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.
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