Positive solutions for a system of fractional $q$-difference equations with generalized $p$-Laplacian operators

1.   Introduction

In this paper, we consider the existence of positive solutions for the following system of fractional $q$-difference equations with generalized $p$-Laplacian operators:

where $0 < q < 1, \ 2 < \alpha < 3, \ 0 < \gamma < 1$, $f, \ g:[0, 1]\times[0, +\infty)\times[0, +\infty)\rightarrow [0, +\infty)$ are continuous, $\eta > 0$ and $\zeta > 0$ are two parameters, $\phi_1$ and $\phi_2$ are generalized $p$-Laplacian operators; $D_q^\gamma$ and $D_q^\alpha$ are the fractional $q$-derivative of the Riemann-Liouville type, $D_q$ is the $q$-derivative.

Due to the extensive application of fractional order equations, many scholars have studied the existence of nontrivial solutions of boundary value problems for fractional order differential equations. In recent years, some authors ^[1,2,3,4,5] have considered the existence of positive solutions for some Riemann-Liouville type, tempered type, Caputo type and Hadamard type fractional order differential equations. The authors ^[6,7] have considered the existence of nontrivial solution of Hadamard-type singular fractional differential equations. Some authors ^{[8,9,10,11,12]} have considered the existence of nontrivial solutions for some Riemann-Liouville type, tempered type, Caputo type and Hadamard type fractional order differential equations with $p$-Laplacian operator. Some authors ^[13,14] have considered the eigenvalue problems of fractional differential equations.

Meanwhile, after Jackson ^[15] introduced the $q$-calculus, Al-Salam ^[16] and Agarwal ^[17] developed the fractional $q$-calculus. Many researchers have studied the existence of nontrivial solutions for fractional $q$-difference equations these years. The commonly used methods include fixed point theorems, lower-upper solution method, monotone iterative technique, and so on. For example, in ^[18], Ferreira studied the following boundary value problem of fractional $q$-difference equation:

where $0 < q < 1$, $2 < \alpha\leq 3$, $f:[0, 1]\times[0, \infty)\rightarrow [0, \infty)$ is continuous; $D_q^\alpha$ is the fractional $q$-derivative of the Riemann-Liouville type, $D_q$ is the $q$-derivative. The author obtained the existence of positive solutions about the boundary value problem (1.2) by using Guo-Krasnosel'skii fixed point theorem.

In ^[19], Zhai and Ren applied iterative algorithm and lower-upper solution method to study the following fractional $q$-difference equation:

where $q\in(0, 1), \ \alpha\in (2, 3)$. Under some conditions, the authors obtained some existence results of positive or negative solutions for the boundary value problem (1.3).

In ^[20], Mao et al. used iterative technique to consider the general fractional $q$-difference equation of the problem (1.3) as followings:

where $q\in(0, 1), \ \alpha\in (2, 3)$, $f$ may be singular at $v = 0, t = 0, 1$. The existence of a unique positive solution of the problem (1.4) has been proved.

In ^[21], Jiang and Zhong studied the following fractional $q$-difference equation with $p$-Laplacian operator:

where $\alpha\in (2, 3), \ \beta, q, \eta, \rho\in (0, 1)$, $\phi_p(s) = |s|^{p-2} s$ is the $p$-Laplacian operator $(p > 1)$. The authors used Banach's contraction principle to prove the existence and uniqueness of nontrivial solution of the problem (1.5), and also used Guo-Krasnosel'skii fixed point theorem to obtain the existence of positive solutions of the problem (1.5).

In ^[22], Li et al. considered the following boundary value problem of nonlinear fractional $q$-difference equation:

where $0 < q < 1$, $2 < \alpha < 3$, $0 < \gamma < 1$, $\phi$ is the generalized $p$-Laplacian operator; $D_q^\gamma$ and $D_q^\alpha$ are the fractional $q$-derivative of the Riemann-Liouville type, $D_q$ is the $q$-derivative. The authors used the fixed point theorem to prove the existence of positive solutions of the boundary value problem (1.6).

