Three-point boundary value problems of fractional functional differential equations with delay

  • Yanan Li1,

    Affiliated with

    • Shurong Sun1Email author,

      Affiliated with

      • Dianwu Yang1 and

        Affiliated with

        • Zhenlai Han1

          Affiliated with

          Boundary Value Problems20132013:38

          DOI: 10.1186/1687-2770-2013-38

          Received: 6 December 2012

          Accepted: 7 February 2013

          Published: 22 February 2013

          Abstract

          In this paper, we study three-point boundary value problems of the following fractional functional differential equations involving the Caputo fractional derivative:

          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equa_HTML.gif

          where D α C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq1_HTML.gif, D β C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq2_HTML.gif denote Caputo fractional derivatives, 2 < α < 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq3_HTML.gif, 0 < β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq4_HTML.gif, η ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq5_HTML.gif, 1 < λ < 1 2 η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq6_HTML.gif. We use the Green function to reformulate boundary value problems into an abstract operator equation. By means of the Schauder fixed point theorem and the Banach contraction principle, some existence results of solutions are obtained, respectively. As an application, some examples are presented to illustrate the main results.

          MSC:34A08, 34K37.

          Keywords

          fractional functional differential equation delay three-point boundary value problems fixed point theorem existence of solutions

          1 Introduction

          Fractional calculus is a branch of mathematics, it is an emerging field in the area of the applied mathematics that deals with derivatives and integrals of arbitrary orders as well as with their applications. The origins can be traced back to the end of the seventeenth century. During the history of fractional calculus, it was reported that the pure mathematical formulations of the investigated problems started to be addressed with more applications in various fields. With the help of fractional calculus, we can describe natural phenomena and mathematical models more accurately. Therefore, fractional differential equations have received much attention and the theory and its application have been greatly developed; see [16].

          Recently, there have been many papers focused on boundary value problems of fractional ordinary differential equations [715] and an initial value problem of fractional functional differential equations [1628]. But the results dealing with the boundary value problems of fractional functional differential equations with delay are relatively scarce [2935]. It is well known that in practical problems, the behavior of systems not only depends on the status just at the present, but also on the status in the past. Thus, in many cases, we must consider fractional functional differential equations with delay in order to solve practical problems. Consequently, our aim in this paper is to study the existence of solutions for boundary value problems of fractional functional differential equations.

          In 2011, Rehman [12] studied the existence and uniqueness of solutions to nonlinear three-point boundary value problems for the following fractional differential equation:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equb_HTML.gif

          where 1 < δ < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq7_HTML.gif, 0 < σ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq8_HTML.gif, α , β R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq9_HTML.gif, α η ( 1 β ) + ( 1 α ) ( t β η ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq10_HTML.gif and D 0 + δ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq11_HTML.gif, D 0 + σ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq12_HTML.gif denote Caputo fractional derivatives. By the Banach contraction principle and the Schauder fixed point theorem, they obtained some new existence and uniqueness results.

          For 0 < r < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq13_HTML.gif, we denote by C r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq14_HTML.gif the Banach space of all continuous functions φ : [ r , 0 ] R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq15_HTML.gif endowed with the sup-norm
          φ [ r , 0 ] : = sup { | φ ( s ) | : s [ r , 0 ] } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equc_HTML.gif
          If u : [ r , 1 ] R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq16_HTML.gif, then for any t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq17_HTML.gif, we denote by u t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq18_HTML.gif the element of C r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq14_HTML.gif defined by
          u t ( θ ) = u ( t + θ ) , for  θ [ r , 0 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equd_HTML.gif
          Enlightened by literature [12], in this paper we study the following three-point boundary value problem for the fractional functional differential equation:
          D α C u ( t ) = f ( t , u t , C D β u ( t ) ) , 0 < t < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ1_HTML.gif
          (1.1)
          where 2 < α < 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq3_HTML.gif, 0 < β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq4_HTML.gif and D α C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq1_HTML.gif, D β C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq2_HTML.gif denote Caputo fractional derivatives, f ( t , u t , C D β u ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq19_HTML.gif is a continuous function associated with the boundary conditions
          u ( 0 ) = 0 , u ( 1 ) = λ u ( η ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ2_HTML.gif
          (1.2)
          and u 0 = φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq20_HTML.gif, where η ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq5_HTML.gif, 1 < λ < 1 2 η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq6_HTML.gif and φ is an element of the space
          C r + ( 0 ) : = { ψ C r | ψ ( s ) 0 , s [ r , 0 ] , ψ ( 0 ) = 0 , C D β ψ ( s ) = 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Eque_HTML.gif

