Ψ -Bielecki-type norm inequalities for a generalized Sturm–Liouville–Langevin differential equation involving Ψ -Caputo fractional derivative

The present research work investigates some new results for a fractional generalized Sturm–Liouville–Langevin (FGSLL) equation involving the Ψ -Caputo fractional derivative with a modiﬁed argument. We prove the uniqueness of the solution using the Banach contraction principle endowed with a norm of the Ψ -Bielecki-type. Meanwhile, the ﬁxed-point theorems of the Leray–Schauder and Krasnoselskii type associated with the Ψ -Bielecki-type norm are used to derive the existence properties by removing some strong conditions. We use the generalized Gronwall-type inequality to discuss Ulam–Hyers (UH), generalized Ulam–Hyers (GUH), Ulam–Hyers–Rassias (UHR), and generalized Ulam–Hyers–Rassias (GUHR) stability of these solutions. Lastly, three examples are provided to show the eﬀectiveness of our main results for diﬀerent cases of (FGSLL)-problem such as Caputo-type Sturm–Liouville, Caputo-type Langevin, Caputo–Erdélyi–Kober-type Langevin problems.

To show the novelty and generality of our BVP, we note that: 1.If χ(z) = 0, for each z ∈ I, the (FGSLL)-problem (1) reduces to the standard form of the fractional Sturm-Liouville (FSL) problem for a nonlinear FDE, which is as follows: and the considered (FSL)-problem (2) contains some problems involving different fractional derivative operators, for various choices of the function Ψ .Among these are interesting extensions: • If Ψ (x) = x, then the (FSL)-problem (2) reduces to the Caputo-type Sturm-Liouville (CSL) problem.• If Ψ (x) = x ν , then the (FSL)-problem (2) becomes the Caputo-Erdélyi-Kober-type Sturm-Liouville (CEKSL) problem.• If Ψ (x) = ln(x), then the (FSL)-problem (2) represents the Caputo-Hadamard-type Sturm-Liouville (CHSL) problem.2. By choosing η(z) ≡ 1, χ(z) ≡ λ (λ ∈ R), for z ∈ I, the (FGSLL)-problem (1) reduces to the standard form of the fractional Langevin (FL) problem for a nonlinear FDE, which is as follows: and the considered (FL)-equation ( 3) contains some problems involving many classical fractional derivative operators, for various choices of a function Ψ .Among these are interesting extensions: • If Ψ (x) = x, then the (FL)-problem (3) reduces to the Caputo-type Langevin (CL) problem.• If Ψ (x) = x ν , then the (FL)-problem (3) represents the Caputo-Erdélyi-Kober-type Langevin (CEKL) problem.• If Ψ (x) = ln(x), then the (FL)-problem (3) becomes the Caputo-Hadamard-type Langevin (CHL) problem.Now, to organize the paper in a standard form for the readers, we arrange it as follows.In Sect.2, we propose some definitions and lemmas that will be used to establish our theorems.In Sect.3, we investigate the existence and uniqueness of the solution for the main (FGSLL)-problem (1) under some Ψ -Bielecki-type norm inequalities, and Sect. 4 presents the study of some stability results for the solutions of the (FGSLL)-problem (1), such as Ulam-Hyers, Ulam-Hyers-Rassias, and their generalizations, with the help of the generalized Gronwall inequality.Our main tools in this study are three fixed point theorems: the Banach contraction principle, Leray-Schauder, and Krasnoselskii theorems under some norm inequalities of the Ψ -Bielecki type.After that we give, in Sect.5, three examples to illustrate our theoretical results.Finally, we complete the paper by a conclusion with some perspectives.

Essential concepts and basic tools
Some concepts are recalled in this section, and also some lemmas are proved.Definition 2.1 ([7]) Let μ > 0, n ∈ N, I = [a, b] with -∞ ≤ a < b ≤ ∞, ϕ : I → R be an integrable function, and Ψ ∈ C 1 (I, R) increasing with Ψ (z) = 0 for any z ∈ I.The Ψ -Riemann-Liouville (R-L) fractional integral of order μ for ϕ that depends on Ψ is given as Definition 2.2 ([7]) Consider an interval I ⊂ R and let μ ∈ (n -1, n).Let also ϕ : I → R be an integrable function and Ψ be as given in Definition 2.1.Then, the Ψ -R-L fractional derivative of the order μ of the function ϕ with respect to Ψ is given as where n = [μ] + 1 and [μ] indicates the integer part of μ.
be functions so that Ψ is increasing and Ψ (z) = 0 for any z ∈ I.The left-sided Ψ -Caputo fractional derivative of order μ for ϕ is defined by where n = [μ] + 1 for μ / ∈ N and n = μ for μ ∈ N.
To simplify the notation, we put ϕ Then, from the definition we can write The properties given below hold: where Now, we define the norms Proof From (4), we derive the following inequality:

