Existence of solutions for some two-point fractional boundary value problems under barrier strip conditions

Abstract

In this paper, we are dedicated to researching the boundary value problems (BVPs) for equation $D^{\alpha }x(t)=f(t,x(t),D^{\alpha -1}x(t))$, with the boundary value conditions to be either: $x(0)=A$, $D^{\alpha -1}x(1)=B$ or $D^{\alpha -1}x(0)=A$, $x(1)=B$. Let the nonlinear term f satisfy some sign conditions, then by making use of the Leray–Schauder nonlinear alternative, some existence results are obtained. In the end, an example is given to verify the main results.

1 Introduction

In the last several years, because the fractional calculus theory has been extensively used in non-Newtonian fluid mechanics, diffusion and transportation theory, engineering, biology, image processing, and other fields [9, 12, 15–19, 21, 24, 25, 35, 36, 38, 41–47], the fractional differential equations (FDEs) have been researched with different methods by many scholars. Many interesting results have been obtained [1–8, 10, 11, 13, 14, 17, 23, 26, 30–32, 35, 36, 40, 48–51].

In 1994, Kelevedjiev [27] investigated the nonlinear second order two-point BVP as follows by the use of the barrier strips argument and the topological transversality theorem [22]:

$$\begin{aligned} & x''(t)=f\bigl(t, x(t), x'(t)\bigr), \quad t\in [0,1], \end{aligned}$$

(1)

$$\begin{aligned} &x(0) = A, \qquad x'(1) =B, \end{aligned}$$

(2)

and got the existence results of solutions.

After that, the barrier strips technique was used by many researchers to study integer-order BVPs and IVPs (initial value problems). For instance, by making use of the barrier strips technique, the existence results for integer p-Laplacian BVP and first order IVP have been obtained by Kelevedjiev and Tersian, see [29] and [28]. Gao and Ma et al. generalized the idea to research the solvability for other integer BVPs such as second order three-point BVP [33], two-point BVP on time scales [34], difference equations BVP with p-Laplacian [20]. But as far as we know, the idea was hardly used to solve fractional BVPs at that time.

Recently, Khalil et al. [30] gave the definitions of conformable fractional derivative, which have many of the basic properties of integer derivatives. These good properties are conducive for scholars to study the BVPs with conformable fractional derivative. Motivated and inspired by the above papers. In 2017, He et al. [23] generalized the idea to research the fractional BVP as follows:

$$\begin{aligned} D^{\alpha }x(t)=f\bigl(t,x(t),D^{\alpha -1}x(t)\bigr), \end{aligned}$$

(3)

with the boundary value conditions to be either

$$\begin{aligned} x(0)=A, \quad\quad D^{\alpha -1}x(1)=B, \end{aligned}$$

(4)

or

$$\begin{aligned} D^{\alpha -1}x(0)=A, \quad\quad x(1)=B, \end{aligned}$$

(5)

where $D^{\alpha }$ is the standard conformable fractional derivative, ${\alpha }\in (1,2]$ is a real number, and $f\in C([0,1]\times \mathbb{R}^{2}, \mathbb{R})$.

Almost at the same time, Song et al. [39] considered the BVP for fractional equation (3) with inhomogeneous Dirichlet boundary conditions. By making use of the barrier strips technique and the fixed-point index theory, they acquired the existence results for the fractional Dirichlet BVP.

In this paper, we are dedicated to researching BVPs (3), (4) and (3), (5). Let the nonlinear term f satisfy certain sign conditions at the origin. Then, by making use of the Leray–Schauder nonlinear alternative [34] together with the barrier strips technique, not only can we get the existence results for BVPs (3), (4) and (3), (5), but also weaken the restrictions imposed on the nonlinear term f in Theorem 11 and Theorem 12 of [23].

The paper is laid out as follows. In Section 2, we present some necessary notions and preliminaries, which play an essential role in our proofs. In Section 3, by applying the technique of barrier strips and the Leray–Schauder nonlinear alternative, our main results are given and proved. Finally, an example is given to verify the main results obtained.

2 Preliminaries and lemmas

We recall some notions and lemmas in this section.

