Existence results for nonlinear multi-term impulsive fractional $q$-integro-difference equations with nonlocal boundary conditions

1.   Introduction

Impulsive systems are found to be of great help in the study of the phenomena exhibiting abrupt changes, such as the passage of a solid body from a given fluid density to another fluid density ^[1], remittent vibrators ^[2], instantaneous disturbances in cellular neural networks ^[3], shock changes in stock exchange ^[4], etc. Evolutionary hereditary processes undergoing abrupt or sudden changes are modeled with the aid of impulsive fractional differential equations ^[5,6]. Such processes naturally occur in bio-medical sciences ^[7,8], thermo-elasticity ^[9], heat conduction ^[10], plasma physics ^[11], and engineering problems ^[12].

The subject of fractional calculus has been extensively studied during the last two decades in view of its vast applications in science and technology. Examples include chaotic synchronization ^[13], immune systems ^[14], neural networks ^[15], fractional diffusion ^[16], ecology ^[17], etc. For the fundamental concepts of fractional calculus, we refer the reader to the text ^[18]. In particular, there has been shown a great interest in investigating the existence, uniqueness and stability of solutions for initial and boundary value problems. One can find an uptodate account of these problems in the book ^[19], while a variety of recent results involving different kinds of fractional derivatives can be found in the articles ^{[20,21,22,23,24,25,26]}. The natural extension of fractional differential equations to fractional $q$-difference equations also received significant attraction. For details and examples of nonlinear fractional $q$-difference equations subject to different kinds of boundary conditions involving $q$-derivatives and $q$-integrals, for instance, the articles ^{[27,28,29,30,31,32,33]} and the references cited therein.

Let us now review some recent results on impulsive fractional and fractional $q$-difference equations. In ^[34], the authors studied a Caputo-Hadamard type fractional impulsive hybrid system with nonlinear fractional integral conditions. Impulsive fractional $q$-integro-difference equations with separated boundary conditions were investigated in ^[35]. Some stability results for abstract fractional differential equations with non-instantaneous impulses can be found in ^[36]. The authors in ^[37] discussed the existence of solutions for an impulsive fractional $q$-difference equation with nonlocal condition. Some existence results for an impulsive fractional $q$-difference equation with antiperiodic boundary conditions were proved in ^[38]. In ^[39], the authors obtained exact solutions for linear Riemann-Liouville fractional differential equations with impulses.

Motivated by the recent development on impulsive fractional $q$-difference equations, in this paper, we introduce and study a nonlinear multi-term impulsive nonlocal boundary value problem involving Caputo type fractional $q$-derivative operators of different orders and the Riemann-Liouville fractional $q$-integral operator. In precise terms, we discuss the existence of solutions for the following problem:

where $^cD_q^\omega$ denotes the Caputo fractional $q$-derivative of order $\omega$, $(\omega = \alpha, \; \beta)$ and $I_q^{\delta}$ denotes the Riemann-Liouville fractional $q$-integral of order $\delta$, $0 < \delta < 1$, $0 < q < 1$, $1 < \alpha < 2$ and $0 < \beta < 1$ such that $\alpha -\beta > 1,$ $0 < \varpi \leq 1$, $\; a, b \in \mathbb{R}^{+}$ and $f, g : \mathcal{J}\times \mathbb{R} \to \mathbb{R}$ are continuous functions. Further, $\mathcal{I}_\sigma, \tilde{\mathcal{I}}_\sigma: \mathbb{R}\rightarrow \mathbb{R}$ are also continuous functions, $\Delta u(t_\sigma) = u(t_\sigma^{+})-u(t_\sigma^{-}), \; \; \Delta u^{'}(t_\sigma) = u^{'}(t_\sigma^{+})-u^{'}(t_\sigma^{-}),$ where $u(t_\sigma^{+})$ and $u(t_\sigma^{-})$ represent the right and left-hand limits of $u(t)$, respectively, at $t = t_{\sigma}\; (\sigma = 1, 2, 3, ..., p)$ and $0 = t_0 < t_1 < t_2 < ... < t_{\sigma} < ... < t_{p} < t_{p+1} = 1, \; \mathcal{J}^{'} = \mathcal{J} - \{t_1, t_2, t_3, ..., t_{\sigma}\}.$

