A new fixed point approach for solutions of a $p$-Laplacian fractional $q$-difference boundary value problem with an integral boundary condition

1.   Introduction

As fractional calculus has grown and developed, numerous investigations have been undertaken on the existence and uniqueness of the solutions for initial and boundary value problems, including fractional equations ^[1,2,3,4]. Recently, with the generalization of fractional operators on arbitrary spaces, the study of fractional initial and boundary value problems (BVPs) has attracted the attention of many scientists ^[5,6]. One of these generalizations led to the definition of the fractional derivative on the time scale interval space $\mathbb{T} = \{q^{k}:k\in\mathbb{N}, 0 < q < 1\}$ and the study on the existence and uniqueness of solutions for initial and boundary value problems incorporating fractional differential equations were conducted on this space ^[7,8,9,10]. However, there are few papers about the investigation of $q$-difference BVPs within the $p$-Laplacian operator ^[11,12,13].

In ^[14], Miao and co-authors used some fixed point theorems on partially or- dered sets and p-Laplacian operators to study the positive solutions of q-difference BVP

where $0 < \upsilon < 1, 2 < \mu < 3, \; 0 < \gamma\xi^{\mu-2} < 1$, $D^\mu_q$ is the Riemann-Liouville derivative and $\phi_p(\zeta) = |\zeta|^{p-2}\zeta$ is $p$-Laplacian operator, $p > 1$.

In ^[15], Mardanov and co-authors used some known fixed point theorems to study the $q$-difference BVP of the form

where $\phi_p(\zeta) = |\zeta|^{p-2}\zeta$ is $p$-Laplacian operator, $p > 1, \phi^{-1}_p(\zeta) = \phi_s(\zeta)$, where $p^{-1}+s^{-1} = 1$ and $0 < \alpha, \beta\leq1$.

In a previous work by Aktuğlu and Özarslan ^[16], they investigated a Caputo-type q-fractional boundary value problem involving the p-Laplacian operator. This problem can be expressed as:

subject to the boundary conditions:

where $a_0$, $a_1\neq0$, $\alpha > 1$ and $f\in C([0, 1]\times \mathbb{R}, \mathbb{R})$. Aktuğlu and Özarslan employed the Banach contraction mapping principle to establish the existence and uniqueness of a solution for this boundary value problem under specific conditions.

Yan and Hou in ^[17] applied the Avery-Peterson fixed point to obtain some existing results for the $q$-difference BVP

where $a, \eta\geq0$, $f\in C([0, 1]\times[0, +\infty)\times\mathbb{R}, [0, +\infty)), h\in C([0, 1]\times[0, +\infty))$ and $g_i$ is nonnegative, integrable and $\int_0^1g_i(\varsigma)d_q\varsigma\in[0, 1), i = 1, 2.$

In 2020, Ragoub and co-authors in ^[18], investigated the q-fractional boundary value problem with the p-Laplace operator

where $aD^\alpha_q, aD^\beta_q$ are the fractional q-derivative of the riemann-Liouville type with $1 < \alpha, \beta < 2$, $0\leq A, B\leq 1, 0 < \xi, \delta < 1, \phi_p(s) = |s|^{p-2}s, p > 1, \phi_p^{-1} = \phi_r, \frac{1}{p}+\frac{1}{r} = 1$, and $Q:[a, b]\to\mathbb{R}$ is a continuous function on $[a, b]$.

Inspired by the aforementioned research, in this research, we examine $q$-difference BVP

where $2 < \nu, \mu < 3$, $D^\mu_q$ is the Riemann-Liouville fractional derivative and $\phi_r(\zeta) = |\zeta|^{r-2}\zeta$ is $p$-Laplacian operator, $r > 1$, $\phi_r^{-1}(\zeta) = \phi_s(\zeta), \; \frac{1}{s}+\frac{1}{r} = 1$, $g:[0, 1]\to[0, \infty)$ be a function such that $\tau^{\mu-1}g(\tau)\in L^1[0, 1]$ and $\lambda$ is a constant such that $\lambda\int_0^1\tau^{\mu-1}g(\tau)d_q\tau < 1.$ Unlike earlier studied equations, Eq (1.2) is formed using the $p$-Laplace operator and a high order fractional derivative, making Green's function analysis challenging. Using a new fixed point theorem that incorporates an $\mathfrak{a}-\eta$-Geraghty contraction, the existence of a positive solution for Eq (1.2) is investigated.

