Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems

  • Chengbo Zhai1Email author and

    Affiliated with

    • Mengru Hao1

      Affiliated with

      Boundary Value Problems20132013:85

      DOI: 10.1186/1687-2770-2013-85

      Received: 22 November 2012

      Accepted: 18 March 2013

      Published: 10 April 2013

      Abstract

      This work is concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problem:

      { D 0 + ν y ( t ) = f ( t , y ( t ) , y ( t ) ) + g ( t , y ( t ) ) , 0 < t < 1 , n 1 < ν n , y ( i ) ( 0 ) = 0 , 0 i n 2 , [ D 0 + α y ( t ) ] t = 1 = 0 , 1 α n 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equa_HTML.gif

      where D 0 + ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq1_HTML.gif is the standard Riemann-Liouville fractional derivative of order ν, and n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq2_HTML.gif, n > 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq3_HTML.gif. Our analysis relies on two new fixed point theorems for mixed monotone operators with perturbation. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate the main result.

      MSC:26A33, 34B18, 34B27.

      Keywords

      Riemann-Liouville fractional derivative fractional differential equation positive solution existence and uniqueness fixed point theorem for mixed monotone operator

      1 Introduction

      In this paper, we investigate the existence and uniqueness of positive solutions for the fractional boundary value problem (FBVP for short) of the form:
      { D 0 + ν y ( t ) = f ( t , y ( t ) , y ( t ) ) + g ( t , y ( t ) ) , 0 < t < 1 , n 1 < ν n , y ( i ) ( 0 ) = 0 , 0 i n 2 , [ D 0 + α y ( t ) ] t = 1 = 0 , 1 α n 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equ1_HTML.gif
      (1.1)

      where D 0 + ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq1_HTML.gif is the standard Riemann-Liouville fractional derivative of order ν, and n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq2_HTML.gif, n > 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq3_HTML.gif.

      Fractional differential equations arise in many fields such as physics, mechanics, chemistry, economics, engineering and biological sciences, etc.; see [16] for example. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Miller and Ross [3], Podlubny [5], Kilbas et al. [6], and the papers [716] and the references therein. In these papers, many authors have investigated the existence of positive solutions for nonlinear fractional differential equation boundary value problems. On the other hand, the uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems has been studied by some authors; see [10, 14, 17] for example.

      In [18], Goodrich utilized the Krasnoselskii’s fixed point theorem to study a FBVP of the form:
      { D 0 + ν y ( t ) = f ( t , y ( t ) ) , 0 < t < 1 , n 1 < ν n , y ( i ) ( 0 ) = 0 , 0 i n 2 , [ D 0 + α y ( t ) ] t = 1 = 0 , 1 α n 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equ2_HTML.gif
      (1.2)
      and established the existence of at least one positive solution for FBVP (1.2). By using the same fixed point theorem, Goodrich [19] considered the existence of a positive solution to the following systems of differential equations of fractional order:
      { D 0 + ν 1 y 1 ( t ) = λ 1 a 1 ( t ) f ( y 1 ( t ) , y 2 ( t ) ) , D 0 + ν 2 y 2 ( t ) = λ 2 a 2 ( t ) g ( y 1 ( t ) , y 2 ( t ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equb_HTML.gif
      where t ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq4_HTML.gif, ν 1 , ν 2 ( n 1 , n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq5_HTML.gif for n > 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq3_HTML.gif and n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq6_HTML.gif, and λ 1 , λ 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq7_HTML.gif, with the following boundary value conditions:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equc_HTML.gif

      under the assumptions that a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq8_HTML.gif, a 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq9_HTML.gif, f, g are nonnegative and continuous. But the uniqueness of positive solutions is not treated in these papers.

      Different from the works mentioned above, motivated by the work [20], we will use two fixed point theorems for mixed monotone operators with perturbation to show the existence and uniqueness of positive solutions for FBVP (1.1). To our knowledge, there are still very few to utilize the fixed point results on mixed monotone operators with perturbation to study the existence and uniqueness of a positive solution for nonlinear fractional differential equation boundary value problems. So, it is worthwhile to investigate FBVP (1.1) by using our new fixed point theorems in [20]. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it.

