Study of new class of q-fractional derivative and its properties

TL;DR

This paper introduces a new Caputo-type q-fractional derivative, explores its properties, and studies related q-difference equations, including existence and uniqueness results using q-approximation methods.

Contribution

It presents a novel Caputo-type q-fractional derivative and analyzes its properties and applications to q-difference equations, expanding fractional calculus in q-calculus context.

Findings

The new q-fractional derivative is bounded.

Existence and uniqueness of solutions to related q-difference equations are established.

Properties of the new operator are systematically investigated.

Abstract

There are several approaches to the fractional differential operator. Generalized q-fractional difference operator was defined in the aid of q-iterated Cauchy integral and q-calculus techniques. We introduce Caputo type derivative related to this operator and some properties of this operator as boundness are investigated. Moreover, related q-difference equation is discussed and in the aid of q-successive approximation , existence and uniqueness are studied.

Equations

$${\left[a\right]}_q=\frac{1-q^a}{1-q}\left(q\ne 1\right);{\left[0\right]}_q!=1;{\left[n\right]}_q!={\left[n\right]}_q{\left[n-1\right]}_{q}n\in \mathbb{N},a\in \mathbb{C}.$$

$${\left(a;q\right)}_0$$

$${\left(a;q\right)}_{\infty }$$

$$\left(\begin{array}[c]{c}n\\ k\end{array}\right)_{q}=\frac{\left(q;q\right)_{n}}{\left(q;q\right)_{n-k}\left(q;q\right)_{k}},$$

$$\int f(x)d_{q}x=(1-q)x\sum _{i=0}^{\infty }{q}^{i}f(q^{i}x)$$

$$\underset{0}{\overset{b}{\int }}f(x)d_{q}x$$

$$\underset{a}{\overset{b}{\int }}f(x)d_{q}x$$

$$\underset{0}{\overset{x}{\int }}\underset{0}{\overset{v}{\int }}{d}_{q}s{d}_{q}v=\underset{0}{\overset{x}{\int }}\underset{qs}{\overset{x}{\int }}{d}_{q}v{d}_{q}s$$

$$\Vert f\Vert =\sum _{i=0}^{n-1}\underset{a\le x\le b}{\max }\left|(D_q^{i}f)(x)\right|\text{ ,}f\in C_q^n[a,b]$$

$$(x-y{)}_{q^p}^{\left(\alpha \right)}=\prod _{k=0}^{\infty }\frac{\left(x-y{\left(q^p\right)}^k\right)}{\left(x-y{\left(q^p\right)}^{k+\alpha }\right)}=\frac{x^{\alpha }{\left(\frac{y}{x};q^p\right)}_{\infty }}{{\left(q^{\alpha \left(p\right)}\frac{y}{x};q^p\right)}_{\infty }}$$

$$(x-a{)}^{\left(\alpha \right)}=x^{\alpha }\prod _{k=0}^{\infty }\frac{\left(1-\frac{x}{a}{q}^k\right)}{\left(1-\frac{x}{a}{q}^{k+\alpha }\right)}=\frac{{\left(\frac{x}{a};q\right)}_{\infty }}{{\left(q^{\alpha }\frac{x}{a};q\right)}_{\infty }}$$

$${}_x{D}_q\left[(x^p-y^p{)}_{q^p}^{\left(\alpha \right)}\right]$$

$${}_y{D}_q\left[(x^p-y^p{)}_{q^p}^{\left(\alpha \right)}\right]$$

$${\Gamma }_q(t)=\frac{(1-q{)}^{\left(t-1\right)}}{(1-q{)}^{t-1}}$$

$$\underset{a}{\overset{x}{\int }}{t}^{p-1}(x^p-{\left(qt\right)}^p{)}_{q^p}^{\left(\alpha -1\right)}(t^p-a^p{)}_{q^p}^{\left(\lambda \right)}{d}_{q}t=\frac{1}{{\left[p\right]}_q}\left(\frac{{\Gamma }_{q^p}(\alpha ){\Gamma }_{q^p}(\lambda +1)}{{\Gamma }_{q^p}(\alpha +\lambda +1)}\right)\left[(x^p-a^p{)}_{q^p}^{\left(\alpha +\lambda \right)}\right]$$

