AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/158436 158436 Research Article Existence of Positive Solutions to Nonlinear Fractional Boundary Value Problem with Changing Sign Nonlinearity and Advanced Arguments Hao Zhaocai Huang Yubo Zhang Xinguang School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165 China qfnu.edu.cn 2014 972014 2014 24 03 2014 19 06 2014 9 7 2014 2014 Copyright © 2014 Zhaocai Hao and Yubo Huang. 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.

We discuss the existence of positive solutions to a class of fractional boundary value problem with changing sign nonlinearity and advanced arguments D α x ( t ) + μ h ( t ) f ( x ( a ( t ) ) ) = 0 , t ( 0,1 ) , 2 < α 3 , μ > 0 , x ( 0 ) = x ( 0 ) = 0 , x ( 1 ) = β x ( η ) + λ [ x ] , β > 0 , and   η ( 0,1 ) , where D α is the standard Riemann-Liouville derivative, f : [ 0 , ) [ 0 , ) is continuous, f ( 0 ) > 0 ,  h  : [ 0,1 ] ( , + ) , and a ( t ) is the advanced argument. Our analysis relies on a nonlinear alternative of Leray-Schauder type. An example is given to illustrate our results.

1. Introduction

Fractional differential equations (FDEs) have been of great interest for the past three decades. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, and so forth (see [1, 2]). Therefore, the theory of FDEs has been developed very quickly. There has been a significant development in fractional differential equations in recent years; see .

In , the author studied existence of positive solutions in case of the nonlinear fractional differential equation as follows: (1) D s u = λ a ( t ) f ( u ) , 0 < t < 1 , u ( 0 ) = 0 , where 0 < s < 1 , D s is the standard Riemann-Liouville fractional derivative, f : [ 0 , ) [ 0 , ) is continuous, and a : [ 0,1 ] R . In , the author applied the Avery-Peterson fixed point theorem to obtain sufficient conditions of the existence of multiple solutions to the following problem: (2) x ′′′ ( t ) + h ( t ) f ( x ( α ( t ) ) ) = 0 , t ( 0,1 ) , x ( 0 ) = x ′′ ( 0 ) = 0 , x ( 1 ) = β x ( η ) + λ [ x ] , β > 0 , η ( 0,1 ) , where f : [ 0 , ) [ 0 , ) is continuous and h ( t ) is a nonnegative continuous function defined on [ 0,1 ] .

Motivated by [5, 10], in this paper, we consider the existence of positive solution of the following boundary value problem for nonlinear fractional differential equation with changing sign nonlinearity and advanced arguments: (3) D α x ( t ) + μ h ( t ) f ( x ( a ( t ) ) ) = 0 , t ( 0,1 ) , 2 < α 3 , μ > 0 , x ( 0 ) = x ( 0 ) = 0 , x ( 1 ) = β x ( η ) + λ [ x ] , β > 0 , η ( 0,1 ) , where λ denotes a linear functional on C [ 0,1 ] given by λ [ x ] = 0 1 x ( t ) d Λ ( t ) involving a Stieltjes integral with a suitable function Λ of bounded variation. It is important to indicate that we did not assume that λ [ x ] is positive to all positive x . The measure d Λ can be a signed measure.

Put J = [ 0,1 ] ; let us introduce the following assumptions:

f : [ 0 , ) [ 0 , ) is continuous, and f ( 0 ) > 0 ;

a C ( J , J ) , and t a ( t ) on J ;

h : [ 0,1 ] ( - , + ) may change sign; h is not identically zero on any subinterval on J ;

0 < β η α - 1 + λ [ p ] < 1 , where p ( t ) = t α - 1 .

2. Basic Definitions and Preliminaries

In this section, we present some preliminaries and lemmas that are useful to the proof of our main results. For convenience, we also present the necessary definitions from fractional calculus theory here. These definitions can be found in the recent literature.

Definition 1.

The fractional integral of order α > 0 of a function x : ( 0 , + ) R is given by (4) I 0 + α x ( t ) = 1 Γ ( α ) 0 t ( t - s ) α - 1 x ( t ) d s , provided that the right-hand side is pointwise defined on ( 0 , + ) .

