# On the Fractional Variable Order Thermostat Model: Existence Theory on Cones via Piece-Wise Constant Functions

Authors: Shahram Rezapour; Mohammed Said Souid; Zoubida Bouazza; Azhar Hussain; Sina Etemad
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

In the current manuscript, we intend to investigate the existence, uniqueness, and the stability of positive solution in relation to a fractional version of variable order thermostat model equipped with nonlocal boundary values in the Caputo sense. In fact, we will get help from the constant piece-wise functions for transforming our variable order model into an auxiliary standard model of thermostat. By Guo-Krasnoselskii’s fixed point theorem on cones, we derive the required conditions ensuring the existence property for positive solutions. An example is illustrated to examine the validity of the observed results.


---

## Body

## 1. Introduction

One of the most significant subbranches of mathematics in other existing computational and applied disciplines is arbitrary order calculus, notably fractional order calculus. Because of its applicability in a variety of fields including applied science and engineering, as well as its flexibility to model different systems, processes, and phenomena with memory effects, an arbitrary order calculus theory is regarded as an important subject of research for most researchers, mathematicians, scientists, and engineers. Fractional derivatives are powerful tools for describing the memory and heredity qualities of a wide range of materials and processes. The experts have contributed significantly to the study of fractional differential equations in recent years, for example, see some papers about the modeling of the Sturm-Liouville-Langevin problem [1], studying the Langevin boundary value problem (BVP) [2], investigation of the combination synchronization of a Caputo-Hadamard system [3], modeling of the Caputo-conformable pantograph problem [4], modeling of the hybrid thermostat [5], impulsive fractional systems [6], and modeling COVID-19 cases in the fractional settings [7], (for more studies, we refer to [8–13]).The discussions of BVPs have attracted the focus of many scholars, and in this direction, valuable results have been obtained. Of course, various methods have been utilized to study fractional BVP such as the Banach contraction map principle [14], fixed point theorems [15], monotone iterative method [16], variational method [17], fixed point index theory and coincidence degree theory [18, 19], and numerical methods [20, 21]. On the other hand, numerous researches and review studies have been done in exploring the existence and stability of solutions to constant order fractional problems, while the subject of existence and stability in relation to the variable order problems is discussed seldom in the newly published papers including investigation of the variable order model of alcoholism [22], multiterm variable order BVPs [23], nonsingular variable order problems [24], initial value problems with conformable variable orders [25], approximate solutions on half-axis for a variable order IVP [26], singular variable order BVPs [27], Ulam-Hyers-Rassias stability for an implicit variable order BVP [28], and the generalized variable order Lyapunov-type inequality [29].Infante and Webb [30] modeled a mechanical device, i.e., thermostat in terms of the following nonlocal second order boundary value problem which is insulated at t=0 with a heat-dissipating controller at t=1 based on the temperature that is measured by a mechanical sensor at the time t=c:
(1)−y′′=Ht,y,t∈I,y′0=0,γy′1+yc=0,I≔0,1.Keeping in view the importance and accuracy of fractional operators to study of boundary value problems, Nieto and Pimentel [31] consider the fractional analogue of thermostat control model as
(2)−cD0αyt=Ht,yt,y′0=0,γcD0α−1y1+yc=0,where γ>0,c∈0,1,α∈1,2, cD0θ denotes the derivative in the form of Caputo of order θ∈α,α−1 and H is continuous on 0,1×0,∞. These two mathematicians discussed the existence of positive solution to the proposed problem using fixed point approach.In 2015, Shen et al. [32] turned to the fractional model of thermostat with a parameter
(3)−cD0αyt=aHt,yt,y′0=0,γcD0α−1y1+yc=0,where γ>0,c∈0,1,α∈1,2. They proved the existence along with the nonexistence of positive solutions for different values of parameter a.On the other side, keeping in view the advancements in studies of fractional calculus, it was observed that the constant order fractional calculus is not the ultimate instrument for the modeling of physical problems. In this regard, Lorenze and Hartley [33], in 1998, proposed the variable order fractional operator to model complex dynamics problems and further study in this field were given by them in [34].In consistent with the theory of variable order fractional calculus, we aim to consider a simple type of the variable order Caputo fractional thermostat model in terminal points which possesses a mathematical structure as(4)−cDutyt=ft,yt,t∈J=0,S,y′0=0,cDut−1yM+y0=0,where ut:J⟶1,2 stands for a function as the variable order, f:J×ℝ⟶ℝ is continuous.There are no research manuscripts about the application of cones on variable order systems. This gap motivates us to investigate the generalized version of thermostat model in the context of fractional variable orders with respect to constant piece-wise functions. Furthermore, the presented structure is expressed in a unique and simple case, allowing us to generalize several typical specific cases previously addressed, and also this structure makes the novelty of our work. The ultimate goal of this study is to explore some results in relation to existence theory for the solutions of variable order BVP for the Caputo fractional thermostat model (4). Note that in this model, we use the Caputo fractional derivative. In fact, the importance of the fractional derivatives is their memory property which can describe the dynamical behavior of a system during a time interval. This kind of derivatives is not local and that is why we can analyze the solutions of a system with respect to various values of fractional orders. Also, we emphasize that our model is a generalization of two constant order thermostat models given in [31, 32] to variable order version.The structure of the manuscript is designed as follows: the following section is based on some basic notions related to the current study. In Section3, to get help of Guo-Krasnoselskii fixed point theorem, a suitable partition of the underlying interval J is proposed. This variable order BVP for the Caputo fractional thermostat model (4) is divided into the finite number of BVPs consisting of the Caputo fractional differential equations of constant orders, and then using the concept of Green’s function, the equivalent solution of the proposed problem is obtained and some properties of Green’s function are discussed. Finally, the existence result for (4) is proved based on the cones in Section 4. In the next section (Section 5), an example is prepared to give the illustrations of the main findings. We end our research by conclusions in Section 6.


