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Ψ-Bielecki-type norm inequalities for a generalized Sturm–Liouville–Langevin differential equation involving Ψ-Caputo fractional derivative
Boundary Value Problems volume 2024, Article number: 81 (2024)
Abstract
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.
1 Introduction
During the last century, fractional differential equations (FDEs) have fascinated the interest of many researchers due to their various applications in many fields of science, like physics, chemistry, biology, economics, engineering, signal processing, electromagnetics, etc. (see [1–3]). In many references, the basic notions and tools of fractional calculus can be observed; see, e.g., [4–6]. Recently, Almeida [7] defined a new fractional derivative called Ψ-Caputo fractional derivative and he published several scientific research works [8, 9]. Afterwards, several mathematicians concentrated their research on the generalized fractional operators; we cite them as examples [10–15]. In this direction, researchers have focused their interests on the investigation of existence, uniqueness, and Ulam stability of FDEs using a number of definitions related to fractional derivatives as can be found in some works like [16–21] and references therein, as well as in [22, 23].
In 1908, Paul Langevin formulated a new equation, named the Langevin equation, to define the evolution of some physical phenomena in fluctuating environments, like Brownian motion [24]. After this, other extensions of the Langevin equation have been studied in the works of many researchers [25–31]. Nowadays, the existence, uniqueness, and stability of solutions for Langevin nonlinear BVPs have been established by many researchers using different kinds of fractional derivatives by applying Banach, Krasnoselskii, Shaefer, and Leray–Schauder classical fixed point theorems. For more information on this topic, the reader is advised to refer to [32–39]. The results on the existence and Ulam–Hyers stability of solutions of Langevin fractional equation have been discussed in [29]. Motivated by the works cited above, several other types of stability will be discussed in this article for an advanced combined differential equation. More precisely, consider the fractional generalized Sturm–Liouville–Langevin (FGSLL) problem:
Here, \(\eta \in C(I,\mathbb{R}^{\star})\), \(\chi \in C(I,\mathbb{R})\), \(f:I\times \mathbb{R}\to \mathbb{R}\) is continuous, \({}^{C}\mathcal{D}_{0^{+}}^{\gamma ,\varPsi} \) is the Ψ-Caputo fractional derivative depending on an increasing function Ψ of order \(\gamma \in \{\alpha _{1},\alpha _{2},\sigma _{1},\sigma _{2}\}\), \(0<\alpha _{1},\alpha _{2}\leq 1\), and \(0<\sigma _{1}\), \(\sigma _{2}<\alpha _{2}\).
To show the novelty and generality of our BVP, we note that:
-
1.
If \(\chi (\mathfrak{z})=0\), for each \(\mathfrak{z} \in I\), the (FGSLL)-problem (1) reduces to the standard form of the fractional Sturm–Liouville (FSL) problem for a nonlinear FDE, which is as follows:
$$ \textstyle\begin{cases} {}^{C}\mathcal{D}_{0^{+}}^{\alpha _{1},\varPsi} (\eta ( \mathfrak{z}) {}^{C}\mathcal{D}_{0^{+}}^{\alpha _{2},\varPsi} u( \mathfrak{z}) ) = f(\mathfrak{z}, u(\mathfrak{z})),\quad \mathfrak{z} \in I, \\ u(0) = 0,\qquad \eta (R) {}^{C}\mathcal{D}_{0^{+}}^{\sigma _{1}, \varPsi} u(R) =0, \end{cases} $$(2)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 \(\varPsi (x)=x\), then the (FSL)-problem (2) reduces to the Caputo-type Sturm–Liouville (CSL) problem.
-
If \(\varPsi (x)=x^{\nu}\), then the (FSL)-problem (2) becomes the Caputo–Erdélyi–Kober-type Sturm–Liouville (CEKSL) problem.
-
If \(\varPsi (x)=\ln (x)\), then the (FSL)-problem (2) represents the Caputo–Hadamard-type Sturm–Liouville (CHSL) problem.
-
-
2.
By choosing \(\eta (\mathfrak{z})\equiv 1\), \(\chi (\mathfrak{z})\equiv \lambda \) (\(\lambda \in \mathbb{R}\)), for \(\mathfrak{z} \in I\), the (FGSLL)-problem (1) reduces to the standard form of the fractional Langevin (FL) problem for a nonlinear FDE, which is as follows:
$$ \textstyle\begin{cases} {}^{C}\mathcal{D}_{0^{+}}^{\alpha _{1},\varPsi} ( {}^{C} \mathcal{D}_{0^{+}}^{\alpha _{2},\varPsi} u(\mathfrak{z}) +\lambda u( \mathfrak{z}) ) = f(\mathfrak{z},u(\mathfrak{z})),\quad \mathfrak{z} \in I, \\ u(0) = 0, \qquad {}^{C}\mathcal{D}_{0^{+}}^{\sigma _{1},\varPsi} u(R) + \lambda {}^{C}\mathcal{D}_{0^{+}}^{\sigma _{2},\varPsi} u(R) =0, \end{cases} $$(3)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 \(\varPsi (x)=x\), then the (FL)-problem (3) reduces to the Caputo-type Langevin (CL) problem.
-
If \(\varPsi (x)=x^{\nu}\), then the (FL)-problem (3) represents the Caputo–Erdélyi–Kober-type Langevin (CEKL) problem.
-
If \(\varPsi (x)=\ln (x)\), 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.
2 Essential concepts and basic tools
Some concepts are recalled in this section, and also some lemmas are proved.
Definition 2.1
([7])
Let \(\mu >0\), \(n\in \mathbb{N} \), \(I=[a,b]\) with \(-\infty \leq a< b\leq \infty \), \(\varphi :I \to \mathbb{R}\) be an integrable function, and \(\varPsi \in \mathcal{C}^{1}(I, \mathbb{R})\) increasing with \(\varPsi '(\mathfrak{z}) \neq 0\) for any \(\mathfrak{z} \in I\). 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 \(I\subset \mathbb{R}\) and let \(\mu \in (n-1,n)\). Let also \(\varphi :I\to \mathbb{R}\) 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 \(n=[\mu ]+1\) and \([\mu ]\) indicates the integer part of μ.
Definition 2.3
([7])
Let \(\mu >0\), \(n\in \mathbb{N} \), \(I=[a,b]\) with \(-\infty \leq a< b\leq \infty \), \(\varPsi , \varphi \in C^{n}(I, \mathbb{R})\) be functions so that Ψ is increasing and \(\varPsi '(\mathfrak{z}) \neq 0\) for any \(\mathfrak{z} \in I\). The left-sided Ψ-Caputo fractional derivative of order μ for φ is defined by
where \(n=[\mu ]+1\) for \(\mu \notin \mathbb{N}\) and \(n=\mu \) for \(\mu \in \mathbb{N}\).
