3. Application of δ�-periodicity of Q-calculus

The following theorem is the modified version of Theorem 8 from [24].

Theorem 5. Assume that for the given time scale T ¼ f g0 ∪q<sup>Z</sup>, while T ∈q<sup>Z</sup>, mesð Þ δþð Þ T; t is equal for each t∈T: In addition to conditions on coefficient functions and

Lemma 1 if Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> a tð ÞΔ<sup>t</sup> � <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ c tð Þ m tð ÞΔ<sup>t</sup> <sup>&</sup>gt; <sup>0</sup> and

$$\left(\frac{\int\_{\mathbb{X}^{+}}^{\mathbb{S}\_{+}(T,\mathbf{x})} a(t) \Delta t - \int\_{\mathbb{X}^{+}}^{\mathbb{S}\_{+}(T,\mathbf{x})} \frac{c(t)}{m(t)} \Delta t}{\int\_{\mathbb{X}^{+}}^{\mathbb{S}\_{+}(T,\mathbf{x})} b(t) \Delta t}\right) \exp\left[-\left(\int\_{\mathbb{X}^{+}}^{\mathbb{S}\_{+}(T,\mathbf{x})} |a(t)| \Delta t + \int\_{\mathbb{X}}^{\mathbb{S}\_{+}(T,\mathbf{x})} a(t) \Delta t\right)\right]$$

$$\cdot \int\_{\mathbb{X}^{+}}^{\mathbb{S}\_{+}(T,\mathbf{x})} f(t) \Delta t - \beta^{u} \left(\int\_{\mathbb{X}^{+}}^{\mathbb{S}\_{+}(T,\mathbf{x})} d(t) \Delta t\right) - \alpha^{u} \left(\int\_{\mathbb{X}^{+}}^{\mathbb{S}\_{+}(T,\mathbf{x})} d(t)\right) \Delta t > 0$$

are satisfied, then there exist at least one δ�-periodic solution.

$$\begin{aligned} \text{Proof.} \ X &\coloneqq \left\{ \begin{bmatrix} u \\ v \end{bmatrix} \in \mathbb{C}\_{\pi l}(\{0\} \cup \eta^{Z}, \mathbb{R}^{2}) \,:\ u(\delta\_{\pm}(T, t)) = u(t), v(\delta\_{\pm}(T, t)) = v(t) \right\} \text{ with the norm:} \\\\ \left\| \begin{bmatrix} u \\ v \end{bmatrix} \right\| &\coloneqq \max\_{t \in [b\_{0}, \delta\_{\mp}(T, b)]\_{\mathbb{T}}} (|u(t)|, |v(t)|) \end{aligned} $$
 
$$ Y \coloneqq \left\{ \begin{bmatrix} u \\ v \end{bmatrix} \in \mathbb{C}\_{\pi l}(\{0\} \cup \eta^{Z}, \mathbb{R}^{2}) : u(\delta\_{\pm}(T, t)) = u(t), v(\delta\_{\pm}(T, t)) = v(t) \right\} \text{ with the norm:} $$
 
$$ \left\| \begin{bmatrix} u \\ v \end{bmatrix} \right\| = \max\_{t \in [b\_{0}, \delta\_{\mp}(T, b)]\_{\mathbb{T}}} (|u(t)|, |v(t)|) $$

Similar steps are used for <sup>v</sup>: <sup>u</sup>

and

VC

Let

and

u v

1 mesð Þ δþð Þ T; κ

1 mesð Þ δþð Þ T; κ

¼

! " #

v

u v � � � �

<sup>¼</sup> <sup>1</sup>

<sup>¼</sup> <sup>1</sup>

mesð Þ δþð Þ T; κ

<sup>κ</sup> u sð ÞΔ<sup>s</sup> � <sup>1</sup>

<sup>κ</sup> v sð ÞΔ<sup>s</sup> � <sup>1</sup>

<sup>κ</sup> �d sð Þþ f sð Þexp u s ð Þ ð Þ

mesð Þ δþð Þ T; κ

projectors U : X ! X and V : Y ! Y such that

U

V

u v � � � �

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

a tð Þ� b tð Þexp u t ð Þ� ð Þ c tð Þexp v t ð Þ ð Þ

<sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> exp u t ð Þþ ð Þ m tð Þexp v t ð Þ ð Þ <sup>¼</sup> <sup>C</sup><sup>2</sup>

KU

�d tð Þþ f tð Þexp u t ð Þ ð Þ

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

0

BBBB@

u v � � � �

The generalized inverse KU ¼ ImL ! DomL ∩KerU is given:

¼

Ðt

Ðt

� �∈<sup>Y</sup> can be written as the summation of an element from Im L

3 5

Quantum Calculus with the Notion δ±-Periodicity and Its Applications

http://dx.doi.org/10.5772/intechopen.74952

193

3 5

1 A:

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> u tð ÞΔt

2 4

2 4

0 @

mesð Þ δþð Þ T; κ

mesð Þ δþð Þ T; κ

<sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> exp u s ð Þþ ð Þ m sð Þexp v s ð Þ ð Þ <sup>Δ</sup><sup>s</sup>

<sup>κ</sup> a sð Þ� b sð Þexp u s ð Þ� ð Þ c sð Þexp v s ð Þ ð Þ

<sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> exp u t ð Þþ ð Þ m tð Þexp v t ð Þ ð Þ <sup>¼</sup> <sup>C</sup><sup>1</sup>

a sð Þ� b sð Þexp u s ð Þ� ð Þ c sð Þexp v s ð Þ ð Þ

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> v tð ÞΔt

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> u tð ÞΔt

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> v tð ÞΔt

> ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

ðt κ u sð ÞΔs

1

CCCCA

ðt κ v sð ÞΔs

<sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> exp u s ð Þþ ð Þ m sð Þexp v s ð Þ ð Þ <sup>Δ</sup><sup>s</sup>

<sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> exp u s ð Þþ ð Þ m sð Þexp v s ð Þ ð Þ <sup>Δ</sup><sup>s</sup> <sup>¼</sup> <sup>C</sup><sup>1</sup>

and an element from Ker L. Also, it is easy to show that any element in Y is uniquely expressed as the summation of an element Ker L and an element from Im L. So, codimImL is also 2, we get the desired result. Hence, L is a Fredholm mapping of index zero. There exist continuous

Let us define the mappings L and C by L : DomL⊂ X ! Y such that

$$L\left(\begin{bmatrix} u \\ v \end{bmatrix}\right) = \begin{bmatrix} u^{\Delta} \\ v^{\Delta} \end{bmatrix}$$

and C : X ! Y such that

$$\mathcal{C}\left(\begin{bmatrix}\mu\\\upsilon\end{bmatrix}\right) = \begin{bmatrix} a(t) - b(t)\exp\left(\mu(t)\right) - \frac{c(t)\exp\left(\upsilon(t)\right)}{\alpha(t) + \beta(t)\exp\left(\mu(t)\right) + m(t)\exp\left(\upsilon(t)\right)}\\ -d(t) + \frac{f(t)\exp\left(\mu(t)\right)}{\alpha(t) + \beta(t)\exp\left(\mu(t)\right) + m(t)\exp\left(\upsilon(t)\right)} \end{bmatrix}$$

Then, KerL <sup>¼</sup> <sup>u</sup> v � � : <sup>u</sup> v � � <sup>¼</sup> <sup>c</sup><sup>1</sup> c2 � � � � , <sup>c</sup><sup>1</sup> and <sup>c</sup><sup>2</sup> are constants.