Lemma 2.6 ([7])
The Ψ -Caputo derivatives of the fractional order are bounded and, for any μ > 0, we have Remark 2.7 From equality (5), we can easily obtain which allows us to conclude that C D μ,Ψ a+ ϕ(a) = 0.
Lemma 2.8 Let μ, θ > 0. We have and Proof By applying the Ψ -R-L fractional operator I μ 1 ,Ψ 0 + to the function z → e θ(Ψ (z)-Ψ (0)) together with the replacement of variables y = Ψ (z) -Ψ (s) and z = θ y, we have For the proof of the inequality (7), we again use the same replacement of variables y = Ψ (z 2 ) -Ψ (s) and z = θ y, and we obtain The proof is now complete.
if and only if it fulfills the integral equation given below: where Proof By applying the Ψ -R-L fractional operators I α 1 ,Ψ 0 + and I α 2 ,Ψ 0 + on both sides of equation (8) and utilizing Lemma 2.4, we obtain two real numbers c 0 and c 1 such that where c 0 and c 1 belong to R.
From the boundary condition ( 9), together with Lemma 2.5, it follows that c 1 = 0, and by using the second boundary condition (10), as well as taking into account the assumption after some computations we obtain Replacing c 0 with its value in (12), we get For the reverse case, taking the Ψ -Caputo operator C D α 2 ,Ψ 0 + on both sides of equation ( 13) and applying again the operator C D α 1 ,Ψ 0 + after multiplying the obtained equation by η, and finally by exploiting Lemma 2.4, we find To examine the boundary conditions, it is trivial to verify them using (13).As a result, u is a solution to the problem (1), and the proof of Lemma 2.9 is now finished.Now, we pay attention to the space C = C(I, R) equipped with the well-known Ψ -Bielecki-type norm u θ,α proposed by previous works (see [40]) defined by where E α indicates the Mittag-Leffler function of one-parameter that is given as , α > 0.
If we take α → 1 in the above norm u θ,α , we obtain and (C, u θ ) is a Banach space.We now focus on the key findings of our study.

Main results
For a good and straightforward continuation of our work, we propose the hypotheses as given below: (H2) For some positive real constant L f , we have (H5) A positive real constant M exists such that Furthermore, to analyze the stability of UHR and GUHR, we adopt the assumption as given below: (H6) A nondecreasing function ϒ ∈ C([0, R], R + ) and a real constant γ ϒ,α 1 +α 2 > 0 exist such that for any z ∈ [0, R], we have In light of Lemma 2.9, we can define the following operator: where Now, we express the operator N as a sum of two operators N 1 and N 2 as follows: To facilitate the reading of the work, we utilize the following notations: and, for more convenience, we put and

Uniqueness of solution by using Banach contraction principle
To prove the results, we first provide the Banach contraction principle as a reminder.
Lemma 3.1 ([40]) Let (U, d) be a complete metric space, and T : U → U a contraction.Then there is a unique fixed point of T in U.
Proof First, we choose r 1 such that Briefly, our aim is to show that N B r 1 ⊆ B r 1 , where is a nonempty, closed, and convex subset of the Banach space C.
For each z ∈ [0, R] and u ∈ B r 1 , we get By using the property ||κ| -| || ≤ |κ + | and taking into consideration Thus, consequently, we get Since J < 1, hence N is a contraction mapping.Consequently, by the Banach contraction principle 3.1, we conclude that N has a unique fixed point in B r 1 .Hence, the (FGSLL)problem ( 1) has a unique solution on [0, R].
Now, we would like to prove Theorem 3.2 using the Ψ -Bielecki-type norm inequalities.
Here, the strong condition J < 1 is removed.Proof Let us choose where θ , ∇ θ , and M f ,θ are three constants defined previously.
closed, and convex subset of the Banach space C.
For each z ∈ [0, R] and u ∈ B r 2 ,θ , we have Using the estimate ||κ| -| || ≤ |κ + | and taking into account By exploiting (6), we get This means that Simple computations give us By using (6), we get Consequently, Hence, we obtain By choosing θ > 0 large enough such that we conclude that the mapping N is a contraction relative to the Ψ -Bielecki norm.Exploiting the Banach fixed point Theorem 3.1, it follows that N has a unique fixed point which is a unique solution to the (FGSLL)-problem (1).

