Existence of solutions for fractional p-Laplacian differential equations with dual periodic boundary conditions

Abstract

This paper introduces a new concept of dual periodic boundary conditions. One of these conditions relates to the endpoints of an interval, while the other one involves two nonlocal positions within the interval. Under these boundary conditions, we establish the criteria for the existence of solutions for a fractional differential equation with p-Laplacian operator by using Mawhin's continuation theorem. An example is constructed for illustrating the obtained results.

Keywords:

• Fractional differential equation
• dual periodic boundary conditions
• p-Laplacian operator
• Mawhin's continuation theorem
• existence

Mathematics Subject Classifications:

• 34A08
• 34B15
• 34B10

1. Introduction

Boundary value problems (BVPs) of ordinary differential equations (ODEs) are a type of definite problems of ODEs. Their characteristic is that the boundary conditions are formed by restricting the values of the solution and its related derivatives at least two points of the independent variable. BVPs of ODEs have rich sources in classical mechanics and electrodynamics, and they are one of the important components of the discipline of ODEs [1–3]. Two-point BVPs of ODEs (such as Dirichlet BVP, Neumann BVP, Robin BVP, Sturm-Liouville BVP and (anti)periodic BVP, etc.). Two-point BVPs of integer-order ODEs have been deeply and extensively studied, and systematic and profound results have been obtained.

In the past 30 years, with the flourishing development of fractional calculus theory, the qualitative analysis of BVPs for fractional differential equations has become an active research field [4–8]. Some important theoretical results have been obtained. The common functional analysis tools for studying these problems are (such as fixed point theorem, upper and lower solution method, monotone iteration method, Leray-Schauder degree theory, critical point theory, etc.). In recent years, the periodic and anti-periodic boundary value problems of fractional differential equations have attracted much attention from scholars and obtained many interesting results [9–15].

In 2018, Benchohra et al. [9] discussed the following nonlinear implicit fractional differential equation with periodic boundary conditions $\{\begin{array}{ll}{}^H{D}_{1+}^{\alpha }y(t)=f(t,y(t),{}^H{D}_{1+}^{\alpha }y(t)),& t\in [1,T],T>1,\\ y(1)=y(T),\end{array}$where ${}^H{D}_{1+}^{\alpha }$ is Hadamard fractional derivative with order α ($0<\alpha \le 1$), $f\in C([1,T]\times {\mathbb{R}}^2,\mathbb{R})$. The authors obtained the existence result of the solution by using the continuation theorem.

Recently, Salim et al. [10] studied the following coupled system of nonlinear k-generalized ψ-Hilfer type implicit fractional differential equations with periodic conditions $\{\begin{array}{l}{}_k^H{D}_{a+}^{{\alpha }_1,{\beta }_1;\psi }x(t)=f(t,x(t),y(t),{}_k^H{D}_{a+}^{{\alpha }_1,{\beta }_1;\psi }x(t),{}_k^H{D}_{a+}^{{\alpha }_2,{\beta }_2;\psi }y(t)),t\in (a,b),\\ {}_k^H{D}_{a+}^{{\alpha }_2,{\beta }_2;\psi }y(t)=g(t,x(t),y(t),{}_k^H{D}_{a+}^{{\alpha }_1,{\beta }_1;\psi }x(t),{}_k^H{D}_{a+}^{{\alpha }_2,{\beta }_2;\psi }y(t)),t\in (a,b),\\ I_{a+}^{k(1-{\delta }_1),k;\psi }x(a)=I_{a+}^{k(1-{\delta }_1),k;\psi }x(b),I_{a+}^{k(1-{\delta }_2),k;\psi }y(a)=I_{a+}^{k(1-{\delta }_2),k;\psi }y(b),\end{array}$where ${}_k^H{D}_{a+}^{{\alpha }_1,{\beta }_1;\psi }$, ${}_k^H{D}_{a+}^{{\alpha }_2,{\beta }_2;\psi }$ are the k-generalized ψ-Hilfer derivatives of order ${\alpha }_i$, i = 1, 2 ($0<{\alpha }_i\le 1$), $f,g\in C([a,b]\times {\mathbb{R}}^4,\mathbb{R})$. The arguments are based on the Mawhin's coincidence degree theory.

In 2020, Ahmad et al. [14] proposed a class of nonlocal antiperiodic boundary value problems (called dual antiperiodic boundary value problems), and considered the following fractional differential-integral equation equipped with dual anti-periodic boundary conditions $\{\begin{array}{l}{}^C{D}^q[({}^C{D}^p+\alpha )y(t)+\beta I^{\sigma }h(t,y(t))]=f(t,y(t)),t\in (x_1,x_2),\\ y(x_1)+y(x_2)=0,y^{\prime }(x_1)+y^{\prime }(x_2)=0,\\ y(\xi )+y(\eta )=0,y^{\prime }(\xi )+y^{\prime }(\eta )=0,\end{array}$where $1<p,q\le 2$, ${}^C{D}_{0+}^{\kappa }$ denotes the Caputo fractional differential derivative of order κ ($\kappa =p,q$), α and β are real numbers, $\sigma >0$, $h,f:[x_1,x_2]\times \mathbb{R}\to \mathbb{R}$ are continuous functions. The authors used the Krasnoselskii, Leray-Schauder fixed point theorems and the Banach contraction mapping principle to obtain the existence and uniqueness results of solutions.

Note that the antiperiodic boundary conditions and the periodic boundary conditions are two important types of boundary conditions for BVPs for ODEs, which have important practical applications. The literature [14] discuss the dual antiperiodic boundary value problems, and a natural idea is to discuss the dual periodic boundary value problems. For this purpose, inspired by the above literature, this paper discusses the following fractional p-Laplacian equation (1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) subject to the dual periodic boundary conditions (2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) where $1<\alpha ,\beta \le 2,0<\xi <\eta <1,{}^C{D}_{0+}^{\rho }$ is the Caputo fractional derivative of order $\rho (\rho =\alpha ,\beta )$, $f\in C([0,1]\times {\mathbb{R}}^2,\mathbb{R})$, ${\varphi }_p(s)=|s{|}^{p-2}s(s\ne 0)$, and ${\varphi }_p(0)=0$ is the p-Laplacian operator.

It should be pointed out that the BVP (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ) is essentially different from the dual antiperiodic BVP mentioned above. The dual antiperiodic BVP is non-resonant, and its existence of solutions can be studied directly by using the fixed point theorem, while the dual periodic BVP (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ) is a resonant problem (A BVP is referred to as a resonant problem if the associated homogeneous BVP yields a non-trivial solution). In fact, the homogeneous equation corresponding to the fractional differential Equation (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) under the dual periodic boundary conditions (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ) $\{\begin{array}{l}{}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=0,t\in (0,1),\\ x(0)=x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )=x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$has a nontrivial solution $x(t)=c,c\in \mathbb{R}$. Since the resonant BVPs cannot be directly applied with the fixed point theorem, this paper uses the Mawhin's continuation theorem to give the existence result of the solutions for problem (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ) under the nonlinear term $f(t,u,v)$ satisfies the p−1 growth conditions. Note that the dual periodic boundary conditions are the nonlocal boundary conditions, while the classical periodic boundary conditions are the local boundary conditions. In particular, taking $\alpha ,\beta \to 2$, p = 2 then problem (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ) degenerates to the classical fourth-order elastic beam equation with dual periodic boundary conditions. Therefore, this paper is a generalization of the classical problem. Moreover, due to the dual periodic boundary conditions being nonlocal boundary conditions, and the p-Laplacian being a quasi-linear operator, it brings some difficulties to the study of the problem.

The structure of this paper is as follows. In Section 2, we recall some definitions and lemmas. In Section 3, based on the Mawhin's continuation theorem, we establish an existence theorem for the problem (1.1). In Section 4, we give an example to illustrate the application of the theorems.

2. Preliminaries

In this section, we recall some basic knowledge of fractional calculus, including its definition and properties, and the coincidence degree theory.