In ^[23], Wang et al. investigated the following boundary value problem of fractional $q$-difference equation with $\phi$-Laplacian:

where $0 < q < 1$, $2 < \alpha\le3$, $1 < \beta\le2$, $\lambda > 0$ is a parameter, and $D_q^\beta$, $D_q^\alpha$ are the standard Riemann-Liouville fractional $q$-derivatives. The existence and nonexistence of positive solutions of the boundary value problem (1.7) was obtained based on Guo-Krasnosel'skii fixed point theorem on cones.

Currently, many other authors have studied fractional $q$-difference equations. Some authors ^[24,25] have considered the existence of multiple positive solutions for some fractional $q$-difference equations. The authors ^{[26,27,28,29,30]} have considered the existence of nontrivial solutions for fractional $q$-difference equations with various boundary conditions.

Meanwhile, many authors have studied the existence of positive solutions of systems of some fractional differential equations with various boundary conditions, see ^{[31,32,33,34,35]} and the references therein. For example, in ^[31], Li et al. investigated the following system of fractional differential equations with $p$-Laplacian operators:

where ${\alpha _i}, {\gamma _i} \in (0, 1], {\beta _i} \in (1, 2], D_{{0^ + }}^{{\alpha _i}}, D_{{0^ + }}^{{\beta _i}}$ and $D_{{0^ + }}^{{\gamma _i}}$ are the standard Riemann-Liouville derivatives, $i = 1, 2$. The authors derived the conditions for the existence of the maximal and minimal solutions, and obtained the existence of extremal solutions of the system (1.8).

In ^[32], He and Song considered the following system of fractional differential equations with $p$-Laplacian operators and two parameters:

where ${\alpha _i}, {\beta _i} \in (1, 2]$, $D_{{0^ + }}^{{\alpha _i}}$ and $D_{{0^ + }}^{{\beta _i}}$ are the standard Riemann-Liouville derivatives, ${\xi _i}, {\eta _i} \in (0, 1), {a_i}, {b_i} \in [0, 1], i = 1, 2$. $\eta$ and $\zeta$ are positive parameters. By using the Banach contraction mapping principle, the authors gave the existence and uniqueness of the solution for the system (1.9).

In ^[33], Hao et al. investigated the following system of fractional boundary value problems with $p$-Laplacian operators and two parameters:

where ${\alpha _i}\in \left({1, 2} \right], \ {\beta _i} \in \left({3, 4} \right], \ D_{{0^ + }}^{{\alpha _i}}$ and $D_{{0^ + }}^{{\beta _i}}$ are the Riemann-Liouville derivatives, ${\varphi _{{p_i}}}\left(s \right) = {\left| s \right|^{{p_i} - 2}}s, \; {p_i} > 1,$ $f, g \in C\left({\left[{0, 1} \right]\times{{\left[{0, + \infty } \right)}\times[0, +\infty)}, \left[{0, + \infty } \right)} \right)$, $\lambda$ and $\mu$ are positive parameters. By means of Guo-Krasnosel'skii fixed point theorem, the authors obtained various existence results of positive solutions of the system (1.10).

To the best of our knowledge, limited attention has been devoted to the investigation of systems of fractional $q$-difference equations. Inspired by the above literatures, in this paper, we consider the existence of positive solutions for the system of fractional $q$-difference equations (1.1) with generalized $p$-Laplacian operators and two parameters. Under some sublinear and superlinear conditions, we establish some existence results of positive solutions for the system (1.1) by using Guo-Krasnosel'skii fixed point theorem.

2.   Preliminaries

In this section, we firstly introduce Guo-Krasonsel'skill fixed point theorem. Secondly, we give some knowledge about fractional $q$-calculus. In the end, we give some lemmas that are used to prove the main results.

Lemma 2.1. (^[36]) Let $E$ be a Banach space, $P \subset E$ be a cone. Assume that ${\Omega _1} \subset {\rm E}$ and ${\Omega _2} \subset {\rm E}$ are bounded open sets with $\theta \in {\Omega_1} \subset {\Omega_2}$, the operator $A:P \cap ({\bar \Omega_2}\backslash{\Omega_1}) \to P$ is completely continuous. If the following conditions are satisfied:

then the operator $A$ has at least one fixed point in $P \cap ({\bar \Omega_2}\backslash{\Omega_1}).$

In the following, we give some definitions and lemmas about fractional $q$-calculus. For the detailed knowledge about fractional $q$-derivative and fractional $q$-integral, we can refer to ^{[15,16,17,18]}.