$$ImL = \left\{ \begin{bmatrix} \boldsymbol{\mu} \\ \boldsymbol{\upsilon} \end{bmatrix} : \begin{bmatrix} \int\_{\kappa}^{\delta\_+(T,\kappa)} \boldsymbol{\mu}(t) \Delta t \\\\ \int\_{\kappa}^{\delta\_+(T,\kappa)} \boldsymbol{\upsilon}(t) \Delta t \end{bmatrix} = \begin{bmatrix} \boldsymbol{0} \\\\ \boldsymbol{0} \end{bmatrix} \right\}.$$

ImL is closed in Y: Its obvious that dimKerL ¼ 2. To show dimKerL ¼ codimImL ¼ 2, we have to prove that KerL ⊕ ImL ¼ Y: It is obvious that when we take an element from Ker L, an element from Im L, we find an element of Y by summing these two elements. If we take an element u v � �<sup>∈</sup> Y, and WLOG taking u tð Þ, we have <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> u tð ÞΔt ¼ I where I is a constant. Let us define a new function <sup>g</sup> <sup>¼</sup> <sup>u</sup> � <sup>I</sup> mesð Þ <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> : Since <sup>I</sup> mesð Þ <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> is constant by Lemma 1, if we take the integral of g from κ to δþð Þ T; κ , we get

$$\int\_{\kappa}^{\mathfrak{s}\_+(T,\kappa)} g(t) \Delta t = \int\_{\kappa}^{\mathfrak{s}\_+(T,\kappa)} \mu(t) \Delta t - I = \mathbf{0}.$$

Similar steps are used for <sup>v</sup>: <sup>u</sup> v � �∈<sup>Y</sup> can be written as the summation of an element from Im L and an element from Ker L. Also, it is easy to show that any element in Y is uniquely expressed as the summation of an element Ker L and an element from Im L. So, codimImL is also 2, we get the desired result. Hence, L is a Fredholm mapping of index zero. There exist continuous projectors U : X ! X and V : Y ! Y such that

$$\mathcal{U}\left(\begin{bmatrix} u \\ v \end{bmatrix}\right) = \frac{1}{m\text{res}\left(\delta\_{+}(T,\kappa)\right)} \begin{bmatrix} \int\_{\kappa}^{\delta\_{+}(T,\kappa)} u(t) \Delta t \\\\ \int\_{\kappa}^{\delta\_{+}(T,\kappa)} v(t) \Delta t \end{bmatrix}.$$

and

Proof. X<sup>≔</sup> <sup>u</sup>

<sup>Y</sup><sup>≔</sup> <sup>u</sup> v � �

v � �

192 Advanced Technologies of Quantum Key Distribution

and C : X ! Y such that

u v � � � �

> v � �

a new function <sup>g</sup> <sup>¼</sup> <sup>u</sup> � <sup>I</sup>

integral of g from κ to δþð Þ T; κ , we get

¼

: <sup>u</sup> v � �

� � � �

<sup>∈</sup> Y, and WLOG taking u tð Þ, we have <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>¼</sup> <sup>c</sup><sup>1</sup> c2

ImL <sup>¼</sup> <sup>u</sup>

< :

v � � :

mesð Þ <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> : Since <sup>I</sup>

g tð ÞΔt ¼

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

C

Then, KerL <sup>¼</sup> <sup>u</sup>

u v � �

<sup>∈</sup> Crd f g<sup>0</sup> <sup>∪</sup>q<sup>Z</sup>; <sup>R</sup><sup>2</sup> � � : <sup>u</sup>ð Þ¼ <sup>δ</sup>�ð Þ <sup>T</sup>; <sup>t</sup> u tð Þ, vð Þ¼ <sup>δ</sup>�ð Þ <sup>T</sup>; <sup>t</sup> v tð Þ � �

� <sup>¼</sup> max<sup>t</sup><sup>∈</sup> ½ � <sup>t</sup>0;δþð Þ <sup>T</sup>;t<sup>0</sup> <sup>T</sup> ð Þ <sup>j</sup>u tð Þj; <sup>j</sup>v tð Þj

� <sup>¼</sup> max<sup>t</sup><sup>∈</sup> ½ � <sup>t</sup>0;δþð Þ <sup>T</sup>;t<sup>0</sup> <sup>T</sup> ð Þ <sup>j</sup>u tð Þj; <sup>j</sup>v tð Þj

<sup>¼</sup> <sup>u</sup><sup>Δ</sup> vΔ � �

αðÞþ t βð Þt exp u t ð Þþ ð Þ m tð Þexp v t ð Þ ð Þ

9 = ;:

<sup>κ</sup> u tð ÞΔt ¼ I where I is a constant. Let us define

mesð Þ <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> is constant by Lemma 1, if we take the

a tð Þ� b tð Þexp u t ð Þ� ð Þ c tð Þexp v t ð Þ ð Þ

αð Þþ t βð Þt exp u t ð Þþ ð Þ m tð Þexp v t ð Þ ð Þ

u tð ÞΔt � I ¼ 0:

, c<sup>1</sup> and c<sup>2</sup> are constants.

� � <sup>8</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> u tð ÞΔt

2 4

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> v tð ÞΔt

ImL is closed in Y: Its obvious that dimKerL ¼ 2. To show dimKerL ¼ codimImL ¼ 2, we have to prove that KerL ⊕ ImL ¼ Y: It is obvious that when we take an element from Ker L, an element from Im L, we find an element of Y by summing these two elements. If we take an element

> ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

u v �� � � � �

u v �� � � � �

� � �

Let us define the mappings L and C by L : DomL⊂ X ! Y such that

� � �

<sup>∈</sup> Crd f g<sup>0</sup> <sup>∪</sup>q<sup>Z</sup>; <sup>R</sup><sup>2</sup> � � : <sup>u</sup>ð Þ¼ <sup>δ</sup>�ð Þ <sup>T</sup>; <sup>t</sup> u tð Þ, vð Þ¼ <sup>δ</sup>�ð Þ <sup>T</sup>; <sup>t</sup> v tð Þ � �

L

�d tðÞþ f tð Þexp u t ð Þ ð Þ

u v � � � � with the norm:

with the norm:

$$V\left(\begin{bmatrix}\boldsymbol{u}\\ \boldsymbol{v}\end{bmatrix}\right) = \frac{1}{\text{mes}(\boldsymbol{\delta}\_{+}(T,\kappa))} \left( \begin{bmatrix} \int\_{\kappa}^{\delta\_{+}(T,\kappa)} \boldsymbol{u}(t) \Delta t\\ \int\_{\kappa}^{\delta\_{+}(T,\kappa)} \boldsymbol{v}(t) \Delta t \end{bmatrix} \right).$$

