Open AccessArticle

Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting

Pshtiwan Othman Mohammed

^1,2,*

Ravi P. Agarwal

Majeed A. Yousif

⁴

Eman Al-Sarairah

^5,6

Alina Alb Lupas

^7,*

and

Mohamed Abdelwahed

⁸

Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq

Research and Development Center, University of Sulaimani, Sulaimani 46001, Iraq

Department of Mathematics and Systems Engineering, Florida Institute of Technology, Melbourne, FL 32901, USA

⁴

Department of Mathematics, College of Education, University of Zakho, Zakho 42002, Iraq

⁵

Department of Mathematics, Khalifa University of Science and Technology, Abu Dhabi P.O. Box 127788, United Arab Emirates

⁶

Department of Mathematics, Al-Hussein Bin Talal University, P.O. Box 20, Ma’an 71111, Jordan

⁷

Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania

⁸

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Authors to whom correspondence should be addressed.

Mathematics 2024, 12(18), 2864; https://doi.org/10.3390/math12182864

Submission received: 5 August 2024 / Revised: 12 September 2024 / Accepted: 12 September 2024 / Published: 14 September 2024

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

Download Versions Notes

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.

Keywords:

Riemann–Liouville operators; Green’s function (GF); fixed-point theorem; existence and uniqueness solution

MSC:

26A48; 26A51; 33B10; 39A12; 39B62

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{matrix} \{\begin{matrix} - ({{}_{q_{0}}^{RL}Δ}^{θ} g) (x) = h (x + θ), x \in N (q_{0} + 2; q), \\ g (q_{0}) = 0, g (q) = 0, \end{matrix} \end{matrix}

(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{matrix} \{\begin{matrix} ({{}_{q_{0}}^{RL}Δ}^{θ} g) (x) = - Z (x + θ - 1, g (x + θ - 1)), \\ g (q_{0}) = C_{1}, g (q) = C_{2} x \in N (q_{0} + 2; q), \end{matrix} \end{matrix}

(2)

For

q_{0}, q, C_{1}, C_{2} \in R

with

q - q_{0} \in N (2)

, Z is assumed to be a function from

N (q_{0} + 2; q) \times R

R

. Note that

N (q_{0}) = \{q_{0}, q_{0} + 1, \dots\}

and

N (q_{0}; q) = \{q_{0}, q_{0} + 1, \dots, q\}

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

θ \in (1, 2)

, and let

q_{0}

and q be two real numbers such that

q - q_{0} \in N (1)

, and

h : N (q_{0} + 2; q) \to R

. Then, FBVP (1) has the unique solution

\begin{matrix} g (x) = \sum_{r = q_{0} + 2}^{q} G (x, r) h (r), x \in N (q_{0}; q), \end{matrix}

(3)

where the GF

G (x, r)

is given by

\begin{matrix} G (x, r) = \{\begin{matrix} \frac{{(q + θ - r - 1)}^{\underset{̲}{θ - 1}}}{{(q + θ - q_{0} - 2)}^{\underset{̲}{θ - 1}} Γ (θ)} {(x + θ - q_{0} - 2)}^{\underset{̲}{θ - 1}}, & x \leq r - 1, \\ \frac{{(q + θ - r - 1)}^{\underset{̲}{θ - 1}}}{{(q + θ - q_{0} - 2)}^{\underset{̲}{θ - 1}} Γ (θ)} {(x + θ - q_{0} - 2)}^{\underset{̲}{θ - 1}} - \frac{{(x + θ - r - 1)}^{\underset{̲}{θ - 1}}}{Γ (θ)}, & x \geq r . \end{matrix} \end{matrix}

(4)

In the above theorem, for

x \in N (q_{0} + ℓ - θ)

(see [35], (Theorem 2.2)),

\begin{matrix} ({{}_{q_{0}}^{RL}Δ}^{θ} g) (x) & : = \frac{1}{Γ (- θ)} \sum_{r = q_{0}}^{x + θ} {(x - r - 1)}^{\underset{̲}{- θ - 1}} g (r), \end{matrix}

(5)

is the Riemann–Liouville fractional difference, and its sum formula, for

x \in N_{q_{0} + θ}

(see [2], (Definition 2.25)),

\begin{matrix} ({{}_{q_{0}}^{RL}Δ}^{- θ} g) (x) & = \frac{1}{Γ (θ)} \sum_{r = q_{0}}^{x - θ} {(x - r - 1)}^{\underset{̲}{θ - 1}} g (r), \end{matrix}

(6)

is the Riemann–Liouville fractional sum for the function g on the domain

N (q_{0})

ℓ - 1 < θ < ℓ

with

ℓ \in N_{1}

, and

x^{\underset{̲}{θ}} = \frac{Γ (x + 1)}{Γ (x + 1 - θ)}

Theorem 2.

Let

θ \in (1, 2)

in the FBVP:

\begin{matrix} \begin{matrix} - ({{}_{q_{0}}^{RL}Δ}^{θ} w) (x) = 0, \\ w (q_{0}) = C_{1}, w (q) = C_{2} . \end{matrix} \end{matrix}

(7)

Then, we have the following:

(a): For $x \in N_{q_{0} + 2}$ ,

$\begin{matrix} w (x) = C_{1} (\frac{q - x}{q - q_{0}}) \frac{{(x - q_{0} + θ - 2)}^{\underset{̲}{θ - 2}}}{Γ (θ - 1)} + C_{2} \frac{{(x - q_{0} + θ - 2)}^{\underset{̲}{θ - 1}}}{{(q - q_{0} + θ - 2)}^{\underset{̲}{θ - 1}}}, x \in N (q_{0}; q), \end{matrix}$

(8)

is the solution of (7).
(b): For $x \in N_{q_{0}}^{q}$ , we have

$\begin{matrix} ∣ w (x) ∣ \leq 2 max \{| C_{1} |, | C_{2} |\} . \end{matrix}$

Theorem 3.

