非线性分数阶微分方程边值问题三个正解的存在性
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α D0+ u(t) + λh(t)f (u(t)) = 0.
The proof is complete. Definition 2.3[6] If P is a cone of the real Banach space E , a mapping α : P → [0, ∞), which is continuous and α(tx + (1 − t)y ) ≥ tα(x) + (1 − t)α(y ), x, y ∈ P, t ∈ [0, 1], (8)
Project supported by the Natural Science Foundation of Shanxi Province (2008011002-1) and Shanxi Datong University Institute (2009-Y-15; 2010-B-01) and the Development Foundation of Higher Education Department of Shanxi Province (20101109; 20111020). † Manuscript received February 5, 2012; Revised March 21, 2012
1
Introduction
The existence of positive solutions to fractional differential equations has been studied extensively ([1-5]). Bai and Lv [6] studied the following two-point boundary value problem of fractional differential equations
α α D0+ u(t) = D0+ λ
(1 − s)α−3 h(s)f (u(s))ds − Γ(α) 0 α α = 0 − λD0+ I0+ h(t)f (u(t)) = −λh(t)f (u(t)). t
1 α−1
t 0
(t − s)α−1 h(s)f (u(s))ds Γ(α)
It shows that
α D0+ f (x) =
1 d Γ(n − α) dx
n 0
x
f (t) dt, (x − t)α−n+1
is called the Riemann-Liouville fractional derivative of order α, where n = [α]+1, [α] denotes the integer part of number α. Lemma 2.1[9] If y (t) ∈ C [0, 1] and y (t) ≥ 0. Then fractional boundary value problem
2
Preliminaries
For convenience, we demonstrate the definitions and lemmas that is useful in proving our main results. Definition 2.1[8] The integral x 1 f (t) α dt, x > 0, I0+ f (x) = Γ(α) 0 (x − t)1−α is called the Riemann-Liouville fractional integral of order α, where α > 0. Definition 2.2[8] For a function f (x) given in the interval [0, ∞), the expression
1
u(t) = λ
0
G(t, s)h(s)f (u(s))ds.
(6)
398
ANN. OF DIFF. EQS.
Vol.28
Proof Let u ∈ P is a solution to the boundary value problem (1) and (2). Applying the method of proving Lemma 2.1 used in [9], we can obtain that u is a solution to (6). Conversely, letting u be a solution to (6), we can obtain
Ann. of Diff. Eqs. 28:4(2012), 396-403
EXISTENCE OF TRIPLE POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION DEPENDING ON A PARAMETER ∗†
1
u(t) = λ
0
G(t, s)h(s)f (u(s))ds
t α−1
=λ
0
t
(1 − s)α−3 − (t − s)α−1 h(s)f (u(s))ds + Γ(α) (1 − s)α−3 h(s)f (u(s))ds − Γ(α)
1 0 t 0
1 α−1
t
t
(1 − s)α−3 h(s)f (u(s))ds Γ(α) (7)
α D0+ u(t) + a(t)f (t, u(t)) = 0,
0 < t < 1, 1 < α ≤ 2,
u(0) = u(1) = 0, and obtained the existence of positive solutions by means of the Krasnoselskii fixed-point theorem and the Leggett-Williams fixed-point theorem. El-Shahed [7] considered the following nonlinear fractional boundary value problem
1 α−1
=λ
0
t
(t − s)α−1 h(s)f (u(s))ds , Γ(α)
t 0
u (t) = λ (α − 1)tα−2
(1 − s)α−3 h(s)f (u(s))ds − (α − 1) Γ(α)
1 0 t
(t − s)α−2 h(s)f (u(s))ds , Γ(α)
u (t) = λ (α − 1)(α − 2)tα−3 −(α − 1)(α − 2)
Abstract
In this paper, we establish the existence of three positive solutions to a nonlinear fractional differential equation by the Leggett-Williams fixed point theorem. Keywords boundary value problem; fractional differential equation; positive solution; cone; fixed point theorem 2000 Mathematics Subject Classification 34K13
α D0+ u(t) + y (t) = 0,
0 < t < 1, 3 < α ≤ 4,
(3) (4)
Baidu Nhomakorabea
u(0) = u (0) = u (0) = u (1) = 0 has a unique positive solution
1
u(t) =
0
G(t, s)y (s)ds,
α−1 t (1 − s)α−3 − (t − s)α−1 , 0 ≤ s ≤ t ≤ 1; Γ(α) G(t, s) = (5) tα−1 (1 − s)α−3 , 0 ≤ t ≤ s ≤ 1. Γ(α) Here G(t, s) is called the Green function of the boundary value problem (3) and (4). Lemma 2.2 Assume that f : [0, ∞] → [0, ∞] is continuous. A function u ∈ P is a solution to the boundary value problem (1) and (2) if and only if it is a solution to the integral equation where
α D0+ u(t) + λa(t)f (t, u(t)) = 0,
0 < t < 1, 2 < α ≤ 3,
u(0) = u (0) = u (1) = 0, and used the Krasnoselskii fixed-point theorem on cone expansion and compression to show the existence and non-existence of positive solutions to the above fractional differential equation boundary value problem. In this article, we are concerned with the existence of three positive solutions to the following nonlinear fractional differential equation boundary value problem:
0
(1 − s)α−3 h(s)f (u(s))ds Γ(α)
(t − s)α−3 h(s)f (u(s))ds . Γ(α) u(0) = u (0) = u (0) = u (1) = 0.
