Uniqueness Analysis of Boundary Value Problems with Variable-order Caputo Fractional Derivatives

2. Preliminaries

This section presents the essential preliminaries and definitions associated with fractional differential equations of variable order.

Definition 2.1 ([5]) The variable-order Riemann-Liouville fractional integral of a function u with order µ(t) is expressed by

$I^{\mu (\text{t})}u(\text{t})={\displaystyle {\int }_{{\rho }_1}^t\frac{{(\text{t}-\theta )}^{\mu (\theta )-1}}{\Gamma (\mu (\theta ))}}u(\theta )d\theta ,\text{ t}>{\rho }_1$

Definition 2.2 ([5]) The Caputo variable-order µ(t) ∈ (0,1) fractional derivative for a function µ is defined as

${}^C{D}^{\mu (\text{t})}u(\text{t})={\displaystyle {\int }_{{\rho }_1}^t\frac{{(\text{t}-\theta )}^{-\mu (\theta )}}{\Gamma (1-\mu (\theta ))}}{u}^{\text{'}}(\theta )d\theta ,0<\mu (\text{t})<1$

If µ(t) = 0, then ^CD_t^µ(t)u(t) = u(t).

Theorem 2.1 ([4]) Consider a Banach space X and a closed subset B𝔽 ⊂ X. Let the operator 𝔽: B𝔽 → B𝔽 satisfy the contraction condition

$\parallel \mathbb{F}(\widehat{u})-\mathbb{F}(u){\parallel }_{\infty }\le L_f\parallel \widehat{u}-u{\parallel }_{\infty },$

for L_f ∈ (0,1) and for all $\widehat{u},u\in B_{\mathbb{F}}$ . Then, has exactly one fixed point $u^{\ast }\in B_{\mathbb{F}}$ .

Assume f (t,u) and µ(t) satisfy conditions ensuring the BVP is well-posed.

(H1) There is a constant G_f > 0 so that |f (t,u)| ≤ G_f |u| for all t ∈ [ρ₁,ρ₂], u ∈ R.

(H2) There is a constant Lf > 0 so that

|f (t,u₁) − f (t,u₂)| ≤ L_f |u₁ − u₂| for all t ∈ [ρ1,ρ2].

(H3) Let α ∈ C([ρ₁,ρ₂]), Then ∥α∥∞ < ∞.

$\begin{array}{l}u(t)=\beta -\frac{1}{\Gamma (\zeta )}{{{\displaystyle \int }}^{\text{}}}_0^{\xi }{(}^{\xi }u(\theta )d\theta +\\ {{{\displaystyle \int }}^{\text{}}}_0^t\frac{{(t-\theta )}^{\mu (\theta )-1}}{\Gamma (\mu (\theta ))}(f(\theta ,u(\theta ))-\alpha (\theta )u(\theta ))d\theta \end{array}$

for every t ∈ [ρ₁,ρ₂], where $u_0=\beta -\frac{1}{\Gamma (\zeta )}{{{\displaystyle \int }}^{\text{}}}_0^{\xi }{(\xi -\theta )}^{\zeta -1}u(\theta )d\theta$ .

3. Uniqueness

In this section, we establish a condition that guarantees exactly one solution for the VO-FDE given in (1). The following theorem provides the main analytical tool for ensuring that the BVP has a single solution.

Let the interval J = [ρ₁,ρ₂] be divided into m consecutive subintervals: ρ1 = t₀ < t₁ < t₂ < ··· < t_m = ρ₂. Each subinterval is denoted by J_i = (t_i-1,t_i], i = 1,2,...,m.

On each subinterval, define a piecewise constant variable fractional-order function µ(t) by $\mu (t)={{{\displaystyle \sum }}^{\text{}}}_{i=1}^m{\mu }_i{1}_{J_i}(t)$ , where µi ∈ (0,1] is the local fractional order on Ji , and 1Ji(t) is the indicator function: $1_{J_i}(t)=\{\begin{array}{ll}1,\hfill & t\in J_i,\hfill \\ 0,\hfill & t\notin J_i.\hfill \end{array}$  Assume that u ∈ C¹(J_i,R) is continuously differentiable.

Then, the Caputo VO-FDE on each subinterval can be written as

${}^C{D}^{{\mu }_i}u(t)={\displaystyle \sum }_{i=1}^m\frac{1}{\Gamma \left(1-{\mu }_i\right)}{\displaystyle {\int }_{J_i{{\displaystyle \cap }}^{\text{}}[0,T]}{(t-\theta )}^{-{\mu }_i}}{u}^{\text{'}}(\theta )d\theta$

Theorem 3.1. Consider the assumptions (H1)-(H3) satisfied. Then, the BVP associated with the Caputo VO-FDE (1) possesses a unique solution whenever k_i < 1.

$k_i=\left[\frac{{\xi }^{\zeta }}{\Gamma (\zeta +1)}+\frac{{\left({\rho }_2-{\rho }_1\right)}^{{\mu }_i}}{\Gamma \left({\mu }_i+1\right)}\left(L_f+\parallel \alpha {\parallel }_{\infty }\right)\right]$

Proof. Define the operator 𝔽: C¹([ρ₁,ρ₂]) → C¹([ρ₁,ρ₁]) by

$\begin{array}{l} (\mathbb{F}u)(t)=\beta -\frac{1}{\Gamma (\zeta )}{\displaystyle {\int }_0^{\xi }{(\xi -\theta )}^{\zeta -1}}u(\theta )d\theta +\\ {\displaystyle {\int }_0^t\frac{{(t-\theta )}^{{\mu }_i-1}}{\Gamma \left({\mu }_i\right)}}(f(\theta ,u(\theta ))-\alpha (\theta )u(\theta ))d\theta \end{array}$

Step 1: Boundedness

Let B𝔽 = {u ∈ C¹([ρ₁,ρ₂]) : ∥u∥∞ ≤ R𝔽} for some R𝔽 > 0. For u ∈ B𝔽,

$\begin{array}{rr}\hfill |(\mathbb{F}u)(t)|\le & \hfill \left|\beta \right|+\frac{1}{\Gamma (\zeta )}\underset{0}{\overset{\xi }{{\displaystyle \int }}}{(\xi -\theta )}^{\zeta -1}|u(\theta )|d\theta +{\displaystyle {\int }_0^{{\rho }_2}\frac{{(t-\theta )}^{{\mu }_i-1}}{\Gamma \left({\mu }_i\right)}}\\ \hfill & \hfill (|f(\theta ,u(\theta ))|+|\alpha (\theta )\left|\right|u(\theta )|)d\theta \\ \hfill \le & \hfill \left[\left|\beta \right|+\frac{{\xi }^{\zeta }}{\Gamma (\zeta +1)}+\frac{{\left({\rho }_2-{\rho }_1\right)}^{{\mu }_i}}{\Gamma \left({\mu }_i+1\right)}\left(G_f+\parallel \alpha {\parallel }_{\infty }\right)\right]\parallel u{\parallel }_{\infty }\\ \hfill \le & \hfill \left[\left|\beta \right|+\frac{{\xi }^{\zeta }}{\Gamma (\zeta +1)}+\frac{{\left({\rho }_2-{\rho }_1\right)}^{{\mu }_i}}{\Gamma \left({\mu }_i+1\right)}\left(G_f+\parallel \alpha {\parallel }_{\infty }\right)\right]R_{\mathbb{F}}\end{array}$

Choose an R𝔽 large enough such that.

$\begin{array}{c}\left|\beta \right|+\frac{R_{\mathbb{F}}{\xi }^{\zeta }}{\Gamma (\zeta +1)}+R_{\mathbb{F}}\frac{{\left({\rho }_2-{\rho }_1\right)}^{{\mu }_i}}{\Gamma \left({\mu }_i+1\right)}\left(G_f+\parallel \alpha {\parallel }_{\infty }\right)\le R_{\mathbb{F}}\\ |(\mathbb{F}u)(t)|\le \left[\left|\beta \right|+\frac{{\xi }^{\zeta }}{\Gamma (\zeta +1)}+\frac{{\left({\rho }_2-{\rho }_1\right)}^{{\mu }_i}}{\Gamma \left({\mu }_i+1\right)}\left(G_f+\parallel \alpha {\parallel }_{\infty }\right)\right]R_{\mathbb{F}}\le R_{\mathbb{F}}\end{array}$

Hence, (BF) ⊆ B𝔽, showing boundedness.

Step 2: Contraction

Let u₁,u₂ ∈ B𝔽 . Then

$\begin{array}{rr}\hfill & \hfill \left|\left(\mathbb{F}{u}_1\right)(t)-\left(\mathbb{F}{u}_2\right)(t)\right|\le \frac{1}{\Gamma (\zeta )}{\displaystyle {\int }_0^{\xi }{(\xi -\theta )}^{\zeta -1}}\left|u_1(\theta )-u_2(\theta )\right|d\theta +{\displaystyle {\int }_0^{{\rho }_2}\frac{{(t-\theta )}^{{\mu }_i-1}}{\Gamma \left({\mu }_i\right)}}\\ \hfill & \hfill \left(\left|f\left(\theta ,u_1(\theta )\right)-f\left(\theta ,u_2(\theta )\right)\right|+|\alpha (\theta )|\left|u_1(\theta )-u_2(\theta )\right|\right)d\theta \\ \hfill & \hfill \le \frac{1}{\Gamma (\zeta )}{\displaystyle {\int }_0^{\xi }{(\xi -\theta )}^{\zeta -1}}\left|u_1(\theta )-u_2(\theta )\right|d\theta \\ \hfill & \hfill +\left(L_f+\parallel \alpha {\parallel }_{\infty }\right){\displaystyle {\int }_0^{{\rho }_2}\frac{{(t-\theta )}^{{\mu }_i-1}}{\Gamma \left({\mu }_i\right)}}\left|u_1(\theta )-u_2(\theta )\right|d\theta \\ \hfill & \hfill \le \left[\frac{{\xi }^{\zeta }}{\text{Γ}(\zeta +1)}+\frac{{\left({\rho }_2-{\rho }_1\right)}^{{\mu }_i}}{\Gamma \left({\mu }_i+1\right)}\left(L_f+\parallel \alpha {\parallel }_{\infty }\right)\right]u_1-u_2{}_{\infty }\\ \hfill & \hfill \left|\left(\mathbb{F}{u}_1\right)(t)-\left(\mathbb{F}{u}_2\right)(t)\right|\le k_i{u}_1-u_2{}_{\infty }\end{array}$

|(𝔽u₁)(t) − (𝔽u₁)(t)|≤ki∥u₁ − u₂∥∞.

If we choose parameters such that $k_i=\left[\frac{{\xi }^{\zeta }}{\Gamma (\zeta +1)}+\frac{{\left({\rho }_2-{\rho }_1\right)}^{{\mu }_i}}{\Gamma \left({\mu }_i+1\right)}\left(L_f+\parallel \alpha {\parallel }_{\infty }\right)\right]<1$ , then

𝔽 is a contraction. By Theorem 2.1, 𝔽 has exactly one fixed point u ∈ B𝔽.

4. Examples

We present illustrative examples to validate the Piecewise Constant Variable Order Fractional BVP discussed in the preceding section.

Consider the BVP on J = [0,0.5]:

${ }^C{D}^{\mu (t)}u(t)+\alpha (t)u(t)=f(t,u(t)),t\in [0,0.5]$

with the condition u(0) + (₀I ζu)(0.4) = β, where β = 1, ζ = 0.4, and the coefficient α(t) = 0.2t.

Solution: Divide J into two subintervals:

J1 = (0,0.25], J₂ = (0.25,0.5].

Define the piecewise constant variable fractional order:

$\mu (t)=\{\begin{array}{ll}0.4,\hfill & t\in J_1,\hfill \\ 0.6,\hfill & t\in J_2.\hfill \end{array}$

Hence, the problem becomes:

$\{\begin{array}{l}{ }^C{D}^{0.6}u(t)+(0.2t)u(t)=0.1t+0.06\text{arctan}(u(t))+0.04\frac{u(t)}{1+u{(t)}^2},t\in (0,0.25]\hfill \\ { }^C{D}^{0.8}u(t)+(0.2t)u(t)=0.1t+0.06\text{arctan}(u(t))+0.04\frac{u(t)}{1+u{(t)}^2},t\in (0.25,0.5]\hfill \\ u(0)+\left({ }_0{I}^{0.4}u\right)(0.4)=1\hfill \end{array}$

Compute the Lipschitz constant with respect to u:

$\begin{array}{rr}\hfill \left|f\left(t,u_1\right)-f\left(t,u_2\right)\right|& \hfill \le 0.06\left|\text{arctan}\left(u_1\right)-\text{arctan}\left(u_2\right)\right|+0.04\left|\frac{u_1}{1+u_1^2}-\frac{u_2}{1+u_2^2}\right|\\ \hfill & \hfill \le (0.06+0.04)\left|u_1-u_2\right|\\ \hfill & \hfill \le L_f\left|u_1-u_2\right|\end{array}$

Hence, the Lipschitz constant is L_f  = 0.1.

Uniqueness:

$k_i=\frac{{\xi }^{\zeta }}{\Gamma (\zeta +1)}+\frac{{\left({\rho }_2-{\rho }_1\right)}^{{\mu }_i}}{\Gamma \left({\mu }_i+1\right)}\left(L_f+\parallel \alpha {\parallel }_{\infty }\right)$

Take ξ = 0.4, ρ₂ − ρ₂ = 0.5, ∥α∥∞ = 0.1, L_f = 0.1:

$\begin{array}{rr}\hfill & \hfill k_1=\frac{{0.4}^{0.4}}{\Gamma (1.4)}+\frac{{0.5}^{0.4}}{\Gamma (1.4)}(0.1+0.1)\approx 0.780+0.171\approx 0.951<1\\ \hfill & \hfill k_2=\frac{{0.4}^{0.4}}{\Gamma (1.4)}+\frac{{0.5}^{0.6}}{\Gamma (1.6)}(0.1+0.1)\approx 0.780+0.148\approx 0.928<1\end{array}$

Hence, the condition k₁,k₂ < 1 is satisfied, guaranteeing the example has exactly one solution (Figures 1-3).

Figure 1: The overall schematic diagram of the system, showing the potential well morphology and particle distribution gradient of the neutron superfluid unit under the local gravitational field.

Figure 2: Variation of the fractional order µ(t) with time t, highlighting the piecewise-constant behavior and its influence on u(t).

Figure 3: Temporal evolution of t as a function of u(t) and u(t), emphasizing the dependency of solution dynamics on the variable order.

The three 3D plots display how u(t) evolves with time and the variable order µ(t). The solution remains smooth and single valued throughout [0, T], providing numerical confirmation of its uniqueness.

5. Conclusion

This study has established a condition that guarantees the uniqueness of the solution for the BVP involving Caputo VO-FDE. By employing the Banach contraction principle, it is shown that under appropriate assumptions on the nonlinear function and the coefficients, the BVP admits exactly one solution. The theoretical findings are further supported by illustrative examples, confirming the practical validity of the uniqueness result.