To simplify the notation, we put \(\varphi _{\varPsi}^{[n]}(\mathfrak{z})= ( \frac {1}{{\varPsi '}(\mathfrak{z})} \frac {\mathrm{d}}{\mathrm{d} \mathfrak{z}} )^{n} \varphi ( \mathfrak{z})\). Then, from the definition we can write
\({ }^{c} \mathcal{D}_{a^{+}}^{\mu ; \varPsi} \varphi (\mathfrak{z})= \begin{cases} \int _{a}^{\mathfrak{z}} \frac {\varPsi '(s)(\varPsi (\mathfrak{z})-\varPsi (s))^{n-\mu -1}}{\Gamma (n-\mu )} \varphi _{\varPsi}^{[n]}(s) \,\mathrm{d} s & \text{ if } \mu \notin \mathbb{N}, \\ \varphi _{\varPsi}^{[n]}(\mathfrak{z}) & \text{ if } \mu \in \mathbb{N}. \end{cases} \)
Lemma 2.4
([7])
Let \(\mu >0\) and \(\varphi :[a,b] \to \mathbb{R}\). The properties given below hold:
-
If \(\varphi \in \mathcal{C} ([a,b])\), then \({}^{C}\mathcal{D}_{a^{+}}^{\mu ,\varPsi} \mathcal{I}_{a^{+}}^{ \mu ,\varPsi} \varphi (\mathfrak{z}) =\varphi (\mathfrak{z})\).
-
If \(\varphi \in \mathcal{C}^{n-1} ([a,b])\), then
$$ \mathcal{I}_{a^{+}}^{\mu ,\varPsi} {}^{C} \mathcal{D}_{a^{+}}^{ \alpha ,\varPsi} \varphi (\mathfrak{z}) = \varphi ( \mathfrak{z}) - \underset{k=0}{\overset{n-1}{\sum }} c_{k} \bigl( \varPsi ( \mathfrak{z}) - \varPsi (a) \bigr)^{k}, $$where \(c_{k} = \frac {\varphi _{\varPsi}^{[k]}(a)}{k!}\), \(\varphi _{\varPsi}^{[k]}(a) = [ \frac {1}{\varPsi ^{ \backprime } (\mathfrak{z})} \frac {d}{d\mathfrak{z}} ]^{k} \varphi (a)\), \(n-1< \mu \leq n\).
In particular, for \(\mu \in (0,1)\), we have \(\mathcal{I}_{a+}^{ \mu , \varPsi C} \mathcal{D}_{a+}^{\mu , \varPsi} \varphi ( \mathfrak{z})=\varphi (\mathfrak{z})-\varphi (a) \).
Now, we define the norms \(\|\cdot \|_{C}:C ( [ a,b ] ) \to \mathbb{R}\) and \(\|\cdot \|_{C_{\varPsi}^{[n]}}:C^{n} ( [ a,b ] ) \to \mathbb{R} \) by
Lemma 2.5
([11])
If \(\varphi :C ( [ a,b ] ) \to \mathbb{R}\), then \(I_{a+}^{\mu , \varPsi} \varphi (\mathfrak{z})\) is well-defined for every \(\mathfrak{z} \in [a, b]\). In addition, we have
-
(i)
\(\mathcal{I}_{a+}^{\mu , \varPsi} \varphi (a)=0\);
-
(ii)
\(\Vert \mathcal{I}_{a+}^{\mu , \varPsi} \varphi \Vert _{C} \leq \frac {(\varPsi (\mathfrak{z})-\varPsi (a))^{\mu}}{\Gamma (\mu +1)} \|\varphi \|_{C} \).
Proof
From (4), we derive the following inequality:
which gives immediately \(\mathcal{I}_{a+}^{\mu , \varPsi} \varphi (a)=0\) and \(\Vert \mathcal{I}_{a+}^{\mu , \varPsi} \varphi \Vert _{C} \leq \frac {(\varPsi (\mathfrak{z})-\varPsi (a))^{\mu}}{\Gamma (\mu +1)} \|\varphi \|_{C} \). □
Lemma 2.6
([7])
The Ψ-Caputo derivatives of the fractional order are bounded and, for any \(\mu >0\), we have
Remark 2.7
From equality (5), we can easily obtain
which allows us to conclude that \({ }^{C} \mathcal{D}_{a+}^{\mu , \varPsi} \varphi (a)=0\).
Lemma 2.8
Let \(\mu , \theta >0\). We have
and
Proof
By applying the Ψ-R–L fractional operator \(\mathcal{I}_{0^{+}}^{\mu _{1},\varPsi} \) to the function \(\mathfrak{z} \mapsto e^{\theta (\varPsi (\mathfrak{z})-\varPsi (0))} \) together with the replacement of variables \(\mathrm{y}=\varPsi (\mathfrak{z})-\varPsi (s)\) and \(z=\theta \mathrm{y}\), we have
For the proof of the inequality (7), we again use the same replacement of variables \(\mathrm{y}=\varPsi (\mathfrak{z}_{2})-\varPsi (s)\) and \(z=\theta \mathrm{y}\), and we obtain
The proof is now complete. □
Lemma 2.9
Let \(0<\alpha _{1},\alpha _{2}\leq 1\), \(\alpha _{3} >0\), and \(0<\sigma _{1}\), \(\sigma _{2}<\alpha _{2}\). Suppose that \(h \in \mathcal{C}(I, \mathbb{R} )\), \(\eta \in \mathcal{C}(I,\mathbb{R}^{\star})\), and \(\chi \in \mathcal{C}(I,\mathbb{R})\). Then, u is a solution of
if and only if it fulfills the integral equation given below:
where
Proof
By applying the Ψ-R–L fractional operators \(\mathcal{I}_{0^{+}}^{\alpha _{1},\varPsi} \) and \(\mathcal{I}_{0^{+}}^{\alpha _{2},\varPsi} \) 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 \(\mathbb{R} \).
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 \({}^{C}\mathcal{D}_{0^{+}}^{\alpha _{2},\varPsi}\) on both sides of equation (13) and applying again the operator \({}^{C}\mathcal{D}_{0^{+}}^{\alpha _{1},\varPsi}\) 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 \(\mathfrak{C}= \mathcal{C} (I, \mathbb{R} )\) equipped with the well-known Ψ-Bielecki-type norm \(\Vert u\Vert _{\theta ,\alpha}\) proposed by previous works (see [40]) defined by
where \(\mathbb{E}_{\alpha} \) indicates the Mittag-Leffler function of one-parameter that is given as
If we take \(\alpha \to 1 \) in the above norm \(\Vert u\Vert _{\theta ,\alpha}\), we obtain
and \(( \mathfrak{C}, \Vert u\Vert _{\theta } ) \) is a Banach space. We now focus on the key findings of our study.
3 Main results
For a good and straightforward continuation of our work, we propose the hypotheses as given below:
(H1) \(f: [0,R] \times \mathbb{R}\to \mathbb{R} \) is continuous.
