Existence and uniqueness criteria for nonlinear quantum difference equations with $p$-Laplacian

1.   Introduction

Quantum calculus initiated at the beginning of the twentieth century, is the calculus without the limits, see ^[1,2,3,4,5]. As well as known, many pratical problems can be reduced to the $q$-difference equations boundary value problems (BVPS), therefore the $q$-difference equations become a very interesting field in difference equations and gain widespread attention and considerable importance. In the last few decades, it has evolved into a multidisciplinary subject and plays an important role in several fields of physics and mathematics such as cosmic strings and black holes, conformal quantum mechanics, nuclear and high energy physics, combinations, orthogoral polynomials, basic hypergeometric functions and so on, see ^[6,7,8] and the reference therein.

At present, the research on solvability of linear $q$-difference equations has made great progress, see ^[3,4,5,9] and the reference therein. Relative to the linear $q$-difference equations, nonlinear $q$-difference equations can describe some phenomenon more accurately in the natural world. In 2010, M. EL-Shanhed and H. A. Hassan firstly studied the existence of positive solutions for nonlinear $q$-difference equations by Kransnoselskii's fixed point theorem in cone ^[10]. In the last years, many scholars have denoted to doing some research on this kind of BVPS, such as local or nonlocal problems for nonlinear $q$-difference equations, impulsive $q$-difference equations, fractional $q$-difference equations and so on, and obtained some results, see ^{[11,12,13,14,15,16,17,18,19,20,21,22,23,24]} and the reference therein. However, many aspects of theory for nonlinear $q$-difference equations BVPS need to be explored.

In 1945, Leibenson ^[25] obtained the following nonlinear diffusion equation when studying the one-dimensional polytropic turbulence problem of gas passing through a porous medium,

when $m > 1$, the equation is the porous medium equation ^[26], when $0 < m < 1$, the equation becomes the rapid diffusion equation, and when $m = 1$, it is the heat equation. However, when $m = 1$ and $\mu \neq 1$, the above equations often applied to the study of non-Newtonian fluids ^[27]. In view of the importance of such equations, they are abstracted into the following $p$-Laplacian equation

where

Obviously, when $p = 2$, the p-Laplacian equation degenerates into a classical second-order differential equation. In ^[28], Anderson et al. studied the existence of at least one positive solution for the time scale delta-nabla dynamic equation $(g(u^{\Delta}))^{\nabla}+c(t)f(u) = 0$, with boundary conditions, $u(a)-B_{0}(u^{\Delta}(v)) = 0$ and $u^{\Delta}(b) = 0$. Here, $g(z) = |z|^{p-2}z$ for $p > 1$, $v\in (a, b).$ In ^[29], Dogan considers the $p$-Laplacian dynamic equation on time scales

or

and obtained the existence of at least three positive solutions for the boundary-value problem by using the Avery and Peterson fixed-point theorem. Here, $\mathbb{T}$ is s a time scale, namely any closed subset of $\mathbb{R}$. Significantly, let $0 < q < 1, \mathbb{T} = I_{q}, T = 1$, then

To the best of our knowledge, limited work has been done in the BVPs nonlinear $q$-difference equations with $p$-Laplacian operator. In particular, there is little research on uniqueness of solutions for BVPS of nonlinear $q$-difference equations with $p$-Laplacian. To fill this gap, we investigate the existence and uniqueness of solutions for nonlinear second-order $q$-difference equations with $p$-Laplacian operator

where $\Phi_p(s) = |s|^{p-2}s(p > 1)$ is an increasing function, $\Phi_{p} ^{-1}(s) = \Phi_{\overline{p}}(s), \; \frac{1}{p}+\frac{1}{\overline{p}} = 1$, $f(t, x, y)\in C(I_q\times R^{2}, R), \; I_q = \{q^{n}:n\in N\}\bigcup \{0, 1\}$, $\alpha\neq0, \; a(t)\in C(I_q, R^{+}), \; R^{+} = [0, \infty).$

Throughout this paper, we adopt the following assumptions.

$(H_{1})$ $f(t, x, y)\in C(I_{q}\times R^{2}, R)$ and there exists $g(t)\in C(I_{q}, R^{+})$ such that

for $t\in I_{q}, \; (x, y)\in R^{2}$.

$(H_{2})$ $f(t, x, y)\in C(I_{q}\times R^{2}, R)$ and there exist $L_{1}(t), L_{2}(t)\in C(I_{q}, R^{+})$ such that

for $t\in I_{q}, \; (x_{1}, y_{1}), (x_{2}, y_{2})\in R^{2}$.

