On the qualitative analysis of the fractional boundary value problem describing thermostat control model via ψ-Hilfer fractional operator

In this research study, we are concerned with the existence and stability of solutions of a boundary value problem (BVP) of the fractional thermostat control model with ψ -Hilfer fractional operator. We verify the uniqueness criterion via the Banach fixed-point principle and establish the existence by using the Schaefer and Krasnoselskii fixed-point results. Moreover, we apply the arguments related to the nonlinear functional analysis to discuss various types of stability in the format of Ulam. Finally, by several examples we demonstrate applications of the main findings.


Introduction
Fractional differential and integral equations have demonstrated high visibility and capability in applications of various topics related to physics, signal processing, mechanics, electromagnetics, economics, biology, and many more [1,2]. Particularly speaking, it has been recognized that fractional integro-differential equations, whose kernels allow much freedom to describe various processes involving memory and hereditary properties, often appear in different fractional models caused by many real-life processes such as phenomena related to electromagnetic waves and heat transfer. Yang et al. [3] implemented a discussion on the steady heat-transfer in the context of fractal media by invoking the local fractional nonlinear integro-differential equations of Volterra type. Furthermore, electromagnetic waves in a wide range of dielectric media including the susceptibility following a fractional power law are formulated in the framework of integro-differential equations [4]. We can see some recent advances and applications of fractional modelings in several newly published researches such as [5][6][7][8]. Also, in some new papers, the advantages and power of mathematical modeling based on fractional operators are illustrated, and that is why in recent years, many researchers prefer studying real processes and phenomena by applying newly defined versions of fractional operators (see, e.g., [9][10][11][12][13][14][15][16][17][18]).
Amongst important physio-electrical models, we are concerned with the thermostat control model. In this context a thermostat is a regulating instrument that measures the temperature of a given physical system and takes actions, provided that its temperature is maintained near a desired level. Thermostats are applied in any industrial system or controlling devices that cool or heat the temperature, examples including central heating, building heating, water heaters, air conditioners, water heaters, and also kitchen equipment such as refrigerators, ovens, and scientific and medical incubators. In [19] the authors demonstrated interest in the investigation of a thermostat control model insulated at ς = 0 with controller at ς = 1 and proposed the following boundary value problem (BVP) for the first time: in which k ∈ (0, 1), and c > 0 is assumed to be an arbitrary parameter. Based on such a second-order mathematical model, the thermostat discharges or adds an amount of the heat with respect to the temperature detected by the existing sensor at ς = k. They proved existence results for (1) by following the fixed point index theory in the context of integral equations of Hammerstein type. Knowing the magnificent advantage of fractional derivatives, the authors in [20] studied the fractional-order thermostat control model ⎧ ⎨ ⎩ C D α x(ς) + f (ς, x(ς)) = 0, ς ∈ (0, 1], where C D α is the Caputo derivative of order α ∈ (1,2]. Based on the hypothesis that the nonlinearity f is assumed to be either superlinear or sublinear, the existence of positive solutions was proved by the help of the obtained Green's function and the Guo-Krasnoselskii fixed-point results. In their recent paper [21] the authors studied the fractional configuration of the thermostat control model subject to a convex-concave source term. They used the fixed point technique to prove the existence and uniqueness of positive solutions and provided an iterative scheme to approximate the obtained solutions. For more details on fractional thermostat control models, the reader can consult [22][23][24][25][26]. Exploring this literature, we can notice that the reported results were restricted to the existence of positive solutions and their properties. However, to the best of the authors' knowledge, no results were observed on the thermostat control model in the frame of generalized fractional operators. Further, the stability of solutions of fractional thermostat control models was not addressed yet. Motivated by the above discussions, we consider a category of ψ-Hilfer nonlinear implicit fractional boundary value problems (FBVPs) describing the thermostat control model of the form . . , m, j = 1, 2, . . . , n, k = 1, . . . , r, ρ ∈ [0, 1], I q;ψ 0 + is the ψ-RL-integral of order q > 0, θ , ε ∈ (0, 1], f ∈ C(J × R 2 , R), and J := [0, T] with T > 0. We establish existence and stability results for (3). We employ fixed point hypotheses to prove the existence results and we use the techniques of nonlinear functional analysis to study the stability in the Ulam sense. We present our results in a general platform, which covers many particular cases for specific values of ρ and ψ. For some relevant results, we refer the reader to recent papers [27][28][29][30][31][32][33].
The remaining parts of the research study adhere to the following plan. In Sect. 2, we define the norms, spaces, and other essential notions and lemmas related to the ψ-Hilfer fractional operator. Further, we derive the equivalent integral representation associated with the linear problem and state some fixed point theorems. We present the existence and uniqueness results in terms of three different fixed-point criteria in Sect. 3. In Sect. 4, we systematically present stabilization analysis of problem (3). In Sect. 5, we construct three particular examples, where the validity of the proposed results is verified. We terminate the investigation by conclusions.

