• 1,119 Views
  • 574 Downloads
ram: Vol. 1
Research Article
Research in Applied Mathematics
Vol. 1 (2017), Article ID 101261, 12 pages
doi:10.11131/2017/101261

Existence of Positive Solutions to a Singular Semipositone Boundary Value Problem of Nonlinear Fractional Differential Systems

Xiaofeng Zhang1 and Hanying Feng1,2

1Department of Mathematics, Shijiazhuang Mechanical Engineering College Shijiazhuang 050003, Hebei, P. R. China

2Department of Mathematics, Nantong Institute of Technology, Nantong 226002, Jiangsu, P. R. China

Received 10 November 2016; Accepted 23 October 2017

Editor: Jianlong Qiu

Copyright © 2017 Xiaofeng Zhang and Hanying Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider the existence of positive solutions to a singular semipositone boundary value problem of nonlinear fractional differential equations. By applying the fixed point index theorem, some new results for the existence of positive solutions are obtained. In addition, an example is presented to demonstrate the application of our main results.

1. Introduction

In this paper, we discuss the following singular semipositone system of nonlinear fractional differential equations:

D 0 + α u ( t ) + f ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , D 0 + α v ( t ) + g ( t , u ( t ) , v ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ' ( 0 ) = u ' ( 1 ) = v ( 0 ) = v ( 1 ) = v ' ( 0 ) = v ' ( 1 ) = 0 , (1)

where 3<α≤4 is a real number, D0+α is the standard Riemann-Liouville fractional derivative, and f, g : (0, 1)×[0, +)×[0, +)arrow(−, +) are given continuous functions. f, g may be singular at t = 0 and/or t = 1 and may take negative values. By using the fixed point index theorem, some new results for the existence of positive solutions are established.

Singular boundary value problems arise from many fields in physics, biology, chemistry and economics, and play a very important role in both theoretical development and application. Recently, some work has been done to study the existence and multiplicity of solutions or positive solutions of nonlinear singular semipositone boundary value problems by the use of techniques of nonlinear analysis such as Leray-Schauder theory, Krasnoselskii's fixed point theorem, etc [1,3,4,7,9,11,12].

In [7], by using the fixed point index theorem, Liu, Zhang and Wu have studied the existence of positive solutions for a nonlinear singular semipositone system:

x ' ' ( t ) = f ( t , y ( t ) , x ( t ) ) + p ( t ) , t ( 0 , 1 ) , y ' ' ( t ) = g ( t , x ( t ) , y ( t ) ) + q ( t ) , t ( 0 , 1 ) , x ( 0 ) = x ( 1 ) = 0 , y ( 0 ) = y ( 1 ) = 0 ,

where f, g : (0, 1)×[0, +)×[0, +)arrow[0, +) are continuous and may be singular at t = 0 and/or t = 1, p and q : (0, 1)arrow(−, +) are Lebesgue integrable and may have finitely many singularities in [0, 1].

In [12], Zhu, Liu and Wu have discussed the existence of positive solutions for the fourth-order singular semipositone system:

x ( 4 ) ( t ) = f ( t , x ( t ) , y ( t ) , x ' ' ( t ) , y ' ' ( t ) ) , t ( 0 , 1 ) , y ( 4 ) ( t ) = g ( t , x ( t ) , y ( t ) , x ' ' ( t ) , y ' ' ( t ) ) , t ( 0 , 1 ) , x ( 0 ) = x ( 1 ) = x ' ' ( t ) = y ' ' ( t ) = 0 , y ( 0 ) = y ( 1 ) = x ' ' ( t ) = y ' ' ( t ) = 0 ,

where f, g : (0, 1)×[0, +)×[0, +)×(−, 0]×(−, 0]arrow(−, +) are given continuous functions. f, g may be singular at t = 0 and/or t = 1 and may take negative values.

