \\& M= \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)\varphi _{q} (\frac{1}{\Gamma (\alpha _{1}+1)} )}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}+ \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)\varphi _{q} (\frac{1}{\Gamma (\alpha _{1}+1)} )}{ \vert \delta _{1} \vert (\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2}))}. \end{aligned}$$

Then problem (1.1) has at least one solution in X.

Proof

For \(u\in \Omega _{1}\), we have \(QN_{\lambda}u(t)=0\). It follows from \((H_{7})\) that there exists \(t_{0}\in (0,1]\), such that \(|t_{0}^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t_{0})| \leq M'_{0}\). By Lemma 3.5, we obtain \(R(u,\lambda )(t)=(I-P)u(t)=u(t)-Pu(t)\). So \(D^{\alpha _{2}-1,\beta _{2}}_{0+}Pu(t)=D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t)-D^{ \alpha _{2}-1,\beta _{2}}_{0+}R(u,\lambda )(t)\). According to the definition of P, we can get

$$\begin{aligned} & \biggl\vert \frac{\delta _{2}D^{\gamma _{2}-2}_{0+}u(0)-\delta _{1}D^{\gamma _{2}-1}_{0+}u(0)}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})} \biggr\vert \\ &\quad \leq \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})} \bigl( \bigl\vert t^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t) \bigr\vert + \bigl\vert t^{- \beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-1,\beta _{2}}_{0+}R(u, \lambda ) (t) \bigr\vert \bigr). \end{aligned}$$
(3.9)

Taking \(t=t_{0}\) into equation (3.9), we have

$$\begin{aligned} &\biggl\vert \frac{\delta _{2}D^{\gamma _{2}-2}_{0+}u(0)-\delta _{1} D^{\gamma _{2}-1}_{0+}u(0)}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+ \delta ^{2}_{1}\Gamma (\gamma _{2})} \biggr\vert \\ &\quad \leq \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})} \bigl(M'_{0}+ \bigl\vert t^{-\beta _{2}(3-\alpha _{2})}_{0}D^{\alpha _{2}-1, \beta _{2}}_{0+}R(u, \lambda ) (t_{0}) \bigr\vert \bigr). \end{aligned}$$

Since

$$\begin{aligned} & \bigl\vert t^{-\beta _{2}(3-\alpha _{2})}_{0}D^{\alpha _{2}-1,\beta _{2}}_{0+}R(u, \lambda ) (t_{0}) \bigr\vert \\ &\quad \leq t^{-\beta _{2}(3-\alpha _{2})}_{0} \int _{0}^{t_{0}}\varphi _{q} \biggl( \int _{s}^{1}\frac{(x-s)^{\alpha _{1}-1}}{\Gamma (\alpha _{1})} \bigl\vert N_{ \lambda}u(x) \bigr\vert \,dx \biggr)\,ds \\ &\quad \quad {}+ \frac{ \Vert T_{1} \Vert _{\infty} \Vert I^{\alpha _{2}}_{0+}\varphi _{q}(I^{\alpha _{1}}_{1-}N_{\lambda}u) \Vert _{X} \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)\Gamma (\gamma _{2})}{(\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2}))\Gamma (\beta _{2}(3-\alpha _{2})+1)} \\ &\quad \leq t^{-\beta _{2}(3-\alpha _{2})}_{0} \int _{0}^{t_{0}}\varphi _{q} \biggl(\frac{ \Vert N_{\lambda}u \Vert _{\infty}}{\Gamma (\alpha _{1}+1)} \biggr)\,ds + \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)\Gamma (\gamma _{2})\varphi _{q} (\frac{ \Vert N_{\lambda}u \Vert _{\infty}}{\Gamma (\alpha _{1}+1)} )}{(\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2}))\Gamma (\beta _{2}(3-\alpha _{2})+1)} \\ &\quad \leq \biggl[\varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1}+1)} \biggr)+ \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)\Gamma (\gamma _{2})\varphi _{q} (\frac{1}{\Gamma (\alpha _{1}+1)} )}{(\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2}))\Gamma (\beta _{2}(3-\alpha _{2})+1)} \biggr] \\ &\quad \quad {}\times \Vert N_{\lambda}u \Vert ^{q-1}_{\infty}, \end{aligned}$$