(H2) For some positive real constant \(L_{f}\), we have
(H3) \(\vert {f(\mathfrak{z},u)} \vert \leq \mathcal{K}_{f}(\mathfrak{z})\), \(\forall (\mathfrak{z},u) \in [0,R] \times \mathbb{R}\), with \(\mathcal{K}_{f} \in \mathcal{C} ([0,R], \mathbb{R}_{+} )\).
(H4) A function \(\mathfrak{g} \in \mathcal{C} ([0,R], \mathbb{R}_{+} )\) and a real constant \(d>0\) exist such that
(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 \(\Upsilon \in \mathcal{C} ([0,R], \mathbb{R}_{+}) \) and a real constant \(\gamma _{\Upsilon ,\alpha _{1}+\alpha _{2}} > 0\) exist such that for any \(\mathfrak{z} \in [0,R ]\), we have
In light of Lemma 2.9, we can define the following operator:
where
Now, we express the operator \(\mathcal{N}\) as a sum of two operators \(\mathcal{N}_{1}\) and \(\mathcal{N}_{2}\) as follows:
To facilitate the reading of the work, we utilize the following notations:
and, for more convenience, we put
and
3.1 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 \(\mathbb{T}: U \rightarrow U\) a contraction. Then there is a unique fixed point of \(\mathbb{T}\) in U.
Theorem 3.2
Suppose that (H1) and (H2) are satisfied. Then the (FGSLL)-problem (1) has a unique solution if \(\mathcal{J} < 1 \), where \(\mathcal{J} \) is defined by (20).
Proof
First, we choose \(r_{1}\) such that
Briefly, our aim is to show that \(\mathcal{N} \mathcal{B}_{r_{1}} \subseteq \mathcal{B}_{r_{1}} \), where
is a nonempty, closed, and convex subset of the Banach space \(\mathfrak{C} \).
For each \(\mathfrak{z}\in [0,R] \) and \(u \in \mathcal{B}_{r_{1}} \), we get
which implies that
Let \(u\in \mathcal{B}_{r_{1}}\), then
By using the property \(\vert \vert \kappa \vert - \vert \ell \vert \vert \le \vert \kappa +\ell \vert \) and taking into consideration
we get
which gives
that is,
which implies that \(\Vert \mathcal{N}u \Vert \leq r_{1} \). Thus, \(\mathcal{N} \) maps \(\mathcal{B}_{r_{1}} \) into itself.
The last step is to show that \(\mathcal{N} \) is a contraction mapping. Letting \(u_{1},u_{2} \in \mathcal{B}_{r_{1}} \) and \(\mathfrak{z}\in [0,R] \), we have
Thus,
consequently, we get
Since \(\mathcal{J}< 1 \), hence \(\mathcal{N} \) is a contraction mapping. Consequently, by the Banach contraction principle 3.1, we conclude that \(\mathcal{N} \) has a unique fixed point in \(\mathcal{B}_{r_{1}}\). 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 \(\mathcal{J}< 1 \) is removed.
Theorem 3.3
Let (H1) and (H2) be satisfied. Then the (FGSLL)-problem (1) has a unique solution on \([0,R]\).
Proof
Let us choose
where \(\Lambda _{\theta}\), \(\nabla _{\theta} \), and \(M_{f,\theta}\) are three constants defined previously.
Claim 1: One has \(\mathcal{N} \mathcal{B}_{r_{2},\theta} \subseteq \mathcal{B}_{r_{2}, \theta} \), where \(\mathcal{B}_{r_{2},\theta}(u) = \{ u \in \mathfrak{C} , \Vert {u} \Vert _{\theta} \leq r_{2} \} \) is a nonempty, closed, and convex subset of the Banach space \(\mathfrak{C} \).
For each \(\mathfrak{z}\in [0,R] \) and \(u \in \mathcal{B}_{r_{2},\theta} \), we have
which implies that
Let \(u \in \mathcal{B}_{r_{2},\theta}\), then
Using the estimate \(\vert \vert \kappa \vert - \vert \ell \vert \vert \le \vert \kappa +\ell \vert \) and taking into account
we obtain
By exploiting (6), we get
which yields
This means that \(\mathcal{N} \) maps \(\mathcal{B}_{r_{2},\theta} \) into itself.
Claim 2: Operator \(\mathcal{N}\) is a contraction mapping.
Let \(u_{1},u_{2} \in \mathcal{B}_{r_{2},\theta} \) and \(\mathfrak{z}\in [0,R] \), we have
Simple computations give us
By using (6), we get
Consequently,
Hence, we obtain
By choosing \(\theta >0\) large enough such that
we conclude that the mapping \(\mathcal{N} \) is a contraction relative to the Ψ-Bielecki norm. Exploiting the Banach fixed point Theorem 3.1, it follows that \(\mathcal{N}\) has a unique fixed point which is a unique solution to the (FGSLL)-problem (1). □
Corollary 3.4
Let (H1) and (H2) be satisfied. Then,
-
If \(\chi (\mathfrak{z})=0\) for \(\mathfrak{z} \in I\), then we have \(\overline{\chi}=0\) and one solution is guaranteed for the (FSL)-problem (2) on I.
-
If \(\eta (\mathfrak{z})= 1\) and \(\chi (\mathfrak{z}) =\lambda \) (\(\lambda \in \mathbb{R}\)) for \(\mathfrak{z} \in I\), then we have \(\underline{\eta} =1\) and \(\overline{\chi}=|\lambda |\), and so the (FL)-problem (3) has a unique solution on I.
3.2 Application of Krasnoselskii’s fixed point theorem for existence results
First, we recall Arzelà–Ascoli and Krasnoselskii theorems and then give our main results.
Lemma 3.5
([40])
A family of functions in \(\mathcal{C}([a_{1},a_{2}])\) is relatively compact if it is both equicontinuous and uniformly bounded on \([a_{1},a_{2}]\).
Lemma 3.6
([40])
Consider a nonempty subset M of a Banach space U that is bounded, closed, and convex. Let \(\mathcal{P}\) and \(\mathcal{Q}\) be operators so that:
-
1.
\(\mathcal{P} x+\mathcal{Q} y \in M\) whenever \(x, y \in M\),
-
2.
\(\mathcal{Q}\) is a contraction,
-
3.
\(\mathcal{P}\) is compact and continuous,
Then there exists \(\varpi \in M\) so that \(\varpi =\mathcal{P} \varpi +\mathcal{Q} \varpi \).
Now, we present the following existence theorem which is proved using the above lemmas.