Here, we emphasize that the problem (1.1) is novel in the sense that it consists of a multi-term fractional $q$-integro-difference equation involving Caputo type fractional $q$-derivative and $q$-Riemann-Liouville integral operators subject to impulsive conditions and nonlocal boundary data. We make use of the fixed-point approach to discuss the existence and uniqueness of solutions for the problem at hand. First of all, we prove an auxiliary lemma dealing with the linear variant of the problem (1.1), which enables us to convert the given nonlinear problem into a fixed-point problem. Then we apply the Schaefer's fixed point theorem to establish an existence result for the problem (1.1), while the uniqueness of its solutions is obtained via Banach's contraction mapping principle. The work accomplished in this paper is not only new in the given configuration but it also accounts for some new results as special cases, for details, see Section 5.

The structure of the rest of the paper is as follows. In section 2, we recall some basic definitions from fractional $q$-calculus and prove a fundamental lemma which plays a key role in the analysis of the problem at hand. Main results concerning the existence and uniqueness of solutions for the problem (1.1) are proved in Section 3, while examples illustrating these results are discussed in Section 4.

2.   Preliminaries

Let $\mathcal{J}_{0} = [0, t_1], \mathcal{J}_{1} = (t_1, t_2], \mathcal{J}_{2} = (t_2, t_3], ...., \mathcal{J}_{\sigma} = (t_\sigma, t_{\sigma+1}], \; \sigma = 1, 2, 3, ..., p$ with $t_{p+1} = 1$ and introduce the space: ${PC(\mathcal{J}, \mathbb{R})} = \{ u: \mathcal{J} \rightarrow \mathbb{R}| u \in {C(\mathcal{J}_{\sigma})}, \;$ $\sigma = 0, 1, ..., p \; {\text{and}}\; u(t_\sigma^{+})\; {\text{exist, for}} \; \sigma = 1, ..., p\}$ endowed with the norm $\Vert u\Vert = \sup_{t\in \mathcal{J}} |u(t)|.$ Obviously, ${PC(\mathcal{J}, \mathbb{R})}$ is a Banach space.

Let us first collect some necessary concepts and definitions from $q$-fractional calculus ^[40,41].

We define a $q$-real number as

The $q$-analogue of the Pochhammer symbol ($q$-shifted factorial) is defined by

The $q$-analogue of the exponent $(x-y)^k$ is given by

The $q$-gamma function $\Gamma_q(y)$ is defined as

where $y \in \mathbb{R} \setminus \{0, -1, -2, ...\}.$ Observe that $\Gamma_q(y+1) = [y]_q \Gamma_q(y).$

Definition 2.1. Let $f$ be a function defined on $[0, b], \, b > 0$ and $a \in (0, b)$ is an arbitrary fixed point. The fractional $q$-integral of the Riemann-Liouville type is defined by

provided the integral exists. Moreover, $(I_{q, a}^{\gamma} I_{q, a}^{\beta}f)(t) = (I_{q, a}^{\beta+\gamma} f)(t); \, \gamma, \beta \in \mathbb{R}^+, \, a \in (0, t).$

We define the $q$-derivative of a function $f$ as follows:

Furthermore, $D_q^0f = f, \, \, \, D_q^nf = D_q (D_q^{n-1}f), \, \, \, n = 1, 2, 3, ....$

Definition 2.2. (^[41]) The fractional $q$-derivative of the Riemann-Liouville type of order $\beta$ is defined as

where $\lceil \beta \rceil$ is the smallest integer greater than or equal to $\beta.$

In passing, we remark that

${\it (i)}$ $(D_{q, a}^{\beta} I_{q, a}^{\beta} f)(t) = f(t), \, \, 0 < a < t.$

${\it (ii)}$ $I_{q, a}^{\beta}\left((x-a)^{(\lambda)}\right) = \frac{\Gamma_q(\lambda+1)}{\Gamma_q(\beta+\lambda+1)}(x-a)^{(\beta+\lambda)}, \; 0 < a < x < b, \beta\in {\mathbb R}^+, \lambda\in (-1, \infty).$

