Open Access

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

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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq4_HTML.gif implies x N y https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq2_HTML.gif and x y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq6_HTML.gif, we write x < y https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq8_HTML.gif, we write x y https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq11_HTML.gif, where Ω ¯ https://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 . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq13_HTML.gif, where Ω i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq14_HTML.gif ( i = 1 , 2 https://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 = https://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 https://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 https://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 ) . https://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 ] https://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 ) ) https://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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equc_HTML.gif

where I = [ a , b ] https://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 } https://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 https://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 https://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 https://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 , ) https://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 https://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 https://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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Eque_HTML.gif

where n = [ α ] + 1 https://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 , ) https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq27_HTML.gif, n = [ α ] https://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 , + ) https://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 < } https://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 ] https://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 . https://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 ) , https://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 https://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 https://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 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq34_HTML.gif, β i > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq35_HTML.gif ( i = 1 , 2 , , m https://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 https://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 ) https://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 , https://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 + ] https://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 } https://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 + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq41_HTML.gif, R + https://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 ] https://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 https://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 < https://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 < https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq47_HTML.gif, as t t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq48_HTML.gif ( t J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq49_HTML.gif).

( H 2 https://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 + ] https://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 + ] https://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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equh_HTML.gif
( H 3 https://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 + ] https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equi_HTML.gif
uniformly for t J https://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 < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equj_HTML.gif
( H 4 https://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 + ] https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equk_HTML.gif
uniformly for t J https://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 < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equl_HTML.gif

( H 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq57_HTML.gif) For any t J https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq49_HTML.gif and r > 0 https://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 } https://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 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq60_HTML.gif.

( H 6 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq62_HTML.gif, 0 < t < t < https://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 + ] https://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 https://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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equn_HTML.gif

where I = [ t , t ] https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq66_HTML.gif.

( H 7 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq68_HTML.gif, 0 < t < t < https://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 + ] https://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 https://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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equp_HTML.gif

where I = [ t , t ] https://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 ) https://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 https://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 https://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 https://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 , https://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 https://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 https://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 ] https://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 ] https://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 . https://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 ] . https://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 , https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif), ( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) and ( H 3 https://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 ] https://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 ] https://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 , https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) and ( H 3 https://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 https://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 https://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 , https://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 } . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq49_HTML.gif, u , v , w P https://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 ) . https://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 ] https://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 ) , https://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 https://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 . https://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 . https://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 . https://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 ] https://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 ] https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq75_HTML.gif ( n https://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 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq77_HTML.gif and u ˜ B r https://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 https://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 . https://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 , https://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 + ] . https://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 https://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 . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq79_HTML.gif ( n https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif), ( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) and ( H 3 https://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 ] https://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 ] https://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 , https://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 ] https://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 ] https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq82_HTML.gif, c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq83_HTML.gif, c 3 https://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 . https://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 https://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 . https://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 ) https://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 . https://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 ) https://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 https://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 . https://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 ] https://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 ) https://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 , https://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 https://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 , ) https://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 . https://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 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq90_HTML.gif. For t η 1 https://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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equu_HTML.gif
0 < s t https://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 , https://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 https://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 , https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq94_HTML.gif, j = 2 , 3 , , m https://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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equx_HTML.gif
η m s https://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 . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif), ( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) and ( H 3 https://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 ] https://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 } https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq98_HTML.gif, there exists τ > 0 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq101_HTML.gif.

Proof For u U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq100_HTML.gif, t 1 < t 2 https://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 ) . https://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 } https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equaa_HTML.gif

for all u U https://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 https://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 https://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 https://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 . https://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 https://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 https://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 , https://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 https://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 . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq106_HTML.gif, s [ 0 , L ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq107_HTML.gif, t [ L , + ) https://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 . https://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 https://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 https://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 . https://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 . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif), ( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq50_HTML.gif) and ( H 3 https://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 ] https://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 ) . https://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 ] https://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 ) < . https://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 https://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 https://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 https://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 https://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 https://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 https://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 } https://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 ] https://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 ) , https://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 } , https://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 ] https://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 ] https://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 https://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 https://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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ33_HTML.gif
(33)