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 \(r_{3} \geq \frac { \Lambda \Vert { \mathcal{K}_{f}} \Vert }{1-\nabla} \) with \(\Vert { \mathcal{K}_{f}} \Vert =\sup_{ \mathfrak{z}\in [0,R]}|\mathcal{K}_{f}(\mathfrak{z})|\), and consider the closed ball \(\mathcal{B}_{r_{3}}(u)=\{u\in \mathfrak{C}, \Vert {u} \Vert \leq r_{3} \} \) which is a convex and nonempty subset of the Banach space \(\mathfrak{C}\). For each \(\mathfrak{z}\in [0,R]\) and any \(x\in \mathcal{B}_{r_{3}}\), we have
which implies that
Claim 1: For \(u,v\in \mathcal{B}_{r_{3}} \) we show that \(\mathcal{N}_{1}u+ \mathcal{N}_{2}v \in \mathcal{B}_{r_{3}}\).
Let \(u \in \mathcal{B}_{r_{3}}\), then
By using \(\vert \vert \kappa \vert - \vert \ell \vert \vert \le \vert \kappa +\ell \vert \), where
we get
which means that
and yields
Similarly, if \(v \in \mathcal{B}_{r_{3}} \), then
This implies that
yielding
Inserting (24) and (25) into (23), we get
which implies that \(\mathcal{N}_{1}u+ \mathcal{N}_{2}v \in \mathcal{B}_{r_{3}}\) for all \(u,v\in \mathcal{B}_{r_{3}}\). Thus assumption 1 of Lemma 3.6 is verified.
Claim 2: We show that \(\mathcal{N}_{2}\) is contraction.
For each \(u_{1},u_{2} \in \mathcal{B}_{r_{3}} \) and \(\mathfrak{z} \in [ 0,R ]\), we have
which yields
Hence, by (22), \(\mathcal{N}_{2}\) is a contraction.
Claim 3: Assumption 3 in Lemma 3.6 holds.
Take a sequence \(\{u_{n}\}_{n\in \mathbb{N}}\) with \(u_{n}\to u \in \mathfrak{C}\) as \(n\to \infty \). For \(\mathfrak{z} \in [ 0,R ] \), we get
with \(\Vert {f_{n} -f } \Vert = \underset{\mathfrak{z}\in [ 0,R ] }{\sup } \vert f( \mathfrak{z} ,u_{n}(\mathfrak{z}))-f(\mathfrak{z} ,u(\mathfrak{z})) \vert \). Thus
where
The Lebesgue’s dominated convergence theorem and continuity of f lead to the conclusion that \(\Vert \mathcal{N}_{1}u_{n} -\mathcal{N}_{1}u \Vert \to 0\) as \(n\to \infty \). Therefore, \(\mathcal{N}_{1}\) is continuous. Furthermore, \(\mathcal{N}_{1} \) is uniformly bounded on \(\mathcal{B}_{r_{3}}\) as \(\Vert {\mathcal{N}_{1}u} \Vert \leq \Lambda \Vert { \mathcal{K}_{f}} \Vert \) due to (24). Also, \(\mathcal{N}_{1} \) is equicontinuous. Indeed, letting \(u \in \mathcal{B}_{r_{3}} \), for \(\mathfrak{z}_{1},\mathfrak{z}_{2}\in [0,R ]\), \(\mathfrak{z}_{1}<\mathfrak{z}_{2}\), we have
i.e.,
Finally, we get
The right-hand side of (27) is clearly independent of u and \(|\mathcal{N}_{1}u(\mathfrak{z}_{2}) -\mathcal{N}_{1}u( \mathfrak{z}_{1}) |\to 0\) as \(\mathfrak{z}_{2}\to \mathfrak{z}_{1}\). Hence, this implies that \(\mathcal{N}_{1} \mathcal{B}_{r_{3}}\) is equicontinuous and \(\mathcal{N}_{1}\) maps bounded subsets into relatively compact subsets, which implies that \(\mathcal{N}_{1} \mathcal{B}_{r_{3}}\) is relatively compact.
Therefore, using Lemma 3.5, we determine that \(\mathcal{N}_{1}\) is compact in \(\mathcal{B}_{r_{3}}\). Then, in view of Lemma 3.6, this guarantees at least one solution for the problem (1) in \([ 0,R ]\). □
Before stating and proving the results via Krasnoselskii and Leray–Schauder fixed point theorems under the Ψ-Bielecki’s norm, we provide an auxiliary lemma which is related to the proof of the equicontinuity property.
Lemma 3.8
For a given \(\eta \in C(I,\mathbb{R}^{\star})\), let (H1) and (H3) hold. For all \(\theta >0\) and with \(0< \alpha _{i} \leq 1\), \(i \in \{1,2\}\), we have
and
Proof
Let \(\mathfrak{z}_{1},\mathfrak{z}_{2} \in [0, R]\) where \(\mathfrak{z}_{1}<\mathfrak{z}_{2}\), we have
By using (7), we get
thus, we have
Similarly, for \(\mathfrak{z}_{1},\mathfrak{z}_{2} \in [0, R]\) where \(\mathfrak{z}_{1}<\mathfrak{z}_{2}\), we get
By using (6) and (7), we obtain
The proof is complete. □
Now, we discuss existence results by using the Krasnoselskii fixed point theorem and some inequalities of the Ψ-Bielecki’s norm-type.
Theorem 3.9
Let (H1) and (H3) hold. Then, at least one solution exists for the (FGSLL)-problem (1) on \([0,R]\).
Proof
We fix \(r_{4} \geq \frac { \Lambda _{\theta} \Vert { \mathcal{K}_{f}} \Vert _{\theta}}{1-\nabla _{\theta}} \), where \(\Lambda _{\theta}\) and \(\nabla _{\theta}\) are constants defined by (18) and (19) and focus on the nonempty closed ball \(\mathcal{B}_{r_{4},\theta}(u)=\{u\in \mathfrak{C}, \Vert { \mathrm{u}} \Vert _{\theta} \leq r_{4} \} \) which is convex in the Banach space \(\mathfrak{C}\).
For each \(\mathfrak{z} \in [ 0,R ] \) and \(x \in \mathcal{B}_{r_{4},\theta} \),
which implies that
Claim 1: For \(u,v\in \mathcal{B}_{r_{4},\theta} \), one has \(\mathcal{N}_{1}u+ \mathcal{N}_{2}v \in \mathcal{B}_{r_{4},\theta}\).
To show this, let \(u \in \mathcal{B}_{r_{4},\theta}\). Then
By using \(\vert \vert a \vert - \vert b \vert \vert \le \vert a+b \vert \) and taking into account
we find
Consequently,
which means that
Similarly, if \(v \in \mathcal{B}_{r_{4},\theta} \), then
implying the following inequality:
This yields
Inserting (31) and (32) into (30) gives
which implies that \(\mathcal{N}_{1}u+ \mathcal{N}_{2}v \in \mathcal{B}_{r_{4},\theta}\) for all \(u,v\in \mathcal{B}_{r_{4},\theta}\), and so assumption 1 of Lemma 3.6 is satisfied.
Claim 2: We show that \(\mathcal{N}_{2}\) is a contraction.