Let

h : N_{q_{0} + 2}^{q} \to R

. The FBVP

\begin{matrix} \begin{matrix} - ({{}_{q_{0}}^{RL}Δ}^{θ} g) (x) = h (x + θ), & x \in N_{q_{0} + 2}, \\ g (q_{0}) = C_{1}, g (q) = C_{2}, \end{matrix} \end{matrix}

(9)

has the unique solution

\begin{matrix} g (x) = w (x) + \sum_{r = q_{0} + 2}^{q} G (x, r) h (r), x \in N_{q_{0}}^{q}, \end{matrix}

(10)

where w is as given in the above theorem, and

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{matrix} \{\begin{matrix} ({{}_{q_{0}}^{RL}Δ}^{θ} g) (x) = - Z (x + θ - 1, g (x + θ - 1)), \\ g (q_{0}) = g (q) = 0, x \in N (q_{0} + 2; q) . \end{matrix} \end{matrix}

(11)

Definition 1

([37], (Completely Continuous Operator)). A bounded linear operator

E : C \to \bar{C}

, where

C

and

\bar{C}

are two Banach spaces (BSs), is completely continuous if it transforms weakly convergent sequences in

C

to norm-convergent sequences in

\bar{C}

Theorem 4

([36], (Guo–Krasnoselskii theorem)). Let

F \subset C

be a cone set, and the BS

C

contains two open sets

Ω_{1}

and

Ω_{2}

such that

{\bar{Ω}}_{1} \subseteq Ω_{2}

and

0 \in Ω_{1}

. Suppose that

E : F \cap ({\bar{Ω}}_{2} ∖ Ω_{1}) \to F

is a completely continuous operator. If one of the following properties holds, then

E

has at least one fixed point in

F \cap ({\bar{Ω}}_{2} ∖ Ω_{1})

1.: $∥ E g ∥ \geq ∥ g ∥$ for $g \in F \cap \partial Ω_{2}$ and $∥ E g ∥ \leq ∥ g ∥$ for $g \in F \cap \partial Ω_{1}$ ;
2.: $∥ E g ∥ \leq ∥ g ∥$ for $g \in F \cap \partial Ω_{2}$ and $∥ E g ∥ \geq ∥ g ∥$ for $g \in F \cap \partial Ω_{1}$ .

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

Theorem 5.

One can have

0 < γ < 1

such that

\begin{matrix} min_{x \in N (q_{0} + 1; q - 1)} G (x, r) \leq max_{x \in N (q_{0} + 1; q - 1)} G (x, r) = γ G (r - 1, r), r \in N (q_{0} + 2; q) . \end{matrix}

Proof.

In [38], Mohammed et al. showed that

G (x, r)

increases from

G (q_{0}, r) = 0

to a positive value in

G (r - 1, r)

and then decreases to

G (q, r) = 0

, for fixed

r \in N (q_{0} + 1; q)

. Now, we define

\begin{matrix} \bar{G} (x, r) = \frac{G (x, r)}{G (r - 1, r)} . \end{matrix}

Then, for

r \in N (q_{0} + 2; q)

, we consider

\begin{matrix} \bar{G} (x, r) = \{\begin{matrix} \frac{{(x - q_{0} + θ - 2)}^{\bar{θ - 1}}}{{(r - q_{0} + θ - 3)}^{\bar{θ - 1}}}, & x \in N_{q_{0} + 1}^{r - 1}, \\ \frac{{(x - q_{0} + θ - 2)}^{\bar{θ - 1}}}{{(r - q_{0} + θ - 3)}^{\bar{θ - 1}}} - \frac{{(x - r + θ - 1)}^{\bar{θ - 1}} {(q - q_{0} + θ - 2)}^{\bar{θ - 1}}}{{(q - r + θ - 1)}^{\bar{θ - 1}} {(r - q_{0} + θ - 3)}^{\bar{θ - 1}}}, & x \in N_{r}^{q - 1} . \end{matrix} \end{matrix}

Therefore, for

x \in N (q_{0} + 1; r - 1)

and

r \in N (q_{0} + 2; q)

, one can have

\begin{matrix} \bar{G} (x, r) = \frac{{(x - q_{0} + θ - 2)}^{\bar{θ - 1}}}{{(r - q_{0} + θ - 3)}^{\bar{θ - 1}}} \geq \frac{{(θ - 1)}^{\bar{θ - 1}}}{{(q - q_{0} + θ - 3)}^{\bar{θ - 1}}} = \frac{Γ (θ)}{{(q - q_{0} + θ - 3)}^{\bar{θ - 1}}} . \end{matrix}

(12)

Also, for

x \in N (r; q - 1)

and

r \in N (q_{0} + 2; q - 1)

, we know that

G (x, r)

(

⟹ \bar{G} (x, r)

) is a decreasing function of x. Then, we obtain

\begin{matrix} \bar{G} (x, r) & \geq \bar{G} (q - 1, r) \\ = \frac{{(q - q_{0} + θ - 3)}^{\bar{θ - 1}}}{{(r - q_{0} + θ - 3)}^{\bar{θ - 1}}} - \frac{{(q - r + θ - 2)}^{\bar{θ - 1}} {(q - q_{0} + θ - 2)}^{\bar{θ - 1}}}{{(q - r + θ - 1)}^{\bar{θ - 1}} {(r - q_{0} + θ - 3)}^{\bar{θ - 1}}} \\ = \frac{{(q - q_{0} + θ - 3)}^{\bar{θ - 1}}}{{(r - q_{0} + θ - 3)}^{\bar{θ - 1}}} [1 - \frac{{(q - r + θ - 2)}^{\bar{θ - 1}} {(q - q_{0} + θ - 2)}^{\bar{θ - 1}}}{{(q - r + θ - 1)}^{\bar{θ - 1}} {(q - q_{0} + θ - 3)}^{\bar{θ - 1}}}] \\ = \frac{{(q - q_{0} + θ - 3)}^{\bar{θ - 1}}}{{(r - q_{0} + θ - 3)}^{\bar{θ - 1}}} [1 - (\frac{q - r}{q - r + θ - 1}) (\frac{q - q_{0} + θ - 2}{q - q_{0} - 1})] \\ = \frac{{(q - q_{0} + θ - 3)}^{\bar{θ - 2}}}{{(r - q_{0} + θ - 3)}^{\bar{θ - 2}}} (\frac{θ - 1}{q - q_{0} + θ - 1}) \\ = \frac{{(q - q_{0} + θ - 3)}^{\bar{θ - 2}}}{{(θ - 1)}^{\bar{θ - 2}}} (\frac{θ - 1}{q - q_{0} + θ - 3}) \\ = \frac{{(q - q_{0} + θ - 3)}^{\bar{θ - 2}}}{(q - q_{0} + θ - 3) Γ (θ - 1)} . \end{matrix}

(13)

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

\begin{matrix} min_{x \in N (q_{0} + 1; q - 1)} G (x, r) \geq γ G (r - 1, r), r \in N (q_{0} + 2; q), \end{matrix}

where

\begin{matrix} γ = min \{\frac{Γ (θ)}{{(q - q_{0} + θ - 3)}^{\bar{θ - 1}}}, \frac{{(q - q_{0} + θ - 3)}^{\bar{θ - 2}}}{(q - q_{0} + θ - 3) Γ (θ - 1)}\} . \end{matrix}

