A Darbo Fixed-Point Approach for Variable-Order Hybrid Caputo–Hadamard Sequential Differential Equations ^†

Abstract

This study examines a boundary value problem involving a variable-order hybrid Caputo–Hadamard sequential fractional differential equation with nonseparated boundary conditions. Variable-order derivatives are used to model memory effects that change over time and cannot be captured by constant-order models. The study establishes an equivalence lemma, providing an explicit integral representation via a boundary-matrix formulation and identifying the corresponding fixed-point operator. Under suitable continuity and growth conditions, Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness are used to prove the existence of at least one solution. Ulam–Hyers and Ulam–Hyers–Rassias stability are demonstrated using estimates from the linear solution operator. An analytical example with fixed parameters verifies the assumptions of the main theorem and illustrates the proposed method.

1. Introduction

Fractional differential equations (FDEs) have become an important mathematical concept for modeling processes characterized by memory and hereditary effects. Conventional fractional derivatives typically assume a constant order of differentiation; however, such an assumption may not adequately describe systems whose dynamic behavior changes over time. To overcome this limitation, variable-order fractional derivatives (VO) have been introduced, providing greater flexibility to represent time-dependent memory phenomena. In addition, hybrid fractional operators that combine the Caputo and Hadamard derivatives provide a more comprehensive description of complex dynamical systems by incorporating distinct structural properties of fractional calculus [1].

Investigating boundary value problems involving hybrid fractional operators of variable order poses considerable analytical challenges, particularly regarding the existence, uniqueness, and stability of solutions. To address these issues, advanced techniques from nonlinear functional analysis are required. In this study, Darbo’s fixed-point theorem [2,3], together with the concept of a measure of noncompactness, is employed to establish sufficient conditions for the existence of solutions. Furthermore, the stability results demonstrate the robustness of the proposed model. Several illustrative examples are provided to verify the applicability and validity of the theoretical findings.

The main objective of this work is to analyze a boundary value problem associated with a variable-order hybrid Caputo–Hadamard (C-H) sequential fractional differential equation. By constructing an appropriate operator framework in a Banach space setting, the study extends existing results on fractional differential equations to a broader class of hybrid variable-order models. Moreover, the results obtained contribute to the theoretical development of nonlinear fractional analysis.

Motivated by recent developments in the literature, we consider a boundary value problem involving a variable-order hybrid Caputo–Hadamard sequential fractional differential equation of the form [4]

$${}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)=f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right),t\in J^{\circ }:=(a,1),J=[a,1],a>0,$$

(1)

where the nonlinear integral operators T and G are defined by

$$Tu\left(t\right)={\int }_a^t\lambda (t,s)g(s,u\left(s\right))ds,$$

$$Gu\left(t\right)={\int }_a^t\delta (t,s)g(s,u\left(s\right))ds,t\in J:=[a,1].$$

Equation (1) is supplemented with the nonseparated boundary conditions.

$$\left\{\begin{array}{c}u\left(a\right)={\eta }_{1}u\left(1\right),\hfill \\ u^{\prime }\left(a\right)={\eta }_2{u}^{\prime }\left(1\right),\hfill \\ {}^{CH}D_{a^{+}}^{\beta \left(a\right)}u\left(a\right)={\eta }_3{}^{CH}D_{a^{+}}^{\beta \left(1\right)}u\left(1\right),\hfill \\ {}^{CH}D_{a^{+}}^{\beta \left(a\right)+1}u\left(a\right)={\eta }_4{}^{CH}D_{a^{+}}^{\beta \left(1\right)+1}u\left(1\right),{}^{CH}D_{a^{+}}^{\beta \left(t\right)+1}u\left(t\right):=\vartheta \left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right),\vartheta :=t{\displaystyle \frac{d}{dt}}.\hfill \end{array}\right.$$

(2)

where the endpoint values are understood as the corresponding one-sided limits and ${\eta }_i\in \mathbb{R}$ for $i=1,2,3,4$. We assume that the variable orders [5] $\alpha ,\beta \in C(J,(1,2\left]\right)$ satisfy

$$1<{\alpha }_{*}\le \alpha \left(t\right)\le {\alpha }^{*}\le 2,1<{\beta }_{*}\le \beta \left(t\right)\le {\beta }^{*}\le 2,t\in J.$$

as well as suitable continuity and boundedness conditions. The sequential operator in (1) is understood as the successive application ${}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right):={}^{C}D_{a^{+}}^{\alpha \left(t\right)}v\left(t\right),v\left(t\right):={}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right),$ provided that both derivatives exist; see in [4]. The kernels $\lambda ,\delta :\Delta \to \mathbb{R}$ are continuous, where $\Delta :=\{(t,s)\in J^2:a\le s\le t\}.$

The boundary value problem considered here is related to that studied in our previous work [6]. However, the present proceedings paper develops a distinct analytical formulation. It introduces a boundary-matrix representation of the equivalent integral equation, defines the fixed-point operator explicitly, and establishes an existence result under continuity and growth assumptions rather than global Lipschitz conditions. In contrast to the previous article, which considered only Ulam–Hyers–Rassias stability, the present work develops both Ulam–Hyers and Ulam–Hyers–Rassias stability results using bounds derived from the linear solution operator. A new example with fixed parameters is also included to verify the hypotheses of the existence and stability theorems and illustrate the boundary-matrix construction.

The problem is investigated in the Banach space $C(J,\mathbb{R})$ equipped with the supremum norm. By transforming the differential equation into an equivalent integral equation, an operator formulation is obtained whose fixed points correspond to solutions of the original problem. This concept enables the application of Darbo’s fixed-point theorem, together with the Kuratowski measure of noncompactness [7], to establish the existence and stability of solutions.

