Existence of solutions for $q$-fractional differential equations with nonlocal Erdélyi-Kober $q$-fractional integral condition

1.   Introduction

The aim of this paper is to establish the existence and uniqueness of solutions for the following nonlinear Riemann-Liouville $q$-fractional differential equation subject to nonlocal Erdélyi-Kober $q$-fractional integral conditions

where $D_{q}^{\alpha}$ and $D_{q}^{\rho}$ are the fractional $q$-derivative of Riemann-Liouville type of order $\alpha$ and $\delta$ on $(0, T)$ respectively, $1 < \alpha < 2, \; 0 < \delta < 1, f\in C([0, T]\times \mathbb{R}\times\mathbb{R}, \mathbb{R})$, $I_{q}^{\eta_{i}, \, \mu_{i}, \, \beta_{i}}$ denotes the Erdélyi-Kober fractional $q$-integral of order $\mu_{i}$ on $(0, T)$, $\mu_{i} > 0, \; \beta_{i} > 0, \; \eta_{i}\in\mathbb{R}$ and $\xi_{i} \in (0, T)$, $a, \; \lambda_{i}(\; i = 1, 2, \cdots, \; n)$ are some given constants.

The $q$-calculus or quantum calculus is an old subject that was initially developed by Jackson ^[1], basic definitions and properties of $q$-calculus can be found in ^[2]. The fractional $q$-calculus had its origin in the works by Al-Salam ^[3] and Agarwal ^[4]. In recent years, considerable interest in $q$-fractional differential equations has been stimulated due to its applicability in mathematical modeling in different branches like engineering, physics and technical, etc. There are many papers and books dealing with the theoretical development of $q$-fractional calcaulus and the existence of solutions of boundary value problems for nonlinear $q$-fractional differential equations, for examples and details, one can see ^{[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]} and references therein.

In ^[9], Zhao, Chen and Zhang considered the following nonlocal $q$-integral boundary value problem of nonlinear fractional $q$-derivatives equation:

where $q\in (0, 1), \, 1 < \alpha\leq 2, \, 0 < \beta\leq 2, \, 0 < \eta < 1$ and $\mu > 0$. $D_{q}^{\alpha}$ is the fractional $q$-derivative of Riemann-Liouville type of order $\alpha$. By using the the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskiis fixed point theorem, the authors obtained some existence results of positive solutions to the above problem.

In ^[10], the authors investigated the $q$-integral boundary value problem for $q$-integro-difference equations involving Riemann-Liouville $q$-derivatives and a $q$-integral of different orders as follows:

where, $q\in (0, 1), \, 1 < \alpha, \, \beta < 2, \, 0 < \delta < 1, \, 0 < \lambda\leq1$ and $0\leq\mu\leq 1, \, \alpha-\beta > 1$. $D_{q}^{\alpha}$ denotes the Riemann-Liouville fractional $q$-derivative of order $\alpha$ and $f, \, g : [0, 1]\times\mathbb{R}\rightarrow \mathbb{R}$ are continuous functions.

In ^[23], the authors considered the existence of solutions for the following nonlinear Riemann-Liouville fractional differential equation with nonlocal Erdélyi-Kober fractional integral conditions

where $1 < q\leq 2, \, D^{q}$ is the Riemann-Liouville fractional derivative of order $q$, $I_{\eta_{i}}^{\gamma_{i}, \, \delta_{i}}$ is the Erdélyi-Kober fractional integral of order $\delta_{i} > 0$ with $\eta_{i} > 0$ and $\gamma_{i}\in \mathbb{R}, \, i = 1, 2, \cdots, m, \, f:[0, T]\times \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function and $\alpha_{i}, \beta_{i}\in \mathbb{R}, \, \xi_{i}\in(0, T), \, i = 1, 2, \cdots, m$ are given constants.

As we all know, few people solve the existence of solutions for a nonlinear Riemann-Liouville $q$-fractional differential equation subject to nonlocal Erdélyi-Kober $q$-fractional integral conditions. Inspired by the paper ^[23], we consider the existence and uniqueness for problem (1.1) by using Banach contraction principle and Schauder's fixed point theorem.

