Fixed point theorem combined with variational methods for a class of nonlinear impulsive fractional problems with derivative dependence

1.   Introduction

Fractional differential equations have received overwhelming interest in the recent years as such equations describe the natural phenomena in a more realistic manner. The $p$-Laplacian operator is found to be of great help in describing certain problems occurring in mechanics, nonlinear dynamics and many other fields. In consequence, the study of fractional differential equations together with the $p$-Laplace operator attracted the attention of many researchers. Let us now dwell on some recent works on $p$-Laplacian fractional boundary value problems.

Liu et al. ^[1] applied the method of lower and upper solutions to study the existence of solutions for the following problem:

where $1 < \alpha, \beta\leq 2, \, r_1, r_2\geq0, \, \phi_p$ is the $p$-Laplacian operator, $p > 1$, $D_{0^+}^{\alpha}$ is the Riemann-Liouville fractional derivative, and ${}^{c}D^{\beta}_{0^+}$ is the Caputo fractional derivative, $f\in C([0, 1]\times[0, +\infty)\times (-\infty, 0], [0, +\infty)).$

In ^[2], Bai investigated the existence of positive solutions with the aid of the properties of Green's functions for the following $p$-Laplacian problem:

where $0 < \beta < 1, \, 2 < \alpha < \beta+2, \, D^{\alpha}_{0+}$ and ${}^cD^{\beta}_{0+}$ are the Riemann-Liouville fractional derivative and the Caputo fractional derivative of order $\alpha$ and $\beta$ respectively, $\phi_p$ is the $p$-Laplacian operator, $p > 1,$ and $f\in C([0, 1]\times \mathbb{R}, \mathbb{R})$.

Recently, Wang and Bai ^[3] discussed the existence and uniqueness of positive solutions to a mixed $p$-Laplacian fractional boundary value problem given by

where $\phi_p(t) = {|t|^{p-2}}\cdot t$, $\frac{1}{p}+\frac{1}{q} = 1, \, p, q > 1, \, 0 < t < 1, \, 1 < \gamma, \delta\leq 2, \, 0 < \mu, \eta < 1, 0\leq r_1 < \frac{1}{\mu^{\beta-1}}, \, 0\leq r_2 < \frac{1}{(1-\eta)}$, ${}^{c}D_{1^-}^{\gamma}$ is the right Caputo fractional derivative and $D^{\delta}_{0^+}$ is the left Riemann-Liouville fractional derivative and $g\in C([0, 1]\times\mathbb{R}^2, \mathbb{R})$. The authors in ^[3] proved the existence and uniqueness of the solutions to the above problem for $1 < p\leq 2$. However, they did not consider the case when $p > 2$.

For some recent results on $p$-Laplacian boundary value problems, for instance, see the articles ^{[4,5,6,7,8,9,10,11]}. The construction of the Green's function together with its properties is a useful tool to investigate the existence of positive solutions to the boundary value problems; for instance, see the text ^[12].

Let us now review some recent works dealing with a modified form of Caputo and Riemann-Liouville fractional derivatives. In ^[13], the authors studied the asymptotic stability of solutions of generalized Caputo fractional differential equations. Caputo modification of the generalized fractional derivatives was discussed in ^[14]. Some existence results for a nonlocal boundary value problem involving generalized Liouville-Caputo derivatives and generalized fractional integral were presented in ^[15]. The authors in ^[16] discussed the existence of solutions for generalized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral boundary conditions. In ^[17], extremal solutions for an integro-initial value problem for generalized Caputo fractional differential equations were obtained. In ^[18], the authors introduced and studied a new class of coupled systems containing both Caputo and Riemann-Liouville generalized fractional derivatives. For some recent works on the problems involving generalized fractional derivatives, for example, see ^{[19,20,21,22,23,24]}.