Define

The $q$-analogue of the power function ${(a-b)^{(\alpha)}}$ with $\alpha\in R$ is

The $q$-gamma function is defined by

and satisfies $\Gamma_q(x+1) = [x]_q\Gamma_q(x)$.

The $q$-derivative of a function $f$ is defined by

and $q$-derivative of higher order by

The $q$-integral of a function $f$ is given by

where $f$ is defined in the interval $[0, b]$.

Definition 2.1. (^[17,18]) The fractional $q$-integral of the Riemann-Liouville type is defined by $(I_0^qf)(x) = f(x)$, and

Definition 2.2. (^[18]) The fractional $q$-derivative of the Riemann-Liouville type order $\alpha \ge 0$ is defined by $(D_q^0f)(x) = f(x)$ and

where $m = [\alpha].$

The paper ^[37] introduced the definition of a generalized $p$-Laplacian operator $\phi$, which included two important cases $\phi(u) = u$ and $\phi(u) = {\left| u \right|^{p - 2}}u(p \ge 1)$. In this paper, we assume that ${\phi_1}$ and ${\phi_2}$ are generalized $p$-Laplacian operators, namely, ${\phi_1}$ and ${\phi_2}$ satisfy the following condition:

$(\mathrm{H}_0)$ ${\phi_i}:R \to R (i = 1, 2)$ is an odd and increasing homeomorphism, and there exist increasing homeomorphisms ${\psi _1}, {\psi _2}, {\psi _3}, {\psi _4}:(0, \infty) \to (0, \infty)$ such that

In the following, we give the other important lemmas. We list the following fractional $q$-difference equation with homogeneous boundary conditions:

where $\varphi(t) = \frac{{\beta {t^{\alpha-1}}}}{{{{[\alpha-1]}_q}}},$ and $\varphi(t)$ is the unique solution of the following fractional $q$-difference equation with nonhomogeneous boundary condition

where $0 < q < 1, 2 < \alpha < 3, 0 < \gamma < 1$, $\phi$ is a generalized $p$-Laplacian operator.

Lemma 2.2. (^[22]) Let $v(t)$ be a solution of the boundary value problem (2.1). Then $u(t) = v(t)+\varphi(t)$ is the solution of the boundary value problem (1.6).

Lemma 2.3. (^[22]) Let $2 < \alpha < 3, 0 < \gamma < 1, y\in C[0, 1]$ be a given function. Then the following boundary value problem of fractional $q$-difference equation

has a unique solution

where

Lemma 2.4. (^[18]) The function $G(t, qs)$ defined by (2.3) has the following properties:

Lemma 2.5. (^[37]) Let $(\mathrm{H}_0)$ hold. Then we have

Let $E = C[0, 1] \times C[0, 1]$ with the norm $\|(x, y)\|_E = \| x\| + \| y\|$, where $\|x\| = \mathop {\max }\limits_{t \in [0, 1]} |x(t)|$ and $\|y\| = \mathop {\max }\limits_{t \in [0, 1]} |y(t)|$. It is obvious that $E$ is a Banach space.

Set $P = \{ (x, y) \in E:x(t) \ge 0, y(t) \ge 0, \mathop {\min }\limits_{t \in [\theta, 1]} (x(t) + y(t)) \ge {\theta ^{\alpha - 1}}\|(x, y)\| _E\}$, where $\theta$ is a real constant and $0 < \theta < 1$.

By ^[22], we define the following operators ${A_\eta}$ and ${A_\zeta }$:

where $G(t, qs)$ is defined by (2.3). Let $A(x, y) = ({A_\eta }(x, y), {A_\zeta }(x, y)), (x, y)\in E$. Then by the literature ^[22], we easily know that the fixed points of the operator $A$ are solutions of the system of fractional $q$-difference equations (1.1).

Lemma 2.6. $A:P\to P$ is completely continuous.

Proof. For $(x, y)\in P$, we easily have $A_\eta(x, y)(t)\geq0, A_\zeta(x, y)(t)\geq0, \forall t \in [0, 1]$.