      With this context in mind, the outline of this paper is as follows. In Section 2 we recall certain results from the theory of fractional calculus and some definitions, notations and results of mixed monotone operators. In Section 3 we provide some conditions, under which the problem FBVP (1.1) has a unique positive solution. Finally, in Section 4, we provide an example, which explicates the applicability of our result.

      2 Preliminaries

      For the convenience of the reader, we present here some definitions, lemmas and basic results that will be used in the proofs of our theorems.

      Definition 2.1 (See [18])

      Let ν > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq10_HTML.gif with ν R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq11_HTML.gif. Suppose that y : [ a , + ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq12_HTML.gif. Then the ν th Riemann-Liouville fractional integral is defined to be
      D a + ν y ( t ) : = 1 Γ ( ν ) a t y ( s ) ( t s ) ν 1 d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equd_HTML.gif
      whenever the right-hand side is defined. Similarly, with ν > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq10_HTML.gif and ν R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq11_HTML.gif, we define the ν th Riemann-Liouville fractional derivative to be
      D a + ν y ( t ) : = 1 Γ ( n ν ) d n d t n a t y ( s ) ( t s ) ν + 1 n d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Eque_HTML.gif

      where n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq2_HTML.gif is the unique positive integer satisfying n 1 ν < n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq13_HTML.gif and t > a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq14_HTML.gif.

      Lemma 2.2 (See [19])

      Let g C [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq15_HTML.gif be given. Then the unique solution to problem D 0 + ν y ( t ) = g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq16_HTML.gif together with the boundary conditions y ( i ) ( 0 ) = 0 = [ D 0 + α y ( t ) ] t = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq17_HTML.gif, where 1 α n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq18_HTML.gif and 0 i n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq19_HTML.gif, is
      y ( t ) = 0 1 G ( t , s ) g ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equ3_HTML.gif
      (2.1)
      where
      G ( t , s ) = { t ν 1 ( 1 s ) ν α 1 ( t s ) ν 1 Γ ( ν ) , 0 s t 1 , t ν 1 ( 1 s ) ν α 1 Γ ( ν ) , 0 t s 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equ4_HTML.gif
      (2.2)

      is the Green function for this problem.

      Lemma 2.3 (See [19])

      Let G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq20_HTML.gif be as given in the statement of Lemma  2.2. Then we have
      1. (i)

        G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq20_HTML.gif is a continuous function on the unit square [ 0 , 1 ] × [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq21_HTML.gif;

         
      2. (ii)

        G ( t , s ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq22_HTML.gif for each ( t , s ) [ 0 , 1 ] × [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq23_HTML.gif.

         
      Lemma 2.4 The function G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq20_HTML.gif defined by (2.2) satisfies the following conditions:
      [ 1 ( 1 s ) α ] ( 1 s ) ν α 1 t ν 1 Γ ( ν ) G ( t , s ) ( 1 s ) ν α 1 t ν 1 , t , s [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equf_HTML.gif
      Proof Evidently, the right inequality holds. So, we only need to prove the left inequality. If 0 s t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq24_HTML.gif, then we have 0 t s t t s = ( 1 s ) t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq25_HTML.gif, and thus
      ( t s ) ν 1 ( 1 s ) ν 1 t ν 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equg_HTML.gif
      Hence,
      Γ ( ν ) G ( t , s ) = t ν 1 ( 1 s ) ν α 1 ( t s ) ν 1 t ν 1 ( 1 s ) ν α 1 t ν 1 ( 1 s ) ν 1 = t ν 1 [ ( 1 s ) ν α 1 ( 1 s ) ν 1 ] = [ 1 ( 1 s ) α ] ( 1 s ) ν α 1 t ν 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equh_HTML.gif
      When 0 t s 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq26_HTML.gif, we have
      Γ ( ν ) G ( t , s ) = t ν 1 ( 1 s ) ν α 1 t ν 1 [ ( 1 s ) ν α 1 ( 1 s ) ν 1 ] = [ 1 ( 1 s ) α ] ( 1 s ) ν α 1 t ν 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equi_HTML.gif

      So, the proof is complete. □

      In the sequel, we present some basic concepts in ordered Banach spaces for completeness and two fixed point theorems which we will be used later. For convenience of readers, we suggest that one refers to [2022] for details.