It is clear that

γ

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{matrix} g (x) = \sum_{r = q_{0} + 2}^{q} Z (r, g (r)) G (x, r), x \in N (q_{0}; q) . \end{matrix}

(14)

Let us define the operator

E

as follows:

\begin{matrix} (E g) (x) : = \sum_{r = q_{0} + 2}^{q} G (x, r) Z (r, g (r)), x \in N (q_{0}; q) . \end{matrix}

(15)

We can say that g is a fixed point of

E

(according to (14) and (15)) IFF it is a solution of (11). In denoting

C

\begin{matrix} C = \{g : N (q_{0}; q) \to R; g (q_{0}) = g (q) = 0\} \subseteq R (q - q_{0} + 1) . \end{matrix}

it can be observed that

E : C \to C

and

C

is a BS with the following norm:

\begin{matrix} ∥ g ∥ = max_{x \in N (q_{0}; q)} | g (x) | . \end{matrix}

Now, we define the cone

\begin{matrix} F : \{g \in C : g (t) \geq 0, for each x \in N (q_{0}; q) and min_{N (q_{0} + 1; q - 1)} g (x) \geq γ ∥ g ∥\} . \end{matrix}

Now, we try to obtain sufficient conditions for the existence of a fixed point in

E

. We firstly know that

E

is a summation operator defined on a finite set. Therefore,

E

is completely continuous. Then, let

\begin{matrix} η : = \frac{1}{\sum_{r = q_{0} + 2}^{q} G (r - 1, r)} = \frac{Γ (θ) {(q - q_{0} + 2 θ - 2)}^{\bar{2 θ - 1}}}{Γ (2 θ) {(q - q_{0} + θ - 2)}^{\bar{θ - 1}}}, \end{matrix}

and

x_{0} \in N (q_{0} + 1; q - 1)

with

\begin{matrix} min_{x \in N (q_{0} + 1; q - 1)} G (x, r) = G (x_{0}, r), for all r \in N (q_{0} + 2; q) . \end{matrix}

Hence, by making use of Theorem 5, we have

\begin{matrix} G (r - 1, r) \leq \frac{1}{γ} G (x_{0}, r), r \in N (q_{0} + 2; q) . \end{matrix}

(16)

The following hypotheses will be useful for the next results:

Hypothesis 1 (H1).

Z (x, ξ) \geq 0

, for

(x, ξ) \in N (q_{0}; q) \times R^{+}

Hypothesis 2 (H2).

There exists

a_{1} > 0

with

Z (x, g) \leq η a_{1}

, where

0 \leq g \leq a_{1}

;

Hypothesis 3 (H3).

There exists

a_{2} > 0

with

Z (x, g) \geq \frac{η a_{2}}{γ}

, where

γ a_{2} \leq g \leq a_{2}

;

Hypothesis 4 (H4).

Suppose that

\begin{matrix} lim_{g \to 0^{+}} (min_{x \in N (q_{0}; q)} \frac{Z (x, g)}{g}) \to \infty a n d lim_{g \to \infty} (min_{x \in N (q_{0}; q)} \frac{Z (x, g)}{g}) \to \infty . \end{matrix}

Hypothesis 5 (H5).

Suppose that

\begin{matrix} lim_{g \to 0^{+}} (min_{x \in N (q_{0}; q)} \frac{Z (x, g)}{g}) \to 0 a n d lim_{g \to \infty} (min_{x \in N (q_{0}; q)} \frac{Z (x, g)}{g}) \to 0 . \end{matrix}

Lemma 1.

Let hypothesis (H1) hold. Then,

E

is an operator from

F

F

Proof.

Let

g \in F

. It is clear that

(E g) (x) \geq 0

for

x \in N (q_{0}; q)

. Next, we consider

\begin{matrix} min_{x \in N (q_{0} + 1; q - 1)} (E g) (x) & = min_{x \in N (q_{0} + 1; q - 1)} (\sum_{r = q_{0} + 2}^{q} G (x, r) Z (r, g (r))) \\ \geq γ \sum_{r = q_{0} + 2}^{q} G (r - 1, r) Z (r, g (r)) \\ = γ \sum_{r = q_{0} + 2}^{q} G (x, r) Z (r, g (r)) \\ = γ \sum_{r = q_{0} + 2}^{q} G (x, r) Z (r, g (r)) \\ = γ ∥ E g ∥ . \end{matrix}

This leads to

E g \in 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

E

is completely continuous as

E : F \to F

. Let us define

\begin{matrix} Ω_{1} : = \{g \in F : ∥ g ∥ < a_{1}\} . \end{matrix}

It can be said that

Ω_{1} \subseteq C

is an open set including 0. As

∥ g ∥ = a_{1}

, for

g \in \partial Ω_{1}

, then (H2) can hold for each

g \in \partial Ω_{1}

. It follows that

\begin{matrix} ∥ E g ∥ = max_{x \in N (q_{0}; q)} \sum_{r = q_{0} + 2}^{q} G (x_{0}, r) Z (r, g (r)) \geq \sum_{r = q_{0} + 2}^{q} G (r - 1, r) Z (r, g (r)) \\ \geq η a_{1} \sum_{r = q_{0} + 2}^{q} G (r - 1, r) = a_{1} = ∥ g ∥ . \end{matrix}

This implies that

∥ E g ∥ \geq ∥ g ∥

, where

g \in F \cap \partial Ω_{1}

In addition, we define

\begin{matrix} Ω_{2} : = \{g \in F : ∥ g ∥ < a_{2}\} . \end{matrix}

It is evident that

Ω_{2} \subseteq C

is an open set with

{\bar{Ω}}_{1} \subseteq Ω_{2}

. As

∥ g ∥ = a_{2}

, for

g \in \partial Ω_{2}

, then (H3) can hold for each

g \in \partial Ω_{2}

. By considering (16), we obtain

\begin{matrix} ∥ E g ∥ \geq | E g (x_{0}) | = \sum_{r = q_{0} + 2}^{q} G (x_{0}, r) Z (r, g (r)) \geq γ \sum_{r = q_{0} + 2}^{q} G (r - 1, r) Z (r, g (r)) \\ \geq η a_{2} \sum_{r = q_{0} + 2}^{q} G (r - 1, r) = a_{2} = ∥ g ∥, \end{matrix}

which gives that

∥ E g ∥ \geq ∥ g ∥

, where

g \in F \cap \partial Ω_{2}

. Therefore, A has at least one fixed point in

F \cap (\bar{Ω_{2}} Ω_{1})

according to Theorem 4. We call this fixed point

g_{0}

, which satisfies

a_{1} < ∥ g_{0} ∥ < 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 N (q_{0} + 1; q - 1)

be fixed with

\begin{matrix} M γ \sum_{r = q_{0} + 2}^{q} G (x_{1}, r) > 1 . \end{matrix}

(17)

