Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions

Fractional differential equations have recently proved to be valuable tools in the modelling of many phenomena in various field of science and applications, such as physics, mechanics, chemistry, biology, economics, control theory, aerodynamics, engineering, etc. See [1–6]. There has been a significant development in the theory of initial and boundary value problems for nonlinear fractional differential equations; see, for example, [7–15].

Ahmad and co-authors have studied the existence and uniqueness of solutions of nonlinear fractional differential and integro-differential equations for a variety of boundary conditions using standard fixed-point theorems and Leray-Schauder degree theory. Ahmad et al. [16] discusses the existence and uniqueness of solutions of fractional integro-differential equations for fractional nonlocal integral boundary conditions. Ahmad et al. [17] and references therein give details of recent work on the properties of solutions of sequential fractional differential equations. Ahmad et al. [18] considers solutions of fractional differential equations with non-separated type integral boundary conditions. In Ahmad et al. [19], the Krasnoselskii fixed point theorem and the contraction mapping principle are used to prove the existence of solutions of the nonlinear Langevin equation with two fractional orders for a number of different intervals. Ahmad et al. [20] discusses the existence and uniqueness of solutions of nonlinear fractional differential equations with three-point integral boundary conditions.

Cabada et al. [21] have also studied properties of solutions of nonlinear fractional differential equations. They used the properties of the associated Green’s function and the Guo-Krasnosellskii fixed-point theorem to investigate the existence of positive solutions of nonlinear fractional differential equations with integral boundary-value conditions.

Motivated by the papers [16] and [20], this article is concerned with the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations with three-point fractional integral boundary conditions given by

(1.1)

(1.2)

We note that if $p=1$, then condition (1.2) reduces to the usual three-point integral condition. In such a case, the boundary condition corresponds to the area under the curve of solutions $x(t)$ from $t=0$ to $t=\eta$.

In this section, we introduce notations, definitions of fractional calculus and prove a lemma before stating our main results.

provided that $f^{(n)}(t)$ exists, where $[q]$ denotes the integer part of the real number q.

provided that such integral exists.

provided that the right-hand side is pointwise defined on $(0,\mathrm{\infty })$.

So, we prefer to use Caputo’s definition which gives better results than those of Riemann-Liouville.

Lemma 2.1 [3]

where n is the smallest integer greater than or equal to q.

Lemma 2.2 [3]

for some $k_i\in \mathbb{R}$, $i=0,1,2,\dots ,n-1$ where n is the smallest integer greater than or equal to q.

Lemma 2.3 Let $\alpha \ne \frac{\mathrm{\Gamma }(p+2)}{{\eta }^{p+1}}$, $1<q\le 2$. Then for $h\in C([0,1],\mathbb{R})$, the problem

(2.1)

(2.2)

for some $c_0,c_1\in \mathbb{R}$.

Substituting the values of $c_0$ and $c_1$ in (2.4), we obtain the solution (2.3). □

Now we are in the position to establish the main results.

Theorem 3.1 Assume that there exists a constant $L>0$ such that

(${\mathrm{H}}_1$) $|f(t,x)-f(t,y)|\le L|x-y|$, for each $t\in [0,1]$, and all $x,y\in \mathbb{R}$.

If $L\mathrm{\Lambda }<1$, where Λ is defined by (2.5), then the BVP (1.1)-(1.2) has a unique solution on $[0,1]$.

Obviously, the fixed points of the operator F are solution of the problem (1.1)-(1.2). We shall use the Banach fixed point theorem to prove that F has a fixed point. We will show that F is a contraction.

Therefore, F is a contraction. Hence, by Banach fixed point theorem, we get that F has a fixed point which is a solution of the problem (1.1)-(1.2). □

The following result is based on Schaefer’s fixed point theorem.

Theorem 3.2 Assume that:

(${\mathrm{H}}_2$) The function $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous.

(${\mathrm{H}}_3$) There exists a constant $M>0$ such that $|f(t,x)|\le M$ for each $t\in J$ and all $x\in \mathbb{R}$.

Then the BVP (1.1)-(1.2) has at least one solution on $[0,1]$.

Proof We shall use Schaefer’s fixed point theorem to prove that F has a fixed point. We divide the proof into four steps.

Step I. Continuity of F.

Since f is continuous function, then $\parallel F(x_n)-F(x)\parallel \to 0$ as $n\to \mathrm{\infty }$. This means that F is continuous.

Step II. F maps bounded sets into bounded sets in $C([0,1],\mathbb{R})$.

Step III. We prove that $F(B_r)$ is equicontinuous with $B_r$ defined as in Step II.

Actually, as $t_1\to t_2$, the right-hand side of the above inequality tends to zero. As a consequence of Steps I to III together with the Arzela-Ascoli theorem, we get that $F:C([0,1],\mathbb{R})\to C([0,1],\mathbb{R})$ is completely continuous.

Step IV. A priori bounds.

is bounded.

This shows that the set E is bounded. As a consequence of Schaefer’s fixed point theorem, we conclude that F has a fixed point which is a solution of the problem (1.1)-(1.2). □

In this section, in order to illustrate our results, we consider two examples.

Example 4.1 Consider the following three-point fractional integral boundary value problem

(4.1)

(4.2)

Hence, by Theorem 3.1, the boundary value problem (4.1)-(4.2) has a unique solution on $[0,1]$.

Example 4.2 Consider the following three-point fractional integral boundary value problem

(4.3)

(4.4)

Set $\eta =1/2$, $q=4/3$, $p=5/3$, $\alpha =1/3\ne \mathrm{\Gamma }(p+2)/{\eta }^{p+1}=\mathrm{\Gamma }(11/3)/{(1/2)}^{\frac{8}{3}}$ and $f(t,x)=t^{\frac{1}{3}}{e}^{-2t}/(1+t{x}^2)$. Clearly $|f(t,x)|\le |t^{\frac{1}{3}}{e}^{-2t}/(1+t{x}^2)|\le 1=M$.

Hence, all the conditions of Theorem 3.2 are satisfied and consequently the problem (4.1)-(4.2) has at least one solution.