In [6], Henderson and Luca have considered the existence of positive solutions for the system of nonlinear fractional differential equations:

D 0 + α u ( t ) + λ f ( t , u ( t ) , v ( t ) ) = 0 , t ( 0 , 1 ) , D 0 + β v ( t ) + μ g ( t , u ( t ) , v ( t ) ) = 0 , t ( 0 , 1 ) ,

with the coupled integral boundary conditions

u ( 0 ) = u ' ( 0 ) = = u ( n 2 ) ( 0 ) = 0 , u ' ( 1 ) = 0 1 v ( s ) d H ( s ) , v ( 0 ) = v ' ( 0 ) = = v ( n 2 ) ( 0 ) = 0 , v ' ( 1 ) = 0 1 u ( s ) d K ( s ) ,

where α∈(n − 1, n], β∈(m − 1, m], n, m, n, m≥3, D0+α, D0+β denote the standard Riemann-Liouville fractional derivatives, f, g are sign-changing continuous functions and may be nonsingular or singular at t = 0 and/or t = 1.

Motivated by the above work, we consider the existence of positive solutions for the system of the fractional order singular semipositone boundary value problem (1).

This paper is organized as follows. In Section 2, we present some basic definitions and properties from the fractional calculus theory. In Section 3, based on the fixed point index theorem, we prove existence theorem of the positive solutions for boundary value problem (1). In section 4, an example is presented to illustrate the main results.

2. Preliminaries

In this section, we present here the necessary definitions and properties from fractional calculus theory. These definitions and properties can be found in the recent literature [2,5,8,10,11,13].

Definition

[see [2]] The Riemann-Liouville fractional integral of order α>0 of a function y : (0, +)arrow is given by

I 0 + α y ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 y ( s ) d s , t > 0 ,

provided the right-hand side is pointwise defined on (0, +).

Definition

[see [2,8]] The Riemann-Liouville fractional derivative of order α>0 for a function y : (0, +)arrow is given by

D 0 + α y ( t ) = ( d d t ) n ( I 0 + n α y ) ( t ) = 1 Γ ( n α ) ( d d t ) n 0 t y ( s ) ( t s ) α n + 1 d s , t > 0 ,

where n = [α] + 1, [α] denotes the integer part of the number α, provided that the right-hand side is pointwise defined on (0, +).

Lemma [see [13]] Let α>0. If we assume uC(0, 1)∩L(0, 1), then the fractional differential equation

D 0 + α u ( t ) = 0

has solutions u(t) = C1tα−1 + C2tα−2 + ⋯+Cntαn, Ci, i = 1, 2, ⋯, n, n = [α] + 1.

Lemma [see [13]] Assume that uC(0, 1)∩L(0, 1) with a fractional derivative of order α(α>0) that belongs to C(0, 1)∩L(0, 1), then

I 0 + α D 0 + α u ( t ) = u ( t ) + C 1 t α 1 + C 2 t α 2 + + C n t α n ,

for some Ci, i = 1, 2, ⋯, n, n = [α] + 1.

In the following, we present Green's function of the fractional differential equation boundary value problem.

Lemma [see [10]] Let yC[0, 1] and 3<α≤4, the unique solution of problem

D 0 + α u ( t ) + y ( t ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ' ( 0 ) = u ' ( 1 ) = 0 , (2)

is

u ( t ) = 0 1 G ( t , s ) y ( s ) d s ,

where

G ( t , s ) = ( t s ) α 1 + ( 1 s ) α 2 t α 2 [ ( s t ) + ( α 2 ) ( 1 t ) s ] Γ ( α ) , 0 s t 1 , t α 2 ( 1 s ) α 2 [ ( s t ) + ( α 2 ) ( 1 t ) s ] Γ ( α ) , 0 t s 1 . (3)

Here G(t, s) is called the Green's function of boundary value problem (2).

Lemma [see [10,11]] The function G(t, s) defined by (4) possesses the following properties:

G(t, s)>0, for t, s∈(0, 1);

G(t, s) = G(1 − s, 1 − t), for t, s∈(0, 1);

tα−2(1 − t)2q(s)≤G(t, s)≤(α − 1)q(s), for t, s∈(0, 1);

tα−2(1 − t)2q(s)≤G(t, s)≤((α − 1)(α − 2)/Γ(α))tα−2(1 − t)2, for t, s∈(0, 1),

where q(s) = ((α − 2)/Γ(α))s2(1 − s)α−2.

Lemma The function q(1 − t) has the property:

max t ( 0 , 1 ) q ( 1 t ) = q ( 2 α ) = 4 ( α 2 ) α 1 Γ ( α ) α α .

Proof

From the Lemma 2.4, we can easy get q ( 1 - t ) = α - 2 Γ ( α ) t α - 2 ( 1 - t ) 2 . Let F(t) = tα−2(1 − t)2, since F'(t) = (1 − t)tα−3[−αt + (α − 2)], for t∈(0, 1), let F'(t) = 0, we get t 0 = α - 2 α .