Motivated by the aforementioned studies on boundary value problems involving a $p$-Laplacian operator and modified versions of fractional derivatives, in this paper, we introduce a new class of mixed $p$-Laplacian fractional boundary value problems involving right-sided and left-sided fractional derivatives and left-sided integral operators with respect to a power function. In precise terms, we investigate the following problem:

where $\phi_p(t) = |t|^{p-2}\cdot t, \, \frac{1}{p}+\frac{1}{q} = 1, \, p, q > 1, \, 0 < t < 1, \, 1 < \alpha, \beta\leq 2, \, \rho > 0, \, \zeta > 0, \, 0 < \mu, \eta < 1, \, 0\leq \lambda_1 < \frac{1}{\mu^{\rho(\beta-1)}}, \, 0\leq \lambda_2 < \frac{1}{(1-\eta^{\rho})^{\alpha-1}}$, ${}^{\rho}D_{1^-}^{\alpha}$ and ${}^{\rho}D^{\beta}_{0^+}$ respectively denote the right and left fractional derivatives of orders $\alpha$ and $\beta$ with respect to a power function (see Definitions 2), ${}^{\rho}I^{\zeta}_{0+}$ is the fractional integral operator of order $\zeta$ with respect to a power function (see Definitions 1), $\nu_1, \nu_2 \in \mathbb{R}$ and $f, g:[0, 1]\times\mathbb{R}^2\to\mathbb{R}$ are continuous functions.

The remainder of the paper is arranged as follows. In Section 2, we present the background material related to our problem and prove some important lemmas that play a key role in the forthcoming analysis. Section 3 contains the main results for the given problem. In Section 4, we illustrate our results with the aid of examples. The paper concludes with certain interesting observations.

2.   Preliminaries

Let us first recall that the concept of fractional calculus of a function with respect to another function can be found in the books by Samko et al. (^[25]; Section 18.2) and Kilbas et al. (^[26]; Section 2.5), while an article by Erdelyi ^[27] contains the first study of fractional integrals with respect to a power function (now mistakenly named after Katugampola).

Definition 2.1. The fractional integral with respect to a power function ($t^\rho/\rho$) of order $\alpha > 0$ for a function $h \in X_c^p(a, b)$ for $-\infty < a < t < b < \infty,$ is defined by

where $X_c^p(a, b)$ denotes the space of all complex-valued Lebesgue measurable functions $\phi$ on $(a, b)$ equipped with the norm:

Note that the integral in (2.1) is called the left-sided fractional integral. Similarly, we can define right-sided fractional integral $^{\rho}I^{\alpha}_{b^-}f$ as

Here we mention that the above definitions of fractional integrals follow from the integrals (2.5.1) and (2.5.2) on pages 99–100 in the text ^[26] by taking the power function $g(x) = x^\rho/\rho$.

Definition 2.2. For $\alpha > 0, \, n = [\alpha]+1$ and $\rho > 0$, the fractional derivatives with respect to a power function ($t^\rho/\rho$), associated with the fractional integrals (2.1) and (2.2) are defined, for $0 \le a < x < b < \infty,$ by

and

Note that the above definitions of fractional derivatives follow from the integrals (2.5.17) and (2.5.18) on page 101 in the text ^[26] by taking the power function $g(x) = x^\rho/\rho$.

Lemma 2.1. ^[28] Let $1 < \alpha\leq 2, \rho > 0$, $u \in X_c^p(0, T)$ and ${}^{\rho}I^{2-\alpha}u\in AC^2_{\rho}$, where $AC^2_\rho([a, b])$ denotes the space of absolutely continuous functions possessing the $t^{1-\rho}\frac{d}{dt}$-derivative defined by

Then the general solution of the fractional differential equation ${}^{\rho}D_{0+}^{\alpha}u(t) = 0$ is

where $c_i\in {\mathbb R}, i = 1, 2.$ Moreover,

Lemma 2.2. For any $\psi\in C([0, 1], \mathbb{R})$, the integral representation of the solution for the following nonlocal $p$-Laplacian boundary value problem involving right-sided and left-sided fractional derivatives with respect to a power function:

is given by

where

and

with

Proof. Letting $-\phi_p\big({}^{\rho}D_{0^+}^{\beta}y(t)\big) = \mathcal{H}(t)$, we decompose the mixed boundary value problem (2.5) as

and

Solving the equation $^{\rho}D_{1^-}^{\alpha}\mathcal{H}(t) = -\psi(t)$, we get

where $c_0$ and $c_1$ are arbitrary constants. Using the condition $\mathcal{H}(1) = 0$ in (2.11) yields $c_0 = 0$. Then, inserting (2.11) with $c_0 = 0$ in the condition: $\mathcal{H}(0) = \lambda_2\mathcal{H}(\eta),$ the value of $c_1$ is found to be

