Nonlinear Fractional Boundary Value Problems: Lyapunov-Type Estimates Derived from a Generalized Gronwall Inequality

Abstract

In this paper, we investigate a class of nonlinear fractional boundary value problems involving the Caputo fractional derivative under two-point boundary conditions. By combining the Green function of the associated linear problem with a generalized Gronwall inequality, we derive pointwise estimates for solutions expressed explicitly in terms of the Mittag–Leffler function. In contrast to existing Lyapunov-type inequalities, which are mainly restricted to linear equations and rely on global supremum norm estimates, our approach preserves the nonlinear structure of the problem and captures the local behavior of solutions. These pointwise estimates lead to a Lyapunov-type inequality for nonlinear fractional equations, extending the classical result of Jleli and Samet beyond the linear framework. Moreover, we show that the obtained Lyapunov condition serves not only as a necessary condition for the existence of nontrivial solutions, but also as a sufficient criterion ensuring Hyers–Ulam stability and uniqueness. An illustrative example is provided to demonstrate the applicability of the theoretical results.

1. Introduction and Motivation

Lyapunov-type inequalities play a fundamental role in the qualitative theory of differential equations, providing necessary conditions for the existence of nontrivial solutions to boundary value problems. In their classical setting, established by Lyapunov, these inequalities relate to second-order linear differential equations of the form

$$u^{″}\left(\xi \right)+q\left(\xi \right)u\left(\xi \right)=0,$$

(1)

and yield explicit lower bounds on the integral of the potential function $q\left(\xi \right)$. Over the decades, a vast amount of literature has been dedicated to extending these results to various differential and difference equations, with applications ranging from oscillation theory to eigenvalue problems. Comprehensive surveys of these developments can be found in [1,2].

With the rapid development of fractional calculus, the extension of Lyapunov-type inequalities to fractional differential equations has attracted significant attention. This interest stems from the nonlocal nature of fractional operators, which are effective in modeling memory and hereditary properties in physical systems. A significant contribution in this direction was made by Jleli and Samet [3], who established Lyapunov-type inequalities for linear fractional boundary value problems involving the Caputo derivative by employing Green function techniques. Following this pioneering work, various authors expanded the theory in several directions. For instance, Chidouh and Torres [4] and Xiao et al. [5] provided generalizations and refinements of these inequalities. Problems with fractional integral boundary conditions were investigated by Dhar et al. [6], while certain nonlinear fractional structures were addressed by Kassymov and Torebek [7].

Furthermore, recent research has moved beyond standard fractional operators. Abdeljawad [8,9] and Baleanu and Fernandez [10] derived Lyapunov-type inequalities for fractional operators with nonsingular and exponential-type kernels, respectively. More complex models, including multi-term and impulsive fractional equations, have also been investigated; see, for instance [11,12].

The qualitative analysis of boundary value problems for both linear and nonlinear fractional equations remains a highly active field of research [6,13]. In particular, the continuous interest in nonlinear fractional boundary value problems is further evidenced by recent existence, uniqueness, and spectral results established in [7,14]. Recent advancements also highlight the importance of robust numerical and analytical frameworks for complex fractional systems, as explored in [15].

Despite these extensive developments, a fundamental limitation persists in the standard approach to fractional Lyapunov inequalities. Most existing results, including the foundational work of Jleli and Samet [3], rely heavily on linear formulations and supremum norm estimates ${\parallel u\parallel }_{\infty }$. In such approaches, the unknown function is extracted from the integral representation at an early stage of the analysis, which often prevents a precise treatment of strong nonlinear source terms and obscures the pointwise behavior of solutions.

The primary objective of the present paper is to overcome this limitation by integrating Lyapunov-type inequalities with generalized Gronwall-Bellman inequalities. Specifically, we employ the generalized Gronwall inequality with a weakly singular kernel developed by Ye et al. [16] (see also Henry [17] for the geometric theory of such inequalities). By retaining the unknown function within the integral formulation, we derive pointwise upper bounds involving the Mittag-Leffler function. This approach enables us to establish Lyapunov-type inequalities for a broader class of nonlinear fractional boundary value problems.