$$\underset{0}{\overset{x}{\int }}{\left(t_1\right)}^{p-1}{d}_q{t}_1\underset{a}{\overset{t_1}{\int }}{\left(t_1\right)}^{p-1}{d}_q{t}_2...\underset{a}{\overset{t_{n-1}}{\int }}{\left(t_n\right)}^{p-1}f(t_n){dt}_n=\frac{1}{{\prod }_{k=1}^{n-1}{\left[kp\right]}_q}\underset{0}{\overset{x}{\int }}{w}^{p-1}f(w)\prod _{k=0}^{n-1}\left(x^p-{\left(wq\right)}^p{q}^{kp}\right)d_{q}w$$

$$J_{p,q}^{\alpha }\left(f(x)\right)$$

$$=\frac{{\left({\left[p\right]}_q\right)}^{1-\alpha }}{{\Gamma }_{q^p}(\alpha )}\underset{0}{\overset{x}{\int }}{w}^{p-1}f(w)(x^p-{\left(wq\right)}^p{)}_{q^p}^{\left(\alpha -1\right)}{d}_{q}w$$

$$(D_{p,q}^{0}f)(x)$$

$$(D_{p,q}^{\alpha }f)(x)$$

$${(}^c{D}_{a^{+},p,q}^{\alpha }f)(t)=(D_{a^{+},p,q}^{\alpha })(f(t)-f(a))=\frac{{\left({\left[p\right]}_q\right)}^{\alpha }}{{\Gamma }_{q^p}(1-\alpha )}(x^{1-p}{D}_q)\underset{a}{\overset{x}{\int }}{w}^{p-1}\left[f(w)-f(a)\right](x^p-{\left(wq\right)}^p{)}_{q^p}^{\left(-\alpha \right)}{d}_{q}w$$

$${(}^c{D}_{a^{+},p,q}^{\alpha }f)(x)=(D_{a^{+},p,q}^{\alpha })(f(x))-(D_{a^{+},p,q}^{\alpha })(f(a))=(D_{a^{+},p,q}^{\alpha })(f(x))-\frac{\left[f(0)\right]{\left({\left[p\right]}_q\right)}^{\alpha }}{{\Gamma }_{q^p}(1-\alpha )}(x^p-a^p{)}_{q^p}^{\left(-\alpha \right)}$$

$${(}^c{D}_{a^{+},p,q}^{\alpha }f)(x)=\frac{{\left({\left[p\right]}_q\right)}^{\alpha }}{{\Gamma }_{q^p}(1-\alpha )}\underset{a}{\overset{x}{\int }}\left[D_{q}f(w)\right](x^p-w^p{)}_{q^p}^{\left(-\alpha \right)}{d}_{q}w$$

$${(}^c{D}_{a^{+},p,q}^{\alpha }f)(x)=\frac{{\left({\left[p\right]}_q\right)}^{\alpha }}{{\left[p(1-\alpha )\right]}_q{\Gamma }_{q^p}(1-\alpha )}(x^{1-p}{D}_q)\underset{a}{\overset{x}{\int }}\left(f(w)-f(a)\right)\left[D_q(x^p-w^p{)}_{q^p}^{\left(1-\alpha \right)}\right]d_{q}w$$

$${(}^c{D}_{a^{+},p,q}^{\alpha }f)(x)=\frac{{\left({\left[p\right]}_q\right)}^{\alpha }}{{\left[p(1-\alpha )\right]}_q{\Gamma }_{q^p}(1-\alpha )}(x^{1-p}{D}_q)\underset{a}{\overset{x}{\int }}(x^p-{\left(wq\right)}^p{)}_{q^p}^{\left(1-\alpha \right)}\left[D_{q}f(w)\right]d_{q}w$$

$${(}^c{D}_{a^{+},p,q}^{\alpha }f)(x)$$

$$=\frac{1}{{\left[p(1-\alpha )\right]}_q}(D_{a^{+},p,q}^{\alpha })\left(w^{1-p}{D}_q\left(f\right)\right)(x)$$

