TransUFold: Unlocking the structural complexity of short and long RNA with pseudoknots

Yunxiang Wang; Hong Zhang; Zhenchao Xu; Shouhua Zhang; Rui Guo; Yunxiang Wang; Hong Zhang; Zhenchao Xu; Shouhua Zhang; Rui Guo

doi:10.3934/mbe.2023854

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 11: 19320-19340. doi: 10.3934/mbe.2023854

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Research article Special Issues

TransUFold: Unlocking the structural complexity of short and long RNA with pseudoknots

1.
School of Cyber Security and Computer, Hebei University, Baoding, Hebei, China
2.
Information Technology and Electrical Engineering, University of Oulu, Oulu, Finland
3.
College of Life Sciences, Institute of Life Science and Green Development, Hebei University, Baoding, China

Academic Editor: Frank Werner

Received: 24 August 2023 Revised: 08 October 2023 Accepted: 12 October 2023 Published: 17 October 2023

The RNA secondary structure is like a blueprint that holds the key to unlocking the mysteries of RNA function and 3D structure. It serves as a crucial foundation for investigating the complex world of RNA, making it an indispensable component of research in this exciting field. However, pseudoknots cannot be accurately predicted by conventional prediction methods based on free energy minimization, which results in a performance bottleneck. To this end, we propose a deep learning-based method called TransUFold to train directly on RNA data annotated with structure information. It employs an encoder-decoder network architecture, named Vision Transformer, to extract long-range interactions in RNA sequences and utilizes convolutions with lateral connections to supplement short-range interactions. Then, a post-processing program is designed to constrain the model's output to produce realistic and effective RNA secondary structures, including pseudoknots. After training TransUFold on benchmark datasets, we outperform other methods in test data on the same family. Additionally, we achieve better results on longer sequences up to 1600 nt, demonstrating the outstanding performance of Vision Transformer in extracting long-range interactions in RNA sequences. Finally, our analysis indicates that TransUFold produces effective pseudoknot structures in long sequences. As more high-quality RNA structures become available, deep learning-based prediction methods like Vision Transformer can exhibit better performance.

Keywords:

Citation: Yunxiang Wang, Hong Zhang, Zhenchao Xu, Shouhua Zhang, Rui Guo. TransUFold: Unlocking the structural complexity of short and long RNA with pseudoknots[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 19320-19340. doi: 10.3934/mbe.2023854

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Abstract

1. Introduction

Widespread applications of fractional calculus significantly contributed to the popularity of the subject. Fractional order operators are nonlocal in nature and give rise to more realistice and informative mathematical modeling of many real world phenomena, in contrast to their integer-order counterparts, for instance, see ^[13,21,31].

Nonlinear fractional order boundary value problems appear in a variety of fields such as applied mathematics, physical sciences, engineering, control theory, etc. Several aspects of these problems, such as existence, uniqueness and stability, have been explored in recent studies ^{[5,6,7,14,22,24,26,28,32]}.

Coupled nonlinear fractional differential equations find their applications in various applied and technical problems such as disease models ^[8,10,29], ecological models ^[18], synchronization of chaotic systems ^[11,33], nonlocal thermoelasticity ^[30], etc. Hybrid fractional differential equations also received significant attention in the recent years, for example, see ^{[2,3,9,15,16,17,19,20,23]}.

The concept of slits-strips conditions introduced by Ahmad et al. in ^[1] is a new idea and has useful applications in imaging via strip-detectors ^[25] and acoustics ^[27].

In ^[1], the authors investigated the following strips-slit problem:

$^cD^px(t) = f_1(t, x(t)), \ n-1 \lt p\le n, \ t\in [0, 1]\\ x(0) = 0, \ x'(0) = 0, \ x^{\prime\prime}(0) = 0, .... \ x^{(n-2)}(0) = 0, \\ x(\xi) = a_1\int_{0}^{\eta}x(s)ds+a_2\int_{\xi_1}^{1}x(s)ds, \ 0 \lt \eta \lt \xi \lt \xi_1 \lt 1,$

where $^cD^p$ denotes the Caputo fractional derivative of order $p, f_1$ is a given continuous function and $a_1, a_2\in\mathfrak{R}$ .

In 2017, Ahmad et al. ^[4] studied a coupled system of nonlinear fractional differential equations

$^cD^{\alpha}x(t) = f_1(t, x(t), y(t)), \ t\in [0, 1], \ 1 \lt \alpha\le 2, \\ ^cD^{\beta}y(t) = f_2(t, x(t), y(t)), \ t\in [0, 1], \ 1 \lt \beta\le 2,$

supplemented with the coupled and uncoupled boundary conditions of the form:

$x(0) = 0, \ x(a_1) = d_1\int_{0}^{\eta}y(s)ds+d_2\int_{\xi_1}^{1}y(s)ds, \ 0 \lt \eta \lt a_1 \lt \xi_1 \lt 1, \\ y(0) = 0, \ y(a_1) = d_1\int_{0}^{\eta}x(s)ds+d_2\int_{\xi_1}^{1}x(s)ds, \ 0 \lt \eta \lt a_1 \lt \xi_1 \lt 1,$

and

$x(0) = 0, \ x(a_1) = d_1\int_{0}^{\eta}x(s)ds+d_2\int_{\xi_1}^{1}x(s)ds, \ 0 \lt \eta \lt a_1 \lt \xi_1 \lt 1, \\ y(0) = 0, \ y(a_1) = d_1\int_{0}^{\eta}y(s)ds+d_2\int_{\xi_1}^{1}y(s)ds, \ 0 \lt \eta \lt a_1 \lt \xi_1 \lt 1,$

where $^cD^{\alpha}$ and $^cD^{\beta}$ denote the Caputo fractional derivatives of orders $\alpha$ and $\beta$ respectively, $f_1, f_2:[0, 1]\times \mathfrak{R}\times \mathfrak{R}\to \mathfrak{R}$ are given continuous functions and $d_1, d_2$ are real constants.