where diam B ( ) https://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 ] https://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 https://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 , + ) . https://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 . https://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 https://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 . https://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 https://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 https://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 https://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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equal_HTML.gif
Hence, for t J https://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 https://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 https://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 . https://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 ) https://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 , https://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 . https://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 ) . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif)-( H 6 https://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 ] https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq123_HTML.gif for t I https://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 ] https://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 ] https://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 ] https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq123_HTML.gif for t I https://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 ] https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq126_HTML.gif ( n = 1 , 2 , 3 , https://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 https://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 https://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 . https://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 , } https://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 ] https://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 , https://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 , } https://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 , } https://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 , } https://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 , } https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq132_HTML.gif for s J https://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 } https://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 https://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 . https://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 , https://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 . https://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 . https://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 ] https://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 } , https://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 https://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 https://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 } https://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 } https://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 . https://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 https://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 https://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 , https://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 . https://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 . https://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 ] https://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 https://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 . https://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 , https://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 . https://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 } . https://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 } https://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 } https://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 . https://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 } https://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 https://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 ] https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq143_HTML.gif, there exists η > 0 https://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 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq72_HTML.gif, u u ¯ B < η https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq146_HTML.gif for t I https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq147_HTML.gif for t I https://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 https://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 https://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 . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq62_HTML.gif and u ¯ ( t ) https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq151_HTML.gif ( δ > 0 https://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 ) , https://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 ) . https://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 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq153_HTML.gif ( j = 1 , 2 , , k https://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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equaw_HTML.gif
where ε j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq155_HTML.gif ( j = 1 , 2 , , k https://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 , https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq156_HTML.gif. Since u 0 θ https://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 https://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 , https://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 ] https://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 https://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 . https://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 https://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 ] https://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 https://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 . https://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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_Equ52_HTML.gif
(52)
Since Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq158_HTML.gif, Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq159_HTML.gif and Ω 3 https://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 ] https://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 ) . https://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 https://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 https://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 ) . https://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 = . https://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 . https://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 https://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 ¯ ) https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq123_HTML.gif for t I https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq163_HTML.gif for t J https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq165_HTML.gif and u https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq44_HTML.gif)-( H 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq57_HTML.gif) and ( H 7 https://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 ] https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq168_HTML.gif for t I https://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 ] https://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 ] https://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 ] https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq170_HTML.gif for t I https://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 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq171_HTML.gif, where u 0 > θ https://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 https://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 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq43_HTML.gif ( U https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq173_HTML.gif). Let u U https://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 https://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 https://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 . https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-79/MediaObjects/13661_2012_Article_340_IEq168_HTML.gif for t I https://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

References

  1. Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.
  2. Guo D, Lakshmikantham V, Liu XZ: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht; 1996.View Article
  3. Arara A, Benchohra M, Hamidi N, Nieto JJ: Fractional order differential equations on an unbounded domain. Nonlinear Anal. 2010, 72: 580-586. 10.1016/j.na.2009.06.106MathSciNetView Article
  4. Babakhani A, Gejji VD: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 2003, 278: 434-442. 10.1016/S0022-247X(02)00716-3MathSciNetView Article
  5. Delbosco D, Rodino L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 1996, 204: 609-625. 10.1006/jmaa.1996.0456MathSciNetView Article
  6. Diethlm K, Ford NJ: Analysis of fractional differential equations. J. Math. Anal. Appl. 2002, 265: 229-248. 10.1006/jmaa.2000.7194MathSciNetView Article
  7. Sayed WGE, Sayed AMAE: On the functional integral equations of mixed type and integro-differential equations of fractional orders. Appl. Math. Comput. 2004, 154: 461-467. 10.1016/S0096-3003(03)00727-6MathSciNetView Article
  8. Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2003.View Article
  9. El-Sayed AMA: On the fractional differential equation. Appl. Math. Comput. 1992, 49: 205-213. 10.1016/0096-3003(92)90024-UMathSciNetView Article
  10. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
  11. Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods, results and problems I. Appl. Anal. 2001, 78: 153-192. 10.1080/00036810108840931MathSciNetView Article
  12. Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods, results and problems II. Appl. Anal. 2002, 81: 435-493. 10.1080/0003681021000022032MathSciNetView Article
  13. Kosmatov N: Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal. 2009, 70: 2521-2529. 10.1016/j.na.2008.03.037MathSciNetView Article
  14. Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Anal. 2008, 69: 3337-3343. 10.1016/j.na.2007.09.025MathSciNetView Article
  15. Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Anal. 2008, 69: 2677-2682. 10.1016/j.na.2007.08.042MathSciNetView Article
  16. Muslim M, Conca C, Nandakumaran AK: Approximate of solutions to fractional integral equation. Comput. Math. Appl. 2010, 59: 1236-1244. 10.1016/j.camwa.2009.06.028MathSciNetView Article
  17. Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York; 1993.
  18. Podlubny I: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego; 1999.
  19. Stojanović M: Existence-uniqueness result for a nonlinear n -term fractional equation. J. Math. Anal. Appl. 2009, 353: 244-245. 10.1016/j.jmaa.2008.11.056MathSciNetView Article
  20. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, Yverdon; 1993.

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© Liu et al.; licensee Springer. 2013

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