          To the best of our knowledge, no one has studied the existence of positive solutions for problem (1.1)-(1.2). The aim of this paper is to fill the gap in the relevant literatures. In this paper, we firstly give the fractional Green function and some properties of the Green function. Consequently, boundary value problem (1.1) and (1.2) is reduced to an equivalent Fredholm integral equation. Then we extend the existence results for boundary value problems of an ordinary fractional differential equation of δ-order ( 1 < δ < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq7_HTML.gif) in [12] to a fractional functional differential equation of α-order ( 2 < α < 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq3_HTML.gif). As an application, some examples are presented to illustrate the main results.

          2 Preliminaries

          For the convenience of the reader, we give the following background material from fractional calculus theory to facilitate the analysis of boundary value problem (1.1) and (1.2). This material can be found in the recent literature; see [1, 2, 36].

          Definition 2.1 ([1])

          The fractional integral of order α ( α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq21_HTML.gif) of a function f : ( t 0 , + ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq22_HTML.gif is given by
          I α f ( t ) = 1 Γ ( α ) t 0 t f ( s ) ( t s ) 1 α d s , t > t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equf_HTML.gif

          where Γ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq23_HTML.gif is the gamma function, provided that the right-hand side is point-wise defined on ( t 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq24_HTML.gif.

          Definition 2.2 ([1])

          The Caputo fractional derivative of order α ( n 1 < α < n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq25_HTML.gif) of a function f : ( t 0 , + ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq22_HTML.gif is given by
          D α C f ( t ) = 1 Γ ( n α ) t 0 t f ( n ) ( s ) ( t s ) α + 1 n d s , t > t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equg_HTML.gif

          where Γ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq23_HTML.gif is the gamma function, provided that the right-hand side is point-wise defined on ( t 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq24_HTML.gif.

          Obviously, the Caputo derivative for every constant function is equal to zero.

          From the definition of the Caputo derivative, we can acquire the following statement.

          Lemma 2.1 ([2])

          Let f ( t ) L 1 [ t 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq26_HTML.gif. Then
          D α C ( I α f ( t ) ) = f ( t ) , t > t 0 and 0 < α < 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equh_HTML.gif

          Lemma 2.2 ([2])

          Let α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq21_HTML.gif. Then
          I α C D α f ( t ) = f ( t ) c 1 c 2 t c n t n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equi_HTML.gif

          for some c i R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq27_HTML.gif, i = 1 , 2 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq28_HTML.gif, where n = [ α ] + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq29_HTML.gif and [ α ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq30_HTML.gif denotes the integer part of α.

          Next, we introduce the Green function of fractional functional differential equations boundary value problems.