This is how the remainder of the paper is structured. In Section 2, we present a few baseline data and relevant resources. In Section 3, the Green function of the problem and its required attributes are calculated. In Section 4, the existence and uniqueness of positive solutions are demonstrated using a new fixed point theorem. Further, we study the existence of the solution to the fractional $q$-difference boundary value issue of the Riemann-Liouville derivative by using a $\mathfrak{a}-\eta$-Geraghty contraction. Finally, a few examples are provided to demonstrate the usefulness of our findings. We concluded the paper by drawing a brief conclusion.

2.   Preliminaries

Prior to delving into the primary notion of this work, a few fundamental ideas must be introduced, and several helpful instruments used in this work should be mentioned (see^[19,20]).

Let $q\in(0, 1)$, we denote

The $q$-analog of the power function $(\nu-\omega)^\kappa$ with $\kappa\in\mathbb N_0 = \{0, 1, 2, \ldots\}$ is

Furthermore, let $\gamma\in\mathbb R$, then

If $\omega = 0$, then $\nu^{(\gamma)} = \nu^\gamma$ and we denote $0^{(\gamma)} = 0$ for $\gamma\geq0$.

The $q$-gamma function is given by

Similar to the gamma function, the following relationship is also established for the $q$-gamma function.

The $q$-integral on $[0, b]$ is given by

Let $a\in[0, b]$ and $f$ be given on $[0, b]$, then its $q$-integral from $a$ to $b$ stated as follows:

Definition 2.1. Let $\upsilon\geq0$ and $f$ be a real function on $[a, b]$. The RL $q$-integral of order $\upsilon$ is given by

Definition 2.2. The fractional $q$-derivative of the RL- type of order $\upsilon \geq 0$ for the function $f$ is defined by

and

where $[\upsilon]$ is the smallest integer greater than or equal to $\upsilon$.

Definition 2.3. Let $\upsilon\geq0$ and the Caputo $q$-derivatives of $f$ be given by

where $[\upsilon]$ is the smallest integer greater than or equal to $\upsilon$.

If $f(\tau) = \tau^{\gamma-1}$ for $\gamma\notin\mathbb N$, then

Lemma 2.4. ^[19] Let $\upsilon, \gamma\geq0$ and $f:[a, b]\to\mathbb R$ is continuous on $[a, b]$. Then

Lemma 2.5. ^[19] Let $f:[a, b]\to\mathbb R$ be differentiable and $p$ be a positive integer. Then

Definition 2.6. We set $\Psi$ be the set of $\psi$ that satisfy the following condition

(1) $\psi:[0, \infty)\to[0, \infty)$ is a continuous and nondecreasing;

(2) $\psi(\tau) > 0$ for all $\tau\in(0, \infty)$;

(3) $\psi(0) = 0$;

(4) $\lim_{\tau\to\infty}\psi(\tau) = \infty$.

Theorem 2.7. ^[21] Suppose $(\mathfrak{E}, d)$ be a partially ordered (with respect the order $\leq$) complete metric space such that satisfy the following conditions

i) If $\{\zeta_n\}$ is a nondecreasing convergent in $\mathfrak{E}$ ($\lim_{n\to\infty}\zeta_n = \zeta$) then $\zeta_n < \zeta, \; n\in\mathbb{N}$;

ii) Let $\psi\in\Psi$, $\digamma:\mathfrak{E}\to\mathfrak{E}$ be a nondecreasing with

and $\zeta_0\leq\digamma(\zeta_0)$.

Then $\digamma$ has a fixed point.

Theorem 2.8. ^[22] By considering the following extra condition,

iii) For each pair $\zeta$ and $\omega$ in $\mathfrak{E}$, there exists a member like $\varpi$ in $\mathfrak{E}$ such that it is comparable to $\zeta$ and $\omega$.

to the assumptions of the previous theorem, we reach the uniqueness of the fixed point.

3.   Green function

This section is dedicated to constructing the Green's function of equations and demonstrating some of its features.

