Global Existence of Solutions for a Fractional Caputo Nonlocal Thermistor Problem

We begin by proving a local existence result for a fractional Caputo nonlocal thermistor problem. Then, additional existence and continuation theorems are obtained, ensuring global existence of solutions.


Introduction
Fractional calculus is acknowledged as an important research tool that opens up many horizons in the field of dynamical systems [1].According to Professor Katsuyuki Nishimoto, "the fractional calculus is the calculus of the XXIst century" [25].This opinion is strengthen by a huge increase of interest in this research tool, expressed by an increase in the number of theoretical developments and basic theory on this subject; see, e.g., [3,12,24,27,28].Recently, it has also been proved that fractional differential equations are significant and essential tools when applied in the study of nonlocal or time-dependent processes and in the modeling of many applications, including chaotic dynamics, material sciences, mechanic of fractal and complex media, quantum mechanics, physical kinetics, chemistry, biology, economics and control theory [21].For instance, a fractional generalization of the Newtonian equation to describe the dynamics of complex phenomena, in both science and engineering, has been proposed in [5]; a fractional Langevin equation, with applications in polymer layers, has been investigated in [6].One can say that real-world problems require definitions of fractional derivatives for initial and boundary value problems [4,26].Fractional mathematical models describing natural phenomena, like shallow water waves and ion acoustic waves in plasma and vibration of large membranes, as well as personal and interpersonal realities, like smoking, romantic relationships and marriages, can be found in [14,36] and [34,35], respectively.Details on the geometric and physical interpretation of fractional differentiation can be found in [27].
Thermistor is a thermo-electric device constructed from a ceramic material whose electrical conductivity depends strongly on the temperature.This makes thermistor problems highly nonlinear [29].They can be used as a switching device in many electronic circuits.A broad application spectrum of thermistor problems in heating processes and current flow can be found in several areas of electronics and its related industries [30].Generally, there are two kinds of thermistors: the first have an electrical conductivity that decreases with the increasing of temperature; the second have an electrical conductivity that increases with the increasing of temperature [15,22].
Here we consider a prototype of electrical conductivity that depends strongly in both time and temperature.Our goal consists to prove global existence of solutions for a fractional Caputo nonlocal thermistor problem.The results are obtained via Schauder's fixed point theorem.Precisely, we consider the following fractional order initial value problem: where C D 2α 0,t is the fractional Caputo derivative operator of order 2α with 0 < α < 1 2 a real parameter.Function u denotes the temperature and λ is a positive real.We shall assume the following hypotheses: In the literature, questions evolving existence and uniqueness of solution for fractional differential equations (FDEs) have been intensely studied by many mathematicians [11,12,16,24].However, much of published papers have been concerned with existence-uniqueness of solutions for FDEs on a finite interval.Since continuation theorems for FDEs are not well developed, results about global existence-uniqueness of solution of FDEs on the half axis [0, +∞), by using directly the results from local existence, have only recently flourished [2,19].
In contrast with our previous works [31,32,33] on fractional nonlocal thermistor problems, which were focused on local existence and numerical methods, here we are concerned with continuation theorems and global existence for the steady state fractional Caputo nonlocal thermistor problem.The paper is organized as follows.In Section 2, we collect some background material and necessary results from fractional calculus.Then, we are concerned in Section 3 with local existence on a finite interval for (1) (Theorem 3.2).Section 4 is devoted to (non)continuation (Theorem 4.1) associated with problem (1), which allows to generalize the main result of Section 3. Our proofs rely on Schauder's fixed point theorem and some extensions of the continuation theorems for ordinary differential equations (ODEs) to the fractional order case.One of the main difficulties lies in handling the nonlocal term 2 , representing a heat source and that depends continuously on time; another one in the fact that electrical conductivity depends on both time and temperature.Based on the results of Section 4, in Section 5 we prove existence of a global solution for (1): see Theorems 5.2 and 5.3.We end with Section 6 of conclusions.

