On the generalized fractional snap boundary problems via G-Caputo operators: existence and stability analysis

This research is conducted for studying some qualitative specifications of solution to a generalized fractional structure of the standard snap boundary problem. We first rewrite the mathematical model of the extended fractional snap problem by means of the G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{G}$\end{document}-operators. After finding its equivalent solution as a form of the integral equation, we establish the existence criterion of this reformulated model with respect to some known fixed point techniques. Then we analyze its stability and further investigate the inclusion version of the problem with the help of some special contractions. We present numerical simulations for solutions of several examples regarding the fractional G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{G}$\end{document}-snap system in different structures including the Caputo, Caputo–Hadamard, and Katugampola operators of different orders.


Introduction
Fractional calculus is one of the most important branches of applied mathematics. The main importance of this field can be observed in many published papers regarding different fractional differential equations and inclusions in recent years. In this direction, different generalizations of derivatives have been introduced by some researchers. For example, recently, Lazreg et al. [1] investigated the Cauchy problem of Caputo-Fabrizio impulsive fractional differential equations , t ∈ I k , k = 0, 1, . . . , m, v(a + k ) = v(ak ) + k (v(ak )), k = 1, 2, . . . , m, v(0) = v 0 , [2] considered the class of terminal value problems of Katugampola implicit differential equations of noninteger orders where the function f : I × R 2 → R is continuous, and K D r 0 + is the Katugampola fractional derivative of order r ∈ (0, 1]. In 2020, Baitiche et al. [3] generalized the fractional settings and studied the existence of solutions of the following ψ-Caputo fractional differential equation: where C D q,ψ a + is the ψ-Caputo fractional derivative of order q ∈ (2, 3], w : J × R → R is a given continuous function, and λ i are real constants satisfying = m i=1 λ i (ψ(η i )-ψ(a)) 2 -(ψ(b)ψ(a)) 2 = 0. Also, Wahash et al. [4] investigated the existence and interval of existence, uniqueness, estimates of solutions, and different types of Ulam stability results of solutions on a subinterval of [0, b] for the nonlinear fractional differential equation involving generalized Caputo fractional derivatives with respect to the function ψ given by In 2019, Pham et al. [5] introduced a chaotic integer-order system, called a snap system, which involves only one quadratic nonlinear term and takes the following mathematical form: where T (v 1 , v 2 , v 3 , v 4 ) = -av 1 -v 2 -v 4 + bv 1 v 3 . Equation (1) can be transformed into a fourth-order differential equation The new equation (2) contains a fourth-order derivative of the variable v 1 , which in physics stands for a second derivative of acceleration in a mechanical system. Equation (2) is called a snap or jounce equation and describes a fourth-order dynamical model. Many researchers have investigated sufficient conditions for the uniqueness, existence, stability, and attractivity of solutions for a wide domain of fractional nonlinear ordinary differential equations (ODEs) or mathematical models containing different fractional derivatives by using numerous types of methods including standard fixed point theory, Tdegree theory, variational methods, monotone iterative approaches, MNC technique, and so on. For more detail, see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. However, to the best of our knowledge, limited results can be found on the existence and stability of solutions of fractional snap systems via the generalized G-Caputo derivative.
The authors in [24] studied the fractional snap model where a = 2, b = 1, and the Caputo fractional order q = 0.95. In view of the above facts, in this paper, we focus our attention on the problem of the existence and uniqueness along with the Hyers-Ulam stability of solutions for different forms of fractional nonlinear snap systems in the G-Caputo sense with initial conditions. Namely, we study the following problem: where c D η;G a + are the G-Caputo derivatives, η belong to {q, p, r, k} such that 0 < q, p, r, k ≤ 1, the increasing function G ∈ C 1 ([a, b]) is such that G (t) = 0, t ∈ [a, b], h ∈ C([a, b] × R 4 , R), and v 0 , v 1 , v 2 , v 3 ∈ R. It is obvious that this system can be rewritten as It is natural that if we set G(t) = t, a = 0, and q = p = r = k = 1, then we obtain the standard 4th-order ODE (2) with initial conditions. Our method in this paper is based on fixed point approaches. Also, we can find more ideas on fractional calculus and its applications in [3,[25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].
The summary of our work in this research is as follows. In Sect. 2, we recall several assembled concepts of fractional calculus, useful lemmas, and some theorems about the fixed points. In Sect. 3, we give the proof of the fundamental theorems of this paper by utilizing fixed point approaches such as Banach's principle and Schauder's theorem. In Sect. 4, we discuss the stability in the context of the Ulam-Hyers stability, its generalized version along with Ulam-Hyers-Rassias stability, and its generalized version for solutions of the fractional G-snap system (4). In Sect. 5, we utilize a special form of contractions to prove the existence results for an inclusion version of (4). Appropriate applications with numerical simulation are provided in Sect. 6 to illustrate and analyze the obtained results. Finally, in Sect. 7, we give the conclusion of our article.

