The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation

: An elastic beam equation (EBEq) described by a fourth-order fractional difference equation is proposed in this work with three-point boundary conditions involving the Riemann–Liouville fractional difference operator. New sufﬁcient conditions ensuring the solutions’ existence and uniqueness of the proposed problem are established. The ﬁndings are obtained by employing properties of discrete fractional equations, Banach contraction, and Brouwer ﬁxed-point theorems. Further, we discuss our problem’s results concerning H yers– U lam ( HU ), generalized H yers– U lam ( GHU ), H yers– U lam– R assias ( HUR ), and generalized H yers– U lam– R assias ( GHUR ) stability. Speciﬁc examples with graphs and numerical experiment are presented to demonstrate the effectiveness of our results.

Fractional calculus (FC) is a generalized form of classical integer-order calculus. Fractional calculus examines the properties of fractional-order derivatives and integrals. Due to its numerous applications in various scientific fields, this research area has gained considerable attention over the past few years. FC can be applicable in several fields of science and engineering, along with aerodynamics, electrical circuits, fluid dynamics, heat conduction, and physics. We refer to the comprehensive works in [7][8][9][10] for a detailed analysis of its applications, and we refer to [11][12][13][14][15] for the latest trends in the area of FC.
Researchers have explored various aspects of fractional difference equations (FDEs). Obviously, the solutions' existence, uniqueness, and stability analysis are some important features of FDEs. Various analytical approaches and fixed-point theory have been used to examine the solutions' existence and stability for FDEs. Several researchers have contributed a number of books and papers in this regard [16]. However, finding the exact solution of nonlinear FDEs is often too difficult; therefore, the stability analysis of solutions plays a crucial role in such investigations. Various kinds of stabilities described in the past are discussed in the literature, such as Lyapunov stability [17], Mittag-Leffler stability [18], and exponential stability [19]. Presumably, the most dependable stabilities are called HU stability. The discussed stability was modified to GHU stability (refer to [20][21][22]). In 1970, Rassias further generalized the aforesaid stability. For FDEs with different BCs concerning Riemann-Liouville and Caputo operators, the addressed fields of existence and stability analysis are well-equipped (see [23][24][25][26][27][28]).
The above findings inspired us in this study concerning the solutions' existence and uniqueness with various types of Ulam stability results for the proposed discrete fractional elastic beam equation (FEBE) that is subject to the three-point BCs as follows: where β ∈ (3, 4] is a fractional order and ζ ∈ N β+n+2 β−1 is constant. Here, we have that is the Riemann-Liouville discrete fractional operator, and n ∈ N 0 . The rest of this research work is structured as follows. Basic background knowledge on DFC is stated in Section 2. The result for a linear version of the BVP Equation (2) is discussed in Section 3. Further, by using this solution, the existence and uniqueness conditions for the proposed discrete FEBE with three-point BCs (Equation (2)) are derived with the help of contraction mapping and the Brouwer fixed-point theorems. Different types of stability results are extensively obtained in Section 4 via the findings of nonlinear analysis. Some illustrative examples with graphs and numerical experiment are presented in Section 5 as applications to provide a better understanding of our findings. Finally, Section 6 concludes our research work.

Essential Preliminaries
Some important notions and preliminary lemmas are stated in this section, which are needed for discussion of our results. Definition 1 ([30]). For β > 0, the βth order fractional sum of G can be defined as for κ ∈ N a+β and σ(i . 30]). Assume that κ and β are any numbers such that κ (β) and κ (β−1) are defined. Then we have ∆κ (β) = βκ (β−1) .