## 2. Preliminaries

The collectionCJ,ℝ of all functions ϰ:J⟶ℝ having the property of continuity will be a Banach space via ϰ=sup∣ϰt∣:t∈J..Definition 1 (see [35, 36]).
The left Riemann-Liouville fractional integral and left Caputo fractional derivative of variable orderut of H are recalled as
(5)I0+utHt=1Γut∫0tt−wut−1Hwdw,t∈J,(6)cD0+utHt=1Γn−ut∫0tt−wn−ut−1Hnwdw,with ut:J⟶n−1,n.Remark 2 (see [37, 38]).
In Equations (5) and (6), by choosing the constant order u, the Caputo fractional derivative and the Riemann-Liouville fractional integral of variable order will be the same conventional of the Caputo fractional derivative and Riemann-Liouville fractional integral of constant order, respectively.Lemma 3 (see [38]).
Supposeα1,b1>0 and n=α1+1, then
(7)Ib1+α1cDb1+α1ψt=ht+∑k=0n−1ψk0k!tk.Lemma 4 (see [39]).
Letα1,α2>0, ψ∈L1b1,b2, and cDb1+α1ψ∈L1b1,b2. Then, the differential equation
(8)cDb1+α1ψ=0has unique solution
(9)ψt=ϖ0+ϖ1t−b1+ϖ2t−b12+⋯+ϖn−1t−b1n−1and
(10)Ib1+α1cDb1+α1ψt=ψt+ϖ0+ϖ1t−b1+ϖ2t−b12+⋯+ϖn−1t−b1n−1under the conditions n−1<α1≤n, ϖj∈ℝ.
Moreover,(11)cDb1+nψt=ψnt,n∈ℕ,cDb1+α1Ib1+α1ψt=ψt,Ib1+α1Ib1+α2ψt=Ib1+α2Ib1+α1ψt=Ib1+α1+α2ψt.Remark 5 (see [26, 27]).
It is observed that the property of semigroup is not consistent for arbitrary functionsut, vt in the role of variable orders, i.e.,
(12)Ib1+utIb1+vtψt≠Ib1+ut+vtψt.Example 1.
Assumeyt≡1 for t∈J=0,3 and ut=t/2 and vt=1,t∈0,12,t∈1,3. Then,
(13)I0+utI0+vtyt=∫0tt−sut−1Γut∫0ss−τvs−1Γvsyτdτds=∫0tt−sut−1Γut∫01s−τ0Γ1dτ+∫1ss−τΓ2dτds=∫0tt−sut−1Γuts22−s+32ds,I0+ut+vtyt=1Γut+vt∫0tt−sut+vt−1ysds.
Fort=2, we write
(14)I0+utI0+vtytt=2=∫022−s0Γ1s22−s+32ds=∫02s22−s+32ds=73,and accordingly,
(15)I0+ut+vtytt=2=∫022−sut+vt−1Γut+vtysds=∫012−s1Γ2ds+∫122−s2Γ3ds=32+16=53.
As a result, the semigroup property is not valid in the generalized case for the Riemann-Liouville fractional integral of variable order.The next results will be used in the main results concerning the Riemann-Liouville fractional integral given in (5).Lemma 6 (see [26]).
Letu∈CJ,1,2, then for
(16)h∈CρJ,ℝ=ht∈CJ,ℝ,tρht∈CJ,ℝ,0≤ρ≤1,we have
(17)∃I0+utht,∀t∈J;I0+utht∈CJ,ℝ.Definition 7 (see [26, 27, 40]).
Consider a subsetI of ℝ:
(a)
A generalized interval is either empty, singleton subset, or an interval(b)
The finite setP of generalized interval is a partition of I whenever every s∈I belongs to exactly one of the generalized intervals E in P(c)
h:I⟶ℝ is piece-wise constant with respect to P of I if ∀E∈P, h is constant on ETheorem 8 (Guo-Krasnoselskii fixed point theorem [41]).
Assume a coneP and bounded subsets B1,B2 of a Banach space E with
(18)0∈B1⊂B1¯⊂B2,and assume a completely continuous operator T:P∩B2¯\B1⟶P such that
(i)
∥Ty∥≥∥y∥,y∈P∩∂B1 and ∥Ty∥≤∥y∥,y∈P∩∂B2; or(ii)
∥Ty∥≤∥y∥,y∈P∩∂B1 and ∥Ty∥≥∥y∥,y∈P∩∂B2.
Then, operatorT possesses a fixed point in P∩B2¯\B1.