We can verify easily that
Derivativing both sides of equation (7) gets
Shugui Kang1 , Yan Li2 , Jianmin Guo1
(1. Applied of Math. Institute, Shanxi Datong University, Datong 037009, Shanxi; 2. Dept. of Math., Shanxi Normal University, Linfen 041000, Shanxi, E-mail: dtkangshugui@126.com (S. Kang))
∗
396
No.4
S.G. Kang, etc., TRIPLE POSITIVE SOLUTIONS
α D0+ u(t) + λh(t)f (u(t)) = 0,
397
(1) (2)
0 < t < 1, 3 < α ≤ 4,
u(0) = u (0) = u (0) = u (1) = 0,
α where λ is a positive parameter and D0+ is the standard Riemann-Liouville fractional deriva1 tive. h : (0, 1) → (0, ∞) is continuous with 0 h(t)dt > 0, and f : [0, ∞) → [0, ∞) is continuous. Based on the Leggett-Williams fixed point theorem, we obtain sufficient conditions for the existence of three positive solutions to the fractional boundary value problem (1) and (2). The paper is organized as follows. In Section 2, we demonstrate basic definitions and present some lemmas in order to prove our main results. In Section 3, we establish some results for the existence of three positive solutions to problem (1) and (2). In Section 4, we present an example to illustrate our results.
The proof is complete. Definition 2.3[6] If P is a cone of the real Banach space E , a mapping α : P → [0, ∞), which is continuous and α(tx + (1 − t)y ) ≥ tα(x) + (1 − t)α(y ), x, y ∈ P, t ∈ [0, 1], (8)
Project supported by the Natural Science Foundation of Shanxi Province (2008011002-1) and Shanxi Datong University Institute (2009-Y-15; 2010-B-01) and the Development Foundation of Higher Education Department of Shanxi Province (20101109; 20111020). † Manuscript received February 5, 2012; Revised March 21, 2012
1
Introduction
The existence of positive solutions to fractional differential equations has been studied extensively ([1-5]). Bai and Lv [6] studied the following two-point boundary value problem of fractional differential equations
α α D0+ u(t) = D0+ λ
(1 − s)α−3 h(s)f (u(s))ds − Γ(α) 0 α α = 0 − λD0+ I0+ h(t)f (u(t)) = −λh(t)f (u(t)). t
1 α−1
t 0
(t − s)α−1 h(s)f (u(s))ds Γ(α)
It shows that
α D0+ f (x) =
1 d Γ(n − α) dx
n 0
x
f (t) dt, (x − t)α−n+1
is called the Riemann-Liouville fractional derivative of order α, where n = [α]+1, [α] denotes the integer part of number α. Lemma 2.1[9] If y (t) ∈ C [0, 1] and y (t) ≥ 0. Then fractional boundary value problem
2
Preliminaries
For convenience, we demonstrate the definitions and lemmas that is useful in proving our main results. Definition 2.1[8] The integral x 1 f (t) α dt, x > 0, I0+ f (x) = Γ(α) 0 (x − t)1−α is called the Riemann-Liouville fractional integral of order α, where α > 0. Definition 2.2[8] For a function f (x) given in the interval [0, ∞), the expression
1
u(t) = λ
0
G(t, s)h(s)f (u(s))ds.
(6)
398
ANN. OF DIFF. EQS.
