Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

Abstract

This paper discusses a kind of coupled nonlocal Laplacian evolution equation with Caputo time-fractional derivatives and proportional delays. Green function and mild solution are firstly established by employing the approach of eigenvalues’ expansions and Fourier analysis technique. By the properties of eigenvalues and Mittag–Leffler functions, several vital estimations of Green functions are presented. In view of these estimations and some appropriate assumptions, the existence and uniqueness of the mild solution are studied by utilizing the Leray–Schauder fixed-point theorem and the Banach fixed-point theorem. Finally, an example is provided to illustrate the effectiveness of our main results.

1. Introduction

It is well known that fractional evolution equations are models abstracted from plentiful practical applications. With this in mind, a class of fractional evolution equations in Banach or Hilbert spaces attracted more and more researchers’ attention in recent years. Byszewski [1] initially considered the existence and uniqueness of mild, strong and classical solutions for a semilinear evolution nonlocal Cauchy problem depicted by

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \dot{\mathbf{u}}\left(t\right)+A\mathbf{u}\left(t\right)=f(t,\mathbf{u}\left(t\right)),}\hfill & t\in (t_0,t_0+a],\hfill \\ \mathbf{u}\left(t_0\right)+g(t_1,\dots ,t_p,\mathbf{u}(·))={\mathbf{u}}_0,\hfill & 0\le t_0<t_1<\dots <t_p\le t_0+a,\hfill \end{array}\right.\end{array}$$

where $a>0$, $-A$ denotes the infinitesimal generator of a $C_0$ semigroup in a Banach space $\mathcal{X}$; $f:[0,a]\times \mathcal{X}\to \mathcal{X}$ and $g:{[0,a]}^p\times \mathcal{X}\to \mathcal{X}$ are given functions. Byszewski [2], Byszewski and Akca [3] then went on to study the existence and uniqueness of mild solutions for different kinds of evolution equations in Banach spaces. Moreover, various existence results for evolution equations and evolution inclusions applying the fixed-point theory and semigroup theory have been achieved [4,5,6,7,8,9,10,11]. For instance, Li et al. [9] learned the following fractional evolution equation in ordered Banach spaces

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^c{D}_0^q\mathbf{u}\left(t\right)+A\mathbf{u}\left(t\right)=f(t,\mathbf{u}\left(t\right)),t>0,}\hfill \\ \mathbf{u}\left(0\right)={\mathbf{u}}_0,\hfill \end{array}\right.\end{array}$$

(1)

where ${}^c{D}_0^q$ represents the Caputo derivative with order $q\in (0,1)$, A is a closed linear operator, and f is a given continuous function. The authors acquired the existence of positive S-asymptotically $\omega$-periodic mild solutions for Equation (1), according to monotone iterative method and fixed-point theorem.

It should be remarked that Equation (1) is turned into time-fractional parabolic equation if A is equal to the Laplacian operator $\Delta$. The study of this kind of nonlocal Laplacian operators has received a great deal of interest, from both a purely mathematical research point of view and a practical application point of view. This type of operator appears in a rather natural way in numerous different settings, e.g., thin barrier problems, optimization, finance, phase transitions, materials science, water waves and population dynamics, please see [12,13,14,15,16]. Additionally, in purely mathematical studies, scholars have obtained numerous interesting results. In fact, Fujita [17] considered a class of Cauchy problems for a semilinear heat equation of the form ${\partial }_t\mathbf{u}=\Delta \mathbf{u}+{\mathbf{u}}^{1+\alpha }$ as early as the 1960s. The existence and other properties of solutions for different types of this problem, moreover, were investigated by various academics. For more details, please refer to [18,19,20,21].

More recently, Ghanmi and Horrigue in [22] replaced the first-order time derivative with the time-fractional derivative and considered the explicit solutions for fractional evolution equations in the form of

$$D_0^{\alpha }\mathbf{u}(x,t)=\Delta \mathbf{u}(x,t),\forall (x,t)\in \mathbb{R}\times {\mathbb{R}}^{+},$$

where $D_0^{\alpha }$ is the Riemann–Liouville or Caputo time-fractional derivative with order $\alpha \in (0,1)$. Zhang and Sun [23] focused on the blow-up and global existence of solutions to Cauchy problems for time-fractional diffusion equation. In [24], the authors investigated the time-fractional Cauchy problem for heat equation with nonlocal nonlinearity and proved the existence and uniqueness of mild solutions by applying the Banach fixed-point theorem. Lately, time–space fractional evolution equations have drawn more and more attention. Kirane et al. [25] learned the following time–space fractional evolution equations

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle {}^c{D}_0^{\alpha }\mathbf{u}+{(-\Delta )}^{\beta /2}\mathbf{u}=h(x,t){\left|\mathbf{u}\right|}^{1+p},}\hfill & \forall (x,t)\in {\mathbb{R}}^N\times {\mathbb{R}}^{+},\hfill \\ \mathbf{u}(x,0)={\mathbf{u}}_0\left(x\right)\ge 0,\hfill & \forall x\in {\mathbb{R}}^N,\hfill \end{array}\right.\end{array}$$

where ${}^c{D}_0^{\alpha }$ is the time-fractional Caputo derivative with order $\alpha \in (0,1)$, ${(-\Delta )}^{\beta /2}$ denotes the $\beta /2$-fractional Laplacian with $\beta \in [1,2]$, $h(x,t)\ge C_h{\left|x\right|}^{\sigma }{t}^{\rho }$ with $C_h>0$ and $p>0$. In [26], based on the finite difference method, the authors researched the Caputo-type parabolic equation with fractional Laplacian depicted by

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle {}^c{D}_0^{\alpha }\mathbf{u}=-{(-\Delta )}^s\mathbf{u}+f,}\hfill & \forall (x,t)\in (0,L)\times {\mathbb{R}}^{+},\hfill \\ \mathbf{u}(x,0)={\mathbf{u}}_0\left(x\right),\hfill & \forall x\in (0,L),\hfill \\ \mathbf{u}(0,t)=\mathbf{u}(L,t)=0,\hfill & \forall t\in {\mathbb{R}}^{+},\hfill \end{array}\right.\end{array}$$

where ${}^c{D}_0^{\alpha }$ represents Caputo derivative with order $\alpha \in (0,1)$, ${(-\Delta )}^s$ is the nonlocal Laplacian operator with order $s\in (0,1)$. Additional content regarding this theme is available in [27,28,29].

However, as far as we know, the existence and uniqueness of solutions for boundary value problems of time-fractional evolution equations with nonlocal Laplacian operators are scarcely discussed. Inspired by above discussions, this paper considers the following coupled nonlocal Laplacian evolution equations with proportional delay

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle {}^c{D}_0^{\alpha }\mathbf{u}+{(-\Delta )}^{s_1}\mathbf{u}=-\theta \mathbf{u}+f(x,t,{\mathbf{u}}_{\tau t},{\mathbf{v}}_{\tau t}),}\hfill & \forall (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ {\displaystyle {}^c{D}_0^{\beta }\mathbf{v}+{(-\Delta )}^{s_2}\mathbf{v}=-\sigma \mathbf{v}+g(x,t,{\mathbf{u}}_{\tau t},{\mathbf{v}}_{\tau t}),}\hfill & \forall (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ \mathbf{u}=\mathbf{u}(x,t)=0,\mathbf{v}=\mathbf{v}(x,t)=0,\hfill & \forall (x,t)\in \partial \mathrm{\Omega }\times [0,T],\hfill \end{array}\right.\end{array}$$

(2)

which is endowed with three-point boundary value conditions (BVCs) below

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle a_1\mathbf{u}(x,0)-b_1\mathbf{u}(x,\eta )=c_1\mathbf{u}(x,T),}\hfill \\ a_2\mathbf{v}(x,0)-b_2\mathbf{v}(x,\xi )=c_2\mathbf{v}(x,T),\hfill \end{array}\right.\end{array}$$

(3)

where α,β ∈ (0, 1), ξ, η ∈ (0, T), x ∈ Ω, Ω ⊂ ${\mathbb{R}}^n$ is an open bounded domain with smooth boundary ∂Ω; u_τt = u(x, τt), v_τt = v(x, τt), τ ∈ (0, 1], ${(-\Delta )}^{s_1}$ and ${(-\Delta )}^{s_2}$ are nonlocal Laplacian operators with orders $s_1,s_2\in (0,1)$, respectively; θ, σ, a_i, b_i, c_i are positive constants, i = 1, 2; f and g are continuous mappings from $\mathrm{\Omega }\times [0,T]\times \mathbb{R}\times \mathbb{R} \mathrm{to} \mathbb{R}$.

Novelties of this article: (1) The Green functions of three-point nonlocal Laplacian BVPs (2) and (3) are established by the approach of eigenvalues’ expansions with respect to nonlocal Laplacian operators ${(-\Delta )}^{s_1}$ and ${(-\Delta )}^{s_2}$. (2) Several crucial estimates of Green functions are learned by virtue of the properties of eigenvalues and Mittag–Leffler functions. (3) A definition of mild solution to boundary value problems (BVPs) (2) and (3) is introduced and the existence and uniqueness of mild solution are investigated by using some fixed-point theorems. (4) Theorems 2 and 3 of this article achieve some easily verifiable conditions compared with the previous works in [6,7,9,11,22,26].

Structure of this article:Section 2 gives some crucial definitions and lemmas of fractional calculus in time and space as well as function spaces and its corresponding norms, and after presents Green functions corresponding to BVPs (2) and (3) by the eigenvalues’ expansions of nonlocal Laplacian operators. In Section 3, a definition of mild solution for BVPs (2) and (3) is introduced firstly and the existence and uniqueness of the mild solution is investigated by employing Leray–Schauder fixed-point theorem and Banach fixed-point theorem. An example is given to illustrate our main results in Section 4. Section 5 draws some conclusions and perspectives.

Notations:$\mathbb{N}$ denotes the set of positive integers; $\mathbb{C}$ is the family of complex numbers; ${\mathbb{R}}^n$ represents the n-dimensional real vector space; ${\mathcal{C}}^n(I,{\mathbb{R}}^n)$ is the set of all functions from I to ${\mathbb{R}}^n$ which have continuous derivatives up to order n.

2. Fractional Calculus and Green Function

This section concentrates on some fundamental definitions and properties regarding fractional calculus and the Green function. Please refer to [14,15,30,31,32] for further details on fractional calculus and more references therein.

2.1. Time Fractional Calculus

We present the definitions and some vital lemmas on Caputo fractional derivative and Mittag–Leffler functions in this section.

Let $\alpha \ge 0$ and $n=\left[\alpha \right]+1$, $n\in \mathbb{N}$. If $f\in {\mathcal{C}}^n([0,T],\mathbb{R})$, then the Caputo time-fractional derivative ${}^c{D}_0^{\alpha }f\left(t\right)$ is defined by

$${}^c{D}_0^{\alpha }f\left(t\right)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}{\int }_0^t\frac{f^{\left(n\right)}\left(s\right)}{{(t-s)}^{\alpha -n+1}}\mathrm{d}s,$$

where $\mathrm{\Gamma }(·)$ is the Euler Gamma function.

