Positive solutions of a fractional thermostat model

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 t=0 with the controller at t=1 adding or discharging heat depending on the temperature detected by the sensor at t=\eta. 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 \beta \ge 1-\eta, 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 \eta =0 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 , {}^{C}D^{\alpha } denotes the Caputo fractional derivative of order α and f\in C([0,1]\times [0,\mathrm{\infty }),[0,\mathrm{\infty })) subject to the boundary conditions

where \beta >0, 0\le \eta \le 1 are given constants.

We point out that for \alpha =2, 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 \alpha >0 of a function g:(0,\mathrm{\infty })\to \mathbb{R} is given by

provided the integral exists.

Definition 2.2 The Riemann-Liouville fractional derivative of order \alpha >0 of a function g:(0,\mathrm{\infty })\to \mathbb{R} is given by

where [\alpha ] denotes the integer part of the real number α.

Definition 2.3 The Caputo derivative of order \alpha >0 of a function g\in A{C}^{n-1}[0,\mathrm{\infty }) is given by

where [\alpha ] denotes the integer part of the real number α.

Lemma 2.1 Let g\in L_1(0,1) and \alpha ,\beta >0.

1. (i)

   If \alpha =n\in \mathbb{N}, then I^{n}g(t)=\frac{1}{(n-1)!}{\int }_0^t{(t-s)}^{n-1}g(s)ds.

2. (ii)

   If \alpha =n\in \mathbb{N}, then {}^{C}D^{n}g(t)=g^{(n)}(t).

3. (iii)

   {}^{C}D^{\alpha }{I}^{\alpha }g(t)=g(t).

4. (iv)

   I^{\alpha }{I}^{\beta }g(t)=I^{\alpha +\beta }g(t).

Remark 2.1 In addition to the above properties, the Caputo derivative of a power function t^k, k\in \mathbb{N}, is given by

where , n=[\alpha ]+1.

Lemma 2.2 For \alpha >0, the general solution of the fractional differential equation {}^{C}D^{\alpha }u(t)=0 is given by

where c_i\in \mathbb{R}, i=0,1,2,\dots ,n-1 (, n=[\alpha ]+1).

Lemma 2.3

for some c_i\in \mathbb{R}, i=0,1,2,\dots ,n-1 (, n=[\alpha ]+1).

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 f\in C[0,1]. A function u\in C[0,1] is a solution of the boundary value problem

if and only if it satisfies the integral equation

where G(t,s) is the Green’s function (depending on α) given by

and for r\in [0,1], H_r:[0,1]\to \mathbb{R} is defined as H_r(s)=\frac{{(r-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )} for s\le r and H_r(s)=0 for s>r.

Proof Using (3) we have, for some constants c_0,c_1\in \mathbb{R},

In view of Lemma 2.1, we obtain

Since u^{\prime }(0)=0, we find that c_1=0.

It also follows that

Using the boundary condition {\beta }^C{D}^{\alpha -1}u(1)+u(\eta )=0, we get

Finally, substituting the values of c_0 and c_1 in (5), we have

where G(t,s) is given by (4). This completes the proof. □

Remark 2.2 We observe that H_r is continuous on [0,1] for any r\in [0,1]. Thus, G(t,s) given by (4) is continuous on [0,1]\times [0,1].

Remark 2.3 By taking \alpha =2, we get

and G(t,s) 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 s\in [0,1], \frac{\partial G}{\partial t}=0 for t\le s and for t>s and deduce that G(t,s) is a decreasing function of t. It then follows that

and

Consequently, by looking at the behavior of G(t,s) with respect to s, we get

and

To establish the existence of positive solutions of problem (1)-(2), we will show that G(t,s) satisfies the following property introduced by Lan and Webb in [17]:

1. (A)

   There exist a measurable function \varphi :[0,1]\to [0,\mathrm{\infty }), a subinterval [a,b]\subseteq [0,1] and a constant \lambda \in [0,1] such that

   |G(t,s)|\le \varphi (s)\mathrm{\forall }t,s\in [0,1]

and

Lemma 2.5 If \beta \mathrm{\Gamma }(\alpha )>{(1-\eta )}^{\alpha -1}, then G(t,s)>0 for all t,s\in [0,1], and G(t,s) satisfies property (A).