The paper is organized as follows. Section 2 presents the preliminary concepts. Section 3 establishes the equivalent formulation and existence result. Section 4 develops the stability analysis. Section 5 provides an illustrative example, and Section 6 concludes the paper.

2. Preliminaries

We recall several fundamental concepts required for the subsequent analysis [8,9].

Definition 1

([2]). Let $0<a<b<\infty$, and let $\alpha :(a,b)\to (0,\infty )$ be a continuous function. For $t\in (a,b)$ and $x\in [a,t)$, $t>x>0$. The left-sided Hadamard fractional integral of variable order $\alpha \left(t\right)$ associated with a function $u:(a,b)\to \mathbb{R}$ is defined as

$${}^{H}I_{a^{+}}^{\alpha \left(t\right)}u\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right(t\left)\right)}}{\int }_a^t{\left(\ln {\displaystyle \frac{t}{x}}\right)}^{\alpha \left(t\right)-1}{\displaystyle \frac{u\left(x\right)}{x}}dx,t>a,$$

(3)

whenever the above integral is well defined. Here, the fractional order depends on the evaluation point t, while x denotes the integration variable.

Definition 2

([10]). Let $0<a<b<\infty$, and let $\alpha :[a,b]\to (n-1,n]$ be continuous for some fixed $n\in \mathbb{N}$. Define $\vartheta :=t{\displaystyle \frac{d}{dt}}.$ Suppose that u is sufficiently regular so that ${\vartheta }^{n}u$ is integrable on $[a,b]$ with respect to $ds/s$. The left-sided variable-order Caputo–Hadamard (CH) fractional derivative of u is defined by

$${}^{CH}D_{a^{+}}^{\alpha \left(t\right)}u\left(t\right):={}^{H}I_{a^{+}}^{n-\alpha \left(t\right)}\left({\vartheta }^{n}u\right)\left(t\right),t\in (a,b].$$

(4)

For $n-1<\alpha \left(t\right)<n$, Definition (4) takes the integral form

$${}^{CH}D_{a^{+}}^{\alpha \left(t\right)}u\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-\alpha (t\left)\right)}}{\int }_a^t{\left(\log {\displaystyle \frac{t}{s}}\right)}^{n-\alpha \left(t\right)-1}{\vartheta }^{n}u\left(s\right){\displaystyle \frac{ds}{s}}.$$

(5)

At every point for which $\alpha \left(t\right)=n$, we adopt the convention

$${}^{H}I_{a^{+}}^0=I,andhence{}^{CH}D_{a^{+}}^{n}u\left(t\right)={\vartheta }^{n}u\left(t\right).$$

In particular, when $\alpha \left(t\right)=2$, we have

$${}^{CH}D_{a^{+}}^{2}u\left(t\right)={\vartheta }^{2}u\left(t\right)={\left(t{\displaystyle \frac{d}{dt}}\right)}^{2}u\left(t\right)=t^2{u}^{″}\left(t\right)+t{u}^{\prime }\left(t\right).$$

(6)

Theorem 1

(Darbo fixed-point theorem [3]). Let Λ be a nonempty, closed, bounded, and convex subset of a Banach space X, and let $\mathcal{Q}$ denote the Kuratowski measure of noncompactness on X. Suppose that $W:\mathsf{\Lambda }\to \mathsf{\Lambda }$ is a continuous operator satisfying

$$\mathcal{Q}\left(W\right(S\left)\right)\le k\mathcal{Q}\left(S\right),forallnonemptyS\subset \mathsf{\Lambda },k\in [0,1).$$

(7)

Then W admits at least one fixed point in Λ.

Proposition 1

([2,3]). Let X be a Banach space, and let $\mathcal{D},{\mathcal{D}}_1,{\mathcal{D}}_2$ be bounded subsets of X. Then the following properties hold:

1.

$\mathcal{Q}\left(\mathcal{D}\right)=0$ if and only if $\mathcal{D}$ is relatively compact.

2.

$\mathcal{Q}(⌀)=0$.

3.

$\mathcal{Q}\left(\mathcal{D}\right)=\mathcal{Q}\left(\overline{\mathcal{D}}\right)=\mathcal{Q}\left(\mathit{conv}\left(\mathcal{D}\right)\right)$.

4.

If ${\mathcal{D}}_1\subset {\mathcal{D}}_2$, then $\mathcal{Q}\left({\mathcal{D}}_1\right)\le \mathcal{Q}\left({\mathcal{D}}_2\right)$.

5.

$\mathcal{Q}({\mathcal{D}}_1+{\mathcal{D}}_2)\le \mathcal{Q}\left({\mathcal{D}}_1\right)+\mathcal{Q}\left({\mathcal{D}}_2\right)$.

6.

$\mathcal{Q}\left(\lambda \mathcal{D}\right)=\left|\lambda \right|\mathcal{Q}\left(\mathcal{D}\right)$, $\lambda \in \mathbb{R}$.

7.

$\mathcal{Q}({\mathcal{D}}_1\cup {\mathcal{D}}_2)=max\{\mathcal{Q}\left({\mathcal{D}}_1\right),\mathcal{Q}\left({\mathcal{D}}_2\right)\}$.

8.

$\mathcal{Q}({\mathcal{D}}_1\cap {\mathcal{D}}_2)\le min\{\mathcal{Q}\left({\mathcal{D}}_1\right),\mathcal{Q}\left({\mathcal{D}}_2\right)\}$.

9.

