The Existence of Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equations

and Applied Analysis 3 Then, for all u ∈ Ω, it is satisfied that |Au (t)| ≤ ∫ 1 0 G (t, s) f (s, u (s)) ds ≤ L∫ 1 0 G (t, s) ds, ∀t ∈ [0, 1] . (15) That is, the set A(Ω) is bounded in E. Finally, we show that A is equicontinuous. For each u ∈ Ω, we have 󵄨 󵄨 󵄨 󵄨 󵄨 (Au) 󸀠 (t) 󵄨 󵄨 󵄨 󵄨 󵄨 = 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 − ∫ t 0 (α − 1) (t − s) α−2 Γ (α) f (s, u (s)) ds +∫ 1 0 (α − 1) [t (1 − s)] α−2 Γ (α) f (s, u (s)) ds 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨


Introduction
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, or polymer rheology; see [1][2][3][4][5].The interest of the study of fractionalorder differential equations lies in the fact that fractionalorder models are more accurate than integer-order models; that is, there are more degrees of freedom in the fractionalorder models.Recently, there are some papers dealing with the existence of solutions (or positive solution) of nonlinear initial value problems of fractional differential equations by the use of techniques of nonlinear analysis (fixed-point theorems, Leray-Schauder theory, lower and upper solution method, Adomian decomposition method, ect.); see [6][7][8][9][10][11][12][13][14][15].
The famous viscous liquid flow problems in the fields of integer-order differential equations can be described by thirdorder ordinary differential equation boundary value problem −  () =  (,  ()) , 0 <  < 1,  (0) =   (0) =   (1) = 0, (1) where  : [0, 1] × [0, +∞) → [0, +∞) is continuous [16][17][18].However, there are only a few exisitng contributions, as far as we know, in the field of fractional-order differential equation.In this paper, we discuss the existence of positive solution for the nonlinear fractional differential equations boundary value problem (BVP) where 2 <  ≤ 3 is a real number,   0 + is the Riemann-Liouville fractional derivative, and  : For a more general case, specially, where  : (0, 1) → [0, +∞) is continuous with ∫ 1 0 () > 0,  ∈ ([0, +∞), [0, +∞)) and   0 + is the Riemann-Liouville fractional derivative; El-Shahed [20] obtained the existence and nonexistence of positive solutions by employing the well-known Guo-Krasnoselskii fixed point theorem of cone extension or compression.The purpose of this paper is to extend this result.Our argument is based on the fixed point index theory, which is more precise than the fixed point theorem of cone extension or compression.We will employ the theory of fixed point index in cones to present some more extensive conditions on  guaranteeing the existence

Preliminaries
In this section, we introduce some preliminary facts which are used throughout this paper.For details, see [19].
In the following, we present the Green's function of fractional differential equation boundary value problem.
It is clear, form Lemma 5, that the nontrivial fixed points of operator  coincide with the positive solutions of BVP (2).
Proof.From the continuity and the nonnegativeness of functions  and  on their domains of definition, we have that if  ∈ , then  ∈  and () ≥ 0 for all  ∈ [0, 1]; from properties (2) and (3) of Lemma 6, for all  ∈ , Hence, () ⊂ .
Next, we show that  is uniformly bounded.
Then, for all  ∈ Ω, it is satisfied that That is, the set (Ω) is bounded in .
Finally, we show that  is equicontinuous.

Define an operator
Clearly,  also is a completely continuous linear operator and () ⊂ .
This implies that The proof is completed.
Hereafter, we use () to denote the spectral radius of the operator .Lemma 9. Suppose that  is defined by (18); then the spectral radius () > 0.
To prove the existence of at least one positive solution of BVP ( 2), we will find the nonzero fixed point of  (defined in (11)) by using the fixed point index theory in cones.
We recall some concepts and conclusions on the fixed point index in cones in [21,22], which will be used in the argument later.Let  be a Banach space and let  ⊂  be a closed convex cone in .Assume that Ω is a bounded open subset of  with boundary Ω and ∩Ω ̸ = 0. Let  : ∩Ω →  be a completely continuous mapping.If  ̸ =  for every  ∈  ∩ Ω, then the fixed point index (,  ∩ Ω, ) is well defined.One important fact is that if (,  ∩ Ω, ) ̸ = 0, then  has a fixed point in  ∩ Ω.

Main Results
In this section we show the existence of positive solutions of BVP (2) by using the fixed point index theory in cones.
Proof.Let  1 be the positive eigenfunction of  corresponding to  1 ; thus  1 =  1  1 , where  is defined by (18).Choose  ∈ (0, ), where  is the constant in assumption (F1).For every  ∈   , from assumption (F1), we have Namely, Suppose that  has no fixed point on   (otherwise, the proof is completed).Now we show that If it is not true, there exist  0 ∈   and  0 > 0 (if  0 = 0, the proof is completed) such that Then, That is, Let It is easy to see that 0 <  0 ≤  * < +∞ and  0 ≥  *  1 .Taking into account the positivity of the Green's function (, ) and definition of the operator , it is easy to know that  is a nondecreasing linear operator, so Therefore by ( 33) which contradicts the definition of  * .Hence (34) holds and we have from Lemma 10 that  (,   , ) = 0.
On the other hand, we choose  >  > 0. Now we show that if  is large enough, then From (F2),  <  1 ; then there exist 0 <  < 1, such that  =  1 .