$(H_{3})$ $f(t, x, y)\in C(I_{q}\times R^{2}, R)$ and there exist two positive constants $L_{1}, \; L_{2}$ such that

for $t\in I_{q}, \; (x_{1}, y_{1}), (x_{2}, y_{2})\in R^{2}$.

$(H_{4})$ $f(t, x, y)\in C(I_{q}\times R^{2}, R)$ and there exist $L_{1}(t), L_{2}(t)\in L^1(I_{q}, R^{+})$ such that

for $t\in I_{q}, \; (x_{1}, y_{1}), (x_{2}, y_{2})\in R^{2}$.

$(H_{5})$ $f(t, x, y)\in C(I_{q}\times R^{2}, R)$ and there exists $h(t)\in C(I_{q}, R^{+})$ and $h\neq 0$ such that

for $t\in I_{q}, \; (x, y)\in R^{2}$.

$(H_{6})$ $f(t, x, y)\in C(I_{q}\times R^{2}, R)$ and there exist three functions $g(t), h(t), r(t)\in L^{1}(I_{q}, R^{+})$ such that

for $t\in I_{q}, \; (x, y)\in R^{2}$.

$(H_{7})$ $f(t, x, y)\in C(I_{q}\times R^{2}, R)$ and there exist three positive real numbers $a, b, c$ such that

for $t\in I_{q}, \; (x, y)\in R^{2}$.

For convenience, we denote

We aim to obtain the existence and uniqueness of solutions for the BVP (1.1) via Banach's Contraction Mapping Principle, Leray-Schauder Nonlinear Alternative theorem and Schaefer's fixed point theorem. The organization of this paper is as follows. In Section 2, we provide some definitions and preliminary lemmas which are key tools for our main results. In Section 3, we give and prove our main results. Finally, in Section 4, we give some examples to illustrate how the main results can be used in practice.

2.   Preliminary results

In this section, we provide some definitions of $q$-calculus ^[9,30] and preliminary lemmas which are key tools for our main results.

Definition 2.1. For $0 < q < 1$, we define the $q$-derivative of a real-value function $f$ as

Note that $\lim\limits_{q\rightarrow 1^{-}}D_{q}f(t) = f'(t).$

Definition 2.2. The higher order $q$-derivatives are defined inductively as

Definition 2.3. The $q$-integral of a function $f$ defined in the interval $[a, b]$ is given by

and for $a = 0$, we denote

Then

Notice that

Remark 2.4. In the limit $q\rightarrow 1^{-}$, the above results correspond to their counterparts in standard calculus.

Definition 2.5. (^[15]) $f:I_{q}\times R^{2}\rightarrow R$ is called an S-Carath$\acute{e}$odory function if and only if

(ⅰ) for each $(u, v)\in R^{2}, t\mapsto f(t, u, v)$ is measurable on $I_{q}$;

(ⅱ) for a.e. $t\in I_{q}, (u, v)\mapsto f(t, u, v)$ is continuous on $R^{2}$;

(ⅲ) for each $r > 0$, there exists $\varphi_{r}(t)\in L^{1}(I_{q}, R^{+})$ with $t\varphi_{r}(t)\in L^{1}(I_{q}, R^{+})$ on $I$ such that $\max\{|u|, |v|\}\leq r$ implies $|f(t, u, v)|\leq \varphi_{r}(t)$, for a.e. $I$, where $L^{1}(I_{q}, R^{+}) = \{u\in C_{q}:\int_{0}^{1}u(t)d_{q}t$ exists$\}$, and normed by $\|u\|_{L^{1}} = \int_{0}^{1}|u(t)|d_{q}t$ for all $u\in L^{1}(I_{q}, R^{+}).$

Theorem 2.6. (Nonlinear alternative for single value maps ^[31]) Let $E$ be a Banach space, $C$ a closed, convex subset of $E$, $U$ an open subset of $C$ and $0\in U$. Suppose that $F: \overline{U}\rightarrow C$ is a continuous, compact (that is $F(\overline{U})$ is a relatively compact subset of $C$) map. Then either

(ⅰ) $F$ has a fixed point in $\overline{U}$, or

(ⅱ) there is a $u\in\partial U$ (the boundary of $U$ in $C$) and $\lambda\in(0, 1)$ with $u = \lambda F(u).$

Theorem 2.7. (Schaefer's fixed point theorem ^[32]) Let $T$ be a continuous and compact mapping of a Banach space $X$ into itself, such that the set $E = \{x\in X:x = \lambda Tx$ for some $0\leq \lambda \leq 1\}$ is bounded. Then $T$ has a fixed point.