By (2.5), for $t\in [\theta, 1]$, we have

Similar to the proof of (2.6), when $t\in[\theta, 1]$, we easily have

By (2.6) and (2.7), we get

From (2.8), we have $A(P)\subset P$.

In the following, we prove $A:P\to P$ is completely continuous. Firstly, we prove $A$ is bounded.

Let $D\subset P$ be bounded. Namely, there exists $K > 0$ such that $\|(x, y)\|_{E} < K, \forall (x, y)\in D$. By the continuity of $f$ and $g$, we know that there exists $M > 0$ such that

By (2.9), (2.10) and Lemma 2.4, for $(x, y)\in D$, we get

By (2.11) and (2.12), we easily know that $A(D)$ is bounded. Secondly, we prove $A$ is equicontinuous on $D$. Namely, for each $(x, y)\in D, \forall \varepsilon > 0, \exists \delta > 0$ such that $|t_{2}-t_{1}| < \delta$, we have

In fact, assume that $0 < t_{1} < t_{2} < 1$, then we have

By (2.13), we have

Similarly, we also have

Hence, by Arzela-Ascoli theorem and the continuity of $f$ and $g$, we have $A:P\to P$ is completely continuous. □

3.   Main results

In the following, we give the denotations that we need in this section.

Let

For ${{f_0}}, {{g_0}}, {{f_\infty }}, {{g_\infty }}\in (0, \infty)$, we denote that

Let

For $\bar{f}_0, \bar{g}_0, \bar{f}_\infty, \bar{g}_\infty \in (0, \infty)$, we give the following denotations:

where

Theorem 3.1. (1) Assume that ${f_0}, {g_0}, {f_\infty }, {g_\infty } \in (0, \infty), {D_1} < {D_2}, {D_3} < {D_4}$, then for each $\eta\in ({D_1}, {D_2})$ and $\zeta \in ({D_3}, {D_4})$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(2) Assume that ${f_0} = 0, {g_0}, {f_{\infty}}, {g_\infty } \in (0, \infty), {D_3} < {D_4}$, then for each $\eta\in ({D_1}, \infty)$ and $\zeta\in ({D_3}, {D_4})$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(3) Assume that ${f_0}, {f_{\infty}}, {g_\infty } \in (0, \infty), {g_0} = 0, {D_1} < {D_2}$, then for each $\eta\in ({D_1}, {D_2})$ and $\zeta\in ({D_3}, \infty)$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(4) Assume that ${f_0} = {g_0} = 0, {f_{\infty}}, {g_\infty } \in (0, \infty)$, then for each $\eta\in ({D_1}, \infty)$ and $\zeta\in ({D_3}, \infty)$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(5) Assume that ${f_0}, {g_0}\in (0, \infty), {f_{\infty}} = \infty$ or ${f_0}, {g_0} \in (0, \infty), {g_{\infty}} = \infty$, then for each $\eta\in (0, {D_2})$ and $\zeta\in (0, {D_4})$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(6) Assume that ${f_0} = 0, {g_0} \in (0, \infty), {g_{\infty}} = \infty$ or ${f_0} = 0, {g_0} \in (0, \infty), {f_{\infty}} = \infty$, then for each $\eta\in (0, \infty)$ and $\zeta\in (0, {D_4})$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(7) Assume that ${f_0} \in (0, \infty), {g_0} = 0, {g_{\infty}} = {\infty}$ or ${f_0} \in (0, \infty), {g_0} = 0, {f_{\infty}} = \infty$, then for each $\eta\in (0, {D_2})$ and $\zeta\in (0, \infty)$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(8) Assume that ${f_0} = {g_0} = 0, {g_{\infty}} = \infty$ or ${f_0} = {g_0} = 0, {f_{\infty}} = \infty$, then for each $\eta\in (0, \infty)$ and $\zeta\in (0, \infty)$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

Proof. In the following, we only prove the Cases (1) and (6).