$${(}^c{D}_{a^{+},p,q}^{\alpha }f)(x)$$

$$=\frac{{\left({\left[p\right]}_q\right)}^{\alpha }}{{\Gamma }_{q^p}(1-\alpha )}(x^{1-p}{D}_q)\underset{a}{\overset{x}{\int }}\underset{qt}{\overset{x}{\int }}\left[D_{q}f(t)\right]w^{p-1}(x^p-{\left(wq\right)}^p{)}_{q^p}^{\left(-\alpha \right)}{d}_{q}w{d}_{q}t$$

$$=\frac{{\left({\left[p\right]}_q\right)}^{\alpha }}{{\left[p(1-\alpha )\right]}_q{\Gamma }_{q^p}(1-\alpha )}(x^{1-p}{D}_q)\underset{a}{\overset{x}{\int }}(x^p-{\left(qt\right)}^p{)}_{q^p}^{\left(1-\alpha \right)}\left[D_{q}f(t)\right]d_{q}t$$

$$\Vert J_{a^{+},p,q}^{\alpha }\left(f\right)(x)\Vert \le A_{p,q,\alpha }\Vert f(x)\Vert$$

$$\left|J_{a^{+},p,q}^{\alpha }\left(f\right)(x)\right|$$

$$=\frac{{\left({\left[p\right]}_q\right)}^{1-\alpha }}{{\left[p\alpha \right]}_q{\Gamma }_{q^p}(\alpha )}{\Vert f\Vert }_C\left|(x^p-a^p{)}_{q^p}^{\left(\alpha \right)}\right|$$

$${}^c{D}_{a^{+},p,q}^{\alpha }\left((J_{a^{+},p,q}^{\alpha }f)(x)\right)$$

$$J_{a^{+},p,q}^{\alpha }\left({(}^c{D}_{a^{+},p,q}^{\alpha }f)(x)\right)$$

Full text

1 Introduction

Since 1695, that a letter related to fractional derivative was written, fractional calculus has created. There are several types of fractional integral and derivative operators that arise from different aspects. Recent appplications of fractional differential equations, in explaining natural phenomena, motivate more and more scientifics to work in this area. One approach to fractional integral operator is using Cauchy integral. Authors in [1] have used $q$-calculus techniques to develop $q$-fractional difference equations. In this paper, we first state some definitions and concepts of $q$-calculus, then, we discuss about the concave operator on Banach space and then, the fixed point theorem will be written based on these conditions. Some properties of $q$-fractional derivative, integral and Caputo type of derivative are discussed.

In the first section, let us introduce some familar concepts of q-calculus that are in use in this paper [2] and [3]. We use $[n]_{q}$ as a $q-$analogue of any complex number. Naturally, we can define $\left[n\right]_{q}!$ as

[TABLE]

The $q$-shifted factorial and $q$-polynomial coefficient are defined by

[TABLE]

[TABLE]

Let the function $|f(x)x^{\alpha}|$ be bounded on the interval $\left(0,A\right]$, for some $0\leq\alpha<1,$ then Jakson integral is defined as [2]

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and it converges to a function $F(x)$ on $\left(0,A\right],$ which is a $q-$antiderivative of $f(x)$. Suppose $0<a<b$, the mentioned $q-$integral is defined as

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In addition, we can interchange the order of double $q$-integral by

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Let $C_{q}^{n}[a,b]$ denotes the space of all continues functions with continuous q-derivatives up to order $n-1$ on the interval $[a,b]$ . Associated norm function of $C_{q}^{n}[a,b]$ is defined as [4]

[TABLE]

The generalize $q$-exponent expression is defined [1]

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where the normal definition can be expressed as [7][6]

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The $q$-derivative of this expression with respect to $x$ and $y$ can be written as

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respectively.

One can express the $q$-Gamma function by using this definition as[9]

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Let us bring the useful lemma proved in [1], which express following integral for $\alpha$ and $\lambda>-1$ ,

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There are several approaches to fractional differential operators. One of the demonstration methods of fractional differential equation is using the itterated Cauchy integrals. In [1], authors calculated

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In the aid of this calculation, generalized $q$-fractional difference integral operator is defined by

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The corresponding inverse operator named as $q$-fractional difference operator is defined by

[TABLE]

for $\alpha\geq 0$ and $n=\left\lfloor\alpha\right\rfloor+1$ and $p>0$.

