Quantum calculus with respect to another function

1.   Introduction

Quantum calculus, also known as $q$-calculus, is the study of calculus without the notion of limits, which presents alternative flavors of calculus. More precisely, instead of defining the derivatives and integrals of real value functions by means of limits, quantum calculus makes use of their respective $q$-analog versions. Quantum calculus is also used to study calculus on discrete or even finite sets. Historically, Euler had obtained some basic formulae in $q$-calculus in the eighteenth century. However, it is Jackson ^[1] who established what is now known as $q$-derivative and $q$-integral. Interests in the research of $q$-calculus are currently flourishing due to its several applications in many areas, such as physics, number theory, orthogonal polynomials, hypergeometric functions, and combinatorics ^[2,3]. Al-Salam ^[4] and Agarwal ^[5] have generalized $q$-derivatives and $q$-integrals into orders other than integers, as seen in the monograph ^[6]. Many interesting results in such areas of research were also presented by several authors in the literature ^{[7,8,9,10,11,12,13,14,15]} and references therein. Tariboon and Ntouyas introduced in ^[16] the notion of quantum calculus on finite intervals. The interested reader is referred to the monograph ^[17] for further details.

The basic aid of the present paper is to give new definitions of quantum calculus on finite intervals with respect to another function. We give the new definitions of the quantum derivative and quantum integral and study their basic properties. Also, we apply the new defined quantum calculus with respect to another function and obtain a new Hermite-Hadamard inequality for a convex function. Our results are new and, in special cases, correspond to the existing results in literature. Finally, an impulsive boundary value problem involving the quantum derivative with respect to another function is studied. An existence and uniqueness result is established via the Banach contraction mapping principle.

The rest of the paper is arranged as follows: In Section 2, we present the new results on quantum calculus with respect to another function. We give the new definitions and investigate their properties. In Section 3, we obtain a new Hermite-Hadamard inequality for a convex function by applying the new quantum calculus, while in Section 4 we study an impulsive quantum boundary value problem. An example illustrating the obtained result is also presented.

2.   Quantum calculus with respect to another function

Let $q\in(0, 1)$ be a quantum number, $a\geq 0$ be a point in real line, and define a $q$-shifting operator by

Let $\psi(t)$ be a strictly increasing function defined on $[a, b]$. We now present a notion of the quantum derivative with respect to $\psi$.

Definition 2.1. The $q$-derivative of a function $f$ with respect to a function $\psi$ on $[a, b]$, is given by

and ${_a}D_{q, \psi}f(a) = \lim_{t\to a}\{{_a}D_{q, \psi}f(t)\}$.

Remark 2.1. $(i)$ If $\psi(t) = t$, then we obtain the Tariboon-Ntouyas q-derivative defined in ^[16] as

Furthermore, if $a = 0$, it is reduced to the classical Jackson q-derivative, which was defined in 1910 by Jackson ^[1] as

$(ii)$ If $\psi(t) = \log t$, where $\log t = \log_e t$ and $t\in [a, b], a > 0$, then we have

which is the q-analog for the derivative of Hadamard sense. To see this, consider as $q\to 1$ (using the L'H$\hat{\rm o}$pital's rule with respect to $q$),

which is a special case of the Hadamard derivative (see [18, Chapters 2.5 and 2.7]).

Corollary 2.1. Let $f(t) = \psi^n(t)$ in (2.2), where $n$ is a positive integer, then the following formula holds:

Proof. From (2.2), we have

which completes the proof. □

Example 2.1. Let us consider two examples for computation of the $q$-derivative with respect to another function.

$(i)$ Let $n > m > 0$ be constants and $\psi(t) = (t-a)^m,$ then

where ${_a}\Phi_q(t)-a = qt+(1-q)a-a = q(t-a)$. If $m = 1$, it is reduced to the $q$-derivative of polynomial formula

where $[n]_q = (1-q^n)/(1-q)$ is a $q$-number of value $n$. When $a = 0$, $m = 1/2,$ and $n = 2,$ we have the special case ${_0}D_{q, \sqrt{t}}t^2 = \left(\frac{1-q^2}{1-\sqrt{q}}\right)t^{\frac{3}{2}}.$

$(ii)$ A special case of (2.4), when $n = 3,$ is

If $a = 1$ and $\psi(t) = \log t,$ then we get

Observe that $\lim_{q\to 1}\left\{{_1}D_{q, \log t}(\log^3t)\right\} = 3\log^2 t$, which is an analogue to the relation $\left(t\frac{d}{dt}\log^3t\right)$$=$$3\log^2 t$.

