Existence of solutions by fixed point theorem of general delay fractional differential equation with $p$-Laplacian operator

1.   Introduction

Fractional calculus (FC) is among the most quickly increasing aspects of research in mathematics. The arbitrary-order generalization of differentiation and integration is FC. Fractional derivatives have been around for nearly three decades. FC is now widely used in the research and engineering to represent an effective and wide range of real-world issues. Biophysics, aerodynamics, capacitor theory, control theory, biology, polymer rheology, geophysics, and many more domains use FDEs. The monograph ^[1,2,3] provides a mathematical perspective and the principles of FC and FDEs. The theory of FDEs and its applications benefited from the contributions of many mathematicians, engineers, and physicists. To address a broad range of natural processes and phenomena, several types of fractional order derivative operators have been developed. More information can be found at ^[4,5].

The EU of solutions of FDEs with regard to fractional boundary value conditions has attracted a lot of interest. Existence results for singular FDEs for mixed and Dirichlet problems was discussed by Agarwal et al. ^[6,7]. EU for FDEs with Caputo derivative with bounded delay was analyzed by Baleanu et al. ^[8]. For instance, see ^[10,11]. For more information see ^[12,13,14]. Also, using a variety of mathematical approaches, several authors have recently investigated FDEs with the operator, $p$-Laplacian. Positive solution for a FDE with integral boundary conditions; At resonance, a boundary value problem for a FDE; and In Banach spaces, solutions of FDE with the operator, $p$-Laplacian is discussed in ^[15,16,17] respectively. For more information, see ^[18].

Also, the stability of FDEs has recently been proven to have a large interset. Stability of the solutions of FDEs is one of the qualitative properties of FDEs. Besides that, many scientists have become interested in the Ulam-Hyers stability concerns. The Hyers-Ulam stability refers to the fact that an FDE's exact solution is extremely close to the approximate solution of the differential equation (DE), and the error is small enough to be calculated. For ABC-FDEs with $p$-Laplacian operator containing spatial singularity, EU of solutions, as well as Hyers-Ulam stability, were examined in ^[19]. For much more information on stability analysis, read the articles ^[22,23]. The stability and existence results for non-linear FDE with singularity have been discussed by Alkhazzan et al. ^[27]. EU findings for delay DE was discussed in ^[28]. Existence results and Hyers Ulam stability analysis for FDE using fixed point theorem was also examined in ^[29]. The problem we have proposed is broader and more complex than the ones previously discussed. For more details, and useful works see ^{[30,31,32,33,34,35,36,37]}. For more results, one may refer ^{[39,40,41,42,43,44,45]}.

To the aimed to contribute, there seems to be no article in the literature which then examines the aforementioned problem. According to the literature, no attempt has been made to investigate the $p$-Laplacian operator in the context of delay fractional DEs. As a direct consequence, there has been no analysis of the operator, $p$-Laplacian in the context of the general class of non-linear delay FDE in the literature. We also discuss the Green's function properties. As a result of the previous work, we look at the Hyers-Ulam stability along with uniqueness and existence and results for non-linear FDEs that involves the $\phi^{*}_{P}$-Laplacian operator:

Here $\mathcal{\; ^{C}\mathfrak{D}}^{\mathcal{\varsigma}}, \mathcal{\mathfrak{D}}^{\mathcal{ \varrho }}$ stands for fractional derivatives of order $\varsigma$ and ${ \varrho }$ in Caputo's and Riemann-Liouville sense respectively, ${\psi^{\ast}}$ is continuous function, $\Lambda(\mathcal{ \omega})$ is non-vanishing function and

The orders $m-1 < \varrho, \; \; \varsigma \le m,$ where $m \in \{3, 4, 5, \cdots\}, \; \mu > 0$ and $\psi^{\ast}\in \mathcal{L}[0, 1]$ and

symbolises the $p$-Laplacian operator and satisfies the

This is how the paper is structured. The remaining of the work is split into four sections. Sections 1 and 2 contain the introducton and fundamental definitions as well as the desired lemmas. Section 3 discusses the properties of Green's functions. Section 4 contains the Existence and uniqueness results. Section 5 discusses HU stability and Section 6 illustrates an application. We'll go over certain definitions, relevant theorems and auxiliary technical lemmas that are necessary to get to the main outcome. Section 7 includes conclusions.