Definition 1

Let $0<a<b<\infty$, $f:[a,b]\rightarrow R$ be a $q$-integrable function, and let $\alpha\in(0,1)$ and  $p>0$ be two fixed real numbers. The Caputo type $q-$ fractional derivative of order $\alpha$ is defined by

[TABLE]

Note that, when $f(a)=0,$ the Caputo and the $q-$ fractional derivatives coincide. Moreover, there exists a relation between these two types of $q-$fractional derivatives. Namely, if both types of $q-$fractional derivative exist, then by considering (3) and (4) we can see that,

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Now, consider the simplification os the Caputo $q$-derivative in the next proposition;

Proposition 2

The Caputo $q$-derivative can be simplified and written as,

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if all $q$-Jackson integrals in (10) be convergent.

Proof.* By using definition (1) and identification (4) we can get;*

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The function $(x^{p}-w^{p})_{q^{p}}^{\left(1-\alpha\right)}$ is q-regular and we can apply q-integral by part to reach

[TABLE]

In the aid of lemma 1.12 of [4], we can apply the differentiation inside of Jackson integral. In fact, the derivative respect to $x$ is defined at (3) and $(\left(xq\right)^{p}-\left(xq\right)^{p})_{q^{p}}^{\left(1-\alpha\right)}=0$ which satisfied the assumption of lemma 1.12, so applying (3) will complete the proof.

Corollary 3

The direct consequence of this proposition can demonstrate the relation between Caputo type and normal fractional derivative operators (10) and (11) as follow

[TABLE]

**

In fact, study of fractional differential operator for the case $\alpha\in\left(0,1\right)$ is more attractive and serves many purposes. First, Abel integral equation which is inspired by a problem of mechanics is one of the first integral equations ever treated and considered the fractional integral operator for $\alpha$ between zero and one.[10] As a second main reason, we can mention the application of fractional differential operator for modelling different phenomena that were used $\alpha\in\left(0,1\right)$ commonly to describe simulation. [12] In addition, the calculations for the case $\alpha\in\left(0,1\right)$ is not too long and complicated and also most of them can be generalized for arbitrary $\alpha.$ Our porpose in this article is introducing this operator clearly such that can be used later for application cases.

On the other hand, Kilbas et. al. substitute $f(a)+{\displaystyle\int\limits_{a}^{t}}f^{\prime}(w)dw$ instead of $f(t)$ to find the formula for derivative operator and this technique can be used for (11) as well to reach $q$-analogue of it.[11] We can apply foundemental theorem of $q$-calculus to see;

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In (17), we interchange the order of iterated $q$-integral and reach to similar result. We can continue this procedure to find more general identity which leads to characterization of the classes of function which fractional operator (7) is defined for them.

2 Some Properties of Integral and Derivative Operator

In this section, some properties of Caputo-type derivative and integral operator are studied. These propeties lead to solving related $q$-fractional difference equation. First, we should state the boundness of $q-$fractional integral operator.

Proposition 4

Let $f(x)$ be a real function which is defined on $q$-geometric set or particularly assume that $x$ can be formed as $bq^{n}$, then $q$-fractional integral operator $J_{a^{+},p,q}^{\alpha}$ is linear and bounded from $C_{q}[a,b]$ to $C_{q}[a,b]$, that means

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Proof.* Assume that $f\in C_{q}[a,b]$ that the space is equipped with maximum norm. In addition, assume that $x$ can be formed as $bq^{n}$ for some $n\in\mathbb{N},$ then we can apply $q$-analogue of Hölder inequality as**[4]*

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*Here, we used identity (4) to calculate the $q$-integral. Since Jackson integral is a linear operator, linearlity is obvious.    *

Let us consider the inverse operator of integral operator, that we introduced it before. In the next proposition, we show that how the Caputo type derivative can be operate on integral operator. Semi-group property of integral operator is investigated[1].