Definition 2.

The fractional derivative of order α > 0 of a continuous function x : ( 0 , + ) R is given by (5) D 0 + α x ( t ) = 1 Γ ( n - α ) ( d d t ) n 0 t ( t - s ) n - α - 1 x ( t ) d s , where n = [ α ] + 1 and [ α ] denotes the integral part of number α , provided that the right-hand side is pointwise defined on ( 0 , + ) .

Lemma 3.

Let α > 0 , x C ( 0,1 ) L ( 0,1 ) ; then (6) I 0 + α D 0 + α x ( t ) = x ( t ) + c 1 t α - 1 + c 2 t α - 2 + + c n t α - n , where c i R ( i = 1,2 , , n ) , n being the smallest integer greater than or equal to α .

Consider the following boundary value problem: (7) D α x ( t ) + y ( t ) = 0 , t ( 0,1 ) , 2 < α 3 , x ( 0 ) = x ( 0 ) = 0 , x ( 1 ) = β x ( η ) + λ [ x ] , β > 0 , η ( 0,1 ) .

Lemma 4.

Assume that β η α - 1 1 and y C ( J , R ) ; then problem (7) has the unique solution given by the following formula: (8) x ( t ) = t α - 1 1 - β η α - 1 λ [ x ] + β t α - 1 1 - β η α - 1 0 1 k ( η , s ) y ( s ) d s + 0 1 k ( t , s ) y ( s ) d s , where (9) k ( t , s ) = 1 Γ ( α ) { [ t ( 1 - s ) ] α - 1 , 0 t s 1 , [ t ( 1 - s ) ] α - 1 - ( t - s ) α - 1 , 0 s t 1 .

Theorem 5.

Let X be a Banach space with C X closed and convex. Assume that U is a relatively open subset of C with 0 C and A : U ¯ C is a continuous, compact map. Then either

A has a fixed point in U ¯ or

there exist u U and τ ( 0,1 ) with u = τ A u .

3. Existence of Positive Solutions

Let us denote by X = C [ 0,1 ] the Banach space of all continuous real functions on [ 0,1 ] endowed with the sup norm and let K be the cone: (10) K = { x X , x ( t ) 0 , t J } .

Lemma 6.

Let assumptions (H 1)–(H 4) hold. Moreover, we assume that assumptions (H5)-(H6) hold with

0 1 d Λ ( t ) 0 ,    0 1 t α - 1 d Λ ( t ) 0 a n d κ ( s ) = 0 1 k ( t , s ) d Λ ( t ) 0 ,

h : [ 0,1 ] ( - , + ) is continuous, h ( 0 ) 0 , and there is σ > 1 such that (11) t α - 1 Δ - ρ ( β ρ Δ 0 1 k ( η , s ) h + ( s ) d s + 0 1 κ ( s ) h + ( s ) d s ) + 0 1 k ( t , s ) h + ( s ) d s + β t α - 1 Δ 0 1 k ( η , s ) h + ( s ) d s σ [ t α - 1 Δ - ρ ( β ρ Δ 0 1 k ( η , s ) h - ( s ) d s + 0 1 κ ( s ) h - ( s ) d s ) + 0 1 k ( t , s ) h - ( s ) d s + β t α - 1 Δ 0 1 k ( η , s ) h - ( s ) d s ] ,

where Δ = 1 - β η α - 1 , ρ = λ [ p ] , h + ( t ) = max { 0 , h ( t ) } a n d h - ( t ) = max { 0 , - h ( t ) } . Then, for every 0 < δ < 1 , there exists a positive number μ ¯ such that, for 0 < μ < μ ¯ , the nonlinear fractional differential equation, (12) D α x ( t ) + μ h + ( t ) f ( x ( a ( t ) ) ) = 0 , f , t ( 0,1 ) , 2 < α 3 , μ > 0 , x ( 0 ) = x ( 0 ) = 0 , x ( 1 ) = β x ( η ) + λ [ x ] , β > 0 , η ( 0,1 ) , has a positive solution x ¯ μ with x ¯ μ 0 as μ 0 and (13) x ¯ μ ( t ) μ δ f ( 0 ) m ( t ) , where (14) m ( t ) = t α - 1 Δ - ρ ( β ρ Δ 0 1 k ( η , s ) h + ( s ) d s + 0 1 κ ( s ) h + ( s ) d s ) + β t α - 1 Δ 0 1 k ( η , s ) h + ( s ) d s + 0 1 k ( t , s ) h + ( s ) d s .