Case (1): Since $\eta\in ({D_1}, {D_2})$ and $\zeta\in ({D_3}, {D_4})$, we easily know that there exists $\varepsilon > 0$ such that

For the above $\varepsilon > 0$ in (3.1) and (3.2), there exists ${r_1} > 0$ such that

Let $W_{1} = \{(x, y)\in E:\|(x, y)\|_{E} < r_{1} \}$. For $(x, y)\in P \cap \partial W_{1}$, we have

By Lemmas 2.4, 2.5 and (3.1)–(3.4), we have

and

By (3.5) and (3.6), we have

For $\varepsilon > 0$ in (3.1) and (3.2), from the definitions of $f_{\infty}$ and $g_{\infty}$, there exists $\bar{r}_2 > 0$ such that

Take $r_{2} = \max\{2r_1, {\theta^{1-\alpha}\bar{r}_2}\}$. Let $W_{2} = \{(x, y)\in E: \|(x, y)\|_{E} < r_{2}\}$. For $(x, y)\in P \cap \partial W_{2}$, we have $\min_{t\in[\theta, 1]}(x(t)+y(t))\geq \theta^{\alpha-1}\|(x, y)\|_{E} = \theta^{\alpha-1}r_{2} \geq \bar{r}_2.$

By (3.8), (3.9) and Lemmas 2.4 and 2.5, we have

and

From (3.10) and (3.11), we have

By (3.7), (3.12) and Lemma 2.1, we know that $A$ has at least one fixed point $(x, y)\in P \cap (\overline{W}_2\backslash W_{1})$. So the system of fractional $q$-difference equations (1.1) has at least one positive solution. The proof of the case (1) is completed.

Case (6): Since $\eta\in (0, \infty)$ and $\zeta\in (0, {D_4})$, we easily know that there exists $\varepsilon > 0$ such that

Since ${f_0} = 0$ and ${g_0} \in (0, \infty)$, for the above $\varepsilon > 0$ in (3.13), we know that there exists ${r_3} > 0$ such that

Let $W_{3} = \{(x, y)\in E:\|(x, y)\|_{E} < r_{3} \}$. By (3.13), (3.14) and Lemma 2.5, for any $(x, y)\in P \cap \partial W_{3}, t \in [0, 1]$, we have

By (3.16), we have

By (3.13), (3.15) and Lemma 2.5, similar to the proof of (3.16), we easily obtain

By (3.17) and (3.18), we have

Since ${g_\infty } = \infty$, for $\varepsilon > 0$ in (3.13), we know that there exists $\bar{r}_4 > 0$ such that

Take $r_{4} = \max\{3r_{3}, \bar{r}_4{{{\theta ^{ 1-\alpha}}}}\}$. Let $W_{4} = \{(x, y)\in E: \|(x, y)\|_{E} < r_{4}\}$. For any $(x, y)\in P \cap \partial W_{4}$, we can easily know that

Hence, by (3.20), (3.21) and Lemma 2.5, for any $(x, y)\in P \cap \partial W_{4}$, we have

By (3.22), we have

Hence, by (3.19), (3.23) and Lemma 2.1, we can obtain that $A$ has at least one fixed point $(x, y) \in P\cap(\overline{W}_4 \backslash {W_3})$. So the system of fractional $q$-difference equations (1.1) has at least one positive solution. □

Theorem 3.2. (1) Assume that $\bar{f}_0, \bar{g}_0, \bar{f}_\infty, \bar{g}_\infty\in (0, \infty),$ and $Z_{1} < Z_{2}, Z_{3} < Z_{4}$, then for each $\eta\in ({Z_1}, {Z_2})$ and $\zeta \in ({Z_3}, Z_{4})$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(2) Assume that $\bar{f}_0, \bar{g}_0, \bar{f}_\infty \in (0, \infty), \bar{g}_\infty = 0,$ and $Z_{1} < Z_{2}$, then for each $\eta\in ({Z_1}, {Z_2})$ and $\zeta \in ({Z_3}, \infty)$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(3) Assume that $\bar{f}_0, \bar{g}_0, \bar{g}_\infty \in (0, \infty), \bar{f}_\infty = 0,$ and $Z_{3} < Z_{4}$, then for each $\eta\in ({Z_1}, \infty)$ and $\zeta \in ({Z_3}, Z_{4})$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(4) Assume that $\bar{f}_0, \bar{g}_0 \in (0, \infty), \bar{f}_\infty = \bar{g}_\infty = 0,$ then for each $\eta\in ({Z_1}, \infty)$ and $\zeta \in ({Z_3}, \infty)$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(5) Assume that $\bar{f}_\infty, \bar{g}_\infty\in (0, \infty), \bar{f}_0 = \infty$ or $\bar{f}_\infty, \bar{g}_\infty \in (0, \infty), \bar{g}_0 = \infty$, then for each $\eta\in (0, Z_{2})$ and $\zeta \in (0, {Z_4})$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(6) Assume that $\bar{f}_0 = \infty, \bar{g}_\infty = 0, \bar{f}_\infty \in (0, \infty)$ or $\bar{f}_\infty \in (0, \infty), \bar{g}_\infty = 0, \bar{g}_0 = \infty$, then for each $\eta\in (0, {Z_2})$ and $\zeta \in (0, \infty)$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(7) Assume that $\bar{f}_0 = \infty, \bar{g}_\infty\in (0, \infty), \bar{f}_\infty = 0$ or $\bar{g}_\infty \in (0, \infty), \bar{g}_0 = \infty, \bar{f}_\infty = 0$, then for each $\eta\in (0, \infty)$ and $\zeta \in (0, Z_{4})$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