          Lemma 2.3 Let 2 < α < 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq3_HTML.gif, 0 < η < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq31_HTML.gif, 1 < λ < 1 2 η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq6_HTML.gif and h : [ 0 , 1 ] R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq32_HTML.gif be continuous. Then the boundary value problem
          D α C u ( t ) = h ( t ) , 0 < t < 1 , u ( 0 ) = u ( 0 ) = 0 , u ( 1 ) = λ u ( η ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ3_HTML.gif
          (2.1)
          has a unique solution
          u ( t ) = 0 1 G ( t , s ) h ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ4_HTML.gif
          (2.2)
          where
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ5_HTML.gif
          (2.3)
          Proof From equation (2.1), we know
          I α C D α u ( t ) = I α h ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equj_HTML.gif
          From Lemma 2.2, we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ6_HTML.gif
          (2.4)
          According to (2.1), we know that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equk_HTML.gif
          By u ( 1 ) = λ u ( η ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq33_HTML.gif, we have
          c 3 = α 1 ( 2 2 λ η ) Γ ( α ) ( 0 1 ( 1 s ) α 2 h ( s ) d s λ 0 η ( η s ) α 2 h ( s ) d s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equl_HTML.gif
          Therefore,
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equm_HTML.gif
          Now, for t η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq34_HTML.gif, we have
          u ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 h ( s ) d s + ( α 1 ) t 2 ( 2 2 λ η ) Γ ( α ) ( ( 0 t + t η + η 1 ) ( 1 s ) α 2 h ( s ) d s λ ( 0 t + t η ) ( η s ) α 2 h ( s ) d s ) = 1 Γ ( α ) 0 t ( ( t s ) α 1 + ( α 1 ) t 2 2 2 λ η ( ( 1 s ) α 2 λ ( η s ) α 2 ) ) h ( s ) d s + 1 Γ ( α ) t η ( α 1 ) t 2 2 2 λ η ( ( 1 s ) α 2 λ ( η s ) α 2 ) h ( s ) d s + 1 Γ ( α ) η 1 ( α 1 ) t 2 2 2 λ η ( 1 s ) α 2 h ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equn_HTML.gif
          For t η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq35_HTML.gif, we have
          u ( t ) = 1 Γ ( α ) ( 0 η + η t ) ( t s ) α 1 h ( s ) d s + ( α 1 ) t 2 ( 2 2 λ η ) Γ ( α ) ( ( 0 η + η t + t 1 ) ( 1 s ) α 2 h ( s ) d s λ 0 η ( η s ) α 2 h ( s ) d s ) = 1 Γ ( α ) 0 η ( ( t s ) α 1 + ( α 1 ) t 2 2 2 λ η ( ( 1 s ) α 2 λ ( η s ) α 2 ) ) h ( s ) d s + 1 Γ ( α ) η t ( ( t s ) α 1 + ( α 1 ) t 2 2 2 λ η ( 1 s ) α 2 ) h ( s ) d s + 1 Γ ( α ) t 1 ( α 1 ) t 2 2 2 λ η ( 1 s ) α 2 h ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equo_HTML.gif
          Hence, we can conclude (2.2) holds, where
          G ( t , s ) = 1 Γ ( α ) { ( t s ) α 1 + ( α 1 ) t 2 2 2 λ η ( ( 1 s ) α 2 λ ( η s ) α 2 ) , 0 s t 1 , s η , ( t s ) α 1 + ( α 1 ) t 2 2 2 λ η ( 1 s ) α 2 , 0 s t 1 , η s , ( α 1 ) t 2 2 2 λ η ( ( 1 s ) α 2 λ ( η s ) α 2 ) , 0 t s 1 , s η , ( α 1 ) t 2 2 2 λ η ( 1 s ) α 2 , 0 t s 1 , η s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equp_HTML.gif

          The proof is completed. □

          Lemma 2.4 ([36] Schauder fixed point theorem)

          Let ( D , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq36_HTML.gif be a complete metric space, U be a closed convex subset of D, and T : D D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq37_HTML.gif be the map such that the set T u : u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq38_HTML.gif is relatively compact in D. Then the operator T has at least one fixed point u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq39_HTML.gif:
          T u = u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equq_HTML.gif

          3 Main results

          In this section, we discuss the existence and uniqueness of solutions for boundary value problem (1.1) and (1.2) by the Schauder fixed point theorem and the Banach contraction principle.

          For convenience, we define the Banach space X = { u | u C [ r , 1 ] , C D β u C [ r , 1 ] , 0 < β < 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq40_HTML.gif. Also, if I is an interval of the real line ℝ, by C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq41_HTML.gif and C 1 ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq42_HTML.gif we denote the set of continuous and continuously differentiable functions on I, respectively. Moreover, for u C ( I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq43_HTML.gif, we define
          u I = max t I | u ( t ) | + max t I | C D β u ( t ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ7_HTML.gif
          (3.1)
          For u 0 = φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq20_HTML.gif, in view of the definitions of u t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq18_HTML.gif and φ, we have
          u 0 = u ( θ ) = φ ( θ ) , for  θ [ r , 0 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equr_HTML.gif
          Thus, we have
          u ( t ) = φ ( t ) , for  t [ r , 0 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equs_HTML.gif
          Since f : [ 0 , 1 ] × C r × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq44_HTML.gif is a continuous function, set f ( t , u t , C D β u ( t ) ) : = h ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq45_HTML.gif in Lemma 2.3. We have by Lemma 2.3 that a function u is a solution of boundary value problem (1.1) and (1.2) if and only if it satisfies
          u ( t ) = { 0 1 G ( t , s ) f ( s , u s , C D β u ( s ) ) d s , t ( 0 , 1 ) , φ ( t ) , t [ r , 0 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equt_HTML.gif
          We define an operator T : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq46_HTML.gif as follows:
          T u ( t ) = u ( t ) = { 0 1 G ( t , s ) f ( s , u s , C D β u ( s ) ) d s , t ( 0 , 1 ) , φ ( t ) , t [ r , 0 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ8_HTML.gif
          (3.2)
          and
          l = max 0 t 1 ( 0 1 | G ( t , s ) g ( s ) | d s ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equu_HTML.gif
          l = max 0 t 1 ( 0 1 | t G ( t , s ) g ( s ) | d s ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equv_HTML.gif
          Q = 1 Γ ( 2 β ) + 1 + λ η α 1 Γ ( 2 β ) ( 1 λ η ) + 1 α + λ η α 1 + 1 2 2 λ η . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equw_HTML.gif