      Suppose that ( E , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq27_HTML.gif is a real Banach space which is partially ordered by a cone P E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq28_HTML.gif, i.e., x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq29_HTML.gif if and only if y x P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq30_HTML.gif. If x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq29_HTML.gif and x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq31_HTML.gif, then we denote x < y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq32_HTML.gif or y > x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq33_HTML.gif. By θ we denote the zero element of E. Recall that a non-empty closed convex set P E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq28_HTML.gif is a cone if it satisfies (i) x P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq34_HTML.gif, λ 0 λ x P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq35_HTML.gif; (ii) x P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq34_HTML.gif, x P x = θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq36_HTML.gif.

      P is called normal if there exists a constant N > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq37_HTML.gif such that, for all x , y E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq38_HTML.gif, θ x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq39_HTML.gif implies x N y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq40_HTML.gif; in this case, N is called the normality constant of P. If x 1 , x 2 E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq41_HTML.gif, the set [ x 1 , x 2 ] = { x E x 1 x x 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq42_HTML.gif is called the order interval between x 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq43_HTML.gif and x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq44_HTML.gif. We say that an operator A : E E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq45_HTML.gif is increasing (decreasing) if x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq29_HTML.gif implies A x A y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq46_HTML.gif ( A x A y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq47_HTML.gif).

      For all x , y E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq38_HTML.gif, the notation x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq48_HTML.gif means that there exist λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq49_HTML.gif and μ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq50_HTML.gif such that λ x y μ x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq51_HTML.gif. Clearly, ∼ is an equivalence relation. Given h > θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq52_HTML.gif (i.e., h θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq53_HTML.gif and h θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq54_HTML.gif), we denote by P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif the set P h = { x E x h } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq56_HTML.gif. It is easy to see that P h P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq57_HTML.gif.

      Definition 2.5 (See [20, 22])

      A : P × P P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq58_HTML.gif is said to be a mixed monotone operator if A ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq59_HTML.gif is increasing in x and decreasing in y, i.e., u i , v i ( i = 1 , 2 ) P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq60_HTML.gif, u 1 u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq61_HTML.gif, v 1 v 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq62_HTML.gif imply A ( u 1 , v 1 ) A ( u 2 , v 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq63_HTML.gif. Element x P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq34_HTML.gif is called a fixed point of A if A ( x , x ) = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq64_HTML.gif.

      Definition 2.6 An operator A : P P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq65_HTML.gif is said to be sub-homogeneous if it is satisfies
      A ( t x ) t A ( x ) , t ( 0 , 1 ) , x P . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equ5_HTML.gif
      (2.3)
      Definition 2.7 Let D = P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq66_HTML.gif and β be a real number with 0 β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq67_HTML.gif. An operator A : D D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq68_HTML.gif is said to be β-concave if it satisfies
      A ( t x ) t β A ( x ) , t ( 0 , 1 ) , x D . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equ6_HTML.gif
      (2.4)

      Lemma 2.8 (See Theorem 2.1 in [20])

      Let h > θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq52_HTML.gif and β ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq69_HTML.gif. A : P × P P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq58_HTML.gif is a mixed monotone operator and satisfies
      A ( t x , t 1 y ) t β A ( x , y ) , t ( 0 , 1 ) , x , y P . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equ7_HTML.gif
      (2.5)
      B : P P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq70_HTML.gif is an increasing sub-homogeneous operator. Assume that
      1. (i)

        there is h 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq71_HTML.gif such that A ( h 0 , h 0 ) P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq72_HTML.gif and B h 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq73_HTML.gif;

         
      2. (ii)

        there exists a constant δ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq74_HTML.gif such that A ( x , y ) δ 0 B x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq75_HTML.gif, x , y P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq76_HTML.gif.

         
      Then:
      1. (1)

        A : P h × P h P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq77_HTML.gif and B : P h P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq78_HTML.gif;

         
      2. (2)
        there exist u 0 , v 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq79_HTML.gif and r ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq80_HTML.gif such that
        r v 0 u 0 < v 0 , u 0 A ( u 0 , v 0 ) + B u 0 A ( v 0 , u 0 ) + B v 0 v 0 ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equj_HTML.gif
         
      3. (3)

        the operator equation A ( x , x ) + B x = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq81_HTML.gif has a unique solution x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq82_HTML.gif in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif;

         
      4. (4)
        for any initial values x 0 , y 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq83_HTML.gif, constructing successively the sequences
        x n = A ( x n 1 , y n 1 ) + B x n 1 , y n = A ( y n 1 , x n 1 ) + B y n 1 , n = 1 , 2 , , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equk_HTML.gif
         

      we have x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq84_HTML.gif and y n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq85_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq86_HTML.gif.