In this article, motivated by aforementioned works, we introduce and study the following hybrid nonlinear fractional differential equations:

$\begin{array}{l} \left. \begin{array}{cl} ^cD^{\gamma}\left[u(t)-h_1(t, u(t), v(t))\right] = \theta_1(t, u(t), v(t)), \ t\in[0, 1], \ 1 \lt \gamma\le 2, \\ ^cD^{\delta}\left[v(t)-h_2(t, u(t), v(t))\right] = \theta_2(t, u(t), v(t)), \ t\in[0, 1], \ 1 \lt \delta\le 2, \end{array} \right. \end{array}$

(1.1)

equipped with coupled slit-strips-type integral boundary conditions:

$\begin{array}{l} \left. \begin{array}{cl} u(0) = 0, \ u(\eta) = \omega_1\int_{0}^{\xi_1}v(s)ds+\omega_2\int_{\xi_2}^{1}v(s)ds, \ 0 \lt \xi_1 \lt \eta \lt \xi_2 \lt 1, \\ v(0) = 0, \ v(\eta) = \omega_1\int_{0}^{\xi_1}u(s)ds+\omega_2\int_{\xi_2}^{1}u(s)ds, \ 0 \lt \xi_1 \lt \eta \lt \xi_2 \lt 1, \end{array} \right. \end{array}$

(1.2)

where $^cD^{\gamma}$ , $^cD^{\delta}$ denote the Caputo fractional derivatives of orders $\gamma$ and $\delta$ respectively, $\theta_i, h_i:[0, 1]\times \mathfrak{R}\times\mathfrak{R}\to\mathfrak{R}$ are given continuous functions with $h_i(0, u(0), v(0)) = 0, \; i = 1, 2$ and $\omega_1, \omega_2$ are real constants.

We arrange the rest of the paper as follows. In section 2, we present some definitions and obtain an auxiliary result, while section 3 contains the main results for the problems (1.1) and (1.2). Section 4 is devoted to the illustrative examples for the derived results.

2. An auxiliary lemma

Let us first recall some related definitions ^[21].

Definition 2.1. For a locally integrable real-valued function $g_1:{[a, \infty)}\longrightarrow \mathbb{R}$ , we define the Riemann-Liouville fractional integral of order $\sigma > 0$ as

$I^{\sigma}g_1(t) = \frac{1}{\Gamma(\sigma)}\int_{0}^{t}\frac{g_1(\tau)}{(t-\tau)^{1-\sigma}}d \tau, \ \sigma \gt 0,$

where $\Gamma$ is the Euler's gamma function.

Definition 2.2. The Caputo derivative of order $\sigma$ for an $n$ -times continuously differentiable function $g_1:[0, \infty)\to \mathfrak{R}$ is defined by

$^cD^{\sigma}g_1(t) = \frac{1}{\Gamma(n-\sigma)}\int_{0}^{t}(t-\tau)^{n-\sigma-1}g_1^{(n)}(\tau)d \tau, \ n-1 \lt \sigma \lt n, \ n = [\sigma]+1$

where $[\sigma]$ is the integer part of a real number.

Lemma 2.1. For $\chi_i, \Phi_i \in C([0, 1], \mathfrak{R})$ with $\chi_i(0) = 0, i = 1, 2,$ the following linear system of equations:

$\begin{array}{l} \left. \begin{array}{cl} ^cD^{\gamma}\left[u(t)-\chi_1(t)\right] = \Phi_1(t), \ t\in[0, 1], \ 1 \lt \gamma\le 2, \\ ^cD^{\delta}\left[v(t)-\chi_2(t)\right] = \Phi_2(t), \ t\in[0, 1], \ 1 \lt \delta\le 2, \end{array} \right. \end{array}$

(2.1)

equipped with coupled slit-strips-type integral boundary conditions (1.2), is equivalent to the integral equations:

$\begin{array}{l} u(t) = \frac{t}{\eta^2-\Delta^2}\Bigg[\eta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\Phi_2(\tau)d\tau+\chi_2(s)\Bigg)ds \\ + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\Phi_2(\tau)d\tau+\chi_2(s)\Bigg)ds -\int_{0}^{\eta}\frac{(\eta-s)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(s)ds-\chi_1(\eta)\Bigg\} \\ + \Delta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(\tau)d\tau+\chi_1(s)\Bigg)ds \\ + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(\tau)d\tau+\chi_1(s)\Bigg)ds -\int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)}\Phi_2(s)ds-\chi_2(\eta)\Bigg\} \Bigg] \\ + \int_{0}^{t}\frac{(t-s)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(s)ds+\chi_1(t), \end{array}$

(2.2)

$\begin{array}{l} v(t) = \frac{t}{\eta^2-\Delta^2}\Bigg[\Delta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\Phi_2(\tau)d\tau+\chi_1(s)\Bigg)ds \\ + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\Phi_2(\tau)d\tau+\chi_1(s)\Bigg)ds -\int_{0}^{\eta}\frac{(\eta-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(s)ds-\chi_2(\eta)\Bigg\} \\ + \eta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(\tau)d\tau+\chi_2(s)\Bigg)ds \\ + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(\tau)d\tau+\chi_2(s)\Bigg)ds -\int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)}\Phi_2(s)ds-\chi_2(\eta)\Bigg\} \Bigg] \\ + \int_{0}^{t}\frac{(t-s)^{\delta-1}}{\Gamma(\delta)}\Phi_2(s)ds+\chi_2(t), \end{array}$

(2.3)

where it is assumed that

$\begin{array}{l} \Delta = \frac{1}{2}\left(\omega_1\xi_1^2+\omega_2(1-\xi_2^2)\right)\neq 0. \end{array}$

(2.4)

Proof. Solving the fractional differential equations in (2.1), we get

$\begin{array}{l} u(t) = c_0+c_1t+\int_{0}^{t}\frac{(t-s)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(s)ds+\chi_1(t) \end{array}$

(2.5)

and

$\begin{array}{l} v(t) = c_2+c_3 t+\int_{0}^{t}\frac{(t-s)^{\delta-1}}{\Gamma(\delta)}\Phi_2(s)ds+\chi_2(t), \end{array}$

(2.6)

where $c_0, c_1, c_2, c_3$ $\in \mathfrak{R}$ are arbitrary constants.

Using the conditions $u(0) = 0$ and $v(0) = 0$ in (2.5) and (2.6), we find that $c_0 = 0$ and $c_2 = 0$ . Thus (2.5) and (2.6) become

$\begin{array}{l} u(t) = c_1 t+\int_{0}^{t}\frac{(t-s)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(s)ds+\chi_1(t), \end{array}$

(2.7)

$\begin{array}{l} v(t) = c_3 t+\int_{0}^{t}\frac{(t-s)^{\delta-1}}{\Gamma(\delta)}\Phi_2(s)ds+\chi_2(t), \end{array}$

(2.8)

Making use of the coupled slit-strips-type integral boundary conditions given by (1.2) in (2.7) and (2.8) together with the notation (2.4), we obtain a system of equations:

$\begin{array}{l} \omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\Phi_2(\tau)d\tau+\chi_2(s)\Bigg)ds+ \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\Phi_2(\tau)d\tau+\chi_2(s)\Bigg)ds \\ - \int_{0}^{\eta}\frac{(\eta-s)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(s)ds-\chi_1(\eta)) = c_1\eta-\Delta c_3, \end{array}$

(2.9)

$\begin{array}{l} \omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(\tau)d\tau+\chi_1(s)\Bigg)ds + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(\tau)d\tau+\chi_1(s)\Bigg)ds \\ - \int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)}\Phi_2(s)ds-\chi_2(\eta) = c_3\eta-\Delta c_1. \end{array}$

