Abstract
In this paper, we investigate the existence and uniqueness of symmetric solutions for fractional differential equations with multi-order fractional integral boundary conditions, by means of standard fixed point theorems. Examples which support our theoretical results are also presented.
Similar content being viewed by others
1 Introduction
In this paper, we study the existence and uniqueness of symmetric solutions for the following boundary value problem for nonlinear fractional differential equations with multi-order fractional integral boundary conditions:
where \({}^{c}D^{\alpha}\) denotes the Caputo fractional derivative of order α, x is symmetric (we recall that a function \(x\in C([0,T], {\mathbb{R}})\) is said to be symmetric on \([0,T]\) if \(x(t)=x(T-t)\), \(t\in[0,T]\)), \(f: [0,T]\times{\mathbb{R}}\to{\mathbb{R}}\) is a continuous function and symmetric with respect to t, that is, \(f(t,x)=f(T-t,x)\), \(\eta_{i}\in(0, T)\), \(\sigma,\lambda_{i}\in{\mathbb{R}}\), for all \(i=1,2,\ldots,m\) and \(I^{\beta_{i}}\) is the Riemann-Liouville fractional integral of order \(\beta_{i}>0\) (\(i=1,2,\ldots,m\)) such that
Fractional calculus has become very useful over the last years because of its many applications in almost all applied sciences. Fractional differential equations have been of great interest and it is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various science such as physics, mechanics, chemistry, and engineering. For details, and some recent results on the subject we refer to [1–17] and references cited therein.
Recently, many authors have focused on the existence of symmetric solutions for ordinary differential equation boundary value problems; for example, see [18–21] and the references therein. To the best of the authors’ knowledge there are no papers dealing with the existence of symmetric solutions for boundary value problems for fractional differential equations. The filling of this gap is the main motivation of this paper. Here we study existence and uniqueness results for symmetric solutions for boundary value problems of nonlinear fractional differential equations with multi-order fractional integral boundary conditions.
Note that the singular case can occur when the left side of (1.2) is equal to zero. For example if \(m=3\), \(\eta_{1}=\eta_{3}=T/2\), \(\eta_{2}=T\), \(\lambda_{1}=\lambda_{3}=1\), \(\lambda_{2}=-1\), \(\sigma=0\), and \(\beta_{i}=1\), \(i=1,2,3\), then the fractional integral condition of (1.1) is reduced to
which is equivalent to the symmetric condition \(x(t)=x(T-t)\). Therefore, the condition (1.2) provides the other ordinary/fractional integral boundary condition which is different from the regular symmetric condition. Now, there are two different conditions which are sufficient to give the existence and uniqueness results for the problem (1.1).
The organization of this paper is as follows: In Section 2 we present some preliminary notations, definitions and lemmas that we need in the sequel. In Section 3 we present the main existence and uniqueness results for the problem (1.1). Several new existence and uniqueness results are proved by using a variety of fixed point theorems (such as Banach’s contraction principle, nonlinear contractions, Krasnoselskii’s fixed point theorem, and Leray-Schauder’s nonlinear alternative). Examples illustrating the obtained results are presented in Section 4.
2 Preliminaries
In this section, we introduce some notations and definitions of fractional calculus [1, 2] and present preliminary results needed in our proofs later.
Definition 2.1
For an at least n-times differentiable function \(g : [0,\infty) \to\mathbb{R}\), the Caputo derivative of fractional order q is defined as
where \([q]\) denotes the integer part of the real number q.
Definition 2.2
The Riemann-Liouville fractional integral of order q is defined as
provided the integral exists.
Lemma 2.1
For \(q >0\), the general solution of the fractional differential equation \({}^{c}D^{q} u(t)=0\) is given by
where \(c_{i} \in\mathbb{R}\), \(i=1, 2,\ldots, n-1\) (\(n=[q]+1\)).
In view of Lemma 2.1, it follows that
for some \(c_{i} \in\mathbb{R}\), \(i=1, 2,\ldots, n-1\) (\(n=[q]+1\)).