      Lemma 2.9 (See Theorem 2.4 in [20])

      Let h > θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq52_HTML.gif and β ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq69_HTML.gif. A : P × P P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq58_HTML.gif is a mixed monotone operator and satisfies
      A ( t x , t 1 y ) t A ( x , y ) , t ( 0 , 1 ) , x , y P . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equ8_HTML.gif
      (2.6)
      B : P P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq70_HTML.gif is an increasing β-concave operator. Assume that
      1. (i)

        there is h 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq71_HTML.gif such that A ( h 0 , h 0 ) P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq72_HTML.gif and B h 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq73_HTML.gif;

         
      2. (ii)

        there exists a constant δ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq74_HTML.gif such that A ( x , y ) δ 0 B x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq87_HTML.gif, x , y P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq76_HTML.gif.

         
      Then:
      1. (1)

        A : P h × P h P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq77_HTML.gif and B : P h P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq78_HTML.gif;

         
      2. (2)
        there exist u 0 , v 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq79_HTML.gif and r ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq80_HTML.gif such that
        r v 0 u 0 < v 0 , u 0 A ( u 0 , v 0 ) + B u 0 A ( v 0 , u 0 ) + B v 0 v 0 ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equl_HTML.gif
         
      3. (3)

        the operator equation A ( x , x ) + B x = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq81_HTML.gif has a unique solution x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq82_HTML.gif in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif;

         
      4. (4)
        for any initial values x 0 , y 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq83_HTML.gif, constructing successively the sequences
        x n = A ( x n 1 , y n 1 ) + B x n 1 , y n = A ( y n 1 , x n 1 ) + B y n 1 , n = 1 , 2 , , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equm_HTML.gif
         

      we have x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq84_HTML.gif and y n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq88_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq89_HTML.gif.

      Remark 2.10 (i) If we take B = θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq90_HTML.gif in Lemma 2.8, then the corresponding conclusion is still true (see Corollary 2.2 in [20]); (ii) if we take A = θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq91_HTML.gif in Lemma 2.9, then the conclusion obtained is also true (see Theorem 2.7 in [23]).

      3 Main results

      In this section, we apply Lemma 2.8 and Lemma 2.9 to study FBVP (1.1), and we obtain some new results on the existence and uniqueness of positive solutions. The method used here is relatively new to the literature and so are the existence and uniqueness results to the fractional differential equations.

      In our considerations, we work in the Banach space C [ 0 , 1 ] = { x : [ 0 , 1 ] R  is  continuous } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq92_HTML.gif with the standard norm x = sup { | x ( t ) | : t [ 0 , 1 ] } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq93_HTML.gif. Notice that this space can be equipped with a partial order given by
      x , y C [ 0 , 1 ] , x y x ( t ) y ( t ) for  t [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equn_HTML.gif

      Set P = { x C [ 0 , 1 ] x ( t ) 0 , t [ 0 , 1 ] } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq94_HTML.gif, the standard cone. It is clear that P is a normal cone in C [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq95_HTML.gif and the normality constant is 1.

      Theorem 3.1 Assume that

      (H1) f : [ 0 , 1 ] × [ 0 , + ) × [ 0 , + ) [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq96_HTML.gif is continuous and g : [ 0 , 1 ] × [ 0 , + ) [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq97_HTML.gif is continuous;

      (H2) f ( t , u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq98_HTML.gif is increasing in u [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq99_HTML.gif for fixed t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq100_HTML.gif and v [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq101_HTML.gif, decreasing in v [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq101_HTML.gif for fixed t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq102_HTML.gif and u [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq103_HTML.gif, and g ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq104_HTML.gif is increasing in u [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq99_HTML.gif for fixed t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq102_HTML.gif;

      (H3) g ( t , 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq105_HTML.gif and g ( t , λ u ) λ g ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq106_HTML.gif for λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq107_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq102_HTML.gif, u [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq103_HTML.gif, and there exists a constant β ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq108_HTML.gif such that f ( t , λ u , λ 1 v ) λ β f ( t , u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq109_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq110_HTML.gif, λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq107_HTML.gif, u , v [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq111_HTML.gif;

      (H4) there exists a constant δ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq74_HTML.gif such that f ( t , u , v ) δ 0 g ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq112_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif, u , v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq114_HTML.gif.