Vol.28
Proof Let u ∈ P is a solution to the boundary value problem (1) and (2). Applying the method of proving Lemma 2.1 used in [9], we can obtain that u is a solution to (6). Conversely, letting u be a solution to (6), we can obtain
Ann. of Diff. Eqs. 28:4(2012), 396-403
EXISTENCE OF TRIPLE POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION DEPENDING ON A PARAMETER ∗†
1
u(t) = λ
0
G(t, s)h(s)f (u(s))ds
t α−1
=λ
0
t
(1 − s)α−3 − (t − s)α−1 h(s)f (u(s))ds + Γ(α) (1 − s)α−3 h(s)f (u(s))ds − Γ(α)
1 0 t 0
1 α−1
t
t
(1 − s)α−3 h(s)f (u(s))ds Γ(α) (7)
α D0+ u(t) + a(t)f (t, u(t)) = 0,
0 < t < 1, 1 < α ≤ 2,
u(0) = u(1) = 0, and obtained the existence of positive solutions by means of the Krasnoselskii fixed-point theorem and the Leggett-Williams fixed-point theorem. El-Shahed [7] considered the following nonlinear fractional boundary value problem
1 α−1
=λ
0
t
(t − s)α−1 h(s)f (u(s))ds , Γ(α)
t 0
u (t) = λ (α − 1)tα−2
(1 − s)α−3 h(s)f (u(s))ds − (α − 1) Γ(α)
1 0 t
(t − s)α−2 h(s)f (u(s))ds , Γ(α)
u (t) = λ (α − 1)(α − 2)tα−3 −(α − 1)(α − 2)
Abstract
In this paper, we establish the existence of three positive solutions to a nonlinear fractional differential equation by the Leggett-Williams fixed point theorem. Keywords boundary value problem; fractional differential equation; positive solution; cone; fixed point theorem 2000 Mathematics Subject Classification 34K13
α D0+ u(t) + y (t) = 0,
0 < t < 1, 3 < α ≤ 4,
(3) (4)
Baidu Nhomakorabea
u(0) = u (0) = u (0) = u (1) = 0 has a unique positive solution
1
u(t) =
0
G(t, s)y (s)ds,
α−1 t (1 − s)α−3 − (t − s)α−1 , 0 ≤ s ≤ t ≤ 1; Γ(α) G(t, s) = (5) tα−1 (1 − s)α−3 , 0 ≤ t ≤ s ≤ 1. Γ(α) Here G(t, s) is called the Green function of the boundary value problem (3) and (4). Lemma 2.2 Assume that f : [0, ∞] → [0, ∞] is continuous. A function u ∈ P is a solution to the boundary value problem (1) and (2) if and only if it is a solution to the integral equation where
α D0+ u(t) + λa(t)f (t, u(t)) = 0,
0 < t < 1, 2 < α ≤ 3,
u(0) = u (0) = u (1) = 0, and used the Krasnoselskii fixed-point theorem on cone expansion and compression to show the existence and non-existence of positive solutions to the above fractional differential equation boundary value problem. In this article, we are concerned with the existence of three positive solutions to the following nonlinear fractional differential equation boundary value problem:
0
(1 − s)α−3 h(s)f (u(s))ds Γ(α)
(t − s)α−3 h(s)f (u(s))ds . Γ(α) u(0) = u (0) = u (0) = u (1) = 0.
We can verify easily that
Derivativing both sides of equation (7) gets
Shugui Kang1 , Yan Li2 , Jianmin Guo1
(1. Applied of Math. Institute, Shanxi Datong University, Datong 037009, Shanxi; 2. Dept. of Math., Shanxi Normal University, Linfen 041000, Shanxi, E-mail: dtkangshugui@126.com (S. Kang))
∗
396
No.4
S.G. Kang, etc., TRIPLE POSITIVE SOLUTIONS
α D0+ u(t) + λh(t)f (u(t)) = 0,
397
(1) (2)
0 < t < 1, 3 < α ≤ 4,
u(0) = u (0) = u (0) = u (1) = 0,
α where λ is a positive parameter and D0+ is the standard Riemann-Liouville fractional deriva1 tive. h : (0, 1) → (0, ∞) is continuous with 0 h(t)dt > 0, and f : [0, ∞) → [0, ∞) is continuous. Based on the Leggett-Williams fixed point theorem, we obtain sufficient conditions for the existence of three positive solutions to the fractional boundary value problem (1) and (2). The paper is organized as follows. In Section 2, we demonstrate basic definitions and present some lemmas in order to prove our main results. In Section 3, we establish some results for the existence of three positive solutions to problem (1) and (2). In Section 4, we present an example to illustrate our results.