Proof.

It is easy to know from (9), (H 5), and (H 6) that m ( t ) > 0 , t ( 0,1 ] . By Lemma 4, (12) has a unique solution in X : (15) x ( t ) = t α - 1 1 - β η α - 1 λ [ x ] + β t α - 1 1 - β η α - 1 μ 0 1 k ( η , s ) h + ( s ) f ( x ( a ( s ) ) ) d s + μ 0 1 k ( t , s ) h + ( s ) f ( x ( a ( s ) ) ) d s . For x C ( J , R + ) , we define two operators T and S by (16) T x ( t ) = t α - 1 Δ λ [ x ] + μ F x ( t ) , S x ( t ) = t α - 1 Δ - ρ μ λ [ F x ] + μ F x ( t ) , where (17) F x ( t ) = β t α - 1 Δ 0 1 k ( η , s ) h + ( s ) f ( x ( a ( s ) ) ) d s + 0 1 k ( t , s ) h + ( s ) f ( x ( a ( s ) ) ) d s , λ [ F x ] = β ρ Δ 0 1 k ( η , s ) h + ( s ) f ( x ( a ( s ) ) ) d s + 0 1 κ ( s ) h + ( s ) f ( x ( a ( s ) ) ) d s . It is easy to show that T : K K and S : K K are completely continuous. We claim that operators T and S have the same fixed points in K . In fact, let x = S x ; then (18) λ [ x ] = ρ Δ - ρ μ λ [ F x ] + μ λ [ F x ] = Δ Δ - ρ μ λ [ F x ] . So (19) x ( t ) = S x ( t ) = t α - 1 Δ - ρ μ λ [ F x ] + μ F x ( t ) = t α - 1 Δ λ [ x ] + μ F x ( t ) = T x ( t ) . Let x = T x ; then λ [ x ] = ( ρ / Δ ) λ [ x ] + μ λ [ F x ] . So λ [ x ] = ( Δ / ( Δ - ρ ) ) μ λ [ F x ] , and hence (20) x ( t ) = T x ( t ) = t α - 1 Δ λ [ x ] + μ F x ( t ) = t α - 1 Δ - ρ μ λ [ F x ] + μ F x ( t ) = S x ( t ) . This shows that fixed points of S are solutions of (12). We will apply the nonlinear alternative of Leray-Schauder type to prove that S has at least one fixed point for small μ .

Let ϵ > 0 be such that (21) f ( x ( a ( t ) ) ) δ f ( 0 ) , 0 x ( a ( t ) ) ϵ , t [ 0,1 ] . Suppose that 0 < μ < ϵ / 2 m f ¯ ( ϵ ) = μ ¯ , where f ¯ ( t ) = max 0 s t f ( s ) ; then (22) f ¯ ( x ) = max 0 | x ( a ( t ) ) | x f ( x ( a ( t ) ) ) , t [ 0,1 ] . Since lim t 0 + ( f ¯ ( t ) / t ) = + , f ¯ ( ϵ ) / ϵ < 1 / 2 μ m , there exists a unique R μ ( 0 , ϵ ) such that (23) f ¯ ( R μ ) R μ = 1 2 μ m .