Some basic properties of the quantum derivative with respect to another function can be presented as follows.

Lemma 2.1. $(i)$ Let $\alpha, \beta$ be constants. The following linear property holds:

$(ii)$ The quantum derivative of the product of two functions can be expressed as

$(iii)$ The quantum derivative of the quotient of two functions can be expressed as

where $g(t)g(qt+(1-q)a)\ne 0$ for all $t\in [a, b]$.

Proof. To prove $(i)$, we see that

Next, the first equation in $(ii)$ follows from

The second equation can be proved similarly. To claim $(iii)$, we have

With that, we are done. □

To derive the quantum integral with respect to another function, let us consider the $q$-shifting operator acting on a function $f(t)$ by

We then can see the $n$-times iteration ${_a}\Phi_q^n f(t) = f({_a}\Phi_q^n(t)) = f(q^nt+(1-q^n)a)$, for all $n\in \mathbb{Z}_0$. To claim this, for $n = 0$, we have ${_a}\Phi{_q^0} f(t) = f(q^0t+(1-q^0)a) = f(t)$. Now, suppose that ${_a}\Phi_q^k f(t) = f(q^kt+(1-q^k)a)$ holds for $n = k$, then we get

By mathematical induction, the $n$-times iteration of the $q$-shifting operator to a function $f(t)$ is well-defined.

Next, from (2.2), we set

in which we can solve for $f(t)$ and obtain

provided that the righthand side is convergent. With this concept of the quantum antiderivative, we give a new definition of quantum integration with respect to another function.

Definition 2.2. The definite q-integral of a function $f$ on $[a, b]$ with respect to the function $\psi$ and quantum number $q$ is defined by

provided that the righthand side exists. In addition, if $c\in (a, b)$, then the definite q-integral can be written as

Remark 2.2. $(i)$ If $\psi(t) = t$, then we have the Tariboon-Ntouyas ^[16] definite q-integral given by

which, for $a = 0,$ gives the Jackson ^[1] definite q-integral as

$(ii)$ For $a > 0$, if $\psi(t) = \log t$, $t\in [a, b]$, then the Hadamard definite q-integral is presented by

Remark 2.3. If $a = 0$ in (2.5), then

which is Eq (19.9) in ^[3], although there were no generalizations into the $q$-integral with respect to other functions as explored in our work.

Example 2.2. If $\psi(t) = (t-a)^m$, $f(t) = (t-a)^n$, when $m, n > 0$, then we have

since ${_a}\Phi_q^j(t)-a = q^j(t-a)$. If $m = 1$, we have the $q$-integral of the power function formula as

If $m = 1/2$ and $n = 2,$ we have the special case ${_a}I_{q, \sqrt{t-a}}(t-a)^2 = (t-a)^{\frac{5}{2}}\left(\frac{1-\sqrt{q}}{1-q^{\frac{5}{2}}}\right).$

Remark 2.4. Since we only have the closed form formula (the geometric series) of infinite series, we can only compute as in Example 2.2 for power function with respect to power function. In addition, it is easy to find that

We now present some $q$-analog of the fundamental theorem of calculus.

Theorem 2.1. The following formulas hold:

$(i)$ ${_a}D_{q, \psi}\left({_a}I_{q, \psi}f\right)(t) = f(t)$.

$(ii)$ ${_a}I_{q, \psi}\left({_a}D_{q, \psi}f\right)(t) = f(t)-f(a)$, where $t\in (a, b]$.

Proof. From the definitions of the $q$-derivative and $q$-integral with respect to another function $\psi$, in (2.2) and (2.5), respectively, we have

which proves $(i).$

Also, we have

with $\lim_{u\to\infty}f(q^ut+(1-q^u)a) = f(a).$ Thus $(ii)$ is proved.□

The next theorem shows that the double definite $q$-integral with respect to another function can be reduced to a single integral.

