Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting

Abstract

This article primarily focuses on examining the existence and uniqueness analysis of boundary fractional difference equations in a class of Riemann–Liouville operators. To this end, we firstly recall the general solution of the homogeneous fractional operator problem. Then, the Green function to the corresponding fractional boundary value problems will be reconstructed, and homogeneous boundary conditions are used to find the unknown constants. Next, the existence of solutions will be studied depending on the fixed-point theorems on the constructed Green’s function. The uniqueness of the problem is also derived via Lipschitz constant conditions.

1. Introduction

The most fundamental and significant notions from integers to fractional difference and differential equations (ordinary or partial) are fractional calculus and discrete fractional calculus [1,2]. In particular, fractional difference models contribute to many work frames with respect to the theory and application in mathematics and physics; see e.g., [3,4,5]. Furthermore, these models and their associated system of difference equations serve as well-suited mathematical models in various areas, such as physics, ecology, social sciences, chemistry, and biology (cf. [6,7,8,9,10]).

Fractional difference operators have played an indispensable role in shaping our understanding of fractional boundary value problems (FBVPs). These FBVPs provided valuable insights into the importance of convergence analysis in discrete fractional calculus theory and the potential impact of dynamical systems; see for example, Refs. [2,11,12,13,14] to be familiar with these operators. In addition, some FBVP models have been proposed and studied in [15,16,17,18,19,20,21] and references therein, and the authors mainly focused on Riemann–Liouville- and Liouville–Caputo-type operators. Furthermore, the existence and uniqueness of FBPV solutions are governed by various fractional operators resulting from a variety of continuous and discrete actions involving AB and CF fractional operators (e.g., [14,22,23,24,25,26,27,28]). In addition, the FBVPs have the effect of various parameters on the bandgap and the feasibility of actively adjusting the bandgap of the system [29,30,31].

In addition, the analysis of the existence/uniqueness of solutions to the fractional difference equations concerning BVPs is paramount in comprehending discrete fractional calculus. These solutions serve as major results in directing and investigating inequalities such as Laypunov-like inequalities. There have been increasing applications of fractional problems in mathematical physics for the existence of positive solutions [32,33]. Recently, Mohammed et al. [34] considered the following FBVP:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-\left({{}_{q_0}{}^{\mathrm{RL}}\Delta }^{\theta }g\right)\left(x\right)=h(x+\theta ),x\in \mathbb{N}(q_0+2;q),\hfill \\ \\ g\left(q_0\right)=0,g\left(q\right)=0,\hfill \end{array}\right.\end{array}$$

(1)

and they obtained the existence of its solutions, as discussed in the next section.

Motivated by [34], we aim to examine, in this current study, the uniqueness of a positive solution for the following FBVP:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left({{}_{q_0}{}^{\mathrm{RL}}\Delta }^{\theta }g\right)\left(x\right)=-Z\left(x+\theta -1,g(x+\theta -1)\right),\hfill \\ \\ g\left(q_0\right)=C_1,g\left(q\right)=C_{2}x\in \mathbb{N}(q_0+2;q),\hfill \end{array}\right.\end{array}$$

(2)

For $q_0,q,C_1,C_2\in \mathbb{R}$ with $q-q_0\in \mathbb{N}\left(2\right)$, Z is assumed to be a function from $\mathbb{N}(q_0+2;q)\times \mathbb{R}$ to $\mathbb{R}$. Note that $\mathbb{N}\left(q_0\right)=\left\{q_0,q_0+1,\dots \right\}$ and $\mathbb{N}(q_0;q)=\left\{q_0,q_0+1,\dots ,q\right\}$.

The remainder of this paper is structured as follows: In the next section, we review the basic theorems about the existence of fractional boundary value problems, focusing on recently published results. In Section 3, we obtain positive solutions in terms of the fundamental systems of solutions on the known fixed-point theorems for solutions of the FBVP and some of their relevant features. Section 4 is devoted to explaining our GF for the corresponding FBVP. Next, in Section 5, the existence of our solutions to the GF formula will be shown by considering the common fixed-point theorems. Two examples are shown in Section 6 as applications of our theoretical results. Finally, our article will end with concluding remarks in Section 7.

2. Review of Results

In this section, we state further results obtained by Mohammed et al. [34].

Theorem 1. 

Let $\theta \in (1,2)$, and let $q_0$ and q be two real numbers such that $q-q_0\in \mathbb{N}\left(1\right)$, and $h:\mathbb{N}(q_0+2;q)\to \mathbb{R}$. Then, FBVP (1) has the unique solution

$$\begin{array}{c}\hfill g\left(x\right)=\sum _{r=q_0+2}^q\mathcal{G}(x,r)h\left(r\right),x\in \mathbb{N}(q_0;q),\end{array}$$

(3)

where the GF $\mathcal{G}(x,r)$ is given by

$$\begin{array}{c}\hfill \mathcal{G}(x,r)=\left\{\begin{array}{cc}{\displaystyle \frac{{(q+\theta -r-1)}^{\underline{\theta -1}}}{{(q+\theta -q_0-2)}^{\underline{\theta -1}}\mathsf{\Gamma }\left(\theta \right)}}{(x+\theta -q_0-2)}^{\underline{\theta -1}},\hfill & x\le r-1,\hfill \\ \\ {\displaystyle \frac{{(q+\theta -r-1)}^{\underline{\theta -1}}}{{(q+\theta -q_0-2)}^{\underline{\theta -1}}\mathsf{\Gamma }\left(\theta \right)}}{(x+\theta -q_0-2)}^{\underline{\theta -1}}-{\displaystyle \frac{{(x+\theta -r-1)}^{\underline{\theta -1}}}{\mathsf{\Gamma }\left(\theta \right)}},\hfill & x\ge r.\hfill \end{array}\right.\end{array}$$

(4)

In the above theorem, for $x\in \mathbb{N}(q_0+\ell -\theta )$ (see [35], (Theorem 2.2)),

$$\begin{array}{cc}\hfill \left({{}_{q_0}{}^{\mathrm{RL}}\Delta }^{\theta }g\right)\left(x\right)& :={\displaystyle \frac{1}{\mathsf{\Gamma }(-\theta )}}\sum _{r=q_0}^{x+\theta }{(x-r-1)}^{\underline{-\theta -1}}g\left(r\right),\hfill \end{array}$$

(5)

is the Riemann–Liouville fractional difference, and its sum formula, for $x\in {\mathbb{N}}_{q_0+\theta }$ (see [2], (Definition 2.25)),

$$\begin{array}{cc}\hfill \left({{}_{q_0}{}^{\mathrm{RL}}\Delta }^{-\theta }g\right)\left(x\right)& ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\theta \right)}}\sum _{r=q_0}^{x-\theta }{\left(x-r-1\right)}^{\underline{\theta -1}}g\left(r\right),\hfill \end{array}$$

(6)

is the Riemann–Liouville fractional sum for the function g on the domain $\mathbb{N}\left(q_0\right)$, $\ell -1<\theta <\ell$ with $\ell \in {\mathbb{N}}_1$, and $x^{\underline{\theta }}={\displaystyle \frac{\mathsf{\Gamma }\left(x+1\right)}{\mathsf{\Gamma }\left(x+1-\theta \right)}}$.

Theorem 2. 

