Abstract
The fixed point theorems on cones are used to investigate the existence of positive solution for higher order fractional differential equation with multipoint boundary condition.
MSC: 26A33; 34B15.
Keywords:
fractional differential equation; fixed point; positive solution; cone1 Introduction
Recently, much attention has been paid to the fractional differential equations due to its wide application in physics, engineering, economics, aerodynamics, and polymer rheology etc. For the basic theory and development of the subject, we refer some contributions on fractional calculus, fractional differential equations, see Delbosco [1], Miller [2], and Lakshmikantham et al. [37]. Especially, there have been some articles dealing with the existence of solutions or positive solutions of boundaryvalue problems for nonlinear fractional differential equations (see [820] and references along this line). For examples, Jiang [16] obtained the existence of positive solution for boundary value problem of fractional differential equation
where denotes the standard RiemannLiouville fractional order derivative.
Agarwal et al. [17] investigated the existence of positive solution of singular problem
where 1 < α < 2, 0 ≤ μ ≤ α  1 and f satisfies the Caratheodory conditions on [0,1] × [0, ∞) × R and f(t, x, y) is singular at x = 0. The existence results of positive solutions are established by using regularization and sequential techniques.
As to the nonlocal problem, Bai [18] established the existence of positive solution for threepoint boundary value problem of fractional differential equation
By using the fixed point theorems on cones, Li et al. [19] established the existence of positive solutions for problem
where 1 < α ≤ 2, 0 ≤ β ≤ 1, 0 ≤ a ≤ 1, ξ ∈ (0, 1) and aξ∈^{αβ}^{2 }≤ 1  β, 0 ≤ α  β  1 and f : [0,1] × [0, ∞) → [0, ∞) satisfies Caratheodory type conditions.
Very recently, Moustafa and Nieto [20] considered the nontrivial solution for following higher order multipoint problem
where n ≥ 2, 0 < η_{i} < 1, β_{i} > 0, i = 1, 2, . . . , m  2, , f ∈ C([0,1] × R, R). The existence of nontrivial solution was established by using the nonlinear alternative of LeraySchauder. But, existence of positive solution for problem (1.1), (1.2), as far as we know, has not been considered before. Considering that problem (1.1) and (1.2) are more general than problems studied before, we believe that it is interesting to investigate the existence of positive solution for this problem.
In this article, we consider the existence and multiplicity of positive solutions for problem (1.1) and (1.2). We obtain some properties of the associated Green's function. By using these properties of Green's function and fixed point theorems on cones, we establish the existence and multiplicity of positive solutions.
2 Preliminaries
For the convenience of the reader, we present here the basic definitions and theory from fractional calculus theory. These definitions and theory can be founded in the literature [1].
Definition 2.1 The fractional integral of order α > 0 of a function u(t): (0, ∞) → R is given by
provided the right side is pointwise defined on (0, ∞).
Definition 2.2 The fractional derivative of order α > 0 of a continuous function u(t): (0, ∞) → R is given by
where n = [α] + 1, provided that the right side is pointwise defined on (0, ∞).
Lemma 2.1 Let α > 0. If we assume u ∈ C(0, 1) ∪ L(0, 1), then problem has solution
for some C_{i }∈ R, i = 1, 2, . . . , N, where N is the smallest integer greater than or equal to α.
Lemma 2.2 Assume that u ∈ C(0, 1) ∪ L(0, 1) with a fractional derivative of order α > 0 that belongs to C(0, 1) ∪ L(0, 1). Then
for some C_{i }∈ R, i = 1, 2, . . . , N.
Lemma 2.3 [21] Let E be a Banach space and K ⊂ E be a cone. Assume Ω_{1}, Ω_{2} are open bounded subsets of E with , and let
be a completely continuous operator such that
Let 0 < a < b be given and let ψ be a nonnegative continuous concave functional on the cone C. Define the convex sets C_{r }and C(ψ , a, b) by
Lemma 2.4 [22] Let be a completely continuous operator and let ψ be a nonnegative continuous concave functional on C such that ψ(u) ≤ u for all . Suppose that there exist 0 < a < b < d ≤ c such that
(S_{1}) {u ∈ C(ψ , b, d) ψ(u) > b}≠ ∅ and ψ(Tu) > b for u ∈ C(ψ , b, d),
(S_{2}) Tu < a for u ≤ a and
(S_{3}) ψ(Tu) > b for u ∈ C(ψ, b, c) with Tu ≥ d.