The generalized inverse KU ¼ ImL ! DomL ∩KerU is given:

$$K\_{\rm II} \left( \begin{bmatrix} \boldsymbol{u} \\ \boldsymbol{\nu} \end{bmatrix} \right) = \begin{bmatrix} \int\_{\kappa}^{t} \boldsymbol{u}(\boldsymbol{s}) \Delta \boldsymbol{s} - \frac{1}{\mathsf{meas}(\boldsymbol{\delta}\_{+}(\boldsymbol{T}, \boldsymbol{\kappa}))} \int\_{\kappa}^{\delta\_{+}(\boldsymbol{T}, \boldsymbol{\kappa})} \int\_{\kappa}^{t} \boldsymbol{u}(\boldsymbol{s}) \Delta \boldsymbol{s} \\\\ \int\_{\kappa}^{t} \boldsymbol{\nu}(\boldsymbol{s}) \Delta \boldsymbol{s} - \frac{1}{\mathsf{meas}(\boldsymbol{\delta}\_{+}(\boldsymbol{T}, \boldsymbol{\kappa}))} \int\_{\kappa}^{\delta\_{+}(\boldsymbol{T}, \boldsymbol{\kappa})} \int\_{\kappa}^{t} \boldsymbol{\nu}(\boldsymbol{s}) \Delta \boldsymbol{s} \end{bmatrix}.$$

$$\begin{split} & V\mathbb{C}\left(\begin{bmatrix} u\\ v \end{bmatrix}\right) = \\ & \frac{1}{m\text{cs}(\delta\_{+}(T,\kappa))} \left( \left[ \int\_{\kappa}^{\delta\_{+}(T,\kappa)} a(s) - b(s) \exp\left(u(s)\right) - \frac{c(s)\exp\left(v(s)\right)}{a(s) + \beta(s)\exp\left(u(s)\right) + m(s)\exp\left(v(s)\right)} \Delta s \right] \right) \\ & \left( \int\_{\kappa}^{\delta\_{+}(T,\kappa)} -d(s) + \frac{f(s)\exp\left(u(s)\right)}{a(s) + \beta(s)\exp\left(u(s)\right) + m(s)\exp\left(v(s)\right)} \Delta s \right) \end{split}$$

Let

$$\begin{aligned} a(t) - b(t) \exp\left(u(t)\right) - \frac{c(t) \exp\left(v(t)\right)}{a(t) + \beta(t) \exp\left(u(t)\right) + m(t) \exp\left(v(t)\right)} &= \mathsf{C}\_1 \\ \mathsf{C}\_2 - d(t) + \frac{f(t) \exp\left(u(t)\right)}{a(t) + \beta(t) \exp\left(u(t)\right) + m(t) \exp\left(v(t)\right)} &= \mathsf{C}\_2 \\ \frac{1}{m \mathrm{cs}(\delta\_+(T, \kappa))} \int\_{\kappa}^{\delta\_+(T, \kappa)} a(s) - b(s) \exp\left(u(s)\right) - \frac{c(s) \exp\left(v(s)\right)}{a(s) + \beta(s) \exp\left(u(s)\right) + m(s) \exp\left(v(s)\right)} &\Delta \mathsf{s} = \overline{\mathsf{C}\_1} \end{aligned}$$

and

$$\begin{split} &\frac{1}{m\kappa(\delta\_{+}(T,\kappa))}\int\_{\kappa}^{\delta\_{+}(T,\kappa)}-d(s)+\frac{f(s)\exp\left(u(s)\right)}{\alpha(s)+\beta(s)\exp\left(u(s)\right)+m(s)\exp\left(v(s)\right)}d\mathbf{s} = \overline{\mathbf{C}}\_{2} \\ &\quad K\_{ll}(I-V)\mathbf{C}\left(\begin{bmatrix} u\\v \end{bmatrix}\right)=K\_{ll}\left(\begin{bmatrix} \mathbf{C}\_{1}-\overline{\mathbf{C}}\_{1} \\ \mathbf{C}\_{2}-\overline{\mathbf{C}}\_{2} \end{bmatrix}\right) \\ &=\left[\int\_{\kappa}^{t}\mathbf{C}\_{1}(s)-\overline{\mathbf{C}}\_{1}(s)\Delta s-\frac{1}{m\kappa(\delta\_{+}(T,\kappa))}\int\_{\kappa}^{\delta\_{+}(T,\kappa)}\int\_{\kappa}^{t}\mathbf{C}\_{1}(s)-\overline{\mathbf{C}}\_{1}(s)\Delta s\right] \\ &\quad \left[\int\_{\kappa}^{t}\mathbf{C}\_{2}(s)-\overline{\mathbf{C}}\_{2}(s)\Delta s-\frac{1}{m\kappa(\delta\_{+}(T,\kappa))}\int\_{\kappa}^{\delta\_{+}(T,\kappa)}\int\_{\kappa}^{t}\mathbf{C}\_{2}(s)-\overline{\mathbf{C}}\_{2}(s)\Delta s\right]. \end{split}$$

Since

x tð Þ is <sup>δ</sup><sup>n</sup>

η1, and η2:

Since Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Then, we get

x y

� �<sup>∈</sup> <sup>X</sup>, then there exist <sup>η</sup><sup>i</sup>

periodic in shifts for any n∈ N on the interval δ<sup>n</sup>

<sup>κ</sup> a tð ÞΔ<sup>t</sup> <sup>≤</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

þð Þ <sup>T</sup>; <sup>ξ</sup><sup>1</sup> and <sup>x</sup>ð Þ¼ <sup>ξ</sup><sup>1</sup> <sup>x</sup> <sup>δ</sup><sup>n</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> b tð ÞΔt > 0, so we get

x η<sup>1</sup> � � ≥ ln

Using the second inequality in Lemma 2, we have

By the first equation of systems (3.2) and (3.5)

Using the first inequality in Lemma 2, we have

x tð Þ ≥ x η<sup>1</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

≥ x η<sup>1</sup>

¼ l<sup>1</sup> � M1≔H<sup>1</sup>

<sup>κ</sup> a tð ÞΔ<sup>t</sup> <sup>≥</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

xð Þ ξ<sup>1</sup> ≤ ln

By the first equation of systems (3.2) and (3.5)

, ξ<sup>i</sup> and i ¼ 1, 2 such that

If ξ<sup>1</sup> is the minimum point of x tð Þ on the interval ½ � κ; δþð Þ T; κ because x tð Þ is a function that is

<sup>κ</sup> b tð Þexp x η<sup>1</sup>

� � � � Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> a tð ÞΔ<sup>t</sup> � <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> b tð ÞΔt

<sup>κ</sup> <sup>∣</sup>x<sup>Δ</sup>ð Þ<sup>t</sup> <sup>∣</sup>Δ<sup>t</sup>

¼ exp xð Þ ð Þ ξ<sup>1</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> a tð ÞΔt Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> b tð ÞΔt

!

