Multiple positive solutions of boundary value problems for fractional order integro-differential equations in a Banach space

  • Ruijuan Liu1, 2Email author,

    Affiliated with

    • Chunhai Kou3 and

      Affiliated with

      • Ran Jin1

        Affiliated with

        Boundary Value Problems20132013:79

        DOI: 10.1186/1687-2770-2013-79

        Received: 25 December 2012

        Accepted: 18 March 2013

        Published: 8 April 2013

        Abstract

        In this paper, we obtain the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space by means of fixed point index theory of completely continuous operators.

        MSC:26A33, 34B15.

        Keywords

        fractional order integro-differential equation measure of noncompactness fixed point index boundary value problem

        1 Introduction

        Fractional differential equations (FDEs) have been of great interest for the last three decades [111]. 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 [12], electrochemistry [13], control, porous media [14], etc. Therefore, the theory of FDEs has been developed very quickly. Many qualitative theories of FDEs have been obtained. Many important results can be found in [1519] and references cited therein.

        In this paper, we shall use the fixed point index theory of completely continuous operators to investigate the multiple positive solutions of a boundary value problem for a class of α order nonlinear integro-differential equations in a Banach space.

        Let E be a real Banach space, P be a cone in E and P 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq1_HTML.gif denote the interior points of P. A partial ordering in E is introduced by x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq2_HTML.gif if and only if y x P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq3_HTML.gif. P is said to be normal if there exists a positive constant N such that θ x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq4_HTML.gif implies x N y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq5_HTML.gif, where θ denotes the zero element of E, and the smallest constant N is called the normal constant of P. P is called solid if P 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq1_HTML.gif is nonempty. If x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq2_HTML.gif and x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq6_HTML.gif, we write x < y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq7_HTML.gif. If P is solid and y x P 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq8_HTML.gif, we write x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq9_HTML.gif. For details on cone theory, see [1].

        For the application in the sequel, we first state the following lemmas and definitions which can be found in [1, 10, 20].

        Lemma 1.1 Let P be a cone in a real Banach space E, and let Ω be a nonempty bounded open convex subset of P. Suppose that A : Ω ¯ P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq10_HTML.gif is completely continuous and A ( Ω ¯ ) Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq11_HTML.gif, where Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq12_HTML.gif denotes the closure of Ω in P. Then the fixed point index
        i ( A , Ω , P ) = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equa_HTML.gif
        Lemma 1.2 Let P be a cone in a real Banach space E, and let Ω = Ω 1 Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq13_HTML.gif, where Ω i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq14_HTML.gif ( i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq15_HTML.gif) are nonempty bounded open convex subsets of P and Ω 1 Ω 2 = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq16_HTML.gif. Suppose that A : Ω ¯ P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq10_HTML.gif is a strict set contraction and A ( Ω ¯ ) Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq11_HTML.gif. Then
        i ( A , Ω , P ) = i ( A , Ω 1 , P ) + i ( A , Ω 2 , P ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equb_HTML.gif
        Lemma 1.3 If U C [ I , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq17_HTML.gif is bounded and equicontinuous, then α E ( U ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq18_HTML.gif is continuous on I, and set
        α C ( U ) = max t I α E ( U ( t ) ) , α E ( I u ( t ) d t : u U ) I α E ( U ( t ) ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equc_HTML.gif

        where I = [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq19_HTML.gif, U ( t ) = { u ( t ) : u U } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq20_HTML.gif.

        Definition 1.1 The fractional integral of order α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq21_HTML.gif of a function f : ( 0 , ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq22_HTML.gif is given by
        I 0 + α f ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 f ( s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equd_HTML.gif

        provided the right-hand side is pointwise defined on ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq23_HTML.gif.

        Definition 1.2 The fractional derivative of order α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq21_HTML.gif of a function f : ( 0 , ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq22_HTML.gif is given by
        D 0 + α f ( t ) = 1 Γ ( n α ) ( d d t ) n 0 t f ( s ) ( t s ) α n + 1 d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Eque_HTML.gif

        where n = [ α ] + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq24_HTML.gif, provided the right-hand side is pointwise defined on ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq25_HTML.gif.

        Lemma 1.4 Let α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq21_HTML.gif, then
        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equf_HTML.gif

        for some c i E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq26_HTML.gif, i = 0 , 1 , 2 , , n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq27_HTML.gif, n = [ α ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq28_HTML.gif.

        In this article, let J = [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq29_HTML.gif, B C [ J , E ] = { u C [ J , E ] : sup t J u ( t ) 1 + t α 1 < } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq30_HTML.gif. It is easy to see that B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq31_HTML.gif is a Banach space with the norm
        u B = sup t J u ( t ) 1 + t α 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equg_HTML.gif
        Consider the boundary value problem (BVP) for a fractional nonlinear integro-differential equation of mixed type in E:
        { D 0 α u ( t ) + f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) = θ t J , u ( 0 ) = u ( 0 ) = θ , D 0 α 1 u ( + ) = i = 1 m β i u ( η i ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ1_HTML.gif
        (1)
        where D 0 α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq32_HTML.gif is the standard Riemann-Liouville fractional derivative of order 2 < α < 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq33_HTML.gif, f C [ J × P × P × P , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq34_HTML.gif, β i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq35_HTML.gif ( i = 1 , 2 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq36_HTML.gif), 0 < η 1 < η 2 < < η m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq37_HTML.gif, i = 1 m β i η i α 1 < Γ ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq38_HTML.gif and
        ( T u ) ( t ) = 0 t K ( t , s ) u ( s ) d s , ( S u ) ( t ) = 0 + H ( t , s ) u ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ2_HTML.gif
        (2)

        K C [ D , R + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq39_HTML.gif, D = { ( t , s ) J × J : t s } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq40_HTML.gif, H C [ J × J , R + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq41_HTML.gif, R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq42_HTML.gif denotes the set of all nonnegative real numbers.

