Analysis of a Nonlinear ψ -Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions

Analysis of a Nonlinear ψ -Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions. Abstract: In this paper, we establish sufﬁcient conditions to approve the existence and uniqueness of solutions of a nonlinear implicit ψ -Hilfer fractional boundary value problem of the cantilever beam model with nonlinear boundary conditions. By using Banach’s ﬁxed point theorem, the uniqueness result is proved. Meanwhile, the existence result is obtained by applying the ﬁxed point theorem of Schaefer. Apart from this, we utilize the arguments related to the nonlinear functional analysis technique to analyze a variety of Ulam’s stability of the proposed problem. Finally, three numerical examples are presented to indicate the effectiveness of our results.


Introduction
During the last few decades, elastic beams (EB) have been prominent in the realm of physical science and engineering problems. In particular, the construction of buildings and bridges requires careful computations of the elastic beam equations (EBEs) to assure the safety of the structure. The equations of the EB problem have been created to represent real situations and their solutions have been provided by different mathematical techniques. EBEs have attracted the interest of many researchers who formulate EBEs in the form of fourth-order ordinary differential equations in various methods. For instance, in 1988, Gupta [1] discussed a fourth-order EBE with two-point boundary conditions as follows: (4) (t) + f (t, x(t)) = 0, t ∈ (0, 1), The problem (1) represents an elastic beam model of length 1 that is restrained at the left end with zero displacement and bending moment, and is free to travel at the right end with a diminishing angular attitude and shear force. Using the Leray-Schauder continuation theorem and Wirtinger-type inequalities, the existence properties of the problem (1) were established. In 2017, Cianciaruso and co-workers [2] studied the fourthorder differential equation of the cantilever beam (CB) model with three-point boundary conditions as follows: (4) (t) + f (t, x(t)) = 0, t ∈ (0, 1), where ξ ∈ (0, 1) is a real constant. They proved the existence, non-existence, localization, and multiplicity of nontrivial solutions for problem (2) with their results by using topological methods. Further, the development of EBEs with linear or nonlinear functions under a variety of boundary conditions is more varied and comprehensive. Many works in the literature deal with linear or nonlinear boundary value problems which consist of two or more points; for example, Zhong and co-workers [3] examined fourth-order nonlinear differential equations with the four-point boundary conditions: (4) (t) + f (t, x(t), x (t)) = 0, t ∈ (0, 1), where c i represents non-negative constants, i = 1, 2, 3, 4, the points ω 1 , ω 2 ∈ [0, 1] with ω 1 < ω 2 , and f ∈ C([0, 1] × [0, ∞) × (−∞, 0], [0, ∞)). By using Krasnoselskii's fixed point theorem, the existence result is obtained. EBEs with a variety of boundary conditions have been studied in recent years; see  and references cited therein.
Fractional calculus generalizes the ordinary differentiation and integration of arbitrary order, which may be non-integer order. It is widely utilized in several areas, including engineering and applied science. Different definitions of fractional derivative and integral operators, such as Riemann-Liouville, Caputo, Hilfer, Katugampola, and others, have been discovered. We refer to the thorough investigations in [26][27][28][29][30][31] for a detailed analysis of applications on fractional calculus. In recent years, several research papers have investigated fractional differential equations, the existence results of solutions, and analyzed system stability. One of the most fascinating aspects of differential equations is existence theory. In the previous several decades, a lot of research has been conducted in this field. Various techniques have been used in the current literature to demonstrate the existence and uniqueness of solutions to differential and integral equations. In addition, one of the most powerful techniques for stability analysis is Ulam's stability, which includes Ulam-Hyers (U H) stability, generalized Ulam-Hyers (GU H) stability, Ulam-Hyers-Rassias (U HR) stability, and generalized Ulam-Hyers-Rassias (GU HR) stability. It is useful because the properties of Ulam's stability guarantee the existence of solutions, and when the problem under consideration is Ulam's stability, it ensures that a close exact solution exists; see [32][33][34][35][36][37][38][39][40][41][42] and references cited therein.
The major goal of this paper is to use well-known fixed point theorems such as Banach's and Schaefer's to show the existence and uniqueness of the solution for the problem of (4). The different types of Ulam's stability, such as U H stability, GU H stability, U HR stability, and GU HR stability, are used to investigate the stability of the solution for problem (4). Finally, we illustrate examples of various functions that are investigated to verify the theoretical results.
The rest of this paper is assembled as follows: In Section 2, we introduce some notations, definitions, lemmas, and essential results. The existence and uniqueness results are obtained by helping the fixed point theorems in Section 3. By using the nonlinear functional analysis technique, we analyzed a variety of Ulam's stabilities for the proposed problem in Section 4. In Section 5, we present examples to guarantee the validity of the obtained results. The conclusion of this paper is presented at the end.

