Discrete relaxation equations of arbitrary order with periodic boundary conditions

Abstract

Two simple nabla fractional relaxation equations with related periodic boundary conditions are addressed in this article. Firstly, we construct the corresponding Green’s functions and obtain some of their properties. Through relevant fixed-point theorems with adequate restrictions, we provide sufficient conditions for the existence of solutions to the problems under consideration. To further illustrate how applicable previous findings are, we also offer a few examples.

1 Introduction

The concept of fractional derivative [1, 2] is a generalization of the classical derivative to an arbitrary noninteger order. Fractional differential equations are applicable in various fields of science and engineering, such as mechanics, economics, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, oscillation theory, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, and many other allied areas. In particular, problems concerning the qualitative analysis of linear and nonlinear fractional differential equations have received the attention of many authors; see [3,4,5,6,7,8] and the references therein.

On the other hand, nabla fractional calculus is an integrated theory of arbitrary order sums and differences in the backward sense. The concept of nabla fractional difference traces back to the works of many famous researchers in the last 2 decades. For a detailed introduction, we refer to the recent monographs [9,10,11] and the references therein.

Since 2010, there has been an increasing interest in analyzing nabla fractional boundary value problems. To name a few notable works, we refer to [12,13,14,15,16,17,18,19]. In this line, we investigate two simple nabla fractional periodic boundary value problems. Specifically, we shall consider the following nabla fractional relaxation equations associated with periodic boundary conditions:

$$\begin{aligned} {\left\{ \begin{array}{ll} \big (\nabla ^{\delta }_{\rho (0)}y\big )(x) = \vartheta y(x) + f(x, y(\rho (x))), \quad x \in {\mathbb {N}}^{T}_{1}, \\ y(0) = y(T), \end{array}\right. } \end{aligned}$$

(1.1)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \big (\nabla ^{\delta }_{0{*}}y\big )(x) = \vartheta y(x) + f(x, y(\rho (x))), \quad x \in {\mathbb {N}}^{T}_{1}, \\ y(0) = y(T), \end{array}\right. } \end{aligned}$$

(1.2)

where $0< \delta < 1$; $-1< \vartheta < 0$; $T \in {\mathbb {N}}_{2}$; $f: {\mathbb {N}}^{T}_{1} \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a continuous function; $\nabla ^{\delta }_{\rho (0)}y$ and $\nabla ^{\delta }_{0{*}}y$ denote the Riemann–Liouville and the Caputo nabla fractional differences of y of order $\delta$, respectively.

The structure of the current article is as follows: Preliminaries for discrete fractional calculus are found in Sect. 2. In Sect. 3, we construct the corresponding Green’s functions and obtain some of their properties. Using relevant fixed-point theorems and some appropriate restrictions on $\vartheta$ and f, we derive sufficient conditions for the existence of solutions to (1.1) and (1.2) in Sect. 4. Additionally, in Sect. 5, we give a few examples to show how the findings in Sect. 4 can be used.

2 Preliminaries

Represent by ${\mathbb {N}}_{\mu } = \{\mu , \mu + 1, \mu + 2, \ldots \}$ and ${\mathbb {N}}^{\nu }_{\mu } = \{\mu , \mu + 1, \mu + 2, \ldots , \nu \}$ for any $\mu$, $\nu \in {\mathbb {R}}$ such that $\nu - \mu \in {\mathbb {N}}_{1}$.

Definition 2.1

[10, 20] For x, $k \in {\mathbb {R}}$ and $t \in {\mathbb {R}} {\setminus } {\mathbb {Z}}^{-}$, define

$$\begin{aligned} H_{t}(x, k) = \frac{(x - k)^{\overline{t}}}{\Gamma (t + 1)} = \frac{\Gamma (x -k + t)}{\Gamma (x - k)\Gamma (t + 1)}, \end{aligned}$$

provided the RHS is well defined. Here, $\Gamma (.)$ denotes the Gamma function.

Definition 2.2

[10] For $y: {\mathbb {N}}_{k + 1} \rightarrow {\mathbb {R}}$ and $\delta > 0$, the $\delta ^{\text {th}}$-order nabla fractional sum of y based at k is given by

$$\begin{aligned} \big (\nabla ^{-\delta }_{k}y\big )(x) = \sum ^{x}_{\xi = k + 1}H_{\delta - 1}(x, \rho (\xi ))y(\xi ), \quad x \in {\mathbb {N}}_{k}, \end{aligned}$$

where $\rho (\xi ) = \xi - 1$.

Definition 2.3

[10] Let $y: {\mathbb {N}}_{k + 1} \rightarrow {\mathbb {R}}$, $\delta > 0$, $n \in {\mathbb {N}}_1$ with $n - 1 < \delta \le n$. The $\delta ^{\text {th}}$-order Riemann–Liouville nabla fractional difference of y based at k is given by

$$\begin{aligned} \big (\nabla ^{\delta }_{k}y\big )(x) = \Big (\nabla ^n\big (\nabla _{k}^{-(n - \delta )}y\big )\Big )(x), \quad x \in {\mathbb {N}}_{k + n}. \end{aligned}$$

Definition 2.4

[10] Let $y: {\mathbb {N}}_{k - n + 1} \rightarrow {\mathbb {R}}$ and $\delta > 0$. The $\delta {\text {th}}$-order Caputo nabla fractional difference of y based at k is given by

$$\begin{aligned} \big (\nabla ^{\delta }_{k*}y\big )(x) = \Big (\nabla _{k}^{-(n - \delta )} \big (\nabla ^n y\big )\Big )(x), \quad x \in {\mathbb {N}}_{k + 1}, \end{aligned}$$

where $n = \lceil \delta \rceil$.

Next is the composition rule of nabla fractional sum, which will be applicable in the following section.