(2.10)

Solving the systems (2.9)–(2.10) for $c_1$ and $c_3$ , we find that

$c_1 = \frac{t}{\eta^2-\Delta^2}\Bigg[\eta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\Phi_2(\tau)d\tau+\chi_2(s)\Bigg)ds \nonumber\\ + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\Phi_2(\tau)d\tau+\chi_2(s)\Bigg)ds -\int_{0}^{\eta}\frac{(\eta-s)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(s)ds-\chi_1(\eta)\Bigg\}\nonumber \\ + \Delta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(\tau)d\tau+\chi_1(s)\Bigg)ds + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(\tau)d\tau+\chi_1(s)\Bigg)ds \nonumber \\ -\int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)}\Phi_2(s)ds-\chi_2(\eta)\Bigg\} \Bigg]$

and

$c_3 = \frac{t}{\eta^2-\Delta^2}\Bigg[\Delta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\Phi_2(\tau)d\tau+\chi_1(s)\Bigg)ds \nonumber\\ + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\Phi_2(\tau)d\tau+\chi_1(s)\Bigg)ds -\int_{0}^{\eta}\frac{(\eta-s)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(s)ds-\chi_2(\eta)\Bigg\} \nonumber \\ + \eta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(\tau)d\tau+\chi_2(s)\Bigg)ds + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\Phi_1(\tau)d\tau+\chi_2(s)\Bigg)ds \nonumber\\ -\int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)}\Phi_2(s)ds-\chi_1(\eta)\Bigg\} \Bigg]$

Inserting the values of $c_1$ and $c_3$ in (2.7) and (2.8) leads to the integral equations (2.2) and (2.3). By direct computation, one can obtain the converse of the lemma. The proof is finished.

3. Main results

Let $\mathcal{W} = \{\tilde{w}(t) : \tilde{w}(t)\in C([0, 1])\}$ be a Banach space equipped with the norm $\left\|\tilde{w}\right\| = \max\{|\tilde{w}(t)|, t\in [0, 1]\}$ , Then the product space $(\mathcal{W}\times\mathcal{W}, \left\|(u, v)\right\|)$ endowed with the norm $\left\|(u, v)\right\| = \left\|u\right\|+\left\|v\right\|$ , $(u, v)\in \mathcal{W}\times\mathcal{W}$ is also a Banach space.

We need the following assumptions to derive the main results.

(A1) Let $\theta_1, \theta_2:[0, 1]\times\mathfrak{R}^2\to\mathfrak{R}$ be continuous and bounded functions and there exists constants $m_i, n_i$ such that, for all $t\in [0, 1]$ and $x_i, y_i\in \mathfrak{R}, i = 1, 2,$

$\left|\theta_1(t, x_1, x_2)-\theta_1(t, y_1, y_2)\right|\le m_1|x_1-y_1|+m_2|x_2-y_2|, \\ \left|\theta_2(t, x_1, x_2)-\theta_2(t, y_1, y_2)\right|\le n_1|x_1-y_1|+n_2|x_2-y_2|.$

(A2) For continuous and bounded functions $h_i,$ i = 1, 2, there exist real constants $\mu_i, \beta_i, \sigma_i > 0$ such that, for all $x_i, y_i \in \mathfrak {R},$ $|h_i(t, x, y)|\le \mu_i$ for all $(t, x, y) \in [0, 1]\times \mathfrak{R}\times\mathfrak{R}$ and

(A3) $\sup\limits_{t\in [0, 1]}\theta_1(t, 0, 0) = \mathcal{N}_1 < \infty \ \ \mbox{and} \, \sup\limits_{t\in [0, 1]}\theta_2(t, 0, 0) = \mathcal{N}_2 < \infty.$

$(A4)$ For the sake of computational convenience, we set

$\mathcal{M}_1 = \frac{1}{\Gamma(\gamma+1)} + \frac{1}{|\eta^2-\Delta^2|}\left[\frac{\eta^{\gamma+1}}{\Gamma(\gamma+1)}+|\Delta||\omega_1|\frac{\xi_1^{\gamma+1}}{\Gamma(\gamma+2)}+\Delta|\omega_2|\frac{1-\xi_2^{\gamma+1}}{\Gamma(\gamma+2) }\right], \\ \mathcal{M}_2 = \frac{1}{|\eta^2-\Delta^2|}\left[\eta|\omega_1|\frac{\xi_1^{\delta+1}}{\Gamma(\delta+2)}+\eta|\omega_2|\frac{1-\xi_2^{\delta+1}}{\Gamma(\delta+2)}+|\Delta|\frac{\eta^{\delta}}{\Gamma(\delta+1)}\right], \\ \mathcal{M}_3 = \frac{1}{|\eta^2-\Delta^2|}\left[\eta|\omega_1|\frac{\xi_1^{\gamma+1}}{\Gamma(\gamma+2)}+\eta|\omega_2|\frac{1-\xi_2^{\gamma+1}}{\Gamma(\gamma+2)}+|\Delta|\frac{\eta^{\gamma}}{\Gamma(\gamma+1)}\right], \\ \mathcal{M}_4 = \frac{1}{\Gamma(\delta+1)} + \frac{1}{|\eta^2-\Delta^2|}\left[\frac{\eta^{\delta+1}}{\Gamma(\delta+1)}+|\Delta||\omega_1|\frac{\xi_1^{\delta+1}}{\Gamma(\delta+2)}+\Delta|\omega_2|\frac{1-\xi_2^{\delta+1}}{\Gamma(\delta+2)}\right], \\ \mathcal{N}_3 = \frac{\eta}{|\eta^2-\Delta^2|}\left[|\omega_1|\xi_1\mu_2+|\omega_2|\mu_2(1-\xi_2)+\mu_1\right]+|\Delta|\left[|\omega_1|\mu_1\xi_1+\mu_1|\omega_2|(1-\xi_2)+\mu_2\right]+\mu_1, \\ \mathcal{N}_4 = \frac{1}{|\eta^2-\Delta^2|}\left[|\Delta||\omega_1|\mu_1\xi_1+|\omega_2|(1-\xi_2)\mu_1+\mu_2\right]+|\eta|\left[|\omega_1|\mu_2\xi_1+|\omega_2|(1-\xi_2)\mu_2+\mu_1\right]+\mu_2, \\ \mathcal{N}_5 = \frac{1}{|\eta^2-\Delta^2|}\left[\eta|\omega_1|\xi_1+|\omega_2|\eta(1-\xi_2)+|\Delta|\right], \\ \mathcal{N}_6 = \frac{1}{|\eta^2-\Delta^2|}\left[|\Delta|\xi_1|\omega_1|+|\Delta||\omega_2|(1-\xi_2)+\eta\right]+1, \\ \mathcal{N}_7 = \frac{1}{|\eta^2-\Delta^2|}\left[|\Delta|+|\eta||\omega_1|\xi_1+|\eta||\omega_2|(1-\xi_2)+|\omega_2|+|\Delta|\right],$

and

$\begin{array}{l} \mathcal{M}_k = \min\Big\{1-(\mathcal{M}_1+\mathcal{M}_3)k_1-(\mathcal{M}_2+\mathcal{M}_4)\lambda_1, \\ 1-(\mathcal{M}_1+\mathcal{M}_3)k_2-(\mathcal{M}_2+\mathcal{M}_4)\lambda_2\Big\}, k_i, \lambda_i\ge 0, \ i = 1, 2. \end{array}$