Definition 2.1

[16]

The (left-sided) Riemann-Liouville fractional integral of order $\gamma >0$ of a function $f:(0,+\mathrm{\infty })\to \mathbb{R}$ is given by $I_{0+}^{\gamma }f(t)=\frac{1}{\mathrm{\Gamma }(\gamma )}{\int }_0^t{(t-s)}^{\gamma -1}f(s)\mathrm{d}s,t>0,$provided the right-hand side is pointwise defined on $(0,+\mathrm{\infty })$.

Definition 2.2

[16]

For $f\in A{C}^n[0,\mathrm{\infty })$, the (left-sided) Caputo fractional derivative of order $\gamma >0$ is defined as ${}^C{D}_{0+}^{\gamma }f(t)=\frac{1}{\mathrm{\Gamma }(n-\gamma )}{\int }_0^t{(t-s)}^{n-\gamma -1}{f}^{(n)}(s)\mathrm{d}s,t>0,$where $n-1<\gamma <n$, $n\in \mathbb{N}$.

Lemma 2.1

see [16]

Let $\gamma >0$ and $n=\lfloor \gamma \rfloor +1$ for $\gamma \notin \mathbb{N}$ and $n=\gamma$ for $\gamma \in \mathbb{N}$, (namely, $n-1<\gamma \le n$). If $f\in A{C}^n[0,1]$, then $I_{0+}^{\gamma }{}^C{D}_{0+}^{\gamma }f(t)=f(t)+\sum _{i=0}^{n-1}{c}_i{t}^i,0<t<1,$where $c_0,\dots ,c_{n-1}\in \mathbb{R}$.

Definition 2.3

[17, 18]

Let $(X,\Vert \cdot {\Vert }_X)$ and $(Y,\Vert \cdot {\Vert }_Y)$ be real Banach spaces, $L:\mathrm{dom}L\subset X\to Y$ is a linear operator. If $\mathrm{Im}L$ is a closed subspace of Y, and $\dim \mathrm{Ker}L=\mathrm{codim}\mathrm{Im}L<\mathrm{\infty }$, then L is called a zero-index Fredholm operator.

Suppose that $L:\mathrm{dom}L\subset X\to Y$ is a zero-index Fredholm operator, then there exist projection operators $P:X\to X$ and $Q:Y\to Y$ such that $\mathrm{Im}P=\mathrm{Ker}L,\mathrm{Im}L=\mathrm{Ker}Q,X=\mathrm{Ker}L\oplus \mathrm{Ker}P,Y=\mathrm{Im}L\oplus \mathrm{Im}Q,$and $L{|}_{\mathrm{dom}L\cap \mathrm{Ker}P}:\mathrm{dom}L\to \mathrm{Im}L$ is invertible, denoted as $K_p=(L{|}_{\mathrm{dom}L\cap \mathrm{Ker}P}{)}^{-1}$. Let Ω be a non-empty bounded open set in X such that $\mathrm{dom}L\cap \overline{\mathrm{\Omega }}\ne \mathrm{\varnothing }$. The mapping $N:X\to Y$ is said to be L-compact in $\overline{\mathrm{\Omega }}$, if $\mathrm{QN}(\overline{\mathrm{\Omega }})$ is bounded, and $K_p(I-Q)N:\overline{\mathrm{\Omega }}\to X$ is compact.

Theorem 2.1

Mawhin's continuation theorem [17, 18]

Suppose that $L:\mathrm{dom}L\subset X\to Y$ is a zero-index Fredholm operator, and $N:X\to Y$ is L-compact in $\overline{\mathrm{\Omega }}$. If the following conditions hold

(i)	$\mathrm{Lu}\ne \mathrm{\lambda Nu}$ for any $u\in (\mathrm{dom}L\mathrm{\setminus }\mathrm{Ker}L)\cap \mathrm{\partial \Omega },\lambda \in (0,1)$.
(ii)	$\mathrm{Nu}\notin \mathrm{Im}L$ for any $u\in \mathrm{Ker}L\cap \mathrm{\partial \Omega }$.
(iii)	$\mathrm{deg}\{\mathrm{QN}{|}_{\mathrm{Ker}L},\mathrm{\Omega }\cap \mathrm{Ker}L,0\}\ne 0$.

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

It is noted that the Mawhin's continuation theorem is only valid for linear operators, while the p-Laplacian operator is a quasi-linear operator, thus the Mawhin's continuation theorem cannot be directly applied to handle BVP (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ). For this reason, we equivalently transform the BVP (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ) into the following system BVP (3) $\{\begin{array}{l}{}^C{D}_{0+}^{\alpha }{x}_1(t)={\varphi }_q(x_2(t)),t\in (0,1),\\ {}^C{D}_{0+}^{\beta }{x}_2(t)=f(t,x_1(t),{\varphi }_q(x_2(t)),t\in (0,1),\\ x_1(0)=x_1(1),x_1(\xi )=x_1(\eta ),\\ x_2(0)=x_2(1),x_2(\xi )=x_2(\eta ),\end{array}$(3) where q>1 with $\frac{1}{p}+\frac{1}{q}=1$. Obviously, if $x=(x_1,x_2)$ is a solution to the BVP (3(3) $\{\begin{array}{l}{}^C{D}_{0+}^{\alpha }{x}_1(t)={\varphi }_q(x_2(t)),t\in (0,1),\\ {}^C{D}_{0+}^{\beta }{x}_2(t)=f(t,x_1(t),{\varphi }_q(x_2(t)),t\in (0,1),\\ x_1(0)=x_1(1),x_1(\xi )=x_1(\eta ),\\ x_2(0)=x_2(1),x_2(\xi )=x_2(\eta ),\end{array}$(3) ), then $x_1$ must be a solution to BVP (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ). Therefore, to prove the existence of solutions to the BVP (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ), it is only necessary to prove the existence of solutions to the system BVP (3(3) $\{\begin{array}{l}{}^C{D}_{0+}^{\alpha }{x}_1(t)={\varphi }_q(x_2(t)),t\in (0,1),\\ {}^C{D}_{0+}^{\beta }{x}_2(t)=f(t,x_1(t),{\varphi }_q(x_2(t)),t\in (0,1),\\ x_1(0)=x_1(1),x_1(\xi )=x_1(\eta ),\\ x_2(0)=x_2(1),x_2(\xi )=x_2(\eta ),\end{array}$(3) ).

3. Main result

Consider the Banach space $Y=C([0,1],\mathbb{R})$ endowed with the norm $||y|{|}_{\mathrm{\infty }}=\underset{t\in \left[0,1\right]}{max}|y(t)|.$Define the space $X=\{x=(x_1,x_2):x_1,x_2\in Y\},$endowed with the norm $||x|{|}_X=max\{||x_1|{|}_{\mathrm{\infty }},||x_2|{|}_{\mathrm{\infty }}\},$It is easy to check that X is a Banach space.

Define the linear operator $L:\mathrm{dom}L\subset X\to X$, (4) $\mathrm{Lx}={(}^C{D}_{0+}^{\alpha }{x}_1{,}^C{D}_{0+}^{\beta }{x}_2),x\in \mathrm{dom}L,$(4) where $\mathrm{dom}L=\{x\in X:{}^C{D}_{0+}^{\alpha }{x}_1,{}^C{D}_{0+}^{\beta }{x}_2\in Y,x_1,x_2\text{satisfy the boundary conditions in}(3)\}.$Define the non-linear operator $N:X\to X$, (5) $\mathrm{Nx}(t)=({\varphi }_q(x_2(t)),f(t,x_1(t),{\varphi }_q(x_2(t))),t\in [0,1].$(5) Then the BVP problem (3(3) $\{\begin{array}{l}{}^C{D}_{0+}^{\alpha }{x}_1(t)={\varphi }_q(x_2(t)),t\in (0,1),\\ {}^C{D}_{0+}^{\beta }{x}_2(t)=f(t,x_1(t),{\varphi }_q(x_2(t)),t\in (0,1),\\ x_1(0)=x_1(1),x_1(\xi )=x_1(\eta ),\\ x_2(0)=x_2(1),x_2(\xi )=x_2(\eta ),\end{array}$(3) ) is equivalent to the operator equation $\mathrm{Lx}=\mathrm{Nx},x\in \mathrm{dom}L.$Below we present the main results of this paper.

