Existence and stability of a q -Caputo fractional jerk differential equation having anti-periodic boundary conditions

In this work, we analyze a q -fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under speciﬁc conditions. To prove the uniqueness of the solution, we employ a Banach ﬁxed point theorem and a mathematical tool for establishing the presence of distinct ﬁxed points. To demonstrate the availability of a solution, we utilize Leray–Schauder’s alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce diﬀerent kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study


Introduction
Recently, a lot of researchers have shown a great interest in the field of q-calculus (QC) and problems involving fractional q-differential equations (q-DEs).The roots of QC can be traced back to 1908 with the work of Jackson in [1].Additionally, q-DEs were developed to characterize the variety of physical processes that emerged, such as discrete stochastic processes, discrete dynamical systems, quantum dynamics, and so on [2].As the theory of QC progressed, some associated ideas have been presented and examined, including q-integral transform theory, q-Mittag-Leffler functions, q-gamma, q-beta functions, q-Laplace transform, and so forth (for more details, see [3][4][5][6][7][8][9]).These concepts find applications in understanding and solving problems related to QC.The reader may refer to [10][11][12][13][14][15][16][17] for more details on QC.
In 1978, Schot [18] introduced the concept of "jerk" J , which is essentially the rate at which acceleration changes.It involves the third derivative of quantity represented by u.The idea of J has proven in several scientific fields, including acoustics, electrical circuits, mechanics, and dynamical processes.It also helps us to understand how acceleration is changing over time, providing valuable insights into the behavior of systems in various applications [19][20][21][22][23][24][25].In three dimensions, a dynamic system can be represented as υ (χ) = a, a (χ) = e, e (χ) = f (υ, a, e), and can be well written in the form of υ = f (υ, υ , υ ).The JE is third order autonomous DE that has found applications in various scientific fields, such as signal processing, secure communication, electrical engineering, control systems, bio-mechanics, and economic systems [17,22].Marcelo and Silva [26] employed the algebraic techniques in 2020 to ascertain the exact structure for a polynomial J function, hence guaranteeing the nonchaotic behavior of the subsequent JE: They also provided the proof for nonchaotic behavior.It can also be useful to investigate the different kinds of ordinary DEs and their nonchaotic behavior.The authors in [27] addressed an initial value problem of nonlinear 3rd order JE: By employing analytical methodologies, the authors were able to enhance the method known as the global error minimization method GEMM to generate estimations using analytical techniques.Their developed approaches were known to be more successful and efficient than previously known current methods when compared to known solutions and accurate numerical ones.The authors in [28] utilized the modified harmonic balance technique for the subsequent nonlinear JE: under conditions υ(0) = 0, D 1 υ(0) = B, and D 2 υ(0) = 0. Sousa et al., by employing fixed point approach, studied stability of the modified impulsive fractional DEs where H D α,β,ψ 0 + (•) is the ψ-Hilfer fractional derivative with α ∈ (0, 1], β ∈ [0, 1], and are prefixed numbers, ξ ∈ C( × R) and τ i ∈ C([t i , s i ] × R) for all i = 1, 2, . . ., m, which are noninstantaneous impulses, here := [0, T] with T > 0 [29].Wang et al. in [30] studied the various forms of Ulam stability (U S ) and existence, uniqueness (EU) for the following nonlinear implicit fractional integro-differential equations involving Caputo derivative (C D) of fractional order: where ν, ζ > 0, 1 < α ≤ 2, 0 ≤ β ≤ 2 and continuous functions are represented as ξ , g : × R × R → R. The authors introduced the ψ-Hilfer pseudo-fractional operator, motivated by the ψ-Hilfer fractional derivative and the theory of pseudo-analysis, and investigated a new class of important and essential results for pseudo-fractional calculus in a semi-ring ([a, b], ⊕, ), and some particular cases were discussed (for more instances, see related research works [31][32][33][34][35][36][37]).Houas et al., by using Riemann-Liouville (RL) and q-fractional C D, examined the EU, Ulam-Hyers (U H ), and Ulam-Hyers-Rassias (U H R) stability of the solution to q-fractional problem (FJP) as follows: , μ ∈ {ω, θ } are the qfractional RL and C Ds respectively [38].The q -FI is I β q having RL type and ξ : × R 3 → R is given an appropriate function [38].
Influenced by the aforementioned works, we present the following q-Caputo fractional JDEs with anti-periodic boundary conditions (ABCs): where 0 < {α, ω, θ } ≤ 1, β ∈ (0, 1], 0 < δ < T, q-fractional C D is C D μ q , μ ∈ {α, ω, θ , β} of order μ on , ξ , g : × R 3 → R are appropriate functions and ν, ζ > 0. We list the important points of this manuscript: 1: We implement Caputo q-fractional JDE having ABCs for the first time in the literature.2: In this manuscript, we established the EU and U S results for the suggested Problem (1).3: Different from previous papers that used nonlinear implicit fractional integrodifferential equations in [30] and RL and q-fractional C D [38], we get better results by employing q-fractional JDE having ABCs.4: We also show the graphical representation of JDE having ABCs.This research article is organized in the following manner: Sect. 2 clarifies some basic ideas in QC and provides related lemmas.In Sect.3, we establish the EU of solution for the proposed system (1) by employing the Leray-Schauder alternative and the Banach fixed point theorem.Various types of U S have been discussed in Sect. 4. In Sect. 5 an example is also presented at the end to verify our results.Finally, conclusion is also provided in Sect.6.

