Abstract
A thermostat model described by a second-order fractional difference equation is proposed in this paper with one sensor and two sensors fractional boundary conditions depending on positive parameters by using the Lipschitz-type inequality. By means of well-known contraction mapping and the Brouwer fixed-point theorem, we provide new results on the existence and uniqueness of solutions. In this work by use of the Caputo fractional difference operator and Hyer–Ulam stability definitions we check the sufficient conditions and solution of the equations to be stable, while most researchers have examined the necessary conditions in different ways. Further, we also establish some results regarding Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stability for our discrete fractional-order thermostat models. To support the theoretical results, we present suitable examples describing the thermostat models that are illustrated by graphical representation.
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1 Introduction
A thermostat is a device that senses a physical system’s temperature and performs actions to maintain the system’s temperature at a desired set point. A thermostat maintains the exact temperature, by controlling the switching on or off of the heating or cooling devices or by controlling the flow of heat-transfer fluid as necessary. In applications, ranging from ambient air control to automotive coolant control, a thermostat may often be the only control unit for a heating or cooling system.
Thermostats are used in an appliance or a system that heats or cools at a set-point temperature, such as house heating, air conditioning, central heating, water heaters, kitchen equipment like stoves and refrigerators, and medical and scientific incubators. Thermostats use various sensor types to measure the temperature. For one type, the mechanical thermostat, a coil-shaped bimetallic strip directly controls electrical contacts that control the source of heating or cooling. Alternatively, electronic thermostats use a thermistor or other semiconductor sensor to monitor the heating or cooling equipment, which includes amplification and processing.
Due to the rapid expansion in the literature of fractional calculus, there are many advanced techniques in the development of fractional-order ordinary and partial differential equations. They were used as excellent sources and methods for modeling many phenomena in the various fields of science, engineering, and technology, see the monographs [1–3]. Furthermore, the thermostat model, Burgers equation, Navier–Stokes equations, or Kirchhoff–Schrodinger-type equations are some of the real-world problems. Thus, different methods and techniques have been suggested for modeling these types of problems [4, 5].
Over the past three decades, many researchers have widely studied the topic of the classical initial boundary value problem (BVP) for ordinary and partial differential equations with integer and fractional order by using different methods. Stability analysis is an important branch of the qualitative theory of differential equations, as we know that sometimes finding the exact solution is quite challenging. Therefore, various numerical techniques were developed to find a solution. The most important type of stability is Ulam–Hyers stability. From a numerical and optimization point of view, Ulam–Hyers stability is essential because it provides a bridge between the exact and numerical solutions. Ulam–Hyers (or Ulam–Hyers–Rasssias) stability has been used extensively to study stability and has found applications in real-life problems such as in economics, biology, population dynamics, etc. [6–21].
However, only a few results have been obtained for linear and nonlinear ordinary and partial differential equations with the Caputo fractional derivative method and nonlocal boundary conditions [22–25]. The Caputo time fractional derivative can be used to model memory systems, since it includes all the context of the past. One of the most important classes of the thermostat models is the fractional thermostat equations that has been discussed and used in various fields of science. As is well known, different types of thermostat models have been studied by several researchers [26–34]. Very recently, Kaabar et al. [35] proved the existence of solutions for the fractional strongly singular thermostat model using nonlinear fixed-point techniques and investigated a hybrid version of the fractional thermostat control model. The study of thermostat models enables the development of efficient equipment used in several mechanical and electronic devices.
In 2006 [31], Infante and Webb developed a thermostat model, insulated at \(\kappa = 0\) with a controller adding or removing heat at \(\kappa = 1\) depending on the temperature detected by a sensor point at η
where \(\eta \in [0,1]\) is a real constant and δ is a positive parameter. By applying the fixed-point index theory on Hammerstein integral equations, they obtained existence results for the BVP. Recently, Nieto and Pimentel [32] extended the fractional thermostat model to the three-point boundary conditions (BCs) of order \(\vartheta \in (1,2]\)
where \({}^{C} D^{\vartheta }\) and \({}^{C} D^{\vartheta -1}\) denote the Caputo fractional derivatives, \(\delta >0\) and \(\eta \in [0,1]\) are real constants.
In recent years, a new field for researchers has become available, which is fractional difference equations (FDE). With the fractional difference operators, some real-world phenomena are being studied, see, e.g., [36]. Nevertheless, quite recently some researchers have developed much interest in the study of discrete fractional calculus (DFC). The study of DFC was initiated by Miller and Ross [37]. The authors [38–48] have recently recorded significant developments in that direction. Further, the existence and uniqueness of solutions and various kinds of Ulam-stability analysis for Caputo fractional difference equations have been established by several authors [49–57]. Motivated by the previously mentioned works [31, 32, 34, 58, 59], in this paper, we aim to investigate the following discrete fractional thermostat model (DFTM) with three-point BCs of the form
for \(\vartheta \in (1,2]\), \(\vartheta -1\in (0,1]\), δ & \(\gamma >0\) are a positive real parameter and a sensor point \(\eta \in \mathbb{N}_{\vartheta -1}^{\vartheta +\ell }\) is a constant, where \({}^{C}\Delta ^{p}\) is the CFDO of order \(p\in \{ \vartheta, \vartheta -1 \} \), \(\mathcal{F}: \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1} \times \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function and \(\ell \in \mathbb{N}_{0}\). Also, we consider various types of Ulam stability for DFTM with four-point BCs
for \(\vartheta \in (1,2]\), \(\vartheta -1\in (0,1]\), δ, β & \(\gamma >0\) and sensor points \(\zeta, \eta \in \mathbb{N}_{\vartheta -1}^{\vartheta +\ell }\) are constants with \(\zeta \leq \eta \). Comparing (3) with (1), we have \(\mathcal{F}(\kappa,u)=-\psi (\kappa,u)\).
This paper is organized as follows. Some definitions and properties of DFC used to establish the main results are provided in Sect. 2. Existence and uniqueness of solutions for a DFTM with three-point BCs (2) are obtained by using a contraction mapping theorem and the Brouwer fixed-point theorem in Sect. 3.1. In Sect. 3.2, we introduce some new results for various forms of Ulam stability analysis of a DFTM with four-point BCs (3). In Sect. 4, suitable examples are discussed as applications to show the applicability of our obtained results, and the paper ends with a conclusion in Sect. 5.
2 Basic preliminaries
This section consists of definitions and preliminary lemmas, which are essential for the discussion of our results.
Definition 2.1
(see [39])
For \(\vartheta > 0\), the ϑth-order fractional sum of \(\mathcal{F}:\mathbb{N}\rightarrow \mathbb{R}\) is defined as
for \(\kappa \in {\mathbb{N}_{a+\vartheta }}\), \(\sigma (\kappa )=\xi +1\) and \(\kappa ^{(\vartheta )}:= \frac{\Gamma (\kappa +1)}{\Gamma (\kappa +1-\vartheta )}\).
Moreover, Composition rules [39] are;
-
Assume \(\mathcal{F}\) is defined on \(\mathbb{N}_{a}\) and \(\mu, \vartheta \) are positive numbers. Then,
$$\begin{aligned} \bigl[\Delta ^{-\mu }_{a+\vartheta } \bigl(\Delta ^{-\vartheta }_{a} \mathcal{F}\bigr) \bigr](\kappa ) = \bigl(\Delta _{a} ^{-(\mu + \vartheta )} \mathcal{F} \bigr) (\kappa ) = \bigl[\Delta _{a+\mu }^{- \vartheta } \bigl( \Delta _{a} ^{-\mu } \mathcal{F}\bigr) \bigr] (\kappa ), \end{aligned}$$for \(\kappa \in \mathbb{N}_{a+\mu +\vartheta }\).
-
Assume \(\mathcal{F}: \mathbb{N}_{a} \to \mathbb{R}\) with \(\vartheta, \mu >0\) and \(0\leq \mathcal{N}-1 < \vartheta \leq \mathcal{N}\). Then,
$$\begin{aligned} \Delta _{a+\mu }^{\vartheta } \Delta _{a} ^{-\mu } \mathcal{F}(\kappa ) = \Delta _{a} ^{\vartheta -\mu } \mathcal{F}(\kappa ), \end{aligned}$$for \(\kappa \in \mathbb{N}_{a+\mu +\mathcal{N}-\vartheta }\) and \(\mathcal{N} \in \mathbb{N}\).
Definition 2.2
(see [38])
For \(\vartheta >0\) and \(\mathcal{F}\) being defined on \(\mathbb{N}_{a}\), the ϑth Caputo fractional difference of \(\mathcal{F}\) is
for \(\kappa \in {\mathbb{N}_{a+\mathcal{N}-\vartheta }}\) and \(\mathcal{N}\in \mathbb{N}\) such that \(0\leq \mathcal{N}-1<\vartheta \leq \mathcal{N}\). If \(\vartheta =\mathcal{N}\), then \({}^{C}\Delta ^{\vartheta } \mathcal{F}(\kappa )=\Delta ^{\mathcal{N}} \mathcal{F}(\kappa )\), for \(\kappa \in \mathbb{N}_{a}\).
Lemma 2.3
Assume \(\kappa, \vartheta >0\) for which \(\kappa ^{(\vartheta )}\), \(\kappa ^{(\vartheta -1)}\) are defined. Then, \(\Delta \kappa ^{(\vartheta )}=\vartheta \kappa ^{(\vartheta -1)}\).
Lemma 2.4
(see [38])
Suppose that \(\vartheta >0\) and \(\mathcal{F}\) is defined on \({\mathbb{N}_{a}}\). Then,
for some \(\mathcal{A}_{i} \in {\mathbb{R}}\), with \(0\leq i \leq \mathcal{N}-1\).
Lemma 2.5
(see [49])
Assume κ, ϑ and ℓ are positive numbers for which \(\kappa ^{(\vartheta )}\) is defined. Then,
-
(a)
\(\sum_{\xi =0}^{\kappa -\vartheta } (\kappa -\sigma ( \xi ) )^{(\vartheta -1)}=\frac{1}{\vartheta }\kappa ^{( \vartheta )} \),
-
(b)
\(\sum_{\xi =0}^{\ell } (\vartheta +\ell -\sigma (\xi ) )^{(\vartheta -1)}=\frac{1}{\vartheta }(\vartheta +\ell )^{( \vartheta )}\).
Lemma 2.6
(see [38])
Let \(\vartheta, j>0\). Then,
3 Main results
3.1 Thermostat model with one sensor
This section studies the existence and uniqueness results to the DFTM with three-point BCs (2). First, we introduce some notations that are used in this paper. Let \(\mathcal{B}\) be a Banach space with norm \(\Vert u \Vert =\max \vert u(\kappa ) \vert \) for \(\kappa \in \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\). Now, we state and prove an important theorem that deals with a linear variant of the solution of DFTM with three-point BCs (2) and we give a representation of the solution.
Theorem 3.1
Let real-valued function \(\mathcal{F}\) be defined on \(\mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\). Then, for \(\kappa \in \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\) the following DFTM
has a unique solution that is obtained by
Proof
Let \(u(\kappa )\) be a solution to (4). Using Lemma 2.4, for some constants \(\mathcal{A}_{i} \in \mathbb{R}\), for \(i=0, 1\), we have
Using the fractional sum of order \(\vartheta \in (1,2]\), we obtain
By applying Δ to the parts of (6), we have
Due to the first boundary condition \(\Delta u(\vartheta -2)=0\) in (7), we obtain \(\mathcal{A}_{1}=0\). Using the CFDO \({}^{C}\Delta ^{\vartheta -1}\) of order \(\vartheta -1\in (0,1]\) on both the sides of (6) with \(\mathcal{A}_{1}=0\), it provides
Here, using the Definition 2.2 that for constant \(\mathcal{A}_{0}\), \({}^{C}\Delta ^{\vartheta -1}\mathcal{A}_{0}= \Delta ^{-(2-\vartheta )} \Delta \mathcal{A}_{0}=\Delta ^{-(2-\vartheta )}(0) = 0\), yields
Using the second boundary condition \(\delta {}^{C}\Delta ^{ \vartheta -1} u(\vartheta +\ell )+\gamma u(\eta )=0\) in (6) and (8), we obtain
Since \(\vartheta -1\leq 1\), we obtain
and
From (9) and (10) in \(\delta {}^{C}\Delta ^{\vartheta -1} u(\vartheta +\ell )+u(\eta )=0\), we arrive at
This leads to
Using the values of \(\mathcal{A}_{i} \in \mathbb{R}\), for \(i=0, 1\) in \(u(\kappa )\), we obtain
for \(\kappa \in \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\). The proof is completed. □
We introduce the notation \(\Phi _{u}^{\vartheta }(\kappa )=\mathcal{F}(\kappa +\vartheta -1, u( \kappa +\vartheta -1))\). To transform the above DFTM with three-point BCs (2) to a fixed-point theorem, we define the operator \(\mathcal{T}:\mathcal{B}\rightarrow \mathcal{B}\) by
for \(\kappa \in \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\). We know that the fixed point of \(\mathcal{T}\) is a solution to (2).
We consider the following hypotheses:
- (\(\mathcal{H}_{1}\)):
-
The Lipschitz-type inequality: There exists \(\mathcal{K}>0\) such that \(\vert \mathcal{F}(\kappa,u)-\mathcal{F}(\kappa,\hat{u}) \vert \leq \mathcal{K} \vert u-\hat{u} \vert \) for all \(u,\hat{u}\in \mathcal{B}\) and each \(\kappa \in \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\).
- \((\mathcal{H}_{2})\):
-
There exists a bounded function \(\mathcal{L}:\mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1} \rightarrow \mathbb{R}\) with \(\vert \mathcal{F}(\kappa,u) \vert \leq \mathcal{L}(\kappa ) \vert u \vert \) for all \(u\in \mathcal{B}\).
Theorem 3.2
If the hypothesis (\(\mathcal{H}_{1}\)) holds, then the DFTM with three-point BCs (2) has a unique solution in \(\mathcal{B}\) provided
Proof
Let \(u,\hat{u}\in \mathcal{B}\). Then, for each \(\kappa \in \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\), we have
where \(\Phi _{u}^{\vartheta },\Phi _{\hat{u}}^{\vartheta }\in \mathcal{C} (\mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}, \mathbb{R} )\) satisfies the functional equations
From the assumption \((\mathcal{H}_{1})\), we obtain
Substituting the inequality (17) into (15), it follows that
In view of Lemma 2.5 of (a), we obtain
therefore, it follows that \(\mathcal{T}\) is a contraction and has a unique fixed point that is the solution of (2). □
Theorem 3.3
The DFTM with three-point BCs (2) has at least one solution under the assumption (\(\mathcal{H}_{2}\)) and the inequality
where \(\mathcal{L}^{*}=\max \{ \mathcal{L}(\kappa ):\mathbb{N}_{ \vartheta -2}^{\vartheta +\ell +1} \} \).
Proof
Suppose that \(\mathfrak{M}>0\) and \(\mathcal{S}_{u}= \{ u(\kappa ) | \mathbb{N}_{\vartheta -2}^{ \vartheta +\ell +1}\rightarrow \mathbb{R}, \Vert u \Vert \leq \mathfrak{M} \} \). We must first show that \(\mathcal{T}\) maps \(\mathcal{S}_{u}\) in \(\mathcal{S}_{u}\).
For \(u(\kappa )\in \mathcal{S}_{u}\), we have
where \(\Phi _{u}^{\vartheta }(\kappa )\) is given in (16). Using (\(\mathcal{H}_{2}\)), we arrive at
Hence, putting the inequality (19) and (20) together, we conclude that
From Lemma 2.5 of (a), we have
In view of (18), we obtained \(\Vert \mathcal{T}u \Vert \leq \mathfrak{M}\). Thus, \(\mathcal{T}\) maps \(\mathcal{S}_{u}\) in \(\mathcal{S}_{u}\) and has at least one fixed point that is a solution to (2), according to the Brouwer fixed-point theorem. □
3.2 Thermostat model with two sensors
This section discusses the stability results for the DFTM with four-point BCs (3).
Theorem 3.4
Assume \(\mathcal{F}: \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1} \rightarrow \mathbb{R}\) is given. A unique solution to the DFTM with four-point BCs
has the form
where \(\kappa \in \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\), \(\mathcal{D}_{1}(\kappa )= \frac{[\delta \mu +\gamma (\eta -\kappa )]}{\mathcal{Q}}\), \(\mathcal{D}_{2}(\kappa )= \frac{[\beta (\zeta -\kappa )-1]}{\mathcal{Q}}\) such that \(\mathcal{Q}=\gamma (\beta \zeta -1)-\beta ( \delta \mu + \gamma \eta )\) and \(\mu =\frac{1}{\Gamma (3-\vartheta )}(\vartheta +\ell )^{(2- \vartheta )}\).
Proof
For the fractional sum of order \(\vartheta \in (1,2]\) for (21) and using Lemma 2.4, we obtain
where \(\mathcal{A}_{i} \in \mathbb{R}\), for \(i=2,3\). Applying the operators Δ and \({}^{C}\Delta ^{\vartheta -1}\) on both sides of (23) together with Definitions 2.1 and 2.2, we obtain
and
In view of \(\Delta u(\vartheta -2)= \beta u(\zeta )\), we obtain
and
From (26) and (27) and employing the first boundary condition (21), we obtain
In view of \(\delta {}^{C}\Delta ^{\vartheta -1} u(\vartheta +\ell )+\gamma u( \eta )=0\), we obtain
and
Since \(\vartheta -1\leq 1\), we arrive at
From (29) and (30) with the help of the second boundary condition (21), we have
The constant \(\mathcal{A}_{3}\) can be obtained by solving equations (28) and (31),
which implies
Substituting \(\mathcal{A}_{3}\) into (28), we have
This implies,
Using the constants \(\mathcal{A}_{i} \in \mathbb{R}\), for \(i=2, 3\) in (23), we obtain u in the form
for \(\kappa \in \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\). □
We assume that \(\mathcal{F}\) is a real-valued continuous function on \({\mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}}\) such that \(\Phi _{\hat{u}}^{\vartheta }(\kappa )=\mathcal{F}(\kappa +\vartheta -1, \hat{u}(\kappa +\vartheta -1))\). Now, we introduce the definitions of Ulam stability for DFC given on the basis of [60, 61].
Definition 3.5
If for every function \(\hat{u}(\kappa )\in \mathbb{B}\) of
where \(\kappa \in {\mathbb{N}_{0}^{\ell +1}}\), \(\epsilon >0\), there exists a solution \(u(\kappa )\in \mathbb{B}\) of (3) and a positive constant \(\mathcal{P}_{1}>0\) such that
Then, the DFTM with four-point BCs (3) is Hyers–Ulam (HU) stable. Equation (3) is also said to be generalized HU stable if we substitute \(\Theta (\epsilon )=\mathcal{P}_{1} \epsilon \) in inequality (34), where \(\Theta (\epsilon )\in \mathbb{C} (\mathbb{R}^{+}, \mathbb{R}^{+} )\) and \(\Theta (0)=0\).
Definition 3.6
Let ∀ \(\hat{u}(\kappa )\in \mathbb{B}\), then the following inequality holds
where \(\kappa \in {\mathbb{N}_{0}^{\ell +1}}\), \(\epsilon >0\), there is a solution \(u(\kappa )\in \mathbb{B}\) of (3) and a positive constant \(\mathcal{P}_{2}>0\) such that
Then, the DFTM with four-point BCs (3) is Hyers–Ulam–Rassias (HUR) stable. Equation (3) is generalized HUR stable if we substitute \(\phi (\kappa +\vartheta -1)= \epsilon \phi (\kappa +\vartheta -1)\) in inequalities (35) and (36).
Remark 3.7
A function \(\hat{u}(\kappa )\in \mathcal{B}\) is a solution to the inequalities (33) and (35) if there exists a function \(f:\mathbb{N}_{\beta -2}^{\beta +\ell +1}\rightarrow \mathbb{R}\) satisfying, for \(\kappa \in \mathbb{N}_{0}^{\ell +1}\)
-
(i)
\(\vert f(\kappa +\vartheta -1) \vert \leq \epsilon \),
-
(ii)
\({}^{C}\Delta ^{\vartheta } \hat{u}(\kappa )=\Phi _{\hat{u}}^{\vartheta }( \kappa )+f(\kappa +\vartheta -1)\),
-
(iii)
\(\vert f(\kappa +\beta -1) \vert \leq \epsilon \phi (\kappa +\beta -1)\),
-
(iv)
\({}^{C} \Delta ^{\beta } \hat{u}(\kappa )=\Phi _{\hat{u}}^{\vartheta }( \kappa )+f(\kappa +\beta -1)\).
Lemma 3.8
If \(\hat{u}(\kappa )\) solves the inequality (33) for \(\kappa \in \mathbb{N}_{0}^{\ell +1}\), then
where \(\mathcal{D}_{1}(\kappa )\) and \(\mathcal{D}_{2}(\kappa )\) are defined in Theorem 3.4.
Proof
If \(\hat{u}(\kappa )\) solves the inequality (33), then from (ii) of Remark 3.7 and Lemma 2.4, the solution to (ii) of Remark 3.7 satisfies
Using (a) of Lemma 2.5 together with (i) of Remark 3.7, we arrive at
This completes the proof. □
Theorem 3.9
Assume that the following inequalities and (\(\mathcal{H}_{1}\)) hold at the same time
then the DFTM with four-point BCs (3) is HU stable and generalized HUR stable.
Proof
From solution (22), for \(\kappa \in \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\), it follows that
where \(\mathcal{D}_{1}(\kappa )\), \(\mathcal{D}_{2}(\kappa )\) are defined in Theorem 3.4 and \(\Phi _{u}^{\vartheta }(\kappa )\), \(\Phi _{\hat{u}}^{\vartheta }(\kappa )\) are given in (16). Using the inequality (17) and Lemma 3.8 along with an application of Lemma 2.5 of (a), implies that
where \(\mathcal{G}_{1}= \vert \frac{\delta \mu +\gamma [\eta -(\vartheta +\ell +1)]}{\mathcal{Q}} \vert \) and \(\mathcal{G}_{2}= \vert \frac{[\beta (\zeta -[\vartheta +\ell +1])-1]}{\mathcal{Q}} \vert \).
Inequality (39) yields \(\Vert \hat{u}-u \Vert \leq \mathcal{P}_{1}\epsilon \), where
Thus, the solution to (3) is HU stable.
Further, by taking \(\Theta (\epsilon )=\mathcal{P}_{1} \epsilon \) with \(\Theta (0)=0\), we have
Hence, the solution to (3) becomes generalized HU stable. □
Finally, we consider the following hypotheses to discuss the HUR stability and generalized HUR stability in the next results.
- (\(\mathcal{H}_{3}\)):
-
For an increasing function \(\phi \in \mathcal{C} ({\mathbb{N}_{\vartheta -2}^{\vartheta + \ell }}, \mathbb{R}^{+} )\), there exists \(\lambda _{\phi }>0\) such that, for \(\kappa \in {\mathbb{N}_{0}^{\ell +1}}\)
-
(i)
\(\frac{\epsilon }{\Gamma (\vartheta )}\sum_{\xi =0}^{\kappa - \vartheta }(\kappa -\sigma (\xi ))^{(\vartheta -1)}\phi (\xi + \vartheta -1)\leq \lambda _{\phi } \epsilon \phi (\kappa +\vartheta -1)\), consequently
-
(ii)
\(\frac{1}{\Gamma (\vartheta )}\sum_{\xi =0}^{\kappa - \vartheta }(\kappa -\sigma (\xi ))^{(\vartheta -1)}\phi (\xi + \vartheta -1)\leq \lambda _{\phi } \phi (\kappa +\vartheta -1)\).
-
(i)
Lemma 3.10
If \(\hat{u}(\kappa )\) solves the inequality (35) for \(\kappa \in \mathbb{N}_{0}^{\ell +1}\), then
where \(\mathcal{D}_{1}(\kappa )\) and \(\mathcal{D}_{2}(\kappa )\) are defined in Theorem 3.4.
Proof
From inequality (35), we obtain a solution to (iv) of Remark 3.7 that satisfies (37). Using (\(\mathcal{H}_{3}\)) of (i), for \(\kappa \in \mathbb{N}_{0}^{\ell +1}\) and Remark 3.7 of (iii), it follows that
This completes the proof. □
Theorem 3.11
If the hypothesis (\(\mathcal{H}_{1}\)) holds with the inequality (38), then the DFTM with four-point BCs (3) is HUR stable and generalized HUR stable.
Proof
From the solution (22), for \(\kappa \in \mathbb{N}_{\vartheta -2}^{\vartheta +\ell +1}\), we obtain
where \(\mathcal{D}_{1}(\kappa )\) and \(\mathcal{D}_{2}(\kappa )\) are defined in Theorem 3.4. Using Lemma 3.10 and the procedure used in Theorem 3.9, we obtain
By an application of Lemma 2.5 of (a), the above inequality becomes
where \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\) are defined in Theorem 3.9. From which, the inequality (40) yields
where \(\mathcal{P}_{2} = \frac{\lambda _{\phi }\Gamma (\vartheta +1)}{\Gamma (\vartheta +1)-\mathcal{K} [(\vartheta +\ell +1)^{(\vartheta )}+\beta \mathcal{G}_{1} \zeta ^{(\vartheta )}+\mathcal{G}_{2} (\gamma \eta ^{(\vartheta )}+\delta (\ell +2)\Gamma (\vartheta +1) ) ]}\).
Hence, the solution of (3) is HUR stable.
Also, by setting \(\phi (\kappa +\vartheta -1)=\epsilon \phi (\kappa +\vartheta -1)\), we have
Therefore, the solution of (3) is generalized HUR stable. □
4 Examples
In this section, we validate the theoretical results by providing examples for discrete fractional thermostat models with three-point BCs (2) and four-point BCs (3) by using CFDO.
Example 4.1
Consider the linear DFTM with three-point BCs (4)
Here, \(\vartheta =1.67\), \(\ell =4\), \(\delta =0.8\), \(\gamma =0.9\) and \(\mathcal{F}(\kappa )=\kappa ^{(8)}\). Applying Theorem 3.1, we obtain that \(u(\kappa )\) is a solution of (41) that is given by
for \(\kappa \in \mathbb{N}_{-0.33}^{6.67}\). Now, solving the solution (42) by using Definition 2.1 and Lemma 2.6, we obtain
Similarly, we obtain
Also, we find
Combining (42), (43), (44), and (45), we obtain a solution to (41) as follows:
Furthermore, we also consider the linear DFTM with four-point BCs (21)
Here, \(\vartheta =1.67\), \(\ell =4\), \(\delta =0.8\), \(\beta =0.2\), \(\gamma =0.9\), and \(\mathcal{F}(\kappa )=\kappa ^{(8)}\). From Theorem 3.4, we obtain \(u(\kappa )\) as a solution to (47) that is given by
where \(\kappa \in \mathbb{N}_{-0.33}^{6.67}\), \(\mathcal{D}_{1}(\kappa )\) and \(\mathcal{D}_{2}(\kappa )\) are defined in Theorem 3.4. From (43), (44), (45), and (48), we obtain a solution to (47) as follows:
Using the solutions (46) and (49) along with \(\zeta =0.67\) and various values of \(\eta =1.67, 2.67\), we obtain different solutions to the corresponding DFTMs with three-point BCs (41) and four-point BCs (47), as seen in Fig. 1 and Table 1. Figure 2 illustrates the solution surface plots over different values of η and κ.
Example 4.2
Let us consider the parameters \(\vartheta =1.6\), \(\ell =0\), \(\delta =0.5\), \(\gamma =0.4\), and \(\eta =0.6\) with \(\mathcal{F}(\kappa,u(\kappa ))=\frac{1}{20} [ \frac{\kappa }{3}\cos ^{2} (\frac{\pi }{2}(\kappa ) )+ \sin (u(\kappa )) ]\). Then, we obtain a DFTM with three-point BCs (2) in the form
We now show that (50) has a unique solution. Since \((\mathcal{H}_{1})\), holds for each \(\kappa \in {\mathbb{N}_{-0.4}^{2.6}}\), we obtain
so for \(\mathcal{K}=\frac{1}{20}\). Thus, for inequality (14), we have
Therefore, from Theorem 3.2 we come to the conclusion that (50) has a unique solution.
Example 4.3
Suppose that \(\vartheta =1.4\), \(\ell =1\), \(\delta =0.4\), \(\gamma =0.3\), \(\mathfrak{M}=150\), and \(\eta = 0.4\) with \(\mathcal{F}(\kappa,u(\kappa ))=\kappa ^{2} e^{-u^{2}(\kappa )}\). Then, we obtain the following Caputo DFTM with three-point BCs (2)
Consider the Banach space \(\mathcal{B}:= \{ u(\kappa ) | \mathbb{N}_{-0.6}^{3.4} \rightarrow \mathbb{R}, \Vert u \Vert \leq 150 \} \). We note that
It is clear that \(\vert \mathcal{F}(\kappa,u(\kappa )) \vert = 11.5600<18.5644\), whenever \(u\in [-150, 150]\). Therefore, having at least one solution for (51) concluded from Theorem 3.3.
Example 4.4
Assume that \(\vartheta =1.5\), \(\ell =1\), \(\delta =0.5\), \(\beta =0.2\), \(\gamma =0.4\), \(\zeta =0.5\), \(\eta =1.5\), and \(\mathcal{F}(\kappa,u(\kappa ))=\frac{2}{\sqrt{\pi }} (\kappa -0.5 )^{(0.5)}+\frac{u(\kappa )}{20}\). Then, for the following Caputo DFTM with four-point BCs (3)
we prove that (52) is HU stable. To begin with, we need to verify that \(\mathcal{F}\) satisfies \((\mathcal{H}_{1})\) for \(\kappa \in {\mathbb{N}_{-0.5}^{3.5}}\), we obtain
Hence, \(\mathcal{K}=\frac{1}{20}\) and \(\mathcal{F}\) is Lipschitz continuous for \(\kappa \in {\mathbb{N}_{-0.5}^{3.5}}\). Since
for \(\mathcal{K}=\frac{1}{20}<0.1120\), we obtain \(\Lambda =0.4465<1\). This shows (52) is HU stable with \(\mathcal{P}_{1}=7.9038\). Further, it is also generalized HU stable. To check this, put \(\epsilon =0.1563\) and \(\hat{u}(\kappa )=\frac{\kappa ^{2}}{2}\), \(\kappa \in {\mathbb{N}_{0}^{2}}\) and also prove that (33) holds. Indeed,
Using (iii) of Lemma 2.6 in (53), we have
Example 4.5
Consider the following Caputo DFTM with four-point BCs (3)
Here, \(\vartheta =1.6\), \(\ell =1\), \(\delta =0.6\), \(\beta =0.3\), \(\gamma =0.2\), \(\zeta =0.6\), \(\eta =1.6\), and \(\mathcal{F}(\kappa,u(\kappa ))=0.1\kappa +0.03\sin (u(\kappa ))\). Now, we prove that (54) is HUR stable. Since \((\mathcal{H}_{1})\) holds for each \(\kappa \in {\mathbb{N}_{-0.4}^{3.6}}\), we obtain
so for \(\mathcal{K}=0.03\). Further, assuming \(\epsilon =0.29\) and \(\phi (\kappa +0.6)=1\), we have
Thus, inequality \((\mathcal{H}_{3})\), of (i), holds with \(\lambda _{\phi }=1.5767\), \(\epsilon =0.29\), and \(\phi (\kappa +0.6)=1\), for \(\kappa \in \mathbb{N}_{0}^{2}\). Since
if \(\mathcal{K}=0.03<0.0882\), from Theorem 3.11, we see that \(\Lambda =0.3400<1\). Hence, HUR stablity of the solution (54) is obtained from \(\mathcal{P}_{2}=2.3890\). To verify this, put \(\epsilon =0.29\), \(\hat{u}(\kappa )=\kappa \) for \(\kappa \in {\mathbb{N}_{0}^{2}}\). We prove that (35) holds. Indeed,
Consequently, it is obviously generalized HUR stable by using Theorem 3.11.
5 Conclusion
It is essential that we enhance our ability to understand complicated discrete fractional thermostat models. One of the strategies is to apply well-known models to various complicated sensor problems. In this paper, we have studied a new form of DFTMs with the three-point and four-point BCs by the Caputo difference operator. Existence and uniqueness results and various forms of HU stability are discussed with the aid of properties of the fractional operator and different fixed-point techniques for the concerned problems. Also, we presented sufficient conditions for stable solutions by using the Caputo difference operator in the discrete case. On the basis of our theoretical findings, we have presented suitable examples with numerical solutions to different values of κ and η supported with graphical illustrations. The findings of this study can be seen as a contribution to the developing area of discrete fractional thermostat models that describe mathematical models of engineering and applied-science applications.
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Acknowledgements
J. Alzabut (ORCID 0000-0002-5262-1138) would like to thank Prince Sultan University and OSTİM Technical University for their endless support. The fifth and sixth authors were supported by Azarbaijan Shahid Madani University. The authors express their gratitude to the dear unknown referees for their helpful suggestions that improved the final version of this paper.
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GMS, RD and ST dealt with the conceptualization, supervision, methodology, investigation, and writing the original draft preparation. JA, MG, and SR made the formal analysis, writing, reviewing, editing and preparing the figures and the table. All authors read and approved the final manuscript.
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Alzabut, J., Selvam, A.G.M., Dhineshbabu, R. et al. A Caputo discrete fractional-order thermostat model with one and two sensors fractional boundary conditions depending on positive parameters by using the Lipschitz-type inequality. J Inequal Appl 2022, 56 (2022). https://doi.org/10.1186/s13660-022-02786-0
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DOI: https://doi.org/10.1186/s13660-022-02786-0