Finally, we list below the following basic properties of the $p$-Laplacian operator which will be used in the sequel.

Lemma 2.8. (^[33]) Assume $\Phi_p(s)$ is a $p$-Laplacian operator, then

and

Lemma 2.9. (^[34]) (1) If $1 < p < 2, \; uv > 0$ and $|u|, \; |v|\geq m > 0$, then

(2) If $p > 2$ and $|u|, \; |v|\leq M$, then

3.   Existence and uniqueness results

In this section, we establish some existence and uniqueness criteria for nonlinear $q-$difference equation (1.1).

Lemma 3.1. Let $y\in C[0, 1]$, then the BVP

has a unique solution

Proof. Integrating the $q$-difference equation from $t$ to $1$, we get

By the BC $D_{q}x(1) = 0$, we have

Integrate (3.2) from $0$ to $t$, we have

Using the BC $\alpha x(0)-\beta D_{q}x(0) = 0$ in (3.3), we find that

Hence, the BVP (3.1) has a unique solution

This completes the proof.

We consider the Banach space $C_{q} = C(I_{q}, R)$ equipped with standard norm $\|x\| = \max\{\|x\|_{\infty}, \|D_{q}x\|_{\infty}\},$ and $\|\cdot\|_{\infty} = \sup\{\|\cdot\|, t\in I_q\}, \; x\in C_{q}$.

Define an integral operator $T:C_{q}\rightarrow C_{q}$ by

Obviously, $T$ is well defined and $x\in C_{q}$ is a solution of BVP (1.1) if and only if $x$ is a fixed point of $T$.

First, we give the uniqueness result based on Banach's Contraction Mapping Principle.

Theorem 3.2. Suppose $1 < p\leq 2$, and $(H_1), (H_2)$ hold. Then the BVP (1.1) has a unique solution provided that $\Gamma < 1$, where $\Gamma = (\Big|\frac{\beta}{\alpha}\Big|+1)\frac{1}{p-1}M^{\frac{2-p}{p-1}}LA$.

Proof. Let us set $\sup_{t\in I_q}|f(t, 0, 0)| = M_{0}$ and choose

where $\Gamma\leq\delta < 1$ and $\Gamma^* = (\Big|\frac{\beta}{\alpha}\Big|+1)\Phi_p^{-1}(M_0A)$. Now, we show that $TB_{r}\subset B_{r}$, where $B_{r} = \{x\in C_{q}:\|x\|\leq r\}$. For each $x\in B_{r}$, we have

and

Hence, we obtain that $\|Tx\|\leq r$, so $TB_{r}\subset B_{r}.$

Now, for $x, y\in C_{q}$ and for each $t\in I_q$, we have

and

Therefore, we obtain that $\|Tx-Ty\| < \|x-y\|,$ so $T$ is a contraction. Thus, the conclusion of the theorem follows by Banach's contraction mapping principle. This completes the proof of Theorem 3.2.

Corollary 3.3. Suppose $1 < p\leq 2$, and $(H_1), (H_3)$ hold. Then the BVP(1.1) has a unique solution provided that $\Gamma_1 < 1$, where $\Gamma_{1} = \Big(\Big|\frac{\beta}{\alpha}\Big|+1\Big)\frac{1}{p-1}M^{\frac{2-p}{p-1}}(L_{1}+L_{2})A$.

Corollary 3.4. Suppose $1 < p\leq 2$, and $(H_1), (H_4)$ hold. Then the BVP(1.1) has a unique solution provided that $\Gamma_{2} < 1$, where $\Gamma_{2} = \Big(\Big|\frac{\beta}{\alpha}\Big|+1\Big)\frac{1}{p-1}M^{\frac{2-p}{p-1}}B$.

Theorem 3.5. Suppose $p > 2$, and $(H_2), (H_5)$ hold. Then the BVP(1.1) has a unique solution provided that $\Gamma_{3} < 1$, where $\Gamma_{3} = \Big(\Big|\frac{\beta}{\alpha}\Big|+1\Big)\frac{1}{p-1}m^{\frac{2-p}{p-1}}LA$.

Proof. It is similar to the proof of Theorem 3.1.

Corollary 3.6. Suppose $p > 2$, and $(H_3), (H_5)$ hold. Then the BVP(1.1) has a unique solution provided that $\Gamma_{4} < 1$, where $\Gamma_{4} = \Big(\Big|\frac{\beta}{\alpha}\Big|+1\Big)\frac{1}{p-1}m^{\frac{2-p}{p-1}}(L_1+L_2)A$.