For convenience we set
Lemma 2.2
Let \(\Omega_{2}\neq0\), \(1<\alpha\leq2\), \(\beta_{i}>0\), \(\eta_{i}\in(0,T)\), for \(i=1,2,\ldots,m\), and \(y\in C([0,T], {\mathbb{R}})\). Then the problem
has a unique solution given by
Proof
Using Lemma 2.1, (2.3) can be expressed as an equivalent integral equation
for arbitrary constants \(c_{1}, c_{2}\in\mathbb{R}\).
Taking the Riemann-Liouville fractional integral of order \(p>0\) for (2.6), we have
From the first condition of (2.4), it follows that
The second condition of (2.4) and (2.7) with \(p=\beta_{i}\) imply that
Putting a constant \(c_{1}\), we have
Substituting constants \(c_{1}\) and \(c_{2}\) into (2.6), we obtain (2.5) as required. □
Next we outline the fixed point theorems that will be used in the proofs of our existence and uniqueness results.
Definition 2.3
Let E be a Banach space and let \(F:E\rightarrow E\) be a mapping. F is said to be a nonlinear contraction if there exists a continuous nondecreasing function \(\Psi:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) such that \(\Psi(0)=0\) and \(\Psi(\varepsilon)<\varepsilon\) for all \(\varepsilon> 0\) with the property:
Lemma 2.3
(Boyd and Wong) [22]
Let E be a Banach space and let \(F:E\rightarrow E\) be a nonlinear contraction. Then F has a unique fixed point in E.
Lemma 2.4
(Krasnoselskii’s fixed point theorem) [23]
Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (a) \(Ax+By \in M\) whenever \(x, y \in M\); (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists \(z \in M\) such that \(z=Az+Bz\).
Lemma 2.5
(Nonlinear alternative for single valued maps) [24]
Let E be a Banach space, C a closed, convex subset of E, X an open subset of C and \(0\in X\). Suppose that \(F:\overline{X}\to C\) is a continuous, compact (that is, \(F(\overline {X})\) is a relatively compact subset of C) map. Then either
-
(i)
F has a fixed point in \(\overline{X}\), or
-
(ii)
there is a \(x\in\partial X\) (the boundary of X in C) and \(\lambda\in(0,1)\) with \(x=\lambda F(x)\).
3 Main results
Let \(\mathcal{C}=C([0, T],\mathbb{R})\) denotes the Banach space of all continuous functions from \([0, T]\) to \(\mathbb{R}\) endowed with the norm defined by \(\|x\|=\sup_{t\in[0, T]}|x(t)|\). Throughout this paper, for convenience, the expression \(I^{a} f(s,x(s))(b)\) means
where \(a\in\{\alpha,\alpha+\beta_{i}\}\) and \(b\in\{t,T,\eta_{i}\}\), \(i=1,2,\dots,m\).
As in Lemma 2.2, we define an operator \(\mathcal{F}:\mathcal {C}\rightarrow\mathcal{C}\) by
It should be noticed that the problem (1.1) has solutions if and only if the operator \(\mathcal{F}\) has fixed points.
In the following subsections we prove existence, as well as existence and uniqueness results, for the boundary value problem (1.1) by using a variety of fixed point theorems.
We set
and
3.1 Existence and uniqueness result via Banach’s fixed point theorem
The first existence and uniqueness result is based on Banach’s contraction mapping principle (Banach’s fixed point theorem).
Theorem 3.1
Assume that \(f:[0,T]\times\mathbb{R}\to\mathbb{R}\) is a symmetric continuous function and
- (H1):
-
there exists a constant \(L>0\) such that \(|f(t,x)-f(t,y)|\leq L|x-y|\), for each \(t\in[0, T]\) and \(x, y\in \mathbb{R}\).
If
where Λ is defined by (3.2), then the boundary value problem (1.1) has a unique symmetric solution on \([0, T]\).
Proof
We transform the problem (1.1) into a fixed point problem, \(x=\mathcal{F}x\), where the operator \(\mathcal {F}\) is defined as in (3.1). Observe that the fixed points of the operator \(\mathcal{F}\) are solutions of the problem (1.1). Applying Banach’s contraction mapping principle, we shall show that \(\mathcal{F}\) has a unique fixed point.
We let \(\sup_{t \in[0,T]}|f(t,0)|=M< \infty\) and choose
where the constant Φ is defined by (3.3).