## 3. Auxiliary BVP and Green’s Function

Before moving to the main results, we first assume the following assertions:Letn∈ℕ, and the finite sequence Sjj=0n of points in a way that 0=S0<Sj<Sn=S,j=1,⋯,n−1. Designate Jj≔Sj−1,Sj,j=1,2,⋯,n. Then, the partition of interval J is P=Jj:j=1,2,⋯,n.Suppose a constant piece-wise maput:J⟶1,2 with respect to P, i.e., ut=∑j=1nujIjt, where uj∈1,2, and Ij stands for the interval Jj,j=1,2,⋯,n:
(19)Ijt=1,fort∈Jj,0,elsewhere.For allj∈ℕ1n, Ej=CJj,ℝ, denotes Banach spaces via ∥y∥Ej=supt∈Jj∣yt∣..To achieve our main findings, let us first reduce the variable order FBVP (4) to a standard constant order system. With the aid of (6), the variable order BVP for the Caputo fractional thermostat model (4) takes the form
(20)−∑k=1j−1∫Sk−1Skt−s1−ukΓ2−uky′′sds−∫Sj−1tt−s1−ujΓ2−ujy′′sds=ft,yt.Ify~∈CJj,ℝ is y~≡0 on 0,Sj−1 and is a solution of (20), then (20) becomes
(21)−cDSj−1+ujy~t=ft,y~t,t∈Jj.Taking the above into consideration∀j∈ℕ1n and with piece-wise functions uj, we take into account the next auxiliary BVP of constant order, the Caputo fractional thermostat model is formulated as
(22)−cDSj−1+ujyt=ft,yt,t∈Jjy′Sj−1=0,cDuj−1ySj+ySj−1=0.In the following, relevant Green’s function is obtained in relation to auxiliary BVP of constant order of the Caputo fractional thermostat model (22).Lemma 9.
Let1<uj≤2,∀j∈ℕ1n. Then, y∈Ej is a solution of the auxiliary BVP of constant order of the Caputo fractional thermostat model
(23)−cDSj−1+ujyt=ft,t∈Jjy′Sj−1=0,cDuj−1ySj+ySj−1=0.if it fulfills
(24)yt=∫Sj−1SjGjt,sfsds,where Green’s function Gjt,s is stated as
(25)Gjt,s=1−t−suj−1Γuj,s≤t1,s>t.Proof.
Using Lemma3, there are c0,c1∈ℝ such that
(26)yt=−Iujft+c0+c1t=−∫Sj−1tt−suj−1Γujfsds+c0+c1t.
In view of Lemma4,
(27)y′t=−∫Sj−1tt−suj−2Γuj−1fsds+c1.
Byy′0=0, we get that c1=0.. Moreover,
(28)CDuj−1yt=−I1yt.