2.   Preliminaries on $q$-calculus and Lemmas

Here we recall some definitions and fundamental results on fractional $q$-integral and fractional $q$-derivative. See the references ^[4,5,6,7] for complete theory.

For $q\in(0, 1)$, define $[a]_{q} = \frac{1-q^{a}}{1-q}, \, \, a\in \mathbb{R}$. The $q$-factorial function is defined as $(a-b)^{(n)} = \prod\limits_{k = 0}^{n-1}(a-bq^{k}), \, \, a, \, b\in \mathbb{R}$, If $n$ is a positive integer. If $\nu$ is not a positive integer, then $(a-b)^{(\nu)} = a^{\nu}\prod\limits_{n = 0}^{\infty}\frac{a-bq^{n}}{a-bq^{\nu+n}}.$ If $b = 0$, then $a^{(\nu)} = a^{\nu}$. The $q$-gamma function is defined by $\Gamma_{q}(\alpha) = \frac{(1-q)^{(\alpha-1)}}{(1-q)^{\alpha-1}}, \, \, \alpha > 0,$ and satisfies $\Gamma_{q}(\alpha+1) = [\alpha]_{q}\Gamma_{q}(\alpha)$.

The $q$-derivative of a function $f$ is defined by $(D_{q}f)(t) = \frac{f(t)-f(qt))}{(1-q)t}, \, (D_{q}f)(0) = \underset{t\to 0}\lim(D_{q}f)(t).$ The $q$-integral of a function $f$ defined on the interval $[0, b]$ is given by $(I_{q}f(t) = \int_{0}^{t}f(s)d_{q}s = (1-q)t\sum\limits_{i = 0}^{\infty}q^{i}f(q^{i}t), \, t\in[0, b].$

Some results about operator $D_{q}$ and $I_{q}$ can be found in references ^[4]. Let us define fractional $q$-derivative and $q$-integral and outline some of their properties ^[4,6,8].

Definition 1 (^[4]) Let $\alpha\geq 0$ and $f$ be a function. The fractional $q$-integral of Riemann-Liouville type is given by $(I_{q}^{0}f)(t) = f(t)$ and

Definition 2 (^[6]) The fractional $q$-derivative fractional of Riemann-Liouville type of order $\nu\geq 0$ is defined by $D_{q}^{0}f(t) = f(t)$ and

where $l$ is the smallest integer greater than or equal to $\nu$.

Definition 3 (^[24]) For $0 < q < 1$, the Erdélyi-Kober fractional $q$-integral of order $\mu > 0$ with $\beta > 0$ and $\eta\in\mathbb{R}$ of a continuous function $f: (0, \infty)\rightarrow \mathbb{R}$ is defined by

provided the right side is pointwise defined on $\mathbb{R}^{+}$.

Remark 1 For $\beta = 1$ the above operator is reduced to the $q$-analogue Kober operator

that is given in ^[4]. For $\eta = 0$ the $q$-analogue Kober operator is reduced to the Riemann-Liouville fractional $q$-integral with a power weight:

Lemma 1 (^[4]) Let $\alpha, \, \beta\in\mathbb{R}^{+}$ and $f$ be a continuous function on $[0, b]$. The Riemann-Liouville fractional $q$-integral has the following semi-group property

Lemma 2 (^[8]) Let $f$ be a $q$-integrable function on $[0, b]$. Then the following equality holds

Lemma 3 (^[4]) Let $\alpha > 0$ and $p$ be a positive integer. Then for $t\in[0, b]$ the following equality holds

Lemma 4 (^[24]) For $f(t) = t^{\lambda}$ and $\beta > 0, \; \mu > 0, \; \eta, \lambda\in\mathbb{R}, \; 0 < q < 1$, then

3.   Main results

In this section, we will give the main results of this paper. Let the space $E = \{x\in C([0, T], \mathbb{R}), D_{q}^{\delta}x\in C([0, T], \mathbb{R})\}$ be endowed with the norm $\| x\| = \underset{t\in [0, T]}{\max}|x(t)|+\underset{t\in [0, T]}{\max}|D_{q}^{\delta}x(t)|$. It is known that the space $E$ is a Banach space. To obtain our main results, we need the following lemma.