      Then:
      1. (1)
        there exist u 0 , v 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq79_HTML.gif and r ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq80_HTML.gif such that r v 0 u 0 < v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq115_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equo_HTML.gif
         
      where h ( t ) = t ν 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq116_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif and G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq20_HTML.gif is given as in (2.2);
      1. (2)

        FBVP (1.1) has a unique positive solution u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq117_HTML.gif in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif;

         
      2. (3)
        for any x 0 , y 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq118_HTML.gif, constructing successively the sequences
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equp_HTML.gif
         

      we have x n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq119_HTML.gif and y n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq120_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq121_HTML.gif.

      Proof To begin with, from Lemma 2.2, FBVP (1.1) has an integral formulation given by
      u ( t ) = 0 1 G ( t , s ) [ f ( s , u ( s ) , u ( s ) ) + g ( s , u ( s ) ) ] d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equq_HTML.gif

      where G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq20_HTML.gif is given as in (2.2).

      Define two operators A : P × P E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq122_HTML.gif and B : P E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq123_HTML.gif by
      A ( u , v ) ( t ) = 0 1 G ( t , s ) f ( s , u ( s ) , v ( s ) ) d s , ( B u ) ( t ) = 0 1 G ( t , s ) g ( s , u ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equr_HTML.gif

      It is easy to prove that u is the solution of FBVP (1.1) if and only if u = A ( u , u ) + B u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq124_HTML.gif. From (H1), we know that A : P × P P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq125_HTML.gif and B : P P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq126_HTML.gif. In the sequel, we check that A, B satisfy all the assumptions of Lemma 2.8.

      Firstly, we prove that A is a mixed monotone operator. In fact, for u i , v i P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq127_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq128_HTML.gif with u 1 u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq129_HTML.gif, v 1 v 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq130_HTML.gif, we know that u 1 ( t ) u 2 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq131_HTML.gif, v 1 ( t ) v 2 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq132_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif, and by (H2) and Lemma 2.3,
      A ( u 1 , v 1 ) ( t ) = 0 1 G ( t , s ) f ( s , u 1 ( s ) , v 1 ( s ) ) d s 0 1 G ( t , s ) f ( s , u 2 ( s ) , v 2 ( s ) ) d s = A ( u 2 , v 2 ) ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equs_HTML.gif

      That is, A ( u 1 , v 1 ) A ( u 2 , v 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq133_HTML.gif.