2.   Preliminaries

Definition 2.1. ^[1] The Riemann-Liouville (RL) fractional integral for an integrable and real valued continuous function, $\psi$ of fractional order $\delta \in\mathbb{R}$ is expressed as on $(0, +\infty)$:

Definition 2.2. ^[1] The Caputo's derivative of fractional order $\delta \in\mathbb{R} \; (\delta > 0)$, for $\psi$ defined on $(0, +\infty)$ which is k-times continuously differentiable real valued function, is defined by:

where $[\delta]$ is used for the greatest-integer part, such that the integral is well defined on $(0, +\infty)$.

Lemma 2.1. ^[2] Let $\mathcal{ \varrho }\in\mathcal{(n-1, n]}$, $\mathcal{\psi(\omega)}\in C^{n-1}$, then

for $\mathcal{d}_{j}\in \mathbb R$; $j = 0, 1, 2, 3, \cdots, n-1.$

Theorem 2.1. ^[25,26] Let us consider a Banach space, ${\varphi}^{\ast}$ and a cone $\mathcal{M^\ast}\in \varphi^{\ast}$. Assuming $\mathcal{S^{\ast}_1, S^{\ast}_2}$ are two bounded subset of $\varphi^{\ast}$ such that $0\in \mathcal{S^{\ast}_1}, \; \mathcal{\overline{S^{\ast}_1}}\subset \mathcal{S^{\ast}_2}$. Then, an operator $\mathcal{T^\ast}$: $\mathcal{M^\ast}\cap(\mathcal{\overline{S^{\ast}_2}} \setminus \mathcal{S^{\ast}_1})\longrightarrow\mathcal{M^\ast},$ is completely continuous and satisfies the following:

or

has a fixed point in $\mathcal{M^\ast}\cap(\mathcal{\overline{S^{\ast}_2}} \setminus \mathcal{S^{\ast}_1})$ (Guo-Krasnosel'skii Theorem^[25,26]).

Lemma 2.2. ^[20,21] The following conditions hold true, for the operator $p$-Laplacian $\phi^{*}_{\mathcal{p}}$:

$(1) \; {{If}}\; \; \vert\gamma_{1}\vert, \vert\gamma_{2}\vert\ge\varrho > 0, \; 1 < \mathcal{p}\le 2, \; \gamma_{1}\gamma_{2} > 0,$ then

$(2)\; {{If}}\; \; \mathcal{p} > 2, \vert\gamma_{1}\vert, \vert\gamma_{2}\vert\le\varrho^{\ast} > 0,$ then

3.   Green's function and its properties

Theorem 3.1. Assuming $\psi^{\ast}\in \mathcal{C}[0, 1]$ satisfies the given FDE (1.1) with $\phi^{*}_\mathcal{p}.$ Then for $\varsigma, \varrho \in (3, n]$ for positive integer $n \geq 4$, the solution of the given FDE with the operator, $\phi^{*}_p$ has a solution equivalent to

where $\mathcal{W}^{ \varrho }(\mathcal{\omega, \mathcal{x}})$, the Green's function is given by

Proof. Utilizing $I^\mathcal{\varsigma}$, the fractional integral operator on (1.1) and by the virtue of Lemma 2.1, the above Eq (1.1) gets the form

From the condition

Using arbitrary constants values, then (3.3) becomes

Using the $p$-Laplacian operator further, (3.4) we obtain the form

Applying the fractional integral operator $I^\mathcal{ \varrho }$ to (3.5) again and imposing Lemma 2.1, (3.5) becomes

where $h_{j}\in \mathbb R$ for $j = 1, 2, \cdots, \varrho-m.$

Making use of boundary conditions, $\mathcal{y(0)} = 0$ and

for $j = 2, 3, 4, \cdots, \varrho-m,$ in (3.6), $\implies h_j = 0$ and

implies that

Putting constants values $h_i$ in (3.6), we have

where $\mathcal{W}^{ \varrho }(\mathcal{\omega, \mathcal{x}})$ is defined in (3.2).