Theorem 2.1

[10] Assume $\delta$, $\gamma > 0$, $y: {\mathbb {N}}_{k + 1} \rightarrow {\mathbb {R}}$, $n \in {\mathbb {N}}_1$ with $n - 1 < \delta \le n$. Then,

• (1) $\displaystyle {\Big (\nabla _{k}^{-\delta } \big (\nabla _{k}^{-\gamma }y\big )\Big )(x) = \big (\nabla _{k}^{-(\delta + \gamma )}y\big )(x), \quad x \in {\mathbb {N}}_{k}.}$

• (2) $\displaystyle {\Big (\nabla _{k}^{\delta } \big (\nabla _{k}^{-\gamma }y\big )\Big )(x) = \big (\nabla _{k}^{\delta - \gamma }y\big )(x), \quad x \in {\mathbb {N}}_{k + n}.}$

Definition 2.5

[10] Let $\alpha > 0$, $\beta \in {\mathbb {R}}$ and $-1< \vartheta < 1$. The discrete Mittag–Leffler function is defined by

$$\begin{aligned} e_{\vartheta , \alpha , \beta }(x, k) = \sum ^{\infty }_{j = 0}\vartheta ^{j} H_{\alpha j + \beta }(x, k), \quad x \in {\mathbb {N}}_{k}. \end{aligned}$$

Clearly,

$$\begin{aligned} e_{\vartheta , \alpha , \beta }(x, \rho (x)) = \frac{1}{1 - \vartheta }, \quad x \in {\mathbb {N}}_{k}. \end{aligned}$$

Theorem 2.2

[10] Assume $0< \delta < 1$ and $-1< \vartheta < 1$. The homogeneous difference equation

$$\begin{aligned} \big (\nabla ^{\delta }_{\rho (k)}y\big )(x) = \vartheta y(x), \quad x \in {\mathbb {N}}_{k + 1}, \end{aligned}$$

(2.1)

has a general solution

$$\begin{aligned} y(x) = C e_{\vartheta , \delta , \delta - 1}(x, \rho (k)), \quad x \in {\mathbb {N}}_{k}. \end{aligned}$$

(2.2)

Here, C is an arbitrary constant.

Theorem 2.3

[21] Assume $0< \delta < 1$ and $-1< \vartheta < 1$. The homogeneous difference equation

$$\begin{aligned} \big (\nabla ^{\delta }_{k^{*}}y\big )(x) = \vartheta y(x), \quad x \in {\mathbb {N}}_{k + 1}, \end{aligned}$$

(2.3)

is given by

$$\begin{aligned} y(x) = C e_{\vartheta , \delta , 0}(x, k), \quad x \in {\mathbb {N}}_{k}. \end{aligned}$$

(2.4)

Here C is an arbitrary constant.

Theorem 2.4

Assume $0< \delta < 1$, $\left| \vartheta \right| < 1$ and h is a real-valued function defined on ${\mathbb {N}}_{k + 1}$. Then, the nonhomogeneous difference equation

$$\begin{aligned} \big (\nabla ^{\delta }_{\rho (k)}y\big )(x) = \vartheta y(x) + h(x), \quad x \in {\mathbb {N}}_{k + 1}, \end{aligned}$$

(2.5)

has a general solution

$$\begin{aligned} y(x) = C e_{\vartheta , \delta , \delta - 1}(x, \rho (k)) + \sum ^{x}_{\xi = k + 1}e_{\vartheta , \delta , \delta - 1}(x, \rho (\xi ))h(\xi ), \quad x \in {\mathbb {N}}_{k}, \end{aligned}$$

(2.6)

where C is an arbitrary constant.

Proof

Denote by

$$\begin{aligned} w(x) = \sum ^{x}_{\xi = k + 1}e_{\vartheta , \delta , \delta - 1}(x, \rho (\xi ))h(\xi ), \quad x \in {\mathbb {N}}_{k}. \end{aligned}$$

We show that w satisfies (2.5), that is,

$$\begin{aligned} \big (\nabla ^{\delta }_{\rho (k)}w\big )(x) = \vartheta w(x) + h(x), \quad x \in {\mathbb {N}}_{k + 1}. \end{aligned}$$

(2.7)

To see this, for $x \in {\mathbb {N}}_{k}$, consider

$$\begin{aligned} w(x)&= \sum ^{x}_{\xi = k + 1}e_{\vartheta , \delta , \delta - 1}(x, \rho (\xi ))h(\xi ) \nonumber \\&= \sum ^{x}_{\xi = k + 1}\left[ \sum ^{\infty }_{j = 0}\vartheta ^{j} H_{\delta j + \delta - 1}(x, \rho (\xi ))\right] h(\xi ) \nonumber \\&= \sum ^{\infty }_{j = 0}\vartheta ^{j} \left[ \sum ^{x}_{\xi = k + 1} H_{\delta j + \delta - 1}(x, \rho (\xi ))h(\xi )\right] \nonumber \\&= \sum ^{\infty }_{j = 0}\vartheta ^{j} \left[ \sum ^{x}_{\xi = k} H_{\delta j + \delta - 1}(x, \rho (\xi ))h(\xi ) \right. \nonumber \\&\quad \left. - H_{\delta j + \delta - 1}(x, \rho (k))h(k)\right] \nonumber \\&= \sum ^{\infty }_{j = 0}\vartheta ^{j} \left[ \big (\nabla ^{- (\delta j + \delta )}_{\rho (k)}h\big )(x)\right] \nonumber \\&\quad - h(k) \sum ^{\infty }_{j = 0}\vartheta ^{j} H_{\delta j + \delta - 1}(x, \rho (k)) \nonumber \\&= \sum ^{\infty }_{j = 0}\vartheta ^{j} \left[ \big (\nabla ^{- (\delta j + \delta )}_{\rho (k)}h\big )(x)\right] - h(k) e_{\vartheta , \delta , \delta - 1}(x, \rho (k)). \end{aligned}$$

(2.8)

Now, for $x \in {\mathbb {N}}_{k + 1}$, consider

$$ \begin{aligned}&\big (\nabla ^{\delta }_{\rho (k)}w\big )(x) \\&\quad = \nabla ^{\delta }_{\rho (k)}\left[ \sum ^{\infty }_{j = 0}\vartheta ^{j} \left[ \big (\nabla ^{- (\delta j + \delta )}_{\rho (k)}h\big )(x)\right] \right. \\&\qquad \left. -\, h(k) e_{\vartheta , \delta , \delta - 1}(x, \rho (k))\right] \quad (\text {By Using }(2.8)) \\&\quad = \sum ^{\infty }_{j = 0}\vartheta ^{j} \left[ \Big ( \nabla ^{\delta }_{\rho (k)} \big (\nabla ^{- (\delta j + \delta )}_{\rho (k)}h \big )\Big ) (x)\right] \\&\qquad - h(k) \left[ \nabla ^{\delta }_{\rho (k)} e_{\vartheta , \delta , \delta - 1}(x, \rho (k))\right] \\&\quad = \sum ^{\infty }_{j = 0}\vartheta ^{j} \left[ \big (\nabla ^{- \delta j}_{\rho (k)}h\big )(x)\right] \\&\qquad - h(k) \left[ \vartheta e_{\vartheta , \delta , \delta - 1}(x, \rho (k))\right] \quad (\text {By Using Theorems }2.1\, \& \,2.2) \\&\quad = h(x) + \sum ^{\infty }_{j = 1}\vartheta ^{j} \left[ \big (\nabla ^{- \delta j}_{\rho (k)}h\big )(x)\right] - \vartheta e_{\vartheta , \delta , \delta - 1}(x, \rho (k)) h(k) \\&\quad = h(x) + \vartheta \left[ \sum ^{\infty }_{j = 0}\vartheta ^{j} \left[ \big (\nabla ^{- (\delta j + \delta )}_{\rho (k)}h\big )(x)\right] \right. \\&\qquad \left. - e_{\vartheta , \delta , \delta - 1}(x, \rho (k)) h(k)\right] \\&\quad = \vartheta w(x) + h(x). \quad (\text {By Using }(2.8)) \end{aligned}$$