Since 3<α≤4, we can know 0<t0<1. So, the function F(t) achieve the maximum when t = α - 2 α .

Therefore

max t ( 0 , 1 ) F ( t ) = F ( α 2 α ) = 4 ( α 2 ) α 2 α α ,

thus,

max t ( 0 , 1 ) q ( 1 t ) = q ( 2 α ) = 4 ( α 2 ) α 1 Γ ( α ) α α .

For convenience, throughout the rest of the paper, we make the following assumptions:

(H1) f, gC((0, 1)×[0, +)×[0, +), (−, +)) and there exist functions pi, ai, kL1((0, 1), [0, +))∩C((0, 1), [0, +)) and hC([0, +)×[0, +), [0, +)) such that

a 1 ( t ) h ( x , y ) f ( t , x , y ) + p 1 ( t ) k ( t ) h ( x , y ) , a 2 ( t ) h ( x , y ) g ( t , x , y ) + p 2 ( t ) k ( t ) h ( x , y ) ,

where ai(t)≥cik(t) a.e. t∈(0, 1), 0<ci≤1, i = 1, 2, ∀(t, x, y)∈(0, 1)×[0, +)×[0, +).

(H2) There exists (a, b)⊂[0, 1] such that

lim x , y a r r o w + min t [ a , b ] f ( t , x , y ) + p 1 ( t ) y = + , or lim x , y a r r o w + min t [ a , b ] g ( t , x , y ) + p 1 ( t ) y = + .

(H3) Assume that

0 1 k ( s ) d s < Γ ( α ) α α r 4 ( α 1 ) ( α 2 ) α 1 M ,

where

M = max x , y [ 0 , r ] h ( x , y ) , r 1 = 0 1 p 1 ( s ) d s , r 2 = 0 1 p 2 ( s ) d s ,

r = max { ( α 1 ) 2 ( α 2 ) r i c i Γ ( α ) , i = 1 , 2 . } .

Lemma For functions pi(t), i = 1, 2 in (H1), then the boundary value problem

D 0 + α u ( t ) + p i ( t ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ' ( 0 ) = u ' ( 1 ) = 0 , (4)

has a unique solution wi(t) = ∫01G(t, s)pi(s)ds with

w i ( t ) ( α 1 ) q ( 1 t ) 0 1 p i ( s ) d s , t [ 0 , 1 ] , i = 1 , 2 . (5)

Proof

By Lemma 2.3 and Lemma 2.4, we have wi(t) = ∫01G(t, s)pi(s)ds is the unique solution of (3) and

w i ( t ) = 0 1 G ( t , s ) p i ( s ) d s ( α 1 ) q ( 1 t ) 0 1 p i ( s ) d s , i = 1 , 2 .

For any xC[0, 1], we define a function [x(·)]* : [0, 1]arrow[0, +) by

[ x ( · ) ] * = x ( t ) , x ( t ) 0 , 0 , x ( t ) < 0 .

In order to overcome the difficulty associated with semipositone, we consider the following approximately singular nonlinear differential system:

D 0 + α u ( t ) + f ( t , [ u ( t ) w 1 ( t ) ] * , [ v ( t ) w 2 ( t ) ] * ) + p 1 ( t ) = 0 , 0 < t < 1 , D 0 + α v ( t ) + g ( t , [ u ( t ) w 1 ( t ) ] * , [ v ( t ) w 2 ( t ) ] * ) + p 2 ( t ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ' ( 0 ) = u ' ( 1 ) = v ( 0 ) = v ( 1 ) = v ' ( 0 ) = v ' ( 1 ) = 0 , (6)

where wi(t)(i = 1, 2) are defined in Lemma 2.6.

It is well-known that the problem (6) can be written equivalently as the following nonlinear system of integral equations

u ( t ) = 0 1 G ( t , s ) [ f ( s , [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) + p 1 ( t ) ] d s , 0 t 1 , v ( t ) = 0 1 G ( t , s ) [ g ( s , [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) + p 2 ( t ) ] d s , 0 t 1 . (7)

We consider the Banach space X = C[0, 1] with the norm u = max 0 t 1 | u ( t ) | , and the Banach space Y = X×X with the norm ∥(u, v)∥ = max{∥u∥, ∥v∥}.