Using the boundary conditionCDα−1ySj+ySj−1=0, we get
(29)c0=∫Sj−1Sjfsds.
At the end, (26) takes the form
(30)yt=∫Sj−1Sjfsds−∫Sj−1tt−suj−1Γujfsds=∫Sj−1SjGjt,sfsds.
The proof is completed.Remark10.
Notice that for each fixeds∈Sj−1,Sj,∂Gj/∂t=0 for t≤s and ∂Gj/∂t<0 for t>s; this turns Gjt,s a decreasing function. Therefore
(31)maxt∈Sj−1,SjGjt,s=GjSj−1,s=1,s>Sj−1,1+Sj−1−suj−1Γuj,s≤Sj−1,mint∈Sj−1,SjGjt,s=GjSj,s=1−Sj−suj−1Γujs>Sj−1,Sj−1−suj−1−Sj−suj−1Γuj+1,s≤Sj−1.
As a result, due to behavior ofGjt,s with respect to s, we have
(32)mint,s∈Sj−1,SjGjt,s=1−Sj−Sj−1uj−1Γuj,maxt,s∈Sj−1,SjGjt,s=1+Sj−1uj−1Γuj.
We shall prove thatGjt,s meets the following property to ensure the existence of positive solution of the variable order Caputo fractional thermostat model (4) proposed by Lan and Webb in [42]:
(H2):∃φ:Sj−1,Sj⟶0,∞ as a measurable mapping, c,d⊆Sj−1,Sj and ξ∈0,1 such that
(33)∣Gjt,s∣≤φs,∀t,s∈Sj−1,Sj,Gjt,s≥ξφs,∀t∈c,d,∀s∈Sj−1,Sj.Lemma 11.
IfΓuj>Sj−Sj−1uj−1, then Gjt,s>0 for all t,s∈Sj−1,Sj, and Gjt,s satisfies (H2).Proof.
Takingc,d=Sj−1,Sj,, then
(34)∣Gjt,s∣=Gjt,s≤1+Sj−1uj−1Γuj≔φs,Gjt,s≥ξφs∀s,t∈Sj−1,Sj,where
(35)ξ=Γuj−Sj−Sj−1uj−1Γuj+Sj−1uj−1.
The proof is completed.Lemma 12.
IfΓuj=Sj−Sj−1uj−1, then Gjt,s≥0 for all t,s∈Sj−1,Sj, and Gjt,s satisfies (H2).Proof.
Takingc,d=Sj−1,d such that Sj−1≤d<Sj, and using the preceding lemma arguments, we obtain
(36)∣Gjt,s∣≤1+Sj−1uj−1Γuj≔φs,∀t,s∈Sj−1,Sj.
Setting(37)ξ=Γuj−d−Sj−1uj−1Γuj+Sj−1uj−1,we get
(38)Gt,s≥ξφs,∀t∈Sj−1,d,∀s∈Sj−1,Sj.
The proof is completed.Lemma 13.
IfΓuj<Sj−Sj−1uj−1, then Gt,s changes sign on Sj−1,Sj×Sj−1,Sj, and Gjt,s satisfies (H2).Proof.
Letc,d=Sj−1,d be such that Sj−1≤d<Sj and Γuj>d−Sj−1uj−1. We have
(39)∣Gjt,s∣≤max1+Sj−1uj−1Γuj,1−Sj−1uj−1Γuj−1≔φs,∀t,s∈Sj−1,Sj,Gt,s≥ξφs,∀t∈Sj−1,d,∀s∈Sj−1,Sj,where
(40)ξ=minΓuj−d−Sj−1uj−1Γuj+Sj−1uj−1,Γuj−−d−Sj−1uj−1Sj−Sj−1uj−1−Γuj.
The proof is completed.


