Investigation of the Fractional Strongly Singular Thermostat Model via Fixed Point Techniques

: Our main purpose in this paper is to prove the existence of solutions for the fractional strongly singular thermostat model under some generalized boundary conditions. In this way, we use some recent nonlinear ﬁxed-point techniques involving α - ψ -contractions and α -admissible maps. Further, we establish the similar results for the hybrid version of the given fractional strongly singular thermostat control model. Some examples are studied to illustrate the consistency of our results.


Introduction
Fractional calculus is one of the most important branches of mathematics that derives and studies many different properties of integration and derivation operators of non-integer orders via singular and nonsingular kernels. These operators are called fractional integrals and derivatives [1][2][3][4]. Because of the importance, potential, high accuracy and flexibility of the mentioned fractional operators, the attention of engineers and applied researchers has been drawn in this direction. One can find several published works regarding applications of this field in mathematical models. For instance, Baleanu et al. [5] designed a novel model of FBVP on glucose graph, or Mohammadi et al. [6] studied a fractional mathematical model of Mumps virus in the context of the Caputo-Fabrizio operators. In [7], Boutiara et al. used the Caputo-Hadamard fractional operators to study the solutions of a three-point BVP-see .
The importance and existing complexity in studying fractional structures having singular points motivate us to investigate the existence and uniqueness of solutions for some real mathematical models in engineering in which solutions possess singular points; therefore, by using main ideas of above mathematical models, in this manuscript, we find a theoretical method to investigate the existence of solutions for the strongly singular fractional model of thermostat control given as with initial conditions x (j) (0) = 0 for j ∈ {0, 1, . . . , n − 1} with j = k coupled with the boundary condition and c D ω displays the Caputo derivative for given order ω. Note that the above boundary problem caused by the singular fractional model of thermostat control is new and it has not been studied in any other paper so far and this guarantees the novelty of the present paper. Further, the used technique to confirm the existence of solutions for such a new singular system is based on the special subclass of operators called α-ψ-contractions and α-admissible maps.
The construction of the paper is organized as: Section 2 is devoted to recalling some basic notions. Section 3 is devoted to deriving a corresponding integral equation for the given singular model of thermostat control (1) and proving the existence of solution by making use of α-ψ-contractions. In Section 4, the hybrid version of the aforementioned strongly singular model of thermostat control is proved by means of the same technique.
Two illustrative examples for both cases are simulated in Section 5 to confirm the correctness of the findings. At last, the conclusion remarks are stated in Section 6.

Basic Notions
Before recalling some basic notions, note that in this article, we apply . 1 for the norm of L 1 [0, 1] and . as the sup-norm for the space X = C([0, 1], R).
In 2012, Samet et al. [37] turned to introduction of a new subclass of special functions, which will be applied in our existence method here.
We introduce via Ψ, the subclass of non-decreasing mappings such as ψ : [37]. In the sequel, regard X as a complete metric space.

Main Results
Here, the existence of solution for the aforesaid strongly singular fractional model of thermostat control (1) is discussed. At first, we provide a key lemma.
Then v as a solution of the linear differential equation c D ω (x(t)) + f(t) = 0 via given BCs is given as where whenever 0 ≤ t ≤ s ≤ 1 and s ≥ η and also ∆ = p (1) + kp(1) + aη k .

Proof.
Let v be the solution of the given linear BVP and satisfies (2). By using Proposition 1, some real constants c 0 , . . . , c n−1 exist provided that Since Hence and for k = 0, we have This implies that On the other hand, in which G(t, s) is given by (3) and the argument is completed.

Remark 1.
In the special case k = 0, Green function is reduced to: Hence, One can simply see that G is continuous by terms of t. Moreover, for k ≥ 1 we have We assume that X = C[0, 1] is furnished with x = sup{|x(t)| : t ∈ [0, 1]}, which will be a Banach space and H : X → X is given by for all t ∈ [0, 1]. In this case, x 0 ∈ X is a solution for the singular fractional model of thermostat control (1) iff x 0 is a fixed point of H.