Let $\theta \in (1,2)$ in the FBVP:

$$\begin{array}{c}\hfill \begin{array}{cc}-\left({{}_{q_0}{}^{\mathrm{RL}}\Delta }^{\theta }w\right)\left(x\right)=0,\hfill & \\ \\ w\left(q_0\right)=C_1,w\left(q\right)=C_2.\hfill & \end{array}\end{array}$$

(7)

Then, we have the following:

(a)

For $x\in {\mathbb{N}}_{q_0+2}$,

$$\begin{array}{c}\hfill w\left(x\right)=C_1\left({\displaystyle \frac{q-x}{q-q_0}}\right){\displaystyle \frac{{(x-q_0+\theta -2)}^{\underline{\theta -2}}}{\mathsf{\Gamma }(\theta -1)}}+C_2{\displaystyle \frac{{(x-q_0+\theta -2)}^{\underline{\theta -1}}}{{(q-q_0+\theta -2)}^{\underline{\theta -1}}}},x\in \mathbb{N}(q_0;q),\end{array}$$

(8)

is the solution of (7).

(b)

For $x\in {\mathbb{N}}_{q_0}^q$, we have

$$\begin{array}{c}\hfill \mid w\left(x\right)\mid \le 2max\left\{|C_1|,|C_2|\right\}.\end{array}$$

Theorem 3. 

Let $h:{\mathbb{N}}_{q_0+2}^q\to \mathbb{R}$. The FBVP

$$\begin{array}{c}\hfill \begin{array}{cc}-\left({{}_{q_0}{}^{\mathrm{RL}}\Delta }^{\theta }g\right)\left(x\right)=h(x+\theta ),\hfill & x\in {\mathbb{N}}_{q_0+2},\hfill \\ \\ g\left(q_0\right)=C_1,g\left(q\right)=C_2,\hfill & \end{array}\end{array}$$

(9)

has the unique solution

$$\begin{array}{c}\hfill g\left(x\right)=w\left(x\right)+\sum _{r=q_0+2}^q\mathcal{G}(x,r)h\left(r\right),x\in {\mathbb{N}}_{q_0}^q,\end{array}$$

(10)

where w is as given in the above theorem, and $\mathcal{G}(x,r)$ is as given in Theorem 1.

3. Positive Solution Results

In view of the Guo–Krasnoselskii theorem (see [36]), we examine the existence of positive solutions for FBVP (2) when $C_1=C_2=0$:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left({{}_{q_0}{}^{\mathrm{RL}}\Delta }^{\theta }g\right)\left(x\right)=-Z\left(x+\theta -1,g(x+\theta -1)\right),\hfill \\ \\ g\left(q_0\right)=g\left(q\right)=0,x\in \mathbb{N}(q_0+2;q).\hfill \end{array}\right.\end{array}$$

(11)

Definition 1 

([37], (Completely Continuous Operator)). A bounded linear operator $\mathcal{E}:\mathcal{C}\to \overline{\mathcal{C}}$, where $\mathcal{C}$ and $\overline{\mathcal{C}}$ are two Banach spaces (BSs), is completely continuous if it transforms weakly convergent sequences in $\mathcal{C}$ to norm-convergent sequences in $\overline{\mathcal{C}}$.

Theorem 4 

([36], (Guo–Krasnoselskii theorem)). Let $\mathcal{F}\subset \mathcal{C}$ be a cone set, and the BS $\mathcal{C}$ contains two open sets ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_2$ such that ${\overline{\mathsf{\Omega }}}_1\subseteq {\mathsf{\Omega }}_2$ and $0\in {\mathsf{\Omega }}_1$. Suppose that $\mathcal{E}:\mathcal{F}\cap ({\overline{\mathsf{\Omega }}}_2\setminus {\mathsf{\Omega }}_1)\to \mathcal{F}$ is a completely continuous operator. If one of the following properties holds, then $\mathcal{E}$ has at least one fixed point in $\mathcal{F}\cap ({\overline{\mathsf{\Omega }}}_2\setminus {\mathsf{\Omega }}_1)$:

1. 

$\parallel \mathcal{E}g\parallel \ge \parallel g\parallel$ for $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_2$ and $\parallel \mathcal{E}g\parallel \le \parallel g\parallel$ for $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_1$;

2. 

$\parallel \mathcal{E}g\parallel \le \parallel g\parallel$ for $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_2$ and $\parallel \mathcal{E}g\parallel \ge \parallel g\parallel$ for $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_1$.

Next, we need a new property of the GF that is as follows.

Theorem 5. 

One can have $0<\gamma <1$ such that

$$\begin{array}{c}\hfill \underset{x\in \mathbb{N}(q_0+1;q-1)}{min}\mathcal{G}(x,r)\le \underset{x\in \mathbb{N}(q_0+1;q-1)}{max}\mathcal{G}(x,r)=\gamma \mathcal{G}(r-1,r),r\in \mathbb{N}(q_0+2;q).\end{array}$$

Proof. 

In [38], Mohammed et al. showed that $\mathcal{G}(x,r)$ increases from $\mathcal{G}(q_0,r)=0$ to a positive value in $\mathcal{G}(r-1,r)$ and then decreases to $\mathcal{G}(q,r)=0$, for fixed $r\in \mathbb{N}(q_0+1;q)$. Now, we define

$$\begin{array}{c}\hfill \overline{\mathcal{G}}(x,r)={\displaystyle \frac{\mathcal{G}(x,r)}{\mathcal{G}(r-1,r)}}.\end{array}$$

Then, for $r\in \mathbb{N}(q_0+2;q)$, we consider

$$\begin{array}{c}\hfill \overline{\mathcal{G}}(x,r)=\left\{\begin{array}{cc}{\displaystyle \frac{{(x-q_0+\theta -2)}^{\overline{\theta -1}}}{{(r-q_0+\theta -3)}^{\overline{\theta -1}}}},\hfill & x\in {\mathbb{N}}_{q_0+1}^{r-1},\hfill \\ \\ {\displaystyle \frac{{(x-q_0+\theta -2)}^{\overline{\theta -1}}}{{(r-q_0+\theta -3)}^{\overline{\theta -1}}}}-{\displaystyle \frac{{(x-r+\theta -1)}^{\overline{\theta -1}}{(q-q_0+\theta -2)}^{\overline{\theta -1}}}{{(q-r+\theta -1)}^{\overline{\theta -1}}{(r-q_0+\theta -3)}^{\overline{\theta -1}}}},\hfill & x\in {\mathbb{N}}_r^{q-1}.\hfill \end{array}\right.\end{array}$$

Therefore, for $x\in \mathbb{N}(q_0+1;r-1)$ and $r\in \mathbb{N}(q_0+2;q)$, one can have

$$\begin{array}{c}\hfill \overline{\mathcal{G}}(x,r)={\displaystyle \frac{{(x-q_0+\theta -2)}^{\overline{\theta -1}}}{{(r-q_0+\theta -3)}^{\overline{\theta -1}}}}\ge {\displaystyle \frac{{(\theta -1)}^{\overline{\theta -1}}}{{(q-q_0+\theta -3)}^{\overline{\theta -1}}}}={\displaystyle \frac{\mathsf{\Gamma }\left(\theta \right)}{{(q-q_0+\theta -3)}^{\overline{\theta -1}}}}.\end{array}$$

(12)