Lemma 3.1. ^[23] Assume $\varrho:[0, 1]\to [0, \infty)$ be a continuous, then fractional q-difference BVP

is equivalent to:

where

Lemma 3.2. Let $h\in C([0, 1]), 2 < \mu\leq 3$, then fractional q-difference BVP

is equivalent to:

where

with

Proof. By integrating of order $\mu$ from Eq (3.4) one can get

Utilizing the conditions $\zeta(0) = 0$, we get $c_3 = 0$. On the other hand by differentiating from relation (3.8) we have

by applying boundary condition $D_q\zeta(0) = 0$, we have $c_2 = 0$, and from the last boundary condition we get

So

If we replace $\zeta(\tau)$ from the relation (3.8) into the relation (3.9) we have

Hence

So

Consequently,

□

Lemma 3.3. Fractional BVP (1.2) is equivalent to:

where $G$ and $H$ were given by (3.3) and (3.6) respectively.

Proof. Let $\varrho(\tau) = \varphi_s(\zeta(\tau))$, $h(\tau) = -\Lambda(\tau, \zeta(\tau))$. Then from Lemma 3.1 we get

Now from Lemma 3.2 we have

□

Lemma 3.4. ^[14,23] Let $\alpha: = \mu$ or $\alpha = \nu$, also let $H(\tau, q\varsigma): = H_\nu(\tau, q\varsigma)$ or $H(\tau, q\varsigma): = H_\mu(\tau, q\varsigma)$, then $H$ satisfies the following conditions

(1) $H$ is continuous and $H(\tau, q\varsigma)\geq 0$ for all $\tau, \varsigma\in[0, 1]$;

(2) $H$ is a strictly increasing function concerning the first variable;

(3) $\tau^{\alpha-1}H(1, q\varsigma)\leq H(\tau, q\varsigma)\leq H(1, q\varsigma)$ for all $\tau, \varsigma\in[0, 1].$

Lemma 3.5. Let $G$ be the function that defined by (3.6), then $G$ satisfies the following conditions.

(1) $G$ is a continuous function and $G(\tau, q\varsigma)\geq 0$ for all $\tau, \varsigma\in[0, 1]$;

(2) $G$ is a strictly increasing function concerning the first variable;

(3) $\tau^{\mu-1}G(1, q\varsigma)\leq G(\tau, q\varsigma)\leq G(1, q\varsigma)$ for all $\tau, \varsigma\in[0, 1].$

The proof is straightforwardly attained from Lemma 3.4.

4.   Main results

Here, we will use all of the items reported in the preceding sections to develop the primary findings of this research.

Let $\mathcal{P}$ be defined as

It is easy to check that $\mathcal{P}$ is a cone and since it is a closed set of $C([0, 1])$, hence it is a complete metric space that is equipped with the meter

also for convenience, we set

Now consider the $\digamma:\mathcal{P}\to\mathcal{P}$ defined by

Since BVP (1.2) is equivalent with (3.10), so the solutions of BVP (1.2) are the fixed points of the operator (4.1). To prove the existence of fixed points of (3.10), we apply Theorems 2.7 and 2.8.

We make use of the following conditions:

(A1) Let $g:[0, 1]\to[0, \infty)$ such that $\tau^{\mu-1}g(\tau)\in L^1[0, 1]$ and $\lambda$ is a constant such that $\lambda\int_0^1\tau^{\mu-1}g(\tau)d_q\tau < 1.$

(A2) $\Lambda\in C([0, 1]\times[0, \infty), [0, \infty))$ and it is non-decreasing respect to the second variable.

(A3) There exists $0 < (\eta+1)\Delta < 1$ such that for all $0\leq\omega\leq\zeta < \infty$ we have

Theorem 4.1. Assume that (A1), (A2) and (A3) hold, then BVP (1.2) has a unique positive solution.

Proof. Given Lemma 3.5 and (A1), it is concluded that $\digamma(\mathcal{P})\subset\mathcal{P}$. Now we check all conditions of Theorems 2.7 and 2.8 for the operator (3.10). Let $\zeta, \omega\in\mathcal{P}$ and $\zeta\geq\omega$, by (A1) we get

That is the operator $\digamma$ is nondecreasing.