Theorem 3.1

Assume that the function $f:[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ is continuous. If the following conditions hold,

(A1)	There exist non-negative functions $a(t),b(t),c(t)\in Y$ such that for $(t,u,v)\in [0,1]\times {\mathbb{R}}^3$, $\left|f\left(t,u,v\right)\right|\le a\left(t\right)+b\left(t\right){\left|u\right|}^{p-1}+c\left(t\right){\left|v\right|}^{p-1},$and $\rho =\frac{4}{\mathrm{\Gamma }(\beta +1)}\left[2^{p-1}{\left(\frac{4}{\mathrm{\Gamma }(\alpha +1)}\right)}^{p-1}||b|{|}_{\mathrm{\infty }}+||c|{|}_{\mathrm{\infty }}\right]<1,$where $b=||b(t)|{|}_{\mathrm{\infty }},c=||c(t)|{|}_{\mathrm{\infty }}$.
(A2)	For any $x=(x_1,x_2)\in \mathrm{dom}L$, there exist constants $B_i>0(i=1,2)$, such that for $t\in [0,1]$, if $|x_1(t)|>B_1$ or $|x_2(t)|$ $>B_2$, then $\mathrm{QNx}\ne (0,0)$.
(A3)	There exists a constant G>0 such that if $|\ell |>G$, then one of the following inequalities holds: (6) $\ell \left[{\int }_0^1{G}^{\beta }(\eta ,s)f(s,\ell ,0)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)f(s,\ell ,0)\mathrm{d}s\right]>0,$(6) or (7) $\ell \left[{\int }_0^1{G}^{\beta }(\eta ,s)f(s,\ell ,0)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)f(s,\ell ,0)\mathrm{d}s\right]<0.$(7)

Then problem (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) has at least one solution.

To prove Theorem 3.1, we first present some auxiliary lemmas.

Lemma 3.1

Assume that the operator L is defined by (4(4) $\mathrm{Lx}={(}^C{D}_{0+}^{\alpha }{x}_1{,}^C{D}_{0+}^{\beta }{x}_2),x\in \mathrm{dom}L,$(4) ), then $\begin{array}{rl}(\mathrm{i})\mathrm{Ker}L& =\{x(t)=(x_1(t),x_2(t))\in X:x(t)=(c_1,c_2)\in {\mathbb{R}}^2,t\in [0,1]\},\\ (\mathrm{ii})\mathrm{Im}L& =\{y(t)=(y_1(t),y_2(t))\in X:{\int }_0^1{G}^{\alpha }(\eta ,s)y_1(s)\mathrm{d}s-{\int }_0^1{G}^{\alpha }(\xi ,s)y_1(s)\mathrm{d}s=0,\\ & {\int }_0^1{G}^{\beta }(\eta ,s)y_2(s)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)y_2(s)\mathrm{d}s=0\},\end{array}$where $G^{\gamma }(t,s)=\{\begin{array}{ll}t(1-s{)}^{\gamma -1}-(t-s{)}^{\gamma -1},& 0\le s\le t<1,\\ t(1-s{)}^{\gamma -1},& 0<t\le s\le 1.\end{array}$

Proof.

By Lemma 2.1, it is easy to verify that (i) holds. We now prove that (ii) holds. In fact, for any $y(t)=(y_1(t),y_2(t))\in \mathrm{Im}L$, there exist $x(t)=(x_1(t),x_2(t))\in \mathrm{dom}L$, such that $(y_1,y_2)=L(x_1,x_2)\in X$. It follows from Lemma 2.1 that $\begin{array}{r}{x}_1(t)=I_{0+}^{\alpha }{y}_1(t)+c_1+c_{2}t,c_1,c_2\in \mathbb{R},\\ x_2(t)=I_{0+}^{\beta }{y}_2(t)+c_3+c_{4}t,c_3,c_4\in \mathbb{R},\end{array}$In view of the boundary conditions $x_1(0)=x_1(1),x_1(\xi )=x_1(\eta )$, we have $\begin{array}{rl}& c_2=-I_{0+}^{\alpha }{y}_1(t){|}_{t=1},\\ & I_{0+}^{\alpha }{y}_1(t){|}_{t=\xi }-I_{0+}^{\alpha }{y}_1(t){|}_{t=\eta }+c_2(\xi -\eta )=0.\end{array}$Then, we obtain $\begin{array}{rl}& {\int }_0^1\eta {(1-s)}^{\alpha -1}{y}_1(s)\mathrm{d}s-{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}{y}_1(s)\mathrm{d}s\\ & -\left[{\int }_0^1\xi {(1-s)}^{\alpha -1}{y}_1(s)\mathrm{d}s-{\int }_0^{\xi }{(\xi -s)}^{\alpha -1}{y}_1(s)\mathrm{d}s\right]=0.\end{array}$It follows that $\begin{array}{rl}& {\int }_0^{\eta }[\eta {(1-s)}^{\alpha -1}-(\eta -s{)}^{\alpha -1}]y_1(s)\mathrm{d}s+{\int }_{\eta }^1\eta {(1-s)}^{\alpha -1}{y}_1(s)\mathrm{d}s\\ & -\left[{\int }_0^{\xi }[\xi {(1-s)}^{\alpha -1}-{(\xi -s)}^{\alpha -1}]y_1(s)\mathrm{d}s+{\int }_{\xi }^1\xi {(1-s)}^{\alpha -1}{y}_1(s)\mathrm{d}s\right]=0,\end{array}$that is, ${\int }_0^1{G}^{\alpha }(\eta ,s)y_1(s)\mathrm{d}s-{\int }_0^1{G}^{\alpha }(\xi ,s)y_1(s)\mathrm{d}s=0.$Similarly, by using boundary conditions $x_2(0)=x_2(1),x_2(\xi )=x_2(\eta )$, we obtain ${\int }_0^1{G}^{\beta }(\eta ,s)y_2(s)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)y_2(s)\mathrm{d}s=0.$That is, $\begin{array}{rl}\mathrm{Im}L& \subset \{y(t)=(y_1(t),y_2(t))\in X:{\displaystyle {\int }_0^1{G}^{\alpha }(\eta ,s)y_1(s)\mathrm{d}s-{\int }_0^1{G}^{\alpha }(\xi ,s)y_1(s)\mathrm{d}s=0,}\\ & {\displaystyle {\int }_0^1{G}^{\beta }(\eta ,s)y_2(s)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)y_2(s)\mathrm{d}s=0}\},\end{array}$On the other hand, let $(y_1,y_2)\in X$ satisfy $\begin{array}{rl}& {\int }_0^1{G}^{\alpha }(\eta ,s)y_1(s)\mathrm{d}s-{\int }_0^1{G}^{\alpha }(\xi ,s)y_1(s)\mathrm{d}s=0,\\ & {\int }_0^1{G}^{\beta }(\eta ,s)y_2(s)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)y_2(s)\mathrm{d}s=0.\end{array}$Take $x_1(t)=I_{0+}^{\alpha }{y}_1(t)-I_{0+}^{\alpha }{y}_1(t){|}_{t=1}t,x_2(t)=I_{0+}^{\beta }{y}_2(t)-I_{0+}^{\beta }{y}_2(t){|}_{t=1}t$, then $(x_1(t),x_2(t))\in \mathrm{dom}L$ and $L(x_1(t),x_2(t))=(y_1(t),y_2(t))$. Therefore, $\begin{array}{rl}& \{y(t)=(y_1(t),y_2(t))\in X:{\displaystyle {\int }_0^1{G}^{\alpha }(\eta ,s)y_1(s)\mathrm{d}s-{\int }_0^1{G}^{\alpha }(\xi ,s)y_1(s)\mathrm{d}s=0,}\\ & {\displaystyle {\int }_0^1{G}^{\beta }(\eta ,s)y_2(s)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)y_2(s)\mathrm{d}s=0}\}\subset \mathrm{Im}L.\end{array}$Hence, (ii) holds. The proof is complete.