        2 Several lemmas

        To establish the existence of multiple positive solutions in B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif of (1), let us list the following assumptions.

        ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif) k = sup t J 0 t K ( t , s ) d s < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq45_HTML.gif, h = sup t J 1 1 + t α 1 0 + H ( t , s ) ( 1 + s α 1 ) d s < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq46_HTML.gif, 0 + ( H ( t , s ) H ( t , s ) ) ( 1 + s α 1 ) d s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq47_HTML.gif, as t t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq48_HTML.gif ( t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq49_HTML.gif).

        ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) There exist a , b C [ J , R + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq51_HTML.gif and g C [ J × J × J , R + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq52_HTML.gif such that
        f ( t , u , v , w ) a ( t ) + b ( t ) g ( u , v , w ) t J , u , v , w P . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equh_HTML.gif
        ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq53_HTML.gif) There exists c C [ J , R + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq54_HTML.gif such that
        f ( t , u , v , w ) c ( t ) ( u + v + w ) 0 , as  u , v , w P , u + v + w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equi_HTML.gif
        uniformly for t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq49_HTML.gif, and
        c = 0 + c ( t ) ( 1 + t α 1 ) d t < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equj_HTML.gif
        ( H 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq55_HTML.gif) There exists d C [ J , R + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq56_HTML.gif such that
        f ( t , ( 1 + t α 1 ) u , ( 1 + t α 1 ) v , ( 1 + t α 1 ) w ) d ( t ) ( u + v + w ) 0 , as  u , v , w P , u + v + w 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equk_HTML.gif
        uniformly for t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq49_HTML.gif, and
        d = 0 + d ( t ) d t < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equl_HTML.gif

        ( H 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq57_HTML.gif) For any t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq49_HTML.gif and r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq58_HTML.gif, f ( t , P r , P r , P r ) = { f ( t , u , v , w ) : u , v , w P r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq59_HTML.gif is relatively compact in E, where P r = { u P : u r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq60_HTML.gif.

        ( H 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq61_HTML.gif) P is normal and solid, and there exist u 0 θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq62_HTML.gif, 0 < t < t < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq63_HTML.gif and σ C [ I , R + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq64_HTML.gif such that
        f ( t , u , v , w ) σ ( t ) u 0 t I , u u 0 , v θ , w θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equm_HTML.gif
        and
        t t γ ( s ) σ ( s ) > 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equn_HTML.gif

        where I = [ t , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq65_HTML.gif, γ ( s ) = min t I G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq66_HTML.gif.

        ( H 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq67_HTML.gif) There exist u 0 > θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq68_HTML.gif, 0 < t < t < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq63_HTML.gif and σ C [ I , R + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq64_HTML.gif such that
        f ( t , u , v , w ) σ ( t ) u 0 t I , u u 0 , v θ , w θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equo_HTML.gif
        and
        t t γ ( s ) σ ( s ) 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equp_HTML.gif

        where I = [ t , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq65_HTML.gif, γ ( s ) = min t I G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq66_HTML.gif.

        Remark 2.1 It is clear that ( H 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq57_HTML.gif) is satisfied automatically when E is finite dimensional.

        Remark 2.2 It is clear that assumption ( H 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq67_HTML.gif) is weaker than assumption ( H 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq61_HTML.gif).

        We shall reduce BVP (1) to an integral equation in E. To this end, we first consider the operator A defined by
        ( A u ) ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s λ Γ ( α ) i = 1 m β i t α 1 0 η i ( η i s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + λ t α 1 0 + f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ3_HTML.gif
        (3)

        where λ = 1 Γ ( α ) i = 1 m β i η i α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq69_HTML.gif.

        In our main results, we make use of the following lemmas.

        Lemma 2.1 Let assumption ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif) be satisfied, then the operators T and S defined by (2) are bounded linear operators from B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq31_HTML.gif into B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif, and
        T k , S h . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ4_HTML.gif
        (4)
        Moreover,
        T : B C [ J , P ] B C [ J , P ] , S : B C [ J , P ] B C [ J , P ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ5_HTML.gif
        (5)
        Proof Inequalities (4) follow from two simple inequalities:
        ( T u ) ( t ) 1 + t α 1 0 t K ( t , s ) 1 + s α 1 1 + t α 1 u ( s ) 1 + s α 1 d s k u B , ( S u ) ( t ) 1 + t α 1 0 + H ( t , s ) 1 + s α 1 1 + t α 1 u ( s ) 1 + s α 1 d s h u B , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equq_HTML.gif

        and (5) is obvious. □

        Lemma 2.2 Let assumptions ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif), ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) and ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq53_HTML.gif) be satisfied, then the operator A defined by (3) is a continuous operator from B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq31_HTML.gif into B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq31_HTML.gif.

        Proof

        Let
        ε 1 = 1 2 ( 1 + k + h ) [ ( 1 Γ ( α ) + λ ) c + λ Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 ( a ( s ) + M b ( s ) ) d s ] 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equr_HTML.gif

        where λ is defined in the operator A.