Lemma 5 Let $h(t)\in C([0, T], \mathbb{R})$. Then for any $t\in [0, T]$, the solution of the following problem

is given by

where $M = aT^{\alpha-1}-\sum\limits_{i = 1}^{n}\lambda_{i}I_{q}^{\eta_{i}, \, \mu_{i}, \, \beta_{i}}\xi_{i}^{\alpha-1} = aT^{\alpha-1}- \sum\limits_{i = 1}^{n}\lambda_{i}\beta_{i}[\frac{1}{\beta_{i}}]_{q}\frac{\Gamma_{q}(\eta_{i}+1+\frac{\alpha-1}{\beta_{i}})}{\Gamma_{q}(\mu_{i}+\eta_{i} +1+\frac{\alpha-1}{\beta_{i}})}\xi_{i}^{\alpha-1}\neq 0$.

Proof. Applying the operator $I_{q}^{\alpha}$ on both sides of the first equation of (3.1) for $t\in (0, T)$ and using Lemma 1 and Lemma 2, we have

Applying the initial value condition $x(0) = 0$, we get $c_{2} = 0.$ By the boundary value condition, we have

that is

Substituting $c_{1}, c_{2}$ to (3.3), we obtain the solution (3.2). This completes the proof.

Using the Lemma 5, we can define an operator $\mathcal{Q}: E\rightarrow E$ as follows:

where

and $\tau\in\{t, T, \xi_{1}, \xi_{2}\cdots, \xi_{n}\}$. Then, the existence of solutions of system (1.1) is equivalent to the problem of fixed point of operator $\mathcal{Q}$ in (3.4).

In the following, we will use some classical fixed point techniques to give our main results.

Theorem 1 Suppose that there exists a function $L(t): [0, T]\rightarrow \mathbb{R}^{+}$ $q$-integrable such that

for each $x, \tilde{x}, y, \tilde{y}\in \mathbb{R}$. Then problem (1.1) has an unique solution on $[0, T]$ if

where $L_{1} = \underset{t\in [0, T]}{\sup}I_{q}^{\alpha}L(t), \, \, L_{2} = \underset{t\in [0, T]}{\sup}I_{q}^{\alpha-1}L(t), \, \, L_{3} = \underset{t\in [0, T]}{\sup}I_{q}^{\alpha-\delta}L(t)$.

Proof. The conclusion will follow once we have shown that the operator $\mathcal{Q}$ defined (3.4) is contractively with respect to a suitable norm on $E$.

For any functions $x, y\in E$, we have

on the other hand

Thus

which implies that

Thus the operator $\mathcal{Q}$ is a contraction in view of the condition (3.5). By Banach's contraction mapping principle, the problem (1.1) has an unique solution on $[0, T]$. This completes the proof.

Corollary 1 Assume that there exists $L_{0} > 0$ such that

for each $t\in[0, T]$ and $x, \tilde{x}, y, \tilde{y}\in \mathbb{R}$. Then the problem (1.1) has an unique solution whenever

where $I_{q}^{\eta_{i}, \, \mu_{i}, \, \xi_{i}}s^{\alpha}(\xi_{i}) = \beta_{i}[\frac{1}{\beta_{i}}]_{q}\frac{\Gamma_{q}(\eta_{i}+1+\frac{\alpha-1}{\beta_{i}})}{\Gamma_{q}(\mu_{i}+\eta_{i} +1+\frac{\alpha-1}{\beta_{i}})}\xi_{i}^{\alpha}$.

In the following Theorem, the existence results for nontrivial solution for problem (1.1) are presented. For convenience, we denote

where $l_{i}(t), i = 1, 2, 3$ are defined in Theorem 2.

Theorem 2 Let $f:[0, T]\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function. Assume there exist three nonnegative continuous functions $l_{i}(t), i = 1, 2, 3$ such that

and $f(t, 0, 0)\neq 0$ for $t\in[0, T]$. Then the problem (1.1) has a nontrivial solution.