Let x K and τ ( 0,1 ) be such that x = τ S x . We claim that x R μ . In fact, (24) x ( t ) = τ t α - 1 Δ - ρ μ ( β ρ Δ 0 1 k ( η , s ) h + ( s ) f ( x ( a ( s ) ) ) d s m m m m m m m m + 0 1 κ ( s ) h + ( s ) f ( x ( a ( s ) ) ) d s β ρ Δ ) + τ μ ( β t α - 1 Δ 0 1 k ( η , s ) h + ( s ) f ( x ( a ( s ) ) ) d s m m m m m m + 0 1 k ( t , s ) h + ( s ) f ( x ( a ( s ) ) ) d s β t α - 1 Δ ) t α - 1 Δ - ρ μ f ¯ ( x ) × ( 0 1 k ( η , s ) h + ( s ) d s + 0 1 κ ( s ) h + ( s ) d s ) + μ f ¯ ( x ) × ( β t α - 1 Δ 0 1 k ( η , s ) h + ( s ) d s + 0 1 k ( t , s ) h + ( s ) d s ) = μ f ¯ ( x ) m ( t ) μ f ¯ ( x ) m . That is,       f ¯ ( x ) / x 1 / μ m , which implies that x R μ . Let U = { x K : x < R μ } . By Theorem 5, S has a fixed point x ¯ μ U ¯ . Moreover, combining (21) with the expression of operator S , we obtain that (25) x ¯ μ ( t ) μ δ f ( 0 ) m ( t ) , t ( 0,1 ] . Hence (12) has a positive solution x ¯ μ ( t ) . Note that R μ 0 as μ 0 ; we get that x ¯ μ 0 as μ 0 .

Theorem 7.

Suppose that (H 1)–(H 6) hold. Then there exists a positive number μ * > 0 such that (3) has at least one positive solution for μ ( 0 , μ * ) .

Proof.

Let (26) ω ( t ) = t α - 1 Δ - ρ ( β ρ Δ 0 1 k ( η , s ) h - ( s ) d s + 0 1 κ ( s ) h - ( s ) d s ) + β t α - 1 Δ 0 1 k ( η , s ) h - ( s ) d s + 0 1 k ( t , s ) h - ( s ) d s . Then ω ( t ) 0 for each t ( 0,1 ] . We have m ( t ) σ ω ( t ) , σ > 1 . Choose c ( 0,1 ) such that σ c > 1 . There is b > 0 such that f ( x ( a ( t ) ) ) σ c f ( 0 ) for x [ 0 , b ] ; then (27) ω ( t ) f ( x ( a ( t ) ) ) c m ( t ) f ( 0 ) for t ( 0,1 ] , x [ 0 , b ] . Fix δ ( c , 1 ) , and let μ * > 0 be such that (28) x ¯ μ + μ δ f ( 0 ) m b , μ ( 0 , μ * ) , where x ¯ μ is given by Lemma 6, and (29) | f ( x 1 ( a ( t ) ) ) - f ( x 2 ( a ( t ) ) ) | f ( 0 ) δ - c 2 , for x 1 , x 2 [ 0 , b ] with | x 1 - x 2 | μ * δ f ( 0 ) m .

Let μ ( 0 , μ * ) . We look for a solution x μ of the form x ¯ μ + v μ , where x ¯ μ is the solution of (12), given by Lemma 6. Thus v μ solves the following equation: (30) D α v μ = μ h + ( t ) ( f 1 - f 2 ) - μ h - ( t ) f 1 , v μ ( 0 ) = v μ ( 0 ) = 0 , v μ ( 1 ) = β v μ ( η ) + λ [ v μ ] , where f 1 = f ( x ¯ μ ( a ( t ) ) + v μ ( a ( t ) ) ) , f 2 = f ( x ¯ μ ( a ( t ) ) ) .