So (2.11) becomes

where $G_2(t, s)$ is given in (2.8). Applying the integral operator ${}^{\rho}I^{\beta}_{0+}$ on both sides of the differential equation in (2.10), we have

where $d_0$ and $d_1$ are arbitrary constants.

Using (2.14) in the boundary conditions of (2.10), we obtain $d_0 = 0$ and

Thus, (2.14) takes the form:

where $G_1(t, s)$ and $\mathcal{H}(.)$ are respectively given in (2.7) and (2.13).

Lemma 2.3. The functions $G_1(t, s)$ and $G_2(t, s)$ given in (2.7) and (2.8) respectively, are continuous and possess the following properties:

$(i)$ $G_1(t, s) > 0, \, G_2(t, s) > 0$, $\forall \, t, s\in(0, 1);$

$(ii)$ $\frac{\mathcal{Q}_1\; \rho^{1-\beta}\; t^{\rho(\beta-1)}\; s^{\rho}\; (1-s^{\rho})^{\beta-1}}{\Gamma(\beta)(1-\lambda_1\mu^{\rho(\beta-1)})}\leq G_1(t, s)\leq \frac{\rho^{1-\beta}\; t^{\rho(\beta-1)}\; (1-s^{\rho})^{\beta-1}}{\Gamma(\beta)(1-\lambda_1\mu^{\rho(\beta-1)})},$ $\forall \, t, s\in(0, 1);$

$(iii)$ $\frac{\mathcal{Q}_2\; \rho^{1-\alpha}\; s^{\rho(\alpha-1)}\; (1-t^{\rho})^{\alpha-1}\; (1-s^{\rho})}{\Gamma(\alpha)(1-\lambda_2(1-\eta^{\rho})^{\alpha-1})}\leq G_2(t, s)\leq \frac{ \rho^{1-\alpha} \; s^{\rho(\alpha-1)}\; (1-t^{\rho})^{\alpha-1}}{\Gamma(\alpha)(1-\lambda_2(1-\eta^{\rho})^{\alpha-1})},$ $\forall \, t, s\in(0, 1),$

where

and

Proof. Let us first prove part $(i)$ with different cases.

Case 1. If $0\leq \tau\leq \min\{t, \mu\} < 1$, then we have $\frac{\tau^\rho}{\mu^\rho}\geq\tau^{\rho}$ since $\mu^{\rho} < 1$, which implies $1-\frac{\tau^{\rho}}{\mu^{\rho}}\leq 1-\tau^{\rho}$. Hence, we find that

which means that

Case 2. For $\mu\leq \tau\leq t$, we have

Thus, $\Lambda_1(1-\tau^{\rho})^{\beta-1}-(t^{\rho}-\tau^{\rho})^{\beta-1}\geq 0$.

Case 3. For $t\leq \tau\leq \mu$, we have

So, $\Lambda_1\big[(1-\tau^{\rho})^{\beta-1}-\lambda_1(\mu^{\rho}-\tau^{\rho})^{\beta-1}\big]\geq0$.

Case 4. For $\max\{t, \mu\}\leq \tau\leq 1$, it is obvious that $\Lambda_1(1-\tau^{\rho})^{\beta-1}\geq 0.$ Consequently, we get $G_1(t, \tau)\geq 0.$ By a similar argument, one can show that $G_2(t, s)\geq 0$. In order to establish $(ii)$, let $\Gamma(\beta)\rho^{\beta-1}(1-\lambda_1\mu^{\rho(\beta-1)})G_1(t, s) = g_1(t, s)$. Then, for $0\leq s\leq \min\{t, \mu\}$, we have

For $\mu\leq s\leq t$, let $\Gamma(\beta)\rho^{\beta-1}(1-\lambda_1\mu^{\rho(\beta-1)})G_1(t, s) = g_2(t, s)$. Then, we get