Corollary 3.7. Suppose $p > 2$, and $(H_4), (H_5)$ hold. Then the BVP(1.1) has a unique solution provided that $\Gamma_{5} < 1$, where $\Gamma_{5} = \Big(\Big|\frac{\beta}{\alpha}\Big|+1\Big)\frac{1}{p-1}m^{\frac{2-p}{p-1}}B$.

The next existence result is based on Leray-Schauder nonlinear alternative theorem.

Lemma 3.8. Let $f:I\times R^{2}\rightarrow R$ be an S-Carathéodory function. Then $T:C_{q}\rightarrow C_{q}$ is completely continuous.

Proof. The proof consists of several steps.

(ⅰ) $T$ maps bounded sets into bounded sets in $C_{q}$. Let $B_{r} = \{x\in C_{q}:\|x\|\leq r\}$ be a bounded set in $C_{q}$ and $x\in B_{r}$. Then we have

and

Thus $\|Tx\|\leq\max\{\|Tx\|_{\infty}, \|D_{q}Tx\|_{\infty}\}\leq\Big(\Big|\frac{\beta}{\alpha}\Big|+1\Big)K.$

(ⅱ) T maps bounded sets into equicontinuous sets of $C_{q}$.

Let $t_{1}, t_{2}\in I_q, t_{1} < t_{2},$ and $B_{r}$ be a bounded set of $C_{q}$ as before. Then for $x\in B_{r}$, we have

and by the Lebesgue dominated convergence theorem, we have

As a consequence of the Arzel$\acute{a}$-Ascoli theorem, we can conclude that $T:C_{q}\rightarrow C_{q}$ is completely continuous. This proof is be completed.

Theorem 3.9. Let $f:I\times R^{2}\rightarrow R$ be an S-Carathéodory function. Suppose further that there exists a real number $N > 0$ such that

holds, where

Then the BVP (1.1) has at least one solution.

Proof. In view of Lemma 3.7, we obtain that $T:C_{q}\rightarrow C_{q}$ is completely continuous. Let $\lambda\in(0, 1)$, and $x = \lambda Tx$. Then, for $t\in I_q$, we have

and

Hence, consequently

Therefore, there exists $N > 0$ such that $\|x\|\neq N$. Let us set $U = \{x\in C_{q}:\|x\| < N \}$. Note that the operator $T:\overline{U}\rightarrow C_{q}$ is completely continuous. From the choice of $U$, there is no $x\in \partial U$ such that $x = \lambda Tx$ for some $\lambda\in(0, 1)$. Consequently, by Theorem 2.6, we deduce that $T$ has a fixed point $x\in \overline{U}$ which is a solution of the problem (1.1). This completes the proof.

Finally, we give the existence result is based on Schaefer's fixed point theorem.

Theorem 3.10. Assume that $(H_6)$ holds. Suppose further $\Lambda < 1$ holds, then BVP(1.1) has at least one solution, where $\Lambda = \Big(\Big|\frac{\beta}{\alpha}\Big|+1\Big)2^{\frac{1}{p-1}}\Phi_p ^{-1}(\int_0^1a(s)[g(s)+h(s)]d_{q}s)$.

Proof. It is easy to know that $T$ is completely continuous. Next, we show that the set

$E = \{x\in C_q:x = \lambda Tx$ for $0\leq\lambda\leq 1\}$ is bounded, which is independent of $\lambda$. Let $x\in E$, then $x(t) = \lambda Tx(t)$ for $0\leq\lambda\leq 1$. By $(H_6)$, for each $t\in I_q$, we have

and

where $\Lambda^* = \Big(\Big|\frac{\beta}{\alpha}\Big|+1\Big)2^{p-1}\Phi_{p}^{-1}\bigg(\int_{0}^{1}a(s)r(s)d_{q}s\bigg)$.

Therefore,

Hence,

This shows that the set $E$ is bounded. By Theorem 2.7, we obtain that BVP(1.1) has at least one solution. This proof is completed.

Corollary 3.11. Assume that $(H_7)$ holds. In addition, if $\Lambda_1 < 1$ holds, where $\Lambda_1 = \Big(\Big|\frac{\beta}{\alpha}\Big|+1\Big)2^{\frac{1}{p-1}}\Phi_{p}^{-1}((a+b)A)$. Then the BVP(1.1) has at least one solution.

4.   Applications

In this section, we give two examples to illustrated our results.

Example 4.1. Consider the following $q$-difference equation BVP:

Here, $f(t, x(t), D_{q}x(t)) = \frac{1}{12}(1+\sqrt{t})\sin(x(t))+\frac{1}{6}\arctan(D_{q}x(t))$, $\; p = \frac{3}{2}, \; q = \frac{1}{4}, \; \alpha = 1, \; \beta = \frac{1}{3}, \; a(t) = \frac{1}{\sqrt{t}}$, $\; g(t) = \frac{1}{12}+\frac{\pi}{12}+\frac{1}{12}\sqrt{t}$.