The two-parameter Mittag–Leffler function is in the form of

$${\mathbf{E}}_{\alpha ,\beta }\left(z\right):=\sum _{k=0}^{\infty }\frac{z^k}{\mathrm{\Gamma }(\alpha k+\beta )},\alpha ,\beta ,z\in \mathbb{C}.$$

In particular,

$${\mathbf{E}}_{\alpha ,1}\left(z\right)={\mathbf{E}}_{\alpha }\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\mathrm{\Gamma }(\alpha k+1)},\alpha ,z\in \mathbb{C},$$

which is known as the one-parameter Mittag–Leffler function.

Let $\alpha ,\beta \in (0,1)$, and the functions ${\mathbf{E}}_{\alpha }$ and ${\mathbf{E}}_{\alpha ,\beta }$ have the following properties.

$\left(E1\right)$

${\mathbf{E}}_{\alpha }\left(0\right)=1$, ${\mathbf{E}}_{\alpha }\left(\lambda t^{\alpha }\right)\in (0,1)$ for $\lambda <0$ and ${\mathbf{E}}_{\alpha }\left(\lambda t^{\alpha }\right)\in (1,\infty )$ for $\lambda >0$, $\forall t>0$, $t\in \mathbb{R}$.

$\left(E2\right)$

${\mathbf{E}}_{\alpha ,\alpha }\left(\lambda t^{\alpha }\right)>0$ for $\lambda \in \mathbb{R}$, $\forall t\ge 0$, $t\in \mathbb{R}$.

$\left(E3\right)$

${\mathbf{E}}_{\alpha }\left(\lambda t_1^{\alpha }\right)\ge {\mathbf{E}}_{\alpha }\left(\lambda t_2^{\alpha }\right)$ and ${\mathbf{E}}_{\alpha ,\alpha }\left(\lambda t_1^{\alpha }\right)\ge {\mathbf{E}}_{\alpha ,\alpha }\left(\lambda t_2^{\alpha }\right)$ for $\lambda <0$, $t_1\le t_2$, $t_1,t_2\in \mathbb{R}$.

$\left(E4\right)$

${\displaystyle {\int }_0^z{s}^{\alpha -1}{\mathbf{E}}_{\alpha ,\alpha }\left(\lambda s^{\alpha }\right)\mathrm{d}s=z^{\alpha }{\mathbf{E}}_{\alpha ,\alpha +1}\left(\lambda z^{\alpha }\right)}$ for $\alpha ,\lambda ,z\in \mathbb{C}$.

$\left(E5\right)$

${\displaystyle {\mathbf{E}}_{\alpha ,\beta }\left(z\right)=\frac{1}{\mathrm{\Gamma }\left(\beta \right)}+z{\mathbf{E}}_{\alpha ,\beta +\alpha }\left(z\right)}$ for $\alpha ,\beta ,z\in \mathbb{C}$.

2.2. Fractional Laplacian

This section presents some fundamental definitions of space-fractional Laplacian.

Definition 1.

Let$s\in (0,1)$and$\mathcal{S}$be the Schwartz space of rapidly decaying${\mathcal{C}}^{\infty }$functions in${\mathbb{R}}^n$; the fractional Laplacian operator${(-\Delta )}^s:\mathcal{S}\to L^2\left({\mathbb{R}}^n\right)$is given by

$$\begin{array}{c}\hfill {(-\Delta )}^s\mathbf{u}\left(x\right)=C(n,s)\underset{\vartheta \to 0^{+}}{lim}{\int }_{{\mathbb{R}}^n\setminus B(x,\vartheta )}\frac{\mathbf{u}\left(x\right)-\mathbf{u}\left(y\right)}{{|x-y|}^{n+2s}}\mathrm{d}y,x\in {\mathbb{R}}^n,\end{array}$$

(4)

where$B(x,\vartheta )$is a ball with center$x\in {\mathbb{R}}^n$and radiusϑ and $C(n,s)$ is the normalization constant expressed by

$$\frac{1}{C(n,s)}={\int }_{{\mathbb{R}}^n}\frac{1-cos{\theta }_1}{{\left|\theta \right|}^{n+2s}}\mathrm{d}\theta ,\theta =({\theta }_1,{\theta }^{\prime }),{\theta }^{\prime }\in {\mathbb{R}}^{n-1}.$$

(5)

However, other authors [14] have provided another two equivalent formulas of (4).

Definition 2.

Let$s\in (0,1)$, for any$\mathbf{u}\in \mathcal{S}$,

$$\begin{array}{c}\hfill {(-\Delta )}^s\mathbf{u}\left(x\right)=-\frac{1}{2}C(n,s){\int }_{{\mathbb{R}}^n}\frac{\mathbf{u}(x+y)+\mathbf{u}(x-y)-2\mathbf{u}\left(x\right)}{{\left|y\right|}^{n+2s}}\mathrm{d}y,\forall x\in {\mathbb{R}}^n,\end{array}$$

whereC(n, s) is defined by (5).

Definition 3.

Let$\varsigma \in {\mathbb{R}}^n$,

$$\begin{array}{c}\hfill {(-\Delta )}^s\mathbf{u}={\mathcal{F}}^{-1}\left({\left|\varsigma \right|}^{2s}\left(\mathcal{F}\mathbf{u}\right)\right),\end{array}$$

where$\mathcal{F}$and${\mathcal{F}}^{-1}$denote Fourier transform and inverse Fourier transform, respectively.

2.3. Function Spaces and Norms

Now, we review some preliminaries on function spaces and corresponding norms.

Define $L^2\left(\mathrm{\Omega }\right)=\{\mathbf{u}:\mathrm{\Omega }\to \mathbb{R}\mathrm{as}\mathrm{the}\mathrm{Lebesgue}\mathrm{measurable}$, ${\int }_{\mathrm{\Omega }}{\left|\mathbf{u}\right|}^2\mathrm{d}x<\infty \}$ with norm ${\parallel \mathbf{u}\parallel }_2=({\int }_{\mathrm{\Omega }}{\left|\mathbf{u}\right|}^2{\mathrm{d}x)}^{\frac{1}{2}}$ and $\left(L^2\left(\mathrm{\Omega }\right),{\parallel ·\parallel }_2\right)$ is a Hilbert space with inner product ${\langle \mathbf{u},\mathbf{v}\rangle }_2={\int }_{\mathrm{\Omega }}\mathbf{u}\mathbf{v}\mathrm{d}x.$ Let ${\mathcal{X}}^s\left(\mathrm{\Omega }\right)$ stand for the linear space of all Lebesgue measurable functions $\mathbf{u}:{\mathbb{R}}^n\to \mathbb{R}$ such that the restriction to Ω of any function $\mathbf{u}$ in ${\mathcal{X}}^s\left(\mathrm{\Omega }\right)$ belongs to $L^2\left(\mathrm{\Omega }\right)$; the norm of ${\mathcal{X}}^s\left(\mathrm{\Omega }\right)$ is depicted as

$${\parallel \mathbf{u}\parallel }_{{\mathcal{X}}^s}={\parallel \mathbf{u}\parallel }_2+{\left({\int }_{\mathrm{\Theta }\times \mathrm{\Theta }}\frac{{|\mathbf{u}\left(x\right)-\mathbf{u}\left(y\right)|}^2}{{|x-y|}^{n+2s}}\mathrm{d}x\mathrm{d}y\right)}^{\frac{1}{2}},\forall \mathbf{u}\in {\mathcal{X}}^s\left(\mathrm{\Omega }\right),$$

where $\mathrm{\Theta }:=({\mathbb{R}}^n\times {\mathbb{R}}^n)\setminus (ð\mathrm{\Omega }\times ð\mathrm{\Omega })$, $ð\mathrm{\Omega }:={\mathbb{R}}^n\setminus \mathrm{\Omega }$. Let ${\mathcal{X}}_0^s\left(\mathrm{\Omega }\right):=\{\mathbf{u}\in {\mathcal{X}}^s\left(\mathrm{\Omega }\right):\mathbf{u}=0a.e.\mathrm{in}ð\mathrm{\Omega }\}$ and the norm of ${\mathcal{X}}_0^s\left(\mathrm{\Omega }\right)$ is in the form of

$${\parallel \mathbf{u}\parallel }_{{\mathcal{X}}_0^s}={\left({\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\frac{{|\mathbf{u}\left(x\right)-\mathbf{u}\left(y\right)|}^2}{{|x-y|}^{n+2s}}\mathrm{d}x\mathrm{d}y\right)}^{\frac{1}{2}},\forall \mathbf{u}\in {\mathcal{X}}_0^s\left(\mathrm{\Omega }\right).$$

$\left({\mathcal{X}}_0^s\left(\mathrm{\Omega }\right),{\parallel ·\parallel }_{{\mathcal{X}}_0^s}\right)$ is a Hilbert space with the inner product

$${\langle \mathbf{u},\mathbf{v}\rangle }_{{\mathcal{X}}_0^s}={\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\frac{\left[\mathbf{u}\right(x)-\mathbf{u}(y\left)\right]\left[\mathbf{v}\right(x)-\mathbf{v}(y\left)\right]}{{|x-y|}^{n+2s}}\mathrm{d}x\mathrm{d}y,\forall \mathbf{u},\mathbf{v}\in {\mathcal{X}}_0^s\left(\mathrm{\Omega }\right).$$

2.4. Eigenvalue Problem

This section considers a class of eigenvalue problem based on the extension of eigenvalues and Fourier analysis method. More precisely, we investigate the problem in the form of

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{(-\Delta )}^s\mathbf{u}=\lambda \mathbf{u},\mathrm{\Omega },\hfill \\ \mathbf{u}=0,\partial \mathrm{\Omega },\hfill \end{array}\right.\end{array}$$

(6)

where $s\in (0,1)$, Ω is an open bounded domain of ${\mathbb{R}}^n$ and $n>2s$. Moreover, the weak formulation of problem (6) is depicted by

$$\begin{array}{c}\hfill {\displaystyle {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\frac{\left[\mathbf{u}\right(x)-\mathbf{u}(y\left)\right]\left[\mathbf{v}\right(x)-\mathbf{v}(y\left)\right]}{{|x-y|}^{n+2s}}\mathrm{d}x\mathrm{d}y=\lambda {\int }_{\mathrm{\Omega }}\mathbf{u}\left(x\right)\mathbf{v}\left(x\right)\mathrm{d}x,\forall \mathbf{u},\mathbf{v}\in {\mathcal{X}}_0^s\left(\mathrm{\Omega }\right).}\end{array}$$

(7)