Lemma 3.1. The function, $\mathcal{W}^{ \varrho }(\mathcal{\omega, \mathcal{x}})$ in (3.2) fulfils the following given considerations:

$(\mathcal{P_1})\; \mathcal{W}^{ \varrho }(\mathcal{\omega, \mathcal{x}}) > 0, \; \; \; \; \forall\; \; \mathcal{ \mathcal{x}, \omega\in(0, 1);}$

$(\mathcal{P_2})\;$The Green function is non-decreasing and $\mathcal{W}^{ \varrho }(\mathcal{1, \mathcal{x}}) = \underset{\omega\in[0, 1]}{\mathit{\text{max}}}\mathcal{W}^{ \varrho }(\mathcal{\omega, \mathcal{x}});$

$(\mathcal{P_3})\; \mathcal{W}^{ \varrho }(\mathcal{1, \mathcal{x}})\ge \mathcal{\omega}^{ \varrho -1}\underset{\mathcal{ \omega}\in[0, 1]}{\mathit{\text{max}}}\mathcal{W}^{ \varrho }(\mathcal{\omega, \mathcal{x}}),$ for $\mathcal{\omega, x}\in(0, 1).$

Proof. The two cases will be used to prove $(\mathcal{P_1})$ $\forall \; \; \mathcal{\omega, \mathcal{x} \in (0, 1)}$.

Case 1. For $0 < \mathcal{x}\le \mathcal{ \omega} < 1$, then $\mathcal{x}\leq \dfrac{\mathcal{ x}}{\mathcal{\omega}}.$ Subsequently,

Case 2. When $\mathcal{ \omega}\le \mathcal{x} < 1$ then $\mathcal{\omega}^{\varrho-1}, (1-\mathcal{ s})^{\varrho-\delta-1}\geq 0.$ Consequently,

From (3.8) and (3.9), $\mathcal{W}^{ \varrho }(\mathcal{\omega, \mathcal{x}}) > 0, \ \ \forall\mathcal{ \mathcal{x}, \omega\in(0, 1).}$ For proving $(\mathcal{P_2})$, suppose that $\forall \mathcal{x}, \mathcal{\omega}\in (0, 1)$.

Case 3. For $\mathcal{x}\le \mathcal{\omega}$. As $\varrho > 3$, then

Case 4. When $\mathcal{ \omega}\le \mathcal{x}$, we find that

From Eqs (3.10) and (3.11), it shows that $\dfrac{\partial}{\partial\mathcal{\omega}}\mathcal{W}^{ \varrho }(\mathcal{\omega, \mathcal{x}}) > 0, \; \text{for all}\; \mathcal{x}, \mathcal{\omega}\in(0, 1),$ subsequently, $\dfrac{\partial}{\partial\mathcal{\omega}}\mathcal{W}^{ \varrho }(\mathcal{\omega, \mathcal{x}})$ be non-decreasing function.

Therefore, as a result we have for $\mathcal{t}\ge \mathcal{x}$,

and for $\mathcal{x}\ge \mathcal{\omega}$

To prove $(\mathcal{P_3})$, suppose

Case 5. When $\mathcal{x}\le \mathcal{\omega}\Longrightarrow \mathcal{x}\leq \dfrac{\mathcal{x}}{\mathcal{\omega}}$, then

Case 6. When $\mathcal{\omega}\le \mathcal{x} < 1$, we obtain

Thus, by Eqs (3.14) and (3.15), condition $\mathcal{P_3}$ is proved.

4.   Main results of existence and uniqueness

4.1.   Existence results

Consider $\varphi^{\ast} = \mathcal{ C[0, 1]}$, Banach space which contains all real valued continuous functions and endowed with

and are defined on ${[0, 1]}$. Assume that $\mathcal{M^\ast}\in \varphi^{\ast}$is a positive, and of the type

cone of functions.

Let

By applying Theorem 3.1, the alternate form of given problem (1.1) is

Let us start by defining an operator $\mathcal{T}^{\ast}:\mathcal{M^\ast}\setminus \{0\}\rightarrow \varphi^{\ast}$ assosiated with problem (1.1), in order for

The solution of our problem given by Eq (1.1) can be a fixed point $\mathcal{y(\omega)}$ of $\mathcal{T}^{\ast}$ using Theorem 3.1 that is,

The following suppositions are essential to reach the existence outcome:

$(\mathcal{B_1})$ $\text{Continuous and non-vanishing function; }\; \; \Lambda(\mathcal{\omega}):(0, 1) \longrightarrow \mathbb{R^{+}}$ with

$(\mathcal{B_2})$ $\mathcal{\psi^{\ast}({\omega}, y(\omega))}, \mathcal{v(\omega, y(\omega))}$: $(0, 1) \times (0, +\infty) \longrightarrow \mathbb{R^{+}}$ are continuous.

$(\mathcal{B_3})$

Where $g_1, g_2, \aleph_1, \aleph_2$ are positive constants and $l_1, l_2\in [0, 1].$

$(\mathcal{B_4})$

Theorem 4.1. Let us consider that conditions $(\mathcal{B_1})$–$(\mathcal{B_3})$ are satisfied. Then, $\mathcal{T^{\ast}}$ is completely continuous.