Application of Krasnoselskii's fixed point theorem for existence results
First, we recall Arzelà-Ascoli and Krasnoselskii theorems and then give our main results.  . P is compact and continuous, Then there exists ∈ M so that = P + Q .Now, we present the following existence theorem which is proved using the above lemmas.

Lemma 3.5 ([40]) A family of functions in C(
Theorem 3.7 Suppose that (H1) and (H3) hold.The (FGSLL)-problem (1) has at least one solution defined on [0, R] under the following condition: Proof We fix r 3 ≥ we get and yields Similarly, if v ∈ B r 3 , then This implies that Inserting ( 24) and ( 25) into (23), we get which implies that Thus assumption 1 of Lemma 3.6 is verified.
Claim 2: We show that N 2 is contraction.
For each u 1 , u 2 ∈ B r 3 and z ∈ [0, R], we have which yields Hence, by (22), N 2 is a contraction.Claim 3: Assumption 3 in Lemma 3.6 holds.Take a sequence {u n } n∈N with u n → u ∈ C as n → ∞.For z ∈ [0, R], we get The Lebesgue's dominated convergence theorem and continuity of f lead to the conclusion that N 1 u n -N 1 u → 0 as n → ∞.Therefore, N 1 is continuous.Furthermore, N 1 is uniformly bounded on B r 3 as N 1 u ≤ K f due to (24).Also, N 1 is equicontinuous.Indeed, letting u ∈ B r 3 , for z 1 , z 2 ∈ [0, R], z 1 < z 2 , we have [ Finally, we get The right-hand side of ( 27) is clearly independent of u and |N 1 u(z 2 ) -N 1 u(z 1 )| → 0 as Hence, this implies that N 1 B r 3 is equicontinuous and N 1 maps bounded subsets into relatively compact subsets, which implies that N 1 B r 3 is relatively compact.
Therefore, using Lemma 3.5, we determine that N 1 is compact in B r 3 .Then, in view of Lemma 3.6, this guarantees at least one solution for the problem (1) in [0, R].
For each z ∈ [0, R] and x ∈ B r 4 ,θ , To show this, let u ∈ B r 4 ,θ .Then By using ||a| -|b|| ≤ |a + b| and taking into account we find Consequently, which means that Similarly, if v ∈ B r 4 ,θ , then implying the following inequality: This yields Inserting ( 31) and ( 32) into (30) gives which implies that N 1 u + N 2 v ∈ B r 4 ,θ for all u, v ∈ B r 4 ,θ , and so assumption 1 of Lemma 3.6 is satisfied.Claim 2: We show that N 2 is a contraction.For each u 1 , u 2 ∈ B r 4 ,θ , z ∈ [0, R], we estimate Then, this gives By choosing θ > 0 large enough so that it follows that N 2 is a contraction.Claim 3: Next, we will verify that condition 3 of Lemma 3.6 holds.Consider a sequence u n so that u n → u ∈ C as n → ∞.For z ∈ [0, R], we get the following inequality: Thus and so The Lebesgue's dominated convergence theorem, along with the continuity of f , leads to the conclusion that By using (28), we get The independence of the right-hand side of ( 33) with respect to u is apparent and Hence, N 1 B r 4 ,θ is equicontinuous and N 1 maps bounded sets to relatively compact sets, so that N 1 B r 4 ,θ is relatively compact.Using the Arzelà-Ascoli theorem, we can conclude that N 1 is compact in B r 4 ,θ .
Then because Lemma 3.6 is verified, this shows that the (FGSLL)-problem ( 1) has at least one solution defined on [0, R].Remark 3.10 The advantage of proving Theorem 3.7 by using the Ψ -Bielecki-type norm is that the strong condition ∇ θ < 1 is removed.