κ

<sup>κ</sup> <sup>j</sup>a tð ÞjΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

� �

<sup>κ</sup> b tð Þexp xð Þ ð Þ ξ<sup>1</sup> Δt

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> b tð ÞΔt:

≔l<sup>2</sup>

� � <sup>¼</sup> max<sup>t</sup> <sup>∈</sup>t<sup>∈</sup> ½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> x tð Þ,

<sup>þ</sup> ð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup>

c tð Þ m tð ÞΔ<sup>t</sup>

κ

1 A≔l<sup>1</sup>

<sup>κ</sup> a tð ÞΔt

(3.6)

� �: We have similar results for the other points for ξ2,

þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> ; <sup>δ</sup><sup>n</sup>þ<sup>1</sup>

� � � � <sup>þ</sup>

<sup>κ</sup> b tð ÞΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

c tð Þ m tð ÞΔ<sup>t</sup>

� �

� � <sup>¼</sup> max<sup>t</sup> <sup>∈</sup>t<sup>∈</sup> ½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> y tð Þ (3.5)

http://dx.doi.org/10.5772/intechopen.74952

195

Quantum Calculus with the Notion δ±-Periodicity and Its Applications

� �, the minimum point of

c tð Þ m tð ÞΔt:

xð Þ¼ ξ<sup>1</sup> min<sup>t</sup>∈<sup>t</sup> <sup>∈</sup>½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> x tð Þ, x η<sup>1</sup>

yð Þ¼ ξ<sup>2</sup> min<sup>t</sup>∈<sup>t</sup> <sup>∈</sup>½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> y tð Þ, y η<sup>2</sup>

þð Þ <sup>T</sup>; <sup>ξ</sup><sup>1</sup>

¼ exp x η<sup>1</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

� � � <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

� � � <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

0 @

Clearly, VC and KUð Þ I � V C are continuous. Here, X and Y are Banach spaces. Since for the given time scale T while T is constant, mesð Þ δþð Þ T; t is equal for each t∈T; then, we can apply Arzela-Ascoli theorem, and by using Arzela-Ascoli theorem, we can find that KUð Þ I � V C Ω � � is compact for any open bounded set Ω ⊂ X: Additionally, VC Ω � � is bounded. Thus, C is Lcompact on Ω with any open bounded set Ω ⊂ X:

To apply the continuation theorem, we investigate the below operator equation:

$$\begin{aligned} \mathbf{x}^{\Delta}(t) &= \lambda \left[ a(t) - b(t) \exp\left(\mathbf{x}(t)\right) - \frac{c(t) \exp\left(y(t)\right)}{a(t) + \beta(t) \exp\left(\mathbf{x}(t)\right) + m(t) \exp\left(y(t)\right)} \right] \\ &\quad \mathbf{y}^{\Delta}(t) = \lambda \left[ -d(t) + \frac{f(t) \exp\left(\mathbf{x}(t)\right)}{a(t) + \beta(t) \exp\left(\mathbf{x}(t)\right) + m(t) \exp\left(y(t)\right)} \right] \end{aligned} \tag{3.1}$$

Let x y � �<sup>∈</sup> <sup>X</sup> be any solution of system (3.1). Integrating both sides of system (3.1) over the interval 0½ � ; w , we obtain

$$\begin{cases} \int\_{\mathbf{x}}^{\mathbf{f}\_{\star}(T,\mathbf{x})} a(t) \Delta t = \int\_{\mathbf{x}}^{\mathbf{f}\_{\star}(T,\mathbf{x})} b(t) \exp\left(\mathbf{x}(t)\right) + \frac{c(t) \exp\left(y(t)\right)}{a(t) + \beta(t) \exp\left(\mathbf{x}(t)\right) + m(t) \exp\left(y(t)\right)} \Delta t \\\\ \int\_{\mathbf{x}}^{\mathbf{f}\_{\star}(T,\mathbf{x})} d(t) \Delta t = \int\_{\mathbf{x}}^{\mathbf{f}\_{\star}(T,\mathbf{x})} \frac{f(t) \exp\left(\mathbf{x}(t)\right)}{a(t) + \beta(t) \exp\left(\mathbf{x}(t)\right) + m(t) \exp\left(y(t)\right)} \Delta t \end{cases} \tag{3.2}$$

From (3.1) and (3.2), we get

$$\begin{split} \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T)} |\Delta^{\mathbb{A}}(t)| \Delta t &\leq \ \lambda \left[ \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T)} |a(t)| \Delta t + \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T)} |b(t) \exp\left(\mathbf{x}(t)\right) + \frac{c(t) \exp\left(y(t)\right)}{a(t) + \beta(t) \exp\left(x(t)\right) + m(t) \exp\left(y(t)\right)} \Delta t \right]. \\ &\leq \ \lambda \left[ \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T,\kappa)} |a(t)| \Delta t + \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T,\kappa)} a(t) \Delta t = \mathcal{M} \right. \\ &\left. \left. \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T,\kappa)} |a(t)| \Delta t \right. \\ &\left. \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T,\kappa)} |a(t)| \Delta t \right. \\ &\leq \lambda \left[ \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T,\kappa)} |d(t)| \Delta t + \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T,\kappa)} d(t) \Delta t \right] \\ &\leq \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T,\kappa)} |d(t)| \Delta t + \int\_{\mathbb{R}} \mathbb{f}\_{\kappa}^{\mathbb{A},(T,\kappa)} d(t) \Delta t \Delta t \end{split} \tag{3.4}$$

Since x y � �<sup>∈</sup> <sup>X</sup>, then there exist <sup>η</sup><sup>i</sup> , ξ<sup>i</sup> and i ¼ 1, 2 such that xð Þ¼ ξ<sup>1</sup> min<sup>t</sup>∈<sup>t</sup> <sup>∈</sup>½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> x tð Þ, x η<sup>1</sup> � � <sup>¼</sup> max<sup>t</sup> <sup>∈</sup>t<sup>∈</sup> ½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> x tð Þ,

$$y(\xi\_2) = \min\_{t \in t \in [\kappa \delta\_+(T, \kappa)]} y(t), \\ y(\eta\_2) = \max\_{t \in t \in [\kappa \delta\_+(T, \kappa)]} y(t)$$
 
$$\dots \dots \dots \dots \dots \xi\_{\kappa \delta\_+(T, \kappa)} y(t), \\ y(\eta\_2) = \max\_{t \in t \in [\kappa \delta\_+(T, \kappa)]} y(t) \tag{3.5}$$

If ξ<sup>1</sup> is the minimum point of x tð Þ on the interval ½ � κ; δþð Þ T; κ because x tð Þ is a function that is periodic in shifts for any n∈ N on the interval δ<sup>n</sup> þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> ; <sup>δ</sup><sup>n</sup>þ<sup>1</sup> <sup>þ</sup> ð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> � �, the minimum point of x tð Þ is <sup>δ</sup><sup>n</sup> þð Þ <sup>T</sup>; <sup>ξ</sup><sup>1</sup> and <sup>x</sup>ð Þ¼ <sup>ξ</sup><sup>1</sup> <sup>x</sup> <sup>δ</sup><sup>n</sup> þð Þ <sup>T</sup>; <sup>ξ</sup><sup>1</sup> � �: We have similar results for the other points for ξ2, η1, and η2:

By the first equation of systems (3.2) and (3.5)

$$\begin{aligned} \int\_{\kappa}^{\delta\_{+}(T,\kappa)} a(t) \Delta t & \leq \quad \int\_{\kappa}^{\delta\_{+}(T,\kappa)} \left[ b(t) \exp\left(\mathbf{x}(\eta\_{1})\right) + \frac{c(t)}{m(t)} \Delta t \right] \\ & = \quad \exp\left(\mathbf{x}(\eta\_{1})\right) \int\_{\kappa}^{\delta\_{+}(T,\kappa)} b(t) \Delta t + \int\_{\kappa}^{\delta\_{+}(T,\kappa)} \frac{c(t)}{m(t)} \Delta t. \end{aligned}$$

Since Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> b tð ÞΔt > 0, so we get

1 mesð Þ δþð Þ T; κ

194 Advanced Technologies of Quantum Key Distribution

KUð Þ I � V C

Ðt

Ðt

¼

Let

x y � �

8 >>><

>>>:

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

interval 0½ � ; w , we obtain

<sup>κ</sup> a tð ÞΔ<sup>t</sup> <sup>¼</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> d tð ÞΔ<sup>t</sup> <sup>¼</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