(3.1)

(A5) $(\mathcal{M}_1+\mathcal{M}_3)(m_1+m_2)+(\mathcal{M}_2+\mathcal{M}_4)(n_1+n_2)+(\mathcal{N}_5+\mathcal{N}_6)(\sigma_1+\sigma_2)+(\mathcal{N}_7+\mathcal{N}_8)(\beta_1+\beta_2) < 1.$

In view of Lemma 1, we define an operator $\mathcal{T}: \mathcal{W}\times\mathcal{W}\to \mathcal{W}\times\mathcal{W}$ associated with the problems (1.1) and (1.2) as follows:

$\begin{array}{l} \mathcal{T}(u, v)(t) = \left( \begin{array}{cl} \mathcal{T}_1(u, v)(t) \\ \mathcal{T}_2(u, v)(t) \end{array} \right), \end{array}$

(3.2)

where

$\mathcal{T}_1(u, v)(t) = \frac{t}{\eta^2-\Delta^2}\Bigg[\eta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\theta_2(\tau, u(\tau), v(\tau))d\tau+h_2(s, u(s), v(s)\Bigg)ds \nonumber\\ + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\theta_2(\tau, u(\tau), v(\tau))d\tau+h_2(s, u(s), v(s)\Bigg)ds \nonumber \\ -\int_{0}^{\eta}\frac{(\eta-s)^{\gamma-1}}{\Gamma(\gamma)}\theta_1(s, u(s), v(s))ds-h_1(\eta, u(\eta), v(\eta))\Bigg\} \nonumber \\ + \Delta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\theta_1(\tau, u(\tau), v(\tau))d\tau+h_1(s, u(s), v(s)\Bigg)ds\nonumber \\ + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\theta_1(\tau, u(\tau), v(\tau))d\tau+h_1(s, u(s), v(s)\Bigg)ds \nonumber\\ -\int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)}\theta_2(s, u(s), v(s))ds-h_2(\eta, u(\eta), v(\eta))\Bigg\} \Bigg]\nonumber \\ + \int_{0}^{t}\frac{(t-s)^{\gamma-1}}{\Gamma(\gamma)}\theta_1(s, u(s), v(s))ds+h_1(t, u(t), v(t))$

and

$\mathcal{T}_2(u, v)(t) = \frac{t}{\eta^2-\Delta^2}\Bigg[\Delta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\theta_2(\tau, u(\tau), v(\tau))d\tau+h_1(s, u(s), v(s)\Bigg)ds \nonumber\\ + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\theta_2(\tau, u(\tau), v(\tau))d\tau+h_1(s, u(s), v(s)\Bigg)ds \nonumber \\ -\int_{0}^{\eta}\frac{(\eta-\tau)^{\gamma-1}}{\Gamma(\gamma)}\theta_1(s, u(s), v(s))ds-h_2(\eta, u(\eta), v(\eta))\Bigg\} \nonumber \\ + \eta\Bigg\{\omega_1\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\theta_1(\tau, u(\tau), v(\tau))d\tau+h_2(s, u(s), v(s)\Bigg)ds\nonumber \\ + \omega_2\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}\theta_1(\tau, u(\tau), v(\tau))d\tau+h_2(s, u(s), v(s)\Bigg)ds \nonumber\\ -\int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)}\theta_2(s, u(s), v(s))ds-h_1(\eta, u(\eta), v(\eta))\Bigg\} \Bigg]\nonumber \\ + \int_{0}^{t}\frac{(t-s)^{\gamma-1}}{\Gamma(\gamma)}\theta_2(s, u(s), v(s))ds+h_2(t, u(t), v(t)).$

Theorem 3.1. Assume that conditions (A1) to (A5) are satisfied. Then there exists a unique solution for the problems (1.1) and (1.2) on $[0, 1]$ .

Proof. In the first step, we establish that $\mathcal{T}\bar{B}_r\subset\bar{B}_r,$ where $\bar{B}_r = \{(u, v)\in\mathcal{W}\times\mathcal{W}:\left\|(u, v)\right\|\le r\}$ is a closed ball with

$r\ge \dfrac{(\mathcal{M}_1+\mathcal{M}_3)\mathcal{N}_1+(\mathcal{M}_2+\mathcal{M}_4)\mathcal{N}_2+\mathcal{N}_3\mu}{1-\left[(\mathcal{M}_1+\mathcal{M}_3)(m_1+m_2)+(\mathcal{M}_2+\mathcal{M}_4)(n_1+n_2) +\mathcal{N}_3(\beta_1+\beta_2)\right]},$

and the operator $\mathcal{T}: \mathcal{W}\times\mathcal{W}\to \mathcal{W}\times\mathcal{W}$ is defined by (3.2). For $(u, v)\in \bar{B}_r$ and $t \in [0, 1]$ , it follows by (A1) that

$\left|\theta_1(t, u(t), v(t))\right| \le \left|\theta_1(t, u(t), v(t))-\theta_1(t, 0, 0)\right| \le m_1||u||+m_2||v||.$

Similarly one can find that $\left|\theta_2(t, u(t), v(t))\right|\le n_1||u||+n_2||v||.$ Then we have