The proof is complete. $\square$

Theorem 2.5

Assume $0< \delta < 1$, $\left| \vartheta \right| < 1$ and h is a real-valued function defined on ${\mathbb {N}}_{k + 1}$. Then, the nonhomogeneous difference equation

$$\begin{aligned} \big (\nabla ^{\delta }_{k*}y\big )(x) = \vartheta y(x) + h(x), \quad x \in {\mathbb {N}}_{k + 1}, \end{aligned}$$

(2.9)

has a general solution

$$\begin{aligned} y(x) = C e_{\vartheta , \delta , 0}(x, k) + \sum ^{x}_{\xi = k + 1}e_{\vartheta , \delta , \delta - 1}(x, \rho (\xi ))h(\xi ), \quad x \in {\mathbb {N}}_{k}, \end{aligned}$$

(2.10)

where C is an arbitrary constant.

Proof

Denote by

$$\begin{aligned} w(x) = \sum ^{x}_{\xi = k + 1}e_{\vartheta , \delta , \delta - 1}(x, \rho (\xi ))h(\xi ), \quad x \in {\mathbb {N}}_{k}. \end{aligned}$$

We show that w satisfies (2.9), that is,

$$\begin{aligned} \big (\nabla ^{\delta }_{k*}w\big )(x) = \vartheta w(x) + h(x), \quad x \in {\mathbb {N}}_{k + 1}. \end{aligned}$$

(2.11)

For $x \in {\mathbb {N}}_{k}$, consider

$$\begin{aligned} w(x)&= \sum ^{x}_{\xi = k + 1}e_{\vartheta , \delta , \delta - 1}(x, \rho (\xi ))h(\xi ) \nonumber \\&= \sum ^{x}_{\xi = k + 1}\left[ \sum ^{\infty }_{j = 0}\vartheta ^{j} H_{\delta j + \delta - 1}(x, \rho (\xi ))\right] h(\xi ) \nonumber \\&= \sum ^{\infty }_{j = 0}\vartheta ^{j} \left[ \sum ^{x}_{\xi = k + 1} H_{\delta j + \delta - 1}(x, \rho (\xi ))h(\xi )\right] \nonumber \\&= \sum ^{\infty }_{j = 0}\vartheta ^{j} \left( \nabla ^{-(\delta j + \delta )}_{k} h \right) (x) . \end{aligned}$$

(2.12)

Now, for $x \in {\mathbb {N}}_{k + 1}$, consider

$$\begin{aligned}&\big (\nabla ^{\delta }_{k*}w\big )(x) = \Big (\nabla ^{-(1-\delta )}_{k} \big (\nabla w\big )\Big )(x) \\&\quad = \nabla ^{-(1-\delta )}_{k} \nabla \left[ \sum ^{\infty }_{j = 0}\vartheta ^{j} \left( \nabla ^{-(\delta j + \delta )}_{k} h \right) (x) \right] \quad (\text {By Using }(2.12)) \\&\quad = \nabla ^{-(1-\delta )}_{k} \left[ \sum ^{\infty }_{j = 0}\vartheta ^{j} \left( \nabla \big (\nabla ^{-(\delta j + \delta )}_{k} h \big )\right) (x) \right] \\&\quad = \nabla ^{-(1-\delta )}_{k} \left[ \sum ^{\infty }_{j = 0}\vartheta ^{j} \left( \nabla ^{(1-\delta j - \delta )}_{k} h\right) (x) \right] \quad (\text {By Using Theorem }2.1) \\&\quad = \sum ^{\infty }_{j = 0}\vartheta ^{j} \left( \nabla ^{-(1-\delta )}_{k} \big (\nabla ^{-(\delta j + \delta - 1)}_{k} h \big )\right) (x) \\&\quad = \sum ^{\infty }_{j = 0}\vartheta ^{j} \left( \nabla ^{-\delta j}_{k} h \right) (x) \quad (\text {By Using Theorem }2.1) \\&\quad = h(x) + \sum ^{\infty }_{j = 1}\vartheta ^{j} \left( \nabla ^{-\delta j}_{k} h \right) (x) \\&\quad = h(x) + \vartheta \left[ \sum ^{\infty }_{j = 0}\vartheta ^{j} \left( \nabla ^{-(\delta j + \delta )}_{k} h \right) (x) \right] \\&\quad = h(x) + \vartheta w(x). \end{aligned}$$

The proof is complete. $\square$

Lemma 2.6

[21,22,23,24,25] Let $0< \delta < 1$ and $-1< \vartheta < 0$. Then,

• (1) $0 < e_{\vartheta , \delta , \delta - 1} (x, \rho (k)) \le 1$;

• (2) $0 < e_{\vartheta , \delta , 0} (x, k) \le 1$,

for $x \in {\mathbb {N}}_{k}$.

3 Green’s functions and their properties

In this section, we construct the Green’s functions for the following linear boundary value problems

$$\begin{aligned} {\left\{ \begin{array}{ll} \big (\nabla ^{\delta }_{\rho (0)}y\big )(x) = \vartheta y(x) + h(x), \quad x \in {\mathbb {N}}^{T}_{1}, \\ y(0) = y(T), \end{array}\right. } \end{aligned}$$

(3.1)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \big (\nabla ^{\delta }_{0*}y\big )(x) = \vartheta y(x) + h(x), \quad x \in {\mathbb {N}}^{T}_{1}, \\ y(0) = y(T), \end{array}\right. } \end{aligned}$$

(3.2)

corresponding to (1.1) and (1.2), respectively, and deduce their properties. Here $h: {\mathbb {N}}^{T}_{1} \rightarrow {\mathbb {R}}$.