Now, we need to prove the existence of v μ . Consider the following equation: (31) D α v = μ h + ( t ) ( f 1 - f 2 ) - μ h - ( t ) f 1 , v ( 0 ) = v ( 0 ) = 0 , v ( 1 ) = β v ( η ) + λ [ v ] , where (32) f 1 = f ( x ¯ μ ( a ( t ) ) + v ( a ( t ) ) ) , f 2 = f ( x ¯ μ ( a ( t ) ) ) . Obviously, (31) is equivalent to the operator equation: (33) S v ( t ) = t α - 1 Δ - ρ μ ( β ρ Δ 0 1 k ( η , s ) h + ( s ) ( f 1 - f 2 ) d s m m m m m m m m + 0 1 κ ( s ) h + ( s ) ( f 1 - f 2 ) d s β ρ Δ ) + μ ( β t α - 1 Δ 0 1 k ( η , s ) h + ( s ) ( f 1 - f 2 ) d s m m m m m m + 0 1 k ( t , s ) h + ( s ) ( f 1 - f 2 ) d s β t α - 1 Δ ) - t α - 1 Δ - ρ μ ( β ρ Δ 0 1 k ( η , s ) h - ( s ) f 1 d s m m m m m m m m m + 0 1 κ ( s ) h - ( s ) f 1 d s β ρ Δ ) - μ ( β t α - 1 Δ 0 1 k ( η , s ) h - ( s ) f 1 d s m m m m m m + 0 1 k ( t , s ) h - ( s ) f 1 d s β t α - 1 Δ ) . It is easy to show that operator S : X X is completely continuous. Let v X and τ ( 0,1 ) such that v = τ S v . That is, (34) v ( t ) = τ t α - 1 Δ - ρ μ ( β ρ Δ 0 1 k ( η , s ) h + ( s ) ( f 1 - f 2 ) d s m m m m m m m m + 0 1 κ ( s ) h + ( s ) ( f 1 - f 2 ) d s β ρ Δ ) + τ μ ( β t α - 1 Δ 0 1 k ( η , s ) h + ( s ) ( f 1 - f 2 ) d s m m m m m m + 0 1 k ( t , s ) h + ( s ) ( f 1 - f 2 ) d s β t α - 1 Δ ) - τ t α - 1 Δ - ρ μ ( β ρ Δ 0 1 k ( η , s ) h - ( s ) f 1 d s m m m m m m m m m + 0 1 κ ( s ) h - ( s ) f 1 d s β ρ Δ ) - τ μ ( β t α - 1 Δ 0 1 k ( η , s ) h - ( s ) f 1 d s m m m m m m + 0 1 k ( t , s ) h - ( s ) f 1 d s β t α - 1 Δ ) . We claim that v μ δ f ( 0 ) m . Suppose on the contrary that v = μ δ f ( 0 ) m . Then, by (28) and (29), we get (35) x ¯ μ + v x ¯ μ + v b , | f 1 - f 2 | f ( 0 ) δ - c 2 . From (27), we get (36) ω ( t ) f ( x ( a ( t ) ) ) c m ( t ) f ( 0 ) , t ( 0,1 ] . Using (34)–(36), for each t ( 0,1 ] , we obtain that (37) | v ( t ) | t α - 1 Δ - ρ μ ( β ρ Δ 0 1 k ( η , s ) h + ( s ) | f 1 - f 2 | d s m m m m m i m + 0 1 κ ( s ) h + ( s ) | f 1 - f 2 | d s β ρ Δ ) + μ ( β t α - 1 Δ 0 1 k ( η , s ) h + ( s ) | f 1 - f 2 | d s m m m m + 0 1 k ( t , s ) h + ( s ) | f 1 - f 2 | d s β t α - 1 Δ ) + t α - 1 Δ - ρ μ ( β ρ Δ 0 1 k ( η , s ) h - ( s ) f 1 d s m m m m i m m m + 0 1 κ ( s ) h - ( s ) f 1 d s β ρ Δ ) + μ ( β t α - 1 Δ 0 1 k ( η , s ) h - ( s ) f 1 d s m m m i m + 0 1 k ( t , s ) h - ( s ) f 1 d s β t α - 1 Δ ) t α - 1 Δ - ρ μ f ( 0 ) δ - c 2 × ( β ρ Δ 0 1 k ( η , s ) h + ( s ) d s + 0 1 κ ( s ) h + ( s ) d s ) + μ f ( 0 ) δ - c 2 ( β t α - 1 Δ 0 1 k ( η , s ) h + ( s ) d s m m m m m m m m m m + 0 1 k ( t , s ) h + ( s ) d s β t α - 1 Δ ) + t α - 1 Δ - ρ μ f ¯ ( b ) ( β ρ Δ 0 1 k ( η , s ) h - ( s ) d s m m m m m m m m m m + 0 1 κ ( s ) h - ( s ) d s β ρ Δ ) + μ f ¯ ( b ) ( β t α - 1 Δ 0 1 k ( η , s ) h - ( s ) d s m m m m m m m + 0 1 k ( t , s ) h - ( s ) d s β t α - 1 Δ ) = μ f ( 0 ) δ - c 2 m ( t ) + μ f ¯ ( b ) ω ( t ) μ f ( 0 ) δ - c 2 m ( t ) + μ c f ( 0 ) m ( t ) = μ f ( 0 ) δ + c 2 m ( t ) . In particular, (38) v μ f ( 0 ) δ + c 2 m < μ f ( 0 ) δ m , which is a contradiction. And so the claim is proved. Let U = { x X : x < μ δ f ( 0 ) m } . By Theorem 5, S has a fixed point v μ U ¯ . Consequently, v μ μ δ f ( 0 ) m . This proves that there exists v μ this is the solution of (30). Hence v μ satisfies (37) and Lemma 6; then we get (39) x μ ( t ) x ¯ μ ( t ) - | v μ ( t ) | μ δ f ( 0 ) m ( t ) - μ f ( 0 ) δ + c 2 m ( t ) = μ f ( 0 ) δ - c 2 m ( t ) > 0 ; that is, x μ is a positive solution of (3). So the proof of Theorem 7 is complete.