$|\mathcal{T}_1(u, v)(t)| \\ \le\max\limits_{t\in[0, 1]}\Bigg[\frac{t}{|\eta^2-\Delta^2|}\Bigg[\eta\Bigg\{|\omega_1|\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)} |\theta_2(\tau, u(\tau), v(\tau))|d\tau+|h_2(s, u(s), v(s)|\Bigg)ds \nonumber\\ + |\omega_2|\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}|\theta_2(\tau, u(\tau), v(\tau))|d\tau+|h_2(s, u(s), v(s)|\Bigg)ds \nonumber \\ +\int_{0}^{\eta}\frac{(\eta-s)^{\gamma-1}}{\Gamma(\gamma)} |\theta_1(s, u(s), v(s))|ds+|h_1(\eta, u(\eta), v(\eta))|\Bigg\} \nonumber \\ + |\Delta|\Bigg\{|\omega_1|\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}|\theta_1(\tau, u(\tau), v(\tau))|d\tau +|h_1(s, u(s), v(s)|\Bigg)ds\nonumber \\ + |\omega_2|\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)}|\theta_1(\tau, u(\tau), v(\tau))|d\tau +|h_1(s, u(s), v(s)|\Bigg)ds \nonumber\\ +\int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)}|\theta_2(s, u(s), v(s))|ds+|h_2(\eta, u(\eta), v(\eta))|\Bigg\} \Bigg]\nonumber \\ + \int_{0}^{t}\frac{(t-s)^{\gamma-1}}{\Gamma(\gamma)}|\theta_1(s, u(s), v(s))|ds+|h_1(t, u(t), v(t))|\Bigg]\nonumber \\ \le \frac{1}{|\eta^2-\Delta^2|}\Bigg[\eta\Bigg\{|\omega_1|\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)} \left(n_1||u||+n_2||v||+\mathcal{N}_2\right)d\tau+\mu_2\Bigg)ds \nonumber\\ + |\omega_2|\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}| \left(n_1||u||+n_2||v||+\mathcal{N}_2\right)d\tau+\mu_2 \Bigg)ds \nonumber \\ +\int_{0}^{\eta}\frac{(\eta-s)^{\gamma-1}}{\Gamma(\gamma)} \left(m_1||u||+m_2||v||+\mathcal{N}_1\right)d\tau+\mu_1 \Bigg\} \nonumber \\ + |\Delta|\Bigg\{|\omega_1|\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)} \left(m_1||u||+m_2||v||+\mathcal{N}_1\right)d\tau+\mu_1 \Bigg)ds\nonumber \\ + |\omega_2|\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)} \left(m_1||u||+m_2||v||+\mathcal{N}_1\right)d\tau+\mu_1 \Bigg)ds \nonumber\\ +\int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)} \left(n_1||u||+n_2||v||+\mathcal{N}_2\right)ds+\mu_2 \Bigg\}\Bigg]\nonumber \\ + \int_{0}^{t}\frac{(t-s)^{\gamma-1}}{\Gamma(\gamma)} \left(m_1||u||+m_2||v||+\mathcal{N}_1\right)ds + \mu_1 \\ le \frac{1}{|\eta^2-\Delta^2|}\Bigg[\eta|\omega_1|\frac{\xi_1^{\delta+1}}{\Gamma(\delta+2)}+\eta|\omega_2|\frac{1-\xi_2^{\delta+1}}{\Gamma(\delta+2)}+|\Delta|\frac{\eta^{\delta}}{\Gamma(\delta+1)}\Bigg] \left(n_1||u||+n_2||v||+\mathcal{N}_2\right) \nonumber\\ + \Bigg[\frac{1}{|\eta^2-\Delta^2|}\Bigg(\frac{\eta^{\gamma+1}}{\Gamma(\gamma+1)}+|\Delta| |\omega_1|\frac{\xi_1^{\gamma+1}}{\Gamma(\gamma+2)}+|\Delta||\omega_2|\frac{1-\xi_2^{\gamma+1}}{\Gamma(\gamma+2)}\Bigg)+\frac{1}{\Gamma(\gamma+1)}\Bigg] \left(m_1||u||+m_2||v||+\mathcal{N}_1\right) \nonumber\\ + \frac{\eta}{|\eta^2-\Delta^2|}\left(|\omega_1|\mu_2\xi_1+|\omega_2|\mu_2(1-\xi_2)+\mu_1\right)+|\Delta|\left(|\omega_1|\mu_1\xi_1+|\omega_2|\mu_1(1-\xi_2)+\mu_2\right)+\mu_1 \nonumber \\ \le \left(\mathcal{M}_2n_1+\mathcal{M}_1m_1+ \mathcal{M}_2n_2+\mathcal{M}_1m_2\right)r+ \mathcal{M}_2\mathcal{N}_2+\mathcal{M}_1\mathcal{N}_1+\mathcal{N}_3.$

Analogously, one can find that

$|\mathcal{T}_2(u, v)(t)| \le \left(\mathcal{M}_4n_1+\mathcal{M}_3m_1+\mathcal{M}_4n_2+\mathcal{M}_3m_2 \right)r+ \mathcal{M}_4\mathcal{N}_2+\mathcal{M}_3\mathcal{N}_1+\mathcal{N}_4.$

From the foregoing estimates for $\mathcal{T}_1$ and $\mathcal{T}_2$ , we obtain $||\mathcal{T}(u, v)(t)||\le r$ .

Next, for $(u_1, v_1), (u_2, v_2)\in \mathcal{W}\times\mathcal{W}$ and $t\in [0, 1]$ , we get

$\left|\mathcal{T}_1(u_2, v_2)(t)-\mathcal{T}_1(u_1, v_1)(t)\right| \nonumber \\ \le \frac{1}{|\eta^2-\Delta^2|}\Bigg[\eta\Bigg\{|\omega_1|\Bigg(\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)} \left|\theta_2(\tau, u_2(\tau), v_2(\tau))-\theta_2(\tau, u_1(\tau), v_1(\tau))\right|\Bigg)d\tau \nonumber \\ \quad +\left|h_2(s, u_2(s), v_2(s))-h_2(s, u_1(s), v_1(s))\right|ds\Bigg) \nonumber\\ + |\omega_2|\Bigg(\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\delta-1}}{\Gamma(\delta)}\left|\theta_2(\tau, u_2(\tau), v_2(\tau))-\theta_2(\tau, u_1(\tau), v_1(\tau))\right|\Bigg)d\tau \nonumber \\ \quad +\left|h_2(s, u_2(s), v_2(s))-h_2(s, u_1(s), v_1(s))\right|ds\Bigg) \nonumber\\ +\int_{0}^{\eta}\frac{(\eta-s)^{\gamma-1}}{\Gamma(\gamma)} \left|\theta_1(\tau, u_2(\tau), v_2(\tau))-\theta_1(\tau, u_1(\tau), v_1(\tau))\right|ds \nonumber \\ \quad +\left|h_1(\eta, u_2(\eta), v_2(\eta))-h_1(\eta, u_1(\eta), v_1(\eta))\right|\Bigg\} \nonumber\\$

