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Existence results for higher order fractional differential equation with multi-point boundary condition

Yang Liu

Author Affiliations

Department of Mathematics, Anhui Normal University, Hefei, Anhui, P. R. China

Boundary Value Problems 2012, 2012:57  doi:10.1186/1687-2770-2012-57

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/57


Received:23 October 2011
Accepted:15 May 2012
Published:15 May 2012

© 2012 Yang; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The fixed point theorems on cones are used to investigate the existence of positive solution for higher order fractional differential equation with multi-point boundary condition.

MSC: 26A33; 34B15.

Keywords:
fractional differential equation; fixed point; positive solution; cone

1 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. [3-7]. Especially, there have been some articles dealing with the existence of solutions or positive solutions of boundary-value problems for nonlinear fractional differential equations (see [8-20] and references along this line). For examples, Jiang [16] obtained the existence of positive solution for boundary value problem of fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M2">View MathML</a> denotes the standard Riemann-Liouville fractional order derivative.

Agarwal et al. [17] investigated the existence of positive solution of singular problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M3">View MathML</a>

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 three-point boundary value problem of fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M4">View MathML</a>

By using the fixed point theorems on cones, Li et al. [19] established the existence of positive solutions for problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M5">View MathML</a>

where 1 < α ≤ 2, 0 ≤ β ≤ 1, 0 ≤ a ≤ 1, ξ ∈ (0, 1) and aξ∈α-β-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 multi-point problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M6">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M7">View MathML</a>

(1.2)

where n ≥ 2, 0 < ηi < 1, βi > 0, i = 1, 2, . . . , m - 2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M8">View MathML</a>, f C([0,1] × R, R). The existence of nontrivial solution was established by using the nonlinear alternative of Leray-Schauder. 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M9">View MathML</a>

provided the right side is point-wise defined on (0, ∞).

Definition 2.2 The fractional derivative of order α > 0 of a continuous function u(t): (0, ∞) → R is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M10">View MathML</a>

where n = [α] + 1, provided that the right side is point-wise defined on (0, ∞).

Lemma 2.1 Let α > 0. If we assume u C(0, 1) ∪ L(0, 1), then problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M11">View MathML</a> has solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M12">View MathML</a>

for some Ci 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M13">View MathML</a>

for some Ci R, i = 1, 2, . . . , N.

Lemma 2.3 [21] Let E be a Banach space and K E be a cone. Assume Ω1, Ω2 are open bounded subsets of E with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M14">View MathML</a>, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M15">View MathML</a>

be a completely continuous operator such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M16">View MathML</a>

then A has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M17">View MathML</a>.

Let 0 < a < b be given and let ψ be a nonnegative continuous concave functional on the cone C. Define the convex sets Cr and C(ψ , a, b) by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M18">View MathML</a>

Lemma 2.4 [22] Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M19">View MathML</a> be a completely continuous operator and let ψ be a nonnegative continuous concave functional on C such that ψ(u) ≤ ||u|| for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M20">View MathML</a>. Suppose that there exist 0 < a < b < d c such that

(S1) {u C(ψ , b, d)| ψ(u) > b}≠ ∅ and ψ(Tu) > b for u C(ψ , b, d),

(S2) ||Tu|| < a for ||u|| ≤ a and

(S3) ψ(Tu) > b for u C(ψ, b, c) with ||Tu|| ≥ d.

Then T has at least three fixed points u1, u2, and u3 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M21">View MathML</a>

Lemma 2.5 Denote η0 = 0, ηm-1 = 1 and β0 = βm-1 = 0. Given y(t) ∈ C[0,1]. The problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M22">View MathML</a>

(3.1)

is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M23">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M24">View MathML</a>

Furthermore, the function G(t, s) is continuous on [0,1] × [0,1] and satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M25">View MathML</a>

Proof. From Lemma 2.1, we get that problem (3.1) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M26">View MathML</a>

The boundary conditions u(0) = u'(0) = ... = u(n-2) = 0 induce that C2 = C3 = ... = Cn = 0.. Considering the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M27">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M28">View MathML</a>

Then for ηi-1 < t < ηi, i = 1, 2,. . . , m - 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M29">View MathML</a>

Furthermore, for ηi-1 s ηi, i = 1, 2, . . . , m-1 and t s

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M30">View MathML</a>

For ηi-1 s ≤ ηi, i = 1, 2, . . . , m-1 and t s

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M31">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M32">View MathML</a>.

Proof (1) For ηi-1 < s < ηi, i = 1, 2, . . . , m-1, Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M33">View MathML</a>

The facts that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M34">View MathML</a>

imply that g1(t, s) is decreasing with respect to t for [ηi-1, s] and g2 (t, s) is increasing with respect to t for [s, ηi], i = 1, 2, . . . , m - 1. Thus one can easily check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M35">View MathML</a>

(2) For ηm-2 < t < 1, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M36">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M59">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M37">View MathML</a>

3 Main results

Let X = C[0,1] be a Banach space endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M38">View MathML</a>

Define the cone P E by P = {u X | u(t) ≥ 0}.

Theorem 3.1 Define the operator T : P X,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M39">View MathML</a>

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 M1 > 0 such that ||u|| ≤ M1 for all u ∈ Ω. Thus for each u ∈ Ω, t1, t2 ∈ [0,1], one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M40">View MathML</a>

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 M2> 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M41">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M42">View MathML</a>

which implies that T is uniformly bounded on the bounded subset of P . Then an application of Ascoli-Arezela ensures that T : P P is completely continuous.

Theorem 3.2 Assume that there exist two positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M43">View MathML</a> such that

(A1) f(t, u) ≤ Mr2, (t, u) ∈ [0, 1] × [0, r2]

(A2) f(t, u) ≥ Nr1, (t, u) ∈ [0, 1] × [0, r1]

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M44">View MathML</a>

Then problem (1.1) and (1.2) has at least one positive solution u such that r1 ≤ ||u|| ≤ r2.

Proof Let Ω2 = {u P | ||u|| ≤ r2}. For u ∈ ∂Ω2, considering assumption (A1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M45">View MathML</a>

Thus ||Tu|| ≤ ||u||, u ∈ ∂Ω2.

Let Ω1 = {u P | ||u|| ≤ r1}. For u ∈ ∂Ω1, considering assumption (A2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M46">View MathML</a>

Thus for t ∈ [ηm-2, 1], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M47">View MathML</a>

which gives that ||Tu|| ≥ ||u||, u ∈ ∂Ω1. 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) ∈ [ηm-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 u1, u2, u3 with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M48">View MathML</a>

Proof Let the nonnegative continuous concave functional θ on the cone P defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M49">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M50">View MathML</a>, then ||u|| ≤ c. Then by condition (A5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M51">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M52">View MathML</a>

which yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M53">View MathML</a>. In the same way, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M60">View MathML</a>

We chose the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M54">View MathML</a>. We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M55">View MathML</a>, which ensures that {u P (θ, b, c)|θ(u) > b} ≠ ∅. And for u P (θ, b, c), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M56">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M57">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/57/mathml/M58">View MathML</a>

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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