Definition 2.3. (^[41]) The fractional $q$-derivative of the Caputo type of order $\beta \in \mathbb{R}^+$ is defined by

Remark 2.1. For $0 < a < t$ and $\beta \in \mathbb{R} \setminus \mathbb{N},$ the following relations hold ^[41]:

${\it (a)}$ $(^{c} D_{q, a}^{\beta+1}f)(t) = (^{c} D_{q, a}^{\beta}D_qf)(t);$

${\it (b)}$ $(^{c} D_{q, a}^{\beta} I_{q, a}^{\beta} f)(t) = f(t);$

${\it (c)}$ $({I_{q, a}^{\beta}}\, ^{c} D_{q, a}^{\beta} f)(t) = f(t)-\sum_{k = 0}^{\lceil \beta\rceil-1}\frac{(D_q^k f)(a)}{\Gamma_q(k+1)} t^k(a/t; q)_k;$

In the following lemma, we solve the linear variant of the problem (1.1), which plays a fundamental role in the forthcoming analysis.

Lemma 2.1. Let $\varrho\in C([0, 1], \mathbb{R})$ and $\eta \in (t_m, t_{m+1}),$ $m$ is a non-negative integer such that $0 \leq m\leq p$. Then the unique solution of the boundary value problem:

is given by

where

Proof. Let $u$ be a solution of the $q$-fractional boundary value problem (2.2). Then, for $t \in \mathcal{J}_{0}$, we have

Operating the $q$-integral operator $I_q^{\alpha}$ to both sides of the above equation, we get

where $a_1, a_2 \in \mathbb{R}$ are arbitrary constants. Differentiating (2.4) with respect to $t$, we obtain

Similarly, for $t \in (t_1, t_2]$, we obtain

where $b_1, \; b_2 \in \mathbb{R}$ are arbitrary constants.

From (2.4)–(2.7), we get

In view of the impulsive conditions: $\Delta u(t_1) = \mathcal{I}_1(u(t_1))$ and $\Delta u^{'}(t_1) = \tilde{\mathcal{I}}_1(u(t_1)),$ we find that

Thus, (2.6) becomes

In general, for $t \in \mathcal{J}_{\sigma}$, we get

Differentiating (2.8) with respect to $t$, we get

Now, using the condition $u(0)+ u'(0) = 0$ after finding $u(0)$ and $u'(0)$ from (2.4) and (2.5), respectively, we obtain $a_1+a_2 = 0$. On the other hand, from (2.8) and (2.9), we have

Making use of the condition: $u(1)+u'(\eta) = 0$ and $a_1+a_2 = 0$, we obtain

Inserting the above values of $a_1$ and $a_2$ in (2.4) and (2.8), we obtain (2.3). Conversely, if (2.3) has a solution $u$, then it can easily be verified that that $u$ satisfies the problem (2.2).

3.   Main results

By Lemma 2.1, we can transform the problem (1.1) into a fixed point problem: $u = {\mathit{F}} u,$ where ${\mathit{F}}: PC(\mathcal{J}, \mathbb{R}) \rightarrow PC(\mathcal{J}, \mathbb{R})$ is defined by

Lemma 3.1. The operator ${\mathit{F}}: PC(\mathcal{J}, \mathbb{R}) \rightarrow PC(\mathcal{J}, \mathbb{R})$ defined by (3.1) is completely continuous.