## 4. Existence and Uniqueness

We now derive the existence result with the aid of Guo-Krasnoselskii’s result [41]. Before starting, set
(41)f0=limy⟶0+mint∈Sj−1,Sjft,yy,f0⋆=limy⟶0+maxt∈Sj−1,Sjft,yy,f∞=limy⟶∞mint∈Sj−1,Sjft,yy,f∞⋆=limy⟶∞maxt∈Sj−1,Sjft,yy.Theorem 14.
Letfs,ys∈CSj−1,Sj×0,∞. Assuming one of the below mentioned cases:
(1)
f0=∞ and f∞=0 (Sublinear case)(2)
f0⋆=0 and f∞⋆=∞ (Superlinear case),if Γuj>Sj−Sj−1uj−1, then there is at least positive solution of the auxiliary BVP of constant order Caputo fractional thermostat model (22).Proof.
DefineT:CSj−1,Sj⟶CSj−1,Sj as
(42)Tyt=∫Sj−1SjGjt,sfs,ysds.
We now define the cone(43)P=y\y∈CSj−1,Sj,yt≥0,mint∈Sj−1,Sjyt≥ξ∥y∥,where ξ is as in (35).
In the start, we prove thatTP⊂P. Since the functions f and Gj are positive and continuous, it follows that if y∈P, then Ty∈CSj−1,Sj and Tyt≥0 for all t∈Sj−1,Sj. We reach to the inequality derived below for a fixed y∈P for all t∈Sj−1,Sj and that Gjt,s satisfies (H2)
(44)Tyt=∫Sj−1SjGjt,sfs,ysds≥ξ∫Sj−1Sjφsfs,ysds≥ξ∫Sj−1Sjmaxt∈Sj−1,SjGjt,sfs,ysds≥ξmaxt∈Sj−1,Sj∫Sj−1SjGjt,sfs,ysds=ξ∥Ty∥,which leads to the first claim. In the next step, we prove the complete continuity of operator T:P⟶P. Because of the continuous behavior of Gj and f, continuity of T:P⟶P is followed immediately. Let B⊂P be bounded. Define
(45)L=maxSj−1≤t≤Sj,0≤y≤M∣ft,y∣+1.
So for ally∈B, we have
(46)∣Tyt∣≤∫Sj−1SjGjt,sfs,ysds≤L∫Sj−1SjGjt,sds,for all t∈Sj−1,Sj. This defines the boundedness of TB. Now, for each y∈B and t1,t2∈Sj−1,Sj such that t1<t2, we write
(47)∣Tyt2−Tyt1∣=∣−∫Sj−1t2t2−suj−1Γujfs,ysds+∫Sj−1t1t1−suj−1Γujfs,ysds∣≤1Γuj∫Sj−1t1t2−suj−1−t1−suj−1∣fs,ys∣ds+1Γuj∫t1t2t2−suj−1∣fs,ys∣ds≤LΓuj∫Sj−1t1t2−suj−1−t1−suj−1ds+∫t1t2t2−suj−1ds=LujΓuj−t2−t1uj+t2uj−t1uj+t2−t1uj=LΓuj+1t2uj−t1uj.
Tendingt1⟶t2 implies that the RHS of the above inequality goes to 0 and thus TB is equicontinuous. At the end, the Arzela-Ascoli theorem confirms complete continuity of operator T:P⟶P.
Next, assume that (2) is true. As f0=∞, a ρ1>0 exists such that ft,y≥δ1y,∀0<y≤ρ1, where δ1 satisfies
(48)δ11−Sj−Sj−1uj−1Γuj≥1.
Takingy∈P with ∥y∥=ρ1, then
(49)Ty=∫Sj−1SjGjt,sfs,ysds≥δ1∫Sj−1SjGjt,sysds≥δ1∥y∥1−Sj−Sj−1uj−1Γuj≥∥y∥.
LetB1=y∈CSj−1,Sj\∥y∥<ρ1. Hence, we have ∥Ty∥≥∥y∥,y∈P∩∂B1.
Due the continuity offt,. on 0,∞, a function f~t,y=maxz∈0,yft,z can be defined which is nondecreasing on 0,∞ by assumption and
(50)limy⟶∞maxt∈Tj−1,Tjf~t,yy=0.
Therefore, there existsρ2>ρ1>0 such that f~t,y≤δ2y,∀y≥ρ2, where δ2 satisfies
(51)δ21+Sj−1uj−1Γuj≤1.
DefineΩ2=y∈CSj−1,Sj\∥y∥<ρ2 and let y∈P such that ∥y∥=ρ2. Then,
(52)Ty=∫Sj−1SjGjt,sfs,ysds≤∫Sj−1SjGjt,sf~s,∥y∥ds≤δ2∥y∥1+Sj−1uj−1Γuj≤∥y∥.
