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 [13] 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 [16] 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 [13]. 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.

Here we present some necessary basic knowledge and definitions for fractional calculus theory that can be found in the literature [1-3].

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 2.1Letand.

(i) If, then.

(ii) If, then.

(iii) .

(iv) .

Remark 2.1 In addition to the above properties, the Caputo derivative of a power function , , is given by

where , .

Lemma 2.2For, the general solution of the fractional differential equationis given by

where, (, ).

Lemma 2.3

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 2.4Suppose. A functionis a solution of the boundary value problem

if and only if it satisfies the integral equation

whereis the Green’s function (depending onα) given by

and for, is defined asforandfor.

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 [13] 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

and

Consequently, by looking at the behavior of with respect to s, we get

and

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 [17]:

(A) There exist a measurable function , a subinterval and a constant such that

and

Lemma 2.5If, thenfor all, andsatisfies property (A).

Proof If , then for all . We choose , and we have

and

where

 □

Lemma 2.6If, thenfor all, andsatisfies property (A).

Proof We choose with . Following the arguments in the previous lemma, we have

Also, by taking

we obtain

 □

Lemma 2.7If, thenchanges sign on, andsatisfies property (A).

Proof We choose with such that . We have

and

where

For the main results, we use the known Guo-Krasnosel’skii fixed point theorem [18]. □

Theorem 2.1LetEbe a Banach space and letbe a cone. Assume, are open bounded subsets ofEsuch that, and letbe a completely continuous operator such that

(i) , and, ; or

(ii) , and, .

Then the operatorPhas a fixed point in.

We set

We now state the main result of this paper.

Theorem 3.1Let. Assume that one of the following conditions is satisfied:

(i) (Sublinear case) and.

(ii) (Superlinear case) and.

If, then problem (1)-(2) admits at least one positive solution.

Theorem 3.2Let. Assume that one of the following conditions is satisfied:

(i) (Sublinear case) and.

(ii) (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:

Hence, .

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.

(i) 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 [19])

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.

(ii) 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 . □

Remark 3.1

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.

Example 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 .

We take

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.

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.

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.

1. Kilbas, A, Srivastava, HM, Trujillo, J: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)

2. Samko, S, Kilbas, A, Maričev, O: Fractional Integrals and Derivatives, Gordon & Breach, New York (1993)

3. Podlubny, I: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego (1999)

4. Ahmad, B, Agarwal, R: On nonlocal fractional boundary value problems. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal.. 18, 535–544 (2011)

5. Ahmad, B, Nieto, JJ: Anti-periodic fractional boundary value problems with nonlinear term depending on lower order derivative. Fract. Calc. Appl. Anal.. 15, 451–462 (2012)

6. Ahmad, B, Nieto, JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl.. 58, 1838–1843 (2009). Publisher Full Text

7. Ahmad, B, Nieto, JJ, Alsaedi, A, El-Shahed, M: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl.. 13(2), 599–606 (2012). Publisher Full Text

8. Ahmad, B, Nieto, JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl.. 2011, Article ID 36 (2011)

9. Bai, Z, Lu, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl.. 311(2), 495–505 (2005). Publisher Full Text

10. Benchohra, M, Cabada, A, Seba, D: An existence result for nonlinear fractional differential equations on Banach spaces. Bound. Value Probl.. 2009, Article ID 628916 (2009)

11. Cabada, A, Wang, G: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl.. 389, 403–411 (2012). Publisher Full Text

12. Zhang, S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ.. 2006, 1–12 (2006). PubMed Abstract

13. Infante, G, Webb, J: Loss of positivity in a nonlinear scalar heat equation. Nonlinear Differ. Equ. Appl.. 13, 249–261 (2006). Publisher Full Text

14. Infante, G, Webb, J: Nonzero solutions of Hammerstein integral equations with discontinuous kernels. J. Math. Anal. Appl.. 272, 30–42 (2002). Publisher Full Text

15. Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc.. 74(3), 673–693 (2006). Publisher Full Text

16. Guidotti, P, Merino, S: Gradual loss of positivity and hidden invariant cones in a scalar heat equation. Differ. Integral Equ.. 13(10/12), 1551–1568 (2000)

17. Lan, K, Webb, J: Positive solutions of semilinear differential equations with singularities. J. Differ. Equ.. 148(2), 407–421 (1998). Publisher Full Text

18. Guo, D, Lakshmikanthan, V: Nonlinear Problems in Abstract Cones, Academic Press, New York (1988)

19. Wang, H: On the number of positive solutions of nonlinear systems. J. Math. Anal. Appl.. 281, 287–306 (2003). Publisher Full Text