On a nonlocal fractional thermostat eigenvalue problem

Imagine you are trying to keep a room at a perfect temperature using a very sophisticated, futuristic thermostat. But this isn't just a simple on/off switch; it's a system that remembers the past, reacts to the whole room at once, and has a "knob" (a parameter) that you can turn to see if the system can find a stable, comfortable state.

This paper is about figuring out when and how that system can find a stable, positive temperature (a "positive solution") and what the exact setting of your knob needs to be to make it work.

Here is a breakdown of the paper's ideas using everyday analogies:

1. The Problem: A "Time-Traveling" Thermostat

In the real world, a thermostat reacts to the current temperature. In this math problem, the thermostat is "fractional."

The Analogy: Imagine a thermostat that doesn't just look at the temperature right now, but also remembers a "fuzzy" version of the temperature from the past few minutes. It's like trying to drive a car where the steering wheel reacts not just to where you are turning it now, but also to where you turned it a second ago, with a delay that gets "fuzzier" the further back you go.
The Goal: The authors want to know: If we turn the "sensitivity knob" (called $\lambda$ ) to a specific number, will the system settle down into a steady, positive temperature? And if so, what is that number?

2. The Twist: The "Sign-Changing" Rule

Most previous studies assumed the thermostat's rules were always "nice" and helpful (mathematically, the "Green's function" was always positive).

The Analogy: Think of a rulebook for the thermostat. In old studies, the rulebook said, "If it's cold, heat it up. If it's hot, cool it down." Always helpful.
The New Discovery: This paper looks at a more chaotic rulebook. Sometimes, the rule says, "If it's cold, heat it up," but at other times or in other parts of the room, the rule might say, "If it's cold, actually cool it down a bit more first."
Why it matters: The authors prove that even when the rules are confusing or "change signs" (sometimes helping, sometimes hindering), the system can still find a stable, positive temperature. They figured out how to handle the "messy" parts of the rulebook.

3. The Tools: The "Cone" and the "Birkhoff-Kellogg" Magic

To solve this, the authors use a specific mathematical tool called a Birkhoff-Kellogg type theorem.

The Analogy: Imagine you are trying to find a specific spot on a giant, bumpy trampoline where a ball will stop rolling.
- The Cone: The authors build a "fence" (a mathematical cone) around the area where they think the ball must be. They only look inside this fence.
- The Magic Theorem: This theorem is like a guarantee that says, "If you push the ball hard enough from the edge of the fence, and the trampoline is shaped just right, the ball must bounce back and land on a specific spot inside the fence."
- The Result: They use this to prove that there is a perfect setting for the knob ( $\lambda$ ) that makes the ball (the temperature solution) land exactly where they want it.

4. The Three Scenarios

The authors break the problem down into three "weather conditions" based on how the thermostat behaves:

Case 1: The Sunny Day (Everything is Positive)
- The rules are always helpful. The math is the easiest here. They prove a solution exists easily.
Case 2: The Cloudy Day (Zeroes appear)
- The rules are mostly helpful, but at one specific point, the effect is zero (neutral). It's like the thermostat pauses for a split second. They show a solution still exists, but you have to be careful about where you look for it.
Case 3: The Stormy Day (Signs Change)
- This is the hardest part. The rules flip back and forth between helpful and unhelpful.
- The Trick: Even though the rules are chaotic, the authors found a small "safe zone" (a sub-interval) where the rules are still helpful. They proved that if the system settles down, it must settle down within this safe zone. This allows them to find the solution even in the storm.

5. The Payoff: Pinpointing the Knob

The paper doesn't just say "a solution exists." It gives you a map.

The Analogy: Instead of just saying, "There is a treasure somewhere in this ocean," they give you a specific coordinate: "The treasure is between 50 and 60 meters deep, and between 100 and 110 meters east."
They calculate specific intervals (ranges of numbers) for the knob ( $\lambda$ ). If you turn your knob to a number inside this range, you are guaranteed to get a stable, positive temperature.

Summary

This paper is a mathematical tour de force that takes a complex, memory-based thermostat model and proves it can work even when the rules are messy and confusing. They built a "safety fence" (a cone) and used a "guarantee theorem" to prove that a stable state exists, and then gave a precise map of exactly where to look for it.

In short: They showed that even in a chaotic, time-traveling system with confusing rules, you can still find the perfect setting to keep things stable and positive.

1. Problem Statement

The paper investigates the existence and localization of positive eigenpairs $(u, \lambda)$ for a parameter-dependent nonlocal boundary value problem (BVP) involving the Caputo fractional derivative. The specific problem is formulated as:

$\begin{cases} {}^C D^\alpha u(t) + \lambda f(t, u(t)) = 0, & t \in (0, 1), \\ u'(0) + \lambda H_1[u] = 0, \\ \beta {}^C D^{\alpha-1} u(1) + u(\eta) = \lambda H_2[u], \end{cases}$

Key Parameters and Conditions:

Order: $1 < \alpha \le 2$ .
Domain: $t \in (0, 1)$ , with $\eta \in (0, 1)$ and $\beta > 0$ .
Nonlinearity: $f: [0, 1] \times \mathbb{R} \to [0, \infty)$ is continuous.
Functionals: $H_1, H_2: C[0, 1] \to [0, \infty)$ are compact, non-negative functionals (generalizing linear or affine boundary conditions).
Parameter: $\lambda > 0$ is the eigenvalue to be determined.

Context:
This problem generalizes the classic fractional thermostat model (Nieto and Pimentel, 2015). While previous studies often focused on cases where the associated Green's function is strictly positive, this work addresses scenarios where the Green's function (and the auxiliary function $\gamma(t)$ ) may change sign depending on the parameter $\beta$ .

2. Methodology

The authors employ a functional analytic approach based on fixed point theory in cones, specifically utilizing a Birkhoff-Kellogg type theorem.

A. Integral Reformulation

The BVP is transformed into a Hammerstein-type integral equation:
$u(t) = \lambda T(u)(t)$
where the operator $T: C[0, 1] \to C[0, 1]$ is defined as:
$T(u)(t) = H_1[u]\gamma(t) + H_2[u] + \int_0^1 K(t, s)f(s, u(s)) \, ds$
Here, $K(t, s)$ is the Green's function associated with the linear part of the problem, and $\gamma(t)$ is a specific function solving a related homogeneous BVP.

B. Analysis of Sign Properties

A crucial contribution of the methodology is the rigorous analysis of the signs of $K(t, s)$ and $\gamma(t)$ based on the parameter $\beta$ . The authors define critical thresholds:

$\beta_K = \frac{(1-\eta)^{\alpha-1}}{\Gamma(\alpha)}$ (Threshold for $K \ge 0$ ).
$\beta_\gamma = (1-\eta)\Gamma(3-\alpha)$ (Threshold for $\gamma \ge 0$ ).

Depending on $\beta$ , the problem is categorized into three distinct cases, requiring different cone constructions:

Case 1 ( $\beta > \max\{\beta_K, \beta_\gamma\}$ ): Both $K$ and $\gamma$ are strictly positive on $[0, 1]$ .
Case 2 ( $\beta = \max\{\beta_K, \beta_\gamma\}$ ): $K$ and $\gamma$ are non-negative but may vanish at specific points (e.g., $t=1$ ).
Case 3 ( $\beta < \min\{\beta_K, \beta_\gamma\}$ ): Both $K$ and $\gamma$ change sign on $[0, 1]$ , though they remain non-negative on an initial sub-interval $[0, t^*]$ .

C. Cone Construction

For each case, the authors construct a specific cone $P_{\sigma_i} \subset C[0, 1]$ to apply the fixed point theorem:

Case 1 & 2: Cones of non-negative functions with a specific "Harnack-type" inequality (minimum value on a sub-interval is bounded below by a fraction of the maximum norm).
Case 3: A cone of functions that are positive on a sub-interval $[0, b]$ but allowed to change sign on the rest of the domain. This extends previous work by Infante and Webb to handle sign-changing kernels.

D. Theoretical Tool

The existence proof relies on the Birkhoff-Kellogg theorem in cones (Krasnosel'skiĭ and Ladyženskiĭ):

If $T$ is a compact operator mapping a cone section into the cone, and $\inf_{x \in \partial U} \|Tx\| > 0$ , then there exists a scalar $\lambda_0 > 0$ and a vector $x_0$ such that $x_0 = \lambda_0 T x_0$ .

3. Key Contributions

Generalization of Boundary Conditions: The paper incorporates nonlinear nonlocal functionals ( $H_1, H_2$ ) in the boundary conditions, moving beyond the linear or affine conditions used in earlier thermostat models.
Handling Sign-Changing Kernels: Unlike most literature that restricts analysis to positive Green's functions, this work explicitly handles cases where the kernel $K$ and the function $\gamma$ change sign. This significantly broadens the applicability of the model to a wider range of physical parameters.
Unified Framework: The authors provide a unified theoretical framework covering three distinct regimes of the parameter $\beta$ , offering existence results for all cases.
Eigenvalue Localization: The paper does not just prove existence; it provides explicit intervals $[L(\rho), U(\rho)]$ that localize the eigenvalue $\lambda$ for a solution with a prescribed norm $\|u\|_\infty = \rho$ .

4. Main Results

Existence Theorems (Theorems 3.2, 3.3, 3.4): Under specific growth conditions on $f$ and lower bounds on the functionals $H_1, H_2$ , the existence of a positive eigenpair $(u_\rho, \lambda_\rho)$ is guaranteed for each of the three cases. The solution $u_\rho$ satisfies specific positivity constraints on sub-intervals depending on the case.
Localization Theorem (Theorem 4.1): Given an eigenpair with norm $\rho$ , the eigenvalue $\lambda_\rho$ is bounded by:
$L(\rho) \le \lambda_\rho \le U(\rho)$
where $L(\rho)$ and $U(\rho)$ are calculated using the upper and lower bounds of the nonlinearity $f$ and the functionals $H_1, H_2$ .
Illustrative Examples: The theory is applied to two concrete examples:
- One with a non-negative kernel (Case 2).
- One with a sign-changing kernel (Case 3).
- The authors use Python to plot the localization regions, visually demonstrating how the eigenvalues and eigenfunctions are constrained.

5. Significance

This work represents a significant advancement in the theory of fractional differential equations with nonlocal boundary conditions.

Mathematical Rigor: By addressing sign-changing kernels, the authors overcome a major limitation in previous fixed-point approaches, which often required strict positivity assumptions that are not always physically realizable.
Modeling Flexibility: The inclusion of nonlinear functionals in the boundary conditions allows for more realistic modeling of complex control systems (thermostats) where the control mechanism is not linear.
Practical Utility: The provision of explicit localization intervals for eigenvalues is highly valuable for numerical simulations and engineering applications, as it allows researchers to predict the range of parameters ( $\lambda$ ) for which stable positive solutions exist without solving the full nonlinear system numerically first.

In summary, Infante and Zeghida extend the classical thermostat model to a more complex, nonlocal, and sign-variable setting, providing robust existence and localization results using advanced cone-theoretic methods.