Preliminaries and basic results
In this section, we collect from the literature [12,16,17,18,20,24] some background material and basic results that will be used in the remainder of the paper.According to the Riemann-Liouville approach to fractional calculus, we introduce the fractional integral of order α, α > 0, as follows.
Definition 2.1.The Riemann-Liouville integral of a function g with order α > 0 is defined by where Γ is the Euler gamma function given by The natural next step, after the notion of fractional integral has been introduced, is to define the fractional derivative of order α, α > 0.
Definition 2.2.The Riemann-Liouville derivative of function g with order α > 0 is defined by Note the remarkable fact that, in the Riemman-Liouville sense, the fractional derivative of the constant function is not zero.We now give an alternative and more restrictive definition of fractional derivative, first introduced by Caputo in the end of 1960's [7,8] and then adopted by Caputo and Mainardi in [9,23].In Caputo sense, the fractional derivative of a constant is zero.
Definition 2.3.The Caputo derivative of function g(t) with order α > 0 is defined by For proving our main results, we make use of the following auxiliary lemmas.Lemma 2.1 (See [19]).Let M be a subset of C([0, T ]).Then M is precompact if and only if the following conditions hold: Lemma 2.2 (Schauder fixed point theorem [19]).Let U be a closed bounded convex subset of a Banach space X.If T : U → U is completely continuous, then T has a fixed point in U .
Finally we recall a generalization of Gronwall's lemma, which is essential for the proof of our Theorem 5.3.[10,37]).Let v : [0, b] → [0, +∞) be a real function and w(•) be a nonnegative, locally integrable function on [0, b].Suppose that there exist a > 0 and

Local existence theorem
In this section, a local existence theorem of solutions for (1) is obtained by applying Schauder's fixed point theorem.In order to transform (1) into a fixed point problem, we give in the following lemma an equivalent integral form of (1).Lemma 3.1.Suppose that (H 1 )-(H 3 ) holds.Then the initial value problem (1) is equivalent to Proof.It is a simple exercise to see that u is a solution of the integral equation ( 2) if and only if it is also a solution of the IVP (1).
Theorem 3.2.Suppose that conditions (H 1 )-(H 3 ) are verified.Then (1) has at least one solution u ∈ C[0, h] for some T ≥ h > 0. Proof.Let where b is a positive constant.Further, put where It is clear that h ≤ T .Note also that D h is a nonempty, bounded, closed, and convex subset of C[0, h].In order to apply Schauder's fixed point theorem, we define the following operator A: It is clear that all solutions of (1) are fixed points of (3).Then, by assumptions (H 1 ) and (H 2 ), we have for any It yields that AD h ⊂ D h .Our next step, in order to prove Theorem 3.2, is to show that the following lemma holds.
We now focus on both right hand terms separately.By hypotheses (H 2 ) and (H 3 ), we have Then, Concerning the second term, we have It follows that Collecting inequalities ( 5) and ( 6) together, and injecting into (4), we have Therefore, Consequently, Au n − Au C[0,h] → 0 as n → +∞, which proves that A is continuous.This ends the proof of Lemma 3.3.
To finish the proof of Theorem 3.2, it remains to show that Lemma 3.4.The operator AD h is continuous.
Proof.Let u ∈ D h and 0 ≤ t 1 ≤ t 2 ≤ h.Then, where we have, by direct calculations, that and The right hand side of inequalities ( 9) and ( 10) do not depend on u and converge to zero as t 2 → t 1 .
Taking into account that AD h ⊂ D h , we infer that AD h is precompact.This implies that A is completely continuous.As a consequence of Schauder's fixed point theorem and Lemma 3.1, we conclude that problem (1) has a local solution.This ends the proof of Theorem 3.2.