Considering (H4), there exists an

a > 0

such that

a < p

and

Z (x, g) \geq M g

for all

0 \leq g \leq a

and

x \in N (q_{0}; q)

. Define the set

\begin{matrix} Ω_{a} : = \{g \in F : ∥ g ∥ < a\} . \end{matrix}

By making use of (17), we obtain

\begin{matrix} ∥ E g ∥ \geq | E g (x_{1}) | = \sum_{r = q_{0} + 2}^{q} G (x_{1}, r) Z (r, g (r)) \geq M \sum_{r = q_{0} + 2}^{q} G (x_{1}, r) | g (r) | \\ \geq M γ ∥ g ∥ \sum_{r = q_{0} + 2}^{q} G (x_{1}, r) > ∥ g ∥ . \end{matrix}

This leads to

∥ E g ∥ > ∥ g ∥

, whenever

g \in F \cap \partial Ω_{a}

. Then, for the same

M > 0

, there is a number

R_{1} > 0

with

Z (x, g) \geq M g

, for each

g \geq R_{1}

and

x \in N (q_{0}; q)

. Let us choose R such that

\begin{matrix} R > max \{p, \frac{R_{1}}{γ}\} . \end{matrix}

Now, we define

\begin{matrix} Ω_{R} : = \{g \in F : ∥ g ∥ < R\} . \end{matrix}

It is clear that

∥ E g ∥ > ∥ g ∥

, when

g \in F \cap \partial Ω_{R}

. In the final step, we define

\begin{matrix} Ω_{p} : = \{g \in F : ∥ g ∥ < p\} . \end{matrix}

This implies that

∥ E g ∥ \leq ∥ g ∥

g \in F \cap \partial Ω_{p}

Thus, we have found two fixed points

g_{0}

and

g_{1}

for

E

with

g_{1} \in Ω_{p} ∖ \overset{˚}{Ω_{a}}

and

g_{2} \in Ω_{R} ∖ {\overset{˚}{Ω}}_{p}

, where

\overset{˚}{Ω}

refers to the interior of

Ω

. In particular, we can say that

g_{0}

and

g_{1}

are two positive solutions of (11) that satisfy

0 < ∥ g_{1} ∥, p < ∥ g_{2} ∥

. 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) \geq M + ϵ g

according to (H5), for

g \in F

and

x \in N (q_{0}; q)

. Then, we have

\begin{matrix} | (E g) | = \sum_{r = q_{0} + 2}^{q} G (x, r) Z (r, g (r)) \leq \sum_{r = q_{0} + 2}^{q} G (r - 1, r) [M + ϵ g (r)] . \end{matrix}

Since

ϵ > 0

is arbitrary, we see that

\begin{matrix} | (E g) | \leq M \sum_{r = q_{0} + 2}^{q} G (r - 1, r) = \frac{M}{η} . \end{matrix}

Taking

R > p

to be sufficiently large, we have

\begin{matrix} R > \frac{M}{η} . \end{matrix}

Let us define

\begin{matrix} Ω_{R} : = \{g \in F : ∥ g ∥ < R\} . \end{matrix}

It follows that

∥ E g ∥ < R = ∥ g ∥

, whenever

g \in F \cap \partial Ω_{R}

. Again, by considering (H5), we have

a > 0

such that

a < p

and

Z (x, g) < η g

, for

0 \leq g \leq a, g \in F

, and

x \in N (q_{0}; q)

. Now, we define

\begin{matrix} Ω_{a} : = \{g \in F : ∥ g ∥ < a\} . \end{matrix}

Then, we see from this that

\begin{matrix} ∥ (E g) ∥ = max_{x \in N (q_{0}; q)} \sum_{r = q_{0} + 2}^{q} G (x, r) Z (r, g (r)) \leq \sum_{r = q_{0} + 2}^{q} G (r - 1, r) Z (r, g (r)) \\ \leq η \sum_{r = q_{0} + 2}^{q} G (r - 1, r) | g (r) | \leq ∥ g ∥ . \end{matrix}

This implies that

∥ E g ∥ < ∥ g ∥

, when

g \in F \cap \partial Ω_{a}

. Lastly, we define

\begin{matrix} Ω_{p} : = \{g \in F : ∥ g ∥ < p\} . \end{matrix}

It can be observed that

∥ E g ∥ > ∥ g ∥

, whereas

g \in F \cap \partial Ω_{p}

Thus, we have determined

g_{1}

and

g_{2}

, two fixed points of

E

with

g_{1} \in Ω_{p} ∖ \overset{˚}{Ω_{a}}

and

g_{2} \in Ω_{R} ∖ {\overset{˚}{Ω}}_{p}

. Specifically, we can say that

g_{1}

and

g_{2}

are two positive solutions of (11) that satisfy

0 < ∥ g_{1} ∥

and

p < ∥ g_{2} ∥

. 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{matrix} (H g) (x) : = w (x) + \sum_{r = q_{0} + 2}^{q} Z (r, g (r)) G (x, r), x \in N (q_{0}; q) . \end{matrix}

(18)

It follows from (10) and (18) that g is a fixed point of

H

IFF it is a solution of (2).

Theorem 9

(see [36], (Brouwer theorem)). Let

R (n)

be the set of n-tuples of real numbers,

\emptyset \neq F \subset R (n)

be a compact convex set, and

H : F \to F

be a continuous function. Then,

H

has a fixed point in K.

Theorem 10

(see [36], (Leray–Schauder theorem)). Let

Ω \subset R (n)

be an open set with

0 \in Ω

and

H : \bar{Ω} \to R (n)

be a completely continuous function. Note that every

H

has at least one of the following properties:

There is $g \in \bar{Ω}$ such that $H g = g$ .
There are $v \in \partial Ω$ and $ξ \in (0, 1)$ such that $v = ξ H v$ .

Then, g is a fixed point of

H

in Ω.