Definition 2.2 ([42])
The q-derivative of a function υ : T → R is expressed by and D q υ(0) = lim χ→0 D q υ(χ).Also the higher q-derivative of function υ is defined by

Definition 2.4 ([43]
) The operator C D μ q is the fractional q-C D of order μ given by and where μ is the smallest integer greater than μ.
We also point out formulas in [14], which will be used in our results. 1).
Lemma 2.8 (Leray-Schauder alternative [44]) Let ρ : F → F be a completely continuous operator (i.e., a map restricted to any bounded set in F is compact).Let Then the set (ρ) is unbounded, or ρ has at least one fixed point.
Lemma 2.9 (Banach fixed point theorem [45]) Let F be a Banach space and mapping ρ : F → F be a contraction on F .Hence ρ has a unique fixed point.
We now examine the U S for the q-FJP (1), as discussed in [46].For x > 0 and h : → R + , we get and for χ ∈ , where Definition 2.10 ( [46]) The q-FJP (1) demonstrates the stability as: 1: In U H sense, if there is a positive real number E * ω,θ > 0 such that there is a solution b of the q-FJP (1) for each x > 0 and for each solution υ of inequality (8) having , if there is a real number E * ω,θ ,h > 0 such that for each x > 0 and for each solution υ of inequality (9) there ∃ a solution υ of q-FJP (1) with

Existence and uniqueness results
In this section, we investigate the 1EU of solution of problem (1).

Lemma 3.1 Consider φ ∈ C( ). Thus, the solution of problem
where φ ∈ F is given as Proof Now, let us consider Applying the operator I α q on both sides of ( 12) and employing Lemma 2.6 with n = 1, we obtain Now, using the operator I ω q , (1) of Lemma 2.5, (a) of Lemma 2.7, and applying the same procedure on both sides of (13), we get It follows that where c j ∈ R, (j = 0, 1, 2).Using boundary constraints Now, using the L.H.S of ( 16) in (15), we obtain Similarly, using the R.H.S of ( 16) in (15), we obtain Thus ( 16) becomes .
Putting all values in (15), we obtain and We define an operator ρ : F → F by applying Lemma 3.1 as follows: The following assumptions will be used in our upcoming results: (H 5 ) ∃ real constants ϕ m ≥ 0 (m = 1, 2, 3) and ϕ 0 > 0 in such a way that for any υ m ∈ R (m = 1, 2, 3) we have ) and ℘ 0 > 0 in such a way that for any (H 7 ) ∃ an increasing ϑ ∈ C( , R + ) and ϑ h > 0, then the following inequality is satisfied.In the following sections, we will employ the fixed point theory to confirm EU of solution of q-fractional J problem outlined in (1).For simplicity, the following notations will be used in our upcoming results: T ω q (ω + 1) Theorem 3.2 Suppose that assumptions (H 2 ), (H 3 ), and (H 4 ) hold.Thus, q-FJP (1) has a unique solution if where i , i = 1, 2, 3, are given by (21).
Using (H 3 ) and (H 4 ), we get * Then we get Now, using (23), we obtain T ω+θ q (ω + θ + 1) Also, we have T ω q (ω + 1) and From the definition of • F , we have which means that ρW ⊂ W .We now demonstrate that the ρ is an operator for a contraction mapping.Now υ, υ ∈ W and χ ∈ , we obtain By (H 3 ) and (H 4 ), we obtain Also, by using (H 3 ) and (H 4 ), we obtain T ω q (ω + 1) Thus, we get We observe that ρ is a contraction operator by using (22).We infer that ρ has a unique fixed point that is a solution of (1) as a result of Lemma 2.9.
By applying Lemma 2.8, we explore certain conditions where q-FJP (1) has at least one solution in Theorem 3.2.