Hence, we have∥Ty∥≤∥y∥,y∈P∩∂B2. In conclusion, there is at least one positive solution by (i) of Theorem 8 for the auxiliary BVP of constant order Caputo fractional thermostat model (22).
Now, assume that (3) is true. For δ2>0 and by assumption, there is r1>0 with ft,y≤δ2y for 0≤y≤r1. Let y∈P such that ∥y∥=r1. Then,
(53)Ty=∫Sj−1SjGjt,sfs,ysds≤δ2∫Sj−1SjGjt,sysds≤δ2∥y∥1+Sj−1uj−1Γuj≤∥y∥.
If we letB1=y∈CSj−1,Sj\∥y∥<r1, then ∥Ty∥≤∥y∥ for y∈P∩∂B1. Again by assumption, there is r>0 such that ft,y≥δ1u,∀y≥r. Define B2=y∈CSj−1,Sj\∥y∥<r2, where r2=max2r1,r/ξ. Then, y∈P and ∥y∥=r2 imply that
(54)minyt≥ξ∥y∥=ξr2≥r,and so we obtain
(55)Ty=∫Sj−1SjGjt,sfs,ysds≥δ1∫Sj−1SjGjt,sysds≥δ1∥y∥1−Sj−Sj−1uj−1Γuj≥∥y∥.
This shows that∥Ty∥≥∥y∥ for y∈P∩∂B2. In conclusion, it is found at least one positive solution, by (ii) of Theorem 8, for the auxiliary BVP of constant order Caputo fractional thermostat model (22) as yj~∈P∩B2¯\B1.Now, we show the uniqueness of solutions for the auxiliary BVP of constant order Caputo fractional thermostat model (22) based on the Banach contraction principle. We consider the following assumption:Letf∈CJ×ℝ,ℝ and there exists a number δ∈0,1 such that tδf∈CJ×ℝ,ℝ and there exists a constant K>0 such that
(56)tδft,y1−ft,y2≤Ky1−y2,foranyy1,y2∈ℝandt∈J.Theorem 15.
Let the conditions (H1) and (H3) be satisfied and the inequality(57)KSj1−δ−Sj−11−δ1−δ1+Sj−1uj−1Γuj<1holds. If Γuj≥Sj−Sj−1uj−1, then the thermostat BVP (22) has a unique positive solution in Ej.Proof.
We shall use the Banach contraction principle to prove thatT has unique fixed point. For xt,yt∈Ej, by Lemma 11 and Lemma 12, we obtain
(58)∣Txt−Tyt∣=∣∫Sj−1SjGjt,sfs,xsds−∫Sj−1SjGjt,sfs,ysds∣≤∫Sj−1SjGjt,s∣fs,xs−fs,ys∣ds≤1+Sj−1uj−1Γuj∫Sj−1Sjs−δK∣xs−ys∣ds≤K1+Sj−1uj−1Γuj∥x−y∥Ej∫Sj−1Sjs−δds≤KSj1−δ−Sj−11−δ1−δ1+Sj−1uj−1Γuj∥x−y∥Ej.
Consequently by (57), the operator T is a contraction. Hence, by Banach’s contraction principal, T has a unique fixed point yj~∈Ej, which is a unique positive solution of the auxiliary BVP of the constant order Caputo fractional thermostat model (22).In the next result, we generalize the existence criteria for the variable order BVP of the main Caputo fractional thermostat model (4).Theorem 16.
Assuming (H1)-(H2), the variable order BVP of the main Caputo fractional thermostat model (4) admits at least one solution in CJ,ℝ.Proof.
We know that the solutionyj~∈Ej fulfills the auxiliary BVP of constant order Caputo fractional thermostat model (22) by Theorem 14 for any j∈1,2,⋯,n. Now, the continuous function on 0,Sj is defined as
(59)yj=0,t∈0,Sj−1,y~j,t∈Jj,is a solution of (20) for t∈Jj, j∈1,2,⋯,n.
Therefore,yt=yjt,t∈Jj,j∈1,2,⋯,n solves the variable order BVP of the main Caputo fractional thermostat model (4). The proof is completed.