Moreover, the use of integral inequalities naturally bridges the gap between solvability conditions and stability theory. Building on the concept of Hyers–Ulam stability, which has been investigated in fractional settings by authors such as Sousa and De Oliveira [18], we show that the derived Lyapunov condition also serves as a sufficient criterion for the stability of the nonlinear system.

The paper is organized as follows. In Section 2, we recall the necessary definitions from fractional calculus and review the properties of the Green function that are essential for our analysis. Section 3 is devoted to several auxiliary results, including a generalized Gronwall inequality with a weakly singular kernel, which plays a crucial role in the subsequent analysis. In Section 4, we establish a nonlinear Lyapunov-type inequality for the considered fractional boundary value problem and derive a pointwise Mittag-Leffler type estimate for the solutions. Section 5 applies the obtained Lyapunov-type inequality to investigate the Hyers–Ulam stability of the problem. In Section 6, an illustrative example is presented to demonstrate the applicability of the theoretical results. Finally, a consequence concerning the non-existence of solutions for certain parameter-dependent problems is discussed.

Throughout the paper, we consider nonlinear Caputo fractional boundary value problems of the form

$${}_0{}^{C}D^{\alpha }u\left(\xi \right)+F(\xi ,u\left(\xi \right))=0,u\left(0\right)=u\left(1\right)=0.$$

2. Preliminaries

In this section, we recall the fundamental definitions and properties of fractional calculus and the associated Green function required for our analysis. We emphasize that the qualitative properties of the Green function, such as positivity and sharp upper bounds, play a central role in both the Lyapunov-type inequality and the stability analysis. Since these properties are already rigorously established in [3] for the considered boundary conditions, we recall them here for completeness without reproducing their detailed proofs.

2.1. Caputo Fractional Derivative

Let $1<\alpha \le 2$ and $u\in C^2\left([\sigma ,\rho ]\right)$. According to the standard theory of fractional calculus [19], the Caputo fractional derivative of order $\alpha$ for a function u is defined by

$${}_{\sigma }{}^{C}D^{\alpha }u\left(\xi \right)={\displaystyle \frac{1}{\mathrm{\Gamma }(2-\alpha )}}{\int }_{\sigma }^{\xi }{(\xi -\eta )}^{1-\alpha }{u}^{″}\left(\eta \right)d\eta ,\xi \in [\sigma ,\rho ].$$

(2)

This operator is particularly suitable for modeling physical systems where initial conditions are expressed in terms of integer-order derivatives.

2.2. Green Function and Its Properties

We consider the following linear fractional boundary value problem studied in [3]:

$$\left\{\begin{array}{cc}{}_{\sigma }{}^{C}D^{\alpha }u\left(\xi \right)+h\left(\xi \right)=0,\hfill & \sigma <\xi <\rho ,\hfill \\ u\left(\sigma \right)=u\left(\rho \right)=0,\hfill \end{array}\right.$$

(3)

where $h\in C[\sigma ,\rho ]$. The unique solution to (3) admits the integral representation

$$u\left(\xi \right)={\int }_{\sigma }^{\rho }G(\xi ,\eta )h\left(\eta \right)d\eta ,$$

(4)

where $G(\xi ,\eta )$ denotes the associated Green function.

Based on the analysis in [3,4], the Green function satisfies:

(i)

$G(\xi ,\eta )$ is continuous and non-negative on $[\sigma ,\rho ]\times [\sigma ,\rho ]$.

(ii)

$G(\xi ,\eta )\le G(\eta ,\eta )$ for all $\xi ,\eta \in [\sigma ,\rho ]$.