The eigenvalues sequence ${\left\{{\lambda }_k\right\}}_{k\in \mathbb{N}}$ of Equation (7) satisfies $0<{\lambda }_1<{\lambda }_2\cdots <{\lambda }_k<{\lambda }_{k+1}<\cdots$ and ${\lambda }_k\to +\infty ,$ as $k\to +\infty$. ${\left\{e_k\right\}}_{k\in \mathbb{N}}$ is the eigenfunction sequence corresponding to the eigenvalues set ${\left\{{\lambda }_k\right\}}_{k\in \mathbb{N}}$. Then, for any $k\in \mathbb{N}$, the eigenvalues can be characterized as follows

$$\begin{array}{c}\hfill {\displaystyle {\lambda }_k=\underset{\mathbf{u}\in \mathrm{span}\{e_1,\dots ,e_k\}\setminus \left\{0\right\}}{max}{\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\frac{{|\mathbf{u}\left(x\right)-\mathbf{u}\left(y\right)|}^2}{{|x-y|}^{n+2s}}\mathrm{d}x\mathrm{d}y{\left[{\int }_{\mathrm{\Omega }}{\left|\mathbf{u}\left(x\right)\right|}^2\mathrm{d}x\right]}^{-1}.}\end{array}$$

(8)

By applying a standard Fourier analysis technique, ${\left\{e_k\right\}}_{k\in \mathbb{N}}$ is an orthonormal basis of $L^2\left(\mathrm{\Omega }\right)$ and is an orthogonal basis of ${\mathcal{X}}_0^s\left(\mathrm{\Omega }\right)$. Therefore, for any $\mathbf{u}\in {\mathcal{X}}_0^s\left(\mathrm{\Omega }\right)$, it has $\mathbf{u}={\sum }_{k=1}^{\infty }\langle \mathbf{u},e_k\rangle e_k$ and ${(-\Delta )}^s\mathbf{u}={\sum }_{k=1}^{\infty }{\lambda }_k\langle \mathbf{u},e_k\rangle e_k$.

From Proposition 9 in [15], let ${\left\{{\lambda }_k\right\}}_{k\in \mathbb{N}}$ and ${\left\{{\tilde{\lambda }}_k\right\}}_{k\in \mathbb{N}}$ be, respectively, the eigenvalues sequences of ${(-\Delta )}^{s_1}\mathbf{u}$ and ${(-\Delta )}^{s_2}\mathbf{v}$. Furthermore, ${\left\{e_k\right\}}_{k\in \mathbb{N}}$ and ${\left\{{\tilde{e}}_k\right\}}_{k\in \mathbb{N}}$ are the corresponding eigenfunctions sequences of eigenvalues sequences, respectively. $(\mathbf{u},\mathbf{v})\in {\mathcal{X}}_0^{s_1}\left(\mathrm{\Omega }\right)\times {\mathcal{X}}_0^{s_2}\left(\mathrm{\Omega }\right)$ is called a weak solution of problem (2), if $(\mathbf{u},\mathbf{v})$ meets

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}^c{D}_0^{\alpha }{\langle \mathbf{u},\phi \rangle }_2+{\langle \mathbf{u},\phi \rangle }_{{\mathcal{X}}_0^{s_1}}=-\theta {\langle \mathbf{u},\phi \rangle }_2+{\langle f(x,t,{\mathbf{u}}_{\tau t},{\mathbf{v}}_{\tau t}),\phi \rangle }_2,\hfill & \mathrm{in}\mathrm{\Omega }\times (0,T),\hfill \\ {}^c{D}_0^{\beta }{\langle \mathbf{v},\varphi \rangle }_2+{\langle \mathbf{u},\varphi \rangle }_{{\mathcal{X}}_0^{s_2}}=-\sigma {\langle \mathbf{v},\varphi \rangle }_2+{\langle g(x,t,{\mathbf{u}}_{\tau t},{\mathbf{v}}_{\tau t}),\varphi \rangle }_2,\hfill & \mathrm{in}\mathrm{\Omega }\times (0,T),\hfill \\ {\langle \mathbf{u},\phi \rangle }_2=0,{\langle \mathbf{v},\varphi \rangle }_2=0,\hfill & \mathrm{in}\partial \mathrm{\Omega }\times [0,T]\hfill \end{array}\right.\end{array}$$

for any $\phi \in {\mathcal{X}}_0^{s_1}\left(\mathrm{\Omega }\right)$ and $\varphi \in {\mathcal{X}}_0^{s_2}\left(\mathrm{\Omega }\right)$.

Additionally, the problem (2) is turned into

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}^c{D}_0^{\alpha }{\langle \mathbf{u},e_k\rangle }_2+{\langle \mathbf{u},e_k\rangle }_{{\mathcal{X}}_0^{s_1}}=-\theta {\langle \mathbf{u},e_k\rangle }_2+{\langle f(x,t,{\mathbf{u}}_{\tau t},{\mathbf{v}}_{\tau t}),e_k\rangle }_2,\hfill & \mathrm{in}\mathrm{\Omega }\times (0,T),\hfill \\ {}^c{D}_0^{\beta }{\langle \mathbf{v},{\tilde{e}}_k\rangle }_2+{\langle \mathbf{u},{\tilde{e}}_k\rangle }_{{\mathcal{X}}_0^{s_2}}=-\sigma {\langle \mathbf{v},{\tilde{e}}_k\rangle }_2+{\langle g(x,t,{\mathbf{u}}_{\tau t},{\mathbf{v}}_{\tau t}),{\tilde{e}}_k\rangle }_2,\hfill & \mathrm{in}\mathrm{\Omega }\times (0,T),\hfill \\ {\langle \mathbf{u},e_k\rangle }_2=0,{\langle \mathbf{v},{\tilde{e}}_k\rangle }_2=0,\hfill & \mathrm{in}\partial \mathrm{\Omega }\times [0,T],\hfill \end{array}\right.\end{array}$$

or equivalently,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^c{D}_0^{\alpha }{\mathbf{u}}_k\left(t\right)=-({\lambda }_k+\theta ){\mathbf{u}}_k\left(t\right)+f_k({\mathbf{u}}_{\tau t},{\mathbf{v}}_{\tau t}),}\hfill \\ {}^c{D}_0^{\beta }{\mathbf{v}}_k\left(t\right)=-({\tilde{\lambda }}_k+\sigma ){\mathbf{v}}_k\left(t\right)+g_k({\mathbf{u}}_{\tau t},{\mathbf{v}}_{\tau t}),\hfill \end{array}\right.\end{array}$$

(9)

where ${\mathbf{u}}_k={\langle \mathbf{u},e_k\rangle }_2$, ${\mathbf{v}}_k={\langle \mathbf{v},{\tilde{e}}_k\rangle }_2,$ $f_k={〈f,e_k〉}_2,$ $g_k={\langle g,{\tilde{e}}_k\rangle }_2$ for all $t\in [0,T]$ and $k\in \mathbb{N}$.

Beyond that, if $(\mathbf{u},\mathbf{v})\in L^2\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)$ fills Equation (3), then

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle a_1{\langle \mathbf{u}(x,0),\phi \rangle }_2-b_1{\langle \mathbf{u}(x,\eta ),\phi \rangle }_2=c_1{\langle \mathbf{u}(x,T),\phi \rangle }_2,}\hfill & \\ a_2{\langle \mathbf{v}(x,0),\varphi \rangle }_2-b_2{\langle \mathbf{v}(x,\xi ),\varphi \rangle }_2=c_2{\langle \mathbf{v}(x,T),\varphi \rangle }_2\hfill \end{array}\right.\end{array}$$

for any $\phi ,\varphi \in L^2\left(\mathrm{\Omega }\right)$, $x\in \mathrm{\Omega }$. Let $\phi =e_k$ and $\varphi ={\tilde{e}}_k$ in the above equations, it produces

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle a_1{\mathbf{u}}_k\left(0\right)-b_1{\mathbf{u}}_k\left(\eta \right)=c_1{\mathbf{u}}_k\left(T\right),}\hfill \\ a_2{\mathbf{v}}_k\left(0\right)-b_2{\mathbf{v}}_k\left(\xi \right)=c_2{\mathbf{v}}_k\left(T\right),\hfill \end{array}\right.\end{array}$$

(10)

where $k\in \mathbb{N}$.

Based on the method of eigenvalues’ expansions and Fourier analysis technique, BVPs (2)–(3) are translated into BVPs (9)–(10), which provides technology for us to construct Green function and mild solutions in the sequel.

2.5. Green Function

We are to propose the Green function for three-point BVPs and study some necessary estimates of Green function.

For convenience, let

$${\mathcal{E}}_t^{\alpha }\left(\lambda \right)={\mathbf{E}}_{\alpha }\left(\lambda t^{\alpha }\right),{\mathcal{E}}_{t,s}^{\alpha ,\beta }\left(\lambda \right)={(t-s)}^{\beta -1}{\mathbf{E}}_{\alpha ,\beta }\left(\lambda {(t-s)}^{\alpha }\right),$$

thus, $\lambda t^{\alpha }{\mathbf{E}}_{\alpha ,\alpha +1}\left(\lambda t^{\alpha }\right)={\mathcal{E}}_t^{\alpha }\left(\lambda \right)-1.$ Additionally, for any $k\in \mathbb{N}$, set

$${\lambda }_k^{\alpha }=-({\lambda }_k+\theta ),{\mathcal{A}}_k^{\alpha }=a_1-b_1{\mathcal{E}}_{\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)-c_1{\mathcal{E}}_T^{\alpha }\left({\lambda }_k^{\alpha }\right),{\mathcal{M}}^{\alpha }=\frac{1}{{\mathcal{A}}_1^{\alpha }}(a_1-b_1-c_1),$$

$${\lambda }_k^{\beta }=-({\tilde{\lambda }}_k+\sigma ),{\mathcal{A}}_k^{\beta }=a_2-b_2{\mathcal{E}}_{\xi }^{\beta }\left({\lambda }_k^{\beta }\right)-c_2{\mathcal{E}}_T^{\beta }\left({\lambda }_k^{\beta }\right),{\mathcal{M}}^{\beta }=\frac{1}{{\mathcal{A}}_1^{\beta }}(a_2-b_2-c_2).$$

Considering the following auxiliary system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^c{D}_0^{\alpha }{\mathbf{u}}_k\left(t\right)=-{\lambda }_k^{\alpha }{\mathbf{u}}_k\left(t\right)+\rho \left(t\right),}\hfill \\ {}^c{D}_0^{\beta }{\mathbf{v}}_k\left(t\right)=-{\lambda }_k^{\beta }{\mathbf{v}}_k\left(t\right)+\varrho \left(t\right),\hfill \end{array}\right.\end{array}$$

(11)

where $\alpha ,\beta \in (0,1)$, $\rho ,\varrho \in \mathcal{C}\left(\right[0,T],\mathbb{R})$, $k\in \mathbb{N}$.

Lemma 1.