For $t \leq s\leq \mu$, let $\Gamma(\beta)\rho^{\beta-1}(1-\lambda_1\mu^{\rho(\beta-1)})G_1(t, s) = g_3(t, s)$. Then, we obtain

Lastly, when $\max\{t, \mu\}\leq s \leq 1$, it is clear that

Thus,

On the other hand, it easy to show that

Now, for $(iii)$, consider $\Gamma(\alpha)\rho^{\alpha-1}(1-\lambda_2(1-\eta^{\rho})^{\alpha-1})G_2(t, s)$. If $0\leq s\leq \min\{t, \eta\}$, then we find that

When $t\leq s\leq \eta$, let $\Gamma(\alpha)\rho^{\alpha-1}(1-\lambda_2(1-\eta^{\rho}))G_2(t, s) = h_1(s, t);$ then we have

When $\eta\leq s\leq t$, let $\Gamma(\alpha)\rho^{\alpha-1}(1-\lambda_2(1-\eta^{\rho}))G_2(t, s) = h_2(t, s);$ then we have

When $\max\{\eta, t\}\leq s\leq 1,$ let $\Gamma(\alpha)\rho^{\alpha-1}(1-\lambda_2(1-\eta^{\rho}))G_2(t, s) = h_3(t, s).$ Then, we obtain

Hence,

On the other hand, it is easy to show that

Thus, the proof is completed.

Remark 2.1. When $\rho\to 1$, (ii) is similar to the result presented in Theorem 1 in ^[29].

Lemma 2.4. Let $\psi\in C([0, 1], \mathbb{R)}$ and

Then, for $t, \tau, s\in [0, 1]$ and $\frac{1}{p}+\frac{1}{q} = 1, \, p, q\geq 1,$ the following results hold:

$(i) \|\omega\|\leq \Omega_1^{q-1}\|\psi\|^{q-1}, \, \, \, \|y\|\leq \Omega_2\Omega_1^{q-1}\|\psi\|^{q-1},$ where

$(ii)$ $\omega(\tau)\leq -m^{q-1}\Omega_3^{q-1}(1-\tau^{\rho})^{(\alpha-1)(q-1)}, \, \, \, y(t)\geq m^{q-1}\Omega_3^{q-1}\Omega_4 t^{\rho(\beta-1)},$ for $\psi(t)\geq m > 0$ and $\forall t\in [0, 1]$, where

Proof. By Lemma 2.3, we have

and

Consequently,

Thus, $\|\omega\|\leq \Omega_1^{q-1}\|\psi\|^{q-1}$. Similarly, we have that $\|y\|\leq \Omega_2\Omega_1^{q-1}\|\psi\|^{q-1}.$ This establishes $(i)$.

For $(ii)$, we have

Likewise, we have that $y(t)\geq m^{q-1}\Omega_3^{q-1}\Omega_4 t^{\rho(\beta-1)}$. This completes the proof.

The following lemma describes some properties of the $p$-Laplace operator which can easily be proved by using the mean value theorem when the function $\phi_p(k) = |k|^{p-2}k$ is differentiable at all $k$ except $k = 0$ and $|\frac{\partial \phi(k)}{\partial k} |$ is bounded by $(p-1)\max|k|^{p-2}$ for $p > 2$ and bounded by $(p-1)\min|k|^{p-2}$ for $1 < p < 2$.

Lemma 2.5. (see (2.1) and (2.2) on page 3268 in ^[30]) The following relations hold for the $p$-Laplace operator:

$(i)$ For $1 < p\leq 2,$ $|k_1|, |k_2|\geq \Delta_1 > 0$ and $k_1 k_2 > 0$, $|\phi_p(k_2)-\phi_p(k_1)|\leq (p-1)\Delta_1^{p-2}|k_2-k_1|;$

$(ii)$ For $p > 2,$ $|k_1|, |k_2|\leq \Delta_2$ and $k_1 k_2 > 0$, $|\phi_p(k_2)-\phi_p(k_1)|\leq (p-1)\Delta_2^{p-2}|k_2-k_1|.$

3.   Existence and uniqueness results

In this section, we discuss the existence and uniqueness of the solutions to the problem (1.1). For a given number $\mathcal{M} > 0,$ let us consider the following set

and denote by $\Gamma[O, \mathcal{M}]$ a closed ball in the space of the continuous function $C[0, 1]$.