then
x ≤r and y ≤r.
We verify in this case, that α(H n x, H n y) ≥ 1. To do this, for each t ∈ [0, 1] and n ≥ n 0 , we may write So H n x ≤r. By the same reason H n y ≤r, consequently α(H n x, H n y) ≥ 1. Further, if x ∈ C, then H n x ∈ C and since C = φ, thus x 0 ∈ C exists so that α(x 0 , H n x 0 ) ≥ 1. Now let α(x, y) = 1. Then x ≤r and y ≤r, x − y ≤ x + y ≤ 2r ≤ δ 0 ( 0 ) and so by using (7), we obtain Therefore, in this case, we have and so H n x − H n y ≤ λQ( x − y ), Since Q is non-decreasing and λ ∈ (0, 1), so accordingly ψ is also nondecreasing and , y)).
If α(x, y) = 0, the last inequality is valid obviously and so H n is α-ψ-contraction. Now by making use of Theorem 1, H n admits a fixed-point in X for all n ≥ n 0 . Now, choose {x n } n≥n 0 so that x n (t) = H n x n (t) and so G(t, s)g(s)f(x n (s))ds.
G is continuous by terms of t on [0, 1]. Hence, for all t ∈ [0, 1]. Thus, {x n } n≥n 0 is equi-continuous and {x n } n≥n 0 admits the relative compactness on X. The Arzela-Ascoli theorem implies the existence of x 0 ∈ X so that lim n→∞ x n = x 0 . For t ∈ [0, 1], put where χ E (s) = 1 when s ∈ E and χ E (s) = 0 when s / ∈ E. Since x n → x 0 , so ∃ n 1 ∈ N such that x n − x 0 < , ∀ n ≥ n 1 . Let n ≥ n 1 ≥ n 0 . By using (7), we have |u n (t, s)| = |χ Hence u n (t, .) ∈ L 1 [0, 1] for any n ≥ n 1 . The Lebesgue dominated theorem yields On the other side, x n → x 0 gives the existence of some n 2 ∈ N such that x n − x 0 < min{δ, δ } for all n ≥ n 2 . Hence, we have for all n ≥ n 2 . Thus, f(x n ) → f(x 0 ) as x → x 0 and so H admits a fixed-point x 0 , which will be a solution for the fractional strongly singular thermostat control BVP (1) and this ends the proof.

Hybrid Version
To follow our study on the strongly singular models, we here consider the hybrid version of the fractional strongly singular thermostat control problem having the form with BCs lim and lim where η ∈ (0, 1), is differentiable in t = 1 and c D ω displays the Caputo derivative with given order ω.
Consider g as a singular function which may admit the strong singularity in the set and 1 g (s) = 0, for all s ∈ [0, 1]\E and 1 g > 0. As an example for such a function g : [0, 1] → R, one can define g(t) = 1 t 2 . Then g involves the strong singularity in t = 0 and 1 g (t) = 0 for all t = 0 and 1 g = 1.
By applying a similar proof given in the Lemma 1, one can immediately conclude that x is a solution for the fractional hybrid strongly singular thermostat control problem (9)- (11) if and only if where G(t, s) is given by (3). Before proceeding to prove the main theorem, we define a new space Y g by It is obvious that Y g = ∅. If y ∈ Y g , then Now, regard the space Y g with the norm · * , where x * = 1 g a x for x ∈ Y g . Lemma 2. The space Y g is Banach with the norm · * defined above.
Proof. Let {y n } be a Cauchy sequence contained in Y g . Then, for every > 0, select some n * ∈ N so that ∀ n, m ≥ n * , we have y n − y m * < . Now, by definition of the space Y g , for j = n, m, take {a y j } in C[0, 1] such that y n (t) = a y n (t) g(t) , and y m (t) = a y m (t) for all t ∈ [0, 1] and 1 g a y n − a y m < . Thus â y n −â y m < 1 1 g for all n, m ≥ n * and so {a y n } is a Cauchy sequence contained in X = C[0, 1]. We select , then y n (t) = a n (t) This means that Y g is a Banach space.
To prove the next theorem, we define H * : Y g → Y g by where G(t, s) is given by (3). Note that in fact, we have H * x(t) = Hx(t) g(t) . One can check, by (12), that x 0 is a fixed-point of H * iff x 0 is a solution for the fractional hybrid strongly singular thermostat control problem (9)-(11).
Ultimately, by Theorem 3, the fractional hybrid strongly singular thermostat control problem (17) and (18) involves a solution.

Conclusions
This work is devoted to studying the existence of solutions for two different strongly singular versions of the thermostat control problem for the first time. In this way, we provided new techniques involving α-ψ-contractive operators, which are considered as the main novelty of the present study. For the hybrid version, we built a Banach space based on a function having strong singularity and proved the relevant results for the mentioned hybrid model of thermostat control. Ultimately, we proposed two illustrated examples for obtained results. This research work clarifies that we are able to investigate some qualitative aspects of more complicated strongly singular FBVPs describing realworld models and this encourages us to study other singular dynamical systems arising in different phenomena in nature and engineering. For future works, we can use these techniques for singular Langevin equations or singular pantograph systems modeled by different fractional operators having singular or non-singular kernels. Data Availability Statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.