          Theorem 3.1 Assume the following:

          (H1) There exists a nonnegative function g L [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq47_HTML.gif such that
          | f ( t , v , w ) | g ( t ) + a | v | k 1 + b | w | k 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equx_HTML.gif

          for each v C r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq48_HTML.gif, w R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq49_HTML.gif, where a , b R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq50_HTML.gif are nonnegative constants and 0 < k 1 , k 2 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq51_HTML.gif; or

          (H2) There exists a nonnegative function g L [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq47_HTML.gif such that
          | f ( t , v , w ) | g ( t ) + a | v | k 1 + b | w | k 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equy_HTML.gif

          for each v C r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq48_HTML.gif, w R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq49_HTML.gif, where a , b R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq50_HTML.gif are nonnegative constants and k 1 , k 2 > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq52_HTML.gif.

          Then boundary value problem (1.1) and (1.2) has a solution.

          Proof Suppose (H1) holds. Choose
          ω max { 3 ( l + l Γ ( 2 β ) ) , ( 3 a Q ) 1 1 k 1 , ( 3 b Q ) 1 1 k 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ9_HTML.gif
          (3.3)

          and define the cone U = { u X | u ω , ω > 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq53_HTML.gif.

          For any u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq54_HTML.gif, we have
          | T u ( t ) | = | 0 1 G ( t , s ) f ( s , u s , C D β u ( s ) ) d s | 0 1 | G ( t , s ) g ( s ) | d s + ( a | ω | k 1 + b | ω | k 2 ) ( 0 t ( t s ) α 1 Γ ( α ) d s + ( α 1 ) t 2 2 2 λ η 0 1 ( 1 s ) α 2 Γ ( α ) d s + ( α 1 ) λ t 2 2 2 λ η 0 η ( η s ) α 2 Γ ( α ) d s ) l + ( a | ω | k 1 + b | ω | k 2 ) ( t α α Γ ( α ) + t 2 ( 2 2 λ η ) Γ ( α ) + λ t 2 η α 1 ( 2 2 λ η ) Γ ( α ) ) l + a | ω | k 1 + b | ω | k 2 Γ ( α ) ( 1 α + λ η α 1 + 1 2 2 λ η ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ10_HTML.gif
          (3.4)
          Also,
          | T u ( t ) | 0 1 | t G ( t , s ) | | f ( s , u s , C D β u ( s ) ) | d s 0 1 | t G ( t , s ) g ( s ) | d s + ( a | ω | k 1 + b | ω | k 2 ) ( 0 t ( t s ) α 2 Γ ( α 1 ) d s + 2 ( α 1 ) t 2 2 λ η 0 1 ( 1 s ) α 2 Γ ( α ) d s + 2 ( α 1 ) λ t 2 2 λ η 0 η ( η s ) α 2 Γ ( α ) d s ) l + ( a | ω | k 1 + b | ω | k 2 ) ( t α 1 Γ ( α ) + t ( 1 λ η ) Γ ( α ) + λ t η α 1 ( 1 λ η ) Γ ( α ) ) l + a | ω | k 1 + b | ω | k 2 Γ ( α ) ( 1 + 1 + λ η α 1 1 λ η ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equz_HTML.gif
          Hence,
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equaa_HTML.gif
          In view of (3.1) and (3.3), we obtain
          T u ( t ) l + l Γ ( 2 β ) + a | ω | k 1 + b | ω | k 2 Γ ( α ) ( 1 Γ ( 2 β ) + 1 + λ η α 1 Γ ( 2 β ) ( 1 λ η ) + 1 α + λ η α 1 + 1 2 2 λ η ) ω 3 + ( a | ω | k 1 + b | ω | k 2 ) Q ω 3 + ω 3 + ω 3 ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ11_HTML.gif
          (3.5)

          which implies that T : U U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq55_HTML.gif. The continuity of the operator T follows from the continuity of f and G.