      Further, it follows from (H2) and Lemma 2.3 that B is increasing. Next we show that A satisfies the condition (2.5). For any λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq107_HTML.gif and u , v P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq134_HTML.gif, by (H3) we have
      A ( λ u , λ 1 v ) ( t ) = 0 1 G ( t , s ) f ( s , λ u ( s ) , λ 1 v ( s ) ) d s λ β 0 1 G ( t , s ) f ( s , u ( s ) , v ( s ) ) d s = λ β A ( u , v ) ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equt_HTML.gif
      That is, A ( λ u , λ 1 v ) λ β A ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq135_HTML.gif for λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq107_HTML.gif, u , v P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq134_HTML.gif. So, the operator A satisfies (2.5). Also, for any λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq107_HTML.gif, u P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq136_HTML.gif, from (H3) we know that
      B ( λ u ) ( t ) = 0 1 G ( t , s ) g ( s , λ u ( s ) ) d s λ 0 1 G ( t , s ) g ( s , u ( s ) ) d s = λ B u ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equu_HTML.gif
      that is, B ( λ u ) λ B u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq137_HTML.gif for λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq138_HTML.gif, u P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq136_HTML.gif. That is, the operator B is sub-homogeneous. Now we show that A ( h , h ) P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq139_HTML.gif and B h P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq140_HTML.gif. On the one hand, from (H1), (H2) and Lemma 2.4, for any t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif, we have
      A ( h , h ) ( t ) = 0 1 G ( t , s ) f ( s , h ( s ) , h ( s ) ) d s = 0 1 G ( t , s ) f ( s , s ν 1 , s ν 1 ) d s 1 Γ ( ν ) h ( t ) 0 1 ( 1 s ) ν α 1 f ( s , 1 , 0 ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equv_HTML.gif
      On the other hand, also from (H1), (H2) and Lemma 2.4, for any t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif, we obtain
      A ( h , h ) ( t ) = 0 1 G ( t , s ) f ( s , h ( s ) , h ( s ) ) d s = 0 1 G ( t , s ) f ( s , s ν 1 , s ν 1 ) d s 1 Γ ( ν ) h ( t ) 0 1 [ 1 ( 1 s ) α ] ( 1 s ) ν α 1 f ( s , 0 , 1 ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equw_HTML.gif
      From (H2), (H4), we have
      f ( s , 1 , 0 ) f ( s , 0 , 1 ) δ 0 g ( s , 0 ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equx_HTML.gif
      Since g ( t , 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq105_HTML.gif, we get
      0 1 f ( s , 1 , 0 ) d s 0 1 f ( s , 0 , 1 ) d s δ 0 0 1 g ( s , 0 ) d s > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equy_HTML.gif
      and in consequence,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equz_HTML.gif
      So, l 2 h ( t ) A ( h , h ) ( t ) l 1 h ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq141_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif; and hence we have A ( h , h ) P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq139_HTML.gif. Similarly,
      1 Γ ( ν ) h ( t ) 0 1 [ 1 ( 1 s ) α ] ( 1 s ) ν α 1 g ( s , 0 ) d s B h ( t ) 1 Γ ( ν ) h ( t ) 0 1 ( 1 s ) ν α 1 g ( s , 1 ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equaa_HTML.gif

      from g ( t , 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq105_HTML.gif, we easily prove B h P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq142_HTML.gif. Hence the condition (i) of Lemma 2.8 is satisfied.

      In the following, we show the condition (ii) of Lemma 2.8 is satisfied. For u , v P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq143_HTML.gif, and any t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif, from (H4),
      A ( u , v ) ( t ) = 0 1 G ( t , s ) f ( s , u ( s ) , v ( s ) ) d s δ 0 0 1 G ( t , s ) g ( s , u ( s ) ) d s = δ 0 B u ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equab_HTML.gif
      Then we get A ( u , v ) δ 0 B u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq144_HTML.gif, for u , v P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq134_HTML.gif. Finally, an application of Lemma 2.8 implies: there exist u 0 , v 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq79_HTML.gif and r ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq80_HTML.gif such that r v 0 u 0 < v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq115_HTML.gif, u 0 A ( u 0 , v 0 ) + B u 0 A ( v 0 , u 0 ) + B v 0 v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq145_HTML.gif; the operator equation A ( u , u ) + B u = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq146_HTML.gif has a unique solution u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq117_HTML.gif in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif; for any initial values x 0 , y 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq83_HTML.gif, constructing successively the sequences
      x n = A ( x n 1 , y n 1 ) + B x n 1 , y n = A ( y n 1 , x n 1 ) + B y n 1 , n = 1 , 2 , , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equac_HTML.gif
      we have x n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq147_HTML.gif and y n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq148_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq86_HTML.gif. That is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equad_HTML.gif
      FBVP (1.1) has a unique positive solution u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq117_HTML.gif in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif; for x 0 , y 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq118_HTML.gif, the sequences
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equae_HTML.gif

      satisfy x n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq119_HTML.gif and y n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq120_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq121_HTML.gif. □

      Theorem 3.2 Assume (H1), (H2) and

      (H5) there exists a constant β ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq108_HTML.gif such that g ( t , λ u ) λ β g ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq149_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq110_HTML.gif, λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq107_HTML.gif, u [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq103_HTML.gif, and f ( t , λ u , λ 1 v ) λ f ( t , u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq150_HTML.gif for λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq107_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq102_HTML.gif, u , v [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq111_HTML.gif;

      (H6) f ( t , 0 , 1 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq151_HTML.gif for t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif and there exists a constant δ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq152_HTML.gif such that f ( t , u , v ) δ 0 g ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq153_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif, u , v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq114_HTML.gif.