Theorem 3.3 Assume that hypotheses (H 5 ) and (H 6 ) hold. If
is satisfied, then the proposed problem described by (1) has at least one solution within the domain .

Stability results
We study the U H and U H R stability [46] of q-FJP in this section.
If we put , we obtain υ -υ F ≤ E * ω,θ x.As a result, the q-FJP (1) is U H stable.

Examples and illustrative results
In this section, we check the correctness of the results by showing several examples.In the first example, we test q-Caputo fractional JDEs with ABCs (1) for the changes of q in the range of zero and one according to the proposed theorems.
The approach is similar to each group of curves in Figs.1a, 1b, and 1c, aligning with each other and reaching a stable value that precisely determines the correctness of the argument.By (22), we get 0.5513, q = 1 5 , 0.5435, q = 2 5 , 0.5373, q = 3 5 , Table 1 Numerical results for and i , i = 1, 2, 3 in Example 5.1 for three cases of q   The numerical values of relation (33) are shown in Table 2.It can be seen that after stabilizing the data of each column, these results are less than one (see Fig. 2).Therefore, the given q-FJP (30) is addressed in Theorem 3.2, asserting that it possesses a unique solution within the interval .Additionally, Theorem 4.1 states that the same q-FJP (30) is U H stable having 1.9020x, q = 1 5 , 1.6398x, q = 2 5 , 1.4460x, q = 3 5 , x > 0.
The next example shows the proven facts for changes in the order of the derivative α.
The data in Table 4 show the values of i , i = 1, 2, 3, for three different values of derivative order α.The approach is similar to each group of curves in Figs.4a, 4b, and 4c, aligning with each other and reaching a stable value that precisely determines the correctness of the argument.By (22), we get The numerical values of relation (36) are shown in Table 5.It can be seen that after stabilizing the data of each column, these results are less than one (see Fig. 5).Therefore, the given q-FJP (35) is addressed in Theorem 3.2, asserting that it possesses a unique solution within the interval .Additionally, Theorem 4.1 states that the same q-FJP (35) is U H stable having 1.7349x, α = 1 8 , 1.6786x, α = 1 6 , 1.4460x, α = 1  3 , x > 0.   Table 5 shows these results.In addition, the curves drawn in Figs.6a and 6b confirm the existence of ϑ h and Ineq.(37) variables.Therefore, condition (H 7 ) is fulfilled with h(χ) = χ ln(3) 5 and ϑ h = 0.097, 0.099, 0.107 whenever α = 1 5 , 2 5 , 3 5 , respectively.Theorem 4.2 indicates that the q-FJP is U H R (35)

Table 2
Numerical results for Eq.(33) in Example 5.1 for three cases of q

Table 3
(34)s these results.In addition, the curves drawn in Figs.3a and 3bconfirm the existence of ϑ h and Ineq.(34)variables.Therefore, condition (H 7 ) is fulfilled with

Table 3
Numerical results of ϑ h in I α+ω+θ

Table 4
Numerical results of i , i = 1, 2, 3, in Example 5.2 for three cases of derivative order α