Now, we show that \(\mathcal{F} B_{r} \subset B_{r}\), where \(B_{r}=\{x \in {\mathcal{C}}: \|x\|\le r \}\). For any \(x \in B_{r}\), we have
which implies that \(\mathcal{F}B_{r}\subset B_{r}\).
Next, we let \(x,y\in\mathcal{C}\). Then, for \(t\in[0,T]\), we have
which implies that \(\|\mathcal{F}x-\mathcal{F}y\|\leq L\Lambda\|x-y\|\). As \(L\Lambda<1\), \(\mathcal{F}\) is a contraction. Therefore, we deduce, by Banach’s contraction mapping principle, that \(\mathcal{F}\) has a fixed point which is the unique symmetric solution of the problem (1.1). The proof is completed. □
3.2 Existence and uniqueness result via Banach’s fixed point theorem and Hölder’s inequality
In this subsection we give another existence and uniqueness theorem for the boundary value problem (1.1) by using Banach’s fixed point theorem and Hölder’s inequality.
Theorem 3.2
Suppose that \(f: [0,T]\times{\mathbb{R}}\to{\mathbb{R}}\) is a symmetric continuous function satisfying the following assumption:
- (H2):
-
\(|f(t,x)-f(t,y)|\leq\delta(t)|x-y|\), for \(t\in[0, T]\), \(x, y\in\mathbb{R}\), and \(\delta\in L^{\frac{1}{\omega}}([0,T],\mathbb {R}^{+})\), \(\omega\in(0,1)\).
Denote \(\|\delta\| = (\int_{0}^{T}|\delta(s)|^{\frac {1}{\omega}}\,ds )^{\omega}\). If
then the boundary value problem (1.1) has a unique symmetric solution on \([0,T]\).
Proof
For \(x,y \in C([0,T],\mathbb{R})\) and for each \(t\in[0,T]\), by Hölder’s inequality, we have
It follows that \(\mathcal{F}\) is a contraction mapping. Hence Banach’s fixed point theorem implies that \(\mathcal{F}\) has a unique fixed point, which is the unique symmetric solution of the boundary value problem (1.1). The proof is completed. □
3.3 Existence and uniqueness result via nonlinear contractions
In this subsection we establish an existence and uniqueness result for the boundary value problem (1.1) by using Boyd and Wong’s fixed point theorem for nonlinear contractions (Lemma 2.3).
Theorem 3.3
Let \(f:[0,T]\times\mathbb{R}\rightarrow \mathbb{R}\) be a symmetric continuous function satisfying the assumption:
- (H3):
-
\(|f(t,x)-f(t,y)|\leq h(t)\frac {|x-y|}{H^{*}+|x-y|}\), \(t\in[0,T]\), \(x,y\geq0\), where \(h:[0,T]\rightarrow\mathbb{R}^{+}\) is continuous and a constant \(H^{*}\) defined by
$$ H^{*}= I^{\alpha}h(T)+ \biggl(\frac{|\Omega_{1}|+|\Omega_{2}|T}{|\Omega_{2}|T} \biggr)I^{\alpha}h(T)+\frac{1}{|\Omega_{2}|}\sum_{i=1}^{m}| \lambda_{i}|I^{\alpha +\beta_{i}}h(\eta_{i}). $$(3.5)
Then the boundary value problem (1.1) has a unique symmetric solution on \([0,T]\).
Proof
We define the operator \(\mathcal{F}:\mathcal{C}\rightarrow \mathcal{C}\) as in (3.1) and a continuous nondecreasing function \(\Psi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) by
Note that the function Ψ satisfies \(\Psi(0)=0\) and \(\Psi (\varepsilon)<\varepsilon\) for all \(\varepsilon>0\).
For any \(x,y \in\mathcal{C}\) and for each \(t\in[0,T]\), we have
This implies that \(\|\mathcal{F}x-\mathcal{F}y\|\leq\Psi(\|x-y\|)\). Therefore \(\mathcal{F}\) is a nonlinear contraction. Hence, by Lemma 2.3 the operator \(\mathcal{F}\) has a unique fixed point which is the unique symmetric solution of the problem (1.1). This completes the proof. □
3.4 Existence result via Krasnoselskii’s fixed point theorem
The next existence theorem is based on Krasnoselskii’s fixed point theorem (Lemma 2.4).