Theorem 2.2. The following formula

holds.

Proof. From Definition 2.5, we have

The proof is completed.□

3.   Quantum Hermite-Hadamard inequality

In this section, we apply quantum calculus with respect to another function to establish a new Hermite-Hadamard inequality for a convex function $f$ on $[a, b].$ Recall that a function $f$ is convex if

for all $\lambda\in[0, 1]$.

Theorem 3.1. Suppose that $f:[a, b]\to \mathbb{R}$ is a convex differentiable function on $(a, b),$ then the $q$-integral of $f$ with respect to $\psi(t) = (t-a)^m$, $m > 0$ satisfies

Proof. Define the point $c: = \frac{b(1-q^m)+aq^m(1-q)}{1-q^{m+1}}$, then $c\in[a, b]$. To see this, note that $\lim_{q\to 0}c = b$ and

for all $m > 0$, using the L'H$\hat{\rm o}$pital rule with respect to $q$.

Since $f$ is a differentiable convex function on $(a, b)$, there exists a tangent line $h(t)$ under the curve of $f(t)$, i.e., for $t\in(a, b)$,

To prove the left-hand side of (3.1), by taking the $q$-integral with respect to a function $\psi(t) = (t-a)^m$, $m > 0$, and applying the formula in Example 2.2 for $n = 0, 1$, we have

On the other side of the inequality, we set the line $k(t)$ that connects the two points $(a, f(a))$ and $(b, f(b))$, i.e., for $t\in(a, b)$,

which implies that $f(t)\leq k(t)$ for all $t\in[a, b]$ by the convexity of $f$. Therefore, we obtain

Combining both cases, we obtain (3.1), and the proof is completed. □

Remark 3.1. If $m = 1$, then (3.1) is presented by

which has appeared in ^[19].

4.   Impulsive boundary value problem

Let $t_0 < t_1 < t_2 < \cdots < t_m < t_{m+1}$ be fixed points in $[0, T]$, where $t_0 = 0,$ $t_{m+1} = T$, $g:[0, T]\times\mathbb{R}\to\mathbb{R}$ be a nonlinear function, and $\{c_i\}$, $\{d_i\}$, $i = 1, 2, \ldots, m$, be given sequences of real numbers. Consider the boundary value problem involving the quantum derivative with respect to another function

with constants $0 < q_i < 1$, $\psi_i(t)$ a strictly increasing function on $[0, T]$, for all $i = 1, 2, \ldots, m$, $\gamma\in \mathbb{R},$ and $g:[0, T]\times \mathbb{R}\to \mathbb{R}$ a given function. Note that if $\gamma = 1$, then (4.1) is reduced to a periodic boundary value problem, and if $\gamma = -1$, it is an anti-periodic boundary value problem.

Our first task is to transform the boundary value problem (4.1) into an integral equation by considering a linear variant of (4.1).

Lemma 4.1. Let $h\in C([0, T], \mathbb{R})$ and $\Lambda = 1-\gamma\prod_{i = 1}^{m}c_i\ne 0,$ then the linear impulsive boundary value problem

has the unique solution

Proof. For the first interval $[0, t_1)$, taking the operator ${_{t_0}}I_{q_0, \psi_0}$ to both sides of the first equation in (4.2), we obtain

The second condition for $i = 1$ yields

The second interval $[t_1, t_2)$ shows

by applying the operator ${_{t_1}}I_{q_1, \psi_1}$ to problem (4.2) for $t\in [t_1, t_2)$. Repeating this process for any $t\in [t_j, t_{j+1})$, it follows that

To prove this, we see that (4.5) holds for $j = 0$ by (4.4). For $t\in[t_{j+1}, t_{j+2})$, we have

which shows that (4.5) is true by mathematical induction.

In particular, $t = T$ implies that $j = m$ and

Solving the boundary condition of problem (4.2) for a constant $x(0)$, we obtain

Substituting a constant $x(0)$ in (4.5), the nonlinear integral Eq (4.3) is proved.