Preliminaries
Here we recall some initial notions, definitions and notations.
Let G : [a, b] → R be increasing via G (t) = 0 for all t. We start this part by defining the G-Riemann-Liouville fractional (G-FRL) integrals and derivatives. In this section, we set Definition 2.1 ([42, 43]) For η > 0, the ηth G-FRL integral of an integrable function v : [a, b] → R with respect to G is given as follows: Let n ∈ N, and let G, v ∈ C n ([a, b], R) be such that G has the same properties mentioned above. The ηth G-FRL derivative of v is defined by where n = [η] + 1 [42,43]. The ηth G-fractional Caputo derivative of v is defined by where n = [η] + 1 for η / ∈ N and n = η for η ∈ N [44]. In other words, Extension (6) gives the Caputo derivative when G(t) = t. Also, in the case G(t) = ln t, it yields the Caputo-Hadamard derivative. If v ∈ C n ([a, b], R), then the ηth G-fractional Caputo derivative of v is specified as [44,Theorem 3] The composition rules for the above G-operators are recalled in the following lemma.
Theorem 2.4 (Banach contraction principle [46]) Let (V, ρ) be a nonempty complete metric space, and let : V → V be a contraction, that is, and for some μ ∈ (0, 1). Then admits a unique fixed point.
Theorem 2.5 (Leray-Schauder [46]) Let V be a Banach space, let be a bounded convex closed subset of V, and let U be an open set contained in with 0 ∈ U. Let : U → be a continuous and compact mapping. Then either (i) admits a fixed point belonging toŪ, or (ii) there exist v ∈ ∂U and μ ∈ (0, 1) such that v = μ (v).
Consider normed space (C, · ). The collection of all closed, bounded, compact and convex subsets of C are denoted by P CL (C), P BN (C), P CP (C), and P CV (C), respectively. Definition 2.6 ([47]) Consider v : R → R as a real-valued function and H as a multifunction. (i) H is u.s.c on C if H(v * ) ∈ P CL (C) for any v * ∈ C, and also there exists a neighbor- where ρ is the metric of M, and [47] . Suppose for H : C → P CL (C) and v 1 , v 2 ∈ M, we have the inequality Then H is said to be (H1) a Lipschitz map if L > 0 and (H2) a contraction if 0 < L < 1 [47].
for all |υ| ≤ and almost all t ∈ [a, b].

Theorem 2.13
Then H has a unique end-point iff H has the (AEP)-property.

Existence and uniqueness results
Here we analyze the existence properties of solutions and their uniqueness for the proposed fractional G-snap problem (4). We need the following lemma, which specifies the corresponding integral equation.
Proof Consider v(t) satisfying the linear fractional G-snap problem (3.1). Applying the kth G-integral operator I k;G a + to both sides of equation (8), by the 4th boundary condition we obtain Similarly, by the 3rd boundary condition, applying the r-th G-integral operator I r;G a + , we get By the 2nd boundary condition, applying the pth G-integral operator I p;G a + , we get and finally, applying the qth G-integral operator I q;G a + to both sides of (10), by the 1st boundary condition, we get v(t) = v 0 + v 1 (G(t) -G(a)) q (q + 1) + v 2 (G(t) -G(a)) q+p (q + p + 1) We see that v(t) fulfills (9), and the proof is complete.
At present, we aim to verify the existence of a unique solution of the fractional G-snap system (4) by relying on Theorem 2.4.
Then the fractional G-snap system (4) admits a unique solution on Proof To prove the desired result, we first let where h * 0 = sup t∈[a,b] |h(t, 0, 0, 0, 0)|, and := |v 0 | + |v 1 | 1 + (G(b) -G(a)) q (q + 1) To apply the Banach principle, we verify that : wherê admits a unique fixed point, which is the same solution of the fractional G-snap BVP (4). First, we show ⊂ , that is, maps into itself. For each v ∈ r , we have Also, and From (16), (17), (18), (19), and (13) which implies that v ≤ for v ∈ , and so ⊂ . Next, we investigate the con- and From (20), (21), (22), and (23) . This, together with Theorem 2.4, guarantees the existence of a unique fixed point for and accordingly the existence of a unique solution for the fractional G-snap BVP (4). The proof is complete. The next existence property for possible solutions of the fractional G-snap BVP (4) is checked based on the hypotheses of Theorem 2.5.

