Abstract
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 secondorder thermostat model to the noninteger case. We base our analysis on the known GuoKrasnosel’skii fixed point theorem on cones.
1 Introduction
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.[13]. For some recent results in fractional differential equations, see [412] and the references therein.
Infante and Webb [13] studied the nonlocal boundary value problem
which models a thermostat insulated at
where
where
We point out that for
2 Preliminaries
Here we present some necessary basic knowledge and definitions for fractional calculus theory that can be found in the literature [13].
Definition 2.1 The RiemannLiouville fractional integral of order
provided the integral exists.
Definition 2.2 The RiemannLiouville fractional derivative of order
where
Definition 2.3 The Caputo derivative of order
where
Lemma 2.1Let
(i) If
(ii) If
(iii)
(iv)
Remark 2.1 In addition to the above properties, the Caputo derivative of a power function
where
Lemma 2.2For
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
if and only if it satisfies the integral equation
where
and for
Proof Using (3) we have, for some constants
In view of Lemma 2.1, we obtain
Since
It also follows that
Using the boundary condition
Finally, substituting the values of
where
Remark 2.2 We observe that
Remark 2.3 By taking
and
Remark 2.4 We observe that for each fixed point
and
Consequently, by looking at the behavior of
and
To establish the existence of positive solutions of problem (1)(2), we will show
that
(A) There exist a measurable function
and
Lemma 2.5If
Proof If
and
where
□
Lemma 2.6If
Proof We choose
Also, by taking
we obtain
□
Lemma 2.7If
Proof We choose
and
where
For the main results, we use the known GuoKrasnosel’skii fixed point theorem [18]. □
Theorem 2.1LetEbe a Banach space and let
(i)
(ii)
Then the operatorPhas a fixed point in
3 Main results
We set
We now state the main result of this paper.
Theorem 3.1Let
(i) (Sublinear case)
(ii) (Superlinear case)
If
Theorem 3.2Let
(i) (Sublinear case)
(ii) (Superlinear case)
If
Proof of Theorem 3.1 Let
We define the operator
where
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 nonnegativity of the functions G and f on their domains of definition that if
For a fixed
Hence,
We now show that
In view of the continuity of the functions G and f, the operator
Let
Then for all
for all
For each
Clearly, the righthand side of the above inequalities tends to 0 as
We now consider the two cases.
(i) Sublinear case (
Since
We take
Let
Since
It is clear that
Therefore, there exists
Define
Hence, we have
Thus, by the first part of the GuoKrasnosel’skii fixed point theorem, we conclude that (1)(2) has at least one positive solution.
(ii) Superlinear case (
Let
Since
If we let
Now, since
Define
and so we obtain
This shows that
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
Example 3.1
Consider the fractional boundary value problem:
which is problem (1)(2) with
First, we note that
Clearly,
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors, JJN and JP, contributed equally and read and approved the final version of the manuscript.
Acknowledgements
Dedicated to Professor Jean Mawhin for his 70th anniversary.
The research has been partially supported by Ministerio de Economía y Competitividad, and FEDER, project MTM201015314.
References

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

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

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

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

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

Ahmad, B, Nieto, JJ, Alsaedi, A, ElShahed, 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

Ahmad, B, Nieto, JJ: RiemannLiouville fractional integrodifferential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl.. 2011, Article ID 36 (2011)

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

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)

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 boundaryvalue 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

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)

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

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

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