Primitive notions
In this section, we give important basic definitions and primitive concepts of fractional calculus, which are useful throughout this paper.
We denote by E = C(J, R) the Banach space of continuous mappings on J with supnorm We also define the space of n times absolutely continuous functions where is the gamma function.
From the first and second boundary conditions in (3) we get the system where 1 , 2 , 3 , and 4 are given by (6), (7), (8), and (9), respectively. Solving the system, it follows that where is given by (10). Hence the solution x follows by inserting c 1 and c 2 into (11). This implies that x(ς) satisfies (5).
On the contrary, it is easy to show by a straightforward procedure that x(ς), which is illustrated by (5), fulfills the given FBVP (3) in terms of supposed boundary conditions. Lemma 2.8 is proved.

Existence results
. . , n. According to Lemma 2.8, we define Q : Note that the proposed ψ-Hilfer FBVP describing thermostat control model (3) involves solutions if and only if Q possesses fixed points. For brevity, we denote

Uniqueness property
In the forthcoming first theorem, we will prove the uniqueness of solution for the ψ-Hilfer FBVP describing thermostat control model (3) for any u l , v l ∈ R, l = 1, 2, and ς ∈ J. If where (·, ·) is defined in (15), then the ψ-Hilfer FBVP describing the thermostat control model (3) has a unique solution x in E.
Step II. The operator Q : E → E is a contraction. Let x, y ∈ E. For each ς ∈ J, we get From (H 1 ) we obtain Then by substituting (23) into (22) we get (Qx)(ς) -(Qy)(ς) ≤ L 1 1 (T, α) + L 2 1 (T, q + α) xy which illustrates that Qx -Qy ≤ (L 1 , L 2 ) xy . In view of the condition (L 1 , L 2 ) < 1, we get that Q is a contraction. Hence by Lemma 3.1 we get that the solution x ∈ E is unique on J for the supposed ψ-Hilfer FBVP describing the thermostat control model (3). The proof of the theorem is completed.