On the other hand, the converse can be proved by direct computation. The proof is finished. □

To prove the existence and uniqueness of solution to the problem (4.1), we define the space of piecewise continuous functions by the fact that $PC([0, T], \mathbb{R}) = \{x:[0, T]\to \mathbb{R}:$ $x(t)$ is continuous everywhere except for some $t_i,$ such that $x(t_i^+)$ and $x(t_i^-)$ exist and $x(t_i^+) = x(t_i)$ for all $i = 1, 2, \ldots, m\}$. $PC([0, T], \mathbb{R})$ is a Banach space with norm $\|x\| = \sup\{x(t):$ $t\in[0, T]\}$.

In view lemma of (4.2), we define an operator $\mathcal{A}:PC([0, T], \mathbb{R})\to PC([0, T], \mathbb{R})$ by

By using the Banach contraction principle we will prove the existence and uniqueness of solutions for the impulsive boundary value problem (4.1). For computational convenience we put

Theorem 4.1. Suppose that the nonlinear function $g:[0, T]\times \mathbb{R}\to \mathbb{R}$ satisfies

If $\mathcal{L}\mathcal{B}\mathcal{E} < 1$, then the boundary value problem (4.1) has a unique solution on $[0, T]$.

Proof. Let us define a ball $B_r = \{x\in PC([0, T], \mathbb{R})\, :\, \|x\|\leq r\}$, where a radius satisfies

when $\mathcal{M}: = \sup\{|g(t, 0)|\, :\, t\in[0, T]\}$. For any $x\in B_r$ and $t\in [0, T],$ we have, by (4.7),

For any $x\in B_r,$ from (2.6), we have

which implies that $\mathcal{A}(B_r)\subset B_r$. Next, we will prove that the operator $\mathcal{A}$ is a contraction. To do this, for any elements $x, y\in B_r$, we have

which means that $\|\mathcal{A}x-\mathcal{A}y\|\leq \mathcal{L}\mathcal{B}\mathcal{E}\|x-y\|$. As $\mathcal{L}\mathcal{B}\mathcal{E} < 1$, this implies that the operator $\mathcal{A}$ is a contraction on $B_r$ and there exists a unique fixed point in $B_r$, which is a unique solution of the problem (4.1) on $[0, T]$. Therefore, the theorem is proved. □

Example 4.1. Consider the following impulsive boundary value problem containing quantum derivative with respect to another function as

From the given problem, we have $q_i = (i+1)/(i+2)$, $\psi_i(t) = \log((i+2)/(i+3)+t)$, $t_i = i/2$, $i = 0, 1, 2, 3, 4$, $m = 4$, $t_{m+1} = T = 5/2,$ $c_i = (i+3)/(i+4)$, $d_i = (i+4)/(i+5)$, $i = 1, 2, 3, 4$, $\gamma = 3/2$ and $g(t, x(t)) = \frac{5}{6(t+7)}\left(\frac{x^2(t)+2|x(t)|}{1+|x(t)|}\right)+\frac{1}{2}, t\in\left[\frac{i}{2}, \frac{i+1}{2}\right).$ This information leads to $\Lambda = 1/4$, $\mathcal{B}\approx1.005099546$, $\mathcal{C}\approx2.818204365$, $\mathcal{D} = 1/2,$ and $\mathcal{E} = 4$. Also, the nonlinear function $g$ satisfies the Lipschitz condition, as

for all $x, y\in \mathbb{R}$ and $t\in[0, 5/2]$. Setting $\mathcal{L} = 5/21$, the following inequality holds

Thus, the Theorem 4.1 quarantees that the boundary value problem (4.8) has a unique solution on $[0, 5/2]$.

5.   Conclusions

In the present paper, we initiated the study of quantum calculus with respect to another function. The definitions of the quantum derivative and integral with respect to another function were given and their basic properties were investigated. A new Hermite-Hadamard type inequality for a convex function was obtained as an application of these newly defined notions. Moreover, an impulsive boundary value problem involving the quantum derivative with respect to another function was studied via the Banach contraction mapping principle. The results are new and enrich the existing results on quantum calculus.

Use of AI tools declaration

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

Acknowledgments

The first author was supported by King Mongkut's University of Technology North Bangkok, Contract No. KMUTNB-PHD-63-02.

Conflict of interest

Sotiris K. Ntouyas is an editorial board member for AIMS Mathematics and was not involved in the editorial review and/or the decision to publish this article.