Problem (11) with BVCs (10) admits a unique solution

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\mathbf{u}}_k\left(t\right)={\int }_0^T{G}_k^{\alpha }(t,s)\rho \left(s\right)\mathrm{d}s,}\hfill \\ {\displaystyle {\mathbf{v}}_k\left(t\right)={\int }_0^T{G}_k^{\beta }(t,s)\varrho \left(s\right)\mathrm{d}s,}\hfill \end{array}\right.\end{array}$$

where Green functions $G_k^{\alpha }(t,s)$ and $G_k^{\beta }(t,s)$ are, respectively, given by

$$\begin{array}{c}\hfill G_k^{\alpha }(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{c_1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right){\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right),}\hfill & 0\le t<\eta \le s\le T,\hfill \\ {\displaystyle \frac{{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right)}{{\mathcal{A}}_k^{\alpha }}\left[b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\right],}\hfill & 0\le t\le s<\eta <T,\hfill \\ {\displaystyle {\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+\frac{{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right)}{{\mathcal{A}}_k^{\alpha }}\left[b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\right],}\hfill & 0\le s<t<\eta <T\hfill \end{array}\right.\end{array}$$

for $0<t<\eta <T$, $k\in \mathbb{N}$;

$$\begin{array}{c}\hfill G_k^{\alpha }(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{c_1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right){\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right),}\hfill & 0<\eta <t\le s\le T,\hfill \\ {\displaystyle {\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+\frac{c_1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right){\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right),}\hfill & 0<\eta \le s<t<T,\hfill \\ {\displaystyle {\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+\frac{{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right)}{{\mathcal{A}}_k^{\alpha }}\left[b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\right],}\hfill & 0\le s<\eta <t<T\hfill \end{array}\right.\end{array}$$

for $0<\eta <t<T$, $k\in \mathbb{N}$;

$$\begin{array}{c}\hfill G_k^{\beta }(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{c_2}{{\mathcal{A}}_k^{\beta }}{\mathcal{E}}_t^{\beta }\left({\lambda }_k^{\beta }\right){\mathcal{E}}_{T,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right),}\hfill & 0\le t<\xi \le s\le T,\hfill \\ {\displaystyle \frac{{\mathcal{E}}_t^{\beta }\left({\lambda }_k^{\beta }\right)}{{\mathcal{A}}_k^{\beta }}\left[b_2{\mathcal{E}}_{\xi ,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right)+c_2{\mathcal{E}}_{T,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right)\right],}\hfill & 0\le t\le s<\xi <T,\hfill \\ {\displaystyle {\mathcal{E}}_{t,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right)+\frac{{\mathcal{E}}_t^{\beta }\left({\lambda }_k^{\beta }\right)}{{\mathcal{A}}_k^{\beta }}\left[b_2{\mathcal{E}}_{\xi ,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right)+c_2{\mathcal{E}}_{T,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right)\right],}\hfill & 0\le s<t<\xi <T\hfill \end{array}\right.\end{array}$$

for $0<t<\xi <T$, $k\in \mathbb{N}$ and

$$\begin{array}{c}\hfill G_k^{\beta }(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{c_2}{{\mathcal{A}}_k^{\beta }}{\mathcal{E}}_t^{\beta }\left({\lambda }_k^{\beta }\right){\mathcal{E}}_{T,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right),}\hfill & 0<\xi <t\le s\le T,\hfill \\ {\displaystyle {\mathcal{E}}_{t,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right)+\frac{c_2}{{\mathcal{A}}_k^{\beta }}{\mathcal{E}}_t^{\beta }\left({\lambda }_k^{\beta }\right){\mathcal{E}}_{T,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right),}\hfill & 0<\xi \le s<t<T,\hfill \\ {\displaystyle {\mathcal{E}}_{t,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right)+\frac{{\mathcal{E}}_t^{\beta }\left({\lambda }_k^{\beta }\right)}{{\mathcal{A}}_k^{\beta }}\left[b_2{\mathcal{E}}_{\xi ,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right)+c_2{\mathcal{E}}_{T,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right)\right],}\hfill & 0\le s<\xi <t<T\hfill \end{array}\right.\end{array}$$

for $0<\xi <t<T$, $k\in \mathbb{N}$.

Proof. 

According to Example $4.9$ in Ref. [30], problem (11) has a unique solution

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{\displaystyle {\mathbf{u}}_k\left(t\right)}\hfill & =\hfill & {\displaystyle {\mathbf{u}}_k\left(0\right){\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right)+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\rho \left(s\right)\mathrm{d}s,}\hfill \\ {\mathbf{v}}_k\left(t\right)\hfill & =\hfill & {\displaystyle {\mathbf{v}}_k\left(0\right){\mathcal{E}}_t^{\beta }\left({\lambda }_k^{\beta }\right)+{\int }_0^t{\mathcal{E}}_{t,s}^{\beta ,\beta }\left({\lambda }_k^{\beta }\right)\varrho \left(s\right)\mathrm{d}s,}\hfill \end{array}\right.\end{array}$$

(12)

where $k\in \mathbb{N}$, $t\in [0,T]$. Moreover,

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{\displaystyle {\mathbf{u}}_k\left(\eta \right)}\hfill & =\hfill & {\displaystyle {\mathbf{u}}_k\left(0\right){\mathcal{E}}_{\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)+{\int }_0^{\eta }{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\rho \left(s\right)\mathrm{d}s,}\hfill \\ {\mathbf{u}}_k\left(T\right)\hfill & =\hfill & {\displaystyle {\mathbf{u}}_k\left(0\right){\mathcal{E}}_T^{\alpha }\left({\lambda }_k^{\alpha }\right)+{\int }_0^T{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\rho \left(s\right)\mathrm{d}s,}\hfill \end{array}\right.\end{array}$$

where $k\in \mathbb{N}$. In accordance with BVCs (10), we yield

$$\begin{array}{ccc}\hfill {\mathbf{u}}_k\left(0\right)& =& \frac{1}{{\mathcal{A}}_k^{\alpha }}\left[b_1{\int }_0^{\eta }{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\rho \left(s\right)\mathrm{d}s+c_1{\int }_0^T{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\rho \left(s\right)\mathrm{d}s\right],\hfill \end{array}$$

where $k\in \mathbb{N}$. Therefore, by an elementary calculation, we obtain

$$\begin{array}{ccc}\hfill {\mathbf{u}}_k\left(t\right)& =& \frac{c_1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right){\int }_0^T{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\rho \left(s\right)\mathrm{d}s\hfill \\ & & {\displaystyle +\frac{b_1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right){\int }_0^{\eta }{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\rho \left(s\right)\mathrm{d}s+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\rho \left(s\right)\mathrm{d}s}\hfill \\ & =& \frac{1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right){\int }_0^t\left[c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\right]\rho \left(s\right)\mathrm{d}s\hfill \\ \hfill & & {\displaystyle +{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\rho \left(s\right)\mathrm{d}s+\frac{1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right){\int }_t^{\eta }[c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)}\hfill \\ & & +b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)]\rho \left(s\right)\mathrm{d}s+\frac{c_1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right){\int }_{\eta }^T{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\rho \left(s\right)\mathrm{d}s\hfill \end{array}$$

(13)

for 0 < t < η < T, k ∈ $\mathbb{N}$.

According to Equation (14), we derive

• ${\displaystyle G_k^{\alpha }(t,s)={\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+\frac{1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right)\left[b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\right]}$ for $0\le s<t$ and $k\in \mathbb{N};$
• ${\displaystyle G_k^{\alpha }(t,s)=\frac{1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right)\left[b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\right]}$ for $t\le s<\eta$ and $k\in \mathbb{N};$
• ${\displaystyle G_k^{\alpha }(t,s)=\frac{c_1}{{\mathcal{A}}_k^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_k^{\alpha }\right){\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)}$ for $\eta \le s\le T$ and $k\in \mathbb{N}.$

With respect to $0<\eta <t<T$, $G_k^{\alpha }$ is received by an equivalent discussion as above for any $k\in \mathbb{N}$. Adopting a similar analysis as $G_k^{\alpha }$, it readily presents $G_k^{\beta }$ for every $k\in \mathbb{N}$. This completes the proof. □

Lemma 2.

Let the condition below hold.

(H1)

$\min \left\{{\mathcal{A}}_1^{\alpha },{\mathcal{A}}_1^{\beta }\right\}=\min \left\{a_1-b_1{\mathcal{E}}_{\eta }^{\alpha }\left({\lambda }_1^{\alpha }\right)-c_1{\mathcal{E}}_T^{\alpha }\left({\lambda }_1^{\alpha }\right),a_2-b_2{\mathcal{E}}_{\xi }^{\beta }\left({\lambda }_1^{\beta }\right)-c_2{\mathcal{E}}_T^{\beta }\left({\lambda }_1^{\beta }\right)\right\}>0.$

Then, the Green functions$G_k^{\alpha }$ and $G_k^{\beta }$ are nonnegative and

$$G_k^{\alpha }\le G_1^{\alpha },G_k^{\beta }\le G_1^{\beta },\forall t,s\in [0,T],k\in \mathbb{N}.$$

Furthermore, for any$t\in [0,T]$,

$${\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s=\frac{1}{{\lambda }_1^{\alpha }}\left[{\mathcal{M}}^{\alpha }{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)-1\right],{\int }_0^T{G}_1^{\beta }(t,s)\mathrm{d}s=\frac{1}{{\lambda }_1^{\beta }}\left[{\mathcal{M}}^{\beta }{\mathcal{E}}_t^{\beta }\left({\lambda }_1^{\beta }\right)-1\right].$$

Proof. 