Also, for $x\in \mathbb{N}(r;q-1)$ and $r\in \mathbb{N}(q_0+2;q-1)$, we know that $\mathcal{G}(x,r)$($⟹\overline{\mathcal{G}}(x,r)$) is a decreasing function of x. Then, we obtain

$$\begin{array}{cc}\hfill \overline{\mathcal{G}}(x,r)& \ge \overline{\mathcal{G}}(q-1,r)\hfill \\ \hfill & ={\displaystyle \frac{{(q-q_0+\theta -3)}^{\overline{\theta -1}}}{{(r-q_0+\theta -3)}^{\overline{\theta -1}}}}-{\displaystyle \frac{{(q-r+\theta -2)}^{\overline{\theta -1}}{(q-q_0+\theta -2)}^{\overline{\theta -1}}}{{(q-r+\theta -1)}^{\overline{\theta -1}}{(r-q_0+\theta -3)}^{\overline{\theta -1}}}}\hfill \\ \hfill & ={\displaystyle \frac{{(q-q_0+\theta -3)}^{\overline{\theta -1}}}{{(r-q_0+\theta -3)}^{\overline{\theta -1}}}}\left[1-{\displaystyle \frac{{(q-r+\theta -2)}^{\overline{\theta -1}}{(q-q_0+\theta -2)}^{\overline{\theta -1}}}{{(q-r+\theta -1)}^{\overline{\theta -1}}{(q-q_0+\theta -3)}^{\overline{\theta -1}}}}\right]\hfill \\ \hfill & ={\displaystyle \frac{{(q-q_0+\theta -3)}^{\overline{\theta -1}}}{{(r-q_0+\theta -3)}^{\overline{\theta -1}}}}\left[1-\left({\displaystyle \frac{q-r}{q-r+\theta -1}}\right)\left({\displaystyle \frac{q-q_0+\theta -2}{q-q_0-1}}\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{{(q-q_0+\theta -3)}^{\overline{\theta -2}}}{{(r-q_0+\theta -3)}^{\overline{\theta -2}}}}\left({\displaystyle \frac{\theta -1}{q-q_0+\theta -1}}\right)\hfill \\ \hfill & ={\displaystyle \frac{{(q-q_0+\theta -3)}^{\overline{\theta -2}}}{{(\theta -1)}^{\overline{\theta -2}}}}\left({\displaystyle \frac{\theta -1}{q-q_0+\theta -3}}\right)\hfill \\ \hfill & ={\displaystyle \frac{{(q-q_0+\theta -3)}^{\overline{\theta -2}}}{(q-q_0+\theta -3)\mathsf{\Gamma }(\theta -1)}}.\hfill \end{array}$$

(13)

Thus, in view of (12) and (13), we see that

$$\begin{array}{c}\hfill \underset{x\in \mathbb{N}(q_0+1;q-1)}{min}\mathcal{G}(x,r)\ge \gamma \mathcal{G}(r-1,r),r\in \mathbb{N}(q_0+2;q),\end{array}$$

where

$$\begin{array}{c}\hfill \gamma =min\left\{{\displaystyle \frac{\mathsf{\Gamma }\left(\theta \right)}{{(q-q_0+\theta -3)}^{\overline{\theta -1}}}},{\displaystyle \frac{{(q-q_0+\theta -3)}^{\overline{\theta -2}}}{(q-q_0+\theta -3)\mathsf{\Gamma }(\theta -1)}}\right\}.\end{array}$$

It is clear that $\gamma$ is between 0 and 1. Thus, the proof is complete. □

By considering Theorem 1, we see that g is a solution of (11) IFF it is a solution of

$$\begin{array}{c}\hfill g\left(x\right)=\sum _{r=q_0+2}^{q}Z(r,g\left(r\right))\mathcal{G}(x,r),x\in \mathbb{N}(q_0;q).\end{array}$$

(14)

Let us define the operator $\mathcal{E}$ as follows:

$$\begin{array}{c}\hfill \left(\mathcal{E}g\right)\left(x\right):=\sum _{r=q_0+2}^q\mathcal{G}(x,r)Z(r,g\left(r\right)),x\in \mathbb{N}(q_0;q).\end{array}$$

(15)

We can say that g is a fixed point of $\mathcal{E}$ (according to (14) and (15)) IFF it is a solution of (11). In denoting $\mathcal{C}$ by

$$\begin{array}{c}\hfill \mathcal{C}=\left\{g:\mathbb{N}(q_0;q)\to \mathbb{R};g\left(q_0\right)=g\left(q\right)=0\right\}\subseteq \mathbb{R}(q-q_0+1).\end{array}$$

it can be observed that $\mathcal{E}:\mathcal{C}\to \mathcal{C}$ and $\mathcal{C}$ is a BS with the following norm:

$$\begin{array}{c}\hfill \parallel g\parallel =\underset{x\in \mathbb{N}(q_0;q)}{max}\left|g\left(x\right)\right|.\end{array}$$

Now, we define the cone

$$\begin{array}{c}\hfill \mathcal{F}:\left\{g\in \mathcal{C}:g\left(t\right)\ge 0,\mathrm{for}\mathrm{each}x\in \mathbb{N}(q_0;q)\mathrm{and}\underset{\mathbb{N}(q_0+1;q-1)}{min}g\left(x\right)\ge \gamma \parallel g\parallel \right\}.\end{array}$$

Now, we try to obtain sufficient conditions for the existence of a fixed point in $\mathcal{E}$. We firstly know that $\mathcal{E}$ is a summation operator defined on a finite set. Therefore, $\mathcal{E}$ is completely continuous. Then, let

$$\begin{array}{c}\hfill \eta :={\displaystyle \frac{1}{{\sum }_{r=q_0+2}^q\mathcal{G}(r-1,r)}}={\displaystyle \frac{\mathsf{\Gamma }\left(\theta \right){(q-q_0+2\theta -2)}^{\overline{2\theta -1}}}{\mathsf{\Gamma }\left(2\theta \right){(q-q_0+\theta -2)}^{\overline{\theta -1}}}},\end{array}$$

and $x_0\in \mathbb{N}(q_0+1;q-1)$ with

$$\begin{array}{c}\hfill \underset{x\in \mathbb{N}(q_0+1;q-1)}{min}\mathcal{G}(x,r)=\mathcal{G}(x_0,r),\mathrm{for}\mathrm{all}r\in \mathbb{N}(q_0+2;q).\end{array}$$

Hence, by making use of Theorem 5, we have

$$\begin{array}{c}\hfill \mathcal{G}(r-1,r)\le {\displaystyle \frac{1}{\gamma }}\mathcal{G}(x_0,r),r\in \mathbb{N}(q_0+2;q).\end{array}$$

(16)

The following hypotheses will be useful for the next results:

Hypothesis 1 (H1). 

$Z(x,\xi )\ge 0$, for $(x,\xi )\in \mathbb{N}(q_0;q)\times {\mathbb{R}}^{+}$.

Hypothesis 2 (H2). 

There exists $a_1>0$ with $Z(x,g)\le \eta a_1$, where $0\le g\le a_1$;

Hypothesis 3 (H3). 

There exists $a_2>0$ with $Z(x,g)\ge {\displaystyle \frac{\eta a_2}{\gamma }}$, where $\gamma a_2\le g\le a_2$;

Hypothesis 4 (H4). 