Theorem 3.1

The boundary value problem (3.1) has a unique solution

$$\begin{aligned} y(x) = \sum ^{T}_{\xi = 1}{\mathcal {G}}_{RL}(x, \xi ) h(\xi ), \quad x \in {\mathbb {N}}^{T}_{0}, \end{aligned}$$

(3.3)

where

$$\begin{aligned} {\mathcal {G}}_{RL}(x, \xi ) = {\left\{ \begin{array}{ll} {\mathcal {G}}_{RL_{1}}(x, \xi ), \quad x \in {\mathbb {N}}^{\rho (\xi )}_{0}, \\ {\mathcal {G}}_{RL_{2}}(x, \xi ), \quad x \in {\mathbb {N}}^{T}_{\xi }. \end{array}\right. } \end{aligned}$$

(3.4)

Here

$$\begin{aligned} {\mathcal {G}}_{RL_{1}}(x, \xi ) = \frac{e_{\vartheta , \delta , \delta - 1}(x, \rho (0))}{\left[ \frac{1}{1-\vartheta } - e_{\vartheta , \delta , \delta - 1}(T, \rho (0))\right] }e_{\vartheta , \delta , \delta - 1}(T, \rho (\xi )), \end{aligned}$$

(3.5)

and

$$\begin{aligned} {\mathcal {G}}_{RL_{2}}(x, \xi ) = {\mathcal {G}}_{RL_{1}}(x, \xi ) + e_{\vartheta , \delta , \delta - 1}(x, \rho (\xi )). \end{aligned}$$

(3.6)

Proof

From Theorem 2.4, the nonhomogeneous difference equation in (3.1) has a general solution

$$\begin{aligned} y(x) = C e_{\vartheta , \delta , \delta - 1}(x, \rho (0)) + \sum ^{x}_{\xi = 1}e_{\vartheta , \delta , \delta - 1}(x, \rho (\xi ))h(\xi ), \quad x \in {\mathbb {N}}_{0}. \end{aligned}$$

(3.7)

Using the boundary condition $y(0) = y(T)$ in (3.7) and rearranging the terms, we get

$$\begin{aligned} C = \frac{1}{\left[ \frac{1}{1-\vartheta } - e_{\vartheta , \delta , \delta - 1}(T, \rho (0))\right] }\sum ^{T}_{\xi = 1}e_{\vartheta , \delta , \delta - 1}(T, \rho (\xi ))h(\xi ). \end{aligned}$$

(3.8)

Substituting the expression for C from (3.8) in (3.7), we obtain (3.3). The proof is complete. $\square$

Theorem 3.2

The boundary value problem (3.2) has a unique solution

$$\begin{aligned} y(x) = \sum ^{T}_{\xi = 1}{\mathcal {G}}_{C}(x, \xi ) h(\xi ), \quad x \in {\mathbb {N}}^{T}_{0}, \end{aligned}$$

(3.9)

where

$$\begin{aligned} {\mathcal {G}}_{C}(x, \xi ) = {\left\{ \begin{array}{ll} {\mathcal {G}}_{C_{1}}(x, \xi ), \quad x \in {\mathbb {N}}^{\rho (\xi )}_{0}, \\ {\mathcal {G}}_{C_{2}}(x, \xi ), \quad x \in {\mathbb {N}}^{T}_{\xi }. \end{array}\right. } \end{aligned}$$

(3.10)

Here

$$\begin{aligned} {\mathcal {G}}_{C_{1}}(x, \xi ) = \frac{e_{\vartheta , \delta , 0}(x, 0)}{\left[ 1 - e_{\vartheta , \delta , 0}(T, 0)\right] }e_{\vartheta , \delta , \delta - 1}(T, \rho (\xi )), \end{aligned}$$

(3.11)

and

$$\begin{aligned} {\mathcal {G}}_{C_{2}}(x, \xi ) = {\mathcal {G}}_{C_{1}}(x, \xi ) + e_{\vartheta , \delta , \delta - 1}(x, \rho (\xi )). \end{aligned}$$

(3.12)

Proof

From Theorem 2.5, the nonhomogeneous difference equation in (3.2) has a general solution

$$\begin{aligned} y(x) = C e_{\vartheta , \delta , 0}(x, 0) + \sum ^{x}_{\xi = 1}e_{\vartheta , \delta , \delta - 1}(x, \rho (\xi ))h(\xi ), \quad x \in {\mathbb {N}}_{0}. \end{aligned}$$

(3.13)

Using the boundary condition $y(0) = y(T)$ in (3.13) and rearranging the terms, we get

$$\begin{aligned} C = \frac{1}{\left[ 1 - e_{\vartheta , \delta , 0}(T, 0)\right] }\sum ^{T}_{\xi = 1}e_{\vartheta , \delta , \delta - 1}(T, \rho (\xi ))h(\xi ). \end{aligned}$$

(3.14)

Substituting the expression for C from (3.14) in (3.13), we obtain (3.9). The proof is complete. $\square$

Remark 1

Note that

$$\begin{aligned} \sum _{\xi =1}^{x} e_{\vartheta , \delta , \delta -1}(x, \rho (\xi ))&= \sum _{\xi =1}^{x} \left[ \sum _{j=0}^{\infty } \vartheta ^{j} H_{\delta j + \delta - 1} (x,\rho (\xi )) \right] \\&= \sum _{j=0}^{\infty } \vartheta ^{j} \left[ \sum _{\xi =1}^{x} H_{\delta j + \delta - 1} (x,\rho (\xi )) \right] \\&= \sum _{j=0}^{\infty } \vartheta ^{j} H_{\delta j + \delta } (x,0) = e_{\vartheta , \delta , \delta }(x, 0). \end{aligned}$$

Lemma 3.3

${\mathcal {G}}_{RL}(x, \xi )$ has the following properties:

• (1) ${\mathcal {G}}_{RL}(x, \xi ) > 0$, $(x,\xi ) \in {\mathbb {N}}^{T}_{0} \times {\mathbb {N}}^{T}_{1};$

• (2) $\sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) = \frac{e_{\vartheta , \delta , \delta - 1}(x, \rho (0))}{\left[ \frac{1}{1-\vartheta } - e_{\vartheta , \delta , \delta - 1}(T, \rho (0))\right] } e_{\vartheta , \delta , \delta } (T, 0)+ e_{\vartheta , \delta , \delta } (x, 0) ,$    $x \in {\mathbb {N}}_{0}^{T}$.