        By virtue of assumptions ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) and ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq53_HTML.gif), there exists an R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq70_HTML.gif such that
        f ( t , u , v , w ) ε 1 c ( t ) ( u + v + w ) t J , u , v , w P , u + v + w > R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ6_HTML.gif
        (6)
        and
        f ( t , u , v , w ) a ( t ) + M b ( t ) t J , u , v , w P , u + v + w R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ7_HTML.gif
        (7)
        where
        M = max { g ( x 1 , x 2 , x 3 ) : 0 x 1 , x 2 , x 3 R } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equs_HTML.gif
        It follows from (6) and (7) that for t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq49_HTML.gif, u , v , w P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq71_HTML.gif, we have
        f ( t , u , v , w ) ε 1 c ( t ) ( u + v + w ) + a ( t ) + M b ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ8_HTML.gif
        (8)
        Let u B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq72_HTML.gif, we have, by (8) and Lemma 2.1,
        f ( t , u , ( T u ) ( t ) , ( S u ) ( t ) ) ε 1 c ( t ) ( 1 + t α 1 ) ( 1 + k + h ) u B + a ( t ) + M b ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ9_HTML.gif
        (9)
        which implies the convergence of the infinite integral
        0 + f ( t , u , ( T u ) ( t ) , ( S u ) ( t ) ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equt_HTML.gif
        and
        0 + f ( t , u , ( T u ) ( t ) , ( S u ) ( t ) ) d s c ε 1 ( 1 + k + h ) u B + a + M b . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ10_HTML.gif
        (10)
        Thus, we have, by (3), (9) and (10),
        ( A u ) ( t ) 1 + t α 1 1 Γ ( α ) 0 t ( t s ) α 1 1 + t α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + λ Γ ( α ) i = 1 m β i t α 1 1 + t α 1 0 η i ( η i s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + λ t α 1 1 + t α 1 0 + f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s ( 1 Γ ( α ) + λ ) 0 + f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + λ Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s ( 1 Γ ( α ) + λ ) ( c ε 1 ( 1 + k + h ) u B + a + M b ) + λ ε 1 Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 ε 1 c ( s ) ( 1 + s α 1 ) ( 1 + k + h ) u B d s + λ Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 ( a ( s ) + M b ( s ) ) d s 1 2 u B + ( 1 Γ ( α ) + λ ) ( a + M b ) + λ Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 ( a ( s ) + M b ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ11_HTML.gif
        (11)
        It follows from (11) that
        A u B 1 2 u B + ( 1 Γ ( α ) + λ ) ( a + M b ) + λ Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 ( a ( s ) + M b ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ12_HTML.gif
        (12)

        Thus, we have A ( B C [ J , E ] ) B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq73_HTML.gif.

        Finally, we show that A is continuous. Let u n , u ˜ B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq74_HTML.gif, u n u ˜ B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq75_HTML.gif ( n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq76_HTML.gif). Then r = sup n u n < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq77_HTML.gif and u ˜ B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq78_HTML.gif. By (3), we have
        ( A u n ) ( t ) 1 + t α 1 ( A u ˜ ) ( t ) 1 + t α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ13_HTML.gif
        (13)
        0 t ( t s ) α 1 1 + t α 1 f ( s , u n ( s ) , ( T u n ) ( s ) , ( S u n ) ( s ) ) f ( s , u ˜ ( s ) , ( T u ˜ ) ( s ) , ( S u ˜ ) ( s ) ) d s + λ Γ ( α ) i = 1 m β i t α 1 1 + t α 1 0 η i ( η i s ) α 1 f ( s , u n ( s ) , ( T u n ) ( s ) , ( S u n ) ( s ) ) f ( s , u ˜ ( s ) , ( T u ˜ ) ( s ) , ( S u ˜ ) ( s ) ) d s + λ t α 1 1 + t α 1 0 + f ( s , u n ( s ) , ( T u n ) ( s ) , ( S u n ) ( s ) ) f ( s , u ˜ ( s ) , ( T u ˜ ) ( s ) , ( S u ˜ ) ( s ) ) d s ( 1 Γ ( α ) + λ ) 0 + f ( s , u n ( s ) , ( T u n ) ( s ) , ( S u n ) ( s ) ) f ( s , u ˜ ( s ) , ( T u ˜ ) ( s ) , ( S u ˜ ) ( s ) ) d s + λ Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 f ( s , u n ( s ) , ( T u n ) ( s ) , ( S u n ) ( s ) ) f ( s , u ˜ ( s ) , ( T u ˜ ) ( s ) , ( S u ˜ ) ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ14_HTML.gif
        (14)
        It is clear that
        f ( t , u n ( t ) , ( T u n ) ( t ) , ( S u n ) ( t ) ) f ( t , u ˜ ( t ) , ( T u ˜ ) ( t ) , ( S u ˜ ) ( t ) ) , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ15_HTML.gif
        (15)
        and by (9),
        f ( t , u n ( t ) , ( T u n ) ( t ) , ( S u n ) ( t ) ) f ( t , u ˜ ( t ) , ( T u ˜ ) ( t ) , ( S u ˜ ) ( t ) ) 2 ε 1 c ( t ) ( 1 + t α 1 ) ( 1 + k + h ) u B + 2 a ( t ) + 2 M b ( t ) = μ ( t ) t J , n = 1 , 2 , 3 , , μ L [ J , R + ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ16_HTML.gif
        (16)
        It follows from (15) and (16) and the dominated convergence theorem that
        lim n 0 + f ( t , u n ( t ) , ( T u n ) ( t ) , ( S u n ) ( t ) ) f ( t , u ˜ ( t ) , ( T u ˜ ) ( t ) , ( S u ˜ ) ( t ) ) d s = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ17_HTML.gif
        (17)
        and
        lim n 0 η i ( η i s ) α 1 f ( t , u n ( t ) , ( T u n ) ( t ) , ( S u n ) ( t ) ) f ( t , u ˜ ( t ) , ( T u ˜ ) ( t ) , ( S u ˜ ) ( t ) ) d s = 0 , i = 1 , 2 , , m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ18_HTML.gif
        (18)

        It follows from (14), (17) and (18) that A u n A u ˜ B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq79_HTML.gif ( n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq76_HTML.gif), and the continuity of A is proved. □

        Lemma 2.3 Let assumptions ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif), ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) and ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq53_HTML.gif) be satisfied, then u B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq80_HTML.gif is a solution of BVP (1) if and only if u B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq80_HTML.gif is a solution of the following integral equation:
        u ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s λ Γ ( α ) i = 1 m β i t α 1 0 η i ( η i s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + λ t α 1 0 + f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ19_HTML.gif
        (19)

        i.e., u is a fixed point of the operator A defined by (3) in B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq31_HTML.gif.