Proof. We shall use Schauder's fixed point theorem to prove our theorem. Define $E_{d} = \{x:x\in E, \parallel x\parallel\leq d\}$, where $d\geq \max\Big\{3(\varsigma_{1}+|a|\sigma_{1}\rho+\varrho_{1}(1+\rho)\zeta), 3(\varsigma_{2}+|a|\sigma_{2}\rho+\varrho_{2}(1+\rho)\zeta)^{\frac{1}{1-r_{1}}}, 3(\varsigma_{3}+|a|\sigma_{3}\rho+\varrho_{3}(1+\rho)\zeta)^{\frac{1}{1-r_{2}}}\Big\}$ and $\rho = \frac{T^{\alpha-1}}{|M|}+\frac{T^{\alpha-\delta-1}}{|M|}\frac{\Gamma_{q}(\alpha)}{\Gamma_{q}(\alpha-\delta-1)}.$ Note that $E_{d}$ is a closed, bounded and convex subset of the Banach space $E$. We now show that $\mathcal{Q}: E_{d}\rightarrow E_{d}$. In fact, for $x\in E_{d}$ we have that

which implies

Thus

From the two inequalities above, we get

Hence, $\mathcal{Q}$ maps $E_{d}$ into $E_{d}$. Also, it is easy to check that $\mathcal{Q}$ is continuous, since $f$ is continuous. For each $x\in E_{d}$ and each $0 < t_{1} < t_{2} < T$, we have

Let $t_{2}\rightarrow t_{1}$, we get $\parallel\mathcal{Q}x(t_{2})-\mathcal{Q}x(t_{2})\parallel\rightarrow 0$. Thus, $\mathcal{Q}$ is uniformly bounded and equicontinuous. The theorem of Arzelá-Ascoli implies that $\mathcal{Q}$ is completely continuous. By Schauder's fixed point theorem, $\mathcal{Q}$ has a fixed point in $E_{d}$. Clearly $x = 0$ is not a fixed point because $f(t, 0, 0)\neq 0$ for $t\in[0, T]$. Hence, the problem (1.1) has at least one nontrivial solution. This proves the theorem.

Remark 2 In the Theorem 2, if $r_{i} > 1, i = 1, 2$, we may choose $l_{2}(t), \; l_{3}(t)$ and $d$ such that

and

For $r_{i} = 1, i = 1, 2$, we have the following theorem.

Theorem 3 Let $f:[0, T]\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function. Assume there exist three nonnegative continuous functions $l_{i}(t), i = 1, 2, 3$ such that

and $\max\Big\{\varsigma_{2}+|a|\sigma_{2}\rho+\varrho_{2}(1+\rho)\zeta), (\varsigma_{3}+|a|\sigma_{3}\rho+\varrho_{3}(1+\rho)\zeta)\Big\} < \frac{1}{3}$. Further, assume that $f(t, 0, 0)\neq 0$ for $t\in[0, T]$. Then the problem (1.1) has a nontrivial solution.

Proof. Let $d > 3\Big(\varsigma_{1}+|a|\sigma_{1}\rho+\varrho_{1}(1+\rho)\zeta\Big),$ The proof is similar to Theorem 2, so it is omitted. This completes the proof.

Although Theorems 2 and Theorem 3 provide some simple conditions on the existence of solution of problem (1.1) and Theorem 1 provides a condition on the existence and uniqueness on the solution of problem (1.1), the following theorem provides an easily verifiable condition for the existence of a nontrivial solution for the problems (1.1).

Theorem 4 Assume that $f:[0, T]\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ with $f(t, 0, 0)\neq 0$ for $t\in[0, T]$ and $x\in E$. Suppose that

holds. Then problem (1.1) has at least one nontrivial solution.