Theorem 3.1. Assume that $1 < p\leq 2$ and there exist positive constants $M_1, \, M_2, \, C_1, \, C_2, \, \kappa_1 and \kappa_2$ such that

$(A_1)$ $|f(t, y, \omega)|\leq M_1, \, |g(t, y, \omega)|\leq M_2\, \, \mathit{\mbox{for}} \, \, \, (t, y, \omega)\in \Upsilon_{\mathcal{M}};$

$(A_2)$ $|f(t, y_1, \omega_1)-f(t, y_2, \omega_2)| \leq C_1 |y_2-y_1|+C_2|\omega_2-\omega_1|, \, \, \mathit{\mbox{for}} \, \, \, (t, y_i, \omega_i)\in \Upsilon_{\mathcal{M}}, \, i = 1, 2;$

$(A_3)$ $|g(t, y_1, \omega_1)-g(t, y_2, \omega_2)| \leq \kappa_1 |y_2-y_1|+\kappa_2|\omega_2-\omega_1|, \, \, \mathit{\mbox{for}} \, \, \, (t, y_i, \omega_i)\in \Upsilon_{\mathcal{M}}, \, i = 1, 2;$

$(A_4)$ $L_1: = (q-1)\mathcal{M}^{q-2}\Omega_1^{q-1}\Bigg\{|\nu_1|\Big(C_1\Omega_2+C_2\Big)+\frac{|\nu_2|}{\rho^{\zeta}\Gamma(\zeta+1)}\Big(\kappa_1\Omega_2+\kappa_2\Big)\Bigg\} < 1$.

Then the mixed boundary value problem (1.1) has a unique solution satisfying the following inequalities:

where $\mathcal{M}\geq|\nu_1|M_1+|\nu_2 |\frac{M_2}{\rho^{\zeta}\Gamma(\zeta+1)}$.

Proof. Define an operator $\mathcal{G}:C[0, 1]\to C[0, 1]$ by

Observe that the continuity of $G_1(t, \tau)$, $G_2(\tau, s)$, $f(t, y, \omega)$ and $g(t, y, \omega)$ leads to that of the operator $\mathcal{G}$. Moreover, if $y(t)$ is a solution to the problem (1.1), then $\psi(t) = {}^{\rho}D_{1^-}^{\alpha}\big(\phi_p\big({}^{\rho}D_{0^+}^{\beta}y(x)\big)\big)$ is the fixed point of the operator $\mathcal{G}$. Conversely, if $\psi(t)$ is a fixed point of the operator $\mathcal{G}$, then

is a solution to the problem (1.1).

Next, we need to show that the operator $\mathcal{G}$ maps $\Gamma[O, \mathcal{M}]$ into itself. Let $\psi\in \Gamma[O, \mathcal{M}]$; then, by Lemma 2.3, we have

Consequently, for any $t\in[0, 1]$, there is $(t, y(t), \omega(t))\in \Upsilon_{\mathcal{M}}.$ So, from $(A_1)$, we have

Therefore, $(\mathcal{G}\psi)(t)\in \Gamma[O, \mathcal{M}].$ Thus, the operator $\mathcal{G}$ maps $\Gamma[O, \mathcal{M}]$ into itself. Now, we show that the operator $\mathcal{G}:\Gamma[O, \mathcal{M}]\to \Gamma[O, \mathcal{M}]$ is a contraction. From $(A_2), \, (A_3)$, Lemma 2.3, Lemma 2.4 and $(ii)$ of Lemma 2.5, there is $\Delta_2: = \Omega_1\mathcal{M}\geq |\int_{0}^{1}s^{\rho-1}G_2(\tau, s)\psi(s)ds|$ for each $\psi_1(t), \psi_2(t)\in \Gamma[O, \mathcal{M}]$, and $1 < p\leq 2$ (that is, $q\geq 2$). Thus, we obtain

which, on taking the norm for $[0, 1],$ yields

Since $L_1 < 1$ by $(A_4)$, the operator $\mathcal{G}$ is a contraction. So, we deduce by Banach's contraction mapping principle that $\psi(t)$ is the unique fixed point of the operator $\mathcal{G}$. Hence, there exists a unique solution to the mixed boundary value problem (1.1) satisfying (3.1). The proof is completed.