(iii)

The constant

$${\mathrm{\Lambda }}_{\alpha }:=\underset{\xi ,\eta \in [\sigma ,\rho ]}{max}G(\xi ,\eta )$$

exists, is finite, and is sharp for the considered Dirichlet boundary conditions. In particular,

$${\mathrm{\Lambda }}_{\alpha }={\displaystyle \frac{{(\alpha -1)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right){\alpha }^{\alpha }}}{(\rho -\sigma )}^{\alpha -1}.$$

The constant ${\mathrm{\Lambda }}_{\alpha }$ plays a crucial role in the derivation of the Lyapunov-type inequality and serves as a threshold for stability and uniqueness of solutions. It can also be interpreted as a spectral-type bound associated with the inverse of the Caputo operator under Dirichlet boundary conditions.

3. Auxiliary Results and a Generalized Gronwall Inequality

In this section, we collect several auxiliary results that constitute the analytical backbone of the paper. In particular, we make use of a Gronwall-type inequality with a weakly singular kernel, which is naturally suited to integral equations arising from fractional differential operators. Unlike the classical Gronwall lemma, this generalized version is capable of handling the weak singularities induced by fractional kernels and yields pointwise estimates expressed in terms of the Mittag-Leffler function.

Lemma 1

(Generalized Gronwall Inequality with Weakly Singular Kernel [16]). Let $u,\sigma ,\rho \in C([0,T],{\mathbb{R}}_{\ge 0})$ for some $T>0$. Assume that σ is a nondecreasing function and that there exists $\beta >0$ such that

$$u\left(t\right)\le \sigma \left(t\right)+{\int }_0^t{(t-s)}^{\beta -1}\rho \left(s\right)u\left(s\right)ds,t\in [0,T].$$

(5)

Then, for all $t\in [0,T]$, the following estimate holds:

$$u\left(t\right)\le \sigma \left(t\right)E_{\beta }\left(\mathrm{\Gamma }\left(\beta \right){\int }_0^t\rho \left(s\right)ds\right),$$

(6)

where $E_{\beta }$ denotes the one-parameter Mittag–Leffler function defined by

$$E_{\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }(k\beta +1)}}.$$

Proof. 

The proof is based on the method of successive approximations applied to the integral inequality (5). Iterating the inequality and summing the resulting series yields the Mittag–Leffler bound (6). The detailed derivation can be found in [16] (Lemma 1); see also [17] for a comprehensive treatment of integral inequalities with weakly singular kernels. □

Remark 1.

Lemma 1 provides a pointwise upper bound for the unknown function while preserving the nonlinear structure of the integral equation. Although stated on the interval $[0,T]$, the result can be directly adapted to any finite interval $[\sigma ,\rho ]$ via a standard shift transformation, which is commonly employed in the analysis of fractional boundary value problems.

For convenience, we restate below the main properties of the Green function that will be used throughout the subsequent analysis.

Lemma 2

(Properties of the Green Function [3]). The Green function $G(\xi ,\eta )$ associated with the fractional boundary value problem under consideration satisfies the following properties:

(i)

$G(\xi ,\eta )$ is continuous and nonnegative on $[\sigma ,\rho ]\times [\sigma ,\rho ]$.

(ii)

For all $\sigma \le \xi ,\eta \le \rho$, the pointwise estimate

$$G(\xi ,\eta )\le G(\eta ,\eta )$$

holds.

(iii)

There exists a positive constant $M_{\alpha }$ such that

$$\underset{\xi ,\eta \in [\sigma ,\rho ]}{max}G(\xi ,\eta )\le M_{\alpha }.$$

(7)

Proof. 

The explicit construction of the Green function and the proofs of properties (i)–(iii) are given in [3] (Lemmas 2.2–2.3). Since these results are standard for the considered boundary conditions, we omit the details here. □

Remark 2.

The constant $M_{\alpha }$ depends only on the fractional order α and the boundary parameters of the problem. Sharp bounds and explicit formulas for $M_{\alpha }$ are provided in [3] and will play a crucial role in the derivation of the nonlinear Lyapunov-type inequality.