Preliminaries
We present a brief overview of the fundamental concepts of ψ-Hilfer fractional calculus as well as essential key results that will be employed in this paper.
Let E = C(J , R) be the Banach space of continuous functions on J equipped with the supnorm x = sup t∈J {|x(t)|}. Let AC n (J , R) be the space of n-times absolutely continuous functions where AC n (J Definition 1 (The ψ-Riemann-Liouville fractional integral operator [43]). Let (a, b) be a finite or infinite interval of the half-axis R + . Let ψ(x) ∈ C 1 (J , R) be an increasing function with ψ (x) = 0 for each t ∈ J . The ψ-Riemann-Liouville fractional integral of order α of a function f depending on the function ψ on J is defined by where Γ(·) represents the (Euler) Gamma function.
On the other hand, it is easy to show by a direct calculation that x(t), which is given by (6), verifies the linear ψ-Hilfer FCB model (5) under the nonlinear boundary conditions.

Existence and Uniqueness Results
For the sake of this paper, we set the notations F . , n. According to Lemma 4, we define T : E → E and Ψ γ (·) Clearly, the ψ-Hilfer fractional boundary value problem (FBVP) describing the CB model (4) has solutions if and only if T has fixed points. For the tightness of calculation in this manuscript, we set the constants

Uniqueness Result
In the first our criteria, we will analyze the uniqueness result of the solution for the ψ-Hilfer FBVP describing the CB model (4) by applying Banach's fixed point theorem (Lemma 5).
Lemma 5 (Banach's fixed point theorem [45]). Let S be a non-empty closed subset of a Banach space E . Then, any contraction mapping Q from E into itself has a unique fixed point.

Proof.
We transform the ψ-Hilfer FBVP describing the CB model (4) into x = T x, where T is defined by (12). Clearly, the fixed points of T are the possible solutions of the ψ-Hilfer FBVP describing the CB model (4). From Lemma 5, we will verify that T has a unique fixed point, which means that the ψ-Hilfer FBVP describing the CB model (4) has a unique solution.
The process of proof will be divided in two steps: Step I. T B r 1 ⊂ B r 1 . Let x ∈ B r 1 and t ∈ J . Then, By applying Proposition 1 (i), we have By applying (A 1 ), (A 2 ), and (19), we have From (20) with Proposition 1 (i), we can compute that Substituting (20)- (25) into (18), we have Thus, Step II. T : E → E is a contraction. Let x, y ∈ E and for any t ∈ J . Then, we have By applying (A 1 ) and (19), we have Hence, by inserting (27) in (26) and using Proposition 1 (i) with (A 2 ), we obtain In view of (16), we find that T is a contraction. Therefore, in accordance with Lemma 5, the ψ-Hilfer FBVP describing the CB model (4) has a unique solution x ∈ E .

Existence Result
The second result is proved by applying Schaefer's fixed point theorem (Lemma 6). Lemma 6. (Schaefer's fixed point theorem [45].) Let E be a Banach space and T : E → E is a completely continuous operator and the set B = {x ∈ E : Then, T has a fixed point in E .
Then, the ψ-Hilfer FBVP describing CB model (4) has at least one solution on J .
Proof. The process will be analyzed in four steps as follows.
Step I. T is continuous. Let x n be a sequence such that x n → x ∈ E . Then, for any t ∈ J , we have The continuity of f implies the continuity of F x . Then, F x n − F x → 0, x n − x → 0, as n → ∞, Hence, T is continuous.
Step II. T maps bounded set into bounded set in E . For For any t ∈ J and x ∈ B r 2 , we obtain Inserting (29)- (31) in (28), we can compute that which implies that Then, T maps bounded set into bounded set in E .
Step III. T maps bounded sets into equicontinuous sets of E . For a ≤ t 1 < t 2 ≤ b and x ∈ B r 2 where B r 2 as defined in Step II, by using the property that f is a bounded on the compact set J × B r 2 , we estimate Note that the right hand-side of the above inequality is independent of the unknown variable x and tends to zero as t 2 → t 1 . Hence, T is equicontinuous. Then, T is relatively compact on B r 2 . We apply the Arzelá-Ascoli theorem, which implies that T is completely continuous.

It follows from
Step II, and for any t ∈ J , that T x ≤ N < ∞. Then, B is a bounded set. Using Theorem 2, we find that there exists N > 0 such that x ≤ N < ∞. Thanks to Lemma 6, T has at least one fixed point, which is the corresponding solution of the ψ-Hilfer FBVP describing the CB model (4).