EB Existence and Uniqueness
The existence and uniqueness of EB is established in this section to the three-point BCs for the proposed discrete FEBE Equation (2). We now introduce the following theorem that deals with a linear variant solution of our proposed BVP Equation (2). → R be given. Then, the linear discrete FEBE with three-point BCs: has the unique solution, for κ ∈ N β+n+3 β−4 , where Proof. By applying the fractional sum ∆ −β of order β ∈ (3, 4] along with Lemma 2 to Equation (3), we have for κ ∈ N β+n+3 β−4 and some constants C j ∈ R, where j = 1, 2, 3, 4. By applying the first BC w(β − 4) = 0 in Equation (6), we obtain By using Definition 1, we obtain Equations (7) and (8) imply C 4 = 0. Using C 4 in Equation (6) provides Using Lemma 1 and taking the operator ∆ on both sides of Equation (9), we obtain From the third BC ∆w(β + n) = 0 in Equation (10), we obtain The operator ∆ is applied on both sides of Equation (10) with the aid of Lemma 1, and we obtain The second BC of Equation (3) implies Again, using Lemma 1 and taking the operator ∆ on both sides of Equation (12), we obtain Using the last BC ∆ 3 w(β + n) + w(ζ) = 0 in Equations (9) and (14) yields and From Equations (15) and (16), and by employing the last BC Equation (3), we obtain Solving Equations (11) and (17), we obtain Now, a constant C 3 is found by solving Equations (13) and (18) as follows: Substituting C 3 into Equation (13), we have By using the value of C 2 and C 3 in Equation (17), we arrive at By using the constants C j for j = 1, 2, 3 in Equation (9), we obtain w(κ) in the form β−4 . Therefore, the theorem's proof is complete.
Assume that B * : C N β+n+3 β−4 , R is a Banach space with a norm defined by To discuss the theorems' existence and uniqueness, we need the following assumptions: Theorem 2. In view of assumption (A 1 ), the discrete FEBE with the three-point BCs in Equation (2) has a unique solution if where such that K is defined in Theorem 1 Proof. Let the operator A : B * → B * be defined as where g w (κ) = G(κ + β − 1, w(κ + β − 1)). Obviously, the fixed point of A is a solution to Equation (2). To show that A is a contraction, let w,ŵ ∈ B * and for each κ ∈ N β+n+3 β−4 , one has where g w (κ), gŵ(κ) ∈ C N β+n+3 β−4 , R satisfies the following functional equations: From which we obtain By the application of Lemma 3, we have and Similarly, by using Lemma 3, we also obtain and By substituting the relations Equations (25)- (28) into Equation (24), we obtain By Equation (19), we obtain Aw − Aŵ < w −ŵ . Hence, A is a contraction. As a result, according to the Banach fixed-point theorem, the three-point BCs for the discrete FEBE Equation (2) has a unique solution.
Theorem 3. If the assumption (A 2 ) holds, then the discrete FEBE with three-point BCs in Equation (2) has at least one solution provided that where Proof. Assume that D > 0 and consider the set V = {w ∈ B * : w ≤ D}. For proving this theorem, let us claim that A maps V in V. Now, for any w ∈ V, one has where g w (κ) is given in Equation (22). Using (A 2 ), we obtain This further implies that Using the relations of Equations (25)- (28) in Equation (30), we obtain By Equation (29), we have Aw ≤ D, which implies that A : V → V. By using the Brouwer fixed-point theorem, let us conclude that three-point BCs for discrete FEBE Equation (2) has at least one solution.

Lemma 5.
According to Remark 1, a functionŵ ∈ B * that corresponds to the discrete FEBE with three-point BCs is expressed as: satisfying the following inequality: where (Aŵ)(κ) is defined in Equation (21).
Proof. By using Theorem 1, the corresponding BVP Equation (35) becomeŝ Using an operator A and taking the modulus on both sides of the above solution along with (A 3 ), we obtain (19), the discrete FEBE Equation (2) is HU stable. (31), and w(κ) is a unique solution to Equation (2), then

Proof. Ifŵ(κ) is any solution of the inequality Equation
By using Lemma 5 in Equation (36), we have This further implies that where . Hence, the solution of Equation (2) is HU stable.
Hence, the solution of Equation (2) is GHU stable.
For our next result, the following hypotheses hold:
Hence, the solution of Equation (2) is GHU R stable.

Applications
Some illustrative examples are provided in this section to demonstrate the applicability of our results in this research work. Example 1. Suppose that β = 3.7, n = 2, and H(κ) = κ (13) with different values of ζ. Then, a linear discrete FEBE with the three-point BCs of Equation (3) becomes We shall apply Theorem 1 to find a solution w(κ) of Equation (37) that can be expressed as: where κ ∈ N 8.7 −0.3 , E 1 (κ) and E 2 (κ) are defined in Theorem 1. With the help of both Definition 1 and Lemma 4, we obtain the expression on right-hand side of Equation (38) as follows: Similarly, we find On one hand, by choosing different values of ζ = 2.7, 3.7, 4.7, 5.7 in Equation (43), we obtain different solutions for this problem, as seen in Figure 1a. On the other hand, Figure 1b shows three-dimensional solution surface plots for various values κ and ζ. In addition, a numerical experiment for our obtained solutions in Example 1 with step size 1 is presented in Table 1.
Furthermore, it is obviously GHU R stable from Remark 3.

Conclusions
Three-point BCs for a discrete FEBE have been investigated in this research work. For our proposed problem involving a Riemann-Liouville discrete fractional operator, some important conditions for the existence and stability theory have been developed. The required findings have been obtained with the help of fixed-point techniques such as the contraction mapping principle and Brouwer fixed-point theorem. Moreover, some new results for various types of Ulam stability of the proposed three-point BCs for a discrete FEBE have been established with the aid of nonlinear analysis. Some suitable examples have been provided and accompanied with numerical experiment for our obtained solutions for various fractional-order values in a graphical representation in order to study the effectiveness and applicability of our theoretical results. All in all, our results are new and interesting for the elastic beam problem arising from mathematical models of engineering and applied science applications.