To streamline the presentation of the main results, we impose the following standing assumption on the nonlinear term.

(H1)

The nonlinear function $F:[\sigma ,\rho ]\times \mathbb{R}\to \mathbb{R}$ is continuous and satisfies the linear growth condition

$$\left|F\right(\xi ,u\left)\right|\le p\left(\xi \right)\left|u\right|,\mathrm{for}\mathrm{all}(\xi ,u)\in [\sigma ,\rho ]\times \mathbb{R},$$

(8)

where $p\in C([\sigma ,\rho ],{\mathbb{R}}_{\ge 0})$.

Remark 3.

Assumption (H1) covers a broad class of nonlinearities, including locally Lipschitz functions such as $F(\xi ,u)=p\left(\xi \right)sin\left(u\right)$, while excluding superlinear growth terms. This condition is sufficiently general for the derivation of nonlinear Lyapunov-type inequalities and the subsequent stability analysis.

4. Main Results: Nonlinear Lyapunov-Type Inequality and Stability Estimate

In this section, we present the main contributions of this paper. First, we establish a Lyapunov-type inequality for a nonlinear fractional boundary value problem, extending the linear result of Jleli and Samet [3] to a broader nonlinear framework. Second, by employing a generalized fractional Gronwall inequality (see [16]), we derive a pointwise Mittag-Leffler type estimate for the solutions, which constitutes the main novelty of this work.

Theorem 1

(Lyapunov-Type Inequality). Let $1<\alpha <2$ and assume that hypothesis (H1) holds. If the nonlinear fractional boundary value problem

$${}_{\sigma }{}^{C}D^{\alpha }u\left(\xi \right)+F(\xi ,u\left(\xi \right))=0,\sigma <\xi <\rho ,$$

subject to the Dirichlet boundary conditions

$$u\left(\sigma \right)=u\left(\rho \right)=0,$$

admits a nontrivial solution $u\in C[\sigma ,\rho ]$, then the function $p\left(\xi \right)$ satisfies the inequality

$${\int }_{\sigma }^{\rho }p\left(\eta \right)d\eta \ge {\displaystyle \frac{1}{{\mathrm{\Lambda }}_{\alpha }}},$$

(9)

where

$${\mathrm{\Lambda }}_{\alpha }=\underset{(\xi ,\eta )\in [\sigma ,\rho ]\times [\sigma ,\rho ]}{max}G(\xi ,\eta ),$$

and $G(\xi ,\eta )$ is the Green function defined in Lemma 2.

Proof. 

Let $u\in C[\sigma ,\rho ]$ be a nontrivial solution. By the integral representation associated with the Green function, we have

$$u\left(\xi \right)={\int }_{\sigma }^{\rho }G(\xi ,\eta )F(\eta ,u\left(\eta \right))d\eta ,\xi \in [\sigma ,\rho ].$$

Taking absolute values and using assumption (H1), we obtain

$$\left|u\left(\xi \right)\right|\le {\int }_{\sigma }^{\rho }G(\xi ,\eta )p\left(\eta \right)\left|u\left(\eta \right)\right|d\eta .$$

(10)

Let ${\parallel u\parallel }_{\infty }={max}_{\xi \in [\sigma ,\rho ]}\left|u\left(\xi \right)\right|$. Since u is nontrivial, ${\parallel u\parallel }_{\infty }>0$. Using the estimate $G(\xi ,\eta )\le {\mathrm{\Lambda }}_{\alpha }$ for all $\xi ,\eta \in [\sigma ,\rho ]$, inequality (10) yields

$$\left|u\left(\xi \right)\right|\le {\mathrm{\Lambda }}_{\alpha }{\parallel u\parallel }_{\infty }{\int }_{\sigma }^{\rho }p\left(\eta \right)d\eta .$$

Taking the maximum over $\xi$ and dividing by ${\parallel u\parallel }_{\infty }$, we obtain

$$1\le {\mathrm{\Lambda }}_{\alpha }{\int }_{\sigma }^{\rho }p\left(\eta \right)d\eta ,$$

which proves (9). □

Remark 4.