Definition 2.1

([26])

Let u be n-order differentiable at $t>0$. The fractional derivative of order $\alpha \in (n,n+1]$ of a function $u: [0,\infty )\rightarrow \mathbb{R}$ is defined as

$$ D^{\alpha }u(t)=\lim_{\epsilon \rightarrow 0}\frac{u^{(n)}(t+ \epsilon t^{n+ 1- \alpha })-u^{(n)}(t)}{\epsilon }, $$

provided the limits of the right-hand side exist.

Lemma 2.1

([26])

Let$t>0$. Function$u(t)$isα-order differentiable if and only ifuis$(n+1)$-order differentiable. Furthermore, the following relation holds:

$$ D^{\alpha }u(t)=t^{n+1-\alpha }u^{(n+1)}(t). $$

Lemma 2.2

([30])

Suppose that$a\geq 0$and$f: [a, b]\rightarrow \mathbb{R}$satisfies the following conditions:

• (i) fis continuous on$[a,b]$,

• (ii) fisα-order differentiable on$(a, b)$.

Then there exists$e\in (a, b)$such that the following relation holds:

$$ f(b)-f(a)=D^{\alpha }f(e)\frac{b^{\alpha }-a^{\alpha }}{\alpha }. $$

Let

$$\begin{aligned}& \begin{aligned} C^{\alpha }[0,1]={}&\bigl\{ u\mid u(t)=J_{0+}^{\alpha }x(t)+C_{n}t^{n}+ \cdots +C_{1}t +C_{0}, \\ & C_{i}\in \mathbb{R}, {i}=0,1,\ldots,n, x\in C[0,1]\bigr\} , \end{aligned} \\& \Vert u \Vert _{\alpha }= { \bigl\Vert {D^{\alpha }}u \bigr\Vert }_{0}+{ \bigl\Vert {D^{\alpha -1}}u \bigr\Vert }_{0}+ \cdots +{ \bigl\Vert {D^{\alpha -n}}u \bigr\Vert }_{0}+ \Vert u \Vert _{0}, \end{aligned}$$

where ${ \Vert u \Vert _{0}}={\max_{t\in [0,1]}{ \vert u(t) \vert }}$.

Lemma 2.3

([23])

${(C^{\alpha }[0,1], \Vert \cdot \Vert _{\alpha })}$is a Banach space.

The next theorem is Leray–Schauder nonlinear alternative, which is crucial in our proofs.

Theorem 2.1

([37])

Suppose thatUis a relatively open subset of a convex setKin Banach space E. Assume that$N: \overline{U}\rightarrow K$is a compact map and$p\in U$. Then either

• (i) Nhas a fixed point inU̅; or

• (ii) there are$\lambda \in (0,1)$and$u\in \partial U$such that$u=\lambda Nu+(1-\lambda )p$.

Let ${C_{B_{0}}^{\alpha }}[0,1]$ be the subspace of $C^{\alpha }[0,1]$ such that boundary condition (4) is satisfied.

Consider the BVPs:

$$\begin{aligned} &D^{\alpha }x(t)=\lambda f\bigl(t,x(t),D^{\alpha -1}x(t) \bigr), \quad t\in (0,1), \end{aligned}$$

(6)

$$\begin{aligned} &x(0)=A, \qquad D^{\alpha -1}x(1)=B, \end{aligned}$$

(7)

where $\lambda \in (0,1)$ is a real number. Define $L: {C_{B_{0}}^{ \alpha }}[0,1]\rightarrow C[0,1]$ by $Lx=D^{\alpha }x$. Obviously, L is one-one mapping. So, the following theorem can be easily obtained by using the nonlinear alternative theorem.

Theorem 2.2

Suppose thatUis an open and bounded neighborhood of$0\in C^{ \alpha -1}[0,1]$and problem (6), (7) has no solutions in∂Ufor$0<\lambda <1$. Then the problem

$$\begin{aligned} &D^{\alpha }x(t)= f\bigl(t,x(t),D^{\alpha -1}x(t)\bigr), \quad t\in (0,1), \end{aligned}$$

(8)

$$\begin{aligned} &x(0)=A, \qquad D^{\alpha -1}x(1)=B \end{aligned}$$

(9)

has at least one solution inU̅.

Therefore, our analysis is simplified to constructing a set U that is open and bounded such that BVP (6), (7) has no solutions in ∂U.