we can obtain

$$\begin{aligned} & \biggl\vert \frac{\delta _{2}D^{\gamma _{2}-2}_{0+}u(0)-\delta _{1}D^{\gamma _{2}-1}_{0+}u(0)}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})} \biggr\vert \\ &\quad \leq \biggl[ \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)\varphi _{q} (\frac{1}{\Gamma (\alpha _{1}+1)} )}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}+ \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)\varphi _{q} (\frac{1}{\Gamma (\alpha _{1}+1)} )}{ \vert \delta _{1} \vert (\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2}))} \biggr] \Vert N_{\lambda}u \Vert ^{q-1}_{\infty} \\ &\quad \quad{}+ \frac{M'_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert Pu \Vert _{X}&\leq \max \biggl\{ \vert \delta _{2} \vert + \vert \delta _{1} \vert , \frac{ \vert \delta _{2} \vert (\beta _{2}(3-\alpha _{2})+1)\Gamma (\gamma _{2}-1)+ \vert \delta _{1} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)}, \frac{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+1)} \biggr\} \\ &\quad{}\times \biggl( \biggl[ \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)\varphi _{q} (\frac{1}{\Gamma (\alpha _{1}+1)} )}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}+ \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)\varphi _{q} (\frac{1}{\Gamma (\alpha _{1}+1)} )}{ \vert \delta _{1} \vert (\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2}))} \biggr] \\ &\quad {}\times \Vert N_{\lambda}u \Vert ^{q-1}_{\infty} + \frac{M'_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})} \biggr) \\ &:= C_{1} \biggl(M \Vert N_{\lambda}u \Vert ^{q-1}_{\infty}+ \frac{M'_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})} \biggr), \end{aligned}$$

where

$$\begin{aligned}& C_{1}=\max \biggl\{ |\delta _{2}|+|\delta _{1}|, \frac{|\delta _{2}|(\beta _{2}(3-\alpha _{2})+1)\Gamma (\gamma _{2}-1)+|\delta _{1}|\Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+2)}, \frac{|\delta _{1}|\Gamma (\gamma _{2})}{\Gamma (\beta _{2}(3-\alpha _{2})+1)} \biggr\} , \\& M= \frac{\Gamma (\beta _{2}(3-\alpha _{2})+1)\varphi _{q} (\frac{1}{\Gamma (\alpha _{1}+1)} )}{|\delta _{1}|\Gamma (\gamma _{2})}+ \frac{\|T_{1}\|_{\infty}|\delta _{2}|\Gamma (\gamma _{2}-1)\varphi _{q} (\frac{1}{\Gamma (\alpha _{1}+1)} )}{|\delta _{1}|(\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2}))}. \end{aligned}$$

According to Lemma 3.4, we can get

$$\begin{aligned} \bigl\Vert R(u,\lambda ) \bigr\Vert _{X} \leq{}& \biggl[\max \biggl\{ 1+ \frac{ \Vert T_{1} \Vert _{\infty}( \vert \delta _{2} \vert \Gamma (\gamma _{2}-1)+ \vert \delta _{1} \vert \Gamma (\gamma _{2}))}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+ \delta ^{2}_{1}\Gamma (\gamma _{2})}, \\ &1+ \frac{ \Vert T_{1} \Vert _{\infty} \vert \delta _{2} \vert \Gamma (\gamma _{2})}{(\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1)) \Gamma (\beta _{2}(3-\alpha _{2})+1)}, \\ &\frac{1}{2}+ \frac{ \Vert T_{1} \Vert _{\infty} ( \vert \delta _{2} \vert \Gamma (\gamma _{2})+ (\beta _{2}(3-\alpha _{2})+1) \vert \delta _{1} \vert \Gamma (\gamma _{2}) )}{(\delta ^{2}_{2}+ \delta ^{2}_{1}(\gamma _{2}-1))\Gamma (\beta _{2}(3-\alpha _{2})+2)} \biggr\} \times \varphi _{q} \biggl(\frac{1}{\Gamma (\alpha _{1}+1)} \biggr) \biggr] \\ &{}\times \Vert N_{\lambda}u \Vert ^{q-1}_{\infty} \\ :={}& C_{2} \Vert N_{\lambda}u \Vert ^{q-1}_{\infty}, \end{aligned}$$