4. An Example

In this section, we give an example to illustrate the result of this paper. Consider the following nonlinear fractional differential equation: (40) D 5 / 2 x ( t ) - μ ( 4 5 - t ) ( x ( 4 ) ( t ) + sin 2 x ( t ) + 1 10 ) = 0 , x ( 0 ) = x ( 0 ) = 0 , x ( 1 ) = 1 2 x ( 1 2 ) + 0 1 x ( t ) ( 3 t - 1 ) d t .

Let f ( x ( a ( t ) ) ) = x ( 4 ) ( t ) + sin 2 x ( t ) + 1 / 10 , a ( t ) = t and h ( t ) = 4 / 5 - t . Obviously, all assumptions (H 1)–(H 3) hold. In the following, we will verify that assumptions (H 4)–(H 6) hold also.

(i) It is obvious that (41) β η α - 1 + λ [ p ] = β η α - 1 + 0 1 t α - 1 ( 3 t - 1 ) d t = β η α - 1 + 2 α - 1 α ( α + 1 ) = 1 4 2 + 16 35 implies (H 4).

(ii) By direct calculation, we have (42) 0 1 ( 3 t - 1 ) d t = 1 2 , 0 1 t α - 1 ( 3 t - 1 ) d t = 16 35 , κ ( s ) = 0 1 k ( t , s ) ( 3 t - 1 ) d t = 1 α ( α + 1 ) ( 1 - s ) α - 1 s ( 3 s + 2 α - 4 ) = 4 35 ( 1 - s ) 3 / 2 s ( 3 s + 1 ) 0 , so assumption (H 5) holds.