## 5. Ulam-Hyers Stability

In this section, we are going to investigate the Ulam-Hyers stability for solutions of the given variable model of thermostat.Definition 17 (see [43]).
The variable order Caputo fractional thermostat model (4) is Ulam-Hyers stable if there exists cf>0 such that for each ε>0 and for every solution z∈CJ,ℝ of the following inequality
(60)∣−cD0+utzt−ft,zt∣≤ε,t∈J,there exists a solution y∈CJ,ℝ of (4) with
(61)∣zt−yt∣≤cfε,t∈J.Theorem 18.
Assume that the conditions (H1) and (H2) to be held. Then, the Caputo fractional thermostat model (4) is Ulam-Hyers stable.Proof.
Letε>0 be an arbitrary number and the function zt belonging to CJ,ℝ satisfies the following inequality
(62)∣−cD0+utzt−ft,zt∣≤ε,t∈J.
For anyj∈1,2,⋯,n, we define the functions z1t≡zt,t∈0,S1 and for j=2,3,⋯,n:(63)zjt=0,t∈0,Sj−1,zt,t∈Jj.
For anyj∈1,2,⋯,n and according to the equality (6), for t∈Jj, we get
(64)cD0+utzjt=∫Sj−1tt−s1−ujΓ2−ujz2sds.
TakingISj−1+uj on both sides of the inequality (62), we obtain
(65)∣−zjt−∫Sj−1SjGjt,sfs,zjsds∣≤ε∫Sj−1tt−suj−1Γujds≤εSj−Sj−1ujΓuj+1.
According to Theorem16, the Caputo fractional thermostat model (4) has a positive solution y∈CJ,ℝ defined by yt=yjt for t∈Jj,j=1,2,⋯,n, where
(66)yj=0,t∈0,Sj−1,y~j,t∈Jj,
andy~j∈Ej is a positive solution of (22). According to Lemma 9, the integral equation
(67)y~jt=∫Sj−1SjGjt,sfs,y~jsds,holds. Let t∈Jj,j=1,2,⋯,n. Then, by Eq (66) and (67) we get
(68)∣zt−yt∣=∣zt−yjt∣=∣zjt−y~jt∣=∣zjt−∫Sj−1SjGjt,sfs,y~jsds∣≤∣zjt−∫Sj−1SjGjt,sfs,zjsds∣+∣∫Sj−1SjGjt,sfs,zjsds−∫Sj−1tGjt,sfs,y~jsds∣≤∣−zjt−∫Sj−1SjGjt,sfs,zjsds∣+∣∫Sj−1SjGjt,sfs,zjsds−∫Sj−1tGjt,sfs,y~jsds∣≤εSj−Sj−1ujΓuj+1+1+Sj−1uj−1Γuj∫Sj−1Sj∣fs,zjsds−fs,y~is∣ds≤εSj−Sj−1ujΓuj+1+1+Sj−1uj−1Γuj∫Sj−1Sjs−δK∣zjs−y~js∣ds≤εSj−Sj−1ujΓuj+1+1+Sj−1uj−1ΓujK∥zi−y~j∥Ej∫Sj−1Sjs−δds≤εSj−Sj−1ujΓuj+1+KSj1−δ−Sj−11−δ1−δ1+Sj−1uj−1Γuj∥zj−y~j∥Ej≤εSj−Sj−1ujΓuj+1+μ∥z−y∥,where
(69)μ=maxj=1,2,⋯,nKSj1−δ−Sj−11−δ1−δ1+Sj−1uj−1Γuj.
Then,(70)∥z−y∥1−μ≤Sj−Sj−1ujΓuj+1ε.
We obtain, for eacht∈Jj(71)∣zt−yt∣≤∥z−y∥≤Sj−Sj−1uj1−μΓuj+1ε≔cfε.
Therefore, by Definition17, the Caputo fractional thermostat model (4) is Ulam-Hyers stable and the proof is completed.