        Proof If u B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq80_HTML.gif is a solution of BVP (1), then by applying Lemma 1.4 we reduce D 0 α u ( t ) + f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) = θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq81_HTML.gif to an equivalent integral equation
        u ( t ) = I 0 + α f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) + c 1 t α 1 + c 2 t α 2 + c 3 t α 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ20_HTML.gif
        (20)
        for some c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq82_HTML.gif, c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq83_HTML.gif, c 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq84_HTML.gif. (20) can be rewritten
        u ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + c 1 t α 1 + c 2 t α 2 + c 3 t α 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ21_HTML.gif
        (21)
        By u ( 0 ) = u ( 0 ) = θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq85_HTML.gif, we have
        c 2 = c 3 = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ22_HTML.gif
        (22)
        By D 0 α 1 u ( + ) = i = 1 m β i u ( η i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq86_HTML.gif, we obtain
        c 1 = λ 0 + f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s λ Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ23_HTML.gif
        (23)

        Now, substituting (22) and (23) into (21), we see that u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq87_HTML.gif satisfies integral equation (19).

        Conversely, if u is a solution of (19), the direct differentiation of (19) gives
        u ( t ) = 1 Γ ( α 1 ) 0 t ( t s ) α 2 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s λ Γ ( α 1 ) i = 1 m β i t α 2 0 η i ( η i s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + λ ( α 1 ) t α 2 0 + f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ24_HTML.gif
        (24)
        and
        D 0 + α 1 u ( t ) = 0 t f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ25_HTML.gif
        (25)

        Consequently, u B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq80_HTML.gif, and by (19), (24) and (25), it is easy to see that u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq87_HTML.gif satisfies BVP (1). □

        Lemma 2.4 Integral equation (19) can be expressed as
        u ( t ) = 0 + G ( t , s ) f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ26_HTML.gif
        (26)
        and G ( t , s ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq88_HTML.gif for any t , s ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq89_HTML.gif, where
        G ( t , s ) = { ( t s ) α 1 ( Γ ( α ) i = 1 m β i η i α 1 ) i = j m β i t α 1 ( η i s ) α 1 + Γ ( α ) t α 1 ( Γ ( α ) i = 1 m β i η i α 1 ) Γ ( α ) , η k 1 t η k , η j 1 s η j , k = 1 , 2 , , m , j = 1 , 2 , , k 1 or η k 1 t η k , s t , k = 1 , 2 , , m ; i = j m β i t α 1 ( η i s ) α 1 + Γ ( α ) t α 1 ( Γ ( α ) i = 1 m β i η i α 1 ) Γ ( α ) , η k 1 t η k , η j 1 s η j , k = 1 , 2 , , m , j = k + 1 , , m or η k 1 t η k , t s , k = 1 , 2 , , m ; ( t s ) α 1 ( Γ ( α ) i = 1 m β i η i α 1 ) + Γ ( α ) t α 1 ( Γ ( α ) i = 1 m β i η i α 1 ) Γ ( α ) , η m s t ; t α 1 Γ ( α ) i = 1 m β i η i α 1 , t η m s or η m t s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ27_HTML.gif
        (27)
        Proof Let h ( t ) = f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq90_HTML.gif. For t η 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq91_HTML.gif, one has
        u ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 h ( s ) d s λ Γ ( α ) β 1 t α 1 ( 0 t ( η 1 s ) α 1 h ( s ) d s + t η 1 ( η 1 s ) α 1 h ( s ) d s ) λ Γ ( α ) β 2 t α 1 ( 0 t ( η 2 s ) α 1 h ( s ) d s + t η 1 ( η 2 s ) α 1 h ( s ) d s + η 1 η 2 ( η 2 s ) α 1 h ( s ) d s ) λ Γ ( α ) β m t α 1 ( 0 t ( η m s ) α 1 h ( s ) d s + t η 1 ( η m s ) α 1 h ( s ) d s + + η m 1 η m ( η m s ) α 1 h ( s ) d s ) + λ t α 1 ( 0 t h ( s ) d s + t η 1 h ( s ) d s + η 1 η 2 h ( s ) d s + + η m 1 η m h ( s ) d s + η m + h ( s ) d s ) = 0 + G ( t , s ) h ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equu_HTML.gif
        0 < s t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq92_HTML.gif
        G ( t , s ) = λ Γ ( α ) [ ( t s ) α 1 ( Γ ( α ) i = 1 m β i η i α 1 ) i = 1 m β i ( η i s ) α 1 t α 1 + Γ ( α ) t α 1 ] λ Γ ( α ) [ t α 1 ( Γ ( α ) i = 1 m β i η i α 1 ) i = 1 m β i ( η i s ) α 1 t α 1 + Γ ( α ) t α 1 ] = λ Γ ( α ) i = 1 m β i ( η i α 1 ( η i s ) α 1 ) t α 1 > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equv_HTML.gif
        0 < t s η 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq93_HTML.gif
        G ( t , s ) = λ Γ ( α ) [ i = 1 m β i ( η i s ) α 1 t α 1 + Γ ( α ) t α 1 ] λ Γ ( α ) ( Γ ( α ) i = 1 m β i η i α 1 ) t α 1 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equw_HTML.gif
        η j 1 s η j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq94_HTML.gif, j = 2 , 3 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq95_HTML.gif
        G ( t , s ) = λ Γ ( α ) [ i = j m β i ( η i s ) α 1 t α 1 + Γ ( α ) t α 1 ] λ Γ ( α ) ( Γ ( α ) i = j m β i η i α 1 ) t α 1 > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equx_HTML.gif
        η m s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq96_HTML.gif
        G ( t , s ) = λ Γ ( α ) t α 1 > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equy_HTML.gif

