Positive solutions of a fractional thermostat model

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.

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].

provided the integral exists.

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

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

where $n-1<\alpha <n$, $n=[\alpha ]+1$.

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

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

We start by solving an auxiliary problem to get an expression for the Green’s function of boundary value problem (1)-(2).

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 $s>r$.

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

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 $[0,1]\times [0,1]$.

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).

 □

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).

 □

Lemma 2.7 If $\beta \mathrm{\Gamma }(\alpha )<{(1-\eta )}^{\alpha -1}$, then $G(t,s)$ changes sign on $[0,1]\times [0,1]$, and $G(t,s)$ satisfies property (A).

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

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

We now state the main result of this paper.

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

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 $\parallel \cdot \parallel$.

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).

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 $t\in [0,1]$.

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.

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

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.

Let ${\mathrm{\Omega }}_1=\{u\in C[0,1]|\parallel u\parallel <{\rho }_1\}$. Hence, we have $\parallel Tu\parallel \ge \parallel u\parallel$, $u\in P\cap \partial {\mathrm{\Omega }}_1$.

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

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

If we let ${\mathrm{\Omega }}_1=\{u\in C[0,1]|\parallel u\parallel <r_1\}$, 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).

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

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 $\beta \mathrm{\Gamma }(\alpha )<{(1-\eta )}^{\alpha -1}$. We skip the rest of the proof as it is similar to the proof of Theorem 3.1.

Example 3.1

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 $\beta \mathrm{\Gamma }(\alpha )-{(1-\eta )}^{\alpha -1}=\frac{2\sqrt{\pi }}{5}-\frac{1}{2}\approx 0.20898>0$.

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.