Lemma 3.2

Assume that the operator L is defined by (4(4) $\mathrm{Lx}={(}^C{D}_{0+}^{\alpha }{x}_1{,}^C{D}_{0+}^{\beta }{x}_2),x\in \mathrm{dom}L,$(4) ), then L is a Fredholm operator with index zero, and the projection operators $P:X\to X$ and $Q:X\to X$ can be defined as $\begin{array}{rl}\mathrm{Px}(t)& =P(x_1(t),x_2(t))=(x_1(0),x_2(0)),t\in [0,1],\\ \mathrm{Qy}(t)& =Q(y_1(t),y_2(t))=(T_{\alpha }{y}_1(t),T_{\beta }{y}_2(t)),t\in [0,1],\end{array}$where $\begin{array}{rl}{T}_{\gamma }{y}_i(t)& =\frac{\gamma }{(\eta -{\eta }^{\gamma })-(\xi -{\xi }^{\gamma })}\\ & \left[{\int }_0^1{G}^{\gamma }(\eta ,s)y_i(s)\mathrm{d}s-{\int }_0^1{G}^{\gamma }(\xi ,s)y_i(s)\mathrm{d}s\right],t\in [0,1],\end{array}$$\gamma =\alpha ,\beta ,i=1,2$. Moreover, the operator $K_p:\mathrm{Im}L\to \mathrm{dom}L\cap \mathrm{Ker}P$ is defined as follows $K_{p}y(t)=(I_{0+}^{\alpha }{y}_1(t)-I_{0+}^{\alpha }{y}_1(t){|}_{t=1}t,I_{0+}^{\beta }{y}_2(t)-I_{0+}^{\beta }{y}_2(t){|}_{t=1}t),t\in [0,1].$

Proof.

By the definition of P, it is easy to see that $\mathrm{Im}P=\mathrm{Ker}L,P^2(x_1,x_2)=P(x_1,x_2)$. For any $(x_1,x_2)\in X$, we have $(x_1,x_2)=((x_1,x_2)-P(x_1,x_2))+P(x_1,x_2)$, that is, $X=\mathrm{Ker}P+\mathrm{Ker}L.$ Moreover, it is easy to prove that $\mathrm{Ker}L\cap \mathrm{Ker}P=\{(0,0)\}$. Therefore, we have $X=\mathrm{Ker}P\oplus \mathrm{Ker}L.$For any $(y_1,y_2)\in X$, by the definition of $T_{\gamma }$, we have $\begin{array}{rl}{T}_{\gamma }{T}_{\gamma }{y}_i(t)& =\frac{\gamma }{(\eta -{\eta }^{\gamma })-(\xi -{\xi }^{\gamma })}\left[{\int }_0^1{G}^{\gamma }(\eta ,s)y_i(s)\mathrm{d}s-{\int }_0^1{G}^{\gamma }(\xi ,s)y_i(s)\mathrm{d}s\right]\\ & =\frac{\gamma }{(\eta -{\eta }^{\gamma })-(\xi -{\xi }^{\gamma })}\left[{\int }_0^1{G}^{\gamma }(\eta ,s)\mathrm{d}s-{\int }_0^1{G}^{\gamma }(\xi ,s)\mathrm{d}s\right]\\ & =T_{\gamma }{y}_i(t),\end{array}$$\gamma =\alpha ,\beta ,i=1,2$. Then, by the definition of Q, we obtain $\begin{array}{rl}{Q}^2(y_1,y_2)& =Q(Q(y_1,y_2))=Q(T_{\alpha }{y}_1(t),T_{\beta }{y}_2(t))\\ & =(T_{\alpha }{T}_{\alpha }{y}_1(t),T_{\beta }{T}_{\beta }{y}_2(t))=(T_{\alpha }{y}_1(t),T_{\beta }{y}_2(t))=Q(y_1,y_2).\end{array}$Let $(y_1,y_2)=((y_1,y_2)-Q(y_1,y_2))+Q(y_1,y_2)$, then $(y_1,y_2)-Q(y_1,y_2)\in \mathrm{Ker}Q=\mathrm{Im}L,Q(y_1,y_2)\in \mathrm{Im}Q$. By $\mathrm{Ker}Q=\mathrm{Im}L$, and $Q^2(y_1,y_2)=Q(y_1,y_2)$, we have $\mathrm{Im}Q\cap \mathrm{Im}L=\{(0,0)\}$. Hence, $X=\mathrm{Im}L\oplus \mathrm{Im}Q.$Therefore, $\dim \mathrm{Ker}L=\dim \mathrm{Im}Q=\mathrm{codim}\mathrm{Im}L=2.$That is, L is a Fredholm operator of index zero. Next, we prove that $K_p$ is the inverse operator of $L{|}_{\mathrm{dom}L\cap \mathrm{Ker}P}$. In fact, for $y=(y_1,y_2)\in \mathrm{Im}L$, by the definition of $K_p$, we get $K_p(y_1,y_2)=(I_{0+}^{\alpha }{y}_1(t)-I_{0+}^{\alpha }{y}_1(t){|}_{t=1}t,I_{0+}^{\beta }{y}_2(t)-I_{0+}^{\beta }{y}_2(t){|}_{t=1}t),t\in [0,1].$Obviously, $K_p(y_1,y_2)\in \mathrm{dom}L$, and $P{K}_p(y_1,y_2)=(0,0)$. Therefore, the definition of $K_p$ is reasonable. Moreover, $L{K}_p(y_1,y_2)=L(I_{0+}^{\alpha }{y}_1(t)-I_{0+}^{\alpha }{y}_1(t){|}_{t=1}t,I_{0+}^{\beta }{y}_2(t)-I_{0+}^{\beta }{y}_2(t){|}_{t=1}t)=(y_1,y_2).$On the other hand, for any $(x_1,x_2)\in \mathrm{dom}L\cap \mathrm{Ker}P$, we have $(0,0)=P(x_1,x_2)=(x_1(0),x_2(0))$, together with $x_1(1)=x_1(0)$, $x_2(1)=x_2(0)$, thus, $\begin{array}{rl}{K}_{p}L(x_1,x_2)& =K_p({}^C{D}_{0+}^{\alpha }{x}_1(t),{}^C{D}_{0+}^{\beta }{x}_2(t))\\ & =(I_{0+}^{\alpha }{}^C{D}_{0+}^{\alpha }{x}_1(t)-I_{0+}^{\alpha }{}^C{D}_{0+}^{\alpha }{x}_1(t){|}_{t=1}t,I_{0+}^{\beta }{}^C{D}_{0+}^{\beta }{x}_2(t)-I_{0+}^{\beta }{}^C{D}_{0+}^{\beta }{x}_2(t){|}_{t=1}t)\\ & =(x_1(t)-x_1(0)-x_1(1)t+x_1(0)t,x_2(t)-x_2(0)-x_2(1)t+x_2(0)t)\\ & =(x_1(t),x_2(t)).\end{array}$Therefore, we can conclude that $K_p=(L{|}_{\mathrm{dom}L\cap \mathrm{Ker}P}{)}^{-1}.$The proof is completed.

Lemma 3.3

Let $\mathrm{\Omega }\subset X$ be a bounded open set and $\mathrm{dom}L\cap \overline{\mathrm{\Omega }}\ne \mathrm{\varnothing }$, then the N defined by (5(5) $\mathrm{Nx}(t)=({\varphi }_q(x_2(t)),f(t,x_1(t),{\varphi }_q(x_2(t))),t\in [0,1].$(5) ) is L-compact on $\overline{\mathrm{\Omega }}$.