      Then:
      1. (1)
        there exist u 0 , v 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq79_HTML.gif and r ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq80_HTML.gif such that r v 0 u 0 < v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq115_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equaf_HTML.gif
         
      where h ( t ) = t ν 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq116_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif and G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq20_HTML.gif is given as in (2.2);
      1. (2)

        FBVP (1.1) has a unique positive solution u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq117_HTML.gif in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif;

         
      2. (3)
        for any x 0 , y 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq118_HTML.gif, constructing successively the sequences
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equag_HTML.gif
         

      we have x n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq119_HTML.gif and y n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq120_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq121_HTML.gif.

      Sketch of the proof Consider two operators A, B defined in the proof of Theorem 3.1. Similarly, from (H1), (H2), we obtain that A : P × P P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq125_HTML.gif is a mixed monotone operator and B : P P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq126_HTML.gif is increasing. From (H5), we have
      A ( λ u , λ 1 v ) λ A ( u , v ) ; B ( λ u ) λ β B u , for  λ ( 0 , 1 ) , u , v P . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equah_HTML.gif
      From (H2), (H6), we have
      g ( s , 0 ) 1 δ 0 f ( s , 0 , 1 ) , f ( s , 1 , 0 ) f ( s , 0 , 1 ) , s [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equai_HTML.gif
      Since f ( t , 0 , 1 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq151_HTML.gif, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equaj_HTML.gif
      and in consequence,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equak_HTML.gif
      So, we can easily prove that A ( h , h ) P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq139_HTML.gif, B h P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq140_HTML.gif. For u , v P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq143_HTML.gif, and any t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif, from (H6),
      A ( u , v ) ( t ) = 0 1 G ( t , s ) f ( s , u ( s ) , v ( s ) ) d s δ 0 0 1 G ( t , s ) g ( s , u ( s ) ) d s = δ 0 B u ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equal_HTML.gif
      Then we get A ( u , v ) δ 0 B u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq154_HTML.gif, for u , v P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq134_HTML.gif. Finally, an application of Lemma 2.9 implies: there exist u 0 , v 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq79_HTML.gif and r ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq80_HTML.gif such that r v 0 u 0 < v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq115_HTML.gif, u 0 A ( u 0 , v 0 ) + B u 0 A ( v 0 , u 0 ) + B v 0 v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq145_HTML.gif; the operator equation A ( u , u ) + B u = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq146_HTML.gif has a unique solution u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq117_HTML.gif in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif; for any initial values x 0 , y 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq83_HTML.gif, constructing successively the sequences
      x n = A ( x n 1 , y n 1 ) + B x n 1 , y n = A ( y n 1 , x n 1 ) + B y n 1 , n = 1 , 2 , , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equam_HTML.gif
      we have x n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq147_HTML.gif and y n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq148_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq86_HTML.gif. That is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equan_HTML.gif
      FBVP (1.1) has a unique positive solution u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq117_HTML.gif in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif; for x 0 , y 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq118_HTML.gif, the sequences
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equao_HTML.gif

      satisfy x n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq119_HTML.gif and y n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq155_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq121_HTML.gif. □

      From Remark 2.10 and similar to the proofs of Theorems 3.1-3.2, we can prove the following conclusions.

      Corollary 3.3 Let g 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq156_HTML.gif. Assume that f satisfies the conditions of Theorem  3.1 and f ( t , 0 , 1 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq157_HTML.gif. Then: (i) there exist u 0 , v 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq79_HTML.gif and r ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq80_HTML.gif such that r v 0 u 0 < v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq115_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equap_HTML.gif
      where h ( t ) = t ν 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq116_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif and G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq20_HTML.gif is given as in (2.2); (ii) the FBVP
      { D 0 + ν y ( t ) = f ( t , y ( t ) , y ( t ) ) , 0 < t < 1 , n 1 < ν n , y ( i ) ( 0 ) = 0 , 0 i n 2 , [ D 0 + α y ( t ) ] t = 1 = 0 , 1 α n 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equaq_HTML.gif
      has a unique positive solution u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq117_HTML.gif in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif; (iii) for any x 0 , y 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq118_HTML.gif, constructing successively the sequences
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equar_HTML.gif

      we have x n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq119_HTML.gif and y n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq120_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq121_HTML.gif.