          Now, if (H2) holds, we choose
          0 < ω min { 3 ( l + l Γ ( 2 β ) ) , ( 1 3 a Q ) 1 1 k 1 , ( 1 3 b Q ) 1 1 k 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ12_HTML.gif
          (3.6)
          and by the same process as above, we obtain
          T u ( t ) l + l Γ ( 2 β ) + a | ω | k 1 + b | ω | k 2 Γ ( α ) ( 1 Γ ( 2 β ) + 1 + λ η α 1 Γ ( 2 β ) ( 1 λ η ) + 1 α + λ η α 1 + 1 2 2 λ η ) ω 3 + ( a | ω | k 1 + b | ω | k 2 ) Q ω 3 + ω 3 + ω 3 ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equab_HTML.gif

          which implies that T : U U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq55_HTML.gif.

          Now, we show that T is a completely continuous operator.

          Let L = max 0 t 1 | f ( t , u t , C D β u ( t ) ) | + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq56_HTML.gif. Then for u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq54_HTML.gif and t 1 , t 2 [ r , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq57_HTML.gif with t 1 < t 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq58_HTML.gif, in view of Lemma 2.3, if 0 t 1 < t 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq59_HTML.gif, then
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equac_HTML.gif
          If r t 1 < t 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq60_HTML.gif, then
          | T u ( t 2 ) T u ( t 1 ) | = | φ ( t 2 ) φ ( t 1 ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equad_HTML.gif
          If r t 1 < 0 < t 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq61_HTML.gif, then
          | T u ( t 2 ) T u ( t 1 ) | | T u ( t 2 ) T u ( 0 ) | + | T u ( 0 ) T u ( t 1 ) | 0 1 | G ( t 2 , s ) G ( 0 , s ) | | f ( s , u s , C D β u ( s ) ) | d s + | φ ( 0 ) φ ( t 1 ) | L Γ ( α ) | t 2 α α + t 2 2 ( 1 λ η α 1 ) 2 2 λ η | + φ ( t 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equae_HTML.gif
          Hence, if 0 t 1 < t 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq59_HTML.gif, we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equaf_HTML.gif
          If r t 1 < t 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq60_HTML.gif, in view of the definition of φ, we have
          | C D β T u ( t 2 ) C D β T u ( t 1 ) | = | C D β φ ( t 2 ) C D β φ ( t 1 ) | = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equag_HTML.gif
          If r t 1 < 0 < t 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq61_HTML.gif, then
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equah_HTML.gif
          Hence, if 0 t 1 < t 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq59_HTML.gif, we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equai_HTML.gif
          If r t 1 < t 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq60_HTML.gif, we have
          T u ( t 2 ) T u ( t 1 ) = φ ( t 2 ) φ ( t 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equaj_HTML.gif
          If r t 1 < 0 < t 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq61_HTML.gif, then
          T u ( t 2 ) T u ( t 1 ) 1 Γ ( α ) | t 2 α α + t 2 2 ( 1 λ η α 1 ) 2 2 λ η | + φ ( t 1 ) + L t 2 1 β Γ ( 2 β ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equak_HTML.gif

          In any case, it implies that T u ( t 2 ) T u ( t 1 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq62_HTML.gif as t 2 t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq63_HTML.gif, i.e., for any ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq64_HTML.gif, there exists δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq65_HTML.gif, independent of t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq66_HTML.gif, t 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq67_HTML.gif and u, such that | T u ( t 2 ) T u ( t 1 ) | ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq68_HTML.gif, whenever | t 2 t 1 | < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq69_HTML.gif. Therefore T : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq46_HTML.gif is completely continuous. The proof is completed. □

          For convenience, we denote
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equal_HTML.gif

          Theorem 3.2 Assume that

          (H3) There exists a constant p > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq70_HTML.gif such that | f ( t , μ , ν ) f ( t , μ ¯ , ν ¯ ) | p ( | μ μ ¯ | + | ν ν ¯ | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq71_HTML.gif for each μ , μ ¯ C r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq72_HTML.gif, ν , ν ¯ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq73_HTML.gif. If
          p < ( M + N ) 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equam_HTML.gif

          then boundary value problem (1.1) and (1.2) has a unique solution.