Proof.

From the continuity of ${\varphi }_q$ and f, we know that $\mathrm{QN}(\overline{\mathrm{\Omega }})$ and $K_p(I-Q)N(\overline{\mathrm{\Omega }})$ are bounded. According to the Arzalà-Ascoli theorem, we only need to prove that $K_p(I-Q)N(\overline{\mathrm{\Omega }})\subset X$ is equicontinuous. In fact, for any $x(t)=(x_1(t),x_2(t))\in \overline{\mathrm{\Omega }},t\in [0,1]$, due to the continuity of ${\varphi }_q$ and f, there exist ${\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2>0$, such that $|{\varphi }_q(x_2(t))|\le {\mathrm{\Lambda }}_1,|f(t,x_1(t),{\varphi }_q(x_2(t)))|\le {\mathrm{\Lambda }}_2.$Therefore, we have $\begin{array}{rl}& |{\varphi }_q(x_2)-T_{\alpha }({\varphi }_q(x_2))|\\ & =|{\varphi }_q(x_2)-\frac{\alpha }{(\eta -{\eta }^{\alpha })-(\xi -{\xi }^{\alpha })}\\ & \times \left[{\int }_0^1{G}^{\alpha }(\eta ,s){\varphi }_q(x_2(s))\mathrm{d}s-{\int }_0^1{G}^{\alpha }(\xi ,s){\varphi }_q(x_2(s))\mathrm{d}s\right]|\\ & \le \frac{2(\eta +{\xi }^{\alpha })}{(\eta -{\eta }^{\alpha })-(\xi -{\xi }^{\alpha })}{\mathrm{\Lambda }}_1:={\tilde{\mathrm{\Lambda }}}_1.\end{array}$Similarly, $\begin{array}{rl}& |f(t,x_1,{\varphi }_q(x_2))-T_{\beta }f(t,x_1,{\varphi }_q(x_2))|\\ & =|f(t,x_1,{\varphi }_q(x_2))-\frac{\beta }{(\eta -{\eta }^{\beta })-(\xi -{\xi }^{\beta })}[{\int }_0^1{G}^{\beta }(\eta ,s)f(s,x_1(s),{\varphi }_q(x_2(s)))\mathrm{d}s\\ & -{\int }_0^1{G}^{\beta }(\xi ,s)f(s,x_1(s),{\varphi }_q(x_2(s)))\mathrm{d}s]|\le \frac{2(\eta +{\xi }^{\beta })}{(\eta -{\eta }^{\beta })-(\xi -{\xi }^{\beta })}{\mathrm{\Lambda }}_2:={\tilde{\mathrm{\Lambda }}}_2.\end{array}$For the convenience of the proof below, denote $\begin{array}{rl}{\mathfrak{h}}_1(t)& ={\varphi }_q(x_2(t))-T_{\alpha }({\varphi }_q(x_2(t))),t\in [0,1],\\ {\mathfrak{h}}_2(t)& =f(t,x_1(t),{\varphi }_q(x_2(t)))-T_{\beta }f(t,x_1(t),{\varphi }_q(x_2(t))),t\in [0,1].\end{array}$For any $(x_1(t),x_2(t))\in \overline{\mathrm{\Omega }}$, $0\le t_1<t_2\le 1$, one has $\begin{array}{rl}& |I_{0+}^{\alpha }{\mathfrak{h}}_1(t){|}_{t=t_2}-I_{0+}^{\alpha }{\mathfrak{h}}_1(t){|}_{t=1}{t}_2-I_{0+}^{\alpha }{\mathfrak{h}}_1(t){|}_{t=t_1}+I_{0+}^{\alpha }{\mathfrak{h}}_1(t){|}_{t=1}{t}_1|\\ & \le |I_{0+}^{\alpha }{\mathfrak{h}}_1(t){|}_{t=t_2}-I_{0+}^{\alpha }{\mathfrak{h}}_1(t){|}_{t=t_1}|+|I_{0+}^{\alpha }{\mathfrak{h}}_1(t){|}_{t=1}|(t_2-t_1)\\ & \le \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{t_1}[{(t_2-s)}^{\alpha -1}-{(t_1-s)}^{\alpha -1}]|{\mathfrak{h}}_1(s)|\mathrm{d}s+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_{t_1}^{t_2}{(t_2-s)}^{\alpha -1}|{\mathfrak{h}}_1(s)|\mathrm{d}s\\ & +\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^1{(1-s)}^{\alpha -1}|{\mathfrak{h}}_1(s)|\mathrm{d}s(t_2-t_1)\\ & \le \frac{{\tilde{\mathrm{\Lambda }}}_1}{\mathrm{\Gamma }(\alpha +1)}(t_2^{\alpha }-t_1^{\alpha }+t_2-t_1)\to 0,t_1\to t_2.\end{array}$Similarly, it can be deduced that $\begin{array}{rl}& |I_{0+}^{\beta }{\mathfrak{h}}_2(t){|}_{t=t_2}-I_{0+}^{\beta }{\mathfrak{h}}_2(t){|}_{t=1}{t}_2-I_{0+}^{\beta }{\mathfrak{h}}_2(t){|}_{t=t_1}+I_{0+}^{\beta }{\mathfrak{h}}_2{|}_{t=1}{t}_1|\\ & \le \frac{{\tilde{\mathrm{\Lambda }}}_2}{\mathrm{\Gamma }(\beta +1)}(t_2^{\beta }-t_1^{\beta }+t_2-t_1)\to 0,t_1\to t_2.\end{array}$Therefore, $K_p(I-Q)N(\overline{\mathrm{\Omega }})\subset X$ is equicontinuous. Hence, $K_p(I-Q)N:\overline{\mathrm{\Omega }}\to X$ is compact. The proof is complete.

Lemma 3.4

Assuming that conditions $(A_1)$ and $(A_2)$ hold, let ${\mathrm{\Omega }}_1=\{x\in \mathrm{dom}L\mathrm{\setminus }\mathrm{Ker}L:\mathrm{Lx}=\mathrm{\lambda Nx},\lambda \in (0,1)\},$then ${\mathrm{\Omega }}_1$ is bounded.

Proof.