By adopting the properties of Mittag–Leffler functions in $\left(E1\right)$–$\left(E3\right)$, we acquire

$${\mathcal{A}}_k^{\alpha }>{\mathcal{A}}_1^{\alpha },{\mathcal{A}}_k^{\beta }>{\mathcal{A}}_1^{\beta },\forall k\in \mathbb{N}.$$

Moreover, in view of condition $\left(H_1\right)$, it is straightforward to deduct that Green functions $G_k^{\alpha }$ and $G_k^{\beta }$ are nonnegative and

$$G_k^{\alpha }\le G_1^{\alpha },G_k^{\beta }\le G_1^{\beta },\forall t,s\in [0,T],k\in \mathbb{N}.$$

Subsequently, the discussion of $G_1^{\alpha }$ can be separated into two cases as below. According to the properties $\left(E4\right)$ and $\left(E5\right)$ of Mittag–Leffler functions, it infers

• if$0<t<\eta <T$, then

  $$\begin{array}{ccc}\hfill {\displaystyle {\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s}& =& \frac{c_1}{{\mathcal{A}}_1^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right){\int }_{\eta }^T{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s\hfill \\ & & {\displaystyle +\frac{{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)}{{\mathcal{A}}_1^{\alpha }}{\int }_t^{\eta }\left[b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)+c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\right]\mathrm{d}s}\hfill \\ & & +{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s+\frac{{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)}{{\mathcal{A}}_1^{\alpha }}{\int }_0^t\left[b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)+c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\right]\mathrm{d}s\hfill \\ & =& {\displaystyle \frac{c_1}{{\mathcal{A}}_1^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right){\int }_0^T{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s}\hfill \\ & & +\frac{b_1}{{\mathcal{A}}_1^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right){\int }_0^{\eta }{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s\hfill \\ & =& \frac{c_1}{{\mathcal{A}}_1^{\alpha }{\lambda }_1^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)\left[{\mathcal{E}}_T^{\alpha }\left({\lambda }_1^{\alpha }\right)-1\right]\hfill \\ & & +\frac{b_1}{{\mathcal{A}}_1^{\alpha }{\lambda }_1^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)\left[{\mathcal{E}}_{\eta }^{\alpha }\left({\lambda }_1^{\alpha }\right)-1\right]+\frac{1}{{\lambda }_1^{\alpha }}\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)-1\right]\hfill \\ & =& \frac{1}{{\lambda }_1^{\alpha }}\left[{\mathcal{M}}^{\alpha }{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)-1\right];\hfill \end{array}$$
• if $0<\eta <t<T$, then

  $$\begin{array}{ccc}\hfill {\displaystyle {\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s}& =& \frac{c_1}{{\mathcal{A}}_1^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right){\int }_t^T{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s\hfill \\ & & +{\int }_{\eta }^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s+\frac{c_1}{{\mathcal{A}}_1^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right){\int }_{\eta }^t{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s\hfill \\ & & +{\int }_0^{\eta }{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s+\frac{{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)}{{\mathcal{A}}_1^{\alpha }}{\int }_0^{\eta }\left[b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)+c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\right]\mathrm{d}s\hfill \\ & =& {\displaystyle \frac{c_1}{{\mathcal{A}}_1^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right){\int }_0^T{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s}\hfill \\ & & +\frac{b_1}{{\mathcal{A}}_1^{\alpha }}{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right){\int }_0^{\eta }{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1^{\alpha }\right)\mathrm{d}s\hfill \\ & =& \frac{1}{{\lambda }_1^{\alpha }}\left[{\mathcal{M}}^{\alpha }{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)-1\right].\hfill \end{array}$$

By the identical procedure, we attain ${\int }_0^T{G}_1^{\beta }(t,s)\mathrm{d}s=\frac{1}{{\lambda }_1^{\beta }}\left[{\mathcal{M}}^{\beta }{\mathcal{E}}_t^{\beta }\left({\lambda }_1^{\beta }\right)-1\right]$ for any $t\in [0,T]$. This completes the proof. □

Remark 1.

Based on the properties$\left(E1\right)$–$\left(E5\right)$ of Mittag–Leffler functions, some vital estimates of Green functions are introduced, which abolishes the restriction of eigenvalues sequence by condition $\left(A\right)$ in Ref. [11]. In addition, the flexibility of Green functions enhances the difficulty of this article.

Remark 2.

By means of the method of eigenvalues’ expansions, the differential equation is converted to the integral form. Taking Equations (6) and (7) into consideration, it demonstrates that the solution of Equation (9) is the weak solution of Equation (2).

3. Main Results

On the basis of the above discussion, the mild solution for Equation (2) with BVCs (3) is introduced first and then the existence and uniqueness of mild solution are researched.

Let $\mathcal{C}([0,T],L^2\left(\mathrm{\Omega }\right))$ denote the set of all continuous functions from $[0,T]$ to $L^2\left(\mathrm{\Omega }\right)$ with the norm ${\parallel ·\parallel }_{\infty }={max}_{t\in [0,T]}{\parallel ·\parallel }_2.$ Define $\mathcal{X}=\mathcal{C}([0,T],L^2\left(\mathrm{\Omega }\right))\times \mathcal{C}([0,T],L^2\left(\mathrm{\Omega }\right))$ endowed with the norm ${\parallel \mathbf{z}\parallel }_{\mathcal{X}}={max\{\parallel \mathbf{u}\parallel }_{\infty }{,\parallel \mathbf{v}\parallel }_{\infty }\}$ for any $\mathbf{z}={(\mathbf{u},\mathbf{v})}^T\in \mathcal{X}$. Hence, $(\mathcal{X},\parallel ·{\parallel }_{\mathcal{X}})$ is a Banach space. Set ${\left|f\right|}_0={max}_{t\in [0,T]}{\parallel f(·,t,0,0)\parallel }_2$, ${\left|g\right|}_0={max}_{t\in [0,T]}{\parallel g(·,t,0,0)\parallel }_2$.

On account of Lemma 1, this paper is devoted to find the mild solution of BVPs (2) and (3), which is defined as below.

Definition 4.

$(\mathbf{u},\mathbf{v})\in L^2\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)$is called a mild solution of BVPs (2) and (3), if

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathbf{u}(x,t)=\sum _{k=1}^{\infty }{\int }_0^T{G}_k^{\alpha }(t,s)f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\mathrm{d}s{e}_k:=\mathbf{F}(\mathbf{u},\mathbf{v}),}\hfill \\ {\displaystyle \mathbf{v}(x,t)=\sum _{k=1}^{\infty }{\int }_0^T{G}_k^{\beta }(t,s)g_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\mathrm{d}s{\tilde{e}}_k:=\mathbf{G}(\mathbf{u},\mathbf{v}),}\hfill \end{array}\right.\end{array}$$

where $(x,t)\in \mathrm{\Omega }\times [0,T]$.

Via Definition 4, define an operator $\mathbf{T}\mathbf{z}={(\mathrm{\Phi }\mathbf{z},\mathrm{\Psi }\mathbf{z})}^T={(\mathbf{F},\mathbf{G})}^T$ for $\mathbf{z}={(\mathbf{u},\mathbf{v})}^T\in \mathcal{X}$. Furthermore,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathrm{\Phi }\mathbf{z}(x,t)=\sum _{k=1}^{\infty }{\int }_0^T{G}_k^{\alpha }(t,s)f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\mathrm{d}s{e}_k,}\hfill \\ {\displaystyle \mathrm{\Psi }\mathbf{z}(x,t)=\sum _{k=1}^{\infty }{\int }_0^T{G}_k^{\beta }(t,s)g_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\mathrm{d}s{\tilde{e}}_k,}\hfill \end{array}\right.\end{array}$$

(14)

where $(x,t)\in \mathrm{\Omega }\times [0,T]$. Obviously, the mild solution of BVPs (2)–(3) is the fixed point of operator T in $\mathcal{X}$.

Definition 5

([33]).Let${\mathbb{E}}_1$, ${\mathbb{E}}_2$be two Banach spaces and$\mathcal{D}\subset {\mathbb{E}}_1$. Operator$\mathbf{T}:\mathcal{D}\to {\mathbb{E}}_2$is completely continuous if—and only if—$\mathbf{T}:\mathcal{D}\to {\mathbb{E}}_2$is continuous and compact.

To begin with, we recall Leray–Schauder fixed-point theorem, which is a generalization of Brouwer fixed-point theorem. Please refer to [34,35] for more information.

Theorem 1.

Let$\mathbb{E}$be a Banach space and$\mathcal{D}\subset \mathbb{E}$be bounded, closed-convex subset. If operator$\mathbf{T}:\mathcal{D}\to \mathcal{D}$is completely continuous, then$\mathbf{T}$possesses at least one point$x\in \mathcal{D}$such that$\mathbf{T}x=x$.

Let ${\mathcal{X}}_M={\{\mathbf{z}\in \mathcal{X}:\parallel \mathbf{z}\parallel }_{\mathcal{X}}\le M\}$ with $M=\frac{{\varpi }_0}{1-\varpi }$,

$${\varpi }_0=max\left\{-\frac{\sqrt{3}{\overline{a}}^f{c}_0^f}{{\lambda }_1^{\alpha }}\sqrt{\left|\mathrm{\Omega }\right|},-\frac{\sqrt{3}{\overline{a}}^g{c}_0^g}{{\lambda }_1^{\beta }}\sqrt{\left|\mathrm{\Omega }\right|}\right\},$$

$$\varpi =max\left\{-\frac{\sqrt{3}{\overline{a}}^f}{{\lambda }_1^{\alpha }}(c_1^f+c_2^f),-\frac{\sqrt{3}{\overline{a}}^g}{{\lambda }_1^{\beta }}(c_1^g+c_2^g)\right\},$$

where $\left|\mathrm{\Omega }\right|$ denotes the Lebesgue measure of Ω and $\overline{a}={max}_{t\in [0,T]}{sup}_{x\in \mathrm{\Omega }}\left|a(x,t)\right|$ for arbitrary $a\in \mathcal{C}(\mathrm{\Omega }\times [0,T],\mathbb{R})$.

Lemma 3.

Let condition$\left(H_1\right)$and the conditions below hold.

(H2)

$f(·,·,·,·)$ and $g(·,·,·,·)$ are Carathéodory functions and for arbitrary $h,\hslash \in \mathbb{R}$, there exist nonnegative real functions $a^f,a^g\in \mathcal{C}(\mathrm{\Omega }\times [0,T],\mathbb{R})$ such that

$$\left|f(·,·,h,\hslash )\right|\le a^f(·,·)(c_0^f+c_1^f\left|h\right|+c_2^f|\hslash |),|g(·,·,h,\hslash )|\le a^g(·,·)(c_0^g+c_1^g\left|h\right|+c_2^g|\hslash |),$$

where $c_i^f$ and $c_i^g$ are positive constants, $i=0,1,2$.

(H3)

$\varpi <1$.

Then$\mathbf{T}:{\mathcal{X}}_M\to {\mathcal{X}}_M$is completely continuous.

Proof. 