$+ |\Delta|\Bigg\{|\omega_1|\Bigg(\int_{0}^{\xi_1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)} \left|\theta_1(\tau, u_2(\tau), v_2(\tau))-\theta_1(\tau, u_1(\tau), v_1(\tau))\right|\Bigg)d\tau \nonumber \\ \quad +\left|h_1(s, u_2(s), v_2(s))-h_1(s, u_1(s), v_1(s))\right|ds\Bigg) \nonumber\\ + |\omega_2|\Bigg(\int_{\xi_2}^{1} \Bigg(\int_{0}^{s}\frac{(s-\tau)^{\gamma-1}}{\Gamma(\gamma)} \left|\theta_1(\tau, u_2(\tau), v_2(\tau))-\theta_1(\tau, u_1(\tau), v_1(\tau))\right|\Bigg)d\tau \nonumber \\ \quad +\left|h_1(s, u_2(s), v_2(s))-h_1(s, u_1(s), v_1(s))\right|ds\Bigg) \nonumber\\ +\int_{0}^{\eta}\frac{(\eta-s)^{\delta-1}}{\Gamma(\delta)} \left|\theta_1(\tau, u_2(\tau), v_2(\tau))-\theta_1(\tau, u_2(\tau), v_2(\tau)\right|ds \nonumber \\ \quad +\left|h_2(\eta, u_2(\eta), v_2(\eta))-h_2(\eta, u_1(\eta), v_1(\eta))\right|ds\Bigg\}\Bigg]\nonumber\\ + \int_{0}^{t}\frac{(t-s)^{\gamma-1}}{\Gamma(\gamma)} \Bigg(\left|\theta_1(s, u_2(s), v_2(s))-\theta_1(s, u_1(s), v_1(s))\right|\Bigg)ds \nonumber \\ \quad +\left|h_1(t, u_2(t), v_2(t))-h_1(t, u_1(t), v_1(t))\right| \nonumber\\ \le \frac{1}{|\eta^2-\Delta^2|} \Bigg[\eta|\omega_1|\frac{\xi_1^{\delta+1}}{\Gamma(\delta+2)}+\eta|\omega_2|\frac{1-\eta^{\delta+1}}{\Gamma(\delta+2)}+|\Delta|\frac{\eta^{\delta}}{\Gamma(\delta+1)}\Bigg]\times \left(n_1||u_2-u_1||+n_2||v_2-v_1||\right) \nonumber\\ + \Bigg(\frac{1}{|\eta^2-\Delta^2|}\Bigg[\frac{\eta^{\gamma+1}}{\Gamma(\gamma+1)}+|\Delta| |\omega_1|\frac{\xi_1^{\gamma+1}}{\Gamma(\gamma+2)}+|\Delta||\omega_2|\frac{1-\xi_2^{\gamma+1}}{\Gamma(\gamma+2)}\Bigg]\nonumber \\ +\frac{1}{\Gamma(\gamma+1)}\Bigg)\times\left(m_1||u_2-u_1||+m_2||v_2-v_1||\right) \nonumber\\ + \frac{1}{|\eta^2-\Delta^2|}\Bigg[(\eta|\omega_1|\xi_1+\eta|\omega_2|(1-\xi_2)+|\Delta|)(\sigma_1||u_2-u_1||+\sigma_2||v_2-v_1||)\\ +(\eta+|\Delta||\omega_1|\xi_1+|\Delta||\omega_2|(1-\xi_2)+1)(\beta_1||u_2-u_1||+\beta_2||v_2-v_1||)\Bigg] \nonumber\\ le \mathcal{M}_2\left(n_1||u_2-u_1||+n_2||v_2-v_1||\right) + \mathcal{M}_1\left(m_1||u_2-u_1||+m_2||v_2-v_1||\right) \nonumber \\ \ + \mathcal{N}_5\left(\sigma_1||u_2-u_1||+\sigma_2||v_2-v_1||\right)+\mathcal{N}_6\left(\beta_1||u_2-u_1||+\beta_2||v_2-v_1||\right) \nonumber \\ = \left(\mathcal{M}_2n_1+\mathcal{M}_1m_1+\mathcal{N}_5\sigma_1+\mathcal{N}_6\beta_1\right)||u_2-u_1|| + \left(\mathcal{M}_2n_2+\mathcal{M}_1m_2+\mathcal{N}_5\sigma_2\right)||v_2-v_1||)$

which implies that

$\begin{array}{l} \left\|\mathcal{T}_1(u_2, v_2)(t)-\mathcal{T}_1(u_1, v_1)(t)\right\| \\ le \left(\mathcal{M}_2n_1+\mathcal{M}_1m_1+\mathcal{N}_5\sigma_1+\mathcal{N}_6\beta_1+\mathcal{M}_2n_2+\mathcal{M}_1m_2+\mathcal{N}_5\sigma_2+\mathcal{N}_6\beta_2\right)\left(||u_2-u_1||+||v_2-v_1||\right). \end{array}$

(3.3)

Likewise, we have

$\begin{array}{l} \left\|\mathcal{T}_2(u_2, v_2)(t)-\mathcal{T}_2(u_1, v_1)(t)\right\| \\ le \left(\mathcal{M}_4n_1+\mathcal{M}_3m_1+\mathcal{N}_6\sigma_1+\mathcal{N}_7\beta_1+\mathcal{M}_4n_2+\mathcal{M}_3m_2+\mathcal{N}_6\sigma_2+\mathcal{N}_7\beta_2\right)\left(||u_2-u_1||+||v_2-v_1||\right). \end{array}$

(3.4)

From (3.3) and (3.4), we deduce that

$\left\|\mathcal{T}(u_2, v_2)(t)-\mathcal{T}(u_1, v_1)(t)\right\| \nonumber \\ \le \left[(\mathcal{M}_1+\mathcal{M}_3)(m_1+m_2)+(\mathcal{M}_2+\mathcal{M}_4)(n_1+n_2)+(\mathcal{N}_7+\mathcal{N}_8)(\beta_1+\beta_2)+(\mathcal{N}_5+\mathcal{N}_6)(\sigma_1+\sigma_2)\right]\\ \times\left(||u_2-u_1||+||v_2-v_1||\right),$

which shows that $\mathcal{T}$ is a contraction by the assumption (A5) and hence it has a unique fixed point by Banach fixed point theorem. This leads to the conclusion that there exists a unique solution for the problems (1.1) and (1.2) on $[0, 1]$ . The proof is complete.

Now, we discuss the existence of solutions for the problems (1.1) and (1.2) by means of Leray-Schauder alternative (^[12], p. 4).

Theorem 3.2. Assume that there exists real constants $\tilde{k}_0 > 0$ , $\tilde{\lambda}_0 > 0$ and $\tilde{k}_i, \tilde{\lambda}_i\ge 0, \ i = 1, 2$ such that, for any $u_i\in \mathfrak{R}, \ i = 1, 2$

$\left|\theta_1(t, u_1, u_2)\right| \le \tilde{k}_0+\tilde{k}_1|u_1|+\tilde{k}_2|u_2|, \, \, \left|\theta_2(t, u_1, u_2)\right| \le \tilde{\lambda}_0+\tilde{\lambda}_1|u_1|+\tilde{\lambda}_2|u_2|.$

In addition,

$(\mathcal{M}_1+\mathcal{M}_3)\tilde{k}_1+(\mathcal{M}_2+\mathcal{M}_4)\tilde{\lambda}_1 \lt 1, \, \, (\mathcal{M}_1+\mathcal{M}_3)\tilde{k}_2+(\mathcal{M}_2+\mathcal{M}_4)\tilde{\lambda}_2 \lt 1,$

where $\mathcal{M}_i, \ i = 1, 2, 3, 4$ are given in (A4). Then the problems (1.1) and (1.2) have at least one solution on $[0, 1]$ .