Proof If \beta \mathrm{\Gamma }(\alpha )>{(1-\eta )}^{\alpha -1}, then G(t,s)>0 for all t,s\in [0,1]. We choose [a,b]=[0,1], and we have

and

where

 □

Lemma 2.6 If \beta \mathrm{\Gamma }(\alpha )={(1-\eta )}^{\alpha -1}, then G(t,s)\ge 0 for all t,s\in [0,1], and G(t,s) satisfies property (A).

Proof We choose [a,b]=[0,b] with . Following the arguments in the previous lemma, we have

Also, by taking

we obtain

 □

Lemma 2.7 If , then G(t,s) changes sign on [0,1]\times [0,1], and G(t,s) satisfies property (A).

Proof We choose [a,b]=[0,b] with such that \beta \mathrm{\Gamma }(\alpha )>{(b-\eta )}^{\alpha -1}. We have

and

where

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

Theorem 2.1 Let E be a Banach space and let P\subset E be a cone. Assume {\mathrm{\Omega }}_1, {\mathrm{\Omega }}_2 are open bounded subsets of E such that 0\in {\mathrm{\Omega }}_1\subset {\overline{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2, and let T:P\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1)\to P be a completely continuous operator such that

1. (i)

   \parallel Tu\parallel \ge \parallel u\parallel, u\in P\cap \partial {\mathrm{\Omega }}_1 and \parallel Tu\parallel \le \parallel u\parallel, u\in P\cap \partial {\mathrm{\Omega }}_2; or

2. (ii)

   \parallel Tu\parallel \le \parallel u\parallel, u\in P\cap \partial {\mathrm{\Omega }}_1 and \parallel Tu\parallel \ge \parallel u\parallel, u\in P\cap \partial {\mathrm{\Omega }}_2.

Then the operator P has a fixed point in P\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1).

We set

We now state the main result of this paper.

Theorem 3.1 Let f(s,u(s))\in C([0,1]\times [0,\mathrm{\infty }),[0,\mathrm{\infty })). Assume that one of the following conditions is satisfied:

1. (i)

   (Sublinear case) f_0=\mathrm{\infty } and f_{\mathrm{\infty }}=0.

2. (ii)

   (Superlinear case) f_0^{\ast }=0 and f_{\mathrm{\infty }}^{\ast }=\mathrm{\infty }.

If \beta \mathrm{\Gamma }(\alpha )>{(1-\eta )}^{\alpha -1}, then problem (1)-(2) admits at least one positive solution.

Theorem 3.2 Let f(s,u(s))\in C([0,1]\times [-\mathrm{\infty },+\mathrm{\infty }),[0,\mathrm{\infty })). Assume that one of the following conditions is satisfied:

1. (i)

   (Sublinear case) f_0=\mathrm{\infty } and f_{\mathrm{\infty }}=0.

2. (ii)

   (Superlinear case) f_0^{\ast }=0 and f_{\mathrm{\infty }}^{\ast }=\mathrm{\infty }.

If \beta \mathrm{\Gamma }(\alpha )\le {(1-\eta )}^{\alpha -1}, then problem (1)-(2) admits a solution which is positive on an interval [0,b]\subset [0,1].

Proof of Theorem 3.1 Let C[0,1] be the Banach space of all continuous real-valued functions on [0,1] endowed with the usual supremum norm \parallel \cdot \parallel.

We define the operator T:C[0,1]\to C[0,1] as

where G(t,s) 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 T(P)\subset P.

It follows from the continuity and the non-negativity of the functions G and f on their domains of definition that if u\in P, then Tu\in C[0,1] and Tu(t)\ge 0 for all t\in [0,1].

For a fixed u\in P and for all t\in [0,1], the fact that G(t,s) satisfies property (A) leads to the following inequalities:

Hence, T(P)\subset P.

We now show that T:P\to P is completely continuous.

In view of the continuity of the functions G and f, the operator T:P\to P is continuous.

Let \mathrm{\Omega }\subset P be bounded, that is, there exists a positive constant M>0 such that {\parallel u\parallel }_{\mathrm{\infty }}\le M for all u\in \mathrm{\Omega }. Define

Then for all u\in \mathrm{\Omega }, we have

for all t\in [0,1]. That is, the set T(\mathrm{\Omega }) is bounded.