We define the cone PY by

P = { ( u , v ) Y | u ( t ) c 1 t α 2 ( 1 t ) 2 α 1 ( u , v ) , v ( t ) c 2 t α 2 ( 1 t ) 2 α 1 ( u , v ) , t [ 0 , 1 ] } .

Define the operators T1, T2 : YarrowX and T : YarrowY as follows:

T 1 ( u , v ) ( t ) = 0 1 G ( t , s ) [ f ( s , [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) + p 1 ( t ) ] d s , 0 t 1 , T 2 ( u , v ) ( t ) = 0 1 G ( t , s ) [ g ( s , [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) + p 2 ( t ) ] d s , 0 t 1 ,

and T(u, v) = (T1(u, v), T2(u, v)), (u, v)∈Y. Thus, the solutions of our problem (6) are the fixed points of the operator T.

Lemma [see [5]] Let E be a real Banach space, P be a cone in E. Ω be a bounded open subset of E with θΩ, and T : Ω ¯ P a r r o w P be a completely continuous operator, then the following conclusions hold:

Suppose that Tuλu, ∀uΩP, λ≥1, then i(T, ΩP, P) = 1.

Suppose that Tuu, ∀uΩP, then i(T, ΩP, P) = 0.

3. Main Results and Proof

Lemma T : ParrowP is a completely continuous operator.

Proof

Let (u, v)∈P be an arbitrary element. From Lemma 2.4 and (H1), we can get

T 1 ( u , v ) = max 0 t 1 | T 1 ( u , v ) ( t ) | 0 1 ( α 1 ) q ( s ) [ f ( s , [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) + p 1 ( t ) ] d s ( α 1 ) 0 1 q ( s ) k ( s ) h ( [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) d s , T 2 ( u , v ) = max 0 t 1 | T 2 ( u , v ) ( t ) | 0 1 ( α 1 ) q ( s ) [ g ( s , [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) + p 2 ( t ) ] d s ( α 1 ) 0 1 q ( s ) k ( s ) h ( [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) d s ,

Hence, we obtain

T ( u , v ) ( α 1 ) 0 1 q ( s ) [ k ( s ) h ( [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) ] d s . (8)

Applying (H1) and (8), we have

T 1 ( u , v ) ( t ) t α 2 ( 1 t ) 2 0 1 q ( s ) [ f ( s , [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) + p 1 ( t ) ] d s t α 2 ( 1 t ) 2 0 1 q ( s ) a 1 ( s ) h ( [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) d s c 1 t α 2 ( 1 t ) 2 0 1 q ( s ) k ( s ) h ( [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) d s c 1 t α 2 ( 1 t ) 2 α 1 T ( u , v ) .

In the similar manner, we deduce

T 2 ( u , v ) ( t ) c 2 t α 2 ( 1 t ) 2 α 1 T ( u , v ) .

Thus T(u, v)∈P, that is T(P)⊂P.

According to the Arzela-Ascoli theorem, we can easily get that T : ParrowP is a completely continuous operator.

Theorem

If (H1)–(H3) hold, then the boundary value problem (1) has at least one positive solution.

Proof

Let Pr = {(u, v)∈P, ∥(u, v)∥<r}, we first prove T(u, v)≠λ(u, v), for ∀(u, v)∈Pr, λ≥1, where the constant r is defined in (H3).

In fact, if not, there exist λ0≥1 and (u, v)∈Pr such that λ0(u, v) = T(u, v), then ( u , v ) = 1 λ 0 T ( u , v ) and 0 < 1 λ 0 1 . Since

u ( t ) c 1 t α 2 ( 1 t ) 2 α 1 ( u , v ) = c 1 t α 2 ( 1 t ) 2 α 1 r , t [ 0 , 1 ] , v ( t ) c 2 t α 2 ( 1 t ) 2 α 1 ( u , v ) = c 2 t α 2 ( 1 t ) 2 α 1 r , t [ 0 , 1 ] ,

and

w 1 ( t ) ( α 1 ) q ( 1 t ) 0 1 p 1 ( s ) d s ( α 1 ) q ( 1 t ) r 1 , w 2 ( t ) ( α 1 ) q ( 1 t ) 0 1 p 2 ( s ) d s ( α 1 ) q ( 1 t ) r 2 ,

for any t∈[0, 1], we get that

u ( t ) w 1 ( t ) c 1 t α 2 ( 1 t ) 2 α 1 r ( α 1 ) q ( 1 t ) r 1 [ c 1 Γ ( α ) r ( α 1 ) ( α 2 ) ( α 1 ) r 1 ] q ( 1 t ) 0 , v ( t ) w 2 ( t ) c 2 t α 2 ( 1 t ) 2 α 1 r ( α 1 ) q ( 1 t ) r 2 [ c 2 Γ ( α ) r ( α 1 ) ( α 2 ) ( α 1 ) r 2 ] q ( 1 t ) 0 .