Proposition 5

Let $0<a<b<\infty$ and $f$ is defined in all following expressions, also assume that $\alpha\in(0,1)$ and  $p>0$ be two fixed reals such that derivative and integral operator are defined, then following identities are true:

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Proof.* proof of the first part is based on interchanging of integral and applying the definitions. This is proved at [1] for derivative opertaor and the procedure is similar for this operator. We prove the second identity by using corollary (3) and semi group property of integral operator, so*

[TABLE]

**

3 Cauchy problem of general q-fractional operator

In this section, Cauchy problem related to the introduced Caputo type derivative is investigated. We construct solution of this $q$-fractional difference equation in the aid of discussed properties in last section. Moreover we apply a fixed point theorem based on $q$-successive Approximations to guarantee existance and uniqueness of this difference equation. We start by direct solution of $q$-difference equation in following lemma:

Lemma 6

let $0<\zeta<1$ and $\alpha\in\left(0,1\right)$, if $f(t)$ is a continuous function on $\left[0,\infty\right)\times\left(\zeta-r,\zeta+r\right)$ where $r$ is a positive real constant and ${}^{c}D_{a^{+},p,q}^{\alpha}u$ is defined , then solution of the following boundary value problem

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can be written as

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Proof.* Apply $J_{a^{+},p,q}^{\alpha}$ in both sides of (22) and use (21), then use initial value to reach the solution.   * **

Following theorem shows the conditions on $f(w,u(w))$ to verify existance and uniqueness for given $q$-fractional difference equation. Moreover, $q$-sucessive approximation based on solution of (22) is constructed. Main theorem of this article determines solution and nature of this approximation. However, the sprit of proof is similar to the discussion on [4]. In the aid of last lemma, we define the sequence of functions as follow

[TABLE]

Theorem 7

Assume that all conditions of last lemma holds true. In addition, assume that $f(w,\varphi(w))$ be a function which is continuous at $w=a$ and verify the Lipschitz’ consition, i.e.

[TABLE]

Where $A$ is a positive constant and $0\leq w$ and $\left|y-\zeta\right|<r$. Then q-fractional differential equation problem (22) with given initial values (23), has a unique solution.

Proof.* Applying Lipschitz’ condition to see*

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We mention that, according to our assumptions $f(w,\varphi(w))$ is continuous at $(a,\varphi(a))=(a,\zeta)$ and for sufficient small $w<\gamma$, $\left|f(w,\zeta)-f(a,\zeta)\right|<1$. Define the nonzero constants $K,h$ to be

[TABLE]

We establish the existance of solution $\left\{\varphi_{n}(w)\right\},$ using the method of succesive approximations. We consider the sequence that is defined by (25), these functions are continuous, because

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Thus, each $\left\{\varphi_{n}(w)\right\}$ is continuous at $w=a$ and (25) is well-defined. In the same manner, if we apply the induction, then we can see

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Now consider following series that is constructed under the above assumption, i.e. general term of series verifies (26).

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The relation (26) supports the uniform convergence of the series by Weierstrass M-test. Therefore, $\varphi_{n}(t)$ uniformly convergence to a function, called \varphi(t)\and we aclaim that $\varphi(t)$ is the solution of (22). Indeed, from Lipschitz’ condition we have:

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Since the right side of above inequality approaches uniformly to zero as $m\rightarrow\infty,$ it follows that

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*Now apply this limit on (25) to verify that $\varphi(x)$ is the solution of (22).    *

4 Example

In this section, we obtain an example of $q$-fractional difference equation which leads to this $q$-analogue of Mittag-leffler function. Indeed, following initial problem is considered:

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We can construct $q$-successive approximation in the aid of (25), then sequence of functions $\varphi_{n}(x)$ can be defined as following recurrence formula

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We can calculate a few terms of $\varphi_{n}(x)$ by using (4) and (6), specificly

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It is easy to see by induction that general solution $\varphi_{m}(x)$ can be presented by

[TABLE]

Definitely, by putting all parameter to suitable values, $\varphi_{n}(x)$ approaches to exponential function and form of solution motivates the definition of Mittag-Leffler. More details about properties and definition of classical Mittag-Leffler can be found at [13][14] and can be compared by $\varphi_{n}(x).$