        By simple calculation, we can prove the rest of the lemma. □

        Lemma 2.5 Let assumptions ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif), ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) and ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq53_HTML.gif) be satisfied, and let U be a bounded subset of B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq31_HTML.gif. Then { ( A u ) ( t ) 1 + t 2 α 1 : u U } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq97_HTML.gif is equicontinuous on any finite subinterval of J, and for any given ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq98_HTML.gif, there exists τ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq99_HTML.gif such that
        A u ( t 1 ) 1 + t 1 α 1 A u ( t 2 ) 1 + t 2 α 1 < ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equz_HTML.gif

        uniformly with respect to u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq100_HTML.gif, as t 1 , t 2 τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq101_HTML.gif.

        Proof For u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq100_HTML.gif, t 1 < t 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq102_HTML.gif, by using (3), we have
        A u ( t 1 ) 1 + t 1 α 1 A u ( t 2 ) 1 + t 2 α 1 1 Γ ( α ) 0 t 1 | ( t 1 s ) α 1 1 + t 1 α 1 ( t 2 s ) α 1 1 + t 2 α 1 | f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + 1 Γ ( α ) t 1 t 2 ( t 2 s ) α 1 1 + t 2 α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + | t 1 α 1 1 + t 1 α 1 t 2 α 1 1 + t 2 α 1 | ( λ 0 + f ( t , u , ( T u ) ( t ) , ( S u ) ( t ) ) d s + 1 Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ28_HTML.gif
        (28)

        This, together with (9) and (10), implies that { A u ( t 1 ) 1 + t 1 α 1 : u U } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq103_HTML.gif are equicontinuous on any finite subinterval of J.

        Now, we are going to prove that for any given ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq98_HTML.gif, there exists sufficiently large τ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq99_HTML.gif, which satisfies
        A u ( t 1 ) 1 + t 1 α 1 A u ( t 2 ) 1 + t 2 α 1 ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equaa_HTML.gif

        for all u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq100_HTML.gif and t 1 , t 2 τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq101_HTML.gif.

        Together with (28), we need only to show that for any given ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq98_HTML.gif, there exists sufficiently large τ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq99_HTML.gif such that
        0 t 1 ( t 1 s ) α 1 1 + t 1 α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s 0 t 2 ( t 2 s ) α 1 1 + t 2 α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s < ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equab_HTML.gif
        It follows from (10) that for any given ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq98_HTML.gif, there exists a sufficiently large L > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq104_HTML.gif such that
        L + f ( t , u , ( T u ) ( t ) , ( S u ) ( t ) ) d s < ε 3 u U , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ29_HTML.gif
        (29)
        and there exists K > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq105_HTML.gif such that
        0 + f ( t , u , ( T u ) ( t ) , ( S u ) ( t ) ) d s K u U . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ30_HTML.gif
        (30)
        On the other hand, let g ( t , s ) = ( t s ) α 1 1 + t α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq106_HTML.gif, s [ 0 , L ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq107_HTML.gif, t [ L , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq108_HTML.gif, then we have
        lim t sup s [ 0 , L ] | g ( t , s ) 1 | lim t g ( t , L ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equac_HTML.gif
        Thus, there exists τ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq99_HTML.gif such that for t 1 , t 2 τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq101_HTML.gif,
        sup s [ 0 , L ] | g ( t 1 , s ) g ( t 2 , s ) | sup s [ 0 , L ] | g ( t 1 , s ) 1 | + sup s [ 0 , L ] | g ( t 2 , s ) 1 | < ε 3 K . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ31_HTML.gif
        (31)
        Therefore, from (29), (30) and (31) we have
        0 t 1 ( t 1 s ) α 1 1 + t 1 α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s 0 t 2 ( t 2 s ) α 1 1 + t 2 α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s 0 L | ( t 1 s ) α 1 1 + t 1 α 1 ( t 2 s ) α 1 1 + t 2 α 1 | f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + L t 1 ( t 1 s ) α 1 1 + t 1 α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + L t 2 ( t 2 s ) α 1 1 + t 2 α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s ε 3 K 0 L f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ε 3 + ε 3 < ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equad_HTML.gif

        Consequently, the proof is complete. □

        Lemma 2.6 Let assumptions ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif), ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) and ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq53_HTML.gif) be satisfied, and let U be a bounded subset of B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq31_HTML.gif. Then
        α B ( A U ) = sup t J α E ( ( A u ) ( t ) 1 + t α 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equae_HTML.gif
        Proof By Lemma 2.2, we know AU is a bounded subset of B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq31_HTML.gif. Thus,
        ϱ = : sup t J α E ( ( A U ) ( t ) 1 + t α 1 ) < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equaf_HTML.gif

        First, we claim that α B ( A U ) ϱ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq109_HTML.gif.

        In fact, by Lemma 2.5, we know that for any given ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq98_HTML.gif, there exists a τ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq99_HTML.gif such that
        ( A u ) ( t 1 ) 1 + t 1 α 1 ( A u ) ( t 2 ) 1 + t 2 α 1 < ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ32_HTML.gif
        (32)

        uniformly with respect to u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq100_HTML.gif and t 1 , t 2 τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq101_HTML.gif.