For $x=(x_1,x_2)\in {\mathrm{\Omega }}_1$, then $x\in \mathrm{dom}L$ and $\mathrm{Nx}\in \mathrm{Im}L=\mathrm{Ker}Q$, then by $(A_2)$, there exist $t_1,t_2\in [0,1]$, such that $|x_1(t_1)|\le B_1,|x_2(t_2)|\le B_2.$Notice that, $\begin{array}{rl}{I}_{0+}^{\alpha }{}^C{D}_{0+}^{\alpha }{x}_1(t)& =x_1(t)+c_1+I{{}_{0+}^{\alpha }}^C{D}_{0+}^{\alpha }{x}_1(t){|}_{t=1}t,c_1\in \mathbb{R},t\in [0,1],\\ I_{0+}^{\beta }{}^C{D}_{0+}^{\beta }{x}_2(t)& =x_2(t)+c_3+I{{}_{0+}^{\alpha }}^C{D}_{0+}^{\alpha }{x}_2(t){|}_{t=1}t,c_3\in \mathbb{R},t\in [0,1].\end{array}$Hence, $\begin{array}{rl}{c}_1& =I_{0+}^{\alpha }{}^C{D}_{0+}^{\alpha }{x}_1(t){|}_{t=t_1}-x_1(t_1)-I{{}_{0+}^{\alpha }}^C{D}_{0+}^{\alpha }{x}_1(t){|}_{t=1}{t}_1,\\ c_3& =I_{0+}^{\beta }{}^C{D}_{0+}^{\beta }{x}_2(t){|}_{t=t_2}-x_2(t_2)-I{{}_{0+}^{\alpha }}^C{D}_{0+}^{\alpha }{x}_2(t){|}_{t=1}{t}_2.\end{array}$Furthermore, (8) $\begin{array}{rl}|x_1(t)|& \le |I_{0+}^{\alpha }{}^C{D}_{0+}^{\alpha }{x}_1(t)|+|c_1|+|I_{0+}^{\alpha }{}^C{D}_{0+}^{\alpha }{x}_1(t){|}_{t=1}|t\\ & \le \frac{4}{\mathrm{\Gamma }(\alpha +1)}||{}^C{D}_{0+}^{\alpha }{x}_1|{|}_{\mathrm{\infty }}+B_1,\end{array}$(8) (9) $\begin{array}{rl}|x_2(t)|& \le |I_{0+}^{\beta }{}^C{D}_{0+}^{\beta }{x}_2(t)|+|c_3|+|I_{0+}^{\alpha }{}^C{D}_{0+}^{\alpha }{x}_1(t){|}_{t=1}|t\\ & \le \frac{4}{\mathrm{\Gamma }(\beta +1)}||{}^C{D}_{0+}^{\beta }{x}_2|{|}_{\mathrm{\infty }}+B_2.\end{array}$(9) From the equation $\mathrm{Lx}=\mathrm{\lambda Nx}$, we know that (10) $\begin{array}{rl}{}^C{D}_{0+}^{\alpha }{x}_1(t)& =\lambda {\varphi }_q(x_2(t)),\end{array}$(10) (11) $\begin{array}{rl}{}^C{D}_{0+}^{\beta }{x}_2(t)& =\mathrm{\lambda f}(t,x_1(t),{\varphi }_q(x_2(t))).\end{array}$(11) From Equation (10(10) $\begin{array}{rl}{}^C{D}_{0+}^{\alpha }{x}_1(t)& =\lambda {\varphi }_q(x_2(t)),\end{array}$(10) ), we can deduce that $||{}^C{D}_{0+}^{\alpha }{x}_1|{|}_{\mathrm{\infty }}\le ||x_2|{|}_{\mathrm{\infty }}^{q-1}.$Combining Equation (8(8) $\begin{array}{rl}|x_1(t)|& \le |I_{0+}^{\alpha }{}^C{D}_{0+}^{\alpha }{x}_1(t)|+|c_1|+|I_{0+}^{\alpha }{}^C{D}_{0+}^{\alpha }{x}_1(t){|}_{t=1}|t\\ & \le \frac{4}{\mathrm{\Gamma }(\alpha +1)}||{}^C{D}_{0+}^{\alpha }{x}_1|{|}_{\mathrm{\infty }}+B_1,\end{array}$(8) ), we obtain that (12) $||x_1|{|}_{\mathrm{\infty }}\le \frac{4}{\mathrm{\Gamma }(\alpha +1)}||x_2|{|}_{\mathrm{\infty }}^{q-1}+B_1.$(12) From Equation (11(11) $\begin{array}{rl}{}^C{D}_{0+}^{\beta }{x}_2(t)& =\mathrm{\lambda f}(t,x_1(t),{\varphi }_q(x_2(t))).\end{array}$(11) ) and condition $(A_1)$, we can infer that (13) $\begin{array}{rl}||{}^C{D}_{0+}^{\beta }{x}_2|{|}_{\mathrm{\infty }}& \le ||a|{|}_{\mathrm{\infty }}+||b|{|}_{\mathrm{\infty }}||x_1|{|}_{\mathrm{\infty }}^{p-1}+||c|{|}_{\mathrm{\infty }}||x_2|{|}_{\mathrm{\infty }}\\ & \le ||a|{|}_{\mathrm{\infty }}+||b|{|}_{\mathrm{\infty }}{\left(\frac{4}{\mathrm{\Gamma }(\alpha +1)}||x_2|{|}_{\mathrm{\infty }}^{q-1}+B_1\right)}^{p-1}+||c|{|}_{\mathrm{\infty }}||x_2|{|}_{\mathrm{\infty }}\\ & \le ||a|{|}_{\mathrm{\infty }}+\left[2^{p-1}{\left(\frac{4}{\mathrm{\Gamma }(\alpha +1)}\right)}^{p-1}||b|{|}_{\mathrm{\infty }}+||c|{|}_{\mathrm{\infty }}\right]\\ & \times ||x_2|{|}_{\mathrm{\infty }}+2^{p-1}{B}_1^{p-1}||b|{|}_{\mathrm{\infty }}\\ & \le ||a|{|}_{\mathrm{\infty }}+\left[2^{p-1}{\left(\frac{4}{\mathrm{\Gamma }(\alpha +1)}\right)}^{p-1}||b|{|}_{\mathrm{\infty }}+||c|{|}_{\mathrm{\infty }}\right]\\ & \times \left(\frac{4}{\mathrm{\Gamma }(\beta +1)}||{}^C{D}_{0+}^{\beta }{x}_2|{|}_{\mathrm{\infty }}+B_2\right)\\ & +2^{p-1}{B}_1^{p-1}||b|{|}_{\mathrm{\infty }}.\end{array}$(13) Due to $\rho <1$, from Equation (13(13) $\begin{array}{rl}||{}^C{D}_{0+}^{\beta }{x}_2|{|}_{\mathrm{\infty }}& \le ||a|{|}_{\mathrm{\infty }}+||b|{|}_{\mathrm{\infty }}||x_1|{|}_{\mathrm{\infty }}^{p-1}+||c|{|}_{\mathrm{\infty }}||x_2|{|}_{\mathrm{\infty }}\\ & \le ||a|{|}_{\mathrm{\infty }}+||b|{|}_{\mathrm{\infty }}{\left(\frac{4}{\mathrm{\Gamma }(\alpha +1)}||x_2|{|}_{\mathrm{\infty }}^{q-1}+B_1\right)}^{p-1}+||c|{|}_{\mathrm{\infty }}||x_2|{|}_{\mathrm{\infty }}\\ & \le ||a|{|}_{\mathrm{\infty }}+\left[2^{p-1}{\left(\frac{4}{\mathrm{\Gamma }(\alpha +1)}\right)}^{p-1}||b|{|}_{\mathrm{\infty }}+||c|{|}_{\mathrm{\infty }}\right]\\ & \times ||x_2|{|}_{\mathrm{\infty }}+2^{p-1}{B}_1^{p-1}||b|{|}_{\mathrm{\infty }}\\ & \le ||a|{|}_{\mathrm{\infty }}+\left[2^{p-1}{\left(\frac{4}{\mathrm{\Gamma }(\alpha +1)}\right)}^{p-1}||b|{|}_{\mathrm{\infty }}+||c|{|}_{\mathrm{\infty }}\right]\\ & \times \left(\frac{4}{\mathrm{\Gamma }(\beta +1)}||{}^C{D}_{0+}^{\beta }{x}_2|{|}_{\mathrm{\infty }}+B_2\right)\\ & +2^{p-1}{B}_1^{p-1}||b|{|}_{\mathrm{\infty }}.\end{array}$(13) ), we know that there exists a constant $M_0>0$ such that $||{}^C{D}_{0+}^{\beta }{x}_2|{|}_{\mathrm{\infty }}\le M_0.$By combining equations (9(9) $\begin{array}{rl}|x_2(t)|& \le |I_{0+}^{\beta }{}^C{D}_{0+}^{\beta }{x}_2(t)|+|c_3|+|I_{0+}^{\alpha }{}^C{D}_{0+}^{\alpha }{x}_1(t){|}_{t=1}|t\\ & \le \frac{4}{\mathrm{\Gamma }(\beta +1)}||{}^C{D}_{0+}^{\beta }{x}_2|{|}_{\mathrm{\infty }}+B_2.\end{array}$(9) ) and (12(12) $||x_1|{|}_{\mathrm{\infty }}\le \frac{4}{\mathrm{\Gamma }(\alpha +1)}||x_2|{|}_{\mathrm{\infty }}^{q-1}+B_1.$(12) ), we get that $\begin{array}{r}||x_2|{|}_{\mathrm{\infty }}\le \frac{4}{\mathrm{\Gamma }(\beta +1)}{M}_0+B_2:=M_1,||x_1|{|}_{\mathrm{\infty }}\le \frac{4}{\mathrm{\Gamma }(\alpha +1)}{M}_1^{q-1}+B_1:=M_2.\end{array}$Hence, $||x|{|}_X\le max\{M_1,M_2\}:=M.$That is, ${\mathrm{\Omega }}_1$ is bounded. The proof is complete.