Proof

The proof of 3.3 follows from Lemma 2.6. To prove 3.3, for $x \in {\mathbb {N}}_{0}^{T}$, consider

$$\begin{aligned}&\sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) = \sum _{\xi =1}^{x} {\mathcal {G}}_{RL_{2}}(x, \xi ) + \sum _{\xi =x+1}^{T} {\mathcal {G}}_{RL_{1}}(x, \xi )\\&\quad = \sum _{\xi =1}^{T} {\mathcal {G}}_{RL_{1}}(x, \xi ) + \sum _{\xi =1}^{x} e_{\vartheta , \delta , \delta -1}(x, \rho (\xi ))\\&\quad = \frac{e_{\vartheta , \delta , \delta - 1}(x, \rho (0))}{\left[ \frac{1}{1-\vartheta } - e_{\vartheta , \delta , \delta - 1}(T, \rho (0))\right] }\\&\sum _{\xi =1}^{T} e_{\vartheta , \delta , \delta - 1} (T, \rho (\xi )) + \sum _{\xi =1}^{x} e_{\vartheta , \delta , \delta - 1} (x, \rho (\xi ))\\&\quad = \frac{e_{\vartheta , \delta , \delta - 1}(x, \rho (0))}{\left[ \frac{1}{1-\vartheta } - e_{\vartheta , \delta , \delta - 1}(T, \rho (0))\right] } \\&\quad e_{\vartheta , \delta , \delta } (T, 0) + e_{\vartheta , \delta , \delta } (x, 0) \quad (\text {By Remark} 1). \end{aligned}$$

The proof is complete. $\square$

Lemma 3.4

${\mathcal {G}}_{C}(x, \xi )$ has the following properties:

• (1) ${\mathcal {G}}_{C}(x, \xi ) > 0$, $(x,\xi ) \in {\mathbb {N}}^{T}_{0} \times {\mathbb {N}}^{T}_{1};$

• (2) $\sum _{\xi =1}^{T} {\mathcal {G}}_{C}(x, \xi ) = \frac{e_{\vartheta , \delta , 0}(x, 0)}{\left[ 1 - e_{\vartheta , \delta , 0}(T, 0)\right] } e_{\vartheta , \delta , \delta } (T, 0) + e_{\vartheta , \delta , \delta } (x, 0) ,$    $x \in {\mathbb {N}}_{0}^{T}.$

Proof

The proof is similar to that of Lemma 3.3. $\square$

Remark 2

Denote by

$$\begin{aligned} \vartheta _{1}= & {} \max \limits _{x \in {\mathbb {N}}^{T}_{0}} \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \nonumber \\= & {} \max \limits _{x \in {\mathbb {N}}^{T}_{0}} \Bigg [ \frac{e_{\vartheta , \delta , \delta - 1}(x, \rho (0))}{\left[ \frac{1}{1-\vartheta } - e_{\vartheta , \delta , \delta - 1}(T, \rho (0))\right] } e_{\vartheta , \delta , \delta } (T, 0) + e_{\vartheta , \delta , \delta } (x, 0) \Bigg ], \nonumber \\ \end{aligned}$$

(3.15)

and

$$\begin{aligned} \vartheta _{2}= & {} \max \limits _{x \in {\mathbb {N}}^{T}_{0}} \sum _{\xi =1}^{T} {\mathcal {G}}_{C}(x, \xi ) \nonumber \\= & {} \max \limits _{x \in {\mathbb {N}}^{T}_{0}} \Bigg [ \frac{e_{\vartheta , \delta , 0}(x, 0)}{\left[ 1 - e_{\vartheta , \delta , 0}(T, 0)\right] } e_{\vartheta , \delta , \delta } (T, 0) + e_{\vartheta , \delta , \delta } (x, 0) \Bigg ]. \end{aligned}$$

(3.16)

4 Existence of solutions

The existence of solutions to (1.1) and (1.2) is established by the sufficient conditions set forth in this section. Theorems 3.1 and 3.2 imply the equivalence between

• (i) the solutions of (1.1) and the solutions of the summation equation

  $$\begin{aligned} y(x) = \sum _{\xi = 1}^{T} {\mathcal {G}}_{RL}(x, \xi ) f(\xi , y(\rho (\xi ))), \quad x \in {\mathbb {N}}_{0}^{T}; \end{aligned}$$

• (ii) the solutions of (1.2) and the solutions of the summation equation

  $$\begin{aligned} y(x) = \sum _{\xi = 1}^{T} {\mathcal {G}}_{C}(x, \xi ) f(\xi , y(\rho (\xi ))), \quad x \in {\mathbb {N}}_{0}^{T}, \end{aligned}$$

respectively. Let ${\mathcal {B}}$ be the set of all real-valued functions defined on ${\mathbb {N}}^{T}_{0}$. Define the operators $S_1$, $S_2: {\mathcal {B}} \rightarrow {\mathcal {B}}$ by

$$\begin{aligned} (S_1y)(x)= & {} \sum _{\xi = 1}^{T} {\mathcal {G}}_{RL}(x, \xi ) f(\xi , y(\rho (\xi ))), \quad x \in {\mathbb {N}}_{0}^{T}, \\ (S_2y)(x)= & {} \sum _{\xi = 1}^{T} {\mathcal {G}}_{C}(x, \xi ) f(\xi , y(\rho (\xi ))), \quad x \in {\mathbb {N}}_{0}^{T}. \end{aligned}$$

Clearly, y is a fixed point of $S_1$ (or $S_2$) if and only if y is a solution of (1.1) [or (1.2)]. Observe that ${\mathcal {B}}$ is equivalent to ${\mathbb {R}}^{T + 1}$. We know that ${\mathcal {B}}$ is a Banach space equipped with the maximum norm defined by

$$\begin{aligned} \left\| y\right\| = \max \limits _{x \in {\mathbb {N}}^{T}_{0}} \left| y(x)\right| . \end{aligned}$$

Let

$$\begin{aligned} K = \left\{ y \in {\mathcal {B}}: y(0) = y(T) ~ \text {and} ~ \left\| y \right\| \le r ~ \text {for all} ~ x \in {\mathbb {N}}^{T}_{0}, ~ r>0 \right\} . \end{aligned}$$

Clearly, K is a nonempty bounded closed convex subset of the finite-dimensional normed space ${\mathcal {B}}$. First, we apply the Brouwer fixed-point theorem [26] to discuss the existence of solutions to (1.1) and (1.2).