        Since { ( A u ) ( t ) 1 + t α 1 : u U } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq110_HTML.gif is equicontinuous on [ 0 , τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq111_HTML.gif, by Lemma 1.3, we know
        α B ( A U | [ 0 , τ ] ) = max t [ 0 , τ ] α E ( ( A u ) ( t ) 1 + t α 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equag_HTML.gif
        where
        A U | [ 0 , τ ] = { u ( t ) : t [ 0 , τ ] , u U } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equah_HTML.gif
        that is, A U | [ 0 , τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq112_HTML.gif is the restriction of AU on [ 0 , τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq111_HTML.gif. Therefore, there exists U 1 , U 2 , , U k U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq113_HTML.gif such that
        U = i = 1 k U i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equai_HTML.gif
        satisfying
        A U | [ 0 , τ ] = i = 1 k A U i | [ 0 , τ ] , diam B ( A U i ) < ϱ + ε , i = 1 , 2 , 3 , , k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ33_HTML.gif
        (33)

        where diam B ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq114_HTML.gif denote the diameters of bounded subsets of B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq31_HTML.gif.

        At the same time, for any A u 1 , A u 2 A U i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq115_HTML.gif, by (32) and (33), we obtain
        ( A u 1 ) ( t ) 1 + t α 1 ( A u 2 ) ( t ) 1 + t α 1 ( A u 1 ) ( t ) 1 + t α 1 ( A u 2 ) ( t ) 1 + t α 1 + ( A u 1 ) ( t ) 1 + t α 1 ( A u 2 ) ( t ) 1 + t α 1 + ( A u 1 ) ( t ) 1 + t α 1 ( A u 2 ) ( t ) 1 + t α 1 ε + ϱ + ε + ε = ϱ + 3 ε t [ τ , + ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ34_HTML.gif
        (34)
        It follows from (33) and (34) that
        diam B ( A U i ) ϱ + 3 ε , i = 1 , 2 , 3 , , k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equaj_HTML.gif
        Then, by using A U = i = 1 k A U i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq116_HTML.gif, we have
        α B ( A U ) ϱ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equak_HTML.gif
        On the other hand, for any given ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq98_HTML.gif, there exist V i U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq117_HTML.gif, i = 1 , 2 , 3 , , l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq118_HTML.gif, such that
        A U = i = 1 l A V i and diam B ( A V i ) α B ( A U ) + ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equal_HTML.gif
        Hence, for t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq119_HTML.gif, u 1 , u 2 U i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq120_HTML.gif, i = 1 , 2 , 3 , , l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq118_HTML.gif, we have
        ( A u 1 ) ( t ) 1 + t α 1 ( A u 2 ) ( t ) 1 + t α 1 A u 1 A u 2 B α B ( A U ) + ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ35_HTML.gif
        (35)
        Since ( A U ) ( t ) = i = 1 l ( A V i ) ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq121_HTML.gif together with (35), we get
        α E ( ( A u ) ( t ) 1 + t α 1 ) α B ( A U ) + ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equam_HTML.gif
        that is,
        sup t J α E ( ( A u ) ( t ) 1 + t α 1 ) α B ( A U ) + ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equan_HTML.gif
        Because ε is arbitrary, we obtain
        sup t J α E ( ( A u 1 ) ( t ) 1 + t α 1 ) α B ( A U ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equao_HTML.gif

        Consequently, the proof is complete. □

        3 Main results

        In this section, we give and prove our main results.

        Theorem 3.1 Let ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif)-( H 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq61_HTML.gif) be satisfied. Then BVP (1) has at least two positive solutions u , u B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq122_HTML.gif such that u ( t ) u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq123_HTML.gif for t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq124_HTML.gif.

        Proof By Lemma 2.2 and Lemma 2.4, the operator A defined by (3) is continuous from B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif into B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif, and by Lemma 2.3, we need only to show that A has two positive fixed points u , u B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq122_HTML.gif such that u ( t ) u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq123_HTML.gif for t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq124_HTML.gif.

        First, we shall prove A is compact.

        Let U = { u n } B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq125_HTML.gif be bounded and u n K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq126_HTML.gif ( n = 1 , 2 , 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq127_HTML.gif). From (9), we can choose a sufficiently large τ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq99_HTML.gif such that for all u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq100_HTML.gif
        τ + f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s < ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ36_HTML.gif
        (36)
        It follows from Lemma 2.5 that
        { ( A u n ) ( t ) 1 + t α 1 : n = 1 , 2 , 3 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ37_HTML.gif
        (37)
        is equicontinuous on [ 0 , τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq111_HTML.gif. Thus, by (3), (36) and (37), we have
        α E ( A U ( t ) 1 + t α 1 ) 1 Γ ( α ) 0 τ α E ( f ( s , U ( s ) , ( T U ) ( s ) , ( S U ) ( s ) ) ) d s + 2 ε + 1 Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 α E ( f ( s , U ( s ) , ( T U ) ( s ) , ( S U ) ( s ) ) ) d s + 0 τ α E ( f ( s , U ( s ) , ( T U ) ( s ) , ( S U ) ( s ) ) ) d s + 2 λ ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ38_HTML.gif
        (38)

        where A U ( t ) 1 + t α 1 = { A u n ( t ) 1 + t α 1 : n = 1 , 2 , 3 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq128_HTML.gif, U ( s ) = { u n ( s ) : n = 1 , 2 , 3 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq129_HTML.gif, ( T U ) ( s ) = { ( T u n ) ( s ) : n = 1 , 2 , 3 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq130_HTML.gif, ( S U ) ( s ) = { ( S u n ) ( s ) : n = 1 , 2 , 3 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq131_HTML.gif.

