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Positive solutions of a fractional thermostat model
Boundary Value Problemsvolume 2013, Article number: 5 (2013)
We study the existence of positive solutions of a nonlinear fractional heat equation with nonlocal boundary conditions depending on a positive parameter. Our results extend the second-order thermostat model to the non-integer case. We base our analysis on the known Guo-Krasnosel’skii fixed point theorem on cones.
Fractional calculus has been studied for centuries mainly as a pure theoretical mathematical discipline, but recently, there has been a lot of interest in its practical applications. In current research, fractional differential equations have arisen in mathematical models of systems and processes in various fields such as aerodynamics, acoustics, mechanics, electromagnetism, signal processing, control theory, robotics, population dynamics, finance, etc. [1–3]. For some recent results in fractional differential equations, see [4–12] and the references therein.
Infante and Webb  studied the nonlocal boundary value problem
which models a thermostat insulated at with the controller at adding or discharging heat depending on the temperature detected by the sensor at . Using fixed point index theory and some results on their work on Hammerstein integral equations [14, 15], they obtained results on the existence of positive solutions of the boundary value problem. In particular, they have shown that if , then positive solutions exist under suitable conditions on f. This type of boundary value problem was earlier investigated by Guidotti and Merino  for the linear case with where they have shown a loss of positivity as β decreases. In the present paper, we consider the following fractional analog of the thermostat model:
where , denotes the Caputo fractional derivative of order α and subject to the boundary conditions
where , are given constants.
We point out that for , we recover the second-order problem of . We use the properties of the corresponding Green’s function and the Guo-Krasnosel’skii fixed point theorem to show the existence of positive solutions of (1)-(2) under the condition that the nonlinearity f is either sublinear or superlinear.
Definition 2.1 The Riemann-Liouville fractional integral of order of a function is given by
provided the integral exists.
Definition 2.2 The Riemann-Liouville fractional derivative of order of a function is given by
where denotes the integer part of the real number α.
Definition 2.3 The Caputo derivative of order of a function is given by
where denotes the integer part of the real number α.
Lemma 2.1 Let and .
If , then .
If , then .
Remark 2.1 In addition to the above properties, the Caputo derivative of a power function , , is given by
where , .
Lemma 2.2 For , the general solution of the fractional differential equation is given by
where , (, ).
for some , (, ).
We start by solving an auxiliary problem to get an expression for the Green’s function of boundary value problem (1)-(2).
Lemma 2.4 Suppose . A function is a solution of the boundary value problem
if and only if it satisfies the integral equation
where is the Green’s function (depending on α) given by
and for , is defined as for and for .
Proof Using (3) we have, for some constants ,
In view of Lemma 2.1, we obtain
Since , we find that .
It also follows that
Using the boundary condition , we get
Finally, substituting the values of and in (5), we have
where is given by (4). This completes the proof. □
Remark 2.2 We observe that is continuous on for any . Thus, given by (4) is continuous on .
Remark 2.3 By taking , we get
and in this case coincides with the one obtained in  for the boundary value problem
Remark 2.4 We observe that for each fixed point , for and for and deduce that is a decreasing function of t. It then follows that
Consequently, by looking at the behavior of with respect to s, we get
To establish the existence of positive solutions of problem (1)-(2), we will show that satisfies the following property introduced by Lan and Webb in :
There exist a measurable function , a subinterval and a constant such that
Lemma 2.5 If , then for all , and satisfies property (A).
Proof If , then for all . We choose , and we have
Lemma 2.6 If , then for all , and satisfies property (A).
Proof We choose with . Following the arguments in the previous lemma, we have
Also, by taking
Lemma 2.7 If , then changes sign on , and satisfies property (A).
Proof We choose with such that . We have
For the main results, we use the known Guo-Krasnosel’skii fixed point theorem . □
Theorem 2.1 Let E be a Banach space and let be a cone. Assume , are open bounded subsets of E such that , and let be a completely continuous operator such that
, and , ; or
, and , .
Then the operator P has a fixed point in .
3 Main results
We now state the main result of this paper.
Theorem 3.1 Let . Assume that one of the following conditions is satisfied:
(Sublinear case) and .
(Superlinear case) and .
If , then problem (1)-(2) admits at least one positive solution.
Theorem 3.2 Let . Assume that one of the following conditions is satisfied:
(Sublinear case) and .
(Superlinear case) and .
If , then problem (1)-(2) admits a solution which is positive on an interval .
Proof of Theorem 3.1 Let be the Banach space of all continuous real-valued functions on endowed with the usual supremum norm .
We define the operator as
where is defined by (4).
It is clear from Lemma 2.4 that the fixed points of the operator T coincide with the solutions of problem (1)-(2).
We now define the cone
where λ is given by (6).
First, we show that .
It follows from the continuity and the non-negativity of the functions G and f on their domains of definition that if , then and for all .
For a fixed and for all , the fact that satisfies property (A) leads to the following inequalities:
We now show that is completely continuous.
In view of the continuity of the functions G and f, the operator is continuous.
Let be bounded, that is, there exists a positive constant such that for all . Define
Then for all , we have
for all . That is, the set is bounded.
For each and such that , we have
Clearly, the right-hand side of the above inequalities tends to 0 as and therefore the set is equicontinuous. It follows from the Arzela-Ascoli theorem that the operator is completely continuous.
We now consider the two cases.
Sublinear case ( and ).
Since , there exists such that for all , where satisfies
We take such that , then we have the following inequalities:
Let . Hence, we have , .
Since is a continuous function on , we can define the function:
It is clear that is non-decreasing on and since , we have (see )
Therefore, there exists such that for all , where satisfies
Define and let such that . Then
Hence, we have , .
Thus, by the first part of the Guo-Krasnosel’skii fixed point theorem, we conclude that (1)-(2) has at least one positive solution.
Superlinear case ( and ).
Let be given as in (8).
Since , there exists a constant such that for . Take such that . Then we have
If we let , we see that for .
Now, since , there exists such that for all , where is as in (7).
Define , where . Then and imply that
and so we obtain
This shows that for . We conclude by the second part of the Guo-Krasnosel’skii fixed point theorem that (1)-(2) has at least one positive solution . □
To prove Theorem 3.2, we use the cone
where b and λ are defined in Lemma 2.6 for the case where , and in Lemma 2.7 for the case where . We skip the rest of the proof as it is similar to the proof of Theorem 3.1.
Consider the fractional boundary value problem:
which is problem (1)-(2) with , , and .
First, we note that is not a solution of (9).
Clearly, and , and we also have .
and consider the cone .
By the first part of Theorem 3.1, we conclude that the boundary value problem (9) has a positive solution in the cone P.
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Dedicated to Professor Jean Mawhin for his 70th anniversary.
The research has been partially supported by Ministerio de Economía y Competitividad, and FEDER, project MTM2010-15314.
The authors declare that they have no competing interests.
Both authors, JJN and JP, contributed equally and read and approved the final version of the manuscript.