Continuation results
Our main contribution of this section is to prove a continuation theorem for the fractional Caputo nonlocal thermistor problem (1).First, we present the definition of noncontinuable solution.
Proof.Let t 1 , t 2 ∈ [2η, β) such that t 1 < t 2 .From ( 8), we have where and Under assumptions (H 1 )-(H 3 ), there exists a positive constant M 3,1 such that Moreover, there exists a positive constant M 3,2 such that and we also have , where In the same manner as in the proof of Lemma 3.4, there exists positive constants M 4,1 and M 4,2 such that We have already proved in (10), for some positive constant M 5 , that Therefore, we conclude that all I i , i = 3, 4, 5, converge to zero when t 2 → t 1 .Thus, from Cauchy's convergence criterion, it yields that lim t→β − u(t) = u * .This finishes the proof of Lemma 4.2.
The second step of the proof of Theorem 4.1 consists to show the following result.
Proof.As S is a closed subset, we can say that (β, u * ) ∈ S. Define u(β) = u * .Hence, u(t) ∈ C[0, β].Then we define the operator K by where and , 1 .Analogously to the proof of Theorem 3.2, we prove β+h] → 0 as n → +∞ and similar arguments to the one above for (7), allow us to declare that there exists a positive constant c h depending on h such that [β,β+h] → 0 as n → +∞, which yields that operator K is continuous.We show that KE h is equicontinuous.For all v ∈ E h , we have (Kv)(β) = u 1 (β) and, in view of the choice of h, it follows from hypotheses (H 1 ) and (H 2 ) that Therefore, we get that KE h ⊂ E h .Furthermore, for any v ∈ E h and β ≤ t 1 ≤ t 2 ≤ β + h, we have (Kv)(t 1 ) − (Kv)(t 2 ) = I 6 + I 7 .
By an analogous early calculation, there exists a positive constant M 6 such that ) An analogous treatment as in ( 8)- (10) yields the existence of a positive constant M 7 such that Since the right side of inequalities (11) and ( 12) go to zero as t 2 → t 1 , we deduce that {(Kv)(t) : v ∈ E h } is equicontinuous.Consequently, K is completely continuous.Then, Schauder's fixed point theorem can be applied to obtain that operator K has a fixed point ũ(t) ∈ E h .On other words, we have where Therefore, according to Lemma 3.1, ũ(t) is a solution of (1) on (0, β + h].This is absurd because u(t) is noncontinuable.This completes the proof of Lemma 4.3.

Global existence
Now we provide two sets of sufficient conditions for the existence of a global solution for (1) (Theorems 5.2 and 5.3).We begin with an auxiliary lemma.
Proof.Follows immediately from the results of Section 4.
Proof.The existence of a local solution u(t) of ( 1) is ensured thanks to Theorem 3.2.We already know, by Lemma 3.1, that u(t) is a also a solution to the integral equation Suppose that the existing interval of the noncontinuable solution u(t) is (0, β), β < +∞.Then, Proof.Suppose that the maximum existing interval of u(t) is (0, β), β < +∞.We claim that u(t) is bounded on [τ, β) for any τ ∈ (0, β).Indeed, we have

Conclusion
In our paper we consider a prototype of electrical conductivity that depends strongly in both time and temperature.The model relates to modern developments of thermistors, where fractional PDEs have a crucial role.It turns out that available computational methods are not theoretically sound in the sense they rely on results of local existence.The main novelty of our paper is that we prove global existence for a nonlocal thermistor problem with fractional differentiation in the Caputo sense.Moreover, we extend some results of continuation and global existence to the fractional order initial value thermistor problem.The proofs rely on Schauder's fixed point theorem.We trust our results will have a positive impact on the development of computer mathematics to fractional thermistor devices.
a Lipschitz continuous function with Lipschitz constant L f with respect to the second variable such that c 1 ≤ f (s, u) ≤ c 2 with c 1 and c 2 two positive constants; (H 2 ) there exists a positive constant M such that f (s, u) ≤ M s 2 ; (H 3 ) |f (s, u) − f (s, v)| ≤ s 2 |u − v| or, in a more general manner, there exists a constant ω ≥ 2 such that |f (s, u) − f (s, v)| ≤ s ω |u − v|.
Let C[a, b] be the Banach space of all real valued continuous functions on [a, b] endowed with the norm x [a,b] = max t∈[a,b] |x(t)|.

Theorem 4 . 3 . 4 . 1 .
1 follows from Lemmas 4.2 and 4.Remark Uniqueness of solution to problem (1) is easily derived from the proof of Theorem 4.1 for a well chosen λ.