For each \(u_{1},u_{2} \in \mathcal{B}_{r_{4},\theta} \), \(\mathfrak{z} \in [ 0,R ]\), we estimate
Then, this gives
By choosing \(\theta >0\) large enough so that
it follows that \(\mathcal{N}_{2}\) is a contraction.
Claim 3: Next, we will verify that condition 3 of Lemma 3.6 holds.
Consider a sequence \(u_{n}\) so that \(u_{n}\to u \in \mathfrak{C}\) as \(n\to \infty \). For \(\mathfrak{z} \in [ 0,R ] \), we get the following inequality:
Thus
and so
with
and
The Lebesgue’s dominated convergence theorem, along with the continuity of f, leads to the conclusion that \(\Vert \mathcal{N}_{1}u_{n} -\mathcal{N}_{1}u \Vert _{\theta} \to 0\) as \(\mathfrak{z}\to \infty \). Therefore, \(\mathcal{N}_{1}\) is continuous. Besides, \(\mathcal{N}_{1}\) is uniformly bounded on \(\mathcal{B}_{r_{4},\theta}\) as \(\Vert {\mathcal{N}_{1}v} \Vert _{\theta} \leq \Lambda \Vert {\mathcal{K}_{f}} \Vert _{\theta}\), due to (31).
Also, \(\mathcal{N}_{1} \) is equicontinuous. Indeed, let \(u \in \mathcal{B}_{r_{4},\theta} \). Then for \(\mathfrak{z}_{1},\mathfrak{z}_{2}\in [0,R ]\), \(\mathfrak{z}_{1}<\mathfrak{z}_{2}\), we have
By using (28), we get
The independence of the right-hand side of (33) with respect to u is apparent and \(|\mathcal{N}_{1}u(\mathfrak{z}_{2}) -\mathcal{N}_{1}u( \mathfrak{z}_{1}) |\to 0\) as \(\mathfrak{z}_{2}\to \mathfrak{z}_{1}\). Hence, \(\mathcal{N}_{1} \mathcal{B}_{r_{4},\theta}\) is equicontinuous and \(\mathcal{N}_{1}\) maps bounded sets to relatively compact sets, so that \(\mathcal{N}_{1} \mathcal{B}_{r_{4},\theta}\) is relatively compact. Using the Arzelà–Ascoli theorem, we can conclude that \(\mathcal{N}_{1}\) is compact in \(\mathcal{B}_{r_{4},\theta}\).
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 \(\nabla _{\theta}< 1 \) is removed.
Corollary 3.11
Let (H1) and (H3) hold. Then
-
If \(\chi (\mathfrak{z}) = 0\) for all \(\mathfrak{z} \in I\), then we get \(\overline{\chi}=0\) and find that the (FSL)-problem (2) has at least one solution defined on I.
-
If \(\eta (\mathfrak{z}) = 1\) and \(\chi (\mathfrak{z}) = \lambda \) (\(\lambda \in \mathbb{R}\)) for \(t \in I\), then we have \(\underline{\eta} =1\) and \(\overline{\chi}=|\lambda |\). We also find that the (FL)-problem (3) has at least one solution defined on I.
3.3 Existence results via Leray–Schauder fixed point theorem
First, we recall Leray–Schauder nonlinear alternative theorem and then give our main results.
Lemma 3.12
([40])
Assume that U is a Banach space, \(\mathcal{C}\) is a convex and closed subset of U, \(\mathcal{M}\) is an open subset of \(\mathcal{C}\), and 0 belongs to \(\mathcal{M}\). Let \(\mathbb{T}: \overline{\mathcal{M}} \rightarrow \mathcal{C}\) be a map that is continuous and compact, i.e., \(\mathbb{T}(\overline{\mathcal{M}})\) is a relatively compact subset of \(\mathcal{C}\). Then either
-
\(\mathbb{T}\) has a fixed point in \(\overline{\mathcal{M}}\), or
-
There exists a point \(x \in \partial \mathcal{M}\), where \(\partial \mathcal{M}\) denotes the boundary of \(\mathcal{M}\) in \(\mathcal{C}\), and then there is a scalar \(\lambda \in (0,1)\) such that \(\lambda \mathbb{T}(x) = x\).
Theorem 3.13
Let (H1) and (H3)–(H5) hold. Then at least one solution exists for the (FGSLL)-problem (1) on \([0,R]\).
Proof
Pay attention to the operator \(\mathcal{N}: \mathfrak{C} \rightarrow \mathfrak{C} \) given by (15).
Claim 1: Operator \(\mathcal{N} \) maps bounded sets to bounded sets in \(\mathfrak{C} \).
For \(r_{5}>0\), assume that \(\mathcal{B}_{r_{5},\theta}(u) = \{ u \in \mathfrak{C}, \Vert {u} \Vert _{\theta} \leq r_{5} \} \) is a bounded set in \(\mathfrak{C} \). Let \(u \in \mathcal{B}_{r_{5},\theta} \), then
By exploiting the well-known inequality \(\vert \vert \kappa \vert - \vert \ell \vert \vert \le \vert \kappa +\ell \vert \) and taking into account
we get
This implies that
which yields
Claim 2: Operator \(\mathcal{N}\) maps bounded sets to equicontinuous sets in \(\mathfrak{C}\).
Assuming that the points \(\mathfrak{z}_{1}, \mathfrak{z}_{2} \in [ 0,R ]\) are arbitrary with \(\mathfrak{z}_{1} < \mathfrak{z}_{2}\) and \(u \in \mathcal{B}_{r_{5},\theta} \), where \(\mathcal{B}_{r_{5},\theta} \) is a bounded set in \(\mathfrak{C}\), we get
By using (28) and (29), we get
Observe that, as \(\mathfrak{z}_{1} \rightarrow \mathfrak{z}_{2} \), the right-hand side goes to zero uniformly. This means that it does not depend on u. Furthermore, by Lemma 3.5, the operator \(\mathcal{N}: \mathfrak{C} \rightarrow \mathfrak{C} \) is completely continuous.
Eventually, we prove that the set of all solutions of the equation \(\lambda \mathcal{N}(u)=u \) is bounded for \(\lambda \in (0,1) \).
Following similar computations as in the first claim, we have
Taking the norm for \(t \in [0,R] \), we have the following:
which leads to
In accordance with (H4), then there exists \(M > 0 \) such that \(\Vert {u} \Vert _{\theta} \neq M \). Define a set
and consider the fact that \(\mathcal{N}: \overline{\mathcal{M}_{\theta}} \rightarrow \mathcal{C}\) is continuous and completely continuous. The choice of \(\mathcal{M}_{\theta}\) gives that there is no \(x \in \partial \mathcal{M}_{\theta}\) such that \(\lambda \mathcal{N}(u)=u \) for some \(\lambda \in (0,1) \). As a result, we conclude by Lemma 3.12 that \(\mathcal{N}\) has a fixed point \(u \in \overline{\mathcal{M}_{\theta}} \) that corresponds to a solution of the (FGSLL)-problem (1). □
Corollary 3.14
Let (H1),(H4), and (H5) hold.