Suppose that

$$\begin{array}{c}\hfill \underset{g\to 0^{+}}{lim}\left(\underset{x\in \mathbb{N}(q_0;q)}{min}{\displaystyle \frac{Z(x,g)}{g}}\right)\to \infty and\underset{g\to \infty }{lim}\left(\underset{x\in \mathbb{N}(q_0;q)}{min}{\displaystyle \frac{Z(x,g)}{g}}\right)\to \infty .\end{array}$$

Hypothesis 5 (H5). 

Suppose that

$$\begin{array}{c}\hfill \underset{g\to 0^{+}}{lim}\left(\underset{x\in \mathbb{N}(q_0;q)}{min}{\displaystyle \frac{Z(x,g)}{g}}\right)\to 0and\underset{g\to \infty }{lim}\left(\underset{x\in \mathbb{N}(q_0;q)}{min}{\displaystyle \frac{Z(x,g)}{g}}\right)\to 0.\end{array}$$

Lemma 1. 

Let hypothesis  (H1)  hold. Then, $\mathcal{E}$ is an operator from $\mathcal{F}$ to $\mathcal{F}$.

Proof. 

Let $g\in \mathcal{F}$. It is clear that $\left(\mathcal{E}g\right)\left(x\right)\ge 0$ for $x\in \mathbb{N}(q_0;q)$. Next, we consider

$$\begin{array}{cc}\hfill \underset{x\in \mathbb{N}(q_0+1;q-1)}{min}\left(\mathcal{E}g\right)\left(x\right)& =\underset{x\in \mathbb{N}(q_0+1;q-1)}{min}\left(\sum _{r=q_0+2}^q\mathcal{G}(x,r)Z(r,g\left(r\right))\right)\hfill \\ \hfill & \ge \gamma \sum _{r=q_0+2}^q\mathcal{G}(r-1,r)Z(r,g\left(r\right))\hfill \\ \hfill & =\gamma \sum _{r=q_0+2}^q\mathcal{G}(x,r)Z(r,g\left(r\right))\hfill \\ \hfill & =\gamma \sum _{r=q_0+2}^q\mathcal{G}(x,r)Z(r,g\left(r\right))\hfill \\ \hfill & =\gamma \parallel \mathcal{E}g\parallel .\hfill \end{array}$$

This leads to $\mathcal{E}g\in \mathcal{F}$. Hence, the proof is complete. □

Theorem 6. 

Let  (H1)–(H3)  hold on Z. Then, one can find at least one positive solution for (11).

Proof. 

It is evident that $\mathcal{E}$ is completely continuous as $\mathcal{E}:\mathcal{F}\to \mathcal{F}$. Let us define

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_1:=\left\{g\in \mathcal{F}:\parallel g\parallel <a_1\right\}.\end{array}$$

It can be said that ${\mathsf{\Omega }}_1\subseteq \mathcal{C}$ is an open set including 0. As $\parallel g\parallel =a_1$, for $g\in \partial {\mathsf{\Omega }}_1$, then (H2) can hold for each $g\in \partial {\mathsf{\Omega }}_1$. It follows that

$$\begin{array}{c}\parallel \mathcal{E}g\parallel =\underset{x\in \mathbb{N}(q_0;q)}{max}\sum _{r=q_0+2}^q\mathcal{G}(x_0,r)Z(r,g\left(r\right))\ge \sum _{r=q_0+2}^q\mathcal{G}(r-1,r)Z(r,g\left(r\right))\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge \eta a_1\sum _{r=q_0+2}^q\mathcal{G}(r-1,r)=a_1=\parallel g\parallel .\hfill \end{array}$$

This implies that $\parallel \mathcal{E}g\parallel \ge \parallel g\parallel$, where $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_1$.

In addition, we define

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_2:=\left\{g\in \mathcal{F}:\parallel g\parallel <a_2\right\}.\end{array}$$

It is evident that ${\mathsf{\Omega }}_2\subseteq \mathcal{C}$ is an open set with ${\overline{\mathsf{\Omega }}}_1\subseteq {\mathsf{\Omega }}_2$. As $\parallel g\parallel =a_2$, for $g\in \partial {\mathsf{\Omega }}_2$, then (H3) can hold for each $g\in \partial {\mathsf{\Omega }}_2$. By considering (16), we obtain

$$\begin{array}{c}\parallel \mathcal{E}g\parallel \ge |\mathcal{E}g\left(x_0\right)|=\sum _{r=q_0+2}^q\mathcal{G}(x_0,r)Z(r,g\left(r\right))\ge \gamma \sum _{r=q_0+2}^q\mathcal{G}(r-1,r)Z(r,g\left(r\right))\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge \eta a_2\sum _{r=q_0+2}^q\mathcal{G}(r-1,r)=a_2=\parallel g\parallel ,\hfill \end{array}$$

which gives that $\parallel \mathcal{E}g\parallel \ge \parallel g\parallel$, where $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_2$. Therefore, A has at least one fixed point in $\mathcal{F}\cap \left(\overline{{\mathsf{\Omega }}_2}{\mathsf{\Omega }}_1\right)$ according to Theorem 4. We call this fixed point $g_0$, which satisfies $a_1<\parallel g_0\parallel <a_2$. This proves our theorem. □

Theorem 7. 

Suppose that Z satisfies (H1)–(H4). Then, there are at least two positive solutions for (11).

Proof. 

Let $M>0$ and $x_1\in \mathbb{N}(q_0+1;q-1)$ be fixed with

$$\begin{array}{c}\hfill M\gamma \sum _{r=q_0+2}^q\mathcal{G}(x_1,r)>1.\end{array}$$

(17)

Considering (H4), there exists an $a>0$ such that $a<p$ and $Z(x,g)\ge Mg$ for all $0\le g\le a$ and $x\in \mathbb{N}(q_0;q)$. Define the set

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_a:=\left\{g\in \mathcal{F}:\parallel g\parallel <a\right\}.\end{array}$$

By making use of (17), we obtain

$$\begin{array}{c}\parallel \mathcal{E}g\parallel \ge |\mathcal{E}g\left(x_1\right)|=\sum _{r=q_0+2}^q\mathcal{G}(x_1,r)Z(r,g\left(r\right))\ge M\sum _{r=q_0+2}^q\mathcal{G}(x_1,r)\left|g\left(r\right)\right|\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge M\gamma \parallel g\parallel \sum _{r=q_0+2}^q\mathcal{G}(x_1,r)>\parallel g\parallel .\hfill \end{array}$$

This leads to $\parallel \mathcal{E}g\parallel > \parallel g\parallel$, whenever $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_a$. Then, for the same $M>0$, there is a number $R_1>0$ with $Z(x,g)\ge Mg$, for each $g\ge R_1$ and $x\in \mathbb{N}(q_0;q)$. Let us choose R such that

$$\begin{array}{c}\hfill R>max\left\{p,{\displaystyle \frac{R_1}{\gamma }}\right\}.\end{array}$$

Now, we define

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_R:=\left\{g\in \mathcal{F}:\parallel g\parallel <R\right\}.\end{array}$$

It is clear that $\parallel \mathcal{E}g\parallel > \parallel g\parallel$, when $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_R$. In the final step, we define

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_p:=\left\{g\in \mathcal{F}:\parallel g\parallel <p\right\}.\end{array}$$

This implies that $\parallel \mathcal{E}g\parallel \le \parallel g\parallel$ as $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_p$.