where

$$\begin{aligned} C_{2}={}&\max \biggl\{ 1+ \frac{\|T_{1}\|_{\infty}(|\delta _{2}|\Gamma (\gamma _{2}-1)+|\delta _{1}|\Gamma (\gamma _{2}))}{\delta ^{2}_{2}\Gamma (\gamma _{2}-1)+\delta ^{2}_{1}\Gamma (\gamma _{2})}, \\ &1+ \frac{\|T_{1}\|_{\infty}|\delta _{2}|\Gamma (\gamma _{2})}{(\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1))\Gamma (\beta _{2}(3-\alpha _{2})+1)}, \\ &\frac{1}{2}+ \frac{\|T_{1}\|_{\infty} (|\delta _{2}|\Gamma (\gamma _{2})+(\beta _{2}(3-\alpha _{2})+1)|\delta _{1}|\Gamma (\gamma _{2}) )}{(\delta ^{2}_{2}+\delta ^{2}_{1}(\gamma _{2}-1))\Gamma (\beta _{2}(3-\alpha _{2})+2)} \biggr\} \times \varphi _{q} (\frac{1}{\Gamma (\alpha _{1}+1)} ). \end{aligned}$$

Using Lemma 3.5 and hypothetical condition \((H_{8})\), we have

$$\begin{aligned} \Vert u \Vert _{X}&\leq \Vert Pu \Vert _{X}+ \bigl\Vert R(u,\lambda ) \bigr\Vert _{X} \\ &\leq (C_{1}M+C_{2}) \Vert N_{\lambda}u \Vert ^{q-1}_{\infty}+ \frac{C_{1}M'_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})} \\ &\leq (C_{1}M+C_{2}) \bigl( \Vert a \Vert _{\infty}+ \Vert b \Vert _{\infty} \Vert u \Vert ^{p-1}_{ \infty}+ \Vert c \Vert _{\infty} \bigl\Vert D^{\alpha _{2}-2,\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{ \infty} \\ &\quad{}+ \Vert d \Vert _{\infty} \bigl\Vert D^{\alpha _{2}-1,\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{ \infty}+ \Vert e \Vert _{\infty} \bigl\Vert D^{\alpha _{2},\beta _{2}}_{0+}u \bigr\Vert ^{p-1}_{\infty} \bigr)^{q-1}+ \frac{C_{1}M'_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}. \end{aligned}$$

If \(1< q\leq 2\), then

$$\begin{aligned} \Vert u \Vert _{X}&\leq (C_{1}M+C_{2}) \Vert a \Vert ^{q-1}_{\infty}+(C_{1}M+C_{2}) \bigl( \Vert b \Vert _{\infty}+ \Vert c \Vert _{\infty}+ \Vert d \Vert _{\infty}+ \Vert e \Vert _{\infty} \bigr)^{q-1} \Vert u \Vert _{X} \\ &\quad{}+ \frac{C_{1}M'_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert u \Vert _{X} \leq \frac{(C_{1}M+C_{2}) \Vert a \Vert ^{q-1}_{\infty}+\frac{C_{1}M'_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}}{1-(C_{1}M+C_{2}) ( \Vert b \Vert _{\infty}+ \Vert c \Vert _{\infty}+ \Vert d \Vert _{\infty}+ \Vert e \Vert _{\infty} )^{q-1}}. \end{aligned}$$