Then T has at least three fixed points u_{1}, u_{2}, and u_{3} such that
Lemma 2.5 Denote η_{0} = 0, η_{m}_{1 }= 1 and β_{0} = β_{m}_{1 }= 0. Given y(t) ∈ C[0,1]. The problem
is equivalent to
where
Furthermore, the function G(t, s) is continuous on [0,1] × [0,1] and satisfies the condition
Proof. From Lemma 2.1, we get that problem (3.1) is equivalent to
The boundary conditions u(0) = u'(0) = ... = u(n2) = 0 induce that C_{2} = C_{3} = ... = C_{n }= 0.. Considering the boundary condition , we get
Then for η_{i}_{1 }< t < η_{i}, i = 1, 2,. . . , m  1,
Furthermore, for η_{i}_{1 }≤ s ≤ η_{i}, i = 1, 2, . . . , m1 and t ≤ s
For η_{i}_{1 }≤ s ≤ η_{i}, i = 1, 2, . . . , m1 and t ≥ s
□
Lemma 2.6 The function G(t, s) satisfies the following conditions:
(1) G(t, s) ≤ G(s, s), t, s ∈ [0, 1],
(2) There exists function γ(s) such that .
Proof (1) For η_{i}_{1 }< s < η_{i}, i = 1, 2, . . . , m1, Denote
The facts that
imply that g_{1}(t, s) is decreasing with respect to t for [η_{i}_{1}, s] and g_{2} (t, s) is increasing with respect to t for [s, η_{i}], i = 1, 2, . . . , m  1. Thus one can easily check that
(2) For η_{m}_{2 }< t < 1, denote
Thus we have
3 Main results
Let X = C[0,1] be a Banach space endowed with the norm
Define the cone P ⊂ E by P = {u ∈ X  u(t) ≥ 0}.
Theorem 3.1 Define the operator T : P → X,
then T : P → P is completely continuous.
Proof From the nonnegative and continuous properties of function f and G(t, s), one can obtain easily that the operator T : P → P and T is continuous. Let Ω be a bounded subset of cone P. That is, there exists a positive constant M_{1} > 0 such that u ≤ M_{1} for all u ∈ Ω. Thus for each u ∈ Ω, t_{1}, t_{2} ∈ [0,1], one has
Then the continuity of function G(t, s) implies that T is equicontinuity on the bounded subset of P . On the other hand, for u ∈ Ω, there exist constant M_{2}> 0 such that
Then
which implies that T is uniformly bounded on the bounded subset of P . Then an application of AscoliArezela ensures that T : P → P is completely continuous.
Theorem 3.2 Assume that there exist two positive constant such that
(A1) f(t, u) ≤ Mr_{2}, (t, u) ∈ [0, 1] × [0, r_{2}]
(A2) f(t, u) ≥ Nr_{1}, (t, u) ∈ [0, 1] × [0, r_{1}]
where
Then problem (1.1) and (1.2) has at least one positive solution u such that r_{1} ≤ u ≤ r_{2}.
Proof Let Ω_{2} = {u ∈ P  u ≤ r_{2}}. For u ∈ ∂Ω_{2}, considering assumption (A1), we have
Thus Tu ≤ u, u ∈ ∂Ω_{2}.
Let Ω_{1} = {u ∈ P  u ≤ r_{1}}. For u ∈ ∂Ω_{1}, considering assumption (A2), we have
Thus for t ∈ [η_{m}_{2}, 1], we get
which gives that Tu ≥ u, u ∈ ∂Ω_{1}. An application of Lemma (2.5) ensures the existence of positive solution u(t) of problem (1.1) and (1.2).
Theorem 3.3 Suppose that there exist constants 0 < a < b < c such that
(A3) f(t, u) < Ma, for (t, u) ∈ [0,1] × [0, a],
(A4) f(t, u) ≥ Nb, for (t, u) ∈ [η_{m}_{2}, 1] × [b, c],
(A5) f(t, u) ≤ Mc, for (t, u) ∈ [0,1] × [0, c],
then problem (1.1) and (1.2) has at least three positive solution u_{1}, u_{2}, u_{3} with
Proof Let the nonnegative continuous concave functional θ on the cone P defined by
If , then u ≤ c. Then by condition (A5), we have
Thus
which yields that . In the same way, we get that
We chose the function . We claim that , which ensures that {u ∈ P (θ, b, c)θ(u) > b} ≠ ∅. And for u ∈ P (θ, b, c), we have
Then
which yields that θ(Tu) > b, for u ∈ P (θ, b, c).
An application of Lemma (2.6) ensures that problem (1.1) and (1.2) has at least three positive solutions with
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
This study was supported by the Anhui Provincial Natural Science Foundation (10040606Q50) and the Natural Science Foundation of Anhui Department of Education (KJ2010A285).
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