Thus, we have found two fixed points $g_0$ and $g_1$ for $\mathcal{E}$ with $g_1\in {\mathsf{\Omega }}_p\setminus \mathring{{\mathsf{\Omega }}_a}$ and $g_2\in {\mathsf{\Omega }}_R\setminus {\mathring{\mathsf{\Omega }}}_p$, where $\mathring{\mathsf{\Omega }}$ refers to the interior of $\mathsf{\Omega }$. In particular, we can say that $g_0$ and $g_1$ are two positive solutions of (11) that satisfy $0 < \parallel g_1\parallel ,p < \parallel g_2\parallel$. This completes our proof. □

Theorem 8. 

Assume that Z satisfies(H1),  (H3), and  (H5). Then, there exist at least two positive solutions for (11).

Proof. 

For any $ϵ>0$, there is an $M>0$ with $Z(x,g)\ge M+ϵg$ according to (H5), for $g\in \mathcal{F}$ and $x\in \mathbb{N}(q_0;q)$. Then, we have

$$\begin{array}{c}\hfill |\left(\mathcal{E}g\right)|=\sum _{r=q_0+2}^q\mathcal{G}(x,r)Z(r,g\left(r\right))\le \sum _{r=q_0+2}^q\mathcal{G}(r-1,r)\left[M+ϵg\left(r\right)\right].\end{array}$$

Since $ϵ>0$ is arbitrary, we see that

$$\begin{array}{c}\hfill |\left(\mathcal{E}g\right)|\le M\sum _{r=q_0+2}^q\mathcal{G}(r-1,r)={\displaystyle \frac{M}{\eta }}.\end{array}$$

Taking $R>p$ to be sufficiently large, we have

$$\begin{array}{c}\hfill R>{\displaystyle \frac{M}{\eta }}.\end{array}$$

Let us define

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_R:=\left\{g\in \mathcal{F}:\parallel g\parallel <R\right\}.\end{array}$$

It follows that $\parallel \mathcal{E}g\parallel <R=\parallel g\parallel$, whenever $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_R$. Again, by considering (H5), we have $a>0$ such that $a<p$ and $Z(x,g)<\eta g$, for $0\le g\le a,g\in \mathcal{F}$, and $x\in \mathbb{N}(q_0;q)$. Now, we define

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_a:=\left\{g\in \mathcal{F}:\parallel g\parallel <a\right\}.\end{array}$$

Then, we see from this that

$$\begin{array}{c}\parallel \left(\mathcal{E}g\right)\parallel =\underset{x\in \mathbb{N}(q_0;q)}{max}\sum _{r=q_0+2}^q\mathcal{G}(x,r)Z(r,g\left(r\right))\le \sum _{r=q_0+2}^q\mathcal{G}(r-1,r)Z(r,g\left(r\right))\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \eta \sum _{r=q_0+2}^q\mathcal{G}(r-1,r)\left|g\left(r\right)\right|\le \parallel g\parallel .\hfill \end{array}$$

This implies that $\parallel \mathcal{E}g\parallel < \parallel g\parallel$, when $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_a$. Lastly, we define

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_p:=\left\{g\in \mathcal{F}:\parallel g\parallel <p\right\}.\end{array}$$

It can be observed that $\parallel \mathcal{E}g\parallel > \parallel g\parallel$, whereas $g\in \mathcal{F}\cap \partial {\mathsf{\Omega }}_p$.

Thus, we have determined $g_1$ and $g_2$, two fixed points of $\mathcal{E}$ with $g_1\in {\mathsf{\Omega }}_p\setminus \mathring{{\mathsf{\Omega }}_a}$ and $g_2\in {\mathsf{\Omega }}_R\setminus {\mathring{\mathsf{\Omega }}}_p$. Specifically, we can say that $g_1$ and $g_2$ are two positive solutions of (11) that satisfy $0 < \parallel g_1\parallel$ and $p < \parallel g_2\parallel$. This concludes our result. □

4. Existence Results

Here, we examine the existence of some solutions by considering some known fixed-point theorems. According to Theorem 3, we can define an operator

$$\begin{array}{c}\hfill \left(\mathcal{H}g\right)\left(x\right):=w\left(x\right)+\sum _{r=q_0+2}^{q}Z\left(r,g\left(r\right)\right)\mathcal{G}(x,r),x\in \mathbb{N}(q_0;q).\end{array}$$

(18)

It follows from (10) and (18) that g is a fixed point of $\mathcal{H}$ IFF it is a solution of (2).

Theorem 9 

(see [36], (Brouwer theorem)). Let $\mathbb{R}\left(n\right)$ be the set of n-tuples of real numbers, $\varnothing \ne \mathcal{F}\subset \mathbb{R}\left(n\right)$ be a compact convex set, and $\mathcal{H}:\mathcal{F}\to \mathcal{F}$ be a continuous function. Then, $\mathcal{H}$ has a fixed point in K.

Theorem 10 

(see [36], (Leray–Schauder theorem)). Let $\mathsf{\Omega }\subset \mathbb{R}\left(n\right)$ be an open set with $0\in \mathsf{\Omega }$ and $\mathcal{H}:\overline{\mathsf{\Omega }}\to \mathbb{R}\left(n\right)$ be a completely continuous function. Note that every $\mathcal{H}$ has at least one of the following properties:

• There is $g\in \overline{\mathsf{\Omega }}$ such that $\mathcal{H}g=g$.
• There are $v\in \partial \mathsf{\Omega }$ and $\xi \in (0,1)$ such that $v=\xi \mathcal{H}v$.

Then, g is a fixed point of $\mathcal{H}$ in Ω.

Theorem 11 

(see [36], (Krasnoselskii–Zabreiko theorem)). Let $\mathcal{H}:\mathbb{R}\left(n\right)\to \mathbb{R}\left(n\right)$ be a completely continuous function and $\ell :\mathbb{R}\left(n\right)\to \mathbb{R}\left(n\right)$. If ℓ is a bounded linear function such that 1 is not its eigenvalue and

$$\begin{array}{c}\hfill \underset{\parallel g\parallel \to \infty }{lim}\left({\displaystyle \frac{\parallel \mathcal{H}g-\ell g\parallel }{\parallel g\parallel }}\right)=0,\end{array}$$

then there exist a fixed point of $\mathcal{H}$ in $\mathbb{R}\left(n\right)$.

Now, we know that $\mathbb{R}(q-q_0+1)$ is a BS with the following norm:

$$\begin{array}{c}\hfill \parallel g\parallel :=\underset{x\in \mathbb{N}(q_0;q)}{max}\left|g\left(x\right)\right|.\end{array}$$

Theorem 12. 

Let $Z(x,g)$ be a continuous function with respect to g for all $x\in \mathbb{N}(q_0;q)$. If there exist $\ell ,M>0$ with

$$\begin{array}{cc}\hfill & max\left\{|C_1|,|C_2|\right\}\le \ell ,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & M=\underset{(x,g)\in \mathbb{N}(q_0;q)\times [-3\ell ,3\ell ]}{max}\left|Z\left(x+\theta -1,g(x+\theta -1)\right)\right|,\hfill \end{array}$$

(20)

and

$$\begin{array}{c}\hfill \delta \le {\displaystyle \frac{\ell }{M}},\end{array}$$

(21)

then, FBVP (2) has a solution.

Proof. 