$\mathcal{Q}(\mathcal{D}+u_0)=\mathcal{Q}\left(\mathcal{D}\right)$ for any $u_0\in X$.

3. Existence Results

In this section, we establish an existence result for (1) and (2).

Lemma 1.

Let $J=[a,1]$, where $0<a<1$, and let $\alpha ,\beta :J\to (1,2]$ be prescribed order functions. Define $\vartheta :=t{\displaystyle \frac{d}{dt}},\left(P_{\alpha }h\right)\left(t\right):={}^{RL}I_{a^{+}}^{\alpha \left(t\right)}h\left(t\right),\left(Q_{\beta }h\right)\left(t\right):={}^{CH}I_{a^{+}}^{\beta \left(t\right)}h\left(t\right).$

Assume the following properties hold on X:

$$P_{\alpha }\left({}^{C}D_{a^{+}}^{\alpha \left(t\right)}v\right)\left(t\right)=v\left(t\right)-v\left(a\right)-v^{\prime }\left(a\right)(t-a),$$

(8)

$$Q_{\beta }\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\right)(t)=u\left(t\right)-d_0-d_1\log \left({\displaystyle \frac{t}{a}}\right),$$

(9)

for suitable constants $d_0,d_1$. Assume additionally that

$${}^{CH}D_{a^{+}}^{\beta \left(t\right)+1}u\left(t\right)=\vartheta \left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right).$$

(10)

For $h\in C(J,\mathbb{R})$, define

$${\mathsf{\Phi }}_h\left(t\right):=Q_{\beta }\left(P_{\alpha }h\right)\left(t\right),$$

and

$${\psi }_1\left(t\right):=Q_{\beta }\left(1\right)\left(t\right),{\psi }_2\left(t\right):=Q_{\beta }[s\mapsto s-a]\left(t\right),$$

$${\psi }_3\left(t\right):=1,{\psi }_4\left(t\right):=\log \left({\displaystyle \frac{t}{a}}\right).$$

Suppose that

$${\mathsf{\Phi }}_h,{\psi }_1,{\psi }_2\in C^1\left(J\right),P_{\alpha }h,\vartheta \left(P_{\alpha }h\right)\in C\left(J\right),$$

and let ${\eta }_i\in \mathbb{R}$, $i=1,2,3,4$, satisfy

$${\eta }_1\ne 1,{\eta }_2\ne a^{-1},{\eta }_3\ne 1,{\eta }_4\ne a.$$

(11)

Then $u\in X$ solves

$${}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)=h\left(t\right),t\in (a,1),$$

(12)

subject to

$$\left\{\begin{array}{c}u\left(a\right)={\eta }_{1}u\left(1\right),\hfill \\ u^{\prime }\left(a\right)={\eta }_2{u}^{\prime }\left(1\right),\hfill \\ {\displaystyle {\left[{}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right]}_{t=a^{+}}={\eta }_3{\left[{}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right]}_{t=1},}\hfill \\ {\displaystyle {\left[{}^{CH}D_{a^{+}}^{\beta \left(t\right)+1}u\left(t\right)\right]}_{t=a^{+}}={\eta }_4{\left[{}^{CH}D_{a^{+}}^{\beta \left(t\right)+1}u\left(t\right)\right]}_{t=1},}\hfill \end{array}\right.$$

(13)

if and only if

$$u\left(t\right)={\mathsf{\Phi }}_h\left(t\right)+{\displaystyle \sum _{j=1}^4}{c}_j\left(h\right){\psi }_j\left(t\right),$$

(14)

where

$$\mathbf{c}\left(h\right):=\left(\begin{array}{c}{c}_1\left(h\right)\\ c_2\left(h\right)\\ c_3\left(h\right)\\ c_4\left(h\right)\end{array}\right)={\mathcal{M}}^{-1}\mathbf{b}\left(h\right),$$

(15)

with

$$\mathcal{M}=\left(\begin{array}{cccc}{\psi }_1\left(a\right)-{\eta }_1{\psi }_1\left(1\right)& {\psi }_2\left(a\right)-{\eta }_1{\psi }_2\left(1\right)& 1-{\eta }_1& -{\eta }_1\log (1/a)\\ {\psi }_1^{\prime }\left(a\right)-{\eta }_2{\psi }_1^{\prime }\left(1\right)& {\psi }_2^{\prime }\left(a\right)-{\eta }_2{\psi }_2^{\prime }\left(1\right)& 0& a^{-1}-{\eta }_2\\ 1-{\eta }_3& -{\eta }_3(1-a)& 0& 0\\ 0& a-{\eta }_4& 0& 0\end{array}\right),$$

(16)

and

$$\mathbf{b}\left(h\right)=\left(\begin{array}{c}{\eta }_1{\mathsf{\Phi }}_h\left(1\right)-{\mathsf{\Phi }}_h\left(a\right)\\ {\eta }_2{\mathsf{\Phi }}_h^{\prime }\left(1\right)-{\mathsf{\Phi }}_h^{\prime }\left(a\right)\\ {\eta }_3\left(P_{\alpha }h\right)\left(1\right)-\left(P_{\alpha }h\right)\left(a\right)\\ {\eta }_4\vartheta \left(P_{\alpha }h\right)\left(1\right)-\vartheta \left(P_{\alpha }h\right)\left(a\right)\end{array}\right).$$

(17)

Proof. 