Ulam's Stability Results
In this section, we analyze the U H stability, GU H stability, U HR stability, and GU HR stability of the solution to the ψ-Hilfer FBVP describing the CB model (4).
there exists a solution x ∈ E of the ψ-Hilfer FBVP describing the CB model (4) such that Definition 5. The ψ-Hilfer FBVP describing the CB model (4) is said to be GU H-stable if there exists a function K ∈ C(R + , R + ) with K(0) = 0 such that, for each solution z ∈ E of there exists a solution x ∈ E of the ψ-Hilfer FBVP describing the CB model (4) such that Definition 6. The ψ-Hilfer FBVP describing the CB model (4) is said to be U HR-stable with respect to K ∈ C(J , R + ) if there exists a positive real number C f ,K > 0 such that for each > 0 and for each a solution z ∈ E of (34) there exists a solution x ∈ E of the ψ-Hilfer FBCP describing the CB model (4) such that Definition 7. The ψ-Hilfer FBVP describing the CB model (4) is said to be GU HR-stable with respect to K ∈ C(J , R + ) if there exists a positive real number there exists a solution x ∈ E of the ψ-HilferFBVP describing CB model (4) such that Remark 1. It is easy to see that (a 1 ) Definition 4 ⇒ Definition 5; (a 2 ) Definition 6 ⇒ Definition 7; (a 3 ) Definition 6 for K(t) = 1 ⇒ Definition 4.

Remark 2.
A function z ∈ E is a solution of (32) if and only if there exists a function v ∈ E (where v depends on solution z) such that:

Remark 3.
A function z ∈ E is a solution of (34) if and only if there exists a function w ∈ E (where w depends on solution z) such that: (i) |w(t)| ≤ K(t), ∀t ∈ J ; (ii) H D α,ρ;ψ

Remark 4.
There exists an increasing function K ∈ C(J , R + ) and there exists a positive constant λ K > 0, such that, for each t ∈ J , we have the integral inequality

The U H and GU H Stability Results
Firstly, we present an important lemma that will be used in the analyses of U H and GU H stability of the ψ-Hilfer FBVP describing the CB model (4).
Proof. Let z be the solution of (32). Thanks to Remark 2 (ii) and Lemma 4, we obtain Then, the solution of (42) can be written as Lemma 7 is obtained. Now, we prove the U H and GU H stability of solution to the ψ-Hilfer FBVP describing the CB model (4).
Theorem 3. Let f : J × R 3 → R be continuous, and let (A 1 )-(A 2 ) be verified with Then, the ψ-Hilfer FBVP describing the CB model (4) is U H and GU H-stable in E .

The U HR and GU HR Stability Results
This lemma will be used in the proofs of U HR and GU HR stability of our results.
Proof. Let z be a solution of (34). Thanks to Remark 3 (ii) and Lemma 4, the solution of can be written in the form: By using Remark 3 (i) with Remark 4, we obtain the following estimation: Lemma 8 is obtained.
Next, we establish the U HR and GU HR stability of the solution to the ψ-Hilfer FBVP describing the CB model (4).
Theorem 4. Let f : J × R 3 → R be a continuous under (A 1 )-(A 2 ) and let (39) be fulfilled. If then the ψ-Hilfer FBVP describing CB model (4) is U HR and GU HR-stable in E .

Proof.
Let z ∈ E be the solution of (34) and x be a unique solution of (4). Thanks to Lemma 8, we obtain where χ x (t) is given by (43).

Conclusions
We analyzed the existence and uniqueness of solutions for a class of a nonlinear implicit ψ-Hilfer fractional integro-differential equation subjected to nonlinear boundary conditions describing the CB model. The uniqueness result is established using Banach's fixed point theorem, while the existence result is established using Schaefer's fixed point theorem, both of which are well-known fixed point theorems. Ulam's stability is also demonstrated in several ways, including U H stability, GU H stability, U HR stability, and GU HR stability. Finally, the numerical examples have been carefully selected to demonstrate how the results can be used. Moreover, the ψ-Hilfer FBVP describing the CB model (4) not only includes the identified previously works about a variety of boundary value problems. As special cases for various values ρ and ψ, the considered problem does cover a large range of many problems as: the Riemann-Liouville-type problem for ρ = 0 and ψ(t) = t, the Caputo-type problem for ρ = 1 and ψ(t) = t, the ψ-Riemann-Liouvilletype problem for ρ = 0, the ψ-Caputo-type problem for ρ = 1, the Hilfer-type problem for ψ(t) = t, the Hilfer-Hadamard-type problem for ψ(t) = log(t), and the Katugampola-type problem for ψ(t) = t q .
As a result, the fixed point technique is a powerful tool to investigate different nonlinear problems, which is very important in various qualitative theories. The present work is innovative and attractive and significantly contributes to the body of knowledge on ψ-Hilfer fractional differential equations and inclusions for researchers. In addition, our results are novel and intriguing for the elastic beam problem emerging from mathematical models of engineering and applied science.