Let us define $\mathcal{F}$ as

$$\begin{array}{c}\hfill \mathcal{F}:=\left\{g:\mathbb{N}(q_0;q)\to \mathbb{R}\mathrm{and}\parallel g\parallel \le 3\ell \right\}.\end{array}$$

We know that $\varnothing \ne \mathcal{F}\subset \mathbb{R}(q-q_0+1)$ is a compact convex set. Now, we claim that

$$\begin{array}{c}\hfill \mathcal{H}:\mathcal{F}\to \mathcal{F}.\end{array}$$

To do this, we suppose that $g\in \mathcal{F}$ and $x\in \mathbb{N}(q_0;q)$. By considering

$$\begin{array}{cc}\hfill \left|\mathcal{H}g\right(x\left)\right|& =\left|w\left(x\right)+\sum _{r=q_0+2}^q\mathcal{G}(x,r)Z(r,g\left(r\right))\right|\hfill \\ \hfill & \le \left|w\left(x\right)|+\sum _{r=q_0+2}^q\mathcal{G}(x,r)|Z(r,g\left(r\right))\right|\hfill \\ \hfill & \le 2max\left\{\right|C_1|,|C_2\left|\right\}+M\sum _{r=q_0+2}^q\mathcal{G}(x,r)\hfill \\ \hfill & \le 2\ell +M\gamma \le 3\ell ,\hfill \end{array}$$

we obtain $\mathcal{H}g\in \mathcal{F}$. Therefore, $\mathcal{H}:\mathcal{F}\to \mathcal{F}$ as claimed. Moreover, $\mathcal{H}$ is trivially continuous on K because it is a summation operator on a discrete finite set. Hence, we can say that $\mathcal{H}$ has a fixed point according to Theorem 9. This concludes that FBVP (2) has a solution, namely $g_0$, such that $\parallel g_0\parallel \le 3\ell$. Hence, the proof is complete. □

Theorem 13. 

Let $Z(x,g)$ be as in the previous theorem, and it is bounded on $\mathbb{N}(q_0;q)\times \mathbb{R}$. Then, there is a solution for FBVP (2).

Proof. 

Let us take

$$\begin{array}{c}\hfill P>\underset{(x,g)\in \mathbb{N}(q_0;q)\times \mathbb{R}}{sup}\left|Z(x,g)\right|.\end{array}$$

Choose ℓ to be as large as

$$\begin{array}{c}\hfill max\left\{\right|C_1|,|C_2\left|\right\}\le \ell \mathrm{and}\delta \le {\displaystyle \frac{\ell }{M}},\end{array}$$

where M is the same as defined in Theorem 12 and $M\le P$ so that

$$\begin{array}{c}\hfill \delta \le {\displaystyle \frac{\ell }{M}}.\end{array}$$

Thus, FBVP (2) has a solution according to Theorem 12. This completes the proof. □

Theorem 14. 

Let Z be as in Theorem 12. If there are two continuous functions ψ and σ such that $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is non-decreasing, $\sigma :\mathbb{N}(q_0;q)\to \mathbb{R}$, and

$$\begin{array}{c}\hfill \left|Z(x,g)\right|\le \sigma \left(x\right)\psi \left(\right|g\left|\right),(x,g)\in \mathbb{N}(q_0;q)\times \mathbb{R}.\end{array}$$

(22)

Furthermore, if there exists $\gamma >0$ with

$$\begin{array}{c}\hfill {\displaystyle \frac{\gamma }{max\left\{\right|C_1|,|C_2\left|\right\}+\delta \parallel \sigma \parallel \psi \left(\gamma \right)}}>1,\end{array}$$

(23)

then FBVP (2) has a solution.

Proof. 

First, we define

$$\begin{array}{c}\hfill \mathsf{\Omega }:=\{g:\mathbb{N}\left(q_0\right)\to \mathbb{R}\mathrm{and}\parallel g\parallel <\gamma \}.\end{array}$$

It is obvious that $\mathsf{\Omega }\subset \mathbb{R}(q-q_0+1)$ is an open set including 0 and $\mathcal{H}:\overline{\mathsf{\Omega }}\to \mathbb{R}(q-q_0+1)$. So, $\mathcal{H}$ is trivially completely continuous on $\overline{\mathsf{\Omega }}$ since $\mathcal{H}$ is as in Theorem 12. On the contrary, we suppose that there are $v\in \mathsf{\Omega }$ and $\xi \in (0,1)$ with

$$\begin{array}{c}\hfill v=\xi \mathcal{H}v.\end{array}$$

(24)

Using the definition of $\mathcal{H}$ and Theorem 2(b) in (24), we can deduce

$$\begin{array}{cc}\hfill \left|v\right(x\left)\right|& \le \left|w\left(x\right)\right|+\sum _{r=q_0+2}^q\mathcal{G}(x,r)\left|Z\left(r,v\left(r\right)\right)\right|\hfill \\ \hfill & \le 2max\left\{\right|C_1|,|C_2\left|\right\}+\sum _{r=q_0+2}^q\mathcal{G}(x,r)\sigma \left(r\right)\psi \left(\left|v\right(r\left)\right|\right)\hfill \\ \hfill & \le 2max\left\{\right|C_1|,|C_2\left|\right\}+\parallel \sigma \parallel \psi (\parallel v\parallel )\sum _{r=q_0+2}^q\mathcal{G}(x,r)\hfill \\ \hfill & \le 2max\left\{\right|C_1|,|C_2\left|\right\}+\delta \parallel \sigma \parallel \psi \left(\gamma \right).\hfill \end{array}$$

This leads to

$$\begin{array}{c}\hfill \parallel v\parallel \le 2max\left\{\right|C_1|,|C_2\left|\right\}+\delta \parallel \sigma \parallel \psi \left(\gamma \right).\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\displaystyle \frac{\gamma }{2max\left\{\right|C_1|,|C_2\left|\right\}+\delta \parallel \sigma \parallel \psi \left(\gamma \right)}}\le 1.\end{array}$$

This contradicts (23). As a consequence, by considering Theorem 10, we see that $\mathcal{H}$ has a fixed point in $\mathsf{\Omega }$. This tells us that FBVP (2) has a solution, namely $g_1$, with $\parallel g_1\parallel <\gamma$. Thus, our proof is complete. □

Theorem 15. 

Let Z be as in Theorem 12. If there exists a continuous function $\varphi :\mathbb{N}(q_0;q)\to \mathbb{R}$ with

$$\begin{array}{c}\hfill \underset{\parallel g\parallel \to \infty }{lim}{\displaystyle \frac{Z(x,g)}{g}}=\varphi \left(x\right),x\in \mathbb{N}(q_0;q),\end{array}$$

(25)

and

$$\begin{array}{c}\hfill \parallel \varphi \parallel <{\displaystyle \frac{1}{\delta }},\end{array}$$

(26)

then FBVP (2) has a solution.

Proof. 