In the following result, we consider a special case of $\Upsilon_\mathcal{M}$. With the aid of Lemma 2.4, for $m > 0$ with $\mathcal{M} > m$, we define the sets

and

The following theorem is concerned with the existence of a unique solution to the problem (1.1) when $p > 2.$

Theorem 3.2. Assume that $p > 2$ and there exist positive numbers $m_1, \, m_2, \, M_1, \, M_2, \, C_1 \, and \, C_2$ such that

$(B_1)$ $m_1\leq f(t, y, \omega)\leq M_1 and \, m_2\leq g(t, y, \omega)\leq M_2, \, \, \mathit{\mbox{for}} \, \, (t, y, \omega)\in \Upsilon_\mathcal{M}^+;$

$(B_2)$ $|f(t, y_1, \omega_1)-f(t, y_2, \omega_2)| \leq C_1 |y_2-y_1|+C_2|\omega_2-\omega_2|, \mathit{\mbox{for}} \, \, (t, y_i, \omega_i)\in \Upsilon_\mathcal{M}^+, \, i = 1, 2;$

$(B_3)$ $|g(t, y_1, \omega_1)-g(t, y_2, \omega_2)| \leq \kappa_1 |y_2-y_1|+\kappa_2|\omega_2-\omega_1|, \, \, \mathit{\mbox{for}} \, \, \, (t, y_i, \omega_i)\in \Upsilon_\mathcal{M}^+, \, i = 1, 2;$

$(B_4)$ $L_2: = (q-1)m^{q-2}\Omega_3^{q-2}\Omega_1\Bigg\{|\nu_1 |\Big(C_1\Omega_2+C_2\Big)+\frac{ |\nu_2|}{\rho^{\zeta}\Gamma(\zeta+1)}\Big(\kappa_1\Omega_2+\kappa_2\Big)\Bigg\} < 1$.

Then the mixed boundary value problem (1.1) has a unique solution satisfying the following inequalities

where $\mathcal{M}\geq|\nu_1|M_1+|\nu_2| \frac{M_2}{\rho^{\zeta}\Gamma(\zeta+1)}$ and $0 < m\leq\nu_1m_1+\nu_2 \frac{m_2}{\rho^{\zeta}\Gamma(\zeta+1)}$.

Proof. As argued in the proof of the last theorem, the operator $\mathcal{G}$ defined by (3.2) is continuous and $\mathcal{G}$ maps any $\psi\in \Gamma_m$ into itself.

Now, from $(B_2), \, (B_3)$, Lemma 2.3, Lemma 2.4, and $(i)$ of Lemma 2.5, there exists $\Delta_1: = m\; \Omega_3 (1-\tau^{\rho})^{\alpha-1}\leq |\int_{0}^{1}s^{\rho-1}G_2(\tau, s)\psi(s)ds|.$ Then, for all $\psi_1, \psi_2\in \Gamma_m$, and $p > 2$ (that is, $1 < q\leq 2$), we find that

which, after taking the norm for $t \in [0, 1],$ takes the form:

with $L_2 < 1$ by the condition $(B_4)$. Consequently, the operator $\mathcal{G}$ is a contraction. Hence, by Banach's contraction mapping principle, $\psi(t)$ is the unique fixed point of the operator $\mathcal{G}$. Therefore, there exists a unique solution to the problem (1.1) satisfying (3.3). This finishes the proof.

4.   Examples

Consider the following problem

where $\alpha = 5/3, \, \beta = 3/2, \, \rho = 1/2, \lambda_1 = 1/2, \, \lambda_2 = 1, \, \mu = 3/4, \, \eta = 1/2, \, \nu_1 = 3/4 \, and \, \nu_2 = -2/3,$ and $p$, $f(t, y, \omega)$ and $g(t, y, \omega)$ will be fixed later.