Theorem 3.3 Let h ∈ C([a, b] × R 4 , R) and assume that:
where * 0 = sup t∈[a,b] | (t)|, and O and are represented in (12) and (14). Then the fractional G-snap system (4) has at least one solution on [a, b]. (15) and the ball N = {v ∈ C([a, b], R) : v ≤ } for some > 0. The continuity of h yields that of the operator . Now by (C2) we have for v ∈ N . In a similar way, we get that and As a consequence, by (25), (26), (27), and (28) where O and are represented by (12) and (14). Hence is uniformly bounded on C([a, b], R). Now let us check the equicontinuity of . Choose arbitrary t, t * ∈ [a, b] with t < t * and v ∈ N . We have By letting Obviously, the right-hand side of (30) does not depend on v and approaches 0 as t * tends to t. In the same way, Again, the right-hand side of (31) goes to zero as t * → t independently of v. Finally, and which independent of v. The right-hand sides of (34) and (33) approach 0 as t * → t. Therefore relations (30), (31), (32), and (34) imply that Thus the equicontinuity of is confirmed. Hence is compact on N by the Arzelá-Ascoli theorem. Until now, we saw that the hypotheses of Theorem 2.5 are fulfilled for the operator . Thus one of two cases (i) or (ii) is valid. By (C3) we build Now we assume the existence of v ∈ ∂U and μ ∈ (0, 1) subject to v = μ v. For such a selection of v and μ, we may write by (34) that a contradiction. Therefore case (ii) does not hold, and by Theorem 2.5 admits a fixed point in U, which is regarded as a solution of the fractional G-snap system (4), and this concludes the proof.

Stability criterion
In this part, we review the stability criterion in the context of the Ulam-Hyers stability, its generalized version along with Ulam-Hyers-Rassias stability, and its generalized version for solutions of the fractional G-snap system (4).

Definition 4.3
The fractional G-snap BVP (4) is Ulam-Hyers-Rassias stable with respect to if there exists 0 < c * h, ∈ R such that for all > 0 and v * ∈ C([a, b], R) satisfying Here we discuss the Ulam-Hyers stability of the fractional G-snap BVP (4).
Let v ∈ C([a, b], R) be the unique solution of the fractional G-snap BVP (4). Then it is given by Also, and From (37), (38), (39), and (40) where O is defined in (12). As a consequence, it follows that If we let c * h = O 1-LO , then the Ulam-Hyers stability is fulfilled. Next, for h (0) = 0, the generalized Ulam-Hyers stability is fulfilled.