It is easy to see that $\mathcal{H}:\mathbb{R}(q-q_0+1)\to \mathbb{R}(q-q_0+1)$. So, $\mathcal{H}$ is trivially completely continuous on $\mathbb{R}(q-q_0+1)$, which is as in Theorem 12. Let us consider a linear bounded mapping ℓ, which is $\ell :\mathbb{R}(q-q_0+1)\to \mathbb{R}(q-q_0+1)$ defined by

$$\begin{array}{c}\hfill \left(\ell g\right)\left(x\right):=\sum _{r=q_0+2}^q\mathcal{G}(x,r)\varphi \left(r\right)g\left(r\right),x\in \mathbb{N}(q_0;q).\end{array}$$

(27)

Clearly, $\parallel \ell g\parallel < \parallel g\parallel$. Then, let $g\in \mathbb{R}\left(n\right)$ and $x\in \mathbb{N}(q_0;q)$, and we consider

$$\begin{array}{cc}\hfill \left(\ell g\right)\left(x\right)& \le \sum _{r=q_0+2}^q\mathcal{G}(x,r)\left|\varphi \left(r\right)\right|\left|g\left(r\right)\right|\hfill \\ \hfill & \le \parallel \varphi \parallel \parallel g\parallel \sum _{r=q_0+2}^q\mathcal{G}(x,r)\le \delta \parallel \varphi \parallel \parallel g\parallel <\parallel g\parallel .\hfill \end{array}$$

This implies that $\parallel \ell g\parallel < \parallel g\parallel$. Thus, 1 is not an eigenvalue of ℓ. By considering (24), we have the following: For every $ϵ>0$, there exists a number N with each $x\in \mathbb{N}(q_0;q)$,

$$\begin{array}{c}\hfill |Z\left(x,g\left(x\right)\right)-\varphi \left(x\right)g\left(x\right)|<ϵ\mathrm{where}\parallel g\parallel >N.\end{array}$$

(28)

Next, for each $x\in \mathbb{N}(q_0;q)$, we have

$$\begin{array}{cc}\hfill |\left(\mathcal{H}g\right)\left(x\right)-\left(\ell g\right)\left(x\right)|& \le \left|w\left(x\right)\right|+\sum _{r=q_0+2}^q\mathcal{G}(x,r)|Z\left(r,g\left(r\right)\right)-\varphi \left(r\right)g\left(r\right)|\hfill \\ \hfill & \le 2max\left\{\right|C_1|,|C_2\left|\right\}+ϵ\sum _{r=q_0+2}^q\mathcal{G}(x,r)\hfill \\ \hfill & \le 2max\left\{\right|C_1|,|C_2\left|\right\}+\delta ϵ.\hfill \end{array}$$

This leads to

$$\begin{array}{c}\hfill {\displaystyle \frac{\parallel \mathcal{H}g-\ell g\parallel }{\parallel g\parallel }}<{\displaystyle \frac{max\left\{\right|C_1|,|C_2\left|\right\}+\delta ϵ}{N}}.\end{array}$$

As a consequence, we obtain

$$\begin{array}{c}\hfill \underset{\parallel g\parallel \to \infty }{lim}{\displaystyle \frac{\parallel \mathcal{H}g-\ell g\parallel }{\parallel g\parallel }}=0.\end{array}$$

Thus, $\mathcal{H}$ has a fixed point in $\mathbb{R}(q-q_0+1)$ by Theorem 11. Hence, FBVP (2) has a solution, as requested. □

5. Uniqueness Results

This part of our article provides the existence of the unique solution of model (2) by considering the Lipschitz condition.

Theorem 16 

(see [36], (Contraction Mapping Theorem)). Let $S\subset \mathbb{R}\left(n\right)$ be closed and $\mathcal{H}:S\to S$ be a contraction function; i.e., $\exists \xi \in [0,1)$ with

$$\begin{array}{c}\hfill \parallel \mathcal{H}g-\mathcal{H}v\parallel \le \xi \parallel g-v\parallel ,\end{array}$$

for each $g,v\in S$. Then, w is a unique fixed point of $\mathcal{H}$ in S.

Theorem 17. 

If $Z(x,g)$ satisfies the Lipschitz condition with respect to g, i.e.,

$$\begin{array}{c}\hfill \parallel Z\left(x,g\right)-Z\left(x,v\right)\parallel \le K\parallel g-v\parallel ,\end{array}$$

(29)

for each $(x,g),(x,v)\in \mathbb{N}(q_0;q)\times \mathbb{R}$, where K is the Lipschitz constant and if

$$\begin{array}{c}\hfill 0<K\delta <1,\end{array}$$

(30)

then there is a unique solution for (2).

Proof. 

Let $g,v\in \mathbb{R}(q-q_0+1)$ and $x\in \mathbb{N}(q_0;q)$. Consider

$$\begin{array}{cc}\hfill \left|\mathcal{H}g\left(x\right)-\mathcal{H}v\left(x\right)\right|& \le \sum _{r=q_0+2}^q\mathcal{G}(x,r)\left|Z\left(r,g\left(r\right)\right)-Z\left(r,v\left(r\right)\right)\right|\hfill \\ \hfill & \le K\sum _{r=q_0+2}^q\mathcal{G}(x,r)|g\left(r\right)-v\left(r\right)|\hfill \\ \hfill & \le K\parallel g-v\parallel \sum _{r=q_0+2}^q\mathcal{G}(x,r)\hfill \\ \hfill & \le K\delta \parallel g-v\parallel ,\hfill \end{array}$$

implying that

$$\begin{array}{c}\hfill \parallel \mathcal{H}g-\mathcal{H}v\parallel \le K\delta \parallel g-v\parallel ,\end{array}$$

Therefore, $\mathcal{H}$ is a contraction on $\mathbb{R}(q-q_0+1)$ according to (30). Thus, $\mathcal{H}$ has a unique fixed point in $\mathbb{R}(q-q_0+1)$ by Theorem 17. This implies that (2) has a unique solution. This ends the proof. □

Theorem 18. 

Let $\eta ,\beta >0$ with

$$\begin{array}{c}\hfill \parallel Z\left(x,g\right)-Z\left(x,v\right)\parallel \le \beta \parallel g-v\parallel ,\end{array}$$

(31)

for each $(x,g),(x,v)\in \mathbb{N}(q_0;q)\times [-\eta ,\eta ]$. We can choose

$$\begin{array}{c}\hfill m_1=\underset{x\in \mathbb{N}(q_0;q)}{max}\left|Z(x,0)\right|,\end{array}$$

(32)

and

$$\begin{array}{c}\hfill m_2=\underset{(x,g)\in \mathbb{N}(q_0;q)\times [-\eta ,\eta ]}{max}\left|Z(x,g)\right|.\end{array}$$

(33)

Moreover, if

$$\begin{array}{c}\hfill 0<\beta \delta <1,\end{array}$$

(34)

and

$$\begin{array}{c}\hfill \eta \ge {\displaystyle \frac{m_1\delta +2max\left\{\right|C_1|,|C_2\left|\right\}}{1-\beta \delta }}⟹2max\left\{\right|C_1|,|C_2\left|\right\}+m_2\delta \le \eta ,\end{array}$$

(35)

then (2) has a unique solution.

Proof. 