Proof. Observe that continuity of ${\mathit{F}}$ follows from that of $f, g, \mathcal{I}_{\sigma}$ and $\tilde{\mathcal{I}}_{\sigma}$. Let $B_{\rho} = \{ u\in PC(\mathcal{J}, \mathbb{R}): \Vert u \Vert \leq \rho\} \subset PC(\mathcal{J}, \mathbb{R})$. Then, there exist positive constants $Q_i > 0 \; (i = 1, 2, 3, 4)$ such that $\vert f(t, u)\vert \leq Q_1, \; \vert g(t, u)\vert \leq Q_2, \; \vert \mathcal{I}_\sigma(u)\vert \leq Q_3$ and $\vert \tilde{\mathcal{I}}_\sigma(u)\vert \leq Q_4, \forall u \in B_{\rho}$. Thus, $\forall u \in B_{\rho}$, we have

which implies that $\Vert {\mathit{F}}u \Vert \leq \Theta$. On the other hand, for $t \in \mathcal{J}_{\sigma}, 0 \leq \sigma \leq p$, we have

Hence, for $t_1, t_2 \in \mathcal{J}_{\sigma}, 0 \leq \sigma \leq p$, we have

which implies that ${\mathit{F}}$ is equicontinuous on all $\mathcal{J}_{\sigma}, \sigma = 0, 1, 2, ...p$. Thus, by the Arzela-Ascoli theorem, the operator ${\mathit{F}}:PC(\mathcal{J}, \mathbb{R}) \rightarrow PC(\mathcal{J}, \mathbb{R})$ is completely continuous.

Theorem 3.1. (Schaefer's fixed-point theorem ^[42]) Let $E$ be a Banach space. Assume that ${\mathit{F}}:E \rightarrow E$ is a completely continuous operator and the set $\Omega = \{u \in E|\; u = \zeta {\mathit{F}}u, \; 0 < \zeta < 1 \}$ is bounded. Then ${\mathit{F}}$ has a fixed point in $E$.

Theorem 3.2. Assume that

$(\mathcal{S}_1)$ There exist nonnegative functions $\mu_1(t), \mu_2(t), \phi_1(t), \phi_2(t) \in L(0, 1)$ and positive constants $Q_i\; (i = 3, 4)$ such that $\vert f(t, u)\vert \leq \mu_1(t) +\mu_2(t) |u|,$ $\vert g(t, u)\vert \leq \phi_1(t) +\phi_2(t) |u|,$ $\vert \mathcal{I}_\sigma(u)\vert \leq Q_3,$ $\vert \tilde{\mathcal{I}}_\sigma(u)\vert \leq Q_4$ for $t \in [0, 1], u \in {\mathbb R}$ and $\sigma = 1, 2, ..., p.$

Then the problem (1.1) has at least one solution on ${\mathcal J}.$

Proof. Let us consider the set $\Omega = \{u \in PC(\mathcal{J}, \mathbb{R})|\; u = \zeta {\mathit{F}}u, \; 0 < \zeta < 1 \}$, where the operator ${\mathit{F}}:PC(\mathcal{J}, \mathbb{R}) \rightarrow PC(\mathcal{J}, \mathbb{R})$ is defined by (3.1) and define a ball $B_\varepsilon = \{u \in PC(\mathcal{J}, \mathbb{R}):\; \Vert u \Vert \leq \varepsilon \}$. We just need to show that the set $\Omega$ is bounded as it has already been proved in Lemma 2 that the operator ${\mathit{F}}$ is completely continuous. Let $u \in \Omega$, then $u = \zeta {\mathit{F}}u, \; 0 < \zeta < 1.$ For any $t \in \mathcal{J}$, we have

which implies that $\Vert u \Vert$ is bounded for any $t \in \mathcal{J}$. So, the set $\Omega$ is bounded. Thus, by the conclusion of Theorem 3.1, the operator ${\mathit{F}}$ has at least one fixed point, which shows that (1.1) has at least one solution on $\mathcal{J}.$

Corollary 3.1. Assume that functions $f, g, \mathcal{I}_\sigma, \tilde{\mathcal{I}}_\sigma \; (\sigma = 1, 2, 3, ...)$ are bounded. Then the nonlinear problem (1.1) has at least one solution $\mathcal{J}.$

Theorem 3.3. (^[42]) Let ${\mathit{F}}: Y \rightarrow Y$ be a contraction on a nonempty closed subset of a Banach space $X$. Then ${\mathit{F}}$ has a unique solution.