Hence by ( u , v ) = 1 λ 0 T ( u , v ) , we obtain that

u ( t ) = 1 λ 0 0 1 G ( t , s ) [ f ( s , u ( s ) w 1 ( s ) , v ( s ) w 2 ( s ) ) + p 1 ( s ) ] d s 0 1 G ( t , s ) k ( s ) h ( s , u ( s ) w 1 ( s ) , v ( s ) w 2 ( s ) ) d s ( α 1 ) q ( 1 t ) 0 1 k ( s ) h ( s , u ( s ) w 1 ( s ) , v ( s ) w 2 ( s ) ) d s ( α 1 ) q ( 1 t ) M 0 1 k ( s ) d s .

Since (u, v)∈Pr, we know ∥u∥ = r or ∥v∥ = r. If ∥u∥ = r, then from Lemma 2.5, we deduce

r = max t [ 0 , 1 ] u ( t ) max t [ 0 , 1 ] { ( α 1 ) q ( 1 t ) M 0 1 k ( s ) d s } 4 ( α 1 ) ( α 2 ) α 1 M Γ ( α ) α α 0 1 k ( s ) d s .

Consequently

0 1 k ( s ) d s Γ ( α ) α α r 4 ( α 1 ) ( α 2 ) α 1 M ,

which is a contradiction to (H3). The proof will be similar when ∥v∥ = r. Therefore, applying Lemma 2.7, we obtain i(T, Pr, P) = 1.

On the other hand, choose a constant L>0 such that

L > ( α 1 ) α α 2 c 2 a α 2 ( 1 b ) 2 ( α 2 ) α 2 a b q ( s ) d s . (9)

From (H2), there exists R1>r such that

f ( t , x , y ) + p 1 ( t ) L y , t [ a , b ] , x , y R 1 . (10)

Taking R 2 R 1 max { α - 1 c i a α - 2 ( 1 - b ) 2 , i = 1 , 2 } . Obviously, R>2R1>2r, thus r R < 1 2 .

Let PR = {(u, v)∈P, ∥(u, v)∥<R}, we will show that T(u, v)≰(u, v), for ∀(u, v)∈PR.

In fact, otherwise, there exists (u, v)∈PR such that T(u, v)≤(u, v). By proceeding as for the proof to get (9) and (H3), we have, for any t∈[a, b],

u ( t ) w 1 ( t ) u ( t ) ( α 1 ) q ( 1 t ) r 1 u ( t ) c 1 Γ ( α ) r ( α 1 ) ( α 2 ) q ( 1 t ) = u ( t ) c 1 t α 2 ( 1 t ) 2 α 1 r u ( t ) u ( t ) R r 1 2 u ( t ) 1 2 R c 1 t α 2 ( 1 t ) 2 α 1 1 2 R c 1 a α 2 ( 1 b ) 2 α 1 R 1 > 0 , v ( t ) w 2 ( t ) v ( t ) ( α 1 ) q ( 1 t ) r 2 v ( t ) c 2 Γ ( α ) r ( α 1 ) ( α 2 ) q ( 1 t ) = v ( t ) c 2 t α 2 ( 1 t ) 2 α 1 r v ( t ) v ( t ) R r 1 2 v ( t ) 1 2 R c 2 t α 2 ( 1 t ) 2 α 1 1 2 R c 2 a α 2 ( 1 b ) 2 α 1 R 1 > 0 .