Lemma 3.5

Assuming that condition $(A_2)$ holds, let ${\mathrm{\Omega }}_2=\{x\in \mathrm{Ker}L:\mathrm{Nx}\in \mathrm{Im}L\},$then ${\mathrm{\Omega }}_2$ is bounded.

Proof.

For $x=(x_1,x_2)\in \mathrm{Ker}L$, then $(x_1,x_2)=(c_1,c_2)\in {\mathbb{R}}^2$, and $\mathrm{Nx}\in \mathrm{Im}L=\mathrm{Ker}Q$, that is, QNx = 0. From $(A_2)$, there exist $t_1^{\prime },t_2^{″}\in [0,1]$ such that $|x_1(t_1^{\prime })|=|c_1|\le B_1,|x_2(t_2^{″})|=|c_2|\le B_2.$Hence, $||x|{|}_X=max\{||x_1|{|}_{\mathrm{\infty }},||x_2|{|}_{\mathrm{\infty }}\}\le max\{B_1,B_2\}.$That is, ${\mathrm{\Omega }}_2$ is bounded. The proof is completed.

Lemma 3.6

Assume that $(A_3)$ holds, let ${\mathrm{\Omega }}_3=\{x\in \mathrm{Ker}L:\mathrm{\vartheta \lambda Jx}+(1-\lambda )\mathrm{QNx}=0,\lambda \in [0,1]\},$where $\vartheta =\{\begin{array}{ll}1,& \text{if}(6)\text{holds},\\ -1,& \text{if}(7)\text{holds},\end{array}$$J:\mathrm{Ker}L\to \mathrm{Im}Q$ is a homeomorphism defined as $J(c_1,c_2)=(c_2,c_1),\mathrm{\forall }{c}_1,c_2\in \mathbb{R}.$Then ${\mathrm{\Omega }}_3$ is bounded.

Proof.

Without loss of generality, suppose that (7(7) $\ell \left[{\int }_0^1{G}^{\beta }(\eta ,s)f(s,\ell ,0)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)f(s,\ell ,0)\mathrm{d}s\right]<0.$(7) ) holds, then for $x=(x_1,x_2)=(c_1,c_2)\in {\mathrm{\Omega }}_3$, $\lambda \in [0,1]$ we have (14) $\begin{array}{rl}\lambda c_2& =\frac{\alpha (1-\lambda )}{(\eta -{\eta }^{\alpha })-(\xi -{\xi }^{\alpha })}\left[{\int }_0^1{G}^{\alpha }(\eta ,s){\varphi }_q(c_2)\mathrm{d}s-{\int }_0^1{G}^{\alpha }(\xi ,s){\varphi }_q(c_2)\mathrm{d}s\right]\\ & =(1-\lambda ){\varphi }_q(c_2),\end{array}$(14) (15) $\begin{array}{rl}\lambda c_1& =\frac{\beta (1-\lambda )}{(\eta -{\eta }^{\beta })-(\xi -{\xi }^{\beta })}\\ & \times \left[{\int }_0^1{G}^{\beta }(\eta ,s)f(s,c_1,{\varphi }_2(c_2))\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)f(s,c_1,{\varphi }_q(c_2))\mathrm{d}s\right].\end{array}$(15) From Equation (14(14) $\begin{array}{rl}\lambda c_2& =\frac{\alpha (1-\lambda )}{(\eta -{\eta }^{\alpha })-(\xi -{\xi }^{\alpha })}\left[{\int }_0^1{G}^{\alpha }(\eta ,s){\varphi }_q(c_2)\mathrm{d}s-{\int }_0^1{G}^{\alpha }(\xi ,s){\varphi }_q(c_2)\mathrm{d}s\right]\\ & =(1-\lambda ){\varphi }_q(c_2),\end{array}$(14) ), it can be inferred that $c_2=0$, substituting into Equation (15(15) $\begin{array}{rl}\lambda c_1& =\frac{\beta (1-\lambda )}{(\eta -{\eta }^{\beta })-(\xi -{\xi }^{\beta })}\\ & \times \left[{\int }_0^1{G}^{\beta }(\eta ,s)f(s,c_1,{\varphi }_2(c_2))\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)f(s,c_1,{\varphi }_q(c_2))\mathrm{d}s\right].\end{array}$(15) ) yields $\lambda c_1=\frac{\beta (1-\lambda )}{(\eta -{\eta }^{\beta })-(\xi -{\xi }^{\beta })}\left[{\int }_0^1{G}^{\beta }(\eta ,s)f(s,c_1,0)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)f(s,c_1,0)\mathrm{d}s\right].$Then, $\lambda c_1^2=\frac{\beta (1-\lambda )}{(\eta -{\eta }^{\beta })-(\xi -{\xi }^{\beta })}{c}_1\left[{\int }_0^1{G}^{\beta }(\eta ,s)f(s,c_1,0)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)f(s,c_1,0)\mathrm{d}s\right]<0,$This is a contradiction! Therefore, $|c_1|\le G$. Hence, $||x|{|}_X=max\{|c_1|,|c_2|\}\le G,$that is, ${\mathrm{\Omega }}_3$ is bounded. The proof is completed.

The proof of Theorem 3.1 is given below.

Proof of Theorem 3.1.

Take Ω to be a bounded set in X satisfying ${\cup }_{i=1}^3{\overline{\mathrm{\Omega }}}_i\subset \mathrm{\Omega }$. By Lemmas 3.2 and 3.3, we know that L is a Fredholm operator of index zero and N is compact with respect to L on $\overline{\mathrm{\Omega }}$. By Lemmas 3.4 and 3.5, we obtain,

1. $\mathrm{Lx}\ne \mathrm{\lambda Nx}$, $x\in (\mathrm{dom}L\mathrm{\setminus }\mathrm{Ker}L)\cap \mathrm{\partial \Omega }$, $\lambda \in (0,1)$;

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

Therefore, it only needs to be shown that condition (iii) in Theorem 2.1 holds. For this, define the homotopy mapping $H(x,\lambda )=\mathrm{\vartheta \lambda Jx}+(1-\lambda )\mathrm{QNx}.$By Lemma 3.6, for $x\in \mathrm{Ker}L\cap \mathrm{\partial \Omega }$, we have $H(x,\lambda )\ne 0$. Using the homotopy invariance of the topological degree, we obtain $\begin{array}{rl}& \mathrm{deg}\left\{\mathrm{QN}{|}_{\mathrm{Ker}L},\mathrm{\Omega }\cap \mathrm{Ker}L,0\right\}\\ & =\mathrm{deg}\left\{H(\cdot ,0),\mathrm{\Omega }\cap \mathrm{Ker}L,0\right\}\\ & =\mathrm{deg}\left\{H(\cdot ,1),\mathrm{\Omega }\cap \mathrm{Ker}L,0\right\}\\ & =\mathrm{deg}\left\{\mathrm{\vartheta I},\mathrm{\Omega }\cap \mathrm{Ker}L,0\right\}\ne 0.\end{array}$In conclusion, by using Theorem 2.1, boundary value problem (3(3) $\{\begin{array}{l}{}^C{D}_{0+}^{\alpha }{x}_1(t)={\varphi }_q(x_2(t)),t\in (0,1),\\ {}^C{D}_{0+}^{\beta }{x}_2(t)=f(t,x_1(t),{\varphi }_q(x_2(t)),t\in (0,1),\\ x_1(0)=x_1(1),x_1(\xi )=x_1(\eta ),\\ x_2(0)=x_2(1),x_2(\xi )=x_2(\eta ),\end{array}$(3) ) has at least one solution in X, that is, the dual periodic boundary value problem (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ) has at least one solution. The proof is completed.