-
If \(\chi (\mathfrak{z}) = 0\) for \(t \in I\), then we get \(\overline{\chi}=0\) and obtain that at least one solution for the (FSL)-problem (2) is guaranteed on I.
-
If \(\eta (\mathfrak{z}) = 1\) and \(\chi (\mathfrak{z}) = \lambda \) for \(t \in I\) and \(\lambda \in \mathbb{R}\), then we have \(\underline{\eta}=1\) and \(\overline{\chi}=|\lambda |\). We also conclude that at least one solution for the (FL)-problem (3) is guaranteed on I.
4 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).
4.1 Ulam stability
Let \(\varepsilon > 0\), \(L_{f}> 0\), and let \(\phi : [0,R ] \to \mathbb{R}_{+} \) be continuous. We will examine the set of inequalities as below (\(\mathfrak{z} \in [0,R ]\)):
Definition 4.1
([34])
The (FGSLL)-problem (1) is UH stable if there exists \(\mathcal{C}_{f}>0\) so that for any \(\varepsilon >0\) and each solution \(\overset{\backsim}{u}\in \mathcal{C}([0,R],\mathbb{R})\) of the inequality (34), there exists \(u \in \mathcal{C}([0,R],\mathbb{R})\) as a solution of the (FGSLL)-problem (1) with
Definition 4.2
([34])
The (FGSLL)-problem (1) has GUH stability if there exists a positive constant \(\mathcal{C}_{f} \) so that for any \(\varepsilon > 0 \) and for any solution \(\overset{\backsim }{u} \in \mathcal{C} ([0,R], \mathbb{R}) \) of the inequality (34), there exists \(u \in \mathcal{C} ([0,R], \mathbb{R}) \) as a solution of the (FGSLL)-problem (1) with
Definition 4.3
([34])
The (FGSLL)-problem (1) is UHR stable asymptotically if and only if there exists \(C>0\) so that for each \(\varepsilon > 0\) and for each solution \(\overset{\backsim}{u}\in \mathcal{C}([0,R],\mathbb{R})\) of the inequality (36), there exists \(u\in \mathcal{C}([0,R],\mathbb{R})\) as a solution of (FGSLL)-problem (1) with
Definition 4.4
([34])
The (FGSLL)-problem (1) is GUHR stable with respect to ϒ if there exists a real number \(\mathcal{C}_{f,\Upsilon}>0 \) so that for any solution \(\overset{\backsim }{u} \in \mathcal{C} ([0,R], \mathbb{R}) \) of the inequality (35), there exists \(u \in \mathcal{C} ([0,R], \mathbb{R}) \) as a solution of the (FGSLL)-problem (1) with
Remark 4.5
(1) Definition 4.2 is implied by Definition 4.1,
(2) Definition 4.4 is implied by Definition 4.3,
(3) Definition 4.1 is implied by Definition 4.3 for \(\Upsilon ( \cdot )=1 \).
Remark 4.6
A continuous function \(\overset{\backsim}{u} \in \mathcal{C}([0,R], \mathbb{R})\) is a solution of the inequality (34) iff there exists \(g \in \mathcal{C}([0,R], \mathbb{R})\), a continuous function depending on \(\overset{\backsim}{u}\) such that
hold.
Remark 4.7
The essential condition for a function \(\overset{\backsim}{u} \in \mathcal{C}([0,R], \mathbb{R})\) to satisfy inequality (36) is the existence of a function \(w \in \mathcal{C}([0,R], \mathbb{R})\) that depends on the solution \(\overset{\backsim}{u}\) and satisfies the following conditions:
The following lemma, a generalized version of Gronwall inequality, plays a crucial role in establishing our main stability results.
Lemma 4.8
([41])
Suppose that u, v are two functions in \(L^{1}([0, R])\) and g in \(\mathcal{C}([0, R])\). Let \(\varPsi \in \mathcal{C}^{1}[0, R]\) be an increasing function so that \({\varPsi '}(\mathfrak{z}) \neq 0\), \(\forall \mathfrak{z} \in [0, R]\). Suppose, in addition, that
(1) u and v are nonnegative;
(2) g is nonnegative and nondecreasing.
If
then
Furthermore, if v is nondecreasing, then
Proof
See [42]. □
Remark 4.9
([41])
Let \(\alpha >0\), \(I=[0, R]\), and \(\varPsi \in \mathcal{C}^{1}(I, \mathbb{R})\) be increasing with \({\varPsi '}(\mathfrak{z}) \neq 0\) for all \(\mathfrak{z} \in I\). 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
Lemma 4.10
Let \(\overset{\backsim }{u} \in \mathcal{C} ([0,R], \mathbb{R}) \) is a solution of the inequality (34) and \(\alpha _{i} \in (0,1]\), \(i \in \{1,2\}\). Then \(\overset{\backsim }{u} \in \mathcal{C} ([0,R], \mathbb{R})\) satisfies
where
with Λ given by (16).
Proof
Let \(\overset{\backsim }{u}\) 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
By using Remark 4.6(1), we acquire
The proof of (37) is finished. □
Theorem 4.11
Let (H1) and (H2) hold. The (FGSLL)-problem (1) is UH stable in \(\mathcal{C}([0,R],\mathbb{R})\).
Proof
Let \(\overset{\backsim }{u} \in \mathcal{C} ([0,R], \mathbb{R})\) be a solution of (34), and \(u \in \mathcal{C} ([0,R], \mathbb{R}) \) be a unique solution of (1). By using Lemma 4.10, it gives
where
Clearly, if \(u(0)=\overset{\backsim }{u}(0)\) and
then we obtain that \(\mathcal{X}(\mathfrak{z})=\mathcal{Z}(\mathfrak{z})\).
By the help of Lemma 4.10 and the known inequality \(\vert u+v \vert \le \vert u \vert + \vert v \vert \) for any \(\mathfrak{z}\in [0,R] \), we get
For simplicity, we take \(\mathcal{C}_{f}:=\Lambda \mathbb{E}_{\alpha _{1}+\alpha _{2}} ( L_{f} ( \varPsi (R) -\varPsi ( 0 ) )^{ \alpha _{1}+ \alpha _{2} } )\). Then (41) becomes
Thus, the (FGSLL)-problem (1) is UH stable. □
Corollary 4.12
Let (H1) and (H2) hold.
-
If \(\chi (\mathfrak{z}) = 0\) for all \(\mathfrak{z} \in I\), then we have \(\overline{\chi}=0\) and the (FSL)-problem (2) is UH stable in \(\mathcal{C} ([0,R], \mathbb{R})\).
-
If \(\eta (\mathfrak{z}) = 1\) and \(\chi (\mathfrak{z}) = \lambda \) (\(\lambda \in \mathbb{R}\)) for \(\mathfrak{z} \in I\), then we have \(\underline{\eta}=1\) and \(\overline{\chi}=|\lambda |\). We also find that the (FL)-problem (3) is UH stable in \(\mathcal{C} ([0,R], \mathbb{R})\).