For every $\mathbf{z}={(\mathbf{u},\mathbf{v})}^T\in {\mathcal{X}}_M$, it readily derives from Lemma 1 and condition $\left(H_2\right)$ that $\mathbf{T}$ is continuous. Subsequently, it will be claim that $\mathbf{T}$ is uniformly bounded for any $\mathbf{z}={(\mathbf{u},\mathbf{v})}^T\in {\mathcal{X}}_M$. Indeed,

$$\begin{array}{ccc}\hfill {\parallel \mathrm{\Phi }\mathbf{z}\parallel }_2& =& {\left[\underset{K\to \infty }{lim}\sum _{k=1}^K{\left|{\int }_0^T{G}_k^{\alpha }(t,s)f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\mathrm{d}s\right|}^2\right]}^{\frac{1}{2}}\hfill \\ & \le & {\left[\underset{K\to \infty }{lim}\sum _{k=1}^K{\int }_0^T{G}_k^{\alpha }(t,s)\mathrm{d}s{\int }_0^T{G}_k^{\alpha }(t,s){\left|f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\right|}^2\mathrm{d}s\right]}^{\frac{1}{2}}\left(\mathrm{H}ö\mathrm{lder}\mathrm{inequality}\right)\hfill \\ & \le & {\left[{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s{\int }_0^T{G}_1^{\alpha }(t,s)\underset{K\to \infty }{lim}\sum _{k=1}^K{\left|f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\right|}^2\mathrm{d}s\right]}^{\frac{1}{2}}\hfill \\ & \le & {\left[{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s{\int }_0^T{G}_1^{\alpha }(t,s){\parallel f(x,s,{\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\parallel }_2^2\mathrm{d}s\right]}^{\frac{1}{2}}\left(\mathrm{Bessel}\mathrm{inequality}\right)\hfill \\ & \le & \sqrt{3}{\overline{a}}^f{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s\underset{s\in [0,T]}{max}{\left[{\left(c_0^f\right)}^2\left|\mathrm{\Omega }\right|+{\left(c_1^f\right)}^2\parallel {\mathbf{u}}_{\tau s}{\parallel }_2^2+{\left(c_2^f\right)}^2{\parallel {\mathbf{v}}_{\tau s}\parallel }_2^2\right]}^{\frac{1}{2}}\hfill \\ & \le & \sqrt{3}{\overline{a}}^f{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s\underset{s\in [0,T]}{max}\left(c_0^f\sqrt{\left|\mathrm{\Omega }\right|}+c_1^f\parallel {\mathbf{u}}_{\tau s}{\parallel }_2+c_2^f{\parallel {\mathbf{v}}_{\tau s}\parallel }_2\right)\left(C_p\mathrm{inequality}\right)\hfill \\ & \le & \sqrt{3}{\overline{a}}^f{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s\left(c_0^f\sqrt{\left|\mathrm{\Omega }\right|}+c_1^f{\parallel \mathbf{u}\parallel }_{\infty }+c_2^f{\parallel \mathbf{v}\parallel }_{\infty }\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill {\parallel \mathrm{\Phi }\mathbf{z}\parallel }_{\infty }& \le \sqrt{3}{\overline{a}}^f\left(c_0^f\sqrt{\left|\mathrm{\Omega }\right|}+c_1^f{\parallel \mathbf{u}\parallel }_{\infty }+c_2^f{\parallel \mathbf{v}\parallel }_{\infty }\right)\underset{t\in [0,T]}{\max }{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s\hfill \\ & \le -\frac{\sqrt{3}{\overline{a}}^f}{{\lambda }_1^{\alpha }}\left(c_0^f\sqrt{\left|\mathrm{\Omega }\right|}+c_1^f{\parallel \mathbf{u}\parallel }_{\infty }+c_2^f{\parallel \mathbf{v}\parallel }_{\infty }\right)\underset{t\in [0,T]}{\max }\left[1-{\mathcal{M}}^{\alpha }{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)\right]\hfill \\ & \le -\frac{\sqrt{3}{\overline{a}}^f}{{\lambda }_1^{\alpha }}{c}_0^f\sqrt{\left|\mathrm{\Omega }\right|}-\frac{\sqrt{3}{\overline{a}}^f}{{\lambda }_1^{\alpha }}(c_1^f+c_2^f){\parallel \mathbf{z}\parallel }_{\mathcal{X}}.\hfill \end{array}$$

(15)

In addition,

$${\parallel \mathrm{\Psi }\mathbf{z}\parallel }_{\infty }\le -\frac{\sqrt{3}{\overline{a}}^g}{{\lambda }_1^{\beta }}{c}_0^g\sqrt{\left|\mathrm{\Omega }\right|}-\frac{\sqrt{3}{\overline{a}}^g}{{\lambda }_1^{\beta }}(c_1^g+c_2^g){\parallel \mathbf{z}\parallel }_{\mathcal{X}},$$

(16)

which indicates that T_Z is well defined in ${\mathcal{X}}_M$. Furthermore, T is uniformly bounded for any $\mathbf{z}={(\mathbf{u},\mathbf{v})}^T\in {\mathcal{X}}_M$. From (15)–(16), we deduce

$${\parallel \mathbf{T}\mathbf{z}\parallel }_{\mathcal{X}}\le {\varpi }_0+\varpi {\parallel \mathbf{z}\parallel }_{\mathcal{X}},\forall \mathbf{z}={(\mathbf{u},\mathbf{v})}^T\in {\mathcal{X}}_M.$$

As a consequence, T is a self-mapping in ${\mathcal{X}}_M$.

Next, we claim that $\mathbf{T}$ is equicontinuous in ${\mathcal{X}}_M$. For simplicity, let

$$I_k^{\alpha }(t_1,t_2)={\int }_0^T\left[G_k^{\alpha }(t_1,s)-G_k^{\alpha }(t_2,s)\right]\mathrm{d}s,\forall t_1,t_2\in [0,T],k\in \mathbb{N}.$$

First of all, in terms of Lemma 2, some crucial estimates are introduced as follows.

(i)

If $0\le t_1<t_2<\eta \le s\le T$, then

$$\begin{array}{ccc}\hfill I_k^{\alpha }(t_1,t_2)& \le & {\displaystyle \frac{c_1}{{\mathcal{A}}_k^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]{\int }_{\eta }^T{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\mathrm{d}s}\hfill \\ & \le & -\frac{c_1}{{\lambda }_k^{\alpha }{\mathcal{A}}_1^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]\left[1-{\mathcal{E}}_{T-\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]\hfill \\ & \le & -\frac{c_1}{{\lambda }_k^{\alpha }{\mathcal{A}}_1^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)\right],\forall k\in \mathbb{N}.\hfill \end{array}$$

(ii)

If $0\le t_1<t_2\le s<\eta <T$, then

$$\begin{array}{ccc}\hfill I_k^{\alpha }(t_1,t_2)& \le & {\displaystyle \frac{1}{{\mathcal{A}}_k^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]{\int }_0^{\eta }\left[b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\right]\mathrm{d}s}\hfill \\ & \le & {\displaystyle -\frac{1}{{\lambda }_k^{\alpha }}\frac{1}{{\mathcal{A}}_1^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]}\hfill \\ & & \times \left[b_1\left(1-{\mathcal{E}}_{\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)\right)+c_1\left({\mathcal{E}}_{T-\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_T^{\alpha }\left({\lambda }_k^{\alpha }\right)\right)\right]\hfill \\ & \le & {\displaystyle -\frac{1}{{\lambda }_k^{\alpha }}\frac{b_1+c_1}{{\mathcal{A}}_1^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)\right],\forall k\in \mathbb{N}.}\hfill \end{array}$$

(iii)

If $0\le s<t_1<t_2<\eta <T$, then

$$\begin{array}{ccc}\hfill I_k^{\alpha }(t_1,t_2)& \le & {\int }_0^{\eta }\left[{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\right]\mathrm{d}s\hfill \\ & & +\frac{1}{{\mathcal{A}}_k^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]{\int }_0^{\eta }\left[b_1{\mathcal{E}}_{\eta ,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)+c_1{\mathcal{E}}_{T,s}^{\alpha ,\alpha }\left({\lambda }_k^{\alpha }\right)\right]\mathrm{d}s\hfill \\ & \le & -\frac{1}{{\lambda }_k^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)+{\mathcal{E}}_{t_1-\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2-\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]\hfill \\ & & -\frac{1}{{\lambda }_k^{\alpha }}\frac{1}{{\mathcal{A}}_1^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]\hfill \\ & & \times \left[b_1\left(1-{\mathcal{E}}_{\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)\right)+c_1\left({\mathcal{E}}_{T-\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_T^{\alpha }\left({\lambda }_k^{\alpha }\right)\right)\right]\hfill \\ & \le & -\frac{1}{{\lambda }_k^{\alpha }}\left\{\frac{{\mathcal{A}}_1^{\alpha }+b_1+c_1}{{\mathcal{A}}_1^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]+\left[{\mathcal{E}}_{t_1-\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2-\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]\right\}\hfill \end{array}$$

for any k ∈ $\mathbb{N}$.

By virtue of $-{\lambda }_k^{\alpha }\to +\infty$ as $k\to \infty$. Accordingly, for any $ϵ>0$, there exists a constant $\mathcal{K}\in \mathbb{N}$ such that

$$\begin{array}{c}\hfill I_k^{\alpha }(t_1,t_2)\le ϵ,\forall k>\mathcal{K}.\end{array}$$

(17)

On the other hand, there exists a constant $\delta =\delta \left(ϵ\right)>0$ with $|t_1-t_2|<\delta$ such that

$$\underset{k\in \{1,2,\dots ,\mathcal{K}\}}{max}\left\{\frac{{\mathcal{A}}_1^{\alpha }+b_1+c_1}{{\mathcal{A}}_1^{\alpha }}\left[{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2}^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]+\left[{\mathcal{E}}_{t_1-\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)-{\mathcal{E}}_{t_2-\eta }^{\alpha }\left({\lambda }_k^{\alpha }\right)\right]\right\}<-{\lambda }_1^{\alpha }ϵ.$$

(18)

To sum up, in conjunction with (i)–(iii), (17) and (18), we conclude

$$I_k^{\alpha }(t_1,t_2)={\int }_0^T\left[G_k^{\alpha }(t_1,s)-G_k^{\alpha }(t_2,s)\right]\mathrm{d}s\le ϵ,\forall k\in \mathbb{N}.$$

(19)

Together with (14) and (19), we derive

$$\begin{array}{ccc}\hfill & & \parallel \mathrm{\Phi }\mathbf{z}(x,t_1)-\mathrm{\Phi }\mathbf{z}(x,t_2){\parallel }_2^2\hfill \\ & =& \underset{K\to \infty }{lim}\sum _{k=1}^K{\left|{\int }_0^T\left[G_k^{\alpha }(t_1,s)-G_k^{\alpha }(t_2,s)\right]f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\mathrm{d}s\right|}^2\hfill \\ & \le & \underset{K\to \infty }{lim}\sum _{k=1}^K{\int }_0^T\left[G_k^{\alpha }(t_1,s)-G_k^{\alpha }(t_2,s)\right]\mathrm{d}s\hfill \\ & & \times {\int }_0^T\left[G_k^{\alpha }(t_1,s)-G_k^{\alpha }(t_2,s)\right]|f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s}){|}^2\mathrm{d}s\left(\mathrm{H}ö\mathrm{lder}\mathrm{inequality}\right)\hfill \\ & \le & {\int }_0^T{G}_1^{\alpha }(t_1,s)\underset{K\to \infty }{lim}\sum _{k=1}^K|f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s}){|}^2\mathrm{d}sϵ\hfill \\ & \le & {\int }_0^T{G}_1^{\alpha }(t_1,s){\parallel f(x,s,{\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\parallel }_2^2\mathrm{d}sϵ\left(\mathrm{Bessel}\mathrm{inequality}\right)\hfill \\ & \le & 3{\left({\overline{a}}^f\right)}^2{\int }_0^T{G}_1^{\alpha }(t_1,s)\mathrm{d}s\left[{\left(c_0^f\right)}^2\left|\mathrm{\Omega }\right|+{\left(c_1^f\right)}^2{\parallel \mathbf{u}\parallel }_{\infty }^2+{\left(c_2^f\right)}^2{\parallel \mathbf{v}\parallel }_{\infty }^2\right]ϵ\left(C_p\mathrm{inequality}\right)\hfill \\ & \le & \frac{3{\left({\overline{a}}^f\right)}^2}{{\lambda }_1^{\alpha }}\left[{\mathcal{M}}^{\alpha }{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_1^{\alpha }\right)-1\right]\left[{\left(c_0^f\right)}^2\left|\mathrm{\Omega }\right|+{\left(c_1^f\right)}^2{\parallel \mathbf{u}\parallel }_{\infty }^2+{\left(c_2^f\right)}^2{\parallel \mathbf{v}\parallel }_{\infty }^2\right]ϵ\hfill \\ & \le & -\frac{3{\left({\overline{a}}^f\right)}^2}{{\lambda }_1^{\alpha }}\left[{\left(c_0^f\right)}^2\left|\mathrm{\Omega }\right|+{\left(c_1^f\right)}^2{M}^2+{\left(c_2^f\right)}^2{M}^2\right]ϵ,\hfill \end{array}$$

which reveals that Φ_z is equicontinuous for any $\mathbf{z}={(\mathbf{u},\mathbf{v})}^T\in {\mathcal{X}}_M$ and so is Ψz. Hence, by Arzelà-Ascoli theorem, T is completely continuous. This completes the proof. □

Theorem 2.