If \(q>2\), then

$$\begin{aligned} \Vert u \Vert _{X}&\leq 2^{q-2}(C_{1}M+C_{2}) \Vert a \Vert ^{q-1}_{\infty} \\ &\quad {}+2^{q-2}(C_{1}M+C_{2}) \bigl( \Vert b \Vert _{\infty}+ \Vert c \Vert _{\infty}+ \Vert d \Vert _{\infty}+ \Vert e \Vert _{\infty} \bigr)^{q-1} \Vert u \Vert _{X} \\ &\quad{}+ \frac{C_{1}M'_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert u \Vert _{X} &\leq \frac{2^{q-2}(C_{1}M+C_{2}) \Vert a \Vert ^{q-1}_{\infty}+\frac{C_{1}M'_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}}{1-2^{q-2}(C_{1}M+C_{2}) ( \Vert b \Vert _{\infty}+ \Vert c \Vert _{\infty}+ \Vert d \Vert _{\infty}+ \Vert e \Vert _{\infty} )^{q-1}}. \end{aligned}$$

Set \(L=\max_{q\in (1,+\infty )}\{1,2^{q-2}\}\), then the above inequality is equivalent to

$$\begin{aligned} \Vert u \Vert _{X} \leq \frac{L(C_{1}M+C_{2}) \Vert a \Vert ^{q-1}_{\infty}+\frac{C_{1}M'_{0}\Gamma (\beta _{2}(3-\alpha _{2})+1)}{ \vert \delta _{1} \vert \Gamma (\gamma _{2})}}{1-L(C_{1}M+C_{2}) ( \Vert b \Vert _{\infty}+ \Vert c \Vert _{\infty}+ \Vert d \Vert _{\infty}+ \Vert e \Vert _{\infty} )^{q-1}}. \end{aligned}$$

This means that \(\Omega _{1}\) is bounded. The remaining proof is similar to Theorem 3.6 and is omitted here. Finally, we can get that boundary value problem (1.1) has at least one solution in X. □

Remark

When the inequality \(|t^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-1,\beta _{2}}_{0+}u(t)|>M'_{0}\) in assumption condition \((H_{7})\) is replaced by \(|t^{-\beta _{2}(3-\alpha _{2})}D^{\alpha _{2}-2,\beta _{2}}_{0+}u(t)|>M''_{0}\), the method of proving the existence of the solution of boundary value problem (1.1) is similar to Theorem 3.9. There is no detailed explanation here.

4 Example

Consider the following boundary value problem at resonance:

$$ \textstyle\begin{cases} D^{\frac{1}{2},\frac{1}{3}}_{1-}\varphi _{\frac{3}{2}}(D^{ \frac{5}{2},\frac{1}{2}}_{0+}u(t))=\frac{1}{45} [5+\sin (\sqrt{ \vert u(t) \vert } )+\sin (\sqrt{ \vert D^{\frac{1}{2},\frac{1}{2}}_{0+}u(t) \vert } )+ \sin (\sqrt{ \vert D^{\frac{3}{2},\frac{1}{2}}_{0+}u(t) \vert } ) \\ \hphantom{D^{\frac{1}{2},\frac{1}{3}}_{1-}\varphi _{\frac{3}{2}}(D^{ \frac{5}{2},\frac{1}{2}}_{0+}u(t))={}}{}+\sin (\sqrt{ \vert D^{\frac{5}{2}, \frac{1}{2}}_{0+}u(t) \vert } ) ], \quad t\in [0,1], \\ u(0)=D^{\frac{5}{2},\frac{1}{2}}_{0+}u(1)=0,\qquad T_{1}(u)=D^{\frac{7}{4}}_{0+}u(1)+D^{ \frac{3}{4}}_{0+}u(1)=0, \\ T_{2}(u)=2D^{\frac{7}{4}}_{0+}u(1)+ \int _{0}^{2}D^{\frac{3}{4}}_{0+}u(t)\,dt =0. \end{cases} $$
(4.1)

