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Existence of solutions to boundary value problem of a class of nonlinear fractional differential equations
Advances in Difference Equations volume 2014, Article number: 174 (2014)
Abstract
In this paper, we study the existence of solutions for the boundary value problem of the following nonlinear fractional differential equation: ${D}_{{0}^{+}}^{\alpha}[\frac{x(t)}{f(t,x(t))}]+g(t,x(t))=0$, $0<t<1$, $x(0)=x(1)={x}^{\prime}(0)=0$, where $2<\alpha \le 3$ is a real number and ${D}_{{0}^{+}}^{\alpha}$ is the RiemannLiouville fractional derivative. By a fixed point theorem in Banach algebra, an existence theorem for the boundary value problem of the above fractional differential equation is proved under both Lipschitz and Carathéodory conditions. Two examples are presented to illustrate the main results.
MSC:34A08, 34B18.
1 Introduction
The theory of fractional derivatives goes back to Leibniz’s note in his list to L’Hospital, dated 30 September 1695, in which the meaning of the derivative of order $1/2$ is discussed. Leibniz’s note led to the appearance of the theory of derivatives and integrals of arbitrary order, which by the end of the 19th century took a more or less final form due primarily to Liouville, Grünwald, Letnikov, and Riemann. Recently, there have been several books on the subject of fractional derivatives and fractional integrals; see [1–4]. For three centuries the theory of fractional derivatives developed mainly as a pure theoretical field of mathematics useful only for mathematicians. However, in the last few decades many authors pointed out that fractional derivatives and fractional integrals are very suitable for the description of properties of various real problems.
Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications. Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in selfsimilar and porous structures, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science. There have appeared lots of works, in which fractional derivatives are used for a better description of considered material properties, mathematical modeling based on enhanced rheological models naturally leads to differential equations of fractional order and to the necessity of the formulation of initial conditions to such equations.
It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear fractional differential equations. Recently, there are some papers dealing with the existence of solutions (or positive solutions) of nonlinear initial (or boundary) fractional differential equation (or system) by the use of techniques of nonlinear analysis (fixed point theorems, LeraySchauder theory, Adomian decomposition method, etc.); see [5–23]. In fact, there has the same requirements for boundary conditions. However, there exist some papers considered the boundary value problems of fractional differential equations; see [10–22].
Yu and Jiang [21] discussed the following twopoint boundary value problem of fractional differential equations:
where $2<\alpha \le 3$ is a real number and ${D}_{{0}^{+}}^{\alpha}$ is the standard RiemannLiouville fractional derivative. By the properties of the Green function, they gave some results of multiple positive solutions for singular and nonsingular boundary value problems by means of the LeraySchauder nonlinear alternative, the fixedpoint theorem on cones, and a mixed monotone method.
In recent years, quadratic perturbations of nonlinear differential equations have attracted much attention. There have been many works on the theory of such differential equations, and we refer the readers to the articles [22–27].
Zhao et al. [23] studied fractional hybrid differential equations involving RiemannLiouville differential operators of order $0<q<1$,
where $f\in C([0,T]\times \mathbb{R},\mathbb{R}\setminus \{0\})$ and $g\in C([0,T]\times \mathbb{R},\mathbb{R})$. They established the existence and uniqueness results and some fundamental differential inequalities for fractional hybrid differential equations, initiating the study of such systems, and proved, utilizing the theory of inequalities, the existence of extremal solutions and a comparison result.
From the above works, we consider the existence of solutions for the boundary value problem of the following nonlinear fractional differential equation:
where $2<\alpha \le 3$ is a real number and ${D}_{{0}^{+}}^{\alpha}$ is the standard RiemannLiouville fractional derivative. Using the fixed point theorem, we give an existence theorem for the boundary value problem of the above nonlinear fractional differential equation under both Lipschitz and Carathéodory conditions. We present two examples to demonstrate our results.
Let ℝ be the real line and $J=[0,1]$ be a bounded interval in ℝ. Let $C(J\times \mathbb{R},\mathbb{R})$ denote the class of continuous functions $f:J\times \mathbb{R}\to \mathbb{R}$ and let $Car(J\times \mathbb{R},\mathbb{R})$ denote the class of functions $g:J\times \mathbb{R}\to \mathbb{R}$ such that