          Proof Consider the operator T : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq46_HTML.gif defined by (3.2). Clearly, the fixed point of the operator T is the solution of boundary value problem (1.1) and (1.2). We will use the Banach contraction principle to prove that T has a fixed point. We first show that T is a contraction. For each t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq17_HTML.gif,
          | T u ( t ) T u ¯ ( t ) | 0 1 | G ( t , s ) | | f ( s , u s , C D β u ( s ) ) f ( s , u ¯ s , C D β u ¯ ( s ) ) | d s p u u ¯ Γ ( α ) ( 0 t ( t s ) α 1 d s + ( α 1 ) t 2 2 2 λ η 0 1 ( 1 s ) α 2 d s + λ ( α 1 ) t 2 2 2 λ η 0 η ( η s ) α 2 d s ) p u u ¯ Γ ( α ) ( t α α + t 2 2 2 λ η + t 2 λ η α 2 2 2 λ η ) p u u ¯ Γ ( α ) ( 1 α + 1 2 2 λ η + λ η α 2 2 2 λ η ) p u u ¯ M . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ13_HTML.gif
          (3.7)
          By a similar method, we get
          | C D β T u ( t ) C D β T u ¯ ( t ) | = | 1 Γ ( 1 β ) 0 t ( t s ) β ( T u ( s ) T u ¯ ( s ) ) d s | 1 Γ ( 1 β ) 0 t ( t s ) β ( 0 1 | s G ( s , z ) | | f ( z , u z , C D β u ( z ) ) f ( z , u ¯ z , C D β u ¯ ( z ) ) | d z ) d s p u u ¯ Γ ( 1 β ) 0 t ( t s ) β ( 0 1 | s G ( s , z ) | d z ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ14_HTML.gif
          (3.8)
          In view of the definition of G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq74_HTML.gif, we obtain
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equan_HTML.gif
          Hence,
          | C D β T u ( t ) C D β T u ¯ ( t ) | p u u ¯ Γ ( 1 β ) 0 t ( t s ) β Γ ( α ) ( 1 + 1 + λ η α 1 1 λ η ) d s p u u ¯ Γ ( 1 β ) 1 Γ ( α ) 1 1 β ( 1 + 1 + λ η α 1 1 λ η ) p u u ¯ Γ ( 2 β ) Γ ( α ) ( 1 + 1 + λ η α 1 1 λ η ) p u u ¯ N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ15_HTML.gif
          (3.9)
          Clearly, for each t [ r , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq75_HTML.gif, we have | T u ( t ) T u ¯ ( t ) | = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq76_HTML.gif. Therefore, by (3.7) and (3.9), we get
          T u T u ¯ p u u ¯ M + p u u ¯ N p u u ¯ ( N + M ) u u ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equao_HTML.gif

          and T is a contraction. As a consequence of the Banach contraction principle, we get that T has a fixed point which is a solution of boundary value problem (1.1) and (1.2). □

          4 Example

          In this section, we will present some examples to illustrate our main results.

          Example 4.1

          Consider boundary value problems of the following fractional functional differential equations:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ16_HTML.gif
          (4.1)
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ17_HTML.gif
          (4.2)

          where D α C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq1_HTML.gif, D β C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq2_HTML.gif denote Caputo fractional derivatives, 2 < α < 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq3_HTML.gif, 0 < β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq4_HTML.gif, t ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq77_HTML.gif.

          Choose λ = 6 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq78_HTML.gif, η = 1 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq79_HTML.gif, ϕ ( t ) = e t 1 36 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq80_HTML.gif, a = e t 110 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq81_HTML.gif, b = e t 143 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq82_HTML.gif and
          f ( t , u t , C D β u ( t ) ) = e t 1 36 + e t 110 | u t | k 1 + e t 143 | C D β u ( t ) | k 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equap_HTML.gif
          Then, for t ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq77_HTML.gif, we have
          | f ( t , u t , C D β u ( t ) ) | ϕ ( t ) + a | u t | k 1 + b | C D β u ( t ) | k 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equaq_HTML.gif

          For 0 < k 1 , k 2 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq51_HTML.gif, (H1) is satisfied and for k 1 , k 2 > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq52_HTML.gif, (H2) is satisfied. Therefore, by Theorem 3.1, boundary value problem (4.1) and (4.2) has a solution.

          Example 4.2

          Consider boundary value problems of the following fractional functional differential equations:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ18_HTML.gif
          (4.3)
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equ19_HTML.gif
          (4.4)

          where D α C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq1_HTML.gif, D β C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq2_HTML.gif denote Caputo fractional derivatives, 2 < α < 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq3_HTML.gif, 0 < β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq4_HTML.gif, t ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq77_HTML.gif.