Theorem 3.4. Let $f, g:[0, 1]\times {\mathbb R} \to {\mathbb R}$ and $\mathcal{I}_\sigma, \tilde{\mathcal{I}}_\sigma: \mathbb{R}\rightarrow \mathbb{R}$ be continuous functions. In addition, the following conditions hold:

$(\mathcal{S}_2)$ There exist positive constants $N_i\; (i = 1, 2, 3, 4)$ such that, for each $t \in [0, 1]$ and $u, v \in {\mathbb R}$,

for $\sigma = 1, 2, ..., p.$

Then the problem (1.1) has a unique solution on $[0, 1]$, provided that

Proof. For $u, v \in PC(\mathcal{J}, \mathbb{R})$ and $t \in \mathcal{J}$, we get

Thus, in view of the condition (3.3), the above inequality takes the form:

As $\Xi < 1$ (by the condition (3.3)), therefore ${\mathit{F}}$ is a contraction. So, by Theorem 3.3, the operator ${\mathit{F}}$ has a unique fixed point. Hence, the problem (1.1) has a unique solution on ${\mathcal J}.$

4.   Examples

In this section, we demonstrate the application of the results obtained in the last section.

Example 4.1. (Illustration of Theorem 3.1). Consider the following boundary value problem of multi-term impulsive fractional $q$-integro-difference equations:

where $\alpha = 1.95$, $q = 0.5$, $\beta = 0.02$, $\varpi = 0.88$, $\delta = 0.25$, $\eta = 0.4$, $a = 0.30$, $b = 0.90$, $p = 1$ and

Clearly, $\mu_1(t) = e^{4t} \cos^{2}(u(t)), \; \mu_2(t) = \sin(t), \; \phi_1(t) = \frac{2}{9}(t^3 +3), \; \phi_2(t) = 3 \cos (t), \; Q_3 = 4, \; Q_4 = 3,$ and the hypothesis of Theorem 3.1 holds true. Thus, the conclusion of Theorem 3.1 applies to the problem (4.1).

Example 4.2. (Illustration of Theorem 3.4). Consider the boundary value problem given by

where $\alpha = 1.95$, $q = 0.5$, $\beta = 0.02$, $\varpi = 0.88$, $\delta = 0.25$, $\eta = 0.4$, $a = 0.30$, $b = 0.90$, $p = 1$ and

It is easy to verify that

Moreover, $\Xi \approx 0.978211 < 1$ ($\Xi$ is given by (3.3)). Thus, all the assumptions of Theorem 3.4 are satisfied. So, by the conclusion of Theorem 3.4, the problem (4.2) has a unique solution $[0, 1].$

5.   Conclusions

We have investigated a new class of nonlinear nonlocal impulsive boundary value problems of multi-term Caputo fractional $q$-difference equations involving both usual and Riemann-Liouville fractional $q$-integral type nonlinearities. The classical fixed point theorems are employed to derive the existence and uniqueness results for the given problem. Our results are indeed new and enrich the related literature on the topic. Moreover, some new results can be recorded as special cases of the present ones by fixing the parameters involved in the governing equation in the problem (1.1). For example, our results reduce to the new ones for the following equations subject to impulsive and nonlocal boundary data:

● $\; ^cD_q^{\alpha}u(t) = a f(t, u(t))+ b I_q^{\delta}g(t, u(t))$ for $\lambda = 1;$

● $\lambda\; ^cD_q^{\alpha}u(t)+ (1-\lambda)\; ^cD_q^{\beta}u(t) = f(t, u(t))$ for $a = 1, b = 0;$

● $\lambda\; ^cD_q^{\alpha}u(t)+ (1-\lambda)\; ^cD_q^{\beta}u(t) = I_q^{\delta}g(t, u(t))$ for $a = 0, b = 1.$

Use of AI tools declaration

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

Acknowledgements

This research work was funded by Institutional Fund Projects under Grant No. (IFPIP: 1444-130-1443). The authors gratefully acknowledge technical and financial support provided by the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia. The authors also thank the reviewers for their constructive remarks on their work.

Conflict of Interest

All authors declare no conflicts of interest in this paper.