Set $v\left(t\right):={}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right).$ Equation (12) becomes

$${}^{C}D_{a^{+}}^{\alpha \left(t\right)}v\left(t\right)=h\left(t\right).$$

(18)

Using (8), we obtain

$$v\left(t\right)=P_{\alpha }h\left(t\right)+c_1+c_2(t-a).$$

(19)

Applying $Q_{\beta }$ and using (9) gives

$$u\left(t\right)={\mathsf{\Phi }}_h\left(t\right)+c_1{\psi }_1\left(t\right)+c_2{\psi }_2\left(t\right)+c_3+c_4\log \left({\displaystyle \frac{t}{a}}\right).$$

(20)

Thus, $u\left(t\right)={\mathsf{\Phi }}_h\left(t\right)+{\sum }_{j=1}^4{c}_j{\psi }_j\left(t\right).$ Define the boundary operator

$$\mathcal{B}u:=\left(\begin{array}{c}u\left(a\right)-{\eta }_{1}u\left(1\right)\\ u^{\prime }\left(a\right)-{\eta }_2{u}^{\prime }\left(1\right)\\ {\left[{}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right]}_{t=a^{+}}-{\eta }_3{\left[{}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right]}_{t=1}\\ {\left[{}^{CH}D_{a^{+}}^{\beta \left(t\right)+1}u\left(t\right)\right]}_{t=a^{+}}-{\eta }_4{\left[{}^{CH}D_{a^{+}}^{\beta \left(t\right)+1}u\left(t\right)\right]}_{t=1}\end{array}\right).$$

(21)

When we substitute the previous representation into $\mathcal{B}u=0$, we get the equation $\mathcal{M}\mathbf{c}=\mathbf{b}\left(h\right)$. The third row comes directly from (19), while, by (10), ${}^{CH}D_{a^{+}}^{\beta \left(t\right)+1}u\left(t\right)=\vartheta \left(P_{\alpha }h\right)\left(t\right)+c_{2}t,$ which gives the fourth row. The form of $\mathcal{M}$ gives $det\left(\mathcal{M}\right)=(1-{\eta }_1)(a^{-1}-{\eta }_2)(1-{\eta }_3)(a-{\eta }_4).$ Hence, (11) implies $det\left(\mathcal{M}\right)\ne 0$, and $\mathbf{c}={\mathcal{M}}^{-1}\mathbf{b}\left(h\right).$ This proves the necessity. Conversely, let u be given by (14) with $\mathbf{c}={\mathcal{M}}^{-1}\mathbf{b}\left(h\right)$. The composition properties (8)–(9) give ${}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)=h\left(t\right).$ Furthermore, $\mathcal{M}\mathbf{c}=\mathbf{b}\left(h\right)$ is exactly the system obtained from the four boundary conditions. Therefore, $\mathcal{B}u=0$, and all the conditions in (13) are satisfied. □

We impose the following assumptions:

(H1)

The function $f:J\times {\mathbb{R}}^3\to \mathbb{R}$ is continuous, and there exist $p,q_i\in C(J,{\mathbb{R}}_{+})$, $i=1,2,3$, such that

$$|f(t,x_1,x_2,x_3)|\le p\left(t\right)+{\displaystyle \sum _{i=1}^3}{q}_i\left(t\right)\left|x_i\right|.$$

(H2)

The function $g:J\times \mathbb{R}\to \mathbb{R}$ is continuous, and there exist $r,\ell \in C(J,{\mathbb{R}}_{+})$ such that

$$\left|g\right(t,x\left)\right|\le r\left(t\right)+\ell \left(t\right)\left|x\right|.$$

(H3)

The kernels $\lambda ,\delta :\Delta \to \mathbb{R}$ are continuous, where

$$\Delta :=\{(t,s)\in J^2:a\le s\le t\}.$$

(H4)

The linear solution operator $\mathcal{S}$ defined below is compact.

Set $M_{\lambda }:={max}_{(t,s)\in \Delta }\left|\lambda (t,s)\right|,M_{\delta }:={max}_{(t,s)\in \Delta }\left|\delta (t,s)\right|$ and $p_{*}:={\parallel p\parallel }_{\infty },q_{i*}:=\parallel q_i{\parallel }_{\infty },r_{*}:={\parallel r\parallel }_{\infty },{\ell }_{*}:={\parallel \ell \parallel }_{\infty }.$ For $u\in C(J,\mathbb{R})$, define $F_u\left(t\right):=f\left(t,u\left(t\right),\left(Tu\right)\left(t\right),\left(Gu\right)\left(t\right)\right).$ From $\left(\mathrm{H}1\right)$–$\left(\mathrm{H}3\right)$, we obtain ${\parallel Tu\parallel }_{\infty }\le (1-a)M_{\lambda }\left(r_{*}+{\ell }_{*}{\parallel u\parallel }_{\infty }\right),$ ${\parallel Gu\parallel }_{\infty }\le (1-a)M_{\delta }\left(r_{*}+{\ell }_{*}{\parallel u\parallel }_{\infty }\right).$ Consequently,

$$\parallel F_u{\parallel }_{\infty }\le B_0+B_1{\parallel u\parallel }_{\infty },$$

(22)

where $B_0:=p_{*}+(1-a)r_{*}\left(q_{2*}{M}_{\lambda }+q_{3*}{M}_{\delta }\right)$ and $B_1:=q_{1*}+(1-a){\ell }_{*}\left(q_{2*}{M}_{\lambda }+q_{3*}{M}_{\delta }\right).$ Let $\mathcal{S}:C(J,\mathbb{R})\to C(J,\mathbb{R})$ be the linear solution operator obtained in Lemma 1, defined by

$$\left(\mathcal{S}h\right)\left(t\right)={\mathsf{\Phi }}_h\left(t\right)+{\displaystyle \sum _{j=1}^4}{\left[{\mathcal{M}}^{-1}\mathbf{b}\left(h\right)\right]}_j{\psi }_j\left(t\right).$$

(23)

Since $\mathcal{S}$ is compact, it is bounded. Set $C_{\mathcal{S}}:=\parallel \mathcal{S}\parallel .$ Thus, ${\parallel \mathcal{S}h\parallel }_{\infty }\le C_{\mathcal{S}}{\parallel h\parallel }_{\infty }.$ Define the nonlinear operator $\mathcal{A}u:=\mathcal{S}\left(F_u\right).$ By Lemma 1, u solves (1) and (2) if and only if $u=\mathcal{A}u.$

Theorem 2.