          Choose λ = 7 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq83_HTML.gif, η = 3 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq84_HTML.gif and
          f ( t , u t , C D β u ( t ) ) = | u t | + | C D β u ( t ) | ( 6 + 9 e t ) ( 1 + | u t | + | C D β u ( t ) | ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equar_HTML.gif
          Set
          f ( t , μ , ν ) = | μ | + | ν | ( 6 + 9 e t ) ( 1 + | μ | + | ν | ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equas_HTML.gif
          Let μ , μ ¯ C r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq72_HTML.gif, ν , ν ¯ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq73_HTML.gif. Then for each t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq17_HTML.gif,
          | f ( t , μ , ν ) f ( t , μ ¯ , ν ¯ ) | = 1 6 + 9 e t | | μ | + | ν | 1 + | μ | + | ν | | μ ¯ | + | ν ¯ | 1 + | μ ¯ | + | ν ¯ | | = | μ + μ ¯ | | ν ν ¯ | ( 6 + 9 e t ) ( 1 + | μ ¯ | + | ν ¯ | ) ( 1 + | μ | + | ν | ) 1 6 + 9 e t ( | μ + μ ¯ | | ν ν ¯ | ) 1 15 ( | μ + μ ¯ | | ν ν ¯ | ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equat_HTML.gif
          For each t [ 1 , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq85_HTML.gif,
          | f ( t , μ , ν ) f ( t , μ ¯ , ν ¯ ) | = 1 6 + 9 e t | | μ | + | ν | 1 + | μ | + | ν | | μ ¯ | + | ν ¯ | 1 + | μ ¯ | + | ν ¯ | | = | μ + μ ¯ | | ν ν ¯ | ( 6 + 9 e t ) ( 1 + | μ ¯ | + | ν ¯ | ) ( 1 + | μ | + | ν | ) 1 6 + 9 e t ( | μ + μ ¯ | | ν ν ¯ | ) 1 6 ( | μ + μ ¯ | | ν ν ¯ | ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equau_HTML.gif
          Thus the condition (H3) holds with p = 1 15 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq86_HTML.gif. For λ = 7 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq83_HTML.gif, η = 3 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq84_HTML.gif, we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equav_HTML.gif
          By 2 < α < 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq3_HTML.gif, 0 < β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_IEq4_HTML.gif, we have
          1 3 1 α < 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equaw_HTML.gif
          and
          M > 1 Γ ( α ) 131 54 , N > 32 9 Γ ( 2 β ) Γ ( α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equax_HTML.gif
          It implies that
          p = 1 15 < 0.167 < ( M + N ) 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-38/MediaObjects/13661_2012_Article_292_Equay_HTML.gif

          Then by Theorem 3.2, boundary value problem (4.3) and (4.4) has a unique solution.

          Declarations

          Acknowledgements

          The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2010AL002, ZR2011AL007), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).

