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:
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.
provided the integral exists.
We start by solving an auxiliary problem to get an expression for the Green’s function of boundary value problem (1)-(2).
if and only if it satisfies the integral equation
In view of Lemma 2.1, we obtain
It also follows that
and in this case coincides with the one obtained in  for the boundary value problem
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 :
Also, by taking
For the main results, we use the known Guo-Krasnosel’skii fixed point theorem . □
3 Main results
We now state the main result of this paper.
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).
We now consider the two cases.
It is clear that is non-decreasing on and since , we have (see )
Thus, by the first part of the Guo-Krasnosel’skii fixed point theorem, we conclude that (1)-(2) has at least one positive solution.
and so we obtain
To prove Theorem 3.2, we use the cone
Consider the fractional boundary value problem:
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.
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)
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
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
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
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
Zhang, S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ.. 2006, 1–12 (2006). PubMed Abstract
Infante, G, Webb, J: Loss of positivity in a nonlinear scalar heat equation. Nonlinear Differ. Equ. Appl.. 13, 249–261 (2006). Publisher Full Text
Infante, G, Webb, J: Nonzero solutions of Hammerstein integral equations with discontinuous kernels. J. Math. Anal. Appl.. 272, 30–42 (2002). Publisher Full Text
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
Lan, K, Webb, J: Positive solutions of semilinear differential equations with singularities. J. Differ. Equ.. 148(2), 407–421 (1998). Publisher Full Text
Wang, H: On the number of positive solutions of nonlinear systems. J. Math. Anal. Appl.. 281, 287–306 (2003). Publisher Full Text