Now, if \(\Upsilon (\varepsilon )= \varepsilon \mathcal{C}_{f}\) with \(\Upsilon (0)=0\), we have a corollary as follows.
Corollary 4.13
Let (H1) and (H2) hold. Then the (FGSLL)-problem (1) is GUH stable in \(\mathcal{C} ([0,R], \mathbb{R})\).
-
If \(\chi (\mathfrak{z}) = 0\) for all \(\mathfrak{z} \in I\), then \(\overline{\chi}=0\) and the (FSL)-problem (2) is GUH stable in \(\mathcal{C} ([0,R], \mathbb{R})\).
-
If \(\eta (\mathfrak{z}) = 1\) and \(\chi (\mathfrak{z}) = \lambda \) (\(\lambda \in \mathbb{R}\)) for \(\mathfrak{z} \in I\), then we have \(\underline{\eta} =1\) and \(\overline{\chi}=|\lambda |\). We also have that the (FL)-problem (3) is GUH stable in \(\mathcal{C} ([0,R], \mathbb{R})\).
In the sequel, we focus on the UHR and generalized UHR stability.
Lemma 4.14
Let \(\alpha _{i} \in (0,1]\), \(i \in \{1,2\}\), and suppose \(\overset{\backsim }{u} \in \mathcal{C} ([0,R], \mathbb{R}) \) is a solution of (34).
Then \(\overset{\backsim }{u} \in \mathcal{C} ([0,R], \mathbb{R})\) satisfies
where
and \(\mathcal{Z}\) is given by (38).
Proof
Assuming that \(\overset{\backsim}{u}\) 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. □
Theorem 4.15
Let (H1), (H2), and (H6) hold. Then the (FGSLL)-problem (1) is UHR stable in \(\mathcal{C}([0,R],\mathbb{R})\).
Proof
Let \(\overset{\backsim }{u} \in \mathcal{C} ([0,R], \mathbb{R})\) be a solution of (36) and u be a unique solution for the (FGSLL)-problem (1). By applying Lemma 4.14, it yields that
where \(\mathcal{X} \) is given by (40). Similarly, if \(u(0)=\overset{\backsim }{u}(0)\) and
then \(\mathcal{X}(\mathfrak{z})=\mathcal{Z}(\mathfrak{z})\).
Applying Lemma 4.14, the triangle inequality, and inequality (14), for any \(t \in [0,R] \), we then may write
Since ϒ is nondecreasing (see condition (H6)), for all \(s \in [0,\mathfrak{z}] \), we obtain \(\Upsilon (s)\le \Upsilon (\mathfrak{z})\) and can write
where Ψ is provided by (42). Thus,
with
Then, the (FGSLL)-problem (1) is UHR stable. □
Corollary 4.16
Let the assumptions (H1), (H2), and (H6) hold.
-
If \(\chi (\mathfrak{z}) = 0\) for all \(\mathfrak{z} \in I\), then \(\overline{\chi}=0\) and the (FSL)-problem (2) is UHR stable in \(\mathcal{C} ([0,R], \mathbb{R})\).
-
If \(\eta (\mathfrak{z}) = 1\) and \(\chi (\mathfrak{z}) = \lambda \) (\(\lambda \in \mathbb{R}\)) for \(\mathfrak{z} \in I\), then we have \(\underline{\eta}=1\) and \(\overline{\chi}=|\lambda |\). Furthermore, the (FL)-problem (3) is UHR stable in \(\mathcal{C} ([0,R], \mathbb{R})\).
Now, we take \(\varepsilon =1\) in \(\vert \overset{\backsim }{u}(\mathfrak{z})-u(\mathfrak{z}) \vert \le \mathcal{C}_{f,\Upsilon} \Upsilon (\mathfrak{z}) \varepsilon \) with \(\Upsilon (0)=0\). Then we have the following.
Corollary 4.17
Suppose that (H1), (H2), and (H6) are fulfilled. Then the (FGSLL)-problem (1) is GUHR stable in \(\mathcal{C} ([0,R], \mathbb{R})\).
-
If \(\chi (\mathfrak{z}) = 0\) for all \(\mathfrak{z} \in I\), then we have \(\overline{\chi}=0\) and the (FSL)-problem (2) is GUHR stable in \(\mathcal{C} ([0,R], \mathbb{R})\).
-
If \(\eta (\mathfrak{z}) = 1\) and \(\chi (\mathfrak{z}) = \lambda \) (\(\lambda \in \mathbb{R}\)) for \(\mathfrak{z} \in I\), then we have \(\underline{\eta} =1\) and \(\overline{\chi}=|\lambda |\). In addition, the (FL)-problem (3) is GUHR stable in \(\mathcal{C} ([0,R], \mathbb{R})\).
5 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).
• First case. We fix \(\alpha _{1}=\frac{1}{3}\), \(\alpha _{2}=\frac{5}{6}\), \(\sigma _{1}=\frac{2}{3}\), \(\sigma _{2}= \frac{1}{2}\), \(\varPsi (\mathfrak{z})=\mathfrak{z}\) for \(\mathfrak{z} \in [0,1]\), \(\eta (\mathfrak{z})=e^{10^{2}}+10^{-3}\mathfrak{z}\), and \(\chi ( \mathfrak{z})= 0\) for \(\mathfrak{z} \in [0,1]\). We have \(\overline{\chi}=\chi (1)=0\), \(\eta (1)=e^{10^{2}}+10^{-3}\) and \(\underline{\eta}=e^{10^{2}}\). In this case, the (FGSLL)-problem (1) is reduced to the (CSL)-problem (Caputo-type Sturm–Liouville)
The conditions (H1) and (H2) are satisfied so that
Then, we have \(L_{f}=\frac {e}{10^{2}}\). Hence,
where \(\mathcal{J}\) is given by (20). Now, all the assumptions of Theorem 3.2 are satisfied. Thus the (CSL)-problem (44) has a unique solution on \([0,1]\).
Similarly, by choosing \(\theta >0\) large enough such that
where \(\Lambda _{\theta} \) and \(\nabla _{\theta}\) are the constants given by (18) and (19), the conditions of Theorem 3.3 are fulfilled. Thus the (CSL)-problem (44) has a unique solution on \([0,1]\). Moreover, we have
Hence, from Theorem 4.11, the (CSL)-problem (44) is UH and GUH stable on \([0,1]\).
By taking \(\Upsilon (\mathfrak{z})= ( \varPsi (\mathfrak{z}) -\varPsi ( 0 ) )^{\frac{1}{6}}=\mathfrak{z}^{\frac{1}{6}} \), it follows that
The inequality (14) is satisfied with
where
Then
Therefore, in view of Theorem 4.15, the (CSL)-problem (44) is UHR and GUHR stable on \([0,1]\).