Assume that conditions$\left(H_1\right)$–$\left(H_3\right)$are fulfilled, then BVPs (2)–(3) admit at least one bounded mild solution.

Proof. 

Apparently, ${\mathcal{X}}_M$ is bounded closed convex subset of $\mathcal{X}$. In view of Lemma 3, we obtain operator $\mathbf{T}$ maps ${\mathcal{X}}_M$ into itself and $\mathbf{T}$ is completely continuous in ${\mathcal{X}}_M$. Therefore, by Leray–Schauder fixed-point theorem, $\mathbf{T}$ owns at least one fixed point ${\mathbf{z}}^{*}={({\mathbf{u}}^{*},{\mathbf{v}}^{*})}^T\in {\mathcal{X}}_M$, which solves BVPs (2) and (3). This completes the proof. □

Remark 3.

The condition on eigenvalues is much weaker than the condition (A) in [11] which required the inverse operator$A^{-1}$of self-adjoint operator A is bounded. The existence of solutions is investigated in [6] by the method of Laplace transform, this paper further considers the uniqueness of solutions.

Next, the uniqueness of mild solution for BVPs (2) and (3) will be stated. Let ${\mathcal{X}}_{\tilde{M}}={\{\mathbf{z}\in \mathcal{X}:\parallel \mathbf{z}\parallel }_{\mathcal{X}}\le \tilde{M}\}$, where $\tilde{M}=\frac{{\kappa }_0}{1-\kappa },$

$${\kappa }_0=max\left\{-\frac{\sqrt{2}}{{\lambda }_1^{\alpha }}{\left|f\right|}_0,-\frac{\sqrt{2}}{{\lambda }_1^{\beta }}{\left|g\right|}_0\right\},$$

$${\displaystyle \kappa =max\left\{-\frac{2}{{\lambda }_1^{\alpha }}\left(L_1^f+L_2^f\right),-\frac{2}{{\lambda }_1^{\beta }}\left(L_1^g+L_2^g\right)\right\}.}$$

Theorem 3.

BVPs (2)–(3) possess a unique bounded mild solution in case condition $\left(H_1\right)$ and conditions below hold.

(H4)

$f(·,·,·,·)$and$g(·,·,·,·)$are Carathéodory functions. For any$\zeta ,\omega ,\tilde{\zeta },\tilde{\omega }\in \mathbb{R}$, there exist nonnegative constants$L_1^f$, $L_2^f$, $L_1^g$and$L_2^g$such that

$$\left|f(·,·,\zeta ,\omega )-f(·,·,\tilde{\zeta },\tilde{\omega })\right|\le L_1^f|\zeta -\tilde{\zeta }|+L_2^f|\omega -\tilde{\omega }|,$$

$$\left|g(·,·,\zeta ,\omega )-g(·,·,\tilde{\zeta },\tilde{\omega })\right|\le L_1^g|\zeta -\tilde{\zeta }|+L_2^g|\omega -\tilde{\omega }|.$$

(H5)

$\kappa <1$.

Proof. 

For any $\mathbf{z}={(\mathbf{u},\mathbf{v})}^T\in {\mathcal{X}}_{\tilde{M}}$, by Hölder inequality and Bessel inequality, we obtain

$$\begin{array}{ccc}\hfill {\parallel \mathrm{\Phi }\mathbf{z}\parallel }_2^2& =& \underset{K\to \infty }{lim}\sum _{k=1}^K{\left|{\int }_0^T{G}_k^{\alpha }(t,s)f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\mathrm{d}s\right|}^2\hfill \\ & \le & \underset{K\to \infty }{lim}\sum _{k=1}^K{\int }_0^T{G}_k^{\alpha }(t,s)\mathrm{d}s{\int }_0^T{G}_k^{\alpha }(t,s){\left|f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\right|}^2\mathrm{d}s\hfill \\ & \le & {\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s{\int }_0^T{G}_1^{\alpha }(t,s)\underset{K\to \infty }{lim}\sum _{k=1}^K{\left|f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\right|}^2\mathrm{d}s\hfill \\ & \le & {\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s{\int }_0^T{G}_1^{\alpha }(t,s){\parallel f(x,s,{\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})\parallel }_2^2\mathrm{d}s\hfill \\ & \le & {\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s{\int }_0^T{G}_1^{\alpha }(t,s)\left[4{\left(L_1^f\right)}^2{\parallel \mathbf{u}\parallel }_2^2+4{\left(L_2^f\right)}^2{\parallel \mathbf{v}\parallel }_2^2+2{\parallel f(x,s,0,0)\parallel }_2^2\right]\mathrm{d}s\hfill \\ & \le & {\left[{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s\right]}^2\left[4{\left(L_1^f\right)}^2{\parallel \mathbf{u}\parallel }_{\infty }^2+4{\left(L_2^f\right)}^2{\parallel \mathbf{v}\parallel }_{\infty }^2+2{\left|f\right|}_0^2\right],\hfill \end{array}$$

which generates

$$\begin{array}{ccc}{\parallel \mathrm{\Phi }\mathbf{z}\parallel }_{\infty }& \le & \underset{t\in [0,T]}{max}{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s{\left[4{\left(L_1^f\right)}^2{\parallel \mathbf{u}\parallel }_{\infty }^2+4{\left(L_2^f\right)}^2{\parallel \mathbf{v}\parallel }_{\infty }^2+2{\left|f\right|}_0^2\right]}^{\frac{1}{2}}\\ & \le & \frac{1}{{\lambda }_1^{\alpha }}\underset{t\in [0,T]}{max}\left[{\mathcal{M}}^{\alpha }{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)-1\right]\left[2L_1^f{\parallel \mathbf{u}\parallel }_{\infty }+2L_2^f{\parallel \mathbf{v}\parallel }_{\infty }+\sqrt{2}{\left|f\right|}_0\right]\hfill \\ & \le & -\frac{1}{{\lambda }_1^{\alpha }}\left[2\left(L_1^f+L_2^f\right){\parallel \mathbf{z}\parallel }_{\mathcal{X}}+\sqrt{2}{\left|f\right|}_0\right].\hfill \end{array}$$

Similarly, it leads to

$${\parallel \mathrm{\Psi }\mathbf{z}\parallel }_{\infty }\le -\frac{1}{{\lambda }_1^{\beta }}\left[2\left(L_1^g+L_2^g\right){\parallel \mathbf{z}\parallel }_{\mathcal{X}}+\sqrt{2}{\left|g\right|}_0\right].$$

Therefore, ${\parallel \mathbf{T}\mathbf{z}\parallel }_{\mathcal{X}}\le \tilde{M}$ for arbitrary $\mathbf{z}={(\mathbf{u},\mathbf{v})}^T\in {\mathcal{X}}_{\tilde{M}}$.

Next, the contraction of $\mathbf{T}:{\mathcal{X}}_{\tilde{M}}\to {\mathcal{X}}_{\tilde{M}}$ ought to be verified. Let $\mathbf{z}={(\mathbf{u},\mathbf{v})}^T$, $\tilde{\mathbf{z}}={(\tilde{\mathbf{u}},\tilde{\mathbf{v}})}^T\in {\mathcal{X}}_{\tilde{M}}$, we deduce

$$\begin{array}{ccc}\hfill \parallel \mathrm{\Phi }\mathbf{z}-\mathrm{\Phi }\tilde{\mathbf{z}}{\parallel }_{\infty }& =& \underset{t\in [0,T]}{max}{\left[\underset{K\to \infty }{lim}\sum _{k=1}^K{\left|{\int }_0^T{G}_k^{\alpha }(t,s)\left(f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})-f_k({\tilde{\mathbf{u}}}_{\tau s},{\tilde{\mathbf{v}}}_{\tau s})\right)\mathrm{d}s\right|}^2\right]}^{\frac{1}{2}}\hfill \\ & \le & \underset{t\in [0,T]}{max}{\left[\underset{K\to \infty }{lim}\sum _{k=1}^K{\int }_0^T{G}_k^{\alpha }(t,s)\mathrm{d}s{\int }_0^T{G}_k^{\alpha }(t,s){\left|f_k({\mathbf{u}}_{\tau t},{\mathbf{v}}_{\tau t})-f_k({\tilde{\mathbf{u}}}_{\tau s},{\tilde{\mathbf{v}}}_{\tau s})\right|}^2\mathrm{d}s\right]}^{\frac{1}{2}}\hfill \\ & \le & \underset{t\in [0,T]}{max}{\left[{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s{\int }_0^T{G}_1^{\alpha }(t,s)\underset{K\to \infty }{lim}\sum _{k=1}^K{\left|f_k({\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})-f_k({\tilde{\mathbf{u}}}_{\tau s},{\tilde{\mathbf{v}}}_{\tau s})\right|}^2\mathrm{d}s\right]}^{\frac{1}{2}}\hfill \\ & \le & \underset{t\in [0,T]}{max}{\left[{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s{\int }_0^T{G}_1^{\alpha }(t,s){\parallel f(x,s,{\mathbf{u}}_{\tau s},{\mathbf{v}}_{\tau s})-f(x,s,{\tilde{\mathbf{u}}}_{\tau s},{\tilde{\mathbf{v}}}_{\tau s})\parallel }_2^2\mathrm{d}s\right]}^{\frac{1}{2}}\hfill \\ & \le & \underset{t\in [0,T]}{max}{\int }_0^T{G}_1^{\alpha }(t,s)\mathrm{d}s{\left[2{\left(L_1^f\right)}^2\parallel \mathbf{u}-\tilde{\mathbf{u}}{\parallel }_{\infty }^2+2{\left(L_2^f\right)}^2{\parallel \mathbf{v}-\tilde{\mathbf{v}}\parallel }_{\infty }^2\right]}^{\frac{1}{2}}\hfill \\ & \le & \frac{\sqrt{2}}{{\lambda }_1^{\alpha }}\underset{t\in [0,T]}{max}\left[{\mathcal{M}}^{\alpha }{\mathcal{E}}_t^{\alpha }\left({\lambda }_1^{\alpha }\right)-1\right]\left(L_1^f+L_2^f\right){\parallel \mathbf{z}-\tilde{\mathbf{z}}\parallel }_{\mathcal{X}}\hfill \\ & \le & -\frac{\sqrt{2}}{{\lambda }_1^{\alpha }}\left(L_1^f+L_2^f\right){\parallel \mathbf{z}-\tilde{\mathbf{z}}\parallel }_{\mathcal{X}}\hfill \\ & \le & \kappa \parallel \mathbf{z}-\tilde{\mathbf{z}}{\parallel }_{\mathcal{X}}.\hfill \end{array}$$

