Global existence of solutions for a fractional Caputo nonlocal thermistor problem
 Moulay Rchid Sidi Ammi^{1}Email author,
 Ismail Jamiai^{1} and
 Delfim F. M. Torres^{2}
https://doi.org/10.1186/s1366201714185
© The Author(s) 2017
Received: 21 July 2017
Accepted: 31 October 2017
Published: 15 November 2017
Abstract
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.
Keywords
MSC
1 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’ [2]. This opinion is strengthened 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., [3–7]. Recently, it has also been proved that fractional differential equations are significant and essential tools when applied in the study of nonlocal or timedependent 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 [8]. 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 [9]; a fractional Langevin equation, with applications in polymer layers, has been investigated in [10]. One can say that realworld problems require the definitions of fractional derivatives for initial and boundary value problems [11, 12]. 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 [13, 14] and [15, 16], respectively. Details of the geometric and physical interpretation of fractional differentiation can be found in [6].
 \((H_{1})\) :

\(f: \mathbb{R}^{+} \times \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) is a Lipschitz continuous function with Lipschitz constant \(L_{f}\) with respect to the second variable such that \(c_{1} \leq f(s,u) \leq c_{2}\) with \(c_{1}\) and \(c_{2}\) two positive constants;
 \((H_{2})\) :

there exists a positive constant M such that \(f(s, u) \leq M s^{2}\);
 \((H_{3})\) :

\(\vert f(s, u) f(s, v) \vert \leq s^{2} \vert uv\vert \) or, in a more general manner, there exists a constant \(\omega \geq 2\) such that \(\vert f(s,u) f(s,v) \vert \leq s^{\omega } \vert uv\vert \).
In the literature, questions involving the existence and uniqueness of solution for fractional differential equations (FDEs) have been intensely studied by many mathematicians [4, 5, 21, 22]. However, much of published papers have been concerned with existenceuniqueness of solutions for FDEs on a finite interval. Since continuation theorems for FDEs are not well developed, results as regards global existenceuniqueness of the solution of FDEs on the half axis \([0,+\infty)\), by using directly the results from local existence, have only recently flourished [23, 24].
In contrast with our previous work [25–27] on fractional nonlocal thermistor problems, which was 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 the (non)continuation (Theorem 4.1) associated with problem (1), which allows one 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 \(\frac{\lambda f(t, u(t))}{(\int_{0}^{t}f(x, u(x))\,dx)^{2}}\), 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 presenting conclusions.
2 Preliminaries and basic results
In this section, we collect from the literature [4, 5, 22, 28–30] some background material and basic results that will be used in the remainder of the paper.
Let \(C[a,b]\) be the Banach space of all real valued continuous functions on \([a,b]\) endowed with the norm \(\Vert x\Vert _{[a,b]}=\max_{t\in [a,b]}\vert x(t)\vert \). According to the RiemannLiouville approach to fractional calculus, we introduce the fractional integral of order α, \(\alpha >0\), as follows.
Definition 2.1
The natural next step, after the notion of fractional integral has been introduced, is to define the fractional derivative of order α, \(\alpha >0\).
Definition 2.2
Note the remarkable fact that, in the RiemannLiouville 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 the 1960s [31, 32] and then adopted by Caputo and Mainardi in [33, 34]. In Caputo sense, the fractional derivative of a constant is zero.
Definition 2.3
For proving our main results, we make use of the following auxiliary lemmas.
Lemma 2.1
(See [24])
 1.
\(\{u(t):u \in M\}\) is uniformly bounded,
 2.
\(\{u(t):u \in M\}\) is equicontinuous on \([0,T]\).
Lemma 2.2
(Schauder fixed point theorem [24])
Let U be a closed bounded convex subset of a Banach space X. If \(T:U\to U\) 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.
Lemma 2.3
(Generalized Gronwall inequality [35, 36])
3 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
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 \(u\in C[0,h]\) for some \(T\geq h>0\).
Proof
Lemma 3.3
The operator A is continuous.
Proof
To finish the proof of Theorem 3.2, it remains to show the following.
Lemma 3.4
The operator \(AD_{h}\) is continuous.
Proof
Taking into account that \(AD_{h}\subset D_{h}\), 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. □
4 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.
Definition 4.1
(See [37])
Let \(u(t)\) on \((0,\beta)\) and \(\tilde{u}(t)\) on \((0,\tilde{\beta })\) be both solutions of (1). If \(\beta <\tilde{\beta }\) and \(u(t)=\tilde{u}(t)\) for \(t\in (0,\beta)\), then we say that \(\tilde{u}(t)\) can be continued to \((0,\tilde{\beta })\). A solution \(u(t)\) is noncontinuable if it has no continuation. The existing interval of the noncontinuable solution \(u(t)\) is called the maximum existing interval of \(u(t)\).
Theorem 4.1
Assume that conditions \((H_{1})\)\((H_{3})\) are satisfied. Then \(u=u(t)\), \(t\in (0,\beta)\), is noncontinuable if and if only for some \(\eta \in ( 0,\frac{\beta }{2} ) \) and any bounded closed subset \(S\subset [\eta,+\infty)\times \mathbb{R}\) there exists a \(t^{\ast }\in [ \eta,\beta)\) such that \((t^{\ast },u(t^{\ast })) \notin S\).
Proof
Lemma 4.2
The limit \(\lim_{t\to \beta^{}}u(t)\) exists.
Proof
The second step of the proof of Theorem 4.1 consists in showing the following result.
Lemma 4.3
Function \(u(t)\) is continuable.
Proof
5 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.
Lemma 5.1
Suppose that conditions \((H_{1})\)\((H_{3})\) hold. Let \(u(t)\) be a solution of (1) on \((0,\beta)\). If \(u(t)\) is bounded on \([\tau,\beta)\) for some \(\tau >0\), then \(\beta =+\infty \).
Proof
The result follows immediately from the results of Section 4. □
Theorem 5.2
Suppose that conditions \((H_{1})\)\((H_{3})\) hold. Then (1) has a solution in \(C([0,+\infty))\).
Proof
Next we give another sufficient condition ensuring global existence for (1).
Theorem 5.3
Suppose that there exist positive constants \(c_{3}\), \(c_{4}\) and \(c_{5}\) such that \(c_{3} \leq \vert f(s, x)\vert \leq c_{4}\vert x\vert + c_{5}\). Then (1) has a solution in \(C([0,+\infty))\).
Proof
6 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 that our results will have a positive impact on the application of computer mathematics to fractional thermistor devices.
Declarations
Acknowledgements
The authors were supported by the Center for Research and Development in Mathematics and Applications (CIDMA) of University of Aveiro, through Fundação para a Ciência e a Tecnologia (FCT), within project UID/MAT/04106/2013. They are grateful to two anonymous referees, for several comments and suggestions of improvement.
Authors’ contributions
All three authors contributed equally to this work. They all read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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