Assume that $\left(\mathrm{H}1\right)$–$\left(\mathrm{H}4\right)$ hold and $C_{\mathcal{S}}{B}_1<1.$ Then problem (1) and (2) has at least one solution in $C(J,\mathbb{R})$.

Proof. 

Choose $R\ge {\displaystyle \frac{C_{\mathcal{S}}{B}_0}{1-C_{\mathcal{S}}{B}_1}}$ and set $B_R:={\{u\in C(J,\mathbb{R}):\parallel u\parallel }_{\infty }\le R\}.$ Clearly, $B_R$ is nonempty, closed, bounded, and convex. For $u\in B_R$, estimate (22) gives ${\parallel \mathcal{A}u\parallel }_{\infty }\le C_{\mathcal{S}}{\parallel F_u\parallel }_{\infty }\le C_{\mathcal{S}}(B_0+B_{1}R)\le R.$ Hence, $\mathcal{A}\left(B_R\right)\subseteq B_R.$ The continuity of $f,g,\lambda$, and $\delta$ implies that $u\mapsto F_u$ is continuous; therefore, $\mathcal{A}$ is continuous. Moreover, (22) yields ${sup}_{u\in B_R}{\parallel F_u\parallel }_{\infty }\le B_0+B_{1}R.$ Thus, $F\left(B_R\right)$ is bounded. By $\left(\mathrm{H}4\right)$, $\mathcal{A}\left(B_R\right)=\mathcal{S}\left(F\left(B_R\right)\right)$ is relatively compact. Let $\mu$ denote the Kuratowski measure of noncompactness. For every bounded set $D\subseteq B_R$, the set $\mathcal{A}\left(D\right)$ is relatively compact, and hence $\mu \left(\mathcal{A}\right(D\left)\right)=0\le q\mu \left(D\right)$ for any $q\in [0,1)$. Therefore, Darbo’s fixed-point theorem ensures that $\mathcal{A}$ has a fixed point $u\in B_R$. By Lemma 1, this fixed point solves (1) and (2). Hence, the problem admits at least one solution. □

Remark 1.

Theorem 2 ensures existence, but not uniqueness. Indeed, the compactness of $\mathcal{A}=\mathcal{S}\circ F_u$ implies $\mu \left(\mathcal{A}\right(D\left)\right)=0$ for every bounded set D; hence, Darbo’s condition holds with $q=0$. Thus, no global Lipschitz assumptions on f and g are needed.

4. Stability Results

In this section, we present the main results on the stability of solutions to the boundary value problem (BVP) (1) and (2).

Consider the nonlinear variable-order hybrid Caputo–Hadamard sequential integro-differential Equations (1) and (2). In this section, we discuss different types of U-H stability for the proposed problem.

We first consider the inequality

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\le \epsilon ,t\in J^{\circ }:=(a,1),$$

(24)

where $\epsilon >0$.

We also consider the inequalities

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\le \varphi \left(t\right),t\in (a,1),$$

(25)

and

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\le \epsilon \varphi \left(t\right),t\in (a,1),$$

(26)

where $\varphi \in C(J,{\mathbb{R}}^{+})$.

Definition 3.

The BVP (1) and (2) is said to be U-H-stable if there exists a constant ${\mu }_f>0$ such that for every $\epsilon >0$ and for each function $u\in C(J,\mathbb{R})$ satisfying

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\le \epsilon ,t\in (a,1),$$

(27)

there exists a solution $u_1\in C(J,\mathbb{R})$ of the problem such that

$$|u\left(t\right)-u_1\left(t\right)|\le {\mu }_f\epsilon ,t\in J.$$

(28)

Definition 4.

The BVP (1) and (2) is said to be generalized U-H-stable if there exists a function ${\chi }_f\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ with ${\chi }_f\left(0\right)=0$ such that for every $\epsilon >0$ and for each function $u\in C(J,\mathbb{R})$ satisfying

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\le \epsilon ,t\in (a,1),$$

(29)

there exists a solution $u_1\in C(J,\mathbb{R})$ of the problem (1) and (2) such that

$$|u\left(t\right)-u_1\left(t\right)|\le {\chi }_f\left(\epsilon \right),t\in J.$$

Definition 5.

The BVP (1) and (2) is said to be U-H-R-stable with respect to ϕ if there exists a constant ${\mu }_{f,\varphi }>0$ such that for every $\epsilon >0$ and for each function $u\in C(J,\mathbb{R})$ satisfying

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\le \epsilon \varphi \left(t\right),t\in (a,1),$$

(30)

there exists a solution $u_1\in C(J,\mathbb{R})$ of the problem such that

$$|u\left(t\right)-u_1\left(t\right)|\le {\mu }_{f,\varphi }\epsilon \varphi \left(t\right),t\in J.$$

Definition 6.