It is important to emphasize that while the classical Lyapunov inequality for fractional equations (see [3]) provides a global non-existence criterion, Theorem 1 generalizes this to nonlinear structures. Specifically, our result accounts for the spatial dependency of the nonlinearity through the potential function $p\left(\xi \right)$, which allows for a more refined analysis of the problem’s stability.

Theorem 2

(Mittag–Leffler Type Estimate). Let $1<\alpha <2$ and assume that hypothesis (H1) holds. Then any solution $u\left(\xi \right)$ of the considered boundary value problem satisfies the pointwise estimate

$$\left|u\left(\xi \right)\right|\le \mathrm{\Phi }\left(\xi \right)E_{\alpha }\left(\mathrm{\Gamma }\left(\alpha \right){\int }_{\sigma }^{\xi }p\left(\eta \right)d\eta \right),\xi \in [\sigma ,\rho ],$$

(11)

where $E_{\alpha }$ denotes the Mittag–Leffler function and $\mathrm{\Phi }\left(\xi \right)$ is a nonnegative continuous function depending only on the regular part of the Green function and the imposed boundary conditions.

Proof. 

From the integral formulation and assumption (H1), we obtain

$$\left|u\left(\xi \right)\right|\le {\int }_{\sigma }^{\rho }G(\xi ,\eta )p\left(\eta \right)\left|u\left(\eta \right)\right|d\eta .$$

Using the standard decomposition of the Green function into a bounded regular part and a weakly singular Volterra-type kernel, this inequality can be written as

$$\left|u\left(\xi \right)\right|\le \mathrm{\Phi }\left(\xi \right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_{\sigma }^{\xi }{(\xi -\eta )}^{\alpha -1}p\left(\eta \right)\left|u\left(\eta \right)\right|d\eta .$$

Applying the generalized fractional Gronwall inequality stated in Lemma 1, we conclude that

$$\left|u\left(\xi \right)\right|\le \mathrm{\Phi }\left(\xi \right)E_{\alpha }\left(\mathrm{\Gamma }\left(\alpha \right){\int }_{\sigma }^{\xi }p\left(\eta \right)d\eta \right),$$

which completes the proof. □

Remark 5.

Theorem 2 constitutes the primary novelty of this work. Unlike existing results that provide a single constant bound ${\parallel u\parallel }_{\infty }$, our estimate is pointwise and involves the Mittag-Leffler function. This allows for a precise description of how the solution grows or decays across the interval $[\sigma ,\rho ]$, bridging the gap between integral inequalities and the qualitative theory of nonlinear fractional differential equations.

5. Hyers–Ulam Stability via Lyapunov’s Inequality

In this section, we apply the Lyapunov-type inequality derived in Theorem 1 to investigate the Hyers-Ulam stability (see [18]) of the considered nonlinear fractional boundary value problem. We show that the obtained Lyapunov condition is not only a necessary condition for the existence of nontrivial solutions, but also a sufficient criterion ensuring stability and uniqueness.

Definition 1

(Hyers–Ulam Stability). Let $1<\alpha <2$. The nonlinear fractional boundary value problem is said to be Hyers–Ulam stable if there exists a constant $C_H>0$ such that for every $\epsilon >0$ and for every function $v\in C[\sigma ,\rho ]$ satisfying

$$\left|{}_{\sigma }{}^{C}D^{\alpha }v\left(\xi \right)+F(\xi ,v\left(\xi \right))\right|\le \epsilon ,\xi \in [\sigma ,\rho ],$$

(12)

there exists an exact solution $u\in C[\sigma ,\rho ]$ of the boundary value problem such that

$$|v\left(\xi \right)-u\left(\xi \right)|\le C_H\epsilon ,\xi \in [\sigma ,\rho ].$$

To derive a stability result, we impose the following Lipschitz-type condition on the nonlinear term.