For each u\in \mathrm{\Omega } and t_1,t_2\in [0,1] such that , we have

Clearly, the right-hand side of the above inequalities tends to 0 as t_1\to t_2 and therefore the set T(\mathrm{\Omega }) is equicontinuous. It follows from the Arzela-Ascoli theorem that the operator T:P\to P is completely continuous.

We now consider the two cases.

1. (i)

   Sublinear case (f_0=\mathrm{\infty } and f_{\mathrm{\infty }}=0).

Since f_0=\mathrm{\infty }, there exists {\rho }_1>0 such that f(t,u)\ge {\delta }_{1}u for all , where {\delta }_1 satisfies

We take u\in P such that \parallel u\parallel ={\rho }_1, then we have the following inequalities:

Let . Hence, we have \parallel Tu\parallel \ge \parallel u\parallel, u\in P\cap \partial {\mathrm{\Omega }}_1.

Since f(t,\cdot ) is a continuous function on [0,\mathrm{\infty }), we can define the function:

It is clear that \tilde{f}(t,u) is non-decreasing on (0,\mathrm{\infty }) and since f_{\mathrm{\infty }}=0, we have (see [19])

Therefore, there exists {\rho }_2>{\rho }_1>0 such that \tilde{f}(t,u)\le {\delta }_{2}u for all u\ge {\rho }_2, where {\delta }_2 satisfies

Define and let u\in P such that \parallel u\parallel ={\rho }_2. Then

Hence, we have \parallel Tu\parallel \le \parallel u\parallel, u\in P\cap \partial {\mathrm{\Omega }}_2.

Thus, by the first part of the Guo-Krasnosel’skii fixed point theorem, we conclude that (1)-(2) has at least one positive solution.

1. (ii)

   Superlinear case (f_0^{\ast }=0 and f_{\mathrm{\infty }}^{\ast }=\mathrm{\infty }).

Let {\delta }_2>0 be given as in (8).

Since f_0^{\ast }=0, there exists a constant r_1>0 such that f(t,u)\le {\delta }_{2}u for 0\le u\le r_1. Take u\in P such that \parallel u\parallel =r_1. Then we have

If we let , we see that \parallel Tu\parallel \le \parallel u\parallel for u\in P\cap \partial {\mathrm{\Omega }}_1.

Now, since f_{\mathrm{\infty }}^{\ast }=\mathrm{\infty }, there exists r>0 such that f(t,u)\ge {\delta }_{1}u for all u\ge r, where {\delta }_1 is as in (7).

Define , where r_2=max(2r_1,\frac{r}{\lambda }). Then u\in P and \parallel u\parallel =r_2 imply that

and so we obtain

This shows that \parallel Tu\parallel \ge \parallel u\parallel for u\in P\cap \partial {\mathrm{\Omega }}_2. We conclude by the second part of the Guo-Krasnosel’skii fixed point theorem that (1)-(2) has at least one positive solution u\in P\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1). □

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 \beta \mathrm{\Gamma }(\alpha )={(1-\eta )}^{\alpha -1}, 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 \alpha =\frac{3}{2}, \beta =\frac{4}{5}, \eta =\frac{3}{4} and f(t,u(t))=t^2{e}^{-u(t)}+\sqrt{u(t)}.

First, we note that u=0 is not a solution of (9).

Clearly, f_0=\mathrm{\infty } and f_{\mathrm{\infty }}=0, and we also have \beta \mathrm{\Gamma }(\alpha )-{(1-\eta )}^{\alpha -1}=\frac{2\sqrt{\pi }}{5}-\frac{1}{2}\approx 0.20898>0.

We take

and consider the cone P=\{u|u\in C[0,1],u(t)\ge 0,{min}_{t\in [0,1]}u(t)\ge \lambda \parallel u\parallel \}.

By the first part of Theorem 3.1, we conclude that the boundary value problem (9) has a positive solution in the cone P.

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.

Affiliations

1. Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain

   • Juan J Nieto
   •  & Johnatan Pimentel

2. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

   • Juan J Nieto

Corresponding author

Correspondence to Juan J Nieto.

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.

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