It can be observed that $\mathcal{H}$ is a contraction on $\mathbb{R}(q-q_0+1)$ according to (34). Let us define

$$\begin{array}{c}\hfill \mathcal{D}=\left\{g:\mathbb{N}(q_0;q)\to \mathbb{R}\mathrm{and}\parallel g\parallel \le \eta \right\}.\end{array}$$

It is clear that $\mathcal{D}\subseteq \mathbb{R}(q-q_0+1)$. Now, let us claim that $\mathcal{H}:\mathcal{D}\to \mathcal{D}$. To prove this, let $g\in \mathcal{D}$ and $x\in \mathbb{N}(q_0;q)$. We assume that (35) holds, and then we consider

$$\begin{array}{cc}\hfill \left|\mathcal{H}\right(0\left)\right(x\left)\right|& =|w\left(x\right)+\sum _{r=q_0+2}^q\mathcal{G}(x,r)Z(r,0)|\hfill \\ \hfill & \le \left|w\left(x\right)\right|+\sum _{r=q_0+2}^q\mathcal{G}(x,r)\left|Z(r,0)\right|\hfill \\ \hfill & \le 2max\left\{\right|C_1|,|C_2\left|\right\}+m_1\sum _{r=q_0+2}^q\mathcal{G}(x,r)\hfill \\ \hfill & \le 2max\left\{\right|C_1|,|C_2\left|\right\}+m_1\delta ,\hfill \end{array}$$

which leads to

$$\begin{array}{c}\hfill \parallel \mathcal{H}\left(0\right)\parallel \le 2max\left\{\right|C_1|,|C_2\left|\right\}+m_1\delta .\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \parallel \mathcal{H}\left(g\right)\parallel & =\parallel \mathcal{H}\left(g\right)+\mathcal{H}\left(0\right)-\mathcal{H}\left(0\right)\parallel \hfill \\ \hfill & \le \parallel \mathcal{H}\left(g\right)-\mathcal{H}\left(0\right)\parallel +\parallel \mathcal{H}\left(0\right)\parallel \hfill \\ \hfill & \le \beta \delta \parallel g-0\parallel +2max\left\{\right|C_1|,|C_2\left|\right\}+m_1\delta \hfill \\ \hfill & \le \beta \delta \eta +2max\left\{\right|C_1|,|C_2\left|\right\}+m_1\delta \le \eta .\hfill \end{array}$$

This implies that $\mathcal{H}:\mathcal{D}\to \mathcal{D}$.

On the other hand, let us assume that (35) holds. Let $g\in \mathcal{D}$ and $x\in \mathbb{N}(q_0;q)$. Then, we consider

$$\begin{array}{cc}\hfill \left|\left(\mathcal{H}g\right)\left(x\right)\right|& =|w\left(x\right)+\sum _{r=q_0+2}^q\mathcal{G}(x,r)Z(r,g\left(r\right))|\hfill \\ \hfill & \le \left|w\left(x\right)\right|+\sum _{r=q_0+2}^q\mathcal{G}(x,r)\left|Z(r,g\left(r\right))\right|\hfill \\ \hfill & \le 2max\left\{\right|C_1|,|C_2\left|\right\}+m_2\sum _{r=q_0+2}^q\mathcal{G}(x,r)\hfill \\ \hfill & \le 2max\left\{\right|C_1|,|C_2\left|\right\}+m_2\delta \le \eta ,\hfill \end{array}$$

which leads to

$$\begin{array}{c}\hfill \parallel \mathcal{H}g\parallel \le \eta ,\end{array}$$

which gives $\mathcal{H}:\mathcal{D}\to \mathcal{D}$. As a consequence, $\mathcal{H}$ has a unique fixed point in $\mathbb{R}(q-q_0+1)$ according to Theorem 16. Finally, we see that (2) has a unique solution, namely $g_2$, and it satisfies $\parallel g_2\parallel \le \eta$. Hence, the proof is complete. □

6. Numerical Examples

The following examples are presented to understand the applicability of the above results.

Example 1. 

In the first example, we consider the FBVP

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & -\left({{}_0{}^{\mathrm{RL}}\Delta }^{{\displaystyle \frac{3}{2}}}g\right)\left(x\right)={\left(g^2+{\left(x+{\displaystyle \frac{1}{2}}\right)}^2+9\right)}^{-1},x\in \mathbb{N}(2;10),\hfill \\ \hfill & g\left(0\right)=1,g\left(10\right)=2.\hfill \end{array}\end{array}$$

(36)

We can observe that

$$\begin{array}{cc}\hfill & q_0=0,q=10,\theta =1.5,Z\left(x+\theta -1,g(x+\theta -1)\right)={\left(g^2+{\left(x+{\displaystyle \frac{1}{2}}\right)}^2+9\right)}^{-1},\hfill \\ \hfill & C_1=1,C_2=2,\ell \ge 2.\hfill \end{array}$$

Moreover,

$$\begin{array}{c}\hfill M=\underset{(x,g)\in {\mathbb{N}}_0^{10}\times [-3\ell ,3\ell ]}{max}\left|Z\left(x+\theta -1,g(x+\theta -1)\right)\right|={\displaystyle \frac{1}{9}},\end{array}$$

and

$$\begin{array}{c}\hfill \delta ={\displaystyle \frac{9}{\left(1.5\right)\mathsf{\Gamma }\left(2.5\right)}}{(12+0.5-1)}^{\overline{0.5}}=15.4738,\end{array}$$

These give that $\delta \le {\displaystyle \frac{\ell }{M}}$. Therefore, FBVP (36) has at least one solution (say $g_0$), which satisfies $|g_0\left(x\right)\le 3\ell |$ for $x\in \mathbb{N}(0;10)$, according to Theorem 12.

Example 2. 

Here, we consider the FBVP

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & -\left({{}_{\rho \left(0\right)}{}^{\mathrm{RL}}\Delta }^{{\displaystyle \frac{3}{2}}}g\right)\left(x\right)=x+{\displaystyle \frac{1}{20}}sin\left(g(x+0.5)\right),x\in \mathbb{N}(2;10),\hfill \\ \hfill & g\left(0\right)=1,g\left(10\right)=2.\hfill \end{array}\end{array}$$

(37)

It is known that $q_0=0,q=10,\theta =1.5,Z\left(x+\theta -1,g(x+\theta -1)\right)=x+\left(0.05\right)sing,C_1=1,$ and $C_2=2$. It is easy to see that Z satisfies the Lipschitz condition with respect to g on $\mathbb{N}(2;10)\times \mathbb{R}$; it has the Lipschitz constant $K=0.05$. Moreover,

$$\begin{array}{c}\hfill \delta ={\displaystyle \frac{9}{\left(1.5\right)\mathsf{\Gamma }\left(2.5\right)}}{\left(12\right)}^{\overline{0.5}}=15.4738,\end{array}$$

which implies that $0<K\delta <1$. Thus, in considering Theorem 17, FBVP (37) has a unique solution on $\mathbb{N}(2;10)\times \mathbb{R}$.

7. Conclusions

We considered the uniqueness of solutions for FBVP (2). We constructed a discrete GF in the sense of Riemann–Liouville operators. In the main results, the minimum value of the GF was found. Furthermore, using five hypotheses (H1)–(H5) together with the Guo–Krasnoselskii theorem, we established the positive solutions of (11). Next, by defining the operator (18) together with the theorems of Brouwer, Leray–Schauder, Krasnoselskii–Zabreiko, the existence of solutions for FBVP (2) was derived. In addition, based on the Contraction Mapping Theorem and Lipschitz constant conditions, we obtained the uniqueness of a solution for (2). Finally, the applicability of the main results was confirmed via two special examples.

An important direction of research that has remained unexplored up to now is related to other types of discrete fractional operators that are continuously used over discrete fractional models (see [13,14]). Hence, new discrete fractional operators should be used to prove existence and uniqueness for fractional boundary value problems. Therefore, these will be welcoming lines of thought for future research.