• Second case. We fix \(\alpha _{1}=\frac {1}{3}\), \(\alpha _{2}=\frac {5}{6}\), \(\sigma _{1}=\frac {2}{3}\), \(\sigma _{2}=\frac {1}{2}\), \(\varPsi (\mathfrak{z})=\mathfrak{z}\), \(\eta (\mathfrak{z}) = 1\) for \(\mathfrak{z} \in [0,1]\), and \(\chi (\mathfrak{z})=10^{-3}\) for \(\mathfrak{z} \in [0,1]\). We get \(\eta (1)=\underline{\eta}=1\) and \(\chi (1)=\overline{\chi}=10^{-3}\). In this case, the (FGSLL)-problem (1) is reduced to (CL)-problem (Caputo-type Langevin)
The conditions (H1) and (H2) are satisfied with \(L_{f}=\frac {e}{10^{2}}\). Hence,
All the assumptions of Theorem 3.2 hold. Hence, the (CL)-problem (45) has a unique solution on \([0,1]\). Similarly, by choosing \(\theta >0\) large enough such that
all the conditions of Theorem 3.3 are fulfilled. Then the (CL)-problem (45) admits one solution uniquely on \([0,1]\). In addition, we have
From Theorem 4.11, it follows that the (CL)-problem (45) is UH and GUH stable on \([0,1]\). Taking \(\Upsilon (\mathfrak{z})= ( \varPsi (\mathfrak{z}) -\varPsi ( 0 ) )^{\frac{1}{6}}=\mathfrak{z}^{\frac{1}{6}} \), we obtain
The inequality (14) is satisfied with
where
Then
Therefore, from Theorem 4.15, the (CL)-problem (45) is UHR and GUHR stable on \([0,1]\).
Example 5.2
We start with the (FGSLL)-problem (1) and choose \(\alpha _{1}=\frac{4}{5}\), \(\alpha _{2}=\frac{\sqrt{5}}{7}\), \(\sigma _{1}=\frac{2}{7}\), \(\sigma _{2}=\frac{1}{4}\), \(\varPsi (x)=x^{3}\). For \(\mathfrak{z} \in [0,1]\), \(\eta (\mathfrak{z})=1\) and \(\chi (\mathfrak{z})=10^{-4}\) for \(\mathfrak{z} \in [0,1]\), we have \(\underline{\eta}=\eta (1)=1\) and \(\chi (1)=\overline{\chi}=10^{-4}\). In this case, the (FGSLL)-problem (1) is reduced to (CEKL)-problem (Caputo–Erdélyi–Kober-type Langevin)
The conditions (H1) and (H3) are satisfied with
Hence,
The assumptions of Theorem 3.7 are met. Hence, the (CEKL)-problem (46) has at least one solution defined on \([0,1]\). Similarly, by choosing \(\theta >0\) large enough such that
and by utilizing Theorem 3.9, we conclude the (CEKL)-problem (46) has at least one solution defined on \([0,1]\).
Example 5.3
Based on the (FGSLL) problem (1), we take \(\alpha _{1}=\frac{3}{4}\), \(\alpha _{2}=\frac{\sqrt{5}}{7}\), \(\sigma _{1}=\frac{2}{7}\), \(\sigma _{2}=\frac{1}{4}\), \(\varPsi (x)=x^{3}\), \(\eta (\mathfrak{z})= 1\) for \(\mathfrak{z} \in [0,1]\), and \(\chi (\mathfrak{z})= 10^{-2}\) for \(\mathfrak{z} \in [0,1]\). We have \(\underline{\eta}=\eta (1)=1\) and \(\chi (1)=\overline{\chi}= 10^{-2}\). In this case, the (FGSLL)-problem (1) is reduced to (CEKL)-problem (Caputo–Erdélyi–Kober-type Langevin)
The conditions (H1), (H3), (H4), and (H5) are satisfied with
and
such that \(\mathfrak{g}(\mathfrak{z})= \frac {1}{3e^{\mathfrak{z}^{2}} \sqrt{1+\mathfrak{z}^{4}}}\), \(d=1\), and \(\Vert \mathfrak{g} \Vert _{\theta}=\frac {1}{3}\), where
and
Hence, from Theorem 3.13, we conclude that the (CEKL)-problem (48) has at least one solution defined on \([0,1]\).
6 Conclusion
We conclude this paper with some useful findings. First, we studied the existence and uniqueness of solutions for a new class generalizing the differential equations of Sturm–Liouville–Langevin (1) including two fractional derivative operators in the Ψ-Caputo sense. When \(\chi (\mathfrak{z})=0\) for \(\mathfrak{z} \in I\), we obtained the (FSL)-differential equation (2) (Sturm–Liouville problem), and if \(\eta (\mathfrak{z}) = 1\) and \(\chi (\mathfrak{z}) = \lambda \) (\(\lambda \in \mathbb{R}\)) for \(\mathfrak{z} \in I\), we obtained the (FL)-differential equation (3) (Langevin problem). The acquired results have been established via Banach’s contraction, Krasnoselskii and Leray–Schauder fixed point theorems using some norm inequalities of the Ψ-Bielecki-type. Moreover, we proved different kinds of stability in the sense of Ulam, such as Ulam–Hyers, Ulam–Hyers–Rassias, generalized Ulam–Hyers and generalized Ulam–Hyers–Rassias. Also, to prove our results, we applied the generalized Gronwall integral inequality.
The second main idea of the current research was to use the Ψ-Bielecki-type norm to reduce the constraints of the (FGSLL)-problem (1) to prove the results of existence and uniqueness. The advantage of this norm (Bielecki’s norm) can be found by comparing the conditions of Theorems 3.2 and 3.3, and by removing the strong condition \(\mathcal{J}<1\) that appeared in proving Theorem 3.2 using the classical supremum norm, while Theorem 3.3 does not require this condition. It is also done by comparing the conditions of Theorems 3.7 and 3.9. In a future work, researchers may consider using the Ψ-Hilfer or other fractional derivative operators, such as the fractal-fractional derivative, to establish the existence, uniqueness, and stability of solutions to the (FGSLL)-problem (1).
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The third and fourth authors would like to thank Azarbaijan Shahid Madani University.
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H.S. and B.T. and S.E. and I.A. dealt with the conceptualization, supervision, methodology, investigation, and writing-original draft preparation. H.S. and B.T. and S.E. and I.A. and S.R. made the formal analysis, writing-review, editing. All authors read and approved the final manuscript.
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Serrai, H., Tellab, B., Etemad, S. et al. Ψ-Bielecki-type norm inequalities for a generalized Sturm–Liouville–Langevin differential equation involving Ψ-Caputo fractional derivative. Bound Value Probl 2024, 81 (2024). https://doi.org/10.1186/s13661-024-01863-1
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DOI: https://doi.org/10.1186/s13661-024-01863-1