Theorem 4.1

Assume that

• (A1) $\left| f(x, y)\right| \le M$ for all $(x, y) \in {\mathbb {N}}^{T}_{0} \times K$.

Choose

$$\begin{aligned} r \ge M \vartheta _{1}. \end{aligned}$$

Then, (1.1) has a solution.

Proof

To show that $S_1: K \rightarrow K$, take $y \in K$, $x \in {\mathbb {N}}_{0}^{T}$ and consider

$$\begin{aligned} \left| (S_1y)(x) \right|&= \left| \sum _{\xi = 1}^{T} {\mathcal {G}}_{RL}(x, \xi ) f(\xi , y(\rho (\xi ))) \right| \\ {}&\le \sum _{\xi = 1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left| f(\xi , y(\rho (\xi ))) \right| \\ {}&\le M \sum _{\xi = 1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \\ {}&\le M \vartheta _{1} \quad (\text {By Remark }2) \\ {}&\le r, \end{aligned}$$

implying that $\left| (S_1y)(x) \right| \le r$ for all $x \in {\mathbb {N}}^{T}_{0}$. Also, $(S_1y)(0) = (S_1y)(T)$. As a result, $S_1: K \rightarrow K$. The continuity of $S_1$ follows from the continuity of f. Thus, by Brouwer fixed-point theorem, (1.1) has a solution y in K. The proof is complete. $\square$

Theorem 4.2

Assume that (A1) holds. Choose

$$\begin{aligned} r \ge M \vartheta _{2}. \end{aligned}$$

Then, (1.2) has a solution.

Proof

The proof is similar to that of Theorem 4.1. $\square$

Next, we apply the Leray–Schauder nonlinear alternative [26] to discuss the existence of solutions to (1.1) and (1.2).

Theorem 4.3

Assume that

• (C 1) There exist ${\hat{\phi }}: {\mathbb {N}}^{T}_{1} \rightarrow [0, \infty )$ and a nondecreasing function ${\hat{\psi }}: [0, \infty ) \rightarrow [0, \infty )$ such that

  $$\begin{aligned} \left| f(x, s)\right| \le {\hat{\phi }}(x) {\hat{\psi }}\left( |s|\right) , \quad (x, s) \in {\mathbb {N}}^{T}_{1} \times {\mathbb {R}}. \end{aligned}$$

• (C 2) There exists $M_{1} > 0$ such that

  $$\begin{aligned} \frac{M_{1}}{\vartheta _{1} {\overline{\Omega }} {\hat{\psi }}\left( M_1\right) } > 1, \end{aligned}$$

  where

  $$\begin{aligned} {\overline{\Omega }} = \max _{x \in {\mathbb {N}}^{T}_{1}} {\hat{\phi }}(x). \end{aligned}$$

Then, the boundary value problem (1.1) has a solution defined on ${\mathbb {N}}^{T}_{0}$.

Proof

We first show that $S_1$ maps bounded sets into bounded sets. By (C 1), for $x \in {\mathbb {N}}^{T}_{0}$ and $y \in K$,

$$\begin{aligned} \left| \big (S_1y\big )(x)\right|&\le \sum _{\xi = 1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left| f(\xi , y(\rho (\xi )))\right| \\&\le \sum ^{T}_{\xi = 1} {\mathcal {G}}_{RL}(x, \xi ) {\hat{\phi }}(\xi ) {\hat{\psi }}\left( |y(\rho (\xi ))|\right) \\&\le {\hat{\psi }}\left( \Vert y\Vert \right) \sum ^{T}_{\xi = 1}{\mathcal {G}}_{RL}(x, \xi ){\hat{\phi }}(\xi ) \\ {}&\le \vartheta _1 {\overline{\Omega }} {\hat{\psi }}\left( r\right) , \end{aligned}$$

implying that

$$\begin{aligned} \left| \big (S_1y\big )(x)\right| \le \vartheta _1 {\overline{\Omega }} {\hat{\psi }}\left( r\right) . \end{aligned}$$

Thus, $S_1$ maps K into a bounded set. Since ${\mathbb {N}}^{T}_{0}$ is a discrete set, it follows immediately that $S_1$ maps K into an equicontinuous set. Therefore, by the Arzela–Ascoli theorem, $S_1$ is completely continuous. Next, we suppose $y \in {\mathcal {B}}$ and that for some $0< \lambda < 1$, $y = \lambda S_1 y$. Then, for $x \in {\mathbb {N}}^{T}_{0}$, and again by (C 1),

$$\begin{aligned} \left| y(x)\right|&= \left| \lambda \big (S_1y\big )(x)\right| \\&\le \sum _{\xi = 1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left| f(\xi , y(\rho (\xi )))\right| \\&\le \sum ^{T}_{\xi = 1} {\mathcal {G}}_{RL}(x, \xi ) {\hat{\phi }}(\xi ) {\hat{\psi }}\left( |y(\rho (\xi ))|\right) \\&\le {\hat{\psi }}\left( \Vert y\Vert \right) \sum ^{T}_{\xi =1} {\mathcal {G}}_{RL}(x, \xi ){\hat{\phi }}(\xi ) \\&\le \vartheta _1 {\overline{\Omega }} {\hat{\psi }}\left( \Vert y\Vert \right) , \end{aligned}$$

implying that

$$\begin{aligned} \frac{\Vert y\Vert }{\vartheta _1 {\overline{\Omega }} {\hat{\psi }}\left( \Vert y\Vert \right) } \le 1. \end{aligned}$$

It follows from (C 2) that $\Vert y\Vert \ne M_{1}$. If we set

$$\begin{aligned} U = \Big \{y \in {\mathcal {B}}: \Vert y\Vert < M_{1} \Big \}, \end{aligned}$$

then the operator $S_1: {\bar{U}} \rightarrow {\mathcal {B}}$ is completely continuous. From the choice of U, there is no $y \in \partial U$ such that $y = \lambda S_1 y$ for some $0< \lambda < 1$. It follows from Leray–Schauder nonlinear alternative that $S_1$ has a fixed point $y_0 \in {\bar{U}}$, which is a desired solution of (1.1). $\square$

Theorem 4.4

Assume that (C 1) and

• (C 3) There exists $M_{2} > 0$ such that

  $$\begin{aligned} \frac{M_{2}}{\vartheta _2 {\overline{\Omega }} {\hat{\psi }}\left( M_2\right) } > 1, \end{aligned}$$

  where

  $$\begin{aligned} {\overline{\Omega }} = \max _{x \in {\mathbb {N}}^{T}_{1}} {\hat{\phi }}(x). \end{aligned}$$

Then, the boundary value problem (1.2) has a solution defined on ${\mathbb {N}}^{T}_{0}$.