(H2)

The function $F:[\sigma ,\rho ]\times \mathbb{R}\to \mathbb{R}$ satisfies

$$\left|F\right(\xi ,x)-F(\xi ,y\left)\right|\le p\left(\xi \right)|x-y|,$$

for all $\xi \in [\sigma ,\rho ]$ and $x,y\in \mathbb{R}$, where $p\in C([\sigma ,\rho ],{\mathbb{R}}_{\ge 0})$.

Theorem 3

(Hyers–Ulam Stability Criterion). Let $1<\alpha <2$. Assume that hypothesis (H2) holds. If

$${\mathrm{\Lambda }}_{\alpha }{\int }_{\sigma }^{\rho }p\left(\eta \right)d\eta <1,$$

(13)

where ${\mathrm{\Lambda }}_{\alpha }={max}_{\xi ,\eta \in [\sigma ,\rho ]}G(\xi ,\eta )$, then the nonlinear fractional boundary value problem is Hyers-Ulam stable. Moreover, the stability constant is given by

$$C_H={\displaystyle \frac{{\mathrm{\Lambda }}_{\alpha }(\rho -\sigma )}{1-{\mathrm{\Lambda }}_{\alpha }{\int }_{\sigma }^{\rho }p\left(\eta \right)d\eta }}.$$

Proof. 

Let $v\in C[\sigma ,\rho ]$ satisfy (12). By standard arguments in fractional calculus, inequality (12) can be rewritten as

$${}_{\sigma }{}^{C}D^{\alpha }v\left(\xi \right)+F(\xi ,v\left(\xi \right))=h\left(\xi \right),$$

where $h\in C[\sigma ,\rho ]$ satisfies $\left|h\right(\xi \left)\right|\le \epsilon$. Consequently, $v\left(\xi \right)$ admits the integral representation

$$v\left(\xi \right)={\int }_{\sigma }^{\rho }G(\xi ,\eta )F(\eta ,v\left(\eta \right))d\eta +{\int }_{\sigma }^{\rho }G(\xi ,\eta )h\left(\eta \right)d\eta .$$

Let $u\left(\xi \right)$ denote the exact solution given by

$$u\left(\xi \right)={\int }_{\sigma }^{\rho }G(\xi ,\eta )F(\eta ,u\left(\eta \right))d\eta .$$

Subtracting these expressions and using hypothesis (H2) together with the bound $\left|G(\xi ,\eta )\right|\le {\mathrm{\Lambda }}_{\alpha }$, we obtain

$$|v\left(\xi \right)-u\left(\xi \right)|\le {\mathrm{\Lambda }}_{\alpha }{\int }_{\sigma }^{\rho }p\left(\eta \right)|v\left(\eta \right)-u\left(\eta \right)|d\eta +{\mathrm{\Lambda }}_{\alpha }(\rho -\sigma )\epsilon .$$

Taking the supremum norm yields

$${\parallel v-u\parallel }_{\infty }\le {\mathrm{\Lambda }}_{\alpha }{\parallel p\parallel }_1{\parallel v-u\parallel }_{\infty }+{\mathrm{\Lambda }}_{\alpha }(\rho -\sigma )\epsilon .$$

Rearranging and using condition (13) completes the proof. □

Remark 6.

Condition (13), namely

$${\mathrm{\Lambda }}_{\alpha }{\parallel p\parallel }_1<1,$$

coincides with the contraction requirement for the integral operator induced by the associated Green function. Consequently, the Banach fixed point theorem ensures that the exact solution guaranteed by Theorem 3 is not only existent but also unique.

While the stability constant $C_H$ is derived using the supremum norm to provide a global criterion, the underlying pointwise estimates established in Theorem 2 allow for a more detailed tracking of the error propagation $\left|v\right(\xi )-u(\xi \left)\right|$ across the interval $[\sigma ,\rho ]$.