3 Existence results

Theorem 3.1

Suppose that$f:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$is continuous. Let$G_{1}$, $G_{2}$be two constants such that$G_{2}< B< G_{1}$and the following conditions are fulfilled:

• (H1) $f(t, x, G_{1})\geq 0$for$(t, x)\in [0, 1]\times [-G, G]$;

• (H2) $f(t, x, G_{2})\leq 0$for$(t, x)\in [0, 1]\times [-G, G]$,

where$G> \vert A \vert + \vert \frac{1}{\alpha -1} \vert \cdot \max {\{ \vert G_{2} \vert , \vert G_{1} \vert \}}$. Then BVP (3), (4) has at least one solution.

Proof

According to the Tietze–Urysohn lemma, we can find a continuous function $\vartriangle : R^{2}\rightarrow [-1, 1]$ such that

$$\begin{aligned} &\vartriangle (x,G_{1}) =1,\quad x\in [-G,G], \end{aligned}$$

(10)

and

$$\begin{aligned} &\vartriangle (x,G_{2}) =-1,\quad x\in [-G,G]. \end{aligned}$$

(11)

For $n\geq 1$, set $f_{n}(t,x,y)=f(t,x,y)+(1/n){\vartriangle (x,y)}$, then

$$\begin{aligned} &f_{n}(t,x,G_{1})>0, \quad x\in [-G,G], \end{aligned}$$

(12)

$$\begin{aligned} &f_{n}(t,x,G_{2})< 0, \quad x\in [-G,G]. \end{aligned}$$

(13)

Consider the BVPs

$$\begin{aligned} &D^{\alpha }x(t)=f_{n}\bigl(t,x(t),D^{\alpha -1}x(t) \bigr), \end{aligned}$$

(14)

$$\begin{aligned} & x(0)=A, \qquad D^{\alpha -1}x(1)=B. \end{aligned}$$

(15)

If we can prove that (14), (15) has a solution $x_{n}$ such that

$$\begin{aligned} -G\leq x_{n}\leq G \quad \text{and} \quad G_{2}\leq D^{\alpha -1}x_{n}\leq G_{1} \end{aligned}$$

(16)

hold for all $n\in N$, then by combining (14), (15), (16) and Arzela–Ascoli theorem, the sequence $\{x_{n}\}$ has a subsequence which converges in $C^{\alpha }[0,1]$-topology to a solution $x_{0}$ for BVP (3), (4).

The set U is defined by

$$ U=\bigl\{ v\in C^{\alpha -1}[0,1]\mid -G< v< G, G_{2}< D^{\alpha -1}v< G_{1}\bigr\} . $$

(17)

In order to prove that (14), (15) has a solution $x_{n}$ such that (16) holds, we only need to demonstrate, according to Theorem 2.1, that if $x\in {C_{{B_{0}}}^{\alpha -1}[0,1]}$ satisfies

$$\begin{aligned} -G\leq x \leq G \quad \text{and} \quad G_{2}\leq D^{\alpha -1}x \leq G_{1}, \end{aligned}$$

(18)

and

$$\begin{aligned} D^{\alpha }x(t)=\lambda f_{n}\bigl(t,x(t),D^{\alpha -1}x(t) \bigr), \end{aligned}$$

(19)

for some $\lambda \in (0,1)$, then $x\in U$, i.e.,

$$\begin{aligned} -G< x < G \quad \text{and} \quad G_{2}< D^{\alpha -1}x < G_{1}. \end{aligned}$$

(20)

Let $x\in {C_{{B_{0}}}^{\alpha -1}[0,1]}$ satisfy (18), (19) for some $\lambda \in [0,1]$. By Lemma 2.1, there exists $d \in (0, t)$ such that the following relation holds:

$$\begin{aligned} x(t)-x(0)=D^{\alpha -1}x(d)\cdot \frac{t^{\alpha -1}}{\alpha -1}, \quad d\in (0,t). \end{aligned}$$

(21)