(i)
the map $t\mapsto g(t,x)$ is measurable for each $x\in \mathbb{R}$, and

(ii)
the map $x\mapsto g(t,x)$ is continuous for each $t\in J$.
The class $Car(J\times \mathbb{R},\mathbb{R})$ is called the Carathéodory class of functions on $J\times \mathbb{R}$ which are Lebesgue integrable when bounded by a Lebesgue integrable function on J.
In this paper, we assume $f\in {C}^{1}([0,1]\times \mathbb{R},\mathbb{R}\setminus \{0\})$ and $g\in Car([0,1]\times \mathbb{R},\mathbb{R})$.
The plan of this paper is as follows. In Section 2, we shall give some definitions and lemmas to prove our main results. In Section 3, we establish the existence of solutions for the boundary value problem (1.1) and (1.2) by the fixed point theorem. Two examples are presented to illustrate the main results in Section 4.
2 Preliminaries
For the convenience of the reader, we give some background materials from fractional calculus theory to facilitate the analysis of problem (1.1) and (1.2). These materials can be found in the recent literature; see [21, 28, 29].
Definition 2.1 ([28])
The RiemannLiouville fractional derivative of order $\alpha >0$ of a continuous function $f:(0,+\mathrm{\infty})\to \mathbb{R}$ is given by
where $n=[\alpha ]+1$, $[\alpha ]$ denotes the integer part of number α, provided that the right side is pointwise defined on $(0,+\mathrm{\infty})$.
Definition 2.2 ([28])
The RiemannLiouville fractional integral of order $\alpha >0$ of a function $f:(0,+\mathrm{\infty})\to \mathbb{R}$ is given by
provided that the right side is pointwise defined on $(0,+\mathrm{\infty})$.
From the definition of the RiemannLiouville derivative, we can obtain the following statement.
Lemma 2.1 ([28])
Let $\alpha >0$. If we assume $x\in C(0,1)\cap L(0,1)$, then the fractional differential equation
has $x(t)={c}_{1}{t}^{\alpha 1}+{c}_{2}{t}^{\alpha 2}+\cdots +{c}_{n}{t}^{\alpha n}$, ${c}_{i}\in \mathbb{R}$, $i=1,2,\dots ,n$, as unique solutions, where n is the smallest integer greater than or equal to α.
Lemma 2.2 ([28])
Assume that $x\in C(0,1)\cap L(0,1)$ with a fractional derivative of order $\alpha >0$ that belongs to $C(0,1)\cap L(0,1)$. Then
for some ${c}_{i}\in \mathbb{R}$, $i=1,2,\dots ,n$, where n is the smallest integer greater than or equal to α.
In the following, we present the Green function of the fractional differential equation boundary value problem.
Lemma 2.3 Let $y\in C[0,1]$ and $2<\alpha \le 3$. The unique solution of the problem
is
where
Here $G(t,s)$ is called the Green function of the boundary value problem (2.1) and (2.2).
Proof We may apply Lemma 2.2 to reduce (2.1) to an equivalent integral equation
for some ${c}_{1},{c}_{2},{c}_{3}\in \mathbb{R}$. Consequently, the general solution of (2.1) is
By $x(0)=0$, we get ${c}_{3}=0$. From (2.4), we have
By ${x}^{\prime}(0)=0$, we obtain ${c}_{2}=0$. From (2.2), we get
Therefore, the unique solution of problem (2.1) and (2.2) is
The proof is complete. □
Lemma 2.4 ([21])
The function $G(t,s)$ defined by (2.3) satisfies the following conditions:

(1)
$G(t,s)=G(1s,1t)$, for $t,s\in (0,1)$;

(2)
${t}^{\alpha 1}(1t)s{(1s)}^{\alpha 1}\le \mathrm{\Gamma}(\alpha )G(t,s)\le (\alpha 1)s{(1s)}^{\alpha 1}$, for $t,s\in (0,1)$;

(3)
$G(t,s)>0$, for $t,s\in (0,1)$;

(4)
${t}^{\alpha 1}(1t)s{(1s)}^{\alpha 1}\le \mathrm{\Gamma}(\alpha )G(t,s)\le (\alpha 1)(1t){t}^{\alpha 1}$, for $t,s\in (0,1)$.
Remark 2.1 Let $q(t)={t}^{\alpha 1}(1t)$, $k(s)=s{(1s)}^{\alpha 1}$. Then we have
Define a supremum norm $\parallel \cdot \parallel $ in $C(J,\mathbb{R})$ by
and a multiplication in $C(J,\mathbb{R})$ by
for $x,y\in C(J,\mathbb{R})$. Clearly $C(J,\mathbb{R})$ is a Banach algebra with respect to above norm and multiplication in it. By ${L}^{1}(J,\mathbb{R})$ denote the space of Lebesgue integrable realvalued functions on J equipped with the norm ${\parallel \cdot \parallel}_{{L}^{1}}$ defined by
The following fixed point theorem in Banach algebra due to Dhage [29] is fundamental in the proofs of our main results.
Lemma 2.5 ([29])
Let S be a nonempty, closed convex and bounded subset of the Banach algebra X and let $A:X\to X$ and $B:S\to X$ be two operators such that