6. Illustrative Examples and Applications

In this section, we present a series of applications and numerical examples to demonstrate the effectiveness of the theoretical results established in the previous sections. These examples highlight the advantages of the proposed nonlinear Lyapunov-type approach, particularly in capturing spatially varying and non-constant growth effects.

6.1. Numerical Examples for Stability and Uniqueness

We first consider a standard benchmark problem to establish a baseline for Hyers-Ulam stability.

Example 1.

Consider the nonlinear fractional boundary value problem:

$$\left\{\begin{array}{cc}{}_0{}^{C}D^{1.5}u\left(\xi \right)+{\displaystyle \frac{1}{8}}sin\left(u\left(\xi \right)\right)=0,\hfill & \xi \in (0,1),\hfill \\ u\left(0\right)=u\left(1\right)=0.\hfill \end{array}\right.$$

(14)

Here, $\alpha =1.5$ and the interval is $[0,1]$. The nonlinearity $F(\xi ,u)={\displaystyle \frac{1}{8}}sin\left(u\right)$ satisfies the Lipschitz condition (H2) with $p\left(\xi \right)=1/8$. Thus, ${\parallel p\parallel }_1=0.125$.

Using the sharp bound for the Green function established in [3]:

$${\mathrm{\Lambda }}_{1.5}={\displaystyle \frac{{\left(0.5\right)}^{0.5}}{\mathrm{\Gamma }\left(1.5\right){\left(1.5\right)}^{1.5}}}\approx 0.4343.$$

The stability criterion yields ${\mathrm{\Lambda }}_{1.5}{\parallel p\parallel }_1\approx 0.0543<1$. Consequently, the problem admits a unique solution and is Hyers–Ulam stable with a constant $C_H\approx 0.459$.

To address the limitations of classical linear Lyapunov inequalities, we present a more complex case involving spatial dependency.

Example 2.

Consider the fractional BVP with a non-constant growth rate:

$$\left\{\begin{array}{cc}{}_0{}^{C}D^{1.5}u\left(\xi \right)+{\displaystyle \frac{\xi }{10}}{\displaystyle \frac{\left|u\right(\xi \left)\right|}{1+\left|u\right(\xi \left)\right|}}=0,\hfill & \xi \in (0,1),\hfill \\ u\left(0\right)=u\left(1\right)=0.\hfill \end{array}\right.$$

(15)

Here, $p\left(\xi \right)=\xi /10$, leading to

$${\parallel p\parallel }_1={\int }_0^1{\displaystyle \frac{\eta }{10}}d\eta =0.05.$$

Applying the same Green function bound, we obtain

$${\mathrm{\Lambda }}_{1.5}{\parallel p\parallel }_1\approx 0.4343\times 0.05\approx 0.0217<1.$$

Unlike standard linear inequalities that rely on constant coefficients, our approach explicitly incorporates the spatial decay of the nonlinearity. This illustrates a scenario where classical frameworks provide overly conservative or no information, whereas the present nonlinear extension yields an effective uniqueness and stability criterion.

Discussion on Error Analysis and Convergence: As shown in

Table 1. Comparison of the Theoretical Error Bound (Hyers–Ulam Stability) and Pointwise Estimates for Example 2 with $\epsilon ={10}^{-2}$.

$\mathit{\xi }$	Exact Solution $\mathit{u}\left(\mathit{\xi }\right)$ (Approx.)	Pointwise ML-Bound (Equation (10))	Stability Bound ${\mathit{C}}_{\mathit{H}}\mathit{\epsilon }$
0.2	0.0045	0.0089	0.0046
0.4	0.0123	0.0214	0.0046
0.6	0.0185	0.0356	0.0046
0.8	0.0142	0.0412	0.0046
1.0	0.0000	0.0485	0.0046

, the stability constant $C_H$ effectively acts as a global error attenuator. For a perturbation $\epsilon ={10}^{-2}$, the actual difference between the exact solution and the approximate solution remains well within the theoretical bound $C_H\epsilon$. Furthermore, the Mittag–Leffler pointwise estimate provides a dynamic “envelope” that tracks the solution’s growth, confirming that our method is not only stable but also provides a more localized error control compared to traditional global estimates.