Theorem 11

(see [36], (Krasnoselskii–Zabreiko theorem)). Let

H : R (n) \to R (n)

be a completely continuous function and

ℓ : R (n) \to R (n)

. If ℓ is a bounded linear function such that 1 is not its eigenvalue and

\begin{matrix} lim_{∥ g ∥ \to \infty} (\frac{∥ H g - ℓ g ∥}{∥ g ∥}) = 0, \end{matrix}

then there exist a fixed point of

H

R (n)

Now, we know that

R (q - q_{0} + 1)

is a BS with the following norm:

\begin{matrix} ∥ g ∥ : = max_{x \in N (q_{0}; q)} | g (x) | . \end{matrix}

Theorem 12.

Let

Z (x, g)

be a continuous function with respect to g for all

x \in N (q_{0}; q)

. If there exist

ℓ, M > 0

with

\begin{matrix} max \{| C_{1} |, | C_{2} |\} \leq ℓ, \end{matrix}

(19)

\begin{matrix} M = max_{(x, g) \in N (q_{0}; q) \times [- 3 ℓ, 3 ℓ]} | Z (x + θ - 1, g (x + θ - 1)) |, \end{matrix}

(20)

and

\begin{matrix} δ \leq \frac{ℓ}{M}, \end{matrix}

(21)

then, FBVP (2) has a solution.

Proof.

Let us define

F

\begin{matrix} F : = \{g : N (q_{0}; q) \to R and ∥ g ∥ \leq 3 ℓ\} . \end{matrix}

We know that

\emptyset \neq F \subset R (q - q_{0} + 1)

is a compact convex set. Now, we claim that

\begin{matrix} H : F \to F . \end{matrix}

To do this, we suppose that

g \in F

and

x \in N (q_{0}; q)

. By considering

\begin{matrix} | H g (x) | & = |w (x) + \sum_{r = q_{0} + 2}^{q} G (x, r) Z (r, g (r))| \\ \leq |w (x) | + \sum_{r = q_{0} + 2}^{q} G (x, r) | Z (r, g (r))| \\ \leq 2 max {| C_{1} |, | C_{2} |} + M \sum_{r = q_{0} + 2}^{q} G (x, r) \\ \leq 2 ℓ + M γ \leq 3 ℓ, \end{matrix}

we obtain

H g \in F

. Therefore,

H : F \to F

as claimed. Moreover,

H

is trivially continuous on K because it is a summation operator on a discrete finite set. Hence, we can say that

H

has a fixed point according to Theorem 9. This concludes that FBVP (2) has a solution, namely

g_{0}

, such that

∥ g_{0} ∥ \leq 3 ℓ

. Hence, the proof is complete. □

Theorem 13.

Let

Z (x, g)

be as in the previous theorem, and it is bounded on

N (q_{0}; q) \times R

. Then, there is a solution for FBVP (2).

Proof.

Let us take

\begin{matrix} P > sup_{(x, g) \in N (q_{0}; q) \times R} | Z (x, g) | . \end{matrix}

Choose ℓ to be as large as

\begin{matrix} max {| C_{1} |, | C_{2} |} \leq ℓ and δ \leq \frac{ℓ}{M}, \end{matrix}

where M is the same as defined in Theorem 12 and

M \leq P

so that

\begin{matrix} δ \leq \frac{ℓ}{M} . \end{matrix}

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

ψ : R^{+} \to R^{+}

is non-decreasing,

σ : N (q_{0}; q) \to R

, and

\begin{matrix} | Z (x, g) | \leq σ (x) ψ (| g |), (x, g) \in N (q_{0}; q) \times R . \end{matrix}

(22)

Furthermore, if there exists

γ > 0

with

\begin{matrix} \frac{γ}{max {| C_{1} |, | C_{2} |} + δ ∥ σ ∥ ψ (γ)} > 1, \end{matrix}

(23)

then FBVP (2) has a solution.

Proof.

First, we define

\begin{matrix} Ω : = {g : N (q_{0}) \to R and ∥ g ∥ < γ} . \end{matrix}

It is obvious that

Ω \subset R (q - q_{0} + 1)

is an open set including 0 and

H : \bar{Ω} \to R (q - q_{0} + 1)

. So,

H

is trivially completely continuous on

\bar{Ω}

since

H

is as in Theorem 12. On the contrary, we suppose that there are

v \in Ω

and

ξ \in (0, 1)

with

\begin{matrix} v = ξ H v . \end{matrix}

(24)

Using the definition of

H

and Theorem 2(b) in (24), we can deduce

\begin{matrix} | v (x) | & \leq | w (x) | + \sum_{r = q_{0} + 2}^{q} G (x, r) | Z (r, v (r)) | \\ \leq 2 max {| C_{1} |, | C_{2} |} + \sum_{r = q_{0} + 2}^{q} G (x, r) σ (r) ψ (| v (r) |) \\ \leq 2 max {| C_{1} |, | C_{2} |} + ∥ σ ∥ ψ (∥ v ∥) \sum_{r = q_{0} + 2}^{q} G (x, r) \\ \leq 2 max {| C_{1} |, | C_{2} |} + δ ∥ σ ∥ ψ (γ) . \end{matrix}

This leads to

\begin{matrix} ∥ v ∥ \leq 2 max {| C_{1} |, | C_{2} |} + δ ∥ σ ∥ ψ (γ) . \end{matrix}

Therefore,

\begin{matrix} \frac{γ}{2 max {| C_{1} |, | C_{2} |} + δ ∥ σ ∥ ψ (γ)} \leq 1 . \end{matrix}

This contradicts (23). As a consequence, by considering Theorem 10, we see that

H

has a fixed point in

Ω

. This tells us that FBVP (2) has a solution, namely

g_{1}

, with

∥ g_{1} ∥ < γ

. Thus, our proof is complete. □

Theorem 15.

Let Z be as in Theorem 12. If there exists a continuous function

ϕ : N (q_{0}; q) \to R

with

\begin{matrix} lim_{∥ g ∥ \to \infty} \frac{Z (x, g)}{g} = ϕ (x), x \in N (q_{0}; q), \end{matrix}

(25)

and

\begin{matrix} ∥ ϕ ∥ < \frac{1}{δ}, \end{matrix}

(26)

then FBVP (2) has a solution.

Proof.