The BVP (1) and (2) is said to be generalized U-H-R-stable with respect to ϕ if there exists a constant ${\mu }_{f,\varphi }>0$ such that for each function $u\in C(J,\mathbb{R})$ satisfying

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\le \varphi \left(t\right),t\in (a,1),$$

(31)

there exists a solution $u_1\in C(J,\mathbb{R})$ of the problem such that

$$|u\left(t\right)-u_1\left(t\right)|\le {\mu }_{f,\varphi }\varphi \left(t\right),t\in J.$$

(32)

Remark 2.

A function $u\in C(J,\mathbb{R})$ satisfies

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\le \epsilon ,t\in (a,1),$$

(33)

if and only if there exists a function $h\in C(J,\mathbb{R})$ such that

$$\left|h\right(t\left)\right|\le \epsilon ,t\in (a,1),$$

and

$${}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)=f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)+h\left(t\right),t\in (a,1).$$

Similarly, u satisfies

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\le \varphi \left(t\right),t\in (a,1),$$

(34)

if and only if there exists a function $h\in C(J,\mathbb{R})$ such that

$$\left|h\right(t\left)\right|\le \varphi \left(t\right),t\in (a,1),$$

and

$${}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)=f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)+h\left(t\right),t\in (a,1).$$

Theorem 3.

Assume that the BVP (1) and (2) admits a unique solution $u_1\in C(J,\mathbb{R})$. Assume also that there exists a constant $L_f>0$ such that, for all $t\in J$ and $x_i,y_i\in \mathbb{R}$, $i=1,2,3$,

$$\begin{array}{c}\hfill |f(t,x_1,x_2,x_3)-f(t,y_1,y_2,y_3)|\le L_f\left(|x_1-y_1|+|x_2-y_2|+|x_3-y_3|\right).\end{array}$$

Assume further that there exists $L_g>0$ such that

$$|g(t,x)-g(t,y)|\le L_g|x-y|,t\in J,x,y\in \mathbb{R}.$$

Suppose that there exist constants $M_T,M_G>0$ such that

$$|\left(Tu\right)\left(t\right)-\left(T{u}_1\right)\left(t\right)|\le M_T{\parallel u-u_1\parallel }_{\infty },$$

and

$$|\left(Gu\right)\left(t\right)-\left(G{u}_1\right)\left(t\right)|\le M_G{\parallel u-u_1\parallel }_{\infty },t\in J,$$

where $\parallel u-u_1{\parallel }_{\infty }:={sup}_{t\in J}|u\left(t\right)-u_1\left(t\right)|.$

Suppose also that there exists a constant $C_{\mathcal{S}}>0$ such that

$$\begin{array}{cc}\hfill \parallel u-u_1{\parallel }_{\infty }& \le C_{\mathcal{S}}\underset{t\in J^{\circ }}{sup}\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)\right.\left.-{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}{u}_1\left(t\right)\right)\right|.\hfill \end{array}$$

Define

$$\kappa :=C_{\mathcal{S}}{L}_f(1+M_T+M_G)<1.$$

(35)

Here, κ is the contraction constant arising in the U-H stability argument. Then the problem (1) and (2) is U-H-stable.

Additionally, by Lemma 1, the boundedness of the linear solution operator $\mathcal{S}$ gives

$$u-u_1=\mathcal{S}(h_u-h_{u_1}),$$

where

$$h_v\left(t\right):={}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}v\left(t\right)\right).$$

Consequently,

$$\begin{array}{cc}\hfill \parallel u-u_1{\parallel }_{\infty }& \le C_{\mathcal{S}}{\parallel h_u-h_{u_1}\parallel }_{\infty }\hfill \\ & =C_{\mathcal{S}}\underset{t\in J^{\circ }}{sup}\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}{u}_1\left(t\right)\right)\right|.\hfill \end{array}$$

Proof. 

Let $u\in C(J,\mathbb{R})$ be any function satisfying

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\le \epsilon ,t\in (a,1).$$

(36)

Let $u_1$ be the unique solution of BVP (1) and (2). Then

$${}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}{u}_1\left(t\right)\right)=f\left(t,u_1\left(t\right),T{u}_1\left(t\right),G{u}_1\left(t\right)\right),t\in (a,1).$$

(37)

Therefore, from Remark 2, we get

$$\begin{array}{cc}\hfill & \left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}{u}_1\left(t\right)\right)\right|\hfill \\ \hfill & \le \left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)\right|\hfill \\ \hfill & +\left|f\left(t,u\left(t\right),Tu\left(t\right),Gu\left(t\right)\right)-f\left(t,u_1\left(t\right),T{u}_1\left(t\right),G{u}_1\left(t\right)\right)\right|\hfill \\ \hfill & \le \epsilon +L_f\left(|u\left(t\right)-u_1\left(t\right)|+|Tu\left(t\right)-T{u}_1\left(t\right)|+|Gu\left(t\right)-G{u}_1\left(t\right)|\right).\hfill \end{array}$$

Using Remark 2 and Theorem 3, for each $t\in J^{\circ }$, we obtain

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}{u}_1\left(t\right)\right)\right|\le \epsilon +L_f(1+M_T+M_G){\parallel u-u_1\parallel }_{\infty }.$$

Therefore, taking the supremum over $t\in J^{\circ }$ yields

$$\begin{array}{cc}\hfill & {∥{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\right)-{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}{u}_1\right)∥}_{\infty }\hfill \\ \hfill & \le \epsilon +L_f(1+M_T+M_G){\parallel u-u_1\parallel }_{\infty }.\hfill \end{array}$$

(38)

Hence,

$$\parallel u-u_1\parallel \le C_{\mathcal{S}}\epsilon +C_{\mathcal{S}}{L}_f(1+M_T+M_G)\parallel u-u_1\parallel .$$