Proof. Choose a constant $A$ such that

where $\kappa = \sum\limits_{i = 1}^{n}|\lambda_{i}|\beta_{i}[\frac{1}{\beta_{i}}]_{q}\frac{\Gamma_{q}(\eta_{i}+1+\frac{\alpha}{\beta_{i}})} {\Gamma_{q}(\mu_{i}+1+\frac{\alpha}{\beta_{i}})}\xi_{i}^{\alpha}$.

By the condition (3.6), there exists a constant $c_{1}$ such that

Since $f:[0, T]\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$, we can find another constant $A_{1} > 0$ such that $|f(t, x(t), D_{q}^{\delta}x(t))| < A_{1},$ for $t\in[0, T]$ and $\parallel x\parallel\leq c_{1}$. Let $c = \max\{ \frac{A_{1}}{A}, c_{1}\}$, then for any $\parallel x\parallel\leq c$, we have $|f(t, x(t), D_{q}^{\delta}x(t))| < A c$. Set

Then for any $x\in E_{c}$, we have

Thus

From (3.9), we obtain $\mathcal{Q}(E_{c})\subset E_{c}$. By Schauder fixed point theorem, $\mathcal{Q}$ has at least one fixed point in $E_{c}$. Clearly, $x = 0$ is not a fixed point because $f(t, 0, 0)\neq 0$. Therefore, problem (1.1) has at least one nontrivial solution, which completes the proof.

4.   Example

In this section, we illustrate the results obtained in the last section.

$Example\; 1$ Consider the following fractional-order boundary value problem involving nonlocal Erdélyi-Kober fractional $q$-integral conditions:

where $q = \frac{1}{2}, \delta = \frac{1}{4}, T = 1, \alpha = \frac{3}{2}, a = 3, \lambda_{1} = \frac{1}{100}, \lambda_{2} = \frac{3}{100}, \beta_{1} = \frac{1}{10}, \beta_{2} = \frac{1}{5}, \eta_{1} = \frac{3}{8}, \eta_{2} = \frac{1}{8}, \xi_{1} = \frac{1}{4}, \xi_{2} = \frac{1}{2}, \mu_{1} = \frac{5}{4}, \mu_{1} = \frac{7}{4}$, and $f(t, , x(t), D_{q}^{\delta} x(t)) = \frac{t^{2}}{1+e^{t}}\Big(\frac{|x(t)+ D_{\frac{1}{2}}^{\frac{1}{2}}x(t)|}{1+2|x(t)+D_{\frac{1}{2}}^{\frac{1}{2}}x(t)|}+\cos t+2\Big)+1$.

By computation, we deduce that

then, the first condition is satisfied with $L(t) = \frac{t^{2}}{2}$.

Hence, by Theorem 1, the boundary value problem (1.1) has an unique solution on [0, 1].

$Example\; 2$ Consider the following fractional-order boundary value problem involving nonlocal Erdélyi-Kober fractional $q$-integral conditions:

Here, $q = \frac{1}{2}, \delta = \frac{1}{4}$. $f(t, x(t), D_{q}^{\rho}x(t)) = \frac{1}{2\pi}\sin(\pi|x|)\frac{(|x|+|D_{q}^{\delta}x|)^{\frac{1}{2}}}{|x|+|D_{q}^{\delta}x|+1}+1$,

Therefore, the conclusion of Theorem 4 implies that problem (1.1) has at least one solution on [0, 1].

5.   Conclusions

In this work, we utilize Banach contraction principle and Schauder's fixed point theorem to research the existence, uniqueness of solutions for a $q$-fractional differential equation with nonlocal Erdélyi-Kober $q$-fractional integral condition and in which the nonlinear term contains a fractional $q$-derivative of Rieman-Liouville type. Some existence and uniqueness results of solutions are obtained, we also provide an easily verifiable condition for the existence of nontrivial solution for the problem (1.1).

Acknowledgments

This research was Supported by Science and Technology Foundation of Guizhou Province (Grant No. [2016]7075, [2019]1162), by the Project for Young Talents Growth of Guizhou Provincial Department of Education under(Grant No. Ky[2017]133), and by the project of Guizhou Minzu University under (Grant No.16yjrcxm002).

Conflict of interest

The authors declare no conflict of interest.