Theorem 3.4
Let \(f : [0,T]\times{\mathbb{R}} \to \mathbb{R}\) be a symmetric continuous function satisfying (H1). In addition we assume that:
- (H4):
-
\(|f(t,x)|\le\phi(t)\), \(\forall(t,x) \in[0,T] \times{\mathbb{R}}\), and \(\phi\in C([0,T], {\mathbb{R}}^{+})\).
Then the boundary value problem (1.1) has at least one symmetric solution on \([0,T]\) provided
Proof
Setting \(\sup_{t\in[0, T]}|\varphi(t)|=\|\varphi\| \) and choosing
(where Λ and Φ are defined by (3.2) and (3.3), respectively), we consider \(B_{\rho}=\{ x\in\mathcal{C}([0, T], \mathbb{R}):\|x\|\leq\rho\}\). We define the operators \(\mathcal{F}_{1}\) and \(\mathcal{F}_{2}\) on \(B_{\rho}\) by
For any \(x,y\in B_{\rho}\), we have
This shows that \(\mathcal{F}_{1}x+\mathcal{F}_{2}y\in B_{\rho}\). It is easy to see using (3.6) that \(\mathcal{F}_{2}\) is a contraction mapping.
Continuity of f implies that the operator \(\mathcal{F}_{1}\) is continuous. Also, \(\mathcal{F}_{1}\) is uniformly bounded on \(B_{\rho}\) as
Now we prove the compactness of the operator \(\mathcal{F}_{1}\).
We define \(\sup_{(t,x) \in[0,T] \times B_{\rho}}|f(t,x)|=\overline{f}< \infty\), and consequently we have
which is independent of x and tends to zero as \(t_{2}-t_{2}\to0\). Thus, \(\mathcal{F}_{1}\) is equicontinuous. So \(\mathcal{F}_{1}\) is relatively compact on \(B_{\rho}\). Hence, by Arzelá-Ascoli’s theorem, \(\mathcal{F}_{1}\) is compact on \(B_{\rho}\). Thus all the assumptions of Lemma 2.4 are satisfied. So the conclusion of Lemma 2.4 implies that the boundary value problem (1.1) has at least one symmetric solution on \([0,T]\). □
3.5 Existence result via Leray-Schauder’s nonlinear alternative
By using Leray-Schauder’s nonlinear alternative (Lemma 2.5) we give in this subsection our last existence theorem.
Theorem 3.5
Assume that:
- (H5):
-
there exist a continuous nondecreasing function \(\psi: [0,\infty) \to (0,\infty)\) and a function \(p \in C([0,T],\mathbb{R}^{+})\) such that
$$\bigl|f(t,x)\bigr|\le p(t)\psi\bigl(|x|\bigr) \quad \textit{for each } (t,x) \in[0,T] \times \mathbb{R}; $$ - (H6):
-
there exists a constant \(M>0\) such that
$$\begin{aligned} \frac{M}{\psi(M)\|p\|\Lambda+\Phi}> 1, \end{aligned}$$
Then the boundary value problem (1.1) has at least one symmetric solution on \([0,T]\).
Proof
Let the operator \(\mathcal{F}\) be defined by (3.1). Firstly, we shall show that \(\mathcal{F}\) maps bounded sets (balls) into bounded sets in \(C([0,T], \mathbb{R})\). For a number \(r>0\), let \(B_{r} = \{x \in C([0,T], \mathbb{R}): \|x\| \le r \}\) be a bounded ball in \(C([0,T], \mathbb{R})\). Then for \(t\in [0,T]\) we have
and consequently,
Next we will show that \(\mathcal{F}\) maps bounded sets into equicontinuous sets of \(C([0,T], \mathbb{R})\) . Let \(\tau_{1}, \tau_{2} \in[0,T]\) with \(\tau_{1}< \tau_{2}\) and \(u \in B_{r}\). Then we have
As \(\tau_{2}-\tau_{1}\rightarrow0\), the right-hand side of the above inequality tends to zero independently of \(x\in B_{r}\). Therefore by Arzelá-Ascoli’s theorem the operator \(\mathcal{F}: C([0,T], \mathbb {R}) \to C([0,T], \mathbb{R})\) is completely continuous.