4. Example

Example 4.1

Consider the following fractional p-Laplacian equation with dual periodic boundary conditions (16) $\{\begin{array}{l}{}^C{D}_{0+}^{7/4}{\varphi }_4({}^C{D}_{0+}^{7/4}x(t))=\frac{1}{70}{x}^3(t)+\mathrm{tsin}(|{}^C{D}_{0+}^{7/4}x(t)|{)}^3-\frac{5}{7},t\in (0,1),\\ x(0)=x(1),{}^C{D}_{0+}^{7/4}x(0)={}^C{D}_{0+}^{7/4}x(1),\\ x(1/8)=x(1/4),{}^C{D}_{0+}^{7/4}x(1/8)={}^C{D}_{0+}^{7/4}x(1/4),\end{array}$(16) For the corresponding BVP (1(1) ${}^C{D}_{0+}^{\beta }{\varphi }_p({}^C{D}_{0+}^{\alpha }x(t))=f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t)),t\in (0,1),$(1) ) and (2(2) $\begin{array}{rl}x(0)& =x(1),{}^C{D}_{0+}^{\alpha }x(0)={}^C{D}_{0+}^{\alpha }x(1),\\ x(\xi )& =x(\eta ),{}^C{D}_{0+}^{\alpha }x(\xi )={}^C{D}_{0+}^{\alpha }x(\eta ),\end{array}$(2) ), here $\alpha =\beta =7/4$, p = 4, $\eta =1/4,\xi =1/8$, $f(t,x(t),{}^C{D}_{0+}^{\alpha }x(t))=\frac{1}{70}{x}^3(t)+\mathrm{tsin}(|{}^C{D}_{0+}^{7/4}x(t)|{)}^3-\frac{5}{7},t\in [0,1],$Choose $a(t)=2$, $b(t)=1/70$, $c(t)=0$, then we have $\begin{array}{rl}\rho & =\frac{4}{\mathrm{\Gamma }(\beta +1)}[2^{p-1}{\left(\frac{4}{\mathrm{\Gamma }(\alpha +1)}\right)}^{p-1}||b|{|}_{\mathrm{\infty }}+||c|{|}_{\mathrm{\infty }}]\\ & =\frac{4^4}{70{[\mathrm{\Gamma }(11/4)]}^3}\approx 0.8929<1,\end{array}$that is, condition $(A_1)$ holds. Take $B_1=20$, $B_2={100}^{3/4}$, let $g(s)=\frac{1}{8}(1-s{)}^{3/4}-{(\frac{1}{4}-s)}^{3/4},s\in [0,\frac{1}{4}],$Obviously, $g(s)$ is monotonically increasing, let $g(s)=0$, we get $s\approx 0.2$. If $x_2>B_2$, we have (17) $\begin{array}{rl}& {\int }_0^1{G}^{\alpha }(\eta ,s){\varphi }_q(x_2(s))\mathrm{d}s-{\int }_0^1{G}^{\alpha }(\xi ,s){\varphi }_q(x_2(s))\mathrm{d}s\\ & ={\int }_0^{1/4}g(s){\varphi }_{4/3}(x_2(s))\mathrm{d}s+{\int }_{1/4}^1\frac{1}{8}{(1-s)}^{3/4}{\varphi }_{4/3}(x_2(s))\mathrm{d}s\\ & +{\int }_0^{1/8}{(\frac{1}{8}-s)}^{3/4}{\varphi }_{4/3}(x_2(s))\mathrm{d}s\\ & \ge {\int }_0^{1/5}g(0){\varphi }_{4/3}(x_2(s))\mathrm{d}s+{\int }_{1/4}^1\frac{1}{8}{(1-s)}^{3/4}{\varphi }_{4/3}(x_2(s))\mathrm{d}s\\ & +{\int }_0^{1/8}{(\frac{1}{8}-s)}^{3/4}{\varphi }_{4/3}(x_2(s))\mathrm{d}s\\ & \ge [\frac{1}{8}-{\left(\frac{1}{4}\right)}^{3/4}]\times 20+\frac{1}{7}\times {\left(\frac{3}{4}\right)}^{7/4}\times 50+\frac{4}{7}\times {\left(\frac{1}{8}\right)}^{7/4}\times 100\\ & \approx 1.2477>0.\end{array}$(17) Similarly, if $x_2<-B_2$, it can be inferred that $T_{\alpha }{\varphi }_q(x_2(t))<0$. Note that $f(t,x_1(t),{\varphi }_q(x_2(t)))=\frac{1}{70}(x_1^3(t)-50)+\mathrm{tsin}(|x_2(t)|),t\in [0,1].$If $|x_1(t)|>B_1$, similar to (17(17) $\begin{array}{rl}& {\int }_0^1{G}^{\alpha }(\eta ,s){\varphi }_q(x_2(s))\mathrm{d}s-{\int }_0^1{G}^{\alpha }(\xi ,s){\varphi }_q(x_2(s))\mathrm{d}s\\ & ={\int }_0^{1/4}g(s){\varphi }_{4/3}(x_2(s))\mathrm{d}s+{\int }_{1/4}^1\frac{1}{8}{(1-s)}^{3/4}{\varphi }_{4/3}(x_2(s))\mathrm{d}s\\ & +{\int }_0^{1/8}{(\frac{1}{8}-s)}^{3/4}{\varphi }_{4/3}(x_2(s))\mathrm{d}s\\ & \ge {\int }_0^{1/5}g(0){\varphi }_{4/3}(x_2(s))\mathrm{d}s+{\int }_{1/4}^1\frac{1}{8}{(1-s)}^{3/4}{\varphi }_{4/3}(x_2(s))\mathrm{d}s\\ & +{\int }_0^{1/8}{(\frac{1}{8}-s)}^{3/4}{\varphi }_{4/3}(x_2(s))\mathrm{d}s\\ & \ge [\frac{1}{8}-{\left(\frac{1}{4}\right)}^{3/4}]\times 20+\frac{1}{7}\times {\left(\frac{3}{4}\right)}^{7/4}\times 50+\frac{4}{7}\times {\left(\frac{1}{8}\right)}^{7/4}\times 100\\ & \approx 1.2477>0.\end{array}$(17) ), it can be inferred that $T_{\beta }f(t,x_1(t),{\varphi }_q(x_2(t)))\ne 0$. Thus, condition $(A_2)$ holds. Take G = 10, if $|c|>G$, we can obtain $\mathrm{cf}(t,c,0)=\frac{1}{70}c(c^3-50)>100,\mathrm{\forall }t\in [0,1].$It is easy to verify that (6(6) $\ell \left[{\int }_0^1{G}^{\beta }(\eta ,s)f(s,\ell ,0)\mathrm{d}s-{\int }_0^1{G}^{\beta }(\xi ,s)f(s,\ell ,0)\mathrm{d}s\right]>0,$(6) ) holds, that is, condition $(A_3)$ holds. In summary, according to Theorem 3.1, the BVP (16(16) $\{\begin{array}{l}{}^C{D}_{0+}^{7/4}{\varphi }_4({}^C{D}_{0+}^{7/4}x(t))=\frac{1}{70}{x}^3(t)+\mathrm{tsin}(|{}^C{D}_{0+}^{7/4}x(t)|{)}^3-\frac{5}{7},t\in (0,1),\\ x(0)=x(1),{}^C{D}_{0+}^{7/4}x(0)={}^C{D}_{0+}^{7/4}x(1),\\ x(1/8)=x(1/4),{}^C{D}_{0+}^{7/4}x(1/8)={}^C{D}_{0+}^{7/4}x(1/4),\end{array}$(16) ) has at least one solution on [0,1].

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No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by the Anhui Provincial Natural Science Foundation (2208085QA05) and National Natural Science Foundation of China (11601007).