(iii) Finally, we check assumption (H 6). It means that there exists ϵ > 0 such that m ( t ) ( 1 + ϵ ) ω ( t ) , t ( 0,1 ] . Note that (43) h + ( t ) = max { 0 , h ( t ) } = { 4 5 - t , 0 t 4 5 , 0 , 4 5 < t 1 , h - ( t ) = max { 0 , - h ( t ) } = { 0 , 0 t 4 5 , t - 4 5 , 4 5 < t 1 . We now verify that there exists ϵ 1 > 0 such that (44) t α - 1 Δ - ρ β ρ Δ 0 1 k ( η , s ) h + ( s ) d s ( 1 + ϵ 1 ) t α - 1 Δ - ρ β ρ Δ 0 1 k ( η , s ) h - ( s ) d s , t ( 0,1 ] ; that is, (45) 0 1 k ( 1 2 , s ) ( 4 5 - s ) d s ϵ 1 4 / 5 1 k ( 1 2 , s ) ( s - 4 5 ) d s , m m m m m i m m m t ( 0,1 ] . By simple calculation, we get (46) 0 1 k ( 1 2 , s ) ( 4 5 - s ) d s = 13 2 700 , 4 / 5 1 k ( 1 2 , s ) ( s - 4 5 ) d s = 2 7 × 5 4 × 10 . Setting ϵ 1 ( 0 , ( 13 × 5 2 × 5 ) / 4 ) , then inequality (44) holds. Similarly, there exists ϵ 2 , ϵ 3 , ϵ 4 > 0 such that (47) t α - 1 Δ - ρ 0 1 κ ( s ) h + ( s ) d s ( 1 + ϵ 2 ) t α - 1 Δ - ρ 0 1 κ ( s ) h - ( s ) d s , t ( 0,1 ] , β t α - 1 Δ 0 1 k ( η , s ) h + ( s ) d s ( 1 + ϵ 3 ) β t α - 1 Δ 0 1 k ( η , s ) h - ( s ) d s , t ( 0,1 ] , 0 1 k ( t , s ) h + ( s ) d s ( 1 + ϵ 4 ) 0 1 k ( t , s ) h - ( s ) d s , t ( 0,1 ] . Let ϵ = min { ϵ 1 , ϵ 2 , ϵ 3 , ϵ 4 } . By (44)–(47), we obtain that there exists ϵ > 0 such that (48) m ( t ) ( 1 + ϵ ) ω ( t ) , t ( 0,1 ] . Thus assumption (H 6) holds. By applying Theorem 7, we know that there exists a number μ * > 0 such that (40) has at least one positive solution for μ ( 0 , μ * ) .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Zhaocai Hao acknowledges the support from NSFC (11371221) and the Education Department of Shandong Province Science and Technology Plan Project (J13LI01). The authors are grateful to the anonymous referees for their helpful suggestions and comments.