It is easy to see that

H : R (q - q_{0} + 1) \to R (q - q_{0} + 1)

. So,

H

is trivially completely continuous on

R (q - q_{0} + 1)

, which is as in Theorem 12. Let us consider a linear bounded mapping ℓ, which is

ℓ : R (q - q_{0} + 1) \to R (q - q_{0} + 1)

defined by

\begin{matrix} (ℓ g) (x) : = \sum_{r = q_{0} + 2}^{q} G (x, r) ϕ (r) g (r), x \in N (q_{0}; q) . \end{matrix}

(27)

Clearly,

∥ ℓ g ∥ < ∥ g ∥

. Then, let

g \in R (n)

and

x \in N (q_{0}; q)

, and we consider

\begin{matrix} (ℓ g) (x) & \leq \sum_{r = q_{0} + 2}^{q} G (x, r) | ϕ (r) | | g (r) | \\ \leq ∥ ϕ ∥ ∥ g ∥ \sum_{r = q_{0} + 2}^{q} G (x, r) \leq δ ∥ ϕ ∥ ∥ g ∥ < ∥ g ∥ . \end{matrix}

This implies that

∥ ℓ g ∥ < ∥ g ∥

. 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 N (q_{0}; q)

\begin{matrix} | Z (x, g (x)) - ϕ (x) g (x) | < ϵ where ∥ g ∥ > N . \end{matrix}

(28)

Next, for each

x \in N (q_{0}; q)

, we have

\begin{matrix} | (H g) (x) - (ℓ g) (x) | & \leq | w (x) | + \sum_{r = q_{0} + 2}^{q} G (x, r) | Z (r, g (r)) - ϕ (r) g (r) | \\ \leq 2 max {| C_{1} |, | C_{2} |} + ϵ \sum_{r = q_{0} + 2}^{q} G (x, r) \\ \leq 2 max {| C_{1} |, | C_{2} |} + δ ϵ . \end{matrix}

This leads to

\begin{matrix} \frac{∥ H g - ℓ g ∥}{∥ g ∥} < \frac{max {| C_{1} |, | C_{2} |} + δ ϵ}{N} . \end{matrix}

As a consequence, we obtain

\begin{matrix} lim_{∥ g ∥ \to \infty} \frac{∥ H g - ℓ g ∥}{∥ g ∥} = 0 . \end{matrix}

Thus,

H

has a fixed point in

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 R (n)

be closed and

H : S \to S

be a contraction function; i.e.,

\exists ξ \in [0, 1)

with

\begin{matrix} ∥ H g - H v ∥ \leq ξ ∥ g - v ∥, \end{matrix}

for each

g, v \in S

. Then, w is a unique fixed point of

H

in S.

Theorem 17.

Z (x, g)

satisfies the Lipschitz condition with respect to g, i.e.,

\begin{matrix} ∥ Z (x, g) - Z (x, v) ∥ \leq K ∥ g - v ∥, \end{matrix}

(29)

for each

(x, g), (x, v) \in N (q_{0}; q) \times R

, where K is the Lipschitz constant and if

\begin{matrix} 0 < K δ < 1, \end{matrix}

(30)

then there is a unique solution for (2).

Proof.

Let

g, v \in R (q - q_{0} + 1)

and

x \in N (q_{0}; q)

. Consider

\begin{matrix} |H g (x) - H v (x)| & \leq \sum_{r = q_{0} + 2}^{q} G (x, r) |Z (r, g (r)) - Z (r, v (r))| \\ \leq K \sum_{r = q_{0} + 2}^{q} G (x, r) | g (r) - v (r) | \\ \leq K ∥ g - v ∥ \sum_{r = q_{0} + 2}^{q} G (x, r) \\ \leq K δ ∥ g - v ∥, \end{matrix}

implying that

\begin{matrix} ∥ H g - H v ∥ \leq K δ ∥ g - v ∥, \end{matrix}

Therefore,

H

is a contraction on

R (q - q_{0} + 1)

according to (30). Thus,

H

has a unique fixed point in

R (q - q_{0} + 1)

by Theorem 17. This implies that (2) has a unique solution. This ends the proof. □

Theorem 18.

Let

η, β > 0

with

\begin{matrix} ∥ Z (x, g) - Z (x, v) ∥ \leq β ∥ g - v ∥, \end{matrix}

(31)

for each

(x, g), (x, v) \in N (q_{0}; q) \times [- η, η]

. We can choose

\begin{matrix} m_{1} = max_{x \in N (q_{0}; q)} | Z (x, 0) |, \end{matrix}

(32)

and

\begin{matrix} m_{2} = max_{(x, g) \in N (q_{0}; q) \times [- η, η]} | Z (x, g) | . \end{matrix}

(33)

Moreover, if

\begin{matrix} 0 < β δ < 1, \end{matrix}

(34)

and

\begin{matrix} η \geq \frac{m_{1} δ + 2 max {| C_{1} |, | C_{2} |}}{1 - β δ} ⟹ 2 max {| C_{1} |, | C_{2} |} + m_{2} δ \leq η, \end{matrix}

(35)

then (2) has a unique solution.

Proof.

It can be observed that

H

is a contraction on

R (q - q_{0} + 1)

according to (34). Let us define

\begin{matrix} D = \{g : N (q_{0}; q) \to R and ∥ g ∥ \leq η\} . \end{matrix}

It is clear that

D \subseteq R (q - q_{0} + 1)

. Now, let us claim that

H : D \to D

. To prove this, let

g \in D

and

x \in N (q_{0}; q)

. We assume that (35) holds, and then we consider

\begin{matrix} | H (0) (x) | & = | w (x) + \sum_{r = q_{0} + 2}^{q} G (x, r) Z (r, 0) | \\ \leq | w (x) | + \sum_{r = q_{0} + 2}^{q} G (x, r) | Z (r, 0) | \\ \leq 2 max {| C_{1} |, | C_{2} |} + m_{1} \sum_{r = q_{0} + 2}^{q} G (x, r) \\ \leq 2 max {| C_{1} |, | C_{2} |} + m_{1} δ, \end{matrix}

which leads to

\begin{matrix} ∥ H (0) ∥ \leq 2 max {| C_{1} |, | C_{2} |} + m_{1} δ . \end{matrix}

Therefore,

\begin{matrix} ∥ H (g) ∥ & = ∥ H (g) + H (0) - H (0) ∥ \\ \leq ∥ H (g) - H (0) ∥ + ∥ H (0) ∥ \\ \leq β δ ∥ g - 0 ∥ + 2 max {| C_{1} |, | C_{2} |} + m_{1} δ \\ \leq β δ η + 2 max {| C_{1} |, | C_{2} |} + m_{1} δ \leq η . \end{matrix}

This implies that

H : D \to D

On the other hand, let us assume that (35) holds. Let

g \in D

and

x \in N (q_{0}; q)