Corresponding to boundary value problem (1.1), we have \(\alpha _{1}=\frac{1}{2}\), \(\alpha _{2}=\frac{5}{2}\), \(\beta _{1}=\frac{1}{3}\), \(\beta _{2}=\frac{1}{2}\), \(\gamma _{1}=\frac{2}{3}\), \(\gamma _{2}=\frac{11}{4}\), \(p={\frac{3}{2}}\), \(k=2\) and

$$\begin{aligned} f\bigl(t,u(t),D^{\frac{1}{2},\frac{1}{2}}_{0+}u(t),D^{\frac{3}{2}, \frac{1}{2}}_{0+}u(t),D^{\frac{5}{2},\frac{1}{2}}_{0+}u(t) \bigr)&= \frac{1}{45} \bigl[5+\sin \bigl(\sqrt{ \bigl\vert u(t) \bigr\vert } \bigr)+\sin \bigl(\sqrt{ \bigl\vert D^{ \frac{1}{2},\frac{1}{2}}_{0+}u(t) \bigr\vert } \bigr) \\ &\quad{}+\sin \bigl(\sqrt{ \bigl\vert D^{\frac{3}{2},\frac{1}{2}}_{0+}u(t) \bigr\vert } \bigr)+\sin \bigl(\sqrt{ \bigl\vert D^{\frac{5}{2},\frac{1}{2}}_{0+}u(t) \bigr\vert } \bigr) \bigr]. \end{aligned}$$

Boundary value problem (4.1) is at resonance with

$$\begin{aligned} \operatorname{Ker}L= \biggl\{ c \biggl(\frac{2t^{\frac{3}{4}}}{\Gamma (\frac{7}{4})}- \frac{t^{\frac{7}{4}}}{\Gamma (\frac{11}{4})} \biggr), c\in R \biggr\} ,\qquad D^{ \frac{7}{4}}_{0+}u(t)=-c,\qquad D^{\frac{3}{4}}_{0+}u(t)=c(2-t). \end{aligned}$$

Thus,

$$\begin{aligned} T_{1}\bigl(t^{\frac{3}{4}}\bigr)=\delta _{1}= \frac{1}{\Gamma (\frac{11}{4})} \neq 0, \qquad T_{1}\bigl(t^{\frac{7}{4}}\bigr)= \delta _{2}= \frac{2}{\Gamma (\frac{7}{4})}\neq 0. \end{aligned}$$

Take \(a(t)=\frac{1}{9}\), \(b(t)=c(t)=d(t)=e(t)=\frac{1}{45}\), and \(q=3\), then

$$\begin{aligned}& C_{1}= \max \{1.715, 3.097, 1.098 \}=3.097, \qquad C_{2}=\max \{2.714, 1.148, 2.219 \}=2.714, \\& M= \biggl(\frac{1}{\Gamma (1.5)} \biggr)^{2}\Gamma (0.25)+ \frac{2 (\frac{1}{\Gamma (1.5)} )^{2}}{\frac{1}{4\Gamma (2.75)} (\frac{4}{\Gamma (1.75)}+\frac{1}{\Gamma (2.75)} )}=8.00025, \qquad L= \max \bigl\{ 1,2^{2} \bigr\} =4. \end{aligned}$$

Therefore,

$$\begin{aligned} &L(C_{1}M+C_{2}) \bigl( \Vert b \Vert _{\infty}+ \Vert c \Vert _{\infty}+ \Vert d \Vert _{\infty}+ \Vert e \Vert _{\infty} \bigr)^{q-1}=0.8688< 1, \\ & \bigl\vert f(t,x,y,z,w) \bigr\vert \leq \frac{1}{9}+ \frac{1}{45}\varphi _{p}\bigl( \vert x \vert \bigr)+ \frac{1}{45}\varphi _{p}\bigl( \vert y \vert \bigr)+ \frac{1}{45}\varphi _{p}\bigl( \vert z \vert \bigr)+ \frac{1}{45}\varphi _{p}\bigl( \vert w \vert \bigr). \end{aligned}$$

That means condition \((H_{8})\) holds.