Proof

The proof is similar to the proof of Theorem 4.3, so we omit it. $\square$

Now, we apply the Banach fixed-point theorem [26] to discuss the existence and uniqueness of solutions to (1.1) and (1.2).

Theorem 4.5

The conditions

• (A2) f is Lipschitz w.r.t. the second variable with ${\overline{\kappa }}$ as the Lipschitz constant on ${\mathbb {N}}^{T}_{0} \times K$;

• (A3) Let

  $$\begin{aligned} \max \limits _{x \in {\mathbb {N}}^{T}_{0}} \left| f(x, 0) \right| = P, \end{aligned}$$

  and

  $$\begin{aligned} \max \limits _{(x, y) \in {\mathbb {N}}^{T}_{0} \times K} \left| f(x,y) \right| = Q. \end{aligned}$$

• (A4) ${\overline{\kappa }} \vartheta _{1} < 1$,

with

$$\begin{aligned} r \ge \frac{P \vartheta _{1}}{1 - {\overline{\kappa }} \vartheta _{1}}, \end{aligned}$$

or

$$\begin{aligned} r \ge Q \vartheta _{1}, \end{aligned}$$

yield a unique solution for (1.1).

Proof

Clearly, $S_1: K \rightarrow {\mathcal {B}}$. To show that $S_1$ is a contraction mapping, take y, $w \in K$, $x \in {\mathbb {N}}_{0}^{T}$ and consider

$$\begin{aligned}&\left| (S_1y)(x) - (S_1w)(x) \right| \\&\quad = \left| \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left[ f(\xi ,y(\rho (\xi ))) - f(\xi ,w(\rho (\xi ))) \right] \right| \\&\quad \le \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left| f(\xi ,y(\rho (\xi ))) - f(\xi ,w(\rho (\xi ))) \right| \\&\quad \le {\overline{\kappa }} \left\| y - w \right\| \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \quad \\&\quad \le {\overline{\kappa }} \vartheta _{1} \left\| y - w \right\| , \end{aligned}$$

implying that

$$\begin{aligned} \left\| S_1y - S_1w \right\| \le {\overline{\kappa }} \vartheta _{1}\left\| y - w \right\| . \end{aligned}$$

Since ${\overline{\kappa }} \vartheta _{1} < 1$, $S_1$ is a contraction mapping. Now, we show that $S_1: K \rightarrow K$. Let $y \in K$, $x \in {\mathbb {N}}_{0}^{T}$ and consider

$$\begin{aligned}&\left| (S_1y)(x) \right| = \left| \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) f(\xi ,y(\rho (\xi ))) \right| \\&\quad \le \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left| f(\xi ,y(\rho (\xi ))) - f(\xi ,0) \right| \\&\qquad + \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left| f(\xi ,0) \right| \\&\quad \le {\overline{\kappa }} \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left| y(\rho (\xi )) \right| + P \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \\&\quad \le ({\overline{\kappa }} r + P) \vartheta _{1} \\&\quad \le r, \end{aligned}$$

implying that $S_1: K \rightarrow K$. Also, consider

$$\begin{aligned} \left| (S_1y)(x) \right|&= \left| \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) f(\xi ,y(\rho (\xi ))) \right| \\&\le \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left| f(\xi ,y(\rho (\xi ))) \right| \\&\le Q\sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \\&\le Q \vartheta _{1} \\ {}&\le r, \end{aligned}$$

implying that $S_1: K \rightarrow K$. So, there exists a unique solution for (1.1) by Banach fixed-point theorem. $\square$

Theorem 4.6

The conditions (A2), (A3) and

• (A5) ${\overline{\kappa }} \vartheta _{2} < 1$,

with

$$\begin{aligned} r \ge \frac{P \vartheta _{2}}{1 - {\overline{\kappa }} \vartheta _{2}}, \end{aligned}$$

or

$$\begin{aligned} r \ge Q \vartheta _{2}, \end{aligned}$$

yield a unique solution for (1.2).

Proof

The proof is similar to the proof of Theorem 4.5, so we omit it. $\square$

Theorem 4.7

The conditions

• (A6) f is Lipschitz w.r.t. the second variable with L as the Lipschitz constant on ${\mathbb {N}}^{T}_{0} \times {\mathcal {B}}$;

• (A7) $L \vartheta _{1} < 1$,

yield a unique solution for (1.1).

Proof

Clearly, $S_1: {\mathcal {B}} \rightarrow {\mathcal {B}}$. To show that $S_1$ is a contraction mapping, take y, $w \in {\mathcal {B}}$, $x \in {\mathbb {N}}_{0}^{T}$ and consider

$$\begin{aligned}&\left| (S_1y)(x) - (S_1w)(x) \right| \\&\quad = \left| \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left[ f(\xi ,y(\rho (\xi ))) - f(\xi ,w(\rho (\xi ))) \right] \right| \\&\quad \le \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \left| f(\xi ,y(\rho (\xi ))) - f(\xi ,w(\rho (\xi ))) \right| \\&\quad \le L \left\| y - w \right\| \sum _{\xi =1}^{T} {\mathcal {G}}_{RL}(x, \xi ) \\&\quad \le L \left\| y - w \right\| \vartheta _{1}, \end{aligned}$$

implying that

$$\begin{aligned} \left\| S_1y - S_1w \right\| \le L \vartheta _{1}\left\| y - w \right\| . \end{aligned}$$

Since $L \vartheta _{1} < 1$, $S_1$ is a contraction mapping. Then, there exists a unique solution for (1.1), by Banach fixed-point theorem. $\square$

Theorem 4.8

The conditions (A6) and

• (A8) $L \vartheta _{2} < 1$,

yield a unique solution for (1.2).