(8) Assume that $\bar{f}_0 = \infty, \bar{f}_\infty = \bar{g}_\infty = 0$ or $\bar{f}_\infty = \bar{g}_\infty = 0, \bar{g}_0 = \infty$, then for each $\eta\in (0, \infty)$ and $\zeta \in (0, \infty)$, the system of fractional $q$-difference equations (1.1) has at least one positive solution.

Proof. We will only prove the Cases (1) and (6). Since the other proofs are similar, so we omit.

We firstly prove the Case (1). Since $\eta\in ({Z_1}, {Z_2})$ and $\zeta \in ({Z_3}, Z_{4})$, there exists $\varepsilon > 0$ such that

From the definitions of $\bar{f}_0$ and $\bar{g}_0$, we easily know that there exists $R_{1} > 0$ such that

Let $W_1 = \{(x, y)\in E: \|(x, y)\|_{E} < R_{1}\}$. By (3.24), (3.25) and Lemma 2.5, for any $(x, y)\in P \cap \partial W_1$, we can get

By (3.24), (3.26) and Lemma 2.5, for any $(x, y) \in P \cap \partial W_1$, we have

By (3.27) and (3.28), we have

Let $F(t, u) = \mathop {\max }\limits_{_{0 \le x + y \le u}} f(t, x + \varphi, y + \varphi)$, $G^*(t, u) = \mathop {\max }\limits_{_{0 \le x + y \le u}} g(t, x + \varphi, y + \varphi)$. Then we have

Similar to the proof of ^[33], we know that

Clearly, we know that there exists $\overline{R}_2 > 0$ such that

Hence, we have

Let $R_{2} =$ max $\{2R_{1}, \overline{R}_2\}$, and $W_2 = \{(x, y)\in E: \|(x, y)\|_{E} < R_{2}\}$, for any $(x, y) \in P \cap \partial W_2$, we get

By (3.30)–(3.32), for any $(x, y)\in P \cap \partial W_2$, we have

and

By (3.33) and (3.34), we have

By (3.29), (3.35) and Lemma 2.1, we know that $A$ has at least one fixed point $(x, y)\in P \cap (\overline{W}_2\setminus W_1)$, so the system of fractional $q$-difference equations (1.1) has at least one positive solution. The proof of the case (1) is completed.

In the following, we prove the Case (6). Since $\bar{f}_0 = \infty, \bar{f}_\infty \in (0, \infty), \bar{g}_\infty = 0$, we can easily know that there exist $\varepsilon > 0$ and ${R_3} > 0$ such that

and

Let $W_{3} = \{(x, y)\in E: \|(x, y)\|_{E} < R_{3}\}$. For $t\in [\theta, 1]$, $(x, y)\in P \cap \partial W_{3}$, we easily know that

By (3.36) and (3.38), we have

So by (3.39), we have

Similar to the proof of ^[33], we obtain

So we know that for above $\varepsilon > 0$ in (3.36) and (3.37), there exists $\overline{R}_4 > 0$ such that

so we have

Let $R_{4} =$ max $\{2R_{3}, \overline{R}_4\}$ and $W_{4} = \{(x, y)\in E: \|(x, y)\|_{E} < R_{4}\}$. We easily have