Therefore, we deduce

R u ( t ) T 1 ( u , v ) ( t ) = 0 1 G ( t , s ) [ f ( s , [ u ( s ) w 1 ( s ) ] * , [ v ( s ) w 2 ( s ) ] * ) + p 1 ( t ) ] d s a b G ( t , s ) [ f ( s , u ( s ) w 1 ( s ) , v ( s ) w 2 ( s ) ) + p 1 ( t ) ] d s L a b G ( t , s ) ( v ( s ) w 2 ( s ) ) d s 1 2 L R c 2 a α 2 ( 1 b ) 2 α 1 a b G ( t , s ) d s 1 2 L R c 2 a α 2 ( 1 b ) 2 α 1 t α 2 ( 1 t ) 2 a b q ( s ) d s , t [ 0 , 1 ] .

Then from Lemma 2.5, we have

R 1 2 L R c 2 a α 2 ( 1 b ) 2 α 1 a b q ( s ) d s max t [ 0 , 1 ] { t α 2 ( 1 t ) 2 } 2 L R c 2 a α 2 ( 1 b ) 2 ( α 2 ) α 2 ( α 1 ) α α a b q ( s ) d s .

Consequently, by (10) we obtain

R 2 L R c 2 a α 2 ( 1 b ) 2 ( α 2 ) α 2 ( α 1 ) α α a b q ( s ) d s > R .

This is a contradiction. Thus from Lemma 2.7, we get i(T, PR, P) = 0.

From the properties of the fixed point index, we have i(T, PRPr, P) = −1. Therefore, T has a fixed point (u0, v0) in P R P ¯ r , with ∥(u0, v0)∥>r. At the same time,

u 0 ( t ) w 1 ( t ) c 1 t α 2 ( 1 t ) 2 α 1 ( u 0 , v 0 ) ( α 1 ) q ( 1 t ) r 1 > [ c 1 Γ ( α ) r ( α 1 ) ( α 2 ) ( α 1 ) r 1 ] q ( 1 t ) 0 , v 0 ( t ) w 2 ( t ) c 2 t α 2 ( 1 t ) 2 α 1 ( u 0 , v 0 ) ( α 1 ) q ( 1 t ) r 2 > [ c 2 Γ ( α ) r ( α 1 ) ( α 2 ) ( α 1 ) r 2 ] q ( 1 t ) 0 .

Then, we known that (u0(t), v0(t)) is a solution of system (6) and wi(t)(i = 1, 2) are solutions of system (3). Thus (u0(t) − w1(t), v0(t) − w2(t)) is a positive solution of the singular semipositone boundary value problem (1).

The proof of Theorem 3.1 is completed.

Remark

The conclusion of Theorem 3.1 is valid if (H2) is replaced by

(H2)* There exists (a, b)⊂[0, 1] such that

lim x , y a r r o w + min t [ a , b ] f ( t , x , y ) + p 1 ( t ) x L ¯ , or lim x , y a r r o w + min t [ a , b ] g ( t , x , y ) + p 1 ( t ) x L ¯ ,

where

L ¯ > ( α 1 ) α α 2 c 1 a α 2 ( 1 b ) 2 ( α 2 ) α 2 a b q ( s ) d s .

4. Example

Now, we present an example to illustrate the main results.

Example

Consider the following system of fractional differential equations

D 0 + 7 2 u ( t ) + 1 10 t 1 5 ( | u ( t ) | 2 + | v ( t ) | 2 ) 1 16 t 1 8 = 0 , 0 < t < 1 , D 0 + 7 2 v ( t ) + 1 5 t 1 5 ( | u ( t ) | 2 + | v ( t ) | 2 ) 1 4 t 1 4 = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ' ( 0 ) = u ' ( 1 ) = v ( 0 ) = v ( 1 ) = v ' ( 0 ) = v ' ( 1 ) = 0 . (11)

In the system (11), α = 7 2 and

f ( t , u , v ) = 1 10 t 1 5 ( | u ( t ) | 2 + | v ( t ) | 2 ) 1 16 t 1 8 , g ( t , u , v ) = 1 5 t 1 5 ( | u ( t ) | 2 + | v ( t ) | 2 ) 1 4 t 1 4 ,

for t∈[0, 1], u, v≥0.

We deduce p 1 ( t ) = 1 16 t - 1 8 , p 2 ( t ) = 1 4 t - 1 4 , k ( t ) = 1 5 t - 1 5 , a i ( t ) = 1 15 t - 1 5 , c i = 1 3 , i = 1 , 2 . h ( u , v ) = | u ( t ) | 2 + | v ( t ) | 2 .