Proof. Utilizing (4.2) and Lemma 3.1, we obtain for any $\mathcal{y}\in (\overline{\mathcal {S^{\ast}}_2(h)}) \setminus \mathcal{x^{\ast}_1}(h)),$

From (4.4) and (4.5), we arrive at

As a result, $\mathcal{T}^{\ast}$: $(\overline{\mathcal {S^{\ast}}_2(h)}) \setminus \mathcal{x^{\ast}_1}(h))\rightarrow \mathcal{M}^{\ast}$.

We then prove: $\vert\mathcal{T}^{\ast}\mathcal{y_{n}(\omega)}-\mathcal{T}^{\ast}\mathcal{y(\omega)}\vert\rightarrow {0}$ as $\mathcal{n}\rightarrow \infty$ to demonstrate that $\mathcal{T}^{\ast}$ map is a continuous map, let us contrive

By means of function continuity, $\mathcal{v}, \psi^{\ast},$ we now have $\vert \mathcal{T}^{\ast}\mathcal{y_{n}(\omega)}-\mathcal{T}^{\ast}\mathcal{y(\omega)}\vert\rightarrow {0}$ as $\mathcal{n}\rightarrow \infty$. This implies, $\mathcal{T}^{\ast}$ map is a continuous map. Here, we need to exemplify that T is uniformly bounded on $(\overline{\mathcal {S^{\ast}}_2(h)}) \setminus \mathcal{x^{\ast}}_1(h).$

By (4.2) and using $(\mathcal{B_1}), (\mathcal{B_3})$, for $\mathcal{\omega}\in [0, 1]$, we have got

where

$\mathcal{T^{\ast}}$ is uniformly bounded by operator $\mathcal{T^{\ast}}$, as a result of this. The equicontinuity of the operator $\mathcal{T^{\ast}}$ is used to demonstrate that operator $\mathcal{T^{\ast}}$ is compact.

When $0 < \mathcal{\omega_1} < \mathcal{\omega_2} < 1,$ we proceed

As a result, $\mathcal{T^{*}}$ is equicontinuous operator on $(\overline{\mathcal {S^{\ast}}_2(h)}) \setminus \mathcal{x^{\ast}_1}(h)$, and $\mathcal{T^{*}}$ is compact according to the Arzela Ascoli theorem on $(\overline{\mathcal {S^{\ast}}_2(h)}) \setminus \mathcal{x^{\ast}_1}(h)$.

Consequently, all the conditions of Theorem 3.1's ^[25] got satisfied. Furthermore,

is completely continuous.

Let us evaluate the height functions for $\psi^{\ast}\mathcal{(\omega, y(\omega))}$, for $h > 0$ and $\forall \; \mathcal{\omega}\in [0, 1]$ now.

Theorem 4.2. Suppose that assumptions $(\mathcal{B_1})$–$(\mathcal{B_3})$ hold and $\exists \mathcal{a}, \mathcal{b}\in \mathbb{R^{+}}$ such that >

and

or

and

Then the given Eq (1.1) has a positive solution $\mathcal{y}\in \mathcal{M^\ast}$ & $\mathcal{a}\le \vert\mathcal{y}\vert \le \mathcal{b}.$

Proof. First, let's look at $(\mathcal{U_1})$. If $\mathcal{y}\in \partial\mathcal {S^{\ast}}(\mathcal{a})$ then $\forall \mathcal{\omega}\in [0, 1], \; \mathcal{\omega}^{ \varrho -1}\mathcal{a}\le\mathcal{y}\le \mathcal{a}$, and $\vert\mathcal{y}\vert = \mathcal{a}$. Using (4.10), $\phi^{*}_{\text{min}}(\mathcal{\omega}, a)\le \psi^{\ast}\mathcal{(\omega, y(\omega))}$, we proceed

Now, $\forall$ $\mathcal{\omega}\in [0, 1], \; \mathcal{\omega}^{ \varrho -1}\mathcal{b}\le\mathcal{y}\le \mathcal{b}$.