Inclusion version of (4)
Here we will derive the existence of solutions to the inclusion version of fractional nonlinear snap system of the G-Caputo sense with initial conditions (4), which takes the form where H ia a multifunction on the product space [a, b] × R 4 . The function v ∈ C := C([a, b], R) is called a solution of system (46) if it satisfies the boundary conditions and there is for all t ∈ [a, b]. For each v ∈ C, we define the set of selections of the operator H as and define the operator U : C → P(C) by where  ([a, b], [0, ∞)) and a nondecreasing function ψ ∈ such that for all t ∈ [a, b] and natural numbers n, then there exists a subsequence {v n j } of {v n } such that for all t ∈ [a, b] and j ≥ 1; (C9) There exist v 0 ∈ C and p ∈ U(v 0 ) such that for all t ∈ [a, b]. Then the inclusion problem (46) has at least one solution.
Proof Obviously, the fixed point of U : C → P(C) is a solution of BVP (46). Since the multivalued map t → H v (t) is closed-valued and measurable for all v ∈ C, H has measurable selection, and S H,v is nonempty. We have to prove that for all t ∈ [a, b]. Since H has compact values, we define a subsequence of {℘ n } (again by the same notation) that converges to ℘ ∈ L 1 ([0, 1]). Hence ℘ ∈ S H,v and v n (t) → v(t) = v 0 + v 1 (G(t) -G(a)) q (q + 1) + v 2 (G(t) -G(a)) q+p (q + p + 1) for all t ∈ [a, b], which gives that v ∈ U(v) and U is closed valued. As H is compact-valued, it is a simple task to affirm the boundedness of U(v) for arbitrary v ∈ C. We have to prove that U is an α-ψ-contraction. For such a goal, we define α(v,v) = 1 whenever for almost all t ∈ [a, b]. Thus there exists ϒ ∈ H v such that . Now let N * : [0, 1] → P(C) be a multivalued map defined as for all t ∈ [a, b]. Let us define * 2 ∈ U(t) by * 2 (t) = v 0 + v 1 (G(t) -G(a)) q (q + 1) + v 2 (G(t) -G(a)) q+p (q + p + 1) Also, and thus for all v,v ∈ C, which implies that U is an α-ψ-contraction. Now, let v ∈ C andv ∈ U(v) be two functions such that α(v,v) ≥ 1. In this case, From this it follows that α(v, ϒ) ≥ 1, which means that the operator U is an α-admissible. Now suppose that v 0 ∈ C andv ∈ U(v 0 ) are such that By hypothesis (C8) there is a subsequence {v n j } of {v n } such that Thus α(v n j , v) ≥ 1(∀j), that is, C has the property C α . Theorem 2.12 guarantees that N has a fixed point, which is the solution of the inclusion BVP (46). (C13) There is φ ∈ C([a, b], [0, ∞)) such that

Theorem 5.2 Consider a multifunction
Then the inclusion BVP (46) has a solution.
Proof We have to prove that U : C → P(C) includes end points. Firstly, we must prove that U(v) is closed for every v ∈ C. Since the mapping is closed-valued and measurable for v ∈ C, it has a measurable selection, and S * H,v = ∅. By applying the same deduction as in the proof of Theorem 5.1, we may simply verify that By the measurability of ℘ 1 and By the same argument as in Theorem 5.1 we get * Hence for all v, ϒ ∈ C. By using hypothesis (xv) we can easily find that U has the (AEP)-property. By Theorem 2.13 there exists v * ∈ C such that U(v * ) = {v * }. This implies that v * satisfies the given problem (46), and the proof is completed.

Numerical applications
Here we give some examples of fractional G-snap systems based on numerical simulations to analyze their solutions.
In these examples, we consider different cases of the function G to cover the Caputo, Caputo-Hadamard, and Katugampola versions. For numerical computations, one can use Algorithms 1, 2 and 3.
In the next example, we examine the correctness of the results caused by Theorem 3.3.

Conclusion
In this paper, we defined a new fractional mathematical model of a BVP consisting of the snap equation in the framework of the generalized sequential G-operators and turned to the investigation of the qualitative behaviors of its solutions including the existence, uniqueness, stability, and inclusion version. To obtain an existence criterion, we used the Leray-Schauder theorem, and to obtain a uniqueness criterion, we utilized the Banach theorem. We studied different kinds of stability criteria based on the standard definitions of these notions. With the help of some special contractions, we established some theorems regarding the inclusion structure of the G-snap problem. In the final step, we designed three examples, and considering different cases of the function G and order q, we obtained numerical results of these two suggested fractional G-snap systems in Caputo, Caputo-Hadamard, and Katugampola versions. Note that in this paper, by assuming G(t) = t and q = p = r = k = 1 we derived the standard 4th-order ODE of snap equation. Therefore we will be able to review other properties of this extended fractional G-snap BVP by designing new generalized models based on nonsingular operators in the future works.