Then, by the inequality $G_{2}\leq D^{\alpha -1}x \leq G_{1}$, there holds

$$ \begin{aligned}[b] \bigl\vert x(t) \bigr\vert & \leq \biggl\vert \frac{t^{\alpha -1}}{\alpha -1} \biggr\vert \cdot \bigl\vert D^{\alpha -1}x(d) \bigr\vert + \vert A \vert \\ &\leq \biggl\vert \frac{t^{\alpha -1}}{\alpha -1} \biggr\vert \cdot \max \bigl\{ \vert G_{2} \vert , \vert G_{1} \vert \bigr\} + \vert A \vert \\ &\leq \biggl\vert \frac{1}{\alpha -1} \biggr\vert \cdot \max \bigl\{ \vert G_{2} \vert , \vert G_{1} \vert \bigr\} + \vert A \vert \\ &< G. \end{aligned} $$

(22)

Relation (22) together with (12) and (13) implies that

$$\begin{aligned} &f_{n}\bigl(t,x(t),G_{1}\bigr)>0, \quad t\in [0,1]; \end{aligned}$$

(23)

$$\begin{aligned} &f_{n}\bigl(t,x(t),G_{2}\bigr)< 0, \quad t\in [0,1]. \end{aligned}$$

(24)

Suppose that $D^{\alpha -1}(t_{0})=G_{1}$ for some $t_{0}\in [0,1]$. We have $t_{0}<1$ since $D^{\alpha -1}x(1)=B$. Hence $D^{\alpha }x(t_{0}) \leq 0$ because $D^{\alpha -1}x(t)$ attains its maximum at $t_{0}$. However, by (23) and (19), there is

$$ \begin{aligned}[b] D^{\alpha }x(t_{0}) &=\lambda f_{n}\bigl(t_{0},x(t_{0}),D^{\alpha -1}x(t _{0})\bigr) \\ &=\lambda f_{n}\bigl(t_{0},x(t_{0}), G_{1}\bigr) \\ &>0. \end{aligned} $$

(25)

This contradiction shows that $D^{\alpha -1}x(t)< G_{1}$. Analogously $D^{\alpha -1}x(t)>G_{2}$. Thus

$$\begin{aligned} G_{2}< D^{\alpha -1}x< G_{1}, \quad t\in [0,1]. \end{aligned}$$

(26)

The theorem is proven. □

Remark 3.1

Theorem 3.1 is a generalization of Theorem 11 in literature [22]. In [22], the conditions imposed on $f(t, x, p)$ are local to the variables t and p and global on x; however, in our Theorem 3.1, the variable x is also localized.

The following theorem can be obtained in a similar way.

Theorem 3.2

Suppose that$f: [0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$is continuous. Let$G_{3}$, $G_{4}$be two constants such that$G_{4}< A< G_{3}$and the following conditions are fulfilled:

• (H3) $f(t, x, G_{3})\leq 0$for$(t, x)\in [0, 1]\times [-G, G]$;

• (H4) $f(t, x, G_{4})\geq 0$for$(t, x)\in [0, 1]\times [-G, G]$,

where$G> \vert B \vert + \vert \frac{1}{\alpha -1} \vert \cdot \max {\{ \vert G_{4} \vert , \vert G_{3} \vert \}}$. Then BVP (3), (5) has at least one solution.

Remark 3.2

By comparison with the above Theorem 3.1 and Theorem 12 of [22], we can know that Theorem 3.2 is not only the “dual” of Theorem 3.1. At the same time, Theorem 3.2 is also a generalization of Theorem 12 of [22].

4 Example

Example 4.1

Consider the fractional BVP

$$\begin{aligned} &D^{\frac{3}{2}}x(t)=x(t)+D^{\frac{1}{2}}x(t)+ \frac{1}{2}{\bigl[D^{ \frac{1}{2}}x(t)\bigr]}^{2}+{ \bigl[D^{\frac{1}{2}}x(t)\bigr]}^{3}-1, \end{aligned}$$

(27)

$$\begin{aligned} & x(0)=-1, \qquad D^{\frac{1}{2}}x(1)=0. \end{aligned}$$

(28)

Choose $G=6$, $G_{1}=2$, $G_{2}=-2$, then

$$\begin{aligned} &f(t,x,G_{1})\geq 0, \quad \text{for } (t,x)\in [0,1]\times [-6, 6]; \\ &f(t,x,G_{2})\leq 0, \quad \text{for } (t,x)\in [0,1]\times [-6, 6]. \end{aligned}$$

By the use of Theorem 3.1, fractional BVP (27), (28) has at least one solution in $C^{\frac{3}{2}}[0,1]$.