(a)
A is Lipschitzian with a Lipschitz constant β,

(b)
B is completely continuous,

(c)
$x=AxBy\Rightarrow x\in S$ for all $y\in S$, and

(d)
$\beta M<1$, where $M=\parallel B(S)\parallel =sup\{\parallel B(x)\parallel :x\in S\}$.
Then the operator equation $AxBx=x$ has a solution in S.
3 Existence result
In this section, we prove the existence results for fractional differential equation (1.1) on the closed and bounded interval $J=[0,1]$ under both Lipschitz and Carathéodory conditions on the nonlinearities involved in it. We place the fractional differential equation (1.1) in the space $C(J,\mathbb{R})$ of continuous realvalued functions defined on J.
For convenience, we denote
We consider the following hypotheses in what follows.
(H_{1}) There exists a constant $\beta >0$ such that
for all $t\in J$ and $x,y\in \mathbb{R}$.
(H_{2}) There exists a function $h\in {L}^{1}(J,{\mathbb{R}}^{+})$ such that
for all $x\in \mathbb{R}$.
Theorem 3.1 Assume that hypotheses (H_{1}) and (H_{2}) hold. Further, if
then the boundary value problem (1.1) and (1.2) has a solution defined on J.
Proof Set $X=C(J,\mathbb{R})$ and define a subset S of X defined by
where $N=\frac{{F}_{0}T{\parallel h\parallel}_{{L}^{1}}}{1\beta T{\parallel h\parallel}_{{L}^{1}}}$ and ${F}_{0}={sup}_{t\in J}f(t,0)$.
Clearly S is a closed, convex, and bounded subset of the Banach space X. By Lemma 2.3, the boundary value problem (1.1) and (1.2) is equivalent to the equation
Define two operators $A:X\to X$ and $B:S\to X$ by
and
Then (3.3) is transformed into the operator equation:
We shall show that the operators A and B satisfy all the conditions of Lemma 2.5.
First, we show that A is a Lipschitz operator on X with the Lipschitz constant β. Let $x,y\in X$. Then by hypothesis (H_{1}),
for all $t\in J$. Taking the supremum over t, we obtain
for all $x,y\in X$.
Next, we show that B is a compact and continuous operator on S into X. First we show that B is continuous on S. Let $\{{x}_{n}\}$ be a sequence in S converging to a point $x\in S$. Then by the Lebesgue dominated convergence theorem,
for all $t\in J$. This shows that B is a continuous operator on S.
Next we show that B is a compact operator on S. It is enough to show that $B(S)$ is a uniformly bounded and equicontinuous set in X. On the one hand, let $x\in S$ be arbitrary. By Lemma 2.4, we have
for all $t\in J$. Taking the supremum over t,
for all $x\in S$. This shows that B is uniformly bounded on S.
On the other hand, given $\u03f5>0$, let
Then for any $x\in S$, ${t}_{1},{t}_{2}\in [0,1]$ with ${t}_{1}<{t}_{2}$, $0<{t}_{2}{t}_{1}<\delta $, we have
In order to estimate ${{t}_{2}}^{\alpha 1}{{t}_{1}}^{\alpha 1}$ and ${{t}_{2}}^{\alpha}{{t}_{1}}^{\alpha}$, we divide the proof into three cases.
Case 1: $0\le {t}_{1}<\delta $, ${t}_{2}<2\delta $.
Case 2: $0<{t}_{1}<{t}_{2}\le \delta $.
Case 3: $\delta \le {t}_{1}<{t}_{2}\le 1$.
Thus, we obtain
for all ${t}_{1},{t}_{2}\in J$ and for all $x\in S$. This shows that $B(S)$ is an equicontinuous set in X. Now the set $B(S)$ is uniformly bounded and equicontinuous set in X, so it is compact by the ArzelaAscoli Theorem. As a result, B is a completely continuous operator on S.
Next, we show that hypothesis (c) of Lemma 2.5 is satisfied. Let $x\in X$ and $y\in S$ be arbitrary such that $x=AxBy$. Then, by assumption (H_{1}), we have
Thus,
Taking the supremum over t,
This shows that hypothesis (c) of Lemma 2.5 is satisfied. Finally, we have
Therefore,
Thus, all the conditions of Lemma 2.5 are satisfied and hence the operator equation $AxBx=x$ has a solution in S. As a result, the boundary value problem (1.1) and (1.2) has a solution defined on J. This completes the proof. □
Remark 3.1 For the special case $f(t,x)\equiv 1$, we can find the corresponding existence results in Yu and Jiang (see [21]).
4 Examples
In this section, we will present two examples to illustrate the main results.
Example 4.1 Consider the boundary value problem
where $\alpha =\frac{5}{2}$.
Let $f(t,x)\equiv 1$, $g(t,x)=cosx$, $h(t)\equiv 1$. Then hypotheses (H_{1}) and (H_{2}) hold. Since
choosing $\beta =1$, then $\beta T{\parallel h\parallel}_{{L}^{1}}=8/35\sqrt{\pi}<1$. Therefore, the boundary value problem (4.1) and (4.2) has a solution.
Example 4.2 Consider the boundary value problem
where $\alpha =\frac{5}{2}$.
Let $f(t,x)=cosx+2$, $g(t,x)=sinx$, $h(t)\equiv 1$. Then hypotheses (H_{1}) and (H_{2}) hold. Since $T=8/35\sqrt{\pi}$, choosing $\beta =1$, then $\beta T{\parallel h\parallel}_{{L}^{1}}=8/35\sqrt{\pi}<1$. Therefore, the boundary value problem (4.3) and (4.4) has a solution.
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Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the National Natural Science Foundation of China (G61374065, G61374002) and the Research Fund for the Taishan Scholar Project of Shandong Province of China.
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Zhao, Y., Wang, Y. Existence of solutions to boundary value problem of a class of nonlinear fractional differential equations. Adv Differ Equ 2014, 174 (2014). https://doi.org/10.1186/168718472014174
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Keywords
 fractional differential equation
 boundary value problem
 fractional Green’s function
 fixed point theorem