Clearly, f, g satisfy conditions (H1) and (H2). Since

r 1 = 0 1 p 1 ( s ) d s = 1 14 , r 2 = 0 1 p 2 ( s ) d s = 1 3 , 0 1 k ( s ) d s = 1 4 .

We have r = max { ( α - 1 ) 2 ( α - 2 ) r i c i Γ ( α ) , i = 1 , 2 } = 5 π and consequently

M = 50 π , Γ ( α ) α α r 4 ( α 1 ) ( α 2 ) α 1 M 1 . 71 .

So

0 1 k ( s ) d s < Γ ( α ) α α r 4 ( α 1 ) ( α 2 ) α 1 M .

It is thus clear that (H3) is satisfied. Hence it follows from Theorem 3.1 that system (11) has at least one positive solution.

Acknowledgement

This work was supported by NNSF of China (11371368) and HEBNSF of China (A2014506016).

Competing Interests

The authors declare no competing interests.

References

  1. R. P. Agarwal and D. O'Regan, “A coupled system of boundary value problems,” Applicable Analysis: An International Journal, vol. 69, no. 3-4, pp. 381–385, 1998. Publisher Full Text | Google Scholar
  2. Z. Bai and H. L\"u, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005. Publisher Full Text | Google Scholar
  3. C. Bai, “Positive solutions for nonlinear fractional differential equations with coefficient that changes sign,” Nonlinear Analysis. Theory, Methods \& Applications. An International Multidisciplinary Journal, vol. 64, no. 4, pp. 677–685, 2006. Publisher Full Text | Google Scholar
  4. X. Feng, H. Feng, H. Tan, and Y. Du, “Positive solutions for systems of a nonlinear fourth-order singular semipositone Sturm-Liouville boundary value problem,” Applied Mathematics and Computation, vol. 41, no. 1-2, pp. 269–282, 2013. Publisher Full Text | Google Scholar
  5. D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1988.
  6. J. Henderson and R. Luca, “Existence of positive solutions for a system of semipositone fractional boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2016, article no. 22, 2016. Publisher Full Text | Google Scholar
  7. L. Liu, X. Zhang, and Y. Wu, “On existence of positive solutions of a two-point boundary value problem for a nonlinear singular semipositone system,” Applied Mathematics and Computation, vol. 192, no. 1, pp. 223–232, 2007. Publisher Full Text | Google Scholar
  8. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  9. Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity,” Nonlinear Analysis. Theory, Methods \& Applications. An International Multidisciplinary Journal, vol. 74, no. 17, pp. 6434–6441, 2011. Publisher Full Text | Google Scholar
  10. X. Xu, D. Jiang, and C. Yuan, “Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation,” Nonlinear Analysis. Theory, Methods \& Applications. An International Multidisciplinary Journal, vol. 71, no. 10, pp. 4676–4688, 2009. Publisher Full Text | Google Scholar
  11. C. Yuan, D. Jiang, and X. Xu, “Singular positone and semipositone boundary value problems of nonlinear fractional differential equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 535209, 2009. Publisher Full Text | Google Scholar
  12. F. Zhu, L. Liu, and Y. Wu, “Positive solutions for systems of a nonlinear fourth-order singular semipositone boundary value problems,” Applied Mathematics and Computation, vol. 216, no. 2, pp. 448–457, 2010. Publisher Full Text | Google Scholar
  13. S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 36, 12 pages, 2006.
Research Article
Research in Applied Mathematics
Vol. 1 (2017), Article ID 101261, 12 pages
doi:10.11131/2017/101261

Existence of Positive Solutions to a Singular Semipositone Boundary Value Problem of Nonlinear Fractional Differential Systems

Xiaofeng Zhang1 and Hanying Feng1,2

1Department of Mathematics, Shijiazhuang Mechanical Engineering College Shijiazhuang 050003, Hebei, P. R. China

2Department of Mathematics, Nantong Institute of Technology, Nantong 226002, Jiangsu, P. R. China

Received 10 November 2016; Accepted 23 October 2017

Editor: Jianlong Qiu

Copyright © 2017 Xiaofeng Zhang and Hanying Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider the existence of positive solutions to a singular semipositone boundary value problem of nonlinear fractional differential equations. By applying the fixed point index theorem, some new results for the existence of positive solutions are obtained. In addition, an example is presented to demonstrate the application of our main results.