      Corollary 3.4 Let f 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq158_HTML.gif. Assume that g satisfies the conditions of Theorem  3.2 and g ( t , 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq105_HTML.gif for t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif. Then: (i) there exist u 0 , v 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq79_HTML.gif and r ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq80_HTML.gif such that r v 0 u 0 < v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq115_HTML.gif and
      u 0 ( t ) 0 1 G ( t , s ) g ( s , u 0 ( s ) ) d s , v 0 ( t ) 0 1 G ( t , s ) g ( s , v 0 ( s ) ) d s , t [ 0 , 1 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equas_HTML.gif
      where h ( t ) = t ν 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq116_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif and G ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq20_HTML.gif is given as in (2.2); (ii) the FBVP
      { D 0 + ν y ( t ) = g ( t , y ( t ) ) , 0 < t < 1 , n 1 < ν n , y ( i ) ( 0 ) = 0 , 0 i n 2 , [ D 0 + α y ( t ) ] t = 1 = 0 , 1 α n 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equat_HTML.gif
      has a unique positive solution u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq117_HTML.gif in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif; (iii) for any x 0 , y 0 P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq118_HTML.gif, constructing successively the sequences
      x n + 1 ( t ) = 0 1 G ( t , s ) g ( s , x n ( s ) ) d s , y n + 1 ( t ) = 0 1 G ( t , s ) g ( s , y n ( s ) ) d s , n = 0 , 1 , 2 , , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equau_HTML.gif

      we have x n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq159_HTML.gif and y n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq120_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq121_HTML.gif.

      4 An example

      We now present one example to illustrate Theorem 3.1.

      Example 4.1

      Consider the following FBVP:
      { D 0 + 6.3 u ( t ) = u 1 4 ( t ) + [ u ( t ) + 2 ] 1 3 + u ( t ) 1 + u ( t ) a ( t ) + b ( t ) + c , 0 < t < 1 , u ( i ) ( 0 ) = 0 , 0 i 5 , [ D 0 + 4.2 u ( t ) ] t = 1 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equ9_HTML.gif
      (4.1)

      where c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq160_HTML.gif is a constant, a , b : [ 0 , 1 ] [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq161_HTML.gif are continuous with a 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq162_HTML.gif.

      Obviously, problem (4.1) fits the framework of FBVP (1.1) with ν = 6.3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq163_HTML.gif, α = 4.2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq164_HTML.gif. (Note that n = 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq165_HTML.gif, therefore, in this case.) In this example, we take 0 < d < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq166_HTML.gif and let
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equav_HTML.gif
      Obviously, a max > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq167_HTML.gif; f : [ 0 , 1 ] × [ 0 , + ) × [ 0 , + ) [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq168_HTML.gif is continuous and g : [ 0 , 1 ] × [ 0 , + ) [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq169_HTML.gif is continuous with g ( t , 0 ) = c d > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq170_HTML.gif. And f ( t , u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq98_HTML.gif is increasing in u [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq171_HTML.gif for fixed t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq102_HTML.gif and v [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq101_HTML.gif, decreasing in v [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq101_HTML.gif for fixed t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq102_HTML.gif and u [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq172_HTML.gif, and g ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq104_HTML.gif is increasing in u [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq99_HTML.gif for fixed t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq102_HTML.gif. Besides, for λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq107_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif, u [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq173_HTML.gif, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equaw_HTML.gif
      Moreover, if we take δ 0 ( 0 , d a max + c d ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq174_HTML.gif, then we obtain
      f ( t , u , v ) = u 1 4 + [ v + 2 ] 1 3 + b ( t ) + d d = d a max + c d ( a max + c d ) δ 0 [ u 1 + u a ( t ) + c d ] = δ 0 g ( t , u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_Equax_HTML.gif

      Hence all the conditions of Theorem 3.1 are satisfied. An application of Theorem 3.1 implies that problem (4.1) has a unique positive solution in P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq55_HTML.gif, where h ( t ) = t ν 1 = t 5.3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq175_HTML.gif, t [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-85/MediaObjects/13661_2012_Article_339_IEq113_HTML.gif.

      Declarations

      Acknowledgements

      The authors are grateful to the anonymous referee for his