          Authors’ Affiliations

          (1)
          School of Mathematical Sciences, University of Jinan

          References

          1. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATH
          2. Kilbas A, Srivastava H, Trujillo J: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.MATH
          3. Oldham K, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.MATH
          4. Miller K, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York; 1993.
          5. Samko S, Kilbas A, Marichev O: Fractional Integral and Derivative, Theory and Applications. Gordon and Breach, Switzerland; 1993.
          6. Chang Y-K, Zhang R, N’Guérékata GM: Weighted pseudo almost automorphic mild solutions to fractional differential equations. Comput. Math. Appl. 2012, 64(10):3160-3170. 10.1016/j.camwa.2012.02.039MATHMathSciNetView Article
          7. Agarwal RP, Benchohra M, Hamani A: Boundary value problems for fractional differential equations. Georgian Math. J. 2009, 16: 401-411.MATHMathSciNet
          8. Zhao Y, Sun S, Han Z, Li Q: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(4):2086-2097. 10.1016/j.cnsns.2010.08.017MATHMathSciNetView Article
          9. Zhao Y, Sun S, Han Z, Li Q: Positive solutions to boundary value problems of nonlinear fractional differential equations. Abstr. Appl. Anal. 2011, 2011: 1-16.MathSciNet
          10. Zhao Y, Sun S, Han Z, Zhang M: Positive solutions for boundary value problems of nonlinear fractional differential equations. Appl. Math. Comput. 2011, 217: 6950-6958. 10.1016/j.amc.2011.01.103MATHMathSciNetView Article
          11. Feng W, Sun S, Han Z, Zhao Y: Existence of solutions for a singular system of nonlinear fractional differential equations. Comput. Math. Appl. 2011, 62(3):1370-1378. 10.1016/j.camwa.2011.03.076MATHMathSciNetView Article
          12. Rehman M, Khan R, Asif N: Three point boundary value problems for nonlinear fractional differential equations. Acta Math. Sci. 2011, 31B(4):1337-1346.MathSciNetView Article
          13. Diethelm K: The Analysis of Fractional Differential Equations. Springer, Berlin; 2010.MATHView Article
          14. Sun S, Zhao Y, Han Z, Li Y: The existence of solutions for boundary value problem of fractional hybrid differential equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 4961-4967. 10.1016/j.cnsns.2012.06.001MATHMathSciNetView Article
          15. Sun S, Zhao Y, Han Z, Xu M: Uniqueness of positive solutions for boundary value problems of singular fractional differential equations. Inverse Probl. Sci. Eng. 2012, 20: 299-309. 10.1080/17415977.2011.603726MATHMathSciNetView Article
          16. Benchohra M, Henderson J, Ntouyas S, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 2008, 338: 1340-1350. 10.1016/j.jmaa.2007.06.021MATHMathSciNetView Article
          17. Lakshmikantham V, Vatsala A: Basic theory of fractional differential equations. Nonlinear Anal. 2008, 69: 2677-2682. 10.1016/j.na.2007.08.042MATHMathSciNetView Article
          18. Lakshmikantham V, Vatsala A: Theory of fractional differential inequalities and applications. Commun. Appl. Anal. 2007, 11: 395-402.MATHMathSciNet
          19. Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Anal. 2008, 69: 3337-3343. 10.1016/j.na.2007.09.025MATHMathSciNetView Article
          20. Lakshmikantham V, Devi J: Theory of fractional differential equations in Banach space. Eur. J. Pure Appl. Math. 2008, 1: 38-45.MATHMathSciNet
          21. Agarwal RP, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 2010, 59: 1095-1100. 10.1016/j.camwa.2009.05.010MATHMathSciNetView Article
          22. Agarwal RP, Zhou Y, Wang J, Xian N: Fractional neutral functional differential equations with causal operators in Banach spaces. Math. Comput. Model. 2011, 54: 1440-1452. 10.1016/j.mcm.2011.04.016MATHView Article
          23. Sun S, Li Q, Li Y: Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations. Comput. Math. Appl. 2012, 64: 3310-3320. 10.1016/j.camwa.2012.01.065MATHMathSciNetView Article
          24. Li Q, Sun S, Zhao P, Han Z: Existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations. Abstr. Appl. Anal. 2012, 2012: 1-14.MATHMathSciNet
          25. Maraaba T, Baleanu D, Jarad F: Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives. J. Math. Phys. 2008, 49: 483-507.MathSciNet
          26. Zhou Y, Jiao F, Li J: Existence and uniqueness for p -type fractional neutral differential equations. Nonlinear Anal. TMA 2009, 71(7-8):2724-2733. 10.1016/j.na.2009.01.105MATHMathSciNetView Article
          27. Zhou Y, Jiao F, Li J: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. TMA 2009, 71(7-8):3249-3256. 10.1016/j.na.2009.01.202MATHMathSciNetView Article
          28. Wang J, Zhou Y, Wei W: A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 4049-4059. 10.1016/j.cnsns.2011.02.003MATHMathSciNetView Article
          29. Zhou Y, Tian Y, He Y: Floquet boundary value problems of fractional functional differential equations. Electron. J. Qual. Theory Differ. Equ. 2010, 50: 1-13.MathSciNet
          30. Bai C: Existence of positive solutions for a functional fractional boundary value problem. Abstr. Appl. Anal. 2010, 2010: 1-13.
          31. Ouyang Z, Chen YM, Zou SL: Existence of positive solutions to a boundary value problem for a delayed nonlinear fractional differential system. Bound. Value Probl. 2011, 2011: 1-17.MathSciNetView Article
          32. Bai C: Existence of positive solutions for boundary value problems of fractional functional differential equations. Electron. J. Qual. Theory Differ. Equ. 2010, 30: 1-14.
          33. Ahmad B, Alsaedi A: Nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions. Bound. Value Probl. 2012, 2012: 1-10. 10.1186/1687-2770-2012-1MathSciNetView Article
          34. Zhao Y, Chen H, Huang L: Existence of positive solutions for nonlinear fractional functional differential equation. Comput. Math. Appl. 2012, 64: 3456-3467. 10.1016/j.camwa.2012.01.081MATHMathSciNetView Article
          35. Su X: Positive solutions to singular boundary value problems for fractional functional differential equations with changing sign nonlinearity. Comput. Math. Appl. 2012, 64: 3425-3435. 10.1016/j.camwa.2012.02.043MATHMathSciNetView Article
          36. Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cone. Academic Press, Orlando; 1988.

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