Proof

The proof is similar to the proof of Theorem 4.7, so we omit it. $\square$

5 Examples

Example 1

Consider (1.1) with $\delta = 0.5$, $\vartheta = -0.1$, $T = 10$ and

$$\begin{aligned} f(x, z) = (0.25)\left( x + \tan ^{-1}z\right) . \end{aligned}$$

Clearly, f is Lipschitz w.r.t. the second variable with $L = 0.25$ as the Lipschitz constant on ${\mathbb {N}}^{10}_{0} \times {\mathcal {B}}$. Table 1 shows the calculations for the evaluation of $\displaystyle {\sum \nolimits _{\xi = 1}^{10} {\mathcal {G}}_{RL}(x, \xi )}$ using Mathematica:

Table 1 Evaluation of $\vartheta _1$

From Table 1, we have

$$\begin{aligned} \vartheta _{1} = \max \limits _{x \in {\mathbb {N}}^{10}_{0}} \sum _{\xi = 1}^{10} {\mathcal {G}}_{RL}(x, \xi ) = 3.07902. \end{aligned}$$

Then, $L \vartheta _{1} < 1$. All assumptions of Theorem 4.7 hold. As a result, there exists a unique solution for (1.1).

Example 2

Consider (1.2) with $\delta = 0.5$, $\vartheta = -0.1$, $T = 10$ and $f(x, z) = \frac{1}{11}\left( x + \tan ^{-1}z\right)$. Clearly, f is Lipschitz w.r.t. the second variable with $L = \frac{1}{11}$ as the Lipschitz constant on ${\mathbb {N}}^{10}_{0} \times {\mathcal {B}}$. Table 2 shows the calculations for the evaluation of $\displaystyle {\sum \nolimits _{\xi = 1}^{10} {\mathcal {G}}_{C}(x, \xi )}$ using Mathematica:

Table 2 Evaluation of $\vartheta _2$

From Table 2, we have

$$\begin{aligned} \vartheta _{2} = \max \limits _{x \in {\mathbb {N}}^{10}_{0}} \sum _{\xi = 1}^{10} {\mathcal {G}}_{C}(x, \xi ) = 10.4132. \end{aligned}$$

Then, $L \vartheta _{2} < 1$. All assumptions of Theorem 4.8 hold. Thus, there exists a unique solution for (1.2).

Example 3

Consider the boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \big (\nabla ^{0.5}_{\rho (0)}y\big )(x) = -\frac{1}{10}y(x) + x y^2(\rho (x)), \quad x \in {\mathbb {N}}^{10}_{1}, \\ y(0) = y(10). \end{array}\right. } \end{aligned}$$

(5.1)

Here $T = 10$, $\delta = 0.5$, $\vartheta = -\frac{1}{10}$ and $f(x, \xi ) = x \xi ^2$. Clearly,

$$\begin{aligned} \left| f(x, \xi )\right| \le {\hat{\phi }}(x) {\hat{\psi }}\left( |\xi |\right) , \quad (x, \xi ) \in {\mathbb {N}}^{10}_{1} \times {\mathbb {R}}, \end{aligned}$$

where

$$\begin{aligned} {\hat{\phi }}(x) = x, \quad x \in {\mathbb {N}}^{10}_{1}, \end{aligned}$$

and

$$\begin{aligned} {\hat{\psi }}\left( |\xi |\right) = |\xi |^2 = \xi ^2, \quad \xi \in {\mathbb {R}}. \end{aligned}$$

Also, ${\hat{\phi }}: {\mathbb {N}}^{10}_{1} \rightarrow [0, \infty )$ and ${\hat{\psi }}: [0, \infty ) \rightarrow [0, \infty )$ is a nondecreasing function. Thus, the assumption (C 1) of Theorem 4.3 holds. Further, we have

$$\begin{aligned} {\overline{\Omega }} = \max _{x \in {\mathbb {N}}^{10}_{1}} {\hat{\phi }}(x) = 10. \end{aligned}$$

Using Mathematica, we found that $\vartheta _1 = 3.07902$. There exists $0< M_1 < \frac{1}{31}$ such that

$$\begin{aligned} \frac{M_1}{(3.07902)(10) M_1^2} > 1, \end{aligned}$$

implying that the assumption (C 2) of Theorem 4.3 holds. Therefore, by Theorem 4.3, the boundary value problem (1.1) has a solution defined on ${\mathbb {N}}^{10}_{0}$.

Example 4

Consider the boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \big (\nabla ^{0.5}_{0*}y\big )(x) = -\frac{1}{10}y(x) + x y^2(\rho (x)), \quad x \in {\mathbb {N}}^{10}_{1}, \\ y(0) = y(10). \end{array}\right. } \end{aligned}$$

(5.2)

Here $T = 10$, $\delta = 0.5$, $\vartheta = -\frac{1}{10}$ and $f(x, \xi ) = x \xi ^2$. Clearly,

$$\begin{aligned} \left| f(x, \xi )\right| \le {\hat{\phi }}(x) {\hat{\psi }}\left( |\xi |\right) , \quad (x, \xi ) \in {\mathbb {N}}^{10}_{1} \times {\mathbb {R}}, \end{aligned}$$

where

$$\begin{aligned} {\hat{\phi }}(x) = x, \quad x \in {\mathbb {N}}^{10}_{1}, \end{aligned}$$

and

$$\begin{aligned} {\hat{\psi }}\left( |\xi |\right) = |\xi |^2 = \xi ^2, \quad \xi \in {\mathbb {R}}. \end{aligned}$$

Also, ${\hat{\phi }}: {\mathbb {N}}^{10}_{1} \rightarrow [0, \infty )$ and ${\hat{\psi }}: [0, \infty ) \rightarrow [0, \infty )$ is a nondecreasing function. Thus, the assumption (C 1) of Theorem 4.4 holds. Further, we have

$$\begin{aligned} {\overline{\Omega }} = \max _{x \in {\mathbb {N}}^{10}_{1}} {\hat{\phi }}(x) = 10. \end{aligned}$$

Using Mathematica, we found that $\vartheta _2 = 10.4132$. There exists $0< M_2 < \frac{1}{101}$ such that

$$\begin{aligned} \frac{M_2}{(10.4132)(10) M_2^2} > 1, \end{aligned}$$

implying that the assumption (C 3) of Theorem 4.4 holds. Therefore, by Theorem 4.4, the boundary value problem (1.2) has a solution defined on ${\mathbb {N}}^{10}_{0}$.

6 Conclusion and future scope

This article considered two simple nabla fractional relaxation equations with related periodic boundary conditions. We provided sufficient conditions for the existence of solutions to the problems under consideration through relevant fixed-point theorems with adequate restrictions. We also offered a few examples to further illustrate the applicability of our findings. To our knowledge, such work has yet to be reported in the case of fractional differences.

The current work can also be extended to obtain sufficient conditions for multiple positive solutions of the considered boundary value problems due to the corresponding Green functions’ positivity.