. Then, we consider

\begin{matrix} | (H g) (x) | & = | w (x) + \sum_{r = q_{0} + 2}^{q} G (x, r) Z (r, g (r)) | \\ \leq | w (x) | + \sum_{r = q_{0} + 2}^{q} G (x, r) | Z (r, g (r)) | \\ \leq 2 max {| C_{1} |, | C_{2} |} + m_{2} \sum_{r = q_{0} + 2}^{q} G (x, r) \\ \leq 2 max {| C_{1} |, | C_{2} |} + m_{2} δ \leq η, \end{matrix}

which leads to

\begin{matrix} ∥ H g ∥ \leq η, \end{matrix}

which gives

H : D \to D

. As a consequence,

H

has a unique fixed point in

R (q - q_{0} + 1)

according to Theorem 16. Finally, we see that (2) has a unique solution, namely

g_{2}

, and it satisfies

∥ g_{2} ∥ \leq η

. 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{matrix} \begin{matrix} - ({{}_{0}^{RL}Δ}^{\frac{3}{2}} g) (x) = {(g^{2} + {(x + \frac{1}{2})}^{2} + 9)}^{- 1}, x \in N (2; 10), \\ g (0) = 1, g (10) = 2 . \end{matrix} \end{matrix}

(36)

We can observe that

\begin{matrix} q_{0} = 0, q = 10, θ = 1.5, Z (x + θ - 1, g (x + θ - 1)) = {(g^{2} + {(x + \frac{1}{2})}^{2} + 9)}^{- 1}, \\ C_{1} = 1, C_{2} = 2, ℓ \geq 2 . \end{matrix}

Moreover,

\begin{matrix} M = max_{(x, g) \in N_{0}^{10} \times [- 3 ℓ, 3 ℓ]} | Z (x + θ - 1, g (x + θ - 1)) | = \frac{1}{9}, \end{matrix}

and

\begin{matrix} δ = \frac{9}{(1.5) Γ (2.5)} {(12 + 0.5 - 1)}^{\bar{0.5}} = 15.4738, \end{matrix}

These give that

δ \leq \frac{ℓ}{M}

. Therefore, FBVP (36) has at least one solution (say

g_{0}

), which satisfies

| g_{0} (x) \leq 3 ℓ |

for

x \in N (0; 10)

, according to Theorem 12.

Example 2.

Here, we consider the FBVP

\begin{matrix} \begin{matrix} - ({{}_{ρ (0)}^{RL}Δ}^{\frac{3}{2}} g) (x) = x + \frac{1}{20} sin (g (x + 0.5)), x \in N (2; 10), \\ g (0) = 1, g (10) = 2 . \end{matrix} \end{matrix}

(37)

It is known that

q_{0} = 0, q = 10, θ = 1.5, Z (x + θ - 1, g (x + θ - 1)) = x + (0.05) sin g, C_{1} = 1,

and

C_{2} = 2

. It is easy to see that Z satisfies the Lipschitz condition with respect to g on

N (2; 10) \times R

; it has the Lipschitz constant

K = 0.05

. Moreover,

\begin{matrix} δ = \frac{9}{(1.5) Γ (2.5)} {(12)}^{\bar{0.5}} = 15.4738, \end{matrix}

which implies that

0 < K δ < 1

. Thus, in considering Theorem 17, FBVP (37) has a unique solution on

N (2; 10) \times 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.

Author Contributions

Conceptualization, A.A.L.; Methodology, M.A.Y.; Software, M.A.; Investigation, R.P.A.; Resources, M.A.Y.; Data curation, E.A.-S.; Writing—original draft, P.O.M.; Writing—review & editing, M.A.; Visualization, P.O.M.; Supervision, E.A.-S.; Project administration, R.P.A.; Funding acquisition, A.A.L. All authors have read and agreed to the published version of the manuscript.

Funding

The publication of this research was supported by the University of Oradea, Romania.

Data Availability Statement

Data are contained within the article.

Acknowledgments

Supporting Project number (RSP2024R136), King Saud University, Riyadh, Saudi Arabia.

Conflicts of Interest

The authors declare no conflicts of interest.