Proof. The proof consists of two steps. First we show that the operator $\mathcal{T}:\mathcal{W}\times\mathcal{W}\to\mathcal{W}\times\mathcal{W}$ defined by (3.2) is completely continuous. Observe that continuity of the operator $\mathcal{T}$ follows from that of $\theta_1$ and $\theta_2$ . Consider a bounded set $\Omega \subset \mathcal{W}\times\mathcal{W}$ so that we can find positive constants $l_1$ and $l_2$ such that $\left|\theta_1(t, u(t), v(t))\right|\le l_1$ and $\left|\theta_2(t, u(t), v(t))\right|\le l_2$ for every $(u, v)\in \Omega$ . Hence, for any $(u, v)\in \Omega$ , we find that

$\left|\mathcal{T}_1(u, v)(t)\right| \le \frac{1}{|\eta^2-\Delta^2|} \Bigg[\eta|\omega_1|\frac{\xi_1^{\delta+1}}{\Gamma(\delta+2)}+\eta|\omega_2|\frac{1-\xi_2^{\delta+1}}{\Gamma(\delta+2)}+|\Delta|\frac{\eta^{\delta}}{\Gamma(\delta+1)}\Bigg]l_2 \nonumber\\ + \Bigg\{\frac{1}{|\eta^2-\Delta^2|}\Bigg[\frac{\eta^{\gamma}}{\Gamma(\gamma+1)}+|\Delta| |\omega_1|\frac{\xi_1^{\gamma+1}}{\Gamma(\gamma+2)}+|\Delta||\omega_2|\frac{1-\xi_2^{\gamma+1}}{\Gamma(\gamma+2)}+\frac{1}{\Gamma(\gamma+1)}\Bigg]\Bigg\}l_1 \nonumber\\ +\frac{1}{|\eta^2-\Delta^2|}\Big\{\eta\left[|\omega_1|\xi_1\mu_2+|\omega_2|\mu_2(1-\xi_2)+\mu_1\right]+\eta\mu_1+|\Delta|\left[\mu_1\xi_1|\omega_1|+\mu_1|\omega_2|(1-\xi_2)+\mu_2\right]\Big\}+\mu_1 \\ = \mathcal{M}_2l_2+\mathcal{M}_1l_1+\mathcal{N}_3.$

Thus we deduce that $\left\|\mathcal{T}_1(u, v)\right\|\le \mathcal{M}_2l_2+\mathcal{M}_1l_1+\mathcal{N}_3.$ In a similar fashion, it can be found that $\left\|\mathcal{T}_2(u, v)\right\|\le \mathcal{M}_4l_2+\mathcal{M}_3l_1+\mathcal{N}_4.$ Hence, it follows from the foregoing inequalities that $\mathcal{T}_1$ and $\mathcal{T}_2$ are uniformly bounded and hence the operator $\mathcal{T}$ is uniformly bounded. In order to show that $\mathcal{T}$ is equicontinuous, we take $0 < r_1 < r_2 < 1$ . Then, for any $(u, v)\in \Omega$ , we obtain

$\left|\mathcal{T}_1(u(r_2), v(r_2))-\mathcal{T}_1(u(r_1), v(r_1))\right| \nonumber \\ \le \frac{l_1}{\Gamma(\gamma)}\int_{0}^{r_1}\left[(r_2-s)^{\gamma-1}-(r_1-s)^{\gamma-1}\right]ds+\frac{l_1}{\Gamma(\gamma)}\int_{r_1}^{r_2}(r_2-s)^{\gamma-1}ds \nonumber \\ + \frac{r_2-r_1}{|\eta^2-\Delta^2|} \Bigg\{\Bigg[\eta|\omega_1|\frac{\xi_1^{\delta+1}}{\Gamma(\delta+2)}+\eta|\omega_2|\frac{1-\xi_2^{\delta+1}}{\Gamma(\delta+2)}+|\Delta|\frac{\eta^{\delta}}{\Gamma(\delta+1)}\Bigg]l_2 \nonumber\\ + \Bigg[\frac{\eta^{\gamma+1}}{\Gamma(\gamma+1)}+|\Delta| |\omega_1|\frac{\xi_1^{\gamma+1}}{\Gamma(\gamma+2)}+|\Delta||\omega_2|\frac{1-\xi_2^{\gamma+1}}{\Gamma(\gamma+2)}+\frac{1}{\Gamma(\gamma+2)}\Bigg]l_1+N_3\Bigg\},$

$\left|\mathcal{T}_2(u(r_2), v(r_2))-\mathcal{T}_2(u(r_1), v(r_1))\right| \nonumber \\ \le \frac{l_2}{\Gamma(\delta)}\int_{0}^{r_1}\left[(r_2-s)^{\delta-1}-(r_1-s)^{\delta-1}\right]ds+\frac{l_2}{\Gamma(\delta)}\int_{r_1}^{r_2}(r_2-s)^{\delta-1}ds \nonumber \\ \ \ + \frac{r_2-r_1}{|\eta^2-\Delta^2|} \Bigg\{\Bigg[\eta|\omega_1|\frac{\xi_1^{\delta+1}}{\Gamma(\delta+2)}+\eta|\omega_2|\frac{1-\xi_2^{\delta+1}}{\Gamma(\delta+2)}+\frac{\eta\dot{.}\eta^{\delta+1}}{\Gamma(\delta+1)}\Bigg]l_2 \nonumber\\ + \Bigg[|\Delta|\frac{\eta^{\gamma}}{\Gamma(\gamma+1)}+\eta |\omega_1|\frac{\xi_1^{\gamma+1}}{\Gamma(\gamma+2)}+\eta|\omega_2|\frac{1-\xi_2^{\gamma+1}}{\Gamma(\gamma+2)}\Bigg]l_1+\mathcal{N}_4\Bigg\},$

which imply that the operator $\mathcal{T}(u, v)$ is equicontinuous. In view of the foregoing arguments, we deduce that operator $\mathcal{T}(u, v)$ is completely continuous.