        Since U ( s ) , ( T U ) ( s ) , ( S U ) ( s ) P r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq132_HTML.gif for s J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq133_HTML.gif, where r = max { r , k r , h r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq134_HTML.gif, we see that, by virtue of assumption ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif),
        α E ( f ( s , U ( s ) , ( T U ) ( s ) , ( S U ) ( s ) ) ) = 0 t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ39_HTML.gif
        (39)
        It follows from (38) and (39) that
        α E ( A U ( t ) 1 + t α 1 ) 2 ( 1 + λ ) ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equap_HTML.gif
        which implies, by virtue of the arbitrariness of ε, that
        α E ( A U ( t ) 1 + t α 1 ) = 0 t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equaq_HTML.gif
        Using Lemma 2.6, we have
        α B ( A U ) = sup t J ( A U ( t ) 1 + t α 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equar_HTML.gif

        Thus, we can conclude that AU is relatively compact in B C [ J , E ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq31_HTML.gif, i.e., A is compact.

        As in the proof of Lemma 2.2, (12) holds. Choose
        R > { 2 u 0 , 2 ( 1 Γ ( α ) + λ ) ( a + M b ) + 2 λ Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 ( a ( s ) + M b ( s ) ) d s } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ40_HTML.gif
        (40)
        where u 0 θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq62_HTML.gif is given in assumption ( H 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq61_HTML.gif), and let Ω 1 = { u B C [ J , P ] : u < R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq135_HTML.gif. Then Ω ¯ 1 = { u B C [ J , P ] : u R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq136_HTML.gif and, by (12) and (40), we have
        A ( Ω ¯ 1 ) Ω 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ41_HTML.gif
        (41)
        By virtue of ( H 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq55_HTML.gif), there exists an r 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq137_HTML.gif such that
        f ( t , ( 1 + t α 1 ) u , ( 1 + t α 1 ) v , ( 1 + t α 1 ) w ) ε 2 d ( t ) ( u + v + w ) t J , u , v , w P , u + v + w r 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ42_HTML.gif
        (42)
        where
        ε 2 = 1 2 ( 1 + k + h ) [ ( 1 Γ ( α ) + λ ) d + λ Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 d ( s ) d s ] 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ43_HTML.gif
        (43)
        Let
        r 2 = r 1 1 + k + h . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equas_HTML.gif
        Then, for u B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq72_HTML.gif with u B r 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq138_HTML.gif, we have by (42)
        f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) = f ( t , ( 1 + t α 1 ) u ( t ) 1 + t α 1 , ( 1 + t α 1 ) ( T u ) ( t ) 1 + t α 1 , ( 1 + t α 1 ) ( S u ) ( t ) 1 + t α 1 ) ε 2 d ( t ) ( u ( t ) 1 + t α 1 + ( T u ) ( t ) 1 + t α 1 + ( S u ) ( t ) 1 + t α 1 ) ε 2 d ( t ) ( 1 + k + h ) u B t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ44_HTML.gif
        (44)
        It follows from (3), (43) and (44) that
        ( A u ) ( t ) 1 + t α 1 1 Γ ( α ) 0 t f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + λ Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + λ 0 + f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s ( 1 Γ ( α ) + λ ) ε 2 d ( 1 + k + h ) u B + ε 2 ( 1 + k + h ) Γ ( α ) i = 1 m β i 0 η i ( η i s ) α 1 d ( s ) d s = 1 2 u B , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equat_HTML.gif
        which implies
        A u B 1 2 u B , u B C [ J , P ] , u B r 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ45_HTML.gif
        (45)
        Choose
        0 < r < min { u 0 N ( 1 + t α 1 ) , r 2 , R } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ46_HTML.gif
        (46)
        Let Ω 2 = { u B C [ J , P ] : u B < r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq139_HTML.gif. Then Ω ¯ 2 = { u B C [ J , P ] : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq140_HTML.gif, and we have, by (45) and (46),
        A ( Ω ¯ 2 ) Ω 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ47_HTML.gif
        (47)
        Let Ω 3 = { u B C [ J , P ] : u B < R , u ( t ) u 0 , t I } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq141_HTML.gif, and we are going to show that Ω 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq142_HTML.gif is an open set of B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif. It is clear that we need only to show the following: for any u ¯ Ω 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq143_HTML.gif, there exists η > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq144_HTML.gif such that u B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq72_HTML.gif, u u ¯ B < η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq145_HTML.gif implies that u ( t ) u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq146_HTML.gif for t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq124_HTML.gif. We have u ¯ ( t ) u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq147_HTML.gif for t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq124_HTML.gif. So, for any s I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq148_HTML.gif, there exists a ε = ε ( s ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq149_HTML.gif such that
        u ¯ ( s ) ( 1 + 3 ε ) u 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ48_HTML.gif
        (48)
        Since u 0 θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq62_HTML.gif and u ¯ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq150_HTML.gif is continuous on J, we can find an open interval I ( s , δ ) = ( s δ , s + δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq151_HTML.gif ( δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq152_HTML.gif) such that
        ε u 0 + [ u ¯ ( t ) u ¯ ( s ) ] θ t I ( s , δ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equau_HTML.gif
        which implies by virtue of (48) that
        u ¯ ( t ) ( 1 + 2 ε ) u 0 t I ( s , δ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equav_HTML.gif
        Since I is compact, there is a finite collection of such intervals { I ( s j , δ j ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq153_HTML.gif ( j = 1 , 2 , , k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq154_HTML.gif) which cover I, and
        u ¯ ( t ) ( 1 + 2 ε j ) u 0 t I ( s j , δ j ) ( j = 1 , 2 , , k ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equaw_HTML.gif
        where ε j > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq155_HTML.gif ( j = 1 , 2 , , k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq154_HTML.gif). Consequently,
        u ¯ ( t ) ( 1 + 2 ε ) u 0 t I , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ49_HTML.gif
        (49)
        where ε = min 1 j k { ε j } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq156_HTML.gif. Since u 0 θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq62_HTML.gif, there exists an η = u 0 2 N ( 1 + t α ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq157_HTML.gif such that
        ε u 0 + [ u ( t ) u ¯ ( t ) ] θ t I , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ50_HTML.gif
        (50)
        whenever u B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq72_HTML.gif satisfying u u ¯ B < η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq145_HTML.gif, which implies by virtue of (49) and (50) that
        u ( t ) ( 1 + ε ) u 0 u 0 , u B C [ J , P ] , u u ¯ B < η . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equax_HTML.gif

        Thus, we have proved that Ω 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq142_HTML.gif is open in B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif.