Hence, for any $t\in[0, 1]$ and $(x, y)\in P \cap \partial {W_{4}}$, we get

and

So by (3.41) and (3.42), we have

By (3.40), (3.43) and Lemma 2.1, we know that $A$ has at least one fixed point $(x, y)\in P \cap (\overline{W}_4\setminus W_{3})$. Hence the system of fractional $q$-difference equations (1.1) has at least one positive solution. □

4.   Applications

Example 4.1. We consider the following system of fractional $q$-difference equations:

where $q = \frac{1}{2}$, ${\phi _1}(u) = u, {\phi _2}(u) = |u|^{-1}u$. Take $f(t, x, y) = t(x+y-2\varphi(t))^2$, $g(t, x, y) = t(x+y)$, where $\varphi (t) = \frac{{4 + \sqrt 2 }}{7}{t^{\frac{3}{2}}}$. By a simple calculation we get

and ${\psi _1}(x) = {\psi _2}(x) = x$, ${\psi _3}(x) = {\psi _4}(x) = 1$, $D_4 \approx0.6465$.

Then, for each $\eta\in (0, \infty)$ and $\zeta\in (0, 0.6465)$, by Theorem 3.1 Case (6) we obtain that the system (4.1) has at least one positive solution.

Example 4.2. We consider the following system of fractional $q$-difference equations:

where $q = \frac{1}{2}$, ${\phi _1}(u) = u, {\phi _2}(u) = |u|^{-1}u$. Take $f(t, x, y) = \frac{{t(x + y - 2\varphi (t))}}{{\arctan (x + y - 2\varphi (t))}}$, $g(t, x, y) = \frac{t}{{x + y}}$, where $\varphi (t) = \frac{{4 + \sqrt 2 }}{7}{t^{\frac{3}{2}}}$. By a simple calculation we get

and ${\psi _1}(x) = {\psi _2}(x) = x$, ${\psi _3}(x) = {\psi _4}(x) = 1$, ${\Gamma _{\frac{1}{2}}}(\frac{5}{2}) \approx 1.1906$, ${\Gamma _{\frac{1}{2}}}(\frac{3}{2}) \approx 0.9209$, $M_1 \le 0.2991$, $Z_2 \ge 2.6259$.

Then, for each $\eta\in (0, 2.6259)$ and $\zeta\in (0, \infty)$, by Theorem 3.2(6) we obtain that the system (4.2) has at least one positive solution.

5.   Conclusions

The system of fractional q-difference equations plays an important role in the study of many fields, such as quantum mechanics, mathematical physics equations and so on, for example, see ^{[16,17,24,35]} and the references therein. In ^[35], by using some classical fixed point theorems, the authors studied the existence of nontrivial solutions of a system of fractional $q$-difference equations with Riemann-Stieltjes integrals conditions. In this paper, we investigate the existence of positive solutions for a system of fractional $q$-difference equations with generalized $p$-Laplacian operators and two parameters. The system in this paper is different from that of ^[35]. We give some assumptions which are combinations of superlinearity and sublinearity of the nonlinear terms $f$ and $g$. Under those assumptions, by using Guo-Krasnosel'skii fixed point theorem, we obtain some existence results of positive solutions in terms of different values of the parameters $\eta$ and $\zeta$. In fact, since the system studied in this paper contains generalized $p$-Laplacian operators, the obtained results in this paper can enrich the relevant knowledge of theories for the system of fractional $q$-difference equations and expand the range of the possible applications. However, this study still has certain limitations, as we only investigated the existence of positive solutions. In the future, some further work can continue to be considered such as the uniqueness and multiplicity of positive solutions and iterative sequences of positive solutions, the case where the nonlinear terms may be changing sign or the generalized $p$-Laplacian operator becomes a $p(t)$ -Laplacian operator, etc.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (11801322; 12371173) and Shandong Natural Science Foundation(ZR2021MA064).

The authors would like to thank reviewers for their valuable comments, which help to enrich the content of this paper.

Conflict of interest

The authors declare there is no conflicts of interest.