Now let $\zeta\geq\omega$, in view of (A2) we obtain

The function $g(\zeta): = \ln(\zeta+1)$ is a nondecreasing function, so by (A2), we have

Now if we set $\psi(\zeta): = \zeta-\ln(\zeta+1)$, then $\psi\in\Psi$. Thus for all $\zeta\geq\omega$ we obtain

Since $G(\tau, q\varsigma)\geq0$, $H_\nu(\tau, q\varsigma)\geq 0$ and $\Lambda\geq0$, so

hence by Theorem 2.7, BVP (1.2) has at least one positive solution. On the other hand since $(\mathcal{P}, \leq)$ satisfies condition (iii) of Theorem 2.8, hence, the BVP (1.2) has a unique positive solution. □

Let $\Theta$ contains all $\theta:\mathbb{R}^+\to[0, 1)$ which satisfy the condition: $\theta(t_n)\rightarrow 1$ implies $t_n\rightarrow 0$.

Definition 4.2. ^[24] Let $(X, d)$ is MS and $\mathfrak{a}, \vartheta:X\times X\to\mathbb{R}^+$ two functions. $g:X\to X$ is said to be an $\mathfrak{a}$-$\vartheta$-Geraghty contraction if there exists $\theta\in \Theta$ such that for $\nu, \omega\in X$,

Definition 4.3. ^[24] Let $g:X\to X$ and $\mathfrak{a}, \vartheta:X\times X\to\mathbb{R}^+$ be given. Then $g$ is called $\mathfrak{a}$-admissible with respect to $\vartheta$, if for $\nu, \omega\in X$,

Theorem 4.4. ^[24] Let $(X, d)$ be a complete metric space and $\varphi:X\rightarrow X$ be a $\mathfrak{a}-\theta$-Geraghty contraction such that

$(i) \; \varphi$ is $\mathfrak{a}$-admissible respect to $\vartheta$;

$(ii) \; \exists \; w_0\in X$ with $\mathfrak{a}(w_0, \varphi w_0)\geq \vartheta(w_0, \varphi w_0)$;

$(iii) \; \varphi$ is continuous.

Then $\varphi$ has a fixed point.

Theorem 4.5. Let $(X, d)$ be a complete metric space and $\varphi:X\rightarrow X$ be a $\mathfrak{a}-\theta$-Geraghty contraction such that

$(i) \; \varphi$ is $\mathfrak{a}$-admissible respect to $\vartheta$;

$(ii) \; \exists \; w_0\in X$ with $\mathfrak{a}(w_0, \varphi w_0)\geq \vartheta(w_0, \varphi w_0)$;

$(iii) \; \{w_n\}\subseteq X$, $w_n\rightarrow u$ in $X$ and $\mathfrak{a}(w_n, w_{n+1})\geq \vartheta(w_n, w_{n+1})$ then $\mathfrak{a}(w_n, w)\geq \vartheta(w_n, w)$.

Then $\varphi$ has a fixed point.

The proof can be concluded by following the same arguments as in the proof of Theorem 2.4 in ^[25].

Theorem 4.6. Suppose that (A1) hold and there exist $\rho:\mathbb R^2\to\mathbb R$ and $\theta\in\Theta$ with the following property:

(i) For all $\zeta, \omega\in C([0, 1])$ and $\tau\in[0, 1]$ we have

(ii) For $\zeta, \omega\in C([0, 1])$, $\rho(\zeta(\tau), \omega(\tau))\geq0$ and there exists $\zeta_0\in C([0, 1])$ with

(iii) If $\rho(\zeta(\xi), \omega(\xi))\geq0$, then

(iv) if $\{\zeta_n\}\subseteq C([0, 1])$, $\zeta_n\to \zeta$ in $C([0, 1])$, and $\rho(\zeta_n, \zeta_{n+1})\geq0$, then $\rho({\zeta}_n, \zeta)\geq0$.

Then Problem (3.1) has at least one solution.