Hilfer R. Applications of Fractional Calculus in Physics 2000 World Scientific Metzler F. Schick W. Kilian H. G. Nonnenmacher T. F. Relaxation in filled polymers: a fractional calculus approach The Journal of Chemical Physics 1995 103 16 7180 7186 10.1063/1.470346 Agarwal R. P. Benchohra M. Slimani B. Existence results for differential equations with fractional order and impulses Georgian Academy of Sciences A: Razmadze Mathematical Institute: Memoirs on Differential Equations and Mathematical Physics 2008 44 1 21 MR2527032 Ahmed E. El-Saka H. A. On fractional order models for Hepatitis C Nonlinear Biomedical Physics 2010 4, article 1 10.1186/1753-4631-4-1 2-s2.0-77952940914 Bai C. Positive solutions for nonlinear fractional differential equations with coefficient that changes sign Nonlinear Analysis: Theory, Methods and Applications 2006 64 4 677 685 10.1016/j.na.2005.04.047 MR2197088 2-s2.0-29244448297 Bai Z. On positive solutions of a nonlocal fractional boundary value problem Nonlinear Analysis: Theory, Methods & Applications 2010 72 2 916 924 10.1016/j.na.2009.07.033 MR2579357 2-s2.0-71649083074 Diethelm K. Freed A. D. Keil F. Mackens W. Voss H. Werther J. On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties 1999 Heidelberg, Germany Springer 217 307 Ding X. Jiang Y. Waveform relaxation methods for fractional functional differential equations Fractional Calculus and Applied Analysis 2013 16 3 573 594 10.2478/s13540-013-0037-4 MR3071202 2-s2.0-84879772044 Gaul L. Klein P. Kemple S. Damping description involving fractional operators Mechanical Systems and Signal Processing 1991 5 2 81 88 10.1016/0888-3270(91)90016-X 2-s2.0-44949282247 Jankowski T. Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions Nonlinear Analysis 2012 75 2 913 923 10.1016/j.na.2011.09.025 MR2847466 2-s2.0-80055016109 Jankowski T. Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions Nonlinear Analysis: Theory, Methods and Applications 2011 74 11 3775 3785 10.1016/j.na.2011.03.022 MR2803102 2-s2.0-79955551764 Kilbas A. A. Srivastava H. M. Trujillo J. J. Theory and Applications of Fractional Differential Equations 2006 204 Amsterdam, The Netherlands Elsevier Science North-Holland Mathematics Studies MR2218073 Kilbas A. A. Trujillo J. J. Differential equations of fractional order: methods, results and problems. I Applicable Analysis 2001 78 1-2 153 192 10.1080/00036810108840931 MR1887959 Kilbas A. A. Trujillo J. J. Differential equations of fractional order: methods, results and problems. II Applicable Analysis 2002 81 2 435 493 10.1080/0003681021000022032 MR1928462 Koeller R. C. Applications of fractional calculus to the theory of viscoelasticity Journal of Applied Mechanics 1984 51 2 299 307 10.1115/1.3167616 MR747787 2-s2.0-0021440218 Kou C. Zhou H. Yan Y. Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis Nonlinear Analysis 2011 74 17 5975 5986 10.1016/j.na.2011.05.074 MR2833368 Kumar P. Agrawal O. P. An approximate method for numerical solution of fractional differential equations Signal Processing 2006 86 10 2602 2610 10.1016/j.sigpro.2006.02.007 ZBLl1172.94436 2-s2.0-33745712076 Lakshmikantham V. Leela S. Vasundhara J. Theory of Fractional Dynamic Systems 2009 Cambridge, UK Cambridge Academic Lakshmikantham V. Vatsala A. S. Basic theory of fractional differential equations Nonlinear Analysis: Theory, Methods & Applications 2008 69 8 2677 2682 10.1016/j.na.2007.08.042 MR2446361 Miller K. S. Ross B. An Introduction to the Fractional Calculus and Differential Equations 1993 New York, NY, USA John Wiley and Sons MR1219954 Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications 1999 San Diego, Calif, USA Academic Press MR1658022 Podlubny I. Geometric and physical interpretation of fractional integration and fractional differentiation Fractional Calculus and Applied Analysis 2002 5 4 367 386 MR1967839 Srivastava H. M. Saxena R. K. Operators of fractional integration and their applications Applied Mathematics and Computation 2001 118 1 1 52 10.1016/S0096-3003(99)00208-8 MR1805158 ZBLl1022.26012 2-s2.0-0012072325 Zhang X. Liu L. Wu Y. Multiple positive solutions of a singular fractional differential equation with negatively perturbed term Mathematical and Computer Modelling 2012 55 3-4 1263 1274 10.1016/j.mcm.2011.10.006 ZBLl1255.34010 2-s2.0-84855198120 Zhang X. Liu L. Wu Y. Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives Applied Mathematics and Computation 2012 219 4 1420 1433 10.1016/j.amc.2012.07.046 2-s2.0-84867579253 Wang Y. Liu L. Wu Y. Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity Nonlinear Analysis: Theory, Methods & Applications 2011 74 17 6434 6441 10.1016/j.na.2011.06.026 MR2833427 2-s2.0-80051599946 Zhang X. Liu L. Wu Y. The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives Applied Mathematics and Computation 2012 218 17 8526 8536 10.1016/j.amc.2012.02.014 ZBLl1254.34016 2-s2.0-84862776655 Zhang X. Liu L. Wu Y. The uniqueness of positive solution for a singular fractional differential system involving derivatives Communications in Nonlinear Science and Numerical Simulation 2013 18 6 1400 1409 10.1016/j.cnsns.2012.08.033 ZBLl1283.34006 2-s2.0-84872405408 Zhang X. Liu L. Wiwatanapataphee B. Wu Y. Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives Abstract and Applied Analysis 2012 2012 16 512127 10.1155/2012/512127 2-s2.0-84862273389 Zhang X. Liu L. Wu Y. Lu Y. The iterative solutions of nonlinear fractional differential equations Applied Mathematics and Computation 2013 219 9 4680 4691 10.1016/j.amc.2012.10.082 2-s2.0-84871234727