Existence property
The second result is obtained by invoking the Schaefer fixed-point theorem (Lemma 3.3).
Proof We divide the proof into four steps.
Step I. Q is continuous. Let a sequence {x n } ⊂ E be such that x n → x in E. Then, for every ς ∈ J, we obtain Since the continuity of f implies the continuity of F x , we obtain and therefore Q is continuous.
Step II. Q maps bounded sets to bounded ones contained in E.
For r 2 > 0, there is N > 0 such that, for every x ∈ B r 2 = {x ∈ E : x ≤ r 2 }, we have Qx ≤ N .
Step III. Q maps bounded sets to equicontinuous ones contained in E. For 0 ≤ ς 1 < ς 2 ≤ T and x ∈ B r 2 , since f is bounded on the compact set J × B r 2 , we have By setting sup (ς ,u,v)∈J×B 2 Clearly, the right-hand side of (26) is independent of the unknown variable x and approaches 0 as ς 2 → ς 1 . Then the operator Q is equicontinuous. So, the operator Q admits the relative compactness on B r 2 , and the Arzelá-Ascoli theorem gives the complete continuity of Q.
Step IV. P = {x ∈ E : x = τ Qx, τ ∈ (0, 1]} is bounded. Let x ∈ P. Then x = τ Qx for some τ ∈ (0, 1]. From (H 2 ), for each ς ∈ J, we get the estimate Step II, for any ς ∈ J, we get that Qx ≤ N < ∞. Hence the set P is bounded. Using Theorem 3.4, we see that we can find N > 0 such that x ≤ N < ∞. By Lemma 3.3 we find at least one fixed-point for Q, which is the corresponding solution of the suggested ψ-Hilfer FBVP describing the thermostat control model (3).
We define Q 1 and Q 2 on B r 3 by Note that Q = Q 1 + Q 2 .
For any x, y ∈ B r 3 , we have This implies that Q 1 x + Q 2 x ∈ B r 3 , which satisfies Lemma 3.5(i). Next, we show that Lemma 3.5(ii) is fulfilled.
Due to the continuity of f , this implies the continuity of F x . Hence we obtain Thus Q 1 is continuous. Also, the set Q 1 B r 3 is uniformly bounded since Now we prove the compactness of Q 1 . Setting sup (ς ,u,v)∈J×B 2 Thus the right-hand side of (28) (independently of the unknown variable x) approaches 0 as ς 2 → ς 1 . Therefore Q 1 is equicontinuous. Thus by the Arzelà-Ascoli theorem Q 1 is relatively compact.
Furthermore, it is easy to compute, utilizing (27), that Q 2 is a contraction, and so Lemma 3.5(iii) holds. Therefore Lemma 3.5 is fulfilled, and thus a solution exists on J for the ψ-Hilfer FBVP describing the thermostat control model (3).

Definition 4.1 ([38])
The ψ-Hilfer FBVP describing the thermostat control model (3) is said to be UH stable if there exists M f ∈ R + such that for any > 0 and for every solution there is a solution x ∈ E of the ψ-Hilfer FBVP describing the thermostat control model there is a solution x ∈ E of ψ-Hilfer FBVP describing the thermostat control model (3) such that

Definition 4.3 ([38])
The ψ-Hilfer FBVP describing the thermostat control model (3) is said to be UHR stable by terms of B ∈ C(J, R + ) if there exists M f ,B ∈ R + such that for each > 0 and for every solution z ∈ E of (31), there is a solution x ∈ C of the ψ-Hilfer FBVP describing the thermostat control model (3) such that there is a solution x ∈ E of the ψ-Hilfer FBVP describing the thermostat control model (3) such that Remark 4.6 z ∈ E is a solution of (29) if there exists f u ∈ E (where u depends on z) such that: Remark 4.7 z ∈ E is a solution of (31) if there exists f v ∈ E (depending on z) such that: Remark 4.8 There exist an increasing function B ∈ C(J, R + ) and a constant n B > 0 such that for each ς ∈ J, we have the following integral inequality:

The Ulam-Hyers stability
First, we give the following lemma, which will be utilized in the arguments on UH and GUH stability of problem (3).
Proof Let z satisfy (29). By Remark 4.6(ii) and Lemma 2.8 we obtain Then the solution of (39) can be written in the form Remark 4.6(i) implies that Inequality (37) is achieved.
Next, we prove the UH and GUH stability of solution to problem (3).
Theorem 4.10 Let f : J × R 2 → R be continuous, and let (H 1 ) be satisfied subject to Then the ψ-Hilfer FBVP describing the thermostat control model (3) is UH and GUH stable in E.

The Ulam-Hyers-Rassias stability
This lemma will is helpful in the arguments on UHR and GUHR stability of our results.
Proof Let z satisfy (31). By Remark 4.7(ii) and Lemma 2.8 the solution of the problem can be written in the following form: From Remarks 4.7(i) and 4.8 we obtain the following estimate: from which we get (41).
Next, we check UHR and GUHR stability of solution to problem (3).

Example
In this section, we provide some examples compatible to the exactitude and applicability of our main results.