Existence results via Leray-Schauder fixed point theorem
First, we recall Leray-Schauder nonlinear alternative theorem and then give our main results.Proof Pay attention to the operator N : C → C given by (15).
Claim 1: Operator N maps bounded sets to bounded sets in C.
For r 5 > 0, assume that B r 5 ,θ (u By exploiting the well-known inequality ||κ| -| || ≤ |κ + | and taking into account This implies that which yields Claim 2: Operator N maps bounded sets to equicontinuous sets in C.
Assuming that the points z 1 , z 2 ∈ [0, are arbitrary with z 1 < z 2 and u ∈ B r 5 ,θ , where B r 5 ,θ is a bounded set in C, we get By using ( 28) and ( 29), we get Observe that, as z 1 → z 2 , the right-hand side goes to zero uniformly.This means that it does not depend on u.Furthermore, by Lemma 3.5, the operator N : C → C is completely continuous.
Eventually, we prove that the set of all solutions of the equation λN (u) = u is bounded for λ ∈ (0, 1).
Following similar computations as in the first claim, we have Taking the norm for t ∈ [0, R], we have the following: which leads to In accordance with (H4), then there exists M > 0 such that u θ = M. Define a set and consider the fact that N : M θ → C is continuous and completely continuous.The choice of M θ gives that there is no x ∈ ∂M θ such that λN (u) = u for some λ ∈ (0, 1).As a result, we conclude by Lemma 3.12 that N has a fixed point u ∈ M θ that corresponds to a solution of the (FGSLL)-problem (1).
• If χ(z) = 0 for t ∈ I, then we get χ = 0 and obtain that at least one solution for the (FSL)-problem ( 2) is guaranteed on I. • If η(z) = 1 and χ(z) = λ for t ∈ I and λ ∈ R, then we have η = 1 and χ = |λ|.We also conclude that at least one solution for the (FL)-problem (3) is guaranteed on I.

Stability analysis
This section analyzes the stability property.In other words, in the present section, we will discuss UH, GUH, UHR, and GUHR stability of the given (FGSLL)-problem (1).
Remark 4.9 ([41]) Let α > 0, I = [0, R], and Ψ ∈ C 1 (I, R) be increasing with Ψ (z) = 0 for all z ∈ I. Assume that v is a nonnegative function with the local integrability on [0, R] and let u be nonnegative and locally integrable on [0, R] with Then ) is a solution of the inequality (34) and where with given by (16).
Proof Let u be a solution of (34).By Lemma 2.9 and Remark 4.6(2), we get and then the solution of problem (39) can be given as Due to Remark 4.6(1), we can write .
In the sequel, we focus on the UHR and generalized UHR stability.
Proof Assuming that u is a solution of (36), we can utilize Lemma 2.9 and Remark 4.7(2) to obtain and then the solution of problem (43) may be given as Thanks to Remark 4.7 (2) and assumption (H6), we have By using Remark 4.7(1), we get In view of inequality (14), it follows that Finally, we conclude that The proof of (4.10) is now complete.Proof Let u ∈ C([0, R], R) be a solution of (36) and u be a unique solution for the (FGSLL)problem (1).By applying Lemma 4.14, it yields that u = X (z) + I α 1 +α 2 ;Ψ where X is given by (40).Similarly, if u(0) = u(0) and Applying Lemma 4.14, the triangle inequality, and inequality ( 14), for any t ∈ [0, R], we then may write Since ϒ is nondecreasing (see condition (H6)), for all s ∈ [0, z], we obtain ϒ(s) ≤ ϒ(z) and can write where Ψ is provided by (42).Thus, Then, the (FGSLL)-problem ( 1) is UHR stable.
• If χ(z) = 0 for all z ∈ I, then we have χ = 0 and the (FSL

Illustrative examples
Here, three test examples are used to show the effectiveness of the proposed techniques.
Example 5.1 Two cases are formulated that require less restrictive conditions for a unique solution.Then we analyze the stability results based on the (FGSLL)-problem (1).
and consider the closed ball B r 3 (u) = {u ∈ C, u ≤ r 3 } which is a convex and nonempty subset of the Banach space C. For each z ∈ [0, R] and any x ∈ B r 3 , we have

Lemma 3 .
12 ([40]) Assume that U is a Banach space, C is a convex and closed subset of U, M is an open subset of C, and 0 belongs to M. Let T : M → C be a map that is continuous and compact, i.e., T(M) is a relatively compact subset of C. Then either • T has a fixed point in M, or • There exists a point x ∈ ∂M, where ∂M denotes the boundary of M in C, and then there is a scalar λ ∈ (0, 1) such that λT(x) = x.Theorem 3.13 Let (H1) and (H3)-(H5) hold.Then at least one solution exists for the (FGSLL)-problem (1) on [0, R].