Let \({M'_{0}=5}\), if \(|t^{-\frac{1}{4}}D^{\frac{3}{2},\frac{1}{2}}_{0+}u(t)|>M'_{0}\) holds for any \(t\in (0,1]\), then

$$\begin{aligned} &f\bigl(t,u(t),D^{\frac{1}{2},\frac{1}{2}}_{0+}u(t),D^{\frac{3}{2}, \frac{1}{2}}_{0+}u(t),D^{\frac{5}{2},\frac{1}{2}}_{0+}u(t) \bigr) \\ &\quad = \frac{1}{45} \bigl[5+\sin \bigl(\sqrt{ \bigl\vert u(t) \bigr\vert } \bigr)+\sin \bigl(\sqrt{ \bigl\vert D^{ \frac{1}{2},\frac{1}{2}}_{0+}u(t) \bigr\vert } \bigr)+\sin \bigl(\sqrt{ \bigl\vert D^{ \frac{3}{2},\frac{1}{2}}_{0+}u(t) \bigr\vert } \bigr)+\sin \bigl(\sqrt{ \bigl\vert D^{ \frac{5}{2},\frac{1}{2}}_{0+}u(t) \bigr\vert } \bigr) \bigr] \\ &\quad >0 \end{aligned}$$

and

$$\begin{aligned} &(T_{2}-kT_{1}) \bigl(I^{\alpha _{2}}_{0+} \varphi _{q}\bigl(I^{\alpha _{1}}_{1-}Nu(t)\bigr) \bigr) \\ &\quad = \frac{1}{\Gamma (\frac{7}{4})} \int _{0}^{2} \biggl[ \int _{0}^{t}(t-s)^{ \frac{3}{4}}\varphi _{3}\bigl(I^{\frac{1}{2}}_{1-}Nu(s)\bigr)\,ds - \int _{0}^{1}(1-s)^{ \frac{3}{4}}\varphi _{3}\bigl(I^{\frac{1}{2}}_{1-}Nu(s)\bigr)\,ds \biggr]\,dt < 0. \end{aligned}$$

Hence, condition \((H_{7})\) holds.

Similarly, let \(B_{1}=24\), \(u(t)=c (\frac{2t^{\frac{3}{4}}}{\Gamma (\frac{7}{4})}- \frac{t^{\frac{7}{4}}}{\Gamma (\frac{11}{4})} )\), \(c\in R\), if \(|c|>B_{1}\), then \(|t^{-\frac{1}{4}}D^{\frac{3}{2},\frac{1}{2}}_{0+}u(t)|=\frac{1}{4}|c|>M'_{0} \). Therefore, \((T_{2}-kT_{1}) (I^{\alpha _{2}}_{0+}\varphi _{q} (I^{\alpha _{1}}_{1-}Nc (\frac{2t^{\frac{3}{4}}}{\Gamma (\frac{7}{4})}- \frac{t^{\frac{7}{4}}}{\Gamma (\frac{11}{4})} ) ) )\neq 0\). Clearly, condition \((H_{6})\) holds. Through the application of Theorem 3.9, we obtain that boundary value problem (4.1) has at least one solution.

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Acknowledgements

The authors are grateful to anonymous referees for their constructive comments and suggestions. The authors would like to thank the handling editors for the help in the processing of the paper.

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This work was supported by the Science Foundation of Hebei Normal University (L2017J01).

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M, J and Z studied the solvability of mixed Hilfer fractional boundary value problems under functional boundary value conditions and were major contributors to the writing of the manuscript. G gives an example of this article. All authors read and approved the final manuscript.

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Correspondence to Lina Zhou.

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Meng, F., Jiang, W., Guo, C. et al. Solvability of mixed Hilfer fractional functional boundary value problems with p-Laplacian at resonance. Bound Value Probl 2022, 81 (2022). https://doi.org/10.1186/s13661-022-01662-6

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