Proof. From Lemma 3.3, $\zeta\in C([0, 1])$ is a solution of (3.1) if and only if is a solution of

Thus we find the fixed point of $\digamma:C([0, 1])\to C([0, 1])$ given by

Let $\zeta, \omega\in C([0, 1])$ with $\rho(\zeta(\tau), \omega(\tau))\geq0$. By (i), we obtain

Let $\mathfrak{a}:C([0, 1])\times C([0, 1])\to\mathbb{R}^+$ be stated as:

Thus, we have

Then $\digamma$ is an $\mathfrak{a}$-$\theta$-contractive. From (iii) and the definition of $\mathfrak{a}$ we get

for $\zeta, \omega\in C([0, 1])$. Thus, $\digamma$ is $\mathfrak{a}$-admissible. By (ii) $\exists \; \zeta_0\in C([0, 1])$ with $\mathfrak{a}(\zeta_0, \digamma \zeta_0)\geq\vartheta(\zeta_0(\tau), \digamma (\zeta_0(\tau)))$. From (iv) and Theorem 4.5, there is $\zeta^*\in C([0, 1])$ with $\zeta^* = \digamma \zeta^*$. Hence $\zeta^*$ is a solution of the problem. □

5.   Examples

The following are two supportive examples that adhere to all theoretical presumptions.

Example 5.1. Consider the $q$-difference BVP

where $\mu = \nu = \frac{5}{2}$, $q = \lambda = \frac{1}{2}$, $g(\tau) = \frac{1}{10}\tau$, $r = \frac{7}{3}$ and $\Lambda(\tau, \zeta(\tau)) = (\frac{1}{100}\sin^2\tau+\frac{1}{2})\ln(2+\zeta(\tau))$. It is easy to see that $\Lambda$ is a continuous function and for $\tau\in[0, 1]$ we have $\Lambda(\tau, \zeta)\neq0$. Also $\Lambda$ is a nondecreasing with respect to the second variable. Now since in this problem $\mu = \nu$, we let $H: = H_\mu = H_\nu$ and due to the fact that

By using Mathematica software, one can calculate the following quantities:

Moreover, we get

So $\eta = 0.51$ and $(\eta+1)\Delta = 1.51\times 0.3794 = 0.5728 < 1$. Consequently all conditions of the Theorem 4.1 hold and the q-difference BVP has a unique positive solution like $\zeta(\tau)$ that satisfies

Example 5.2. Let $\theta(\tau) = {(\cos (\tau))^{\frac{3}{4}}}$, $\rho(y, z) = yz$, $\zeta_n(\tau) = \dfrac{\tau}{n^2+1}$. Consider $\Lambda:\mathcal{I}\times C(\mathcal{I})\to[0, \infty]$ and the BVP

One can easily see that $\Lambda(\tau, \zeta(\tau)) = \frac{1}{4}\sin ^{2}2(\zeta(\tau))$, $(\tau, v(\tau))\in \mathcal{I}\times [1, \infty)$. $\theta(t_n)\rightarrow 1$ implies $t_n\rightarrow 0$, hence $\theta\in \Theta$.

Furthermore, we get

when $\tau\in \mathcal{I}$ and $\zeta(\tau), \omega(\tau)\in [1, \infty)$ with $\rho(\zeta(\tau), \omega(\tau))\geq 0$. So the condition (i) from Theorem 4.6 hold.

If $\zeta_0(\tau) = \tau$, then

for $\tau\in \mathcal{I}$. Further, $\rho(\zeta(\tau), \omega(\tau)) = \zeta(\tau)\omega(\tau)\geq 0$ implies that

It is obvious that condition (iv) in Theorem 4.6 hold. Hence, the all conditions of Theorem 4.6 are satisfied. Thus, Eq (1.2) has at least one solution.

6.   Conclusions

There are few papers in the literature about $p$-Laplacian $q$-fractional boundary value problems, thus we investigated a class of $p$-Laplacian $q$-fractional boundary value problems with an integral boundary condition. The Green function of the problem was computed and some properties of the Green function were determined. By using a new fixed point theorem that involves a $\mathfrak{a}$-$\eta$-Geraghty contraction, the existence of positive solutions was proved. Two examples are provided to support the theoretical findings.

The technique applied in this paper is different and may be used effectively to verify the existence of solutions to many sorts of equations. Moreover, one can use this technique to verify the existence of positive solutions for some boundary value problems including a system of $q$-fractional differential equations in the future.

Author contributions

H. Afshari: Conceptualization, Investigation; A. Ahmadkhanlu: Conceptualization, Formal analysis, Writing –review; J. Azabut: Writing–original draft, Editing. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

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

Acknowledgments

J. Alzabut is thankful to Prince Sultan University and OSTİM Technical University for their endless support.

Conflict of interest

The authors declare no conflicts of interest.