The identical discussion contributes to $\parallel \mathrm{\Psi }\mathbf{z}-\mathrm{\Psi }\tilde{\mathbf{z}}{\parallel }_{\infty }\le \kappa \parallel \mathbf{z}-\tilde{\mathbf{z}}{\parallel }_{\mathcal{X}}$. Hence, $\parallel \mathbf{T}\mathbf{z}-\mathbf{T}\tilde{\mathbf{z}}{\parallel }_{\mathcal{X}}\le \kappa {\parallel \mathbf{z}-\tilde{\mathbf{z}}\parallel }_{\mathcal{X}}$, which demonstrates that operator T is contractive with condition (H₅). According to Banach fixed-point theorem, T possesses a fixed point ${\mathbf{z}}^{\star }={({\mathbf{u}}^{\star },{\mathbf{v}}^{\star })}^T\in {\mathcal{X}}_{\tilde{M}}$, which is a unique bounded mild solution of BVPs (2)–(3). This completes the proof. □

Remark 4.

If$s_1=s_2=1$, then Equation (2) is turned into Caputo-type fractional derivative in the literature [6,9,11,23,24]. If all the orders equal 1, then Equation (2) is changed into the classical parabolic equation [17,21]. From this point of view, we extend and supplement the existing results in [6,9,11,17,21,23,24] to some extent.

Remark 5.

In [26,27], authors studied the time–space fractional evolution equations with nonlocal Laplacian by difference methods. We obtain the mild solution by the technology of eigenvalues’ expansions about nonlocal Laplacian operators. Therefore, we provide a different approach to research this type of evolution equations in this paper.

4. Illustrative Example

In this section, a numerical illustration is given to verify the validity and feasibility of the results we develop.

Example 1.

Considering the Caputo-type coupled nonlocal Laplacian evolution equations with proportional delay depicted by

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle {}^c{D}_0^{0.2}\mathbf{u}+{(-\Delta )}^{s_1}\mathbf{u}=-\mathbf{u}+f(x,t,{\mathbf{u}}_{0.3t},{\mathbf{v}}_{0.3t}),}\hfill & \mathrm{in}{\mathbf{B}}_2\left(0\right)\times (0,1);\hfill \\ {\displaystyle {}^c{D}_0^{0.85}\mathbf{v}+{(-\Delta )}^{s_2}\mathbf{v}=-1.5\mathbf{v}+g(x,t,{\mathbf{u}}_{0.3t},{\mathbf{v}}_{0.3t}),}\hfill & \mathrm{in}{\mathbf{B}}_2\left(0\right)\times (0,1);\hfill \\ \mathbf{u}=\mathbf{u}(x,t)=0,\mathbf{v}=\mathbf{v}(x,t)=0,\hfill & \mathrm{on}\partial {\mathbf{B}}_2\left(0\right)\times [0,1],\hfill \end{array}\right.\end{array}$$

(20)

which is endowed with three-point BVCs, as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle 8\mathbf{u}(x,0)-\mathbf{u}(x,0.4)=3\mathbf{u}(x,1),}\hfill \\ 5\mathbf{v}(x,0)-2\mathbf{v}(x,0.75)=\mathbf{v}(x,1),\hfill \end{array}\right.\end{array}$$

(21)

where$x\in {\mathbf{B}}_2\left(0\right)\subset {\mathbb{R}}^2$, ${\mathbf{B}}_2\left(0\right)$is a circle centered at the original point with radius 2, ${(-\Delta )}^{s_1}$ and ${(-\Delta )}^{s_2}$ are nonlocal Laplacian operators with orders $s_1,s_2\in (0,1)$, respectively,

$$f(x,t,{\mathbf{u}}_{0.3t},{\mathbf{v}}_{0.3t})=\frac{x}{1+\sqrt{t}}+\frac{1}{3}sin(5t-1)sin\left|{\mathbf{u}}_{0.3t}\right|+\frac{1}{8}sin(2t+3)\frac{{\mathbf{v}}_{0.3t}^2}{1+{\mathbf{v}}_{0.3t}},$$

$$g(x,t,{\mathbf{u}}_{0.3t},{\mathbf{v}}_{0.3t})=\frac{t}{4+x}+\frac{1}{2}cos(7t-2){\mathbf{u}}_{0.3t}+\frac{1}{7}sin(t+8)sin\left|{\mathbf{v}}_{0.3t}\right|,$$

where $(x,t)\in {\mathbf{B}}_2\left(0\right)\times [0,1]$.

Based on Equation (8), ${\lambda }_1$ and ${\tilde{\lambda }}_1$ are eigenvalues corresponding to ${(-\Delta )}^{s_1}$ and ${(-\Delta )}^{s_2}$, respectively, which can be characterized as follows

$${\lambda }_1=\underset{\mathbf{u}\in {\mathcal{X}}_0^{s_1}\left(\mathrm{\Omega }\right)}{min}{\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\frac{{|\mathbf{u}\left(x\right)-\mathbf{u}\left(y\right)|}^2}{{|x-y|}^{n+2s}}\mathrm{d}x\mathrm{d}y{\left[{\int }_{\mathrm{\Omega }}{\left|\mathbf{u}\left(x\right)\right|}^2\mathrm{d}x\right]}^{-1},$$

$${\tilde{\lambda }}_1=\underset{\mathbf{v}\in {\mathcal{X}}_0^{s_2}\left(\mathrm{\Omega }\right)}{min}{\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\frac{{|\mathbf{v}\left(x\right)-\mathbf{v}\left(y\right)|}^2}{{|x-y|}^{n+2s}}\mathrm{d}x\mathrm{d}y{\left[{\int }_{\mathrm{\Omega }}{\left|\mathbf{v}\left(x\right)\right|}^2\mathrm{d}x\right]}^{-1}.$$

In view of BVCs. (21), ${\lambda }_1^{0.2}=-({\lambda }_1+1)$ and ${\lambda }_1^{0.85}=-({\tilde{\lambda }}_1+1.5)$, we yield

$${\mathcal{A}}_1^{0.2}=8-{\mathbf{E}}_{0.2}\left(0.83{\lambda }_1^{0.2}\right)-3{\mathbf{E}}_{0.2}\left({\lambda }_1^{0.2}\right)>0,$$

$${\mathcal{A}}_1^{0.85}=5-2{\mathbf{E}}_{0.85}\left(0.78{\lambda }_1^{0.85}\right)-{\mathbf{E}}_{0.85}\left({\lambda }_1^{0.85}\right)>0.$$

Hence, condition $\left(H_1\right)$ is valid.

Furthermore,

$$\left|f\right(x,t,{\mathbf{u}}_{0.3t},{\mathbf{v}}_{0.3t}|\le 2+\frac{1}{3}|{\mathbf{u}}_{0.3t}|+\frac{1}{8}|{\mathbf{v}}_{0.3t}|,$$

$$\left|g\right(x,t,{\mathbf{u}}_{0.3t},{\mathbf{v}}_{0.3t}|\le 0.8532+\frac{1}{2}|{\mathbf{u}}_{0.3t}|+\frac{1}{7}|{\mathbf{v}}_{0.3t}|.$$

Correspondingly,

$$\varpi =max\left\{\frac{11}{24}\frac{\sqrt{3}}{{\lambda }_1+1},\frac{9}{14}\frac{\sqrt{3}}{{\tilde{\lambda }}_1+1.5}\right\}<0.7938.$$

Apparently, conditions$\left(H_2\right)$–$\left(H_3\right)$are all fulfilled. By Theorem 2, BVPs (20)–(21) admits at least one mild solution.

In addition, for any$\mathbf{u},\mathbf{v},\tilde{\mathbf{u}},\tilde{\mathbf{v}}\in \mathbb{R}$, it acquires

$$\left|f(x,t,{\mathbf{u}}_{0.3t},{\mathbf{v}}_{0.3t})-f(x,t,{\tilde{\mathbf{u}}}_{0.3t},{\tilde{\mathbf{v}}}_{0.3t})\right|\le \frac{1}{3}|{\mathbf{u}}_{0.3t}-{\tilde{\mathbf{u}}}_{0.3t}|+\frac{1}{8}|{\mathbf{v}}_{0.3t}-{\tilde{\mathbf{v}}}_{0.3t}|,$$

$$\left|g(x,t,{\mathbf{u}}_{0.3t},{\mathbf{v}}_{0.3t})-g(x,t,{\tilde{\mathbf{u}}}_{0.3t},{\tilde{\mathbf{v}}}_{0.3t})\right|\le \frac{1}{2}|{\mathbf{u}}_{0.3t}-{\tilde{\mathbf{u}}}_{0.3t}|+\frac{1}{7}|{\mathbf{v}}_{0.3t}-{\tilde{\mathbf{v}}}_{0.3t}|$$

and

$${\displaystyle \kappa =max\left\{\frac{11}{12}\frac{1}{{\lambda }_1+1},\frac{9}{7}\frac{1}{{\tilde{\lambda }}_1+1.5}\right\}<0.9167.}$$

Consequently, conditions$\left(H_4\right)$ and $\left(H_5\right)$ are constructed. From Theorem 3, BVPs (20) and (21) possess a unique bounded solution.

5. Conclusions and Perspectives

This article considered a class of BVPs of a coupled nonlinear time–space fractional differential equations with three-point boundary conditions. To begin with, we utilized the Fourier analysis technique and the method of eigenvalues’ expansions to transform the differential form into the integral case, and calculated the corresponding Green functions to nonlocal three-point BVPs. Furthermore, the mild solution corresponding to nonlocal three-point BVPs was proposed. In the end, the existence and uniqueness of the mild solution for nonlocal three-point BVPs was investigated by employing Leray–Schauder fixed-point theorem and Banach fixed-point theorem.

Based on the current works, some issues below could be studied further.

(1) 

The order $\alpha \in (1,2)$ could be further learned.

(2) 

Caputo fractional derivative could be replaced by other fractional derivatives.

(3) 

Variable order fractional derivatives should be taken into consideration.

(4) 

Other types of boundary value conditions could be investigated.