From the given data, we have

and

which satisfy the properties expressed in Lemma 2.3. Moreover, for $t, s\in(0, 1)$, we have $G_1(t, s) > 0, G_2(t, s) > 0$ and

with $\mathcal{Q}_1\approx 0.465302$ and $\mathcal{Q}_2\approx 0.4410348$.

For illustrating Theorem 3.1, let us take

and $p = 3/2$ (that is, $q = 3$). Using the given values, it is found that $\Omega_1 = 3.7750084, \, \Omega_2 = 3.9792441, \, \Omega_3 = 0.6243413 and \, \Omega_4 = 0.2557131$. Also, $\mathcal{M}$ satisfies the following relations:

with $\|y\|\leq 56.706968\; \mathcal{M}^2$ and $\|\omega\|\leq 14.250688\; \mathcal{M}^2$. Choosing $M_1 = 0.3, \, M_2 = 0.08$ and $\mathcal{M} = 0.4$, it can easily be verified that the functions $f(t, y, \omega)$ and $g(t, y, \omega)$ given by (4.2) and (4.3) respectively, satisfy the condition $(A_1)$. Furthermore, on the domain:

we find that

Obviously the conditions $(A_2)$ and $(A_3)$ are satisfied with $C_1 = 0.01680206459, \, C_2 = 0.001055606548, \, \kappa_1 = 0.0016 and \, \kappa_2 = 0.00729635246$. Also, $L_1 = 0.7169476783 < 1$. Thus all of the conditions of Theorem 3.1 are satisfied and hence the problem (4.1) has a unique solution on $\Upsilon_{0.4}$.

We illustrate Theorem 3.2 by choosing

and $p = 4$ (that is, $q = 4/3$). Here the values of $\Omega_1, \, \Omega_2 \, and \, \Omega_3$ are the same as those found in the first example and $\Omega_4 = 0.592399$. Letting $m = 0.05 and \mathcal{M} = 0.4$, as argued in the first example, we find that the functions $f(t, y, \omega)$ and $g(t, y, \omega)$ given by (4.4) and (4.5) respectively, satisfy the condition $(B_1)$ in the following domain: $\Upsilon_{0.4}^+: = \{(t, y, \omega), 0\leq t\leq1, 0.1865284535\; t^{1/4}\leq y(t)\leq 4.565200828, -1.147253270\leq\omega\leq - 0.3148695697\; (1-\sqrt{t})^{2/9}\}$.

Moreover, the conditions $(B_2)$ and $(B_3)$ hold true with $C_1 = 1/120, \, C_2 = 0.01912088784, \, \kappa_1 = 1/90, \, \kappa_2 = 0.005247826157$ and $L_2 = 0.3153939012 < 1$. Thus all of the conditions of Theorem 3.2 are satisfied and hence the problem (4.1) has a unique solution on $\Upsilon^+_{0.4}$.

5.   Conclusions

In this paper, we have investigated the criteria for ensuring the uniqueness of positive solutions for a class of fractional integro-differential equations with a $p$-Laplacian operator, complemented with nonlocal boundary conditions involving fractional derivatives and the $p$-Laplacian operator. Using a method employed in ^[31] together with the properties of the associated Green's functions established for the given problem, we proved two uniqueness results for the cases $1 < p\le 2$ and $p > 2$, respectively. Illustrative examples demonstrating application of the obtained results are presented. It is worthwhile to note that our results are new in the given configuration and enrich the literature on $p$-Laplacian fractional boundary value problems involving right-sided and left-sided fractional derivative operators, as well as left-sided fractional integral operators with respect to the power function.

Acknowledgements

The Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia has funded this project, under grant no. (KEP-PhD: 35-130-1443). The authors thank the Editor for indicating the correct terminology and references for the concepts of fractional calculus used in this paper. The authors also thank the reviewers for their constructive remarks on their work.

Conflict of interest

The authors declare that there is no conflict of interest.