Thus,

$$\left(1-C_{\mathcal{S}}{L}_f(1+M_T+M_G)\right)\parallel u-u_1\parallel \le C_{\mathcal{S}}\epsilon .$$

Since $C_{\mathcal{S}}{L}_f(1+M_T+M_G)<1$, it follows that

$$\parallel u-u_1\parallel \le {\displaystyle \frac{C_{\mathcal{S}}}{1-C_{\mathcal{S}}{L}_f(1+M_T+M_G)}}\epsilon .$$

Consequently,

$$|u\left(t\right)-u_1\left(t\right)|\le {\mu }_f\epsilon ,t\in J,$$

where

$${\mu }_f={\displaystyle \frac{C_{\mathcal{S}}}{1-C_{\mathcal{S}}{L}_f(1+M_T+M_G)}}.$$

Therefore, the problem considered (1) and (2) is U-H-stable. □

Corollary 1.

Under the assumptions of Theorem 3, the problem (1) and (2) is generalized U-H-stable with

$${\chi }_f\left(\epsilon \right)={\mu }_f\epsilon .$$

Theorem 4.

Assume that all the hypotheses of Theorem 3 hold. Let $\varphi \in C(J,(0,\infty \left)\right)$ and set

$$m_{\varphi }:=\underset{t\in J}{min}\varphi \left(t\right)>0.$$

If $u\in C(J,\mathbb{R})$ satisfies

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),\left(Tu\right)\left(t\right),\left(Gu\right)\left(t\right)\right)\right|\le \epsilon \varphi \left(t\right),t\in J^{\circ }=(a,1),$$

(39)

then there exists a solution $u_1$ of the problem such that

$$|u\left(t\right)-u_1\left(t\right)|\le {\mu }_{f,\varphi }\epsilon \varphi \left(t\right),t\in J,$$

for some constant ${\mu }_{f,\varphi }>0$. Hence, the problem is U-H-R-stable with respect to ϕ.

Proof. 

Let u satisfy

$$\left|{}^{C}D_{a^{+}}^{\alpha \left(t\right)}\left({}^{CH}D_{a^{+}}^{\beta \left(t\right)}u\left(t\right)\right)-f\left(t,u\left(t\right),\left(Tu\right)\left(t\right),\left(Gu\right)\left(t\right)\right)\right|\le \epsilon \varphi \left(t\right),t\in J^{\circ }.$$

(40)

By Theorem 3, we obtain

$$\parallel u-u_1{\parallel }_{\infty }\le C_{\mathcal{S}}{\epsilon \parallel \varphi \parallel }_{\infty }+C_{\mathcal{S}}{L}_f(1+M_T+M_G){\parallel u-u_1\parallel }_{\infty },\kappa :=C_{\mathcal{S}}{L}_f(1+M_T+M_G)<1$$

(41)

Then

$$\parallel u-u_1{\parallel }_{\infty }\le {\displaystyle \frac{C_{\mathcal{S}}{\parallel \varphi \parallel }_{\infty }}{1-\kappa }}\epsilon .$$

Since $m_{\varphi }={min}_{t\in J}\varphi \left(t\right)>0$, for every $t\in J$,

$$\begin{array}{cc}\hfill |u\left(t\right)-u_1\left(t\right)|& \le {\displaystyle \frac{C_{\mathcal{S}}{\parallel \varphi \parallel }_{\infty }}{1-\kappa }}\epsilon \hfill \\ & \le {\displaystyle \frac{C_{\mathcal{S}}{\parallel \varphi \parallel }_{\infty }}{(1-\kappa )m_{\varphi }}}\epsilon \varphi \left(t\right).\hfill \end{array}$$

Thus, we may take

$${\mu }_{f,\varphi }:={\displaystyle \frac{C_{\mathcal{S}}{\parallel \varphi \parallel }_{\infty }}{(1-\kappa )m_{\varphi }}}.$$

Therefore, problem (1) and (2) is U-H-R-stable with respect to $\varphi$. □

Corollary 2.

According to Definition 6 and Theorem 4, the proposed BVP problem (1) and (2) is generalized U-H-R-stable with respect to ϕ.

5. Illustrative Examples

To demonstrate the applicability of the theoretical results, we present illustrative examples of variable-order hybrid CH sequential fractional BVPs.

Example 1.

Let $a=1/2$, $J=[1/2,1]$, and $\alpha \left(t\right)=\beta \left(t\right)={\displaystyle \frac{3}{2}}.$ Consider

$${}^{C}D_{{(1/2)}^{+}}^{3/2}\left({}^{CH}D_{{(1/2)}^{+}}^{3/2}u\left(t\right)\right)=f\left(t,u\left(t\right),\left(Tu\right)\left(t\right),\left(Gu\right)\left(t\right)\right),t\in J^{\circ },$$

(42)

subject to

$$\left\{\begin{array}{c}{\displaystyle u(1/2)=\frac{1}{2}u\left(1\right),}\hfill \\ {\displaystyle u^{\prime }(1/2)=\frac{1}{3}{u}^{\prime }\left(1\right),}\hfill \\ {\displaystyle {\left[{}^{CH}D_{{(1/2)}^{+}}^{3/2}u\left(t\right)\right]}_{t={(1/2)}^{+}}=\frac{1}{4}{\left[{}^{CH}D_{{(1/2)}^{+}}^{3/2}u\left(t\right)\right]}_{t=1},}\hfill \\ {\displaystyle {\left[\vartheta \left({}^{CH}D_{{(1/2)}^{+}}^{3/2}u\left(t\right)\right)\right]}_{t={(1/2)}^{+}}=\frac{1}{5}{\left[\vartheta \left({}^{CH}D_{{(1/2)}^{+}}^{3/2}u\left(t\right)\right)\right]}_{t=1}.}\hfill \end{array}\right.$$

(43)

Since $a=1/2$, the nonresonance conditions of Lemma 1 are satisfied. Hence, $det\left(\mathcal{M}\right)\ne 0$. Define $\lambda (t,s):=t-s,\delta (t,s):=ts,g(t,x):={\displaystyle \frac{x}{1+\left|x\right|}}.$ Then $M_{\lambda }={\displaystyle \frac{1}{2}},M_{\delta }=1,$ and $\left|g\right(t,x)-g(t,y\left)\right|\le |x-y|.$ Thus, $L_g=1$.