Next, we consider a set $\mathcal{P} = \{(u, v)\in \mathcal{W}\times\mathcal{W} : (u, v) = \lambda\mathcal{T}(u, v), \ 0\le \lambda\le 1\}$ and show that it is bounded. Let us take $(u, v)\in \mathcal{P}$ and $t\in [0, 1]$ . Then it follows from $u(t) = \lambda \mathcal{T}_1(u, v)(t)$ and $v(t) = \lambda \mathcal{T}_2(u, v)(t)$ , together with the given assumptions that

$\|u\|\le \mathcal{M}_1\left(\tilde{k}_0+\tilde{k}_1||u||+\tilde{k}_2||v||\right) +\mathcal{M}_2\left(\tilde{\lambda}_0+\tilde{\lambda}_1||u||+\tilde{\lambda}_2||v||\right)+\mathcal{N}_3, \\ |v\| \le \mathcal{M}_3\left(\tilde{k}_0+\tilde{k}_1||u||+\tilde{k}_2||v||\right)+\mathcal{M}_4\left(\tilde{\lambda}_0+\tilde{\lambda}_1||u||+\tilde{\lambda}_2||v||\right)+\mathcal{N}_4,$

which lead to

$\|u\|+\|v\| \le \left[(\mathcal{M}_1+\mathcal{M}_3)\tilde{k}_0+(\mathcal{M}_2+\mathcal{M}_4)\tilde{\lambda}_0 +\mathcal{N}_3+\mathcal{N}_4\right] \nonumber \\ + \left[(\mathcal{M}_1+\mathcal{M}_3)\tilde{k}_1+(\mathcal{M}_2+\mathcal{M}_4)\tilde{\lambda}_1\right]\|u\|+ \left[(\mathcal{M}_1+\mathcal{M}_3)\tilde{k}_2+(\mathcal{M}_2+\mathcal{M}_4)\tilde{\lambda}_2\right]\|v\|.$

Thus

$\|(u, v)\| \le \frac{(\mathcal{M}_1+\mathcal{M}_3)\tilde{k}_0+(\mathcal{M}_2+\mathcal{M}_4)\tilde{\lambda}_0+\mathcal{N}_3+\mathcal{N}_4}{\mathcal{M}_k},$

where $\mathcal{M}_{k}$ is defined by (3.1). Consequently the set $\mathcal {P}$ is bounded. Hence, it follows by Leray-Schauder alternative (^[12], p. 4) that the operator $\mathcal{T}$ has at least one fixed point. Therefore, the problems (1.1) and (1.2) have at least one solution on $[0, 1].$ This finishes the proof.

4. Examples

Example 4.1. Consider a coupled boundary value problem of fractional differential equations with slit-strips-type conditions given by

$\begin{array}{l} ^cD^{3/2}\left(u(t)-\frac{\sin t|u(t)|}{2(2+|u(t)|)}\right) = \frac{1}{56}u(t)+\frac{2}{7}\frac{v(t)} {1+v(t)}+\frac{5}{7}, \\ ^cD^{5/4}\left(v(t)-\frac{\sin t|v(t)|} {2(2+|v(t)|)}\right) = \frac{1}{39}\frac{|\cos u(t)|} {1+|\cos u(t)|}+\frac{1}{28}\sin v(t)+\frac{3}{7}, \end{array}$

(4.1)

$\begin{array}{l} u(0) = 0, \ u\left(\frac{1}{2}\right) = \int_{0}^{1/5}v(s)ds+\int_{4/5}^{1}v(s)ds, \\ v(0) = 0, \ v\left(\frac{1}{2}\right) = \int_{0}^{1/5}u(s)ds+\int_{4/5}^{1}u(s)ds. \end{array}$

(4.2)

Here $\gamma = \frac{3}{2}$ , $\delta = \frac{5}{4}$ , $\omega_1 = 1$ , $\omega_2 = 1$ , $\eta = \frac{1}{2}$ , $\xi_1 = \frac{1}{5}$ , $\xi_2 = \frac{4}{5}$ . From the given data, we find that $\Delta = -0.11$ , $m_1 = \frac{1}{56}$ , $m_2 = \frac{2}{71}$ , $n_1 = \frac{1}{39}$ , $n_2 = \frac{1}{28}$ , $\mathcal{M}_1\simeq 1.44716$ , $\mathcal{M}_2\simeq 0.51905$ , $\mathcal{M}_3\simeq 0.4046$ , $\mathcal{M}_4\simeq 2.51887$ , $\mathcal{N}_5\simeq 2.94238$ , $\mathcal{N}_6\simeq 7.3223$ , $\mathcal{N}_7\simeq 5.6164$ , $\mathcal{N}_8\simeq 5.2206$ , and $(\mathcal{M}_1+\mathcal{M}_3)(m_1+m_2)+(\mathcal{M}_2+\mathcal{M}_4)(n_1+n_2)+(\mathcal{N}_5+\mathcal{N}_6)(\sigma_1+\sigma_2)+(\mathcal{N}_7+\mathcal{N}_8)(\beta_1+\beta_2)\simeq 0.8030305 < 1$ .

Clearly all the conditions of Theorem 3.1 are satisfied. In consequence, the conclusion of Theorem 3.1 applies to the problems (4.1)–(4.2).

Example 4.2. We consider the problems (4.1)–(4.2) with

$\begin{array}{l} \theta_1(t, u(t), v(t))\ = \frac{1}{2}+\frac{2}{39}tanu(t)+\frac{2}{41}v(t), \, \, \theta_2(t, u(t), v(t))\ = \frac{2}{5}+\frac{1}{9} sinu(t)+\frac{1}{17}v(t). \end{array}$

(4.3)

Observe that

$\left|\theta_1(t, u, v)\right| \le \tilde{k}_0+\tilde{k}_1|u|+\tilde{k}_2|v|, \, \, \left|\theta_2(t, u, v)\right| \le \tilde{\lambda}_0+\tilde{\lambda}_1|u|+\tilde{\lambda}_2|v|$

with $\tilde{k}_0 = \frac{1}{2}$ , $\tilde{k}_1 = \frac{2}{39}$ , $\tilde{k}_2 = \frac{2}{41}$ $\tilde{\lambda}_0 = \frac{2}{5}$ , $\tilde{\lambda}_1 = \frac{1}{9}$ , $\tilde{\lambda}_2 = \frac{1}{17}$ . Furthermore,

$(\mathcal{M}_1+\mathcal{M}_3)\tilde{k}_1+(\mathcal{M}_2+\mathcal{M}_4)\tilde{\lambda}_1 \simeq 0.432507777 \lt 1, (\mathcal{M}_1+\mathcal{M}_3)\tilde{k}_2+(\mathcal{M}_2+\mathcal{M} _4)\tilde{\lambda}_2 \simeq 0.269030756 \lt 1.$

Thus all the conditions of Theorem 3.2 hold true and hence there exists at least one solution for the problems (4.1)–(4.2) with $\theta_1(t, u, v)$ and $\theta_2(t, u, v)$ given by (4.3).

Acknowledgment

The authors thank the reviewers for their useful remarks on our paper.

Conflict of interest

All authors declare no conflicts of interest in this paper.

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