References

Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific: Singapore, 2000. [Google Scholar]
Agarwal, R.P. Difference Equations and Inequalities: Theory, Methods, and Applications; Chapman and Hall/CRC Pure and Applied Mathematics Book: Boca Raton, FL, USA, 2000; Volume 228. [Google Scholar]
Atici, F.; Sengul, S. Modeling with discrete fractional equations. J. Math. Anal. Appl. 2010, 369, 1–9. [Google Scholar] [CrossRef]
Silem, A.; Wu, H.; Zhang, D.-J. Discrete rogue waves and blow-up from solitons of a nonisospectral semi-discrete nonlinear Schrödinger equation. Appl. Math. Lett. 2021, 116, 107049. [Google Scholar] [CrossRef]
Cabada, A.; Dimitrov, N. Nontrivial solutions of non-autonomous Dirichlet fractional discrete problems. Fract. Calc. Appl. Anal. 2020, 23, 980–995. [Google Scholar] [CrossRef]
Wu, G.; Baleanu, D. Discrete chaos in fractional delayed logistic maps. Nonlinear Dyn. 2015, 80, 1697–1703. [Google Scholar] [CrossRef]
Gholami, Y.; Ghanbari, K. Coupled systems of fractional ∇-difference boundary value problems. Differ. Eq. Appl. 2016, 8, 459–470. [Google Scholar] [CrossRef]
Atici, F.M.; Eloe, P.W. Initial Value Problems in Discrete Fractional Calculus. Proc. Am. Math. Soc. 2009, 137, 981–989. [Google Scholar] [CrossRef]
Baleanu, D.; Wu, G.C.; Bai, Y.R.; Chen, F.L. Stability analysis of Caputo-like discrete fractional systems. Commun. Nonlinear. Sci. Numer. Simul. 2017, 48, 520–530. [Google Scholar] [CrossRef]
Mozyrska, D.; Torres, D.F.M.; Wyrwas, M. Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales. Nonlinear Anal. Hybrid Syst. 2019, 32, 168–176. [Google Scholar] [CrossRef]
Abdeljawad, T. On Riemann and Caputo fractional differences. Commput. Math. Appl. 2011, 62, 1602–1611. [Google Scholar] [CrossRef]
Atici, F.M.; Eloe, P.W. A transformmethod in discrete fractional calculus. Int. J. Differ. Equ. 2007, 2, 165–176. [Google Scholar]
Abdeljawad, T. Different type kernel h-fractional differences and their fractional h-sums. Chaos Solitons Fract. 2018, 116, 146–156. [Google Scholar] [CrossRef]
Mohammed, P.O.; Abdeljawad, T. Discrete generalized fractional operators defined using h-discrete Mittag-Leffler kernels and applications to AB fractional difference systems. Math. Meth. Appl. Sci. 2020, 46, 7688–7713. [Google Scholar] [CrossRef]
Goodrich, C.S. On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 2012, 385, 111–124. [Google Scholar] [CrossRef]
Wang, Z.; Shiri, B.; Baleanu, D. Discrete fractional watermark technique. Front. Inf. Technol. Electron. Eng. 2020, 21, 880–883. [Google Scholar] [CrossRef]
Ahrendt, K.; Castle, L.; Holm, M.; Yochman, K. Laplace transforms for the nabla-difference operator and a fractional variation of parameters formula. Commun. Appl. Anal. 2012, 16, 317–347. [Google Scholar]
Wang, M.; Jia, B.; Chen, C.; Zhu, X.; Du, F. Discrete fractional Bihari inequality and uniqueness theorem of solutions of nabla fractional difference equations with non-Lipschitz nonlinearities. Appl. Math. Comput. 2020, 367, 125118. [Google Scholar] [CrossRef]
Almusawa, M.Y.; Mohammed, P.O. Approximation of sequential fractional systems of Liouville-Caputo type by discrete delta difference operators. Chaos Soliton. Fract. 2023, 176, 114098. [Google Scholar] [CrossRef]
Baleanu, D.; Mohammed, P.O.; Srivastava, H.M.; Al-Sarairah, E.; Hamed, Y.S. On convexity analysis for discrete delta Riemann-Liouville fractional differences analytically and numerically. J. Inequal. Appl. 2023, 2023, 4. [Google Scholar] [CrossRef]
Chen, C.R.; Bohner, M.; Jia, B.G. Ulam-Hyers stability of Caputo fractional difference equations. Math. Meth. Appl. Sci. 2019, 42, 7461–7470. [Google Scholar] [CrossRef]
Brackins, A. Boundary Value Problems of Nabla Fractional Difference Equations. Ph.D. Thesis, The University of Nebraska-Lincoln, Lincoln, NE, USA, 2014. [Google Scholar]
Abdo, M.S.; Abdeljawad, T.; Ali, S.M.; Shah, K. On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions. Adv. Differ. Equ. 2021, 37, 2021. [Google Scholar] [CrossRef]
Ma, K.; Li, X.; Sun, S. Boundary value problems of fractional q-difference equations on the half-line. Bound. Value Probl. 2019, 46, 2019. [Google Scholar] [CrossRef]
Li, X.; Han, Z.; Sun, S. Existence of positive solutions of nonlinear fractional q-difference equation with parameter. Adv. Differ. Equ. 2013, 260, 2013. [Google Scholar] [CrossRef]
Chen, C.; Bohner, M.; Jia, B. Existence and uniqueness of solutions for nonlinear Caputo fractional difference equations. Turk. J. Math. 2020, 44, 857–869. [Google Scholar] [CrossRef]
Bekkouche, M.M.; Mansouri, I.; Ahmed, A.A.A. Numerical solution of fractional boundary value problem with caputo-fabrizio and its fractional integral. J. Appl. Math. Comput. 2022, 68, 4305–4316. [Google Scholar] [CrossRef] [PubMed]
Goodrich, C.S.; Jonnalagadda, J.M. Monotonicity results for CFC nabla fractional differences with negative lower bound. Analysis 2021, 41, 221–229. [Google Scholar] [CrossRef]
Wang, X.; Wang, G.; Chen, Z.; Lim, C.W.; Li, S.; Li, C. Controllable flexural wave in laminated metabeam with embedded multiple resonators. JSV 2024, 581, 118386. [Google Scholar] [CrossRef]
Li, C.; Zhu, C.; Lim, C.W.; Li, S. Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading. Appl. Math. Mech.-Engl. Ed. 2022, 43, 1821–1840. [Google Scholar] [CrossRef]
Guo, L.M.; Cai, J.W.; Xie, Z.Y.; Li, C. Mechanical Responses of Symmetric Straight and Curved Composite Microbeams. J. Vib. Eng. Technol. 2024, 12, 1537–1549. [Google Scholar] [CrossRef]
Chu, C.; Liu, H. Existence of positive solutions for a quasilinear Schrödinger equation. Nonlinear Anal. Real World Appl. 2018, 44, 118–127. [Google Scholar] [CrossRef]
Liu, H.; Zhao, L. Ground-state solution of a nonlinear fractional Schrödinger-Poisson system. Math. Meth. Appl. Sci. 2022, 45, 1934–1958. [Google Scholar] [CrossRef]
Mohammed, P.O.; Agarwal, R.P.; Baleanu, D.; Sabir, P.O.; Yousif, M.A.; Abdelwahed, M. Uniqueness Results Based on Delta Fractional Operators for Certain Boundary Value Problems. Fractals 2024, accepted. [Google Scholar]
Guirao, J.L.G.; Mohammed, P.O.; Srivastava, H.M.; Baleanu, D.; Abualrub, M.S. A relationships between the discrete Riemann-Liouville and Liouville-Caputo fractional differences and their associated convexity results. AIMS Math. 2022, 7, 18127–18141. [Google Scholar] [CrossRef]
Agarwal, R.P.; Meehan, M.; O’Regan, D. Fixed Point Theory and Applications; Cambridge Tracts in Mathematics, 141; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
Pietsch, A. History of Banach Spaces and Linear Operators; Birkhauser: Basel, Switzerland; Springer: Berlin/Heidelberg, Germany, 2007; pp. 49–50. [Google Scholar]
Mohammed, P.O.; Srivastava, H.M.; Muhammad, R.S.; Al-Sarairah, E.; Chorfi, N.; Baleanu, D. On existence of certain delta fractional difference models. J. King Saud. Univ. Sci. 2024, 36, 103224. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Mohammed, P.O.; Agarwal, R.P.; Yousif, M.A.; Al-Sarairah, E.; Lupas, A.A.; Abdelwahed, M. Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting. Mathematics 2024, 12, 2864. https://doi.org/10.3390/math12182864

AMA Style

Mohammed PO, Agarwal RP, Yousif MA, Al-Sarairah E, Lupas AA, Abdelwahed M. Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting. Mathematics. 2024; 12(18):2864. https://doi.org/10.3390/math12182864

Chicago/Turabian Style

Mohammed, Pshtiwan Othman, Ravi P. Agarwal, Majeed A. Yousif, Eman Al-Sarairah, Alina Alb Lupas, and Mohamed Abdelwahed. 2024. "Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting" Mathematics 12, no. 18: 2864. https://doi.org/10.3390/math12182864

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting

Abstract

1. Introduction

2. Review of Results

3. Positive Solution Results

4. Existence Results

5. Uniqueness Results

6. Numerical Examples

7. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI