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

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.


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][2][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][7][8][9][10][11][12][13][14][15][16][17][18][19][20][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][23][24][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][27][28][29][30][31][32][33][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 κ = 0 with a controller adding or removing heat at κ = 1 depending on the temperature detected by a sensor point at η where η ∈ [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 ϑ ∈ (1, 2] where C D ϑ and C D ϑ-1 denote the Caputo fractional derivatives, δ > 0 and η ∈ [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][39][40][41][42][43][44][45][46][47][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][50][51][52][53][54][55][56][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 ϑ ∈ (1, 2], ϑ -1 ∈ (0, 1], δ & γ > 0 are a positive real parameter and a sensor point η ∈ N ϑ+ ϑ-1 is a constant, where C p is the CFDO of order p ∈ {ϑ, ϑ -1}, F : N ϑ+ +1 ϑ-2 × R → R is a continuous function and ∈ N 0 . Also, we consider various types of Ulam stability for DFTM with four-point BCs for ϑ ∈ (1, 2], ϑ -1 ∈ (0, 1], δ, β & γ > 0 and sensor points ζ , η ∈ N ϑ+ ϑ-1 are constants with 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.

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 B be a Banach space with norm u = max |u(κ)| for κ ∈ N ϑ+ +1 ϑ-2 . 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.

This leads to
Using the values of A i ∈ R, for i = 0, 1 in u(κ), we obtain for κ ∈ N ϑ+ +1 ϑ-2 . The proof is completed.
We introduce the notation ϑ u (κ) = F(κ + ϑ -1, u(κ + ϑ -1)). To transform the above DFTM with three-point BCs (2) to a fixed-point theorem, we define the operator T : for κ ∈ N ϑ+ +1 ϑ-2 . We know that the fixed point of T is a solution to (2). We consider the following hypotheses: Proof Let u,û ∈ B. Then, for each κ ∈ N ϑ+ +1 ϑ-2 , we have From the assumption (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 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 (H 2 ) and the inequality
where where ϑ u (κ) is given in (16). Using (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 T u ≤ M. Thus, T maps S u in S u and has at least one fixed point that is a solution to (2), according to the Brouwer fixed-point theorem.

Thermostat model with two sensors
This section discusses the stability results for the DFTM with four-point BCs (3).
Proof For the fractional sum of order ϑ ∈ (1, 2] for (21) and using Lemma 2.4, we obtain where A i ∈ R, for i = 2, 3. Applying the operators and C ϑ-1 on both sides of (23) together with Definitions 2.1 and 2.2, we obtain and In view of u(ϑ -2) = βu(ζ ), we obtain and From (26) and (27) and employing the first boundary condition (21), we obtain In view of δ C ϑ-1 u(ϑ + ) + γ u(η) = 0, we obtain and Since ϑ -1 ≤ 1, we arrive at From (29) and (30) with the help of the second boundary condition (21), we have The constant A 3 can be obtained by solving equations (28) and (31), which implies Substituting A 3 into (28), we have This implies, Using the constants A i ∈ R, for i = 2, 3 in (23), we obtain u in the form We assume that F is a real-valued continuous function on N ϑ+ +1 ϑ-2 such that θ u (κ) = F(κ + ϑ -1,û(κ + ϑ -1)). Now, we introduce the definitions of Ulam stability for DFC given on the basis of [60,61].
Proof Ifû(κ) solves the inequality (33), then from (ii) of Remark 3.7 and Lemma 2.4, the solution to (ii) of Remark 3.7 satisfieŝ Using (a) of Lemma 2.5 together with (i) of Remark 3.7, we arrive at This completes the proof.
Finally, we consider the following hypotheses to discuss the HUR stability and generalized HUR stability in the next results. (
Proof From inequality (35), we obtain a solution to (iv) of Remark 3.7 that satisfies (37). Using (H 3 ) of (i), for κ ∈ N +1 0 and Remark 3.7 of (iii), it follows that This completes the proof.

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.
Therefore, from Theorem 3.2 we come to the conclusion that (50) has a unique solution.
Consequently, it is obviously generalized HUR stable by using Theorem 3.11.

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.