For convenience, set $v\left(t\right):={\left(t-{\displaystyle \frac{1}{2}}\right)}^2{(1-t)}^2.$ This gives

$${}^{C}D_{{(1/2)}^{+}}^{3/2}v\left(t\right)={\displaystyle \frac{1}{\sqrt{\pi }}}\left[{\left(t-{\displaystyle \frac{1}{2}}\right)}^{1/2}-8{\left(t-{\displaystyle \frac{1}{2}}\right)}^{3/2}+{\displaystyle \frac{64}{5}}{\left(t-{\displaystyle \frac{1}{2}}\right)}^{5/2}\right].$$

(44)

Therefore, choose

$$f(t,x_1,x_2,x_3):={\displaystyle \frac{1}{\sqrt{\pi }}}\left[{\left(t-{\displaystyle \frac{1}{2}}\right)}^{1/2}-8{\left(t-{\displaystyle \frac{1}{2}}\right)}^{3/2}+{\displaystyle \frac{64}{5}}{\left(t-{\displaystyle \frac{1}{2}}\right)}^{5/2}\right].$$

(45)

The function f is continuous and independent of $x_1,x_2,x_3$. Hence, hypothesis $\left(\mathrm{H}1\right)$ holds with $p\left(t\right):=\left|f(t,0,0,0)\right|,q_1=q_2=q_3=0.$ Moreover, $B_0={\parallel p\parallel }_{\infty }<\infty ,B_1=0.$

Consequently,

$$C_{\mathcal{S}}{B}_1=0<1.$$

(46)

Therefore, Theorem 2 ensures that (42) and (43) has at least one solution.

Furthermore, f satisfies the Lipschitz condition with $L_f=0$. Thus, $\kappa =C_{\mathcal{S}}{L}_f(1+M_T+M_G)=0<1.$ The fixed-point operator $\mathcal{A}u=\mathcal{S}\left(F_u\right)$ is constant because $F_u=f$ for every u. Hence, its fixed point is unique.

Finally, if u satisfies the same boundary conditions and

$$\left|{}^{C}D_{{(1/2)}^{+}}^{3/2}\left({}^{CH}D_{{(1/2)}^{+}}^{3/2}u\left(t\right)\right)-f\left(t\right)\right|\le \epsilon ,t\in J^{\circ },$$

then Lemma 1 gives

$$\parallel u-u_1{\parallel }_{\infty }\le C_{\mathcal{S}}\epsilon ,$$

(47)

where $u_1$ is the exact solution. Therefore, the problem is U-H-stable with stability constant ${\mu }_f=C_{\mathcal{S}}.$

Example 2.

Let $a=1/2$, $J=[1/2,1]$, and choose $\alpha \left(t\right):={\displaystyle \frac{3}{2}}+{\displaystyle \frac{t}{4}},\beta \left(t\right):={\displaystyle \frac{7}{4}}-{\displaystyle \frac{t}{4}}.$ Then ${\displaystyle \frac{13}{8}}\le \alpha \left(t\right)\le {\displaystyle \frac{7}{4}},{\displaystyle \frac{3}{2}}\le \beta \left(t\right)\le {\displaystyle \frac{13}{8}},$ so $1<\alpha \left(t\right),\beta \left(t\right)<2$ on J.

Consider problem (1) and (2) with ${\eta }_1={\displaystyle \frac{1}{2}},{\eta }_2={\displaystyle \frac{1}{3}},{\eta }_3={\displaystyle \frac{1}{4}},{\eta }_4={\displaystyle \frac{1}{5}},$ and $\lambda (t,s)=t-s,\delta (t,s)=ts,g(t,x)={\displaystyle \frac{x}{1+\left|x\right|}}.$ Let $f(t,x_1,x_2,x_3):={\displaystyle \frac{t^2}{10}}.$ Then f is continuous and satisfies $\left(\mathrm{H}1\right)$ with $p\left(t\right)={\displaystyle \frac{t^2}{10}},q_1=q_2=q_3=0.$

Hence, $B_0=\left|p\right|\infty ={\displaystyle \frac{1}{10}},B_1=0,$ and therefore $C_{\mathcal{S}}{B}_1=0<1.$ The nonresonance conditions are also satisfied. Consequently, Theorem 2 ensures that the problem in Example 2 has at least one solution.

6. Conclusions

This work investigates a boundary value problem for variable-order hybrid Caputo–Hadamard sequential fractional differential equations under nonseparated boundary conditions. An equivalence lemma provides an explicit integral representation and defines the associated fixed-point operator. Using Darbo’s fixed-point theorem [2,3], the existence of at least one solution is established, while Ulam–Hyers and Ulam–Hyers–Rassias stability are also used to analyze the behavior of solutions under perturbations. The examples support the applicability of the theoretical results. The proposed approach may also be extended to related hybrid models involving other fractional operators, such as the Prabhakar and Katugampola operators, to investigate complex systems with evolving memory effects.