## 6. Example

We here simulate the simple form of our variable order BVP of the main Caputo fractional thermostat model (4) by giving an example numerically.Example 2.
Consider the nonlinear function(72)ft,y=et+1y,on t,y∈0,2×0,+∞ and
(73)ut=1.4,t∈J1≔0,1,1.5,t∈J2≔1,2.
Then, in consistent with (22) and corresponding to the variable order BVP of fractional differential equation
(74)−cDutyt=et+1yt,t∈J≔0,2,y′0=0,cDut−1y2+y0=0,the constant order auxiliary BVPs are
(75)−cD0+1.4yt=et+1yt,t∈J1,y′0=0,cD0+0.4y1+y0=0,(76)cD1+1.5yt=et+1yt,t∈J2,y′1=0,cD1+0.5y2+y1=0.
Clearly,f0=∞ and f∞=0.
Forj=1, we get Γ1.4−1≈−0.11274<0, we take
(77)ξ1=minΓuj−d−Sj−1uj−1Γuj+Sj−1uj−1,Γuj−d−Sj−1uj−1Sj−Sj−1uj−1−Γuj=Γ1.4−duj−1Γ1.4
and define the coneP1=y\y∈C0,1,mint∈0,1yt≥ξ1∥y∥. By Theorem 14, it is deduced that that the auxiliary BVP of constant order Caputo fractional thermostat model (75) possesses a positive solution y~1∈P1.
Forj=2, we get Γ1.5−1≈−0.11377<0, we take
(78)ξ2=minΓuj−d−Sj−1uj−1Γuj+Sj−1uj−1,Γuj−d−Sj−1uj−1Sj−Sj−1uj−1−Γuj=Γ1.5−d−11.5Γ1.5+1
and consider the coneP2=y\y∈C1,2,mint∈1,2yt≥ξ2∥y∥.
According to Theorem14, the auxiliary BVP of constant order Caputo fractional thermostat model (76) admits a positive solution y~2∈P2, and by Theorem 16, the BVP for variable order Caputo fractional thermostat model (74) has a solution
(79)yt=y~1t,t∈J1,y2t,t∈J2,where
(80)y2t=0,t∈J1,y~2t,t∈J2.


## 7. Conclusion

In science and technology, fractional differential equations are utilized to model and describe a variety of natural processes. In connection with standard fractional models, variable fractional models appear to be more important for complex natural phenomena. The focus of this study was to analyze the BVP of variable order Caputo fractional thermostat model (4) and to explore its solutions’ existence utilizing techniques from fixed point theory on cones. To do such a method, we first noticed the invalidity of semigroup property for the Riemann-Liouville fractional integral of variable order. In order to obtain the solution of problem and solve this issue, we considered a partition of the interval J and the corresponding auxiliary constant order BVP of the Caputo fractional thermostat model (22) was derived from the variable order one. Some properties of relevant Green’s function were reviewed. For the solutions’ existence, utilizing fixed point attributed to Guo-Krasnoselskii’s on cones, the main theorems (Theorem 14 and 16) were deduced. An example to confirm the validity of theoretical findings was provided. This technique can be used to consider various physical models with variable order fractional operators. Moreover, with the help of our results in this research paper, investigations on this open research problem can be also possible and one can extend the proposed BVP to other complicated fractional models. In the future, we want to study these boundary value problems with different conditions involving integral conditions or integroderivative conditions or nonlocal conditions along with infinite delay.



---

*Source: 8053620-2022-03-24.xml*