If $\mathcal{y}\in \partial\mathcal {S^{\ast}}(\mathcal{b})$ then $\vert\mathcal{y}\vert = \mathcal{b}$. Using (4.10), we have $\psi^{\ast}\mathcal{(\omega, y(\omega))} \leq \phi^{*}_{\text{max}}(\mathcal{\omega}, b)$. This implies

Thus using Theorem 2.2, $\mathcal{T^{*}}$ has a fixed point, i.e. $\mathcal{y}\in(\overline{\mathcal {S^{\ast}}(\mathcal{b})}) \setminus \mathcal{x^{\ast}}(\mathcal{a})$. Theorem 3.1 and Lemma 3.1 are used, for $\mathcal{\omega}\in (0, 1)$ & $\mathcal{a}\le\vert\mathcal{y}\vert\le\mathcal{b}$, we have $\mathcal{y(\omega)} \ge \mathcal{\omega}^{ \varrho -1}\vert\mathcal{y(\omega)}\vert\ge \mathcal{a\omega}^{ \varrho -1} > 0.$ Therefore, the solution $\mathcal{y(\omega)}$ is positive.

4.2.   Uniqueness of solution

Theorem 4.3. Let us consider assumptions $\mathcal{(B_1)}, \mathcal{(B_2)\; \mathit{\text{and}}\; (B_4)}$ are satisfied. Then, for the given Eq (1.1) on ${[0, 1]}$, there is a unique solution, if

Proof. We show that the result for $\mathcal{p}\ge2$ is unique.

By (4.10) and $\forall$ $\mathcal{\omega}\in[0, 1]$,

For each $\mathcal{y}\in(\overline{\mathcal {S^{\ast}}({h})}) \setminus \mathcal{x^{\ast}}({h})$, and using (4.15) we get

but in $(4.14)$ assumed that $\Delta^{\ast} < 1$. $\mathcal{T^{*}}$ is a contraction map, as evidenced by this. According to the Banach mapping principle of contraction, $\mathcal{T^{*}}$ thus possesses a unique fixed point. As a result, on ${[0, 1]}$ the provided equation (1.1) has unique solution.

5.   HU stability

Definition 5.1. ^[24] If there is a positive constant $\Delta^{\ast}$, the I.E. (4.1) is stated as Hyers-Ulam stable if it fulfills the following conditions for some fixed positive constant $\gamma^{\ast} > 0$.

If,

Then $\exists$$\mathcal{w(\omega)}$, a continuous function that fulfils the following equation:

with

Theorem 5.1. For $\mathcal{p} > 2$, if $\mathcal{(R_1), (R_2)\; \mathit{\text{and}}\; (R_4)}$ are satisfied, the FDE (1.1) with the non-linear $p$-Laplacian opertor is HU stable.

Proof. For problem (1.1), we show that Eq (4.1), with suppositions $\mathcal{(R_1), \; (R_2), \; \text{and}\; (R_4)}$ is HU stable. Consequently, we have

As a conclusion, the Eq (4.1) is Hyers-Ulam stable by using (5.4). So, the FDE (1.1) has achieved HU stability.

6.   Application

Furthermore, we give an illustration of our findings using an application.

Consider the following FDE

where $\omega \in [0, 1],$

$q = 2.5, \mathcal{ \varrho } = 4, {\mathcal{\varsigma}} = 3.5, {\mathcal{ \nu }} = 1.5, \mu = 0.3,$

Height functions are given by

Then, $\omega \in (0, 1)$, we have

Therefore, all the results of the above proved theorems holds, which implies solution exists and $\dfrac{1}{10000}\leq \vert\vert y^*\vert\vert\leq1.$

7.   Conclusions

The focus of this paper is on two fascinating and crucial elements of FC: EU and Hyers-Ulam stability. Green's function was used to convert the proposed problem into an I.E. in order to achieve this objective. The Green's function was then tested on (0, 1) to see if it was increasing or decreasing, and whether it was positive or negative. Following that, Guo-Krasnoselskii's fixed point theorem is used to construct existence results for general FDE including the operator, $p$-Laplacian. Following that, the proposed problem's Hyers-Ulam stability was taken into account. Further, a significant example is provided to represent the conclusion. The proposed problem for multiplicity findings can also be studied using other mathematical methodologies by the readers. For future research work, for the proposed problem, we suggest qualitative properties/analysis and evaluation of the issue for multiplicity and exponential stability of the solution of the FDEs.

Acknowledgments

The first author, Kirti Kaushik would like to thank University Grants Commission and Central University of Punjab for this research work. The authors Aziz Khan and Thabet Abdeljawad would like to thank Prince Sultan University for paying the APC and the support through TAS research lab.

Competing interests

There does not exist any kind of competing interest.