        On the other hand, Lemma 2.4 and assumption ( H 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq61_HTML.gif) imply
        ( A u ) ( t ) t t G ( t , s ) f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s t t G ( t , s ) σ ( s ) d s u 0 t t γ ( s ) σ ( s ) d s u 0 u 0 t I . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ51_HTML.gif
        (51)
        Hence
        A ( Ω ¯ 3 ) Ω 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ52_HTML.gif
        (52)
        Since Ω 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq158_HTML.gif, Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq159_HTML.gif and Ω 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq142_HTML.gif are nonempty bounded convex open subsets of B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif, we see that (41), (47) and (52) imply by virtue of Lemma 1.1 the fixed point indices
        i ( A , Ω i , B C [ J , P ] ) = 1 ( i = 1 , 2 , 3 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ53_HTML.gif
        (53)
        On the other hand, for u Ω 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq160_HTML.gif, we have u ( t ) u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq146_HTML.gif, and so
        u B u ( t ) 1 + t α 1 u 0 N ( 1 + t α 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equay_HTML.gif
        Consequently,
        Ω 2 Ω 1 B C [ J , P ] , Ω 3 Ω 1 B C [ J , P ] , Ω 2 Ω 3 = . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ54_HTML.gif
        (54)
        By (53), (54) and the additivity of the fixed point index (Lemma 1.2), we can obtain
        i ( A , Ω 1 / ( Ω 2 Ω 3 ¯ ) , B C [ J , P ] ) = i ( A , Ω 1 , B C [ J , P ] ) i ( A , Ω 2 , B C [ J , P ] ) i ( A , Ω 3 , B C [ J , P ] ) = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ55_HTML.gif
        (55)

        Finally, (53), (54) and (55) imply that A has two fixed points u Ω 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq161_HTML.gif and u Ω 1 / ( Ω 2 Ω 3 ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq162_HTML.gif. We have, by (51), u ( t ) u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq123_HTML.gif for t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq124_HTML.gif. The proof is complete. □

        Remark 3.1 Assumption ( H 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq67_HTML.gif) and the continuity of f imply that f ( t , θ , θ , θ ) = θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq163_HTML.gif for t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq49_HTML.gif. Hence, under the assumptions of the theorem, BVP (1) has the trivial solution u ( t ) θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq164_HTML.gif besides two positive solutions u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq165_HTML.gif and u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq166_HTML.gif.

        Theorem 3.2 Let ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif)-( H 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq57_HTML.gif) and ( H 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq67_HTML.gif) be satisfied. Then BVP (1) has at least one positive solution u ˜ ( t ) B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq167_HTML.gif such that u ˜ ( t ) u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq168_HTML.gif for t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq124_HTML.gif.

        Proof By Lemma 2.2, Lemma 2.4 and the proof of Theorem 3.1, the operator A defined by (3) is completely continuous from B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif into B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif, and by Lemma 2.3, we need only to show that A has one positive fixed point u ˜ B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq169_HTML.gif such that u ˜ ( t ) u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq170_HTML.gif for t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq124_HTML.gif.

        As in the proof of Lemma 2.2, (12) holds. Choose R satisfying (40) and let U = { u B C [ J , P ] : u R , u ( t ) u 0 t I } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq171_HTML.gif, where u 0 > θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq68_HTML.gif is given by assumption ( H 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq67_HTML.gif). It is clear that U is a nonempty bounded closed convex subset in B C [ J , P ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif ( U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq172_HTML.gif because 2 u 0 U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq173_HTML.gif). Let u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq100_HTML.gif, by (40), we have A u R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq174_HTML.gif. On the other hand, as in the proof of Theorem 3.1, Lemma 2.4 and assumption ( H 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq67_HTML.gif) imply
        ( A u ) ( t ) t t G ( t , s ) f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s t t G ( t , s ) σ ( s ) d s u 0 t t γ ( s ) σ ( s ) d s u 0 u 0 t I . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ56_HTML.gif
        (56)

        Hence, A u W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq175_HTML.gif, and therefore A U U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq176_HTML.gif. Thus, the Schauder fixed point theorem implies that A has a fixed point u ˜ U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq177_HTML.gif, and by (56) u ˜ ( t ) u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq168_HTML.gif for t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq124_HTML.gif. The proof is complete. □

        4 Conclusion

        In this paper, the issue on the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space has been addressed for the first time. Taking advantage of the fixed point index theory of completely continuous operators, the existence conditions for such boundary value problems have been established.

        Declarations

        Acknowledgements

        This work was supported by the Natural Science Foundation of China under grant No. 11271248 and the Science and Technology Research Program of Zhejiang Province under grant No. 2011C21036.

        Authors’ Affiliations

        (1)
        College of Information Science and Technology, Donghua University
        (2)
        College of Fundamental Studies, Shanghai University of Engineering Science
        (3)
        Department of Applied mathematics, Donghua University

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