Let x be a solution. Then, for \(t\in[0,T]\), and the following similar computations to the first step, we have
which leads to
In view of (H6), there exists M such that \(\|x\|\ne M\). Let us set
We see that the operator \(\mathcal{F}:\overline{X}\rightarrow C([0, T], \mathbb{R})\) is continuous and completely continuous. From the choice of X, there is no \(x\in\partial X\) such that \(x=\nu\mathcal{F}x\) for some \(\nu\in(0,1)\). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.5), we deduce that \(\mathcal{F}\) has a fixed point \(x\in\overline{X}\) which is a symmetric solution of the boundary value problem (1.1). This completes the proof. □
4 Examples
In this section, we present some examples to illustrate our results.
Example 4.1
Consider the following nonlinear fractional differential equation with multi-order fractional integral conditions:
Here \(\alpha=3/2\), \(T=4\), \(m=5\), \(\sigma=2\), \(\beta_{1}=1/2\), \(\beta _{2}=\sqrt{3}\), \(\beta_{3}=1/2\), \(\beta_{4}=1/3\), \(\beta_{5}=\sqrt{2}\), \(\lambda _{1}=1/5\), \(\lambda_{2}=1/3\), \(\lambda_{3}=2/3\), \(\lambda_{4}=\sqrt{2}\), \(\lambda_{5}=1/5\), \(\eta_{1}=1/2\), \(\eta_{2}=1\), \(\eta_{3}=3/2\), \(\eta_{4}=1\), \(\eta_{5}=1/2\), and \(f(t,x)=(\sin ((t-2)^{2})/19)((|x|/(3+|x|))+1)|x|+(3/4)\). Since \(|f(t,x)-f(t,y)| \leq (4/57)|x-y|\), then (H1) is satisfied with \(L=4/57\). We can show that \(\Omega_{2}=2.934752823\neq0\) and
Thus \(L\Lambda=0.9648829050<1\). Hence, by Theorem 3.1, the problem (4.1) has a unique symmetric solution on \([0,4]\).
Example 4.2
Consider the following nonlinear fractional differential equation with multi-order fractional integral conditions:
Here \(\alpha=3/2\), \(T=1\), \(m=5\), \(\sigma=3/2\), \(\beta_{1}=1/9\), \(\beta _{2}=1/5\), \(\beta_{3}=1/7\), \(\beta_{4}=1/2\), \(\beta_{5}=1/8\), \(\lambda_{1}=1/8\), \(\lambda_{2}=1/34\), \(\lambda_{3}=1/37\), \(\lambda_{4}=1/49\), \(\lambda_{5}=1/25\), \(\eta_{1}=1/2\), \(\eta_{2}=1/3\), \(\eta_{3}=1/5\), \(\eta_{4}=2/7\), \(\eta_{5}=\pi /11\), and \(f(t,x)=(e^{((1/2)-t)^{2}}/18)(|x|/(1+|x|))+(1/2)\). We choose \(h(t) = e^{t^{2}}/9\) and we obtain \(\Omega_{2}=0.2195131282\neq0\),
Clearly,
Hence, by Theorem 3.3, the problem (4.2) has a unique symmetric solution on \([0,1]\).
Example 4.3
Consider the following nonlinear fractional differential equation with multi-order fractional integral conditions:
Here \(\alpha=3/2\), \(T=1\), \(m=5\), \(\sigma=3\), \(\beta_{1}=1/2\), \(\beta _{2}=1/3\), \(\beta_{3}=1/4\), \(\beta_{4}=1/3\), \(\beta_{5}=1/3\), \(\lambda _{1}=1/199\), \(\lambda_{2}=1/256\), \(\lambda_{3}=1/10\), \(\lambda_{4}=1/189\), \(\lambda_{5}=1/191\), \(\eta_{1}=1/3\), \(\eta_{2}=2/3\), \(\eta_{3}=1/5\), \(\eta _{4}=1/4\), \(\eta_{5}=1/6\), and \(f(t,x)=(e^{-(t-(1/2))^{2}}\sin ((t-(1/2))^{2})/2)(|x|/(1+|x|))+2t(1-t)\). Since \(|f(t,x)-f(t,y)| \leq (1/2)|x-y|\), (H1) is satisfied with \(L=1/2\). We find that \(\Omega _{2}=0.08783388964\neq0\),
Clearly,
Hence, by Theorem 3.4, the problem (4.3) has at least one symmetric solution on \([0,1]\).