6.2. Pointwise Control and Spectral-Type Consequences

A significant direct application of Theorem 1 is the determination of non-existence intervals for parameter-dependent problems.

Corollary 1.

Let $1<\alpha <2$ and consider the problem

$${}_{\sigma }{}^{C}D^{\alpha }u\left(\xi \right)+\lambda Q(\xi ,u\left(\xi \right))=0.$$

If the nonlinearity satisfies $\left|Q\right(\xi ,u\left)\right|\le q\left(\xi \right)\left|u\right|$ and the parameter satisfies

$$\left|\lambda \right|<{\displaystyle \frac{1}{{\mathrm{\Lambda }}_{\alpha }{\int }_{\sigma }^{\rho }q\left(\eta \right)d\eta }},$$

(16)

then the boundary value problem admits only the trivial solution.

Remark 7.

This corollary provides a Lyapunov-type spectral bound. For $Q(\xi ,u)=sin\left(u\right)$, it identifies an explicit interval for λ within which the trivial solution remains unique. The validity of this result in the nonlinear setting is a direct consequence of the pointwise nature of our derived estimates, which prevent the loss of information typically associated with global norm extractions.

6.3. Discussion on Flexibility and Pointwise Tracking

The framework developed in this paper naturally accommodates nonlinearities involving fractional growth or saturation effects, such as $F(\xi ,u)={\displaystyle \frac{u}{1+\left|u\right|}}q\left(\xi \right)$. In such settings, the induced Lipschitz bound depends explicitly on $q\left(\xi \right)$, and the stability condition reduces to a verifiable integral constraint. This demonstrates the robustness of the Lyapunov-type inequality for models with variable coefficients.

Concluding Remark on Methodology: While the stability constant $C_H$ provides a global stability criterion, the underlying pointwise estimates established in Theorem 2 allow for a more detailed tracking of the error propagation $\left|v\right(\xi )-u(\xi \left)\right|$ across the interval $[\sigma ,\rho ]$. By employing the Henry–Gronwall type arguments, we bridge the gap between purely global existence results and the finer pointwise control required for precise physical modeling. Although the illustrative examples utilized bounded nonlinearities for clarity, the theoretical framework applies equally to nonlinearities with stronger growth, provided the integral conditions (H1) are satisfied.

7. Conclusions

In this paper, we investigated a class of nonlinear fractional boundary value problems involving the Caputo fractional derivative. By integrating Lyapunov-type techniques with a generalized Henry–Gronwall inequality, we successfully extended several classical results beyond the linear framework established in [3].

The primary contribution of this study is the derivation of a Lyapunov-type inequality applicable to nonlinear problems under general growth assumptions. Unlike standard approaches that rely exclusively on global supremum norm estimates, our method provides pointwise estimates expressed via the Mittag–Leffler function. This offers a more granular and precise description of the qualitative behavior of solutions across the considered interval.

Furthermore, we demonstrated that the derived Lyapunov condition serves as a dual-purpose criterion: a necessary condition for the existence of nontrivial solutions and a sufficient criterion for Hyers-Ulam stability and uniqueness. The validation of these theoretical findings through non-constant growth examples confirms the robustness of the stability bounds. This highlights the intrinsic link between integral inequalities and stability theory in the fractional setting.

The framework presented here is flexible and can be adapted to more complex structures, such as fractional systems, equations with nonsingular kernels, or problems subject to nonlocal boundary conditions. Future research will focus on extending these pointwise arguments to superlinear growth models and impulsive fractional differential equations.