From (3.1) and (3.2), we get

<sup>κ</sup> <sup>∣</sup>x<sup>Δ</sup>ð Þ<sup>t</sup> <sup>∣</sup>Δ<sup>t</sup> <sup>≤</sup> <sup>λ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

κ

≤ λ Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

≤ Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> <sup>∣</sup>y<sup>Δ</sup>ð Þ<sup>t</sup> <sup>∣</sup>Δ<sup>t</sup> <sup>≤</sup> <sup>λ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> <sup>j</sup>a tð ÞjΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> <sup>j</sup>a tð ÞjΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> <sup>∣</sup>a tð Þ∣Δ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

≤ λ Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

≤ Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

h i

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

> u v

<sup>κ</sup> <sup>C</sup>1ð Þ� <sup>s</sup> <sup>C</sup>1ð Þ<sup>s</sup> <sup>Δ</sup><sup>s</sup> � <sup>1</sup>

<sup>κ</sup> <sup>C</sup>2ð Þ� <sup>s</sup> <sup>C</sup>2ð Þ<sup>s</sup> <sup>Δ</sup><sup>s</sup> � <sup>1</sup>

compact on Ω with any open bounded set Ω ⊂ X:

¼ KU

is compact for any open bounded set Ω ⊂ X: Additionally, VC Ω

! " #

�d sð Þþ f sð Þexp u s ð Þ ð Þ

Clearly, VC and KUð Þ I � V C are continuous. Here, X and Y are Banach spaces. Since for the given time scale T while T is constant, mesð Þ δþð Þ T; t is equal for each t∈T; then, we can apply Arzela-Ascoli theorem, and by using Arzela-Ascoli theorem, we can find that KUð Þ I � V C Ω

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

� �

∈ X be any solution of system (3.1). Integrating both sides of system (3.1) over the

<sup>y</sup><sup>Δ</sup>ðÞ¼ <sup>t</sup> <sup>λ</sup> �d tð Þþ f tð Þexp x t ð Þ ð Þ

<sup>κ</sup> b tð Þexp x t ð Þþ ð Þ c tð Þexp y t ð Þ ð Þ

<sup>κ</sup> a tð ÞΔt

<sup>κ</sup> a tð ÞΔt≔M<sup>1</sup>

κ

<sup>κ</sup> d tð ÞΔt

<sup>κ</sup> d tð ÞΔt≔M<sup>2</sup>

<sup>κ</sup> <sup>j</sup>d tð ÞjΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> <sup>j</sup>d tð ÞjΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> <sup>∣</sup>d tð Þ∣Δ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

h i

f tð Þexp x t ð Þ ð Þ

ðt κ

ðt κ

C<sup>1</sup> � C<sup>1</sup> C<sup>2</sup> � C<sup>2</sup>

mesð Þ δþð Þ T; κ

mesð Þ δþð Þ T; κ

To apply the continuation theorem, we investigate the below operator equation:

<sup>x</sup><sup>Δ</sup>ðÞ¼ <sup>t</sup> <sup>λ</sup> a tð Þ� b tð Þexp x t ð Þ� ð Þ c tð Þexp y t ð Þ ð Þ

! " #

<sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> exp u s ð Þþ ð Þ m sð Þexp v s ð Þ ð Þ <sup>Δ</sup><sup>s</sup> <sup>¼</sup> <sup>C</sup><sup>2</sup>

C1ð Þ� s C1ð Þs Δs

� � is bounded. Thus, C is L-

� �

(3.2)

(3.3)

(3.4)

,

C2ð Þ� s C2ð Þs Δs

αð Þþ t βð Þt exp x t ð Þþ ð Þ m tð Þexp y t ð Þ ð Þ

� � (3.1)

<sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> exp x t ð Þþ ð Þ m tð Þexp y t ð Þ ð Þ <sup>Δ</sup>t ,

f tð Þexp x t ð Þ ð Þ <sup>α</sup>ðÞþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> exp x t ð Þþ ð Þ m tð Þexp y t ð Þ ð Þ <sup>Δ</sup><sup>t</sup>

<sup>α</sup>ðÞþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> exp x t ð Þþ ð Þ m tð Þexp y t ð Þ ð Þ <sup>Δ</sup><sup>t</sup>

<sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> exp x t ð Þþ ð Þ m tð Þexp y t ð Þ ð Þ <sup>Δ</sup>t ,

<sup>κ</sup> b tð Þexp x t ð Þþ ð Þ c tð Þexp y t ð Þ ð Þ

� �

� �

αð Þþ t βð Þt exp x t ð Þþ ð Þ m tð Þexp y t ð Þ ð Þ

$$\propto \left( \eta\_1 \right) \geq \ln \left( \frac{\int\_{\kappa}^{\delta\_+ \left( T, \kappa \right)} a(t) \Delta t - \int\_{\kappa}^{\delta\_+ \left( T, \kappa \right)} \frac{c(t)}{m(t)} \Delta t}{\int\_{\kappa}^{\delta\_+ \left( T, \kappa \right)} b(t) \Delta t} \right) \coloneqq l\_1.$$

Using the second inequality in Lemma 2, we have

$$\begin{array}{llll} \mathbf{x}(t) & \succeq & \mathbf{x}(\eta\_{1}) - \int\_{\mathbf{x}}^{\delta\_{+}(T,\mathbf{x})} |\mathbf{x}^{\Delta}(t)| \Delta t \\ & \succeq & \mathbf{x}(\eta\_{1}) - \left(\int\_{\mathbf{x}}^{\delta\_{+}(T,\mathbf{x})} |a(t)| \Delta t + \int\_{\mathbf{x}}^{\delta\_{+}(T,\mathbf{x})} a(t) \Delta t\right) \\ & = & l\_{1} - M\_{1} \coloneqq H\_{1} \end{array} \tag{3.6}$$

By the first equation of systems (3.2) and (3.5)

$$\begin{aligned} \int\_{\mathbb{K}}^{\delta\_+(T,\kappa)} a(t) \Delta t & \quad \ge \quad \int\_{\kappa}^{\delta\_+(T,\kappa)} b(t) \exp\left(\mathfrak{x}(\xi\_1)\right) \Delta t \\ & = \quad \exp\left(\mathfrak{x}(\xi\_1)\right) \int\_{\kappa}^{\delta\_+(T,\kappa)} b(t) \Delta t. \end{aligned}$$

Then, we get

$$\mathfrak{a}(\xi\_1) \le \ln \left( \frac{\int\_{\kappa}^{\delta\_+(T,\kappa)} a(t) \Delta t}{\int\_{\kappa}^{\delta\_+(T,\kappa)} b(t) \Delta t} \right) \coloneqq l\_2.$$

Using the first inequality in Lemma 2, we have

$$\begin{aligned} \mathbf{x}(t) &\leq \quad \mathbf{x}(\xi\_1) + \int\_{\mathbf{x}}^{\delta\_+(T,\mathbf{x})} |\mathbf{x}^\Delta(t)| \Delta t \\ &\leq \quad \mathbf{x}(\xi\_1) + \left(\int\_{\mathbf{x}}^{\delta\_+(T,\mathbf{x})} |a(t)| \Delta t + \int\_{\mathbf{x}}^{\delta\_+(T,\mathbf{x})} a(t) \Delta t\right) \\ &= \quad l\_2 + M\_1 \coloneqq H\_2 \end{aligned} \tag{3.7}$$

exp y η<sup>2</sup> � � � � ≥

Using the assumption of the Theorem 5, we obtain

f tð ÞΔ<sup>t</sup> � <sup>β</sup><sup>u</sup>

mu

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

y η<sup>2</sup>

By using the second inequality in Lemma 2

� � <sup>≥</sup> ln <sup>1</sup>

y tð Þ ≥ y η<sup>2</sup>

≥ y η<sup>2</sup>

e H<sup>1</sup>

and

<sup>Ω</sup> <sup>¼</sup> <sup>x</sup>

VC

x y

y �� � � � �

� �∈KerL <sup>∩</sup> <sup>∂</sup>Ω, <sup>x</sup>

x y ! " #

� � � � <sup>∈</sup> <sup>X</sup> : <sup>x</sup>

y

¼

y �� � � � �

0

BBBB@

" #

where J : ImV ! KerL is the identity operator.