Example 4.4
Consider the following nonlinear fractional differential equation with multi-order fractional integral conditions:
Here \(\alpha=4/3\), \(T=6\), \(m=5\), \(\sigma=1/225\), \(\beta_{1}=1/2\), \(\beta _{2}=1/\sqrt{2}\), \(\beta_{3}=1/5\), \(\beta_{4}=1/3\), \(\beta_{5}=1/\sqrt{3}\), \(\lambda_{1}=1/2\), \(\lambda_{2}=1/3\), \(\lambda_{3}=3/11\), \(\lambda_{4}=1/7\), \(\lambda_{5}=1/5\), \(\eta_{1}=1/2\), \(\eta_{2}=1/4\), \(\eta_{3}=2/7\), \(\eta _{4}=1/5\), \(\eta_{5}=1/2\), and \(f(t,x)=(1/225)(t-3)^{2}((x^{2}/(1+|x|))+((|x|+2)/(3+|x|)))\). Then, we get \(\Omega_{2}=1.011549229\neq0\),
and
Clearly,
Choosing \(p(t) = (1/225)(t-3)^{2}\) and \(\psi(|x|) = |x| + 1\), we can show that
implies that \(M>3.099548142\). Hence, by Theorem 3.5, the problem (4.3) has at least one symmetric solution on \([0,6]\).
References
Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)
Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon (1993)
Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Sabatier, J, Agrawal, OP, Machado, JAT (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)
Lakshmikantham, V, Leela, S, Vasundhara Devi, J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009)
Agarwal, RP, Zhou, Y, He, Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59, 1095-1100 (2010)
Ahmad, B, Nieto, JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011, Article ID 36 (2011)
Ahmad, B, Nieto, JJ: Boundary value problems for a class of sequential integrodifferential equations of fractional order. J. Funct. Spaces Appl. 2013, Article ID 149659 (2013)
Ahmad, B, Ntouyas, SK, Alsaedi, A: New existence results for nonlinear fractional differential equations with three-point integral boundary conditions. Adv. Differ. Equ. 2011, Article ID 107384 (2011)
Ahmad, B, Ntouyas, SK: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 15, 362-382 (2012)
Ahmad, B, Ntouyas, SK, Alsaedi, A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013, Article ID 320415 (2013)
Ahmad, B, Ntouyas, SK: Nonlocal fractional boundary value problems with slit-strips integral boundary conditions. Fract. Calc. Appl. Anal. 18, 261-280 (2015)
Choudhary, S, Daftardar-Gejji, V: Nonlinear multi-order fractional differential equations with periodic/anti-periodic boundary conditions. Fract. Calc. Appl. Anal. 17, 333-347 (2014)
Liu, X, Jia, M, Ge, W: Multiple solutions of a p-Laplacian model involving a fractional derivative. Adv. Differ. Equ. 2013, Article ID 126 (2013)
O’Regan, D, Stanek, S: Fractional boundary value problems with singularities in space variables. Nonlinear Dyn. 71, 641-652 (2013)
Zhang, L, Ahmad, B, Wang, G, Agarwal, RP: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 249, 51-56 (2013)
Sun, Y, Zhang, X: Existence of symmetric positive solutions for an m-point boundary value problem. Bound. Value Probl. 2007, Article ID 79090 (2007)
Kosmatov, N: Symmetric solutions of a multi-point boundary value problem. J. Math. Anal. Appl. 309, 25-36 (2005)
Zhao, J, Miao, C, Ge, W, Zhang, J: Multiple symmetric positive solutions to a new kind of four point boundary value problem. Nonlinear Anal. 71, 9-18 (2009)
Pang, H, Tong, Y: Symmetric positive solutions to a second-order boundary value problem with integral boundary conditions. Bound. Value Probl. 2013, Article ID 150 (2013)
Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458-464 (1969)
Krasnoselskii, MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 10, 123-127 (1955)
Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2003)
Acknowledgements
The research of J Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in this article. They read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Aphithana, A., Ntouyas, S.K. & Tariboon, J. Existence and uniqueness of symmetric solutions for fractional differential equations with multi-order fractional integral conditions. Bound Value Probl 2015, 68 (2015). https://doi.org/10.1186/s13661-015-0329-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-015-0329-1