Obviously, $\forall t\in I_q, \; (x_1, y_1), \; (x_2, y_2)\in R^2,$ we have

Then, $L_{1} = L_{2} = \frac{1}{6}$, $A = \int_0^1 a(s)d_qs = \frac{3}{2}$, $M = \int_0^1a(s)g(s)d_qs = \frac{1+\pi}{12}\cdot\frac{3}{2}+\frac{1}{12} = \frac{5}{24}+\frac{3\pi}{24} = \frac{5+3\pi}{24}$, and

By Corollary 3.3, we obtain that BVP (4.1) has a unique solution.

Example 4.2. Consider the following $q$-difference equation BVP:

Here, $a(t) = \frac{e^{-qt}-e^{-t}}{t}$, $f(t, x(t), D_{q}x(t)) = \frac{1}{8}\sin(|x(t)|)+\frac{1}{8}\ln(1+|D_qx(t)|)+1$, $\; p = 3, \; q = \frac{1}{2}, \; \alpha = \beta = 1$. Obviously, $\forall t\in I_q, \; (x, y)\in {R_+}^2,$ we have

Then $a = b = \frac{1}{8}$, $A = \int_0^1 a(s)d_qs = \frac{1}{2}(1-e^{-1})$, and $\Lambda_1 = \sqrt{1-e^{-1}} < 1$. By Corollary 3.11, we obtain that BVP (4.2) has at least one solution.

5.   Concluding remarks and observations

Basic (or $q$-) series and basic (or $q$-) polynomials, especially the basic (or $q$-) hypergeometric functions and basic (or $q$-) hypergeometric polynomials, are known to have widespread applications, in particular, in several areas of combinatorial analysis and number theory such as (for example) the theory of partitions. In fact, $q$-calculus is widely used in many fields, such as finite vector spaces, Lie theory, particle physics, non-linear electric circuit theory, mechanical engineering, cosmology and so on. We now choose to indicate some obvious connection between the classical $q$-analysis and the so-called $(p, q)$-analysis. Srivastava et al. have observed the fact that the results for the $q$-analogues, which we have considered in this article for $0 < q < 1$, can easily (and possibly trivially) be translated into the corresponding results for the $(p, q)$-analogues (with $0 < q < p\leq 1$) by applying some obvious parametric and argument variations, the additional parameter $p$ being redundant, see Page 340 of the work ^[35] and Pages 1511–1512 of the work ^[36].

The so-called $(p, q)$-number $[n]_{p, q}$ is given (for $0 < q < p\leq 1)$ by

where, for the classical $q$-number $[n]_{q}$, we have

Furthermore, the so-called $(p, q)$-derivative or the so-called $(p, q)$-difference of a suitably constrained function $f(z)$ is denoted by $(D_{p, q}f)(z)$ and defined, in a given subset of $\mathbb{C}$, by

Thus, clearly, we have the following connection with the familiar $q$-derivative or the $q$-difference $(D_{p, q}f)(z)$ given by the definition 5.3:

From the Eqs (5.1)–(5.4), we can find that the subject for $0 < q < 1$ can easily (and possibly trivially) be translated into the corresponding $(p, q)$-analogues (with $0 < q < p\leq1$) by applying some straightforward parametric and argument variations of the types indicated above, the additional parameter $p$ being redundant. Therefore, it is worthwhile to draw the attention of the interested readers toward.

In the last few decades, quantum difference equation has evolved into a multidisciplinary subject and plays an important role in several fields of physics and mathematics. In this paper, we obtain existence and uniqueness of solutions for nonlinear second-order quantum difference equation BVPs with $p$-Laplacian by means of the Leray-Schauder nonlinear alternative and some standard fixed point theorems. The results are novel, enrich the theories for $q$-difference equations with $p$-Laplacian, and provide the theoretical guarantee for the application of fractional $q$-difference equations in every field. In the future, we will use bifurcation theory, critical point theory, variational method, and other methods to continue our works in this area.

Acknowledgments

The author is very grateful to the referees for their very helpful comments and suggestions. The research project is supported by the National Natural Science Foundation of China (11772007), Beijing Natural Science Foundation (1172002, Z180005), the Natural Science Foundation of Hebei Province (A2015208114), and the Foundation of Hebei Education Department (QN2017063).

Conflict of interest

The authors declare that they have no competing interests.