6¼ <sup>0</sup> 0 � � � � < M

� � is a constant with

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

1 mu

! !

e<sup>H</sup><sup>1</sup>

e<sup>H</sup><sup>1</sup>

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

> Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> f tð ÞΔt

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

� � � <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

� � � <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

¼ L<sup>2</sup> � M2≔H4:

are both independent of λ: Let M ¼ B<sup>1</sup> þ B<sup>2</sup> þ 1. Then, max<sup>t</sup><sup>∈</sup> ½ � <sup>t</sup>0;δþð Þ <sup>T</sup>;t<sup>0</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> f tð ÞΔt

d tð ÞΔt

<sup>κ</sup> d tð ÞΔ<sup>t</sup> � <sup>β</sup>ue

<sup>κ</sup> <sup>∣</sup>y<sup>Δ</sup>ð Þ<sup>t</sup> <sup>∣</sup>Δ<sup>t</sup>

By Eq. (3.8) and (3.9), we have max<sup>t</sup> <sup>∈</sup>½ � <sup>t</sup>0;δþð Þ <sup>T</sup>;t<sup>0</sup> ∣y tð Þ∣ ≤ maxf g jH3j; jH4j ≔B2. Obviously, B<sup>1</sup> and B<sup>2</sup>

x y �� � � � �

<sup>κ</sup> �d sð Þþ f sð Þexp xð Þ

x y � � � �

JVC

Let us define the homotopy such that H<sup>ν</sup> ¼ νð Þþ JVC ð Þ 1 � ν G where

� � �

� �; then, <sup>Ω</sup> verifies the requirement (a) in Theorem 4. When

<sup>κ</sup> a sð Þ� b sð Þexp xð Þ� c sð Þexp yð Þ

¼ VC

� <sup>¼</sup> M, ; then,

<sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> exp xð Þþ m sð Þexp yð Þ <sup>Δ</sup><sup>t</sup>

x y � � � �

<sup>κ</sup> <sup>j</sup>d tð ÞjΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

� �

! !

<sup>κ</sup> d tð ÞΔ<sup>t</sup> � <sup>β</sup>ue

!

� <sup>α</sup><sup>u</sup>

<sup>H</sup><sup>1</sup> � <sup>α</sup><sup>u</sup>

Quantum Calculus with the Notion δ±-Periodicity and Its Applications

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

<sup>H</sup><sup>1</sup> � <sup>α</sup><sup>u</sup>

<sup>κ</sup> d tð ÞΔt

<sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> exp xð Þþ m sð Þexp yð ÞΔ<sup>t</sup>

:

http://dx.doi.org/10.5772/intechopen.74952

d tð ÞΔt !

≔L2:

> 0

x y �� � � � �

� � � � (3.9)

197

< M: Let

1

CCCCA

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

By Eq. (3.6) and (3.7), max<sup>t</sup>∈½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> ∣x tð Þ∣ ≤ maxf g jH1j; jH2j ≔B1: From the second equation of system (3.2) and the second equation of system (3.6), we can derive that

$$\begin{split} \int\_{\kappa}^{\delta\_{+}(T,\kappa)} d(t) \Delta t &\quad \leq \quad \int\_{\kappa}^{\delta\_{+}(T,\kappa)} \frac{f(t) \exp\left(\mathbf{x}(t)\right)}{\beta^{l} \exp\left(\mathbf{x}(t)\right) + m^{l} \exp\left(y(t)\right)} \Delta t \\ &\leq \quad \quad \int\_{\kappa}^{\delta\_{+}(T,\kappa)} \frac{f(t) \varepsilon^{H\_{2}}}{\beta^{l} e^{H\_{2}} + m^{l} \exp\left(y(\xi\_{2})\right)} \Delta t \\ &= \quad \quad \quad \frac{e^{H\_{2}}}{\beta^{l} e^{H\_{2}} + m^{l} \exp\left(y(\xi\_{2})\right)} \int\_{\kappa}^{\delta\_{+}(T,\kappa)} f(t) \Delta t. \end{split}$$

Therefore,

$$\exp\left(y(\xi\_2)\right) \le \frac{1}{m^l} \left(\frac{e^{H\_2} \int\_{\kappa}^{\delta\_+(T,\kappa)} f(t) \Delta t}{\int\_{\kappa}^{\delta\_+(T,\kappa)} d(t) \Delta t} - \beta^l e^{H\_2}\right).$$

By the assumption of the Theorem 5, we get,

$$\int\_{\kappa}^{\delta\_+(T,\kappa)} f(t)\Delta t - \beta^l \left(\int\_{\kappa}^{\delta\_+(T,\kappa)} d(t)\right) \Delta t > 0 \text{ and}$$

$$y(\xi\_2) \le \ln \left(\frac{1}{m^l} \left(\frac{e^{H\_2} \int\_{\kappa}^{\delta\_+(T,\kappa)} f(t) \Delta t}{\int\_{\kappa}^{\delta\_+(T,\kappa)} d(t) \Delta t} - \beta^l e^{H\_2}\right)\right) := L\_1.$$

Hence, by using the first inequality in Lemma 2 and the second equation of system (3.2)

$$\begin{split} \left| y(t) \right| &\leq \left. y(\xi\_2) + \int\_{\kappa}^{\delta\_+(T,\kappa)} |y^\Lambda(t)| \Delta t \\ &\leq \left. y(\xi\_2) + \left( \int\_{\kappa}^{\delta\_+(T,\kappa)} |d(t)| \Delta t + \int\_{\kappa}^{\delta\_+(T,\kappa)} d(t) \Delta t \right) \right| \\ &\leq \left. L\_1 + M\_2 \coloneqq H\_3. \end{split} \tag{3.8}$$

Again, using the second equation of system (3.2), we obtain

$$\begin{split} \int\_{\kappa}^{\delta\_{+}(T,\kappa)} d(t) \Delta t &\quad \geq \quad \int\_{\kappa}^{\delta\_{+}(T,\kappa)} \frac{f(t) \exp\left(\chi(t)\right)}{\alpha^{u} + \beta^{u} \exp\left(\chi(t)\right) + m^{u} \exp\left(\chi(t)\right)} \Delta t \\ &\quad \geq \quad \int\_{\kappa}^{\delta\_{+}(T,\kappa)} \frac{f(t) e^{H\_{1}}}{\alpha^{u} + \beta^{u} e^{H\_{1}} + m^{u} \exp\left(\chi(\eta\_{2})\right)} \Delta t \\ &= \quad \quad \frac{e^{H\_{1}}}{\alpha^{u} + \beta^{u} e^{H\_{1}} + m^{u} \exp\left(\chi(\eta\_{2})\right)} \int\_{\kappa}^{\delta\_{+}(T,\kappa)} f(t) \Delta t \end{split}$$

Quantum Calculus with the Notion δ±-Periodicity and Its Applications http://dx.doi.org/10.5772/intechopen.74952 197

$$\exp\left(y\left(\eta\_{2}\right)\right) \geq \frac{1}{m^{\mu}} \left(\frac{e^{H\_{1}}\int\_{\kappa}^{\delta\_{+}(T,\kappa)} f(t)\Delta t}{\int\_{\kappa}^{\delta\_{+}(T,\kappa)} d(t)\Delta t} - \beta^{\mu}e^{H\_{1}} - \alpha^{\mu}\right).$$

Using the assumption of the Theorem 5, we obtain

$$\left(\varepsilon^{H\_1} \left(\int\_{\kappa}^{\delta\_+(T,\kappa)} f(t) \Delta t - \beta^u \left(\int\_{\kappa}^{\delta\_+(T,\kappa)} d(t) \Delta t\right)\right) - \alpha^u \left(\int\_{\kappa}^{\delta\_+(T,\kappa)} d(t) \Delta t\right) > 0\right)$$

and

x tð Þ ≤ xð Þþ ξ<sup>1</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

196 Advanced Technologies of Quantum Key Distribution

By the assumption of the Theorem 5, we get,

Therefore,

≤ xð Þþ ξ<sup>1</sup>

¼ l<sup>2</sup> þ M1≔H<sup>2</sup>

system (3.2) and the second equation of system (3.6), we can derive that

κ

≤ Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

> 1 ml

f tð ÞΔ<sup>t</sup> � <sup>β</sup><sup>l</sup>

e<sup>H</sup><sup>2</sup>

ml

<sup>¼</sup> <sup>e</sup><sup>H</sup><sup>2</sup> βl

e<sup>H</sup><sup>2</sup>

<sup>κ</sup> d tð ÞΔ<sup>t</sup> <sup>≤</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

exp yð Þ ð Þ ξ<sup>2</sup> ≤

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

<sup>y</sup>ð Þ <sup>ξ</sup><sup>2</sup> <sup>≤</sup> ln <sup>1</sup>

y tð Þ ≤ yð Þþ ξ<sup>2</sup>

Again, using the second equation of system (3.2), we obtain

<sup>κ</sup> d tð ÞΔ<sup>t</sup> <sup>≥</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

≤ yð Þþ ξ<sup>2</sup>

≤ L<sup>1</sup> þ M2≔H3:

κ

≥ Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

<sup>¼</sup> <sup>e</sup><sup>H</sup><sup>1</sup>

<sup>α</sup><sup>u</sup> <sup>þ</sup> <sup>β</sup>ueH<sup>1</sup> <sup>þ</sup> mu exp y <sup>η</sup><sup>2</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> <sup>∣</sup>x<sup>Δ</sup>ð Þ<sup>t</sup> <sup>∣</sup>Δ<sup>t</sup>

By Eq. (3.6) and (3.7), max<sup>t</sup>∈½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> ∣x tð Þ∣ ≤ maxf g jH1j; jH2j ≔B1: From the second equation of

βl

eH<sup>2</sup> þ ml exp yð Þ ð Þ ξ<sup>2</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> f tð ÞΔt

<sup>κ</sup> d tð ÞΔ<sup>t</sup> � <sup>β</sup><sup>l</sup>

d tð Þ !

!

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

<sup>κ</sup> d tð ÞΔ<sup>t</sup> � <sup>β</sup><sup>l</sup>

<sup>κ</sup> <sup>j</sup>d tð ÞjΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

� �

f tð Þexp x t ð Þ ð Þ <sup>α</sup><sup>u</sup> <sup>þ</sup> <sup>β</sup><sup>u</sup> exp x t ð Þþ ð Þ mu exp y t ð Þ ð Þ <sup>Δ</sup><sup>t</sup>

� � � �

f tð Þe<sup>H</sup><sup>1</sup> <sup>α</sup><sup>u</sup> <sup>þ</sup> <sup>β</sup>ueH<sup>1</sup> <sup>þ</sup> mu exp y <sup>η</sup><sup>2</sup>

! !

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> f tð ÞΔt

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Hence, by using the first inequality in Lemma 2 and the second equation of system (3.2)

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> <sup>∣</sup>y<sup>Δ</sup>ð Þ<sup>t</sup> <sup>∣</sup>Δ<sup>t</sup>

<sup>κ</sup> <sup>j</sup>a tð ÞjΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

� �

f tð Þexp x t ð Þ ð Þ <sup>β</sup><sup>l</sup> exp x t ð Þþ ð Þ ml exp y t ð Þ ð Þ

f tð Þe<sup>H</sup><sup>2</sup>

eH<sup>2</sup> þ ml exp yð Þ ð Þ ξ<sup>2</sup>

<sup>κ</sup> a tð ÞΔt

Δt

Δt

f tð ÞΔt:

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

> e H<sup>2</sup>

Δt > 0 and

≔L<sup>1</sup>

<sup>κ</sup> d tð ÞΔt

� � � � <sup>Δ</sup><sup>t</sup>

f tð ÞΔt,

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

e H<sup>2</sup> (3.7)

(3.8)

$$\log \left( \eta\_2 \right) \ge \ln \left( \frac{1}{m^u} \left( \frac{e^{H\_1} \int\_{\kappa}^{\delta\_+(T,\kappa)} f(t) \Delta t}{\int\_{\kappa}^{\delta\_+(T,\kappa)} d(t) \Delta t} - \beta^u e^{H\_1} - \alpha^u \right) \right) := L\_2 \dots$$

By using the second inequality in Lemma 2

$$\begin{split} \left| y(t) \right| &\geq \quad y(\eta\_2) - \int\_{\kappa}^{\delta\_+(T,\kappa)} \left| y^\Lambda(t) \right| \Delta t \\ &\geq \quad y(\eta\_2) - \left( \int\_{\kappa}^{\delta\_+(T,\kappa)} \left| d(t) \right| \Delta t + \int\_{\kappa}^{\delta\_+(T,\kappa)} d(t) \Delta t \right) \\ &= \quad L\_2 - M\_2 \coloneqq H\_4. \end{split} \tag{3.9}$$

By Eq. (3.8) and (3.9), we have max<sup>t</sup> <sup>∈</sup>½ � <sup>t</sup>0;δþð Þ <sup>T</sup>;t<sup>0</sup> ∣y tð Þ∣ ≤ maxf g jH3j; jH4j ≔B2. Obviously, B<sup>1</sup> and B<sup>2</sup> are both independent of λ: Let M ¼ B<sup>1</sup> þ B<sup>2</sup> þ 1. Then, max<sup>t</sup><sup>∈</sup> ½ � <sup>t</sup>0;δþð Þ <sup>T</sup>;t<sup>0</sup> x y �� � � � � � � � � < M: Let <sup>Ω</sup> <sup>¼</sup> <sup>x</sup> y �� � � � � � � � � <sup>∈</sup> <sup>X</sup> : <sup>x</sup> y �� � � � � � � � � < M � �; then, <sup>Ω</sup> verifies the requirement (a) in Theorem 4. When x y � �∈KerL <sup>∩</sup> <sup>∂</sup>Ω, <sup>x</sup> y � � is a constant with x y �� � � � � � � � � <sup>¼</sup> M, ; then, VC x y ! " # ¼ Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> a sð Þ� b sð Þexp xð Þ� c sð Þexp yð Þ <sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> exp xð Þþ m sð Þexp yð ÞΔ<sup>t</sup> Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> �d sð Þþ f sð Þexp xð Þ <sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> exp xð Þþ m sð Þexp yð Þ <sup>Δ</sup><sup>t</sup> 2 6 6 6 6 4 3 7 7 7 7 5 0 BBBB@ 1 CCCCA 6¼ <sup>0</sup> 0 " # JVC x y � � � � ¼ VC x y � � � �

where J : ImV ! KerL is the identity operator.

Let us define the homotopy such that H<sup>ν</sup> ¼ νð Þþ JVC ð Þ 1 � ν G where

$$G\left(\begin{bmatrix}\mathbf{x}\\\mathbf{y}\end{bmatrix}\right) = \begin{bmatrix}\int\_{\kappa}^{\delta\_{+}(\overline{T},\mathbf{x})} a(\mathbf{s}) - b(\mathbf{s}) \exp\left(\mathbf{x}\right) \Delta t\\ \int\_{\kappa}^{\delta\_{+}(\overline{T},\mathbf{x})} d(\mathbf{s}) - \frac{f(\mathbf{s}) \exp\left(\mathbf{x}\right)}{a(\mathbf{s}) + \beta(\mathbf{s}) \exp\left(\mathbf{x}\right) + m(\mathbf{s}) \exp\left(\mathbf{y}\right)} \Delta t \end{bmatrix}.$$

As a result, it is seen that one can define a periodicity notion that is applicable to the structure of the quantum calculus. Additionally, it is shown that this notion is useful for different

Quantum Calculus with the Notion δ±-Periodicity and Its Applications

http://dx.doi.org/10.5772/intechopen.74952

199

There are many studies about the predator–prey dynamic systems on time scale calculus such as [14, 19, 27, 28]. All of these cited studies are about the periodic solutions of the considered system on a periodic time scale. However, in the world, there are many different species. While investigating the periodicity notion of the different life cycle of the species, the w-periodic time scales could be a little bit restricted. Therefore, if the life cycle of this kind of species is appropriate to the Beddington-DeAngelis functional response, then the results that we have

In addition to these, the δ-periodic solutions for predator–prey dynamic systems with Holling-type functional response, semiratio-dependent functional response, and monotype functional response can be also taken into account for future studies. In that dynamic systems, delay conditions and impulsive conditions can also be added for the new investigations.

A major portion of the chapter is borrowed from the publication "Behavior of the solutions for predator-prey dynamic systems with Beddington-DeAngelis-type functional response on peri-

Faculty of Science, Department of Mathematics, Ondokuz Mayis University, Samsun, Turkey

[1] Exton H. q-Hypergeometric Functions and Applications. New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN0853124914, ISBN0470274530, ISBN9780470274538

[2] Kac V, Cheung P. Quantum Calculus. Springer Science and Business Media; 2001

applications. One of its applications is analyzed in this study with an example.

found in that study are becoming more useful and important.

This is a joint work with Ayse Feza Guvenilir and Billur Kaymakcalan.

Neslihan Nesliye Pelen\*, Ayşe Feza Güvenilir and Billur Kaymakçalan

\*Address all correspondence to: nesliyeaykir@gmail.com

5. Discussion

Acknowledgements

odic time scales in shifts" [24].

Author details

References

Take DJG as the determinant of the Jacobian of G: Since x y � �∈KerL, then Jacobian of <sup>G</sup> is

$$\begin{bmatrix} -\varepsilon^{\mathbf{r}} \int\_{\mathbf{x}}^{\mathbf{t}\_{+}(T,\mathbf{x})} b(\mathbf{s}) \Delta t & 0\\\\ \int\_{\mathbf{x}}^{\mathbf{t}\_{+}(T,\mathbf{x})} \frac{-\varepsilon^{\mathbf{r}} f(\mathbf{s})}{a(\mathbf{s}) + \beta(\mathbf{s})\varepsilon^{\mathbf{r}} + m(\mathbf{s})\varepsilon^{\mathbf{r}}} \Delta t + \int\_{\mathbf{x}}^{\mathbf{t}\_{+}(T,\mathbf{x})} \frac{(\varepsilon^{\mathbf{r}})^{2} f(\mathbf{s}) \delta(\mathbf{s})}{\left(a(\mathbf{s}) + \beta(\mathbf{s})\varepsilon^{\mathbf{r}} + m(\mathbf{s})\varepsilon^{\mathbf{r}}\right)^{2}} \Delta t & -\int\_{\mathbf{x}}^{\mathbf{t}\_{+}(T,\mathbf{x})} \frac{\varepsilon^{\mathbf{r}} \varepsilon^{\mathbf{r}} f(\mathbf{s}) m(\mathbf{s})}{\left(a(\mathbf{s}) + \beta(\mathbf{s})\varepsilon^{\mathbf{r}} + m(\mathbf{s})\varepsilon^{\mathbf{r}}\right)^{2}} \Delta t \end{bmatrix}$$

All the functions in Jacobian of G is positive; then, signDJG is always positive. Hence,

$$\deg(\text{UC}, \Omega \cap \text{Ker}L, 0) = \deg(\text{G}, \Omega \cap \text{Ker}L, 0) = \sum\_{\begin{bmatrix} x \\ y \end{bmatrix} \in \mathcal{G}^{-1}} \text{sign} Dl\_{\mathbb{G}} \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) \neq 0.1$$

Thus, all the conditions of Theorem 4 are satisfied. Therefore, system (2.1) has at least a positive <sup>δ</sup>�-periodic solution. □

Example 1 Let T ¼ f g0 ∪q<sup>Z</sup>: δ�ð Þ q; t is the shift operator and t<sup>0</sup> ¼ 1:

$$\begin{split} \mathbf{x}^{\Delta}(t) &= \left( (-1)^{\frac{\ln|t|}{\ln|t|}} + 4 \right) - \left( (-1)^{\frac{\ln|t|}{\ln|t|}} + 0.5 \right) \exp\left( \mathbf{x}(t) \right) - \frac{\exp\left( y(t) \right)}{\exp\left( \mathbf{x}(t) \right) + 2 \exp\left( y(t) \right)}, \\ \mathbf{y}^{\Delta}(t) &= -0.3 + \frac{\left( (-1)^{\frac{\ln|t|}{\ln|t|}} + 7 \right) \exp\left( \mathbf{x}(t) \right)}{\exp\left( \mathbf{x}(t) \right) + 2 \exp\left( y(t) \right)}, \end{split} \tag{3.10}$$

Each function in system (12) is <sup>δ</sup>� <sup>q</sup><sup>2</sup>; <sup>t</sup> � � periodic and satisfies Theorem 1; then, the system has at least one <sup>δ</sup>� <sup>q</sup><sup>2</sup>; <sup>t</sup> � � periodic solution. Here, mes <sup>δ</sup><sup>þ</sup> <sup>q</sup><sup>2</sup>; <sup>t</sup> � � � � <sup>¼</sup> <sup>2</sup>:

#### 4. Conclusion

The important results of this study are:


As a result, it is seen that one can define a periodicity notion that is applicable to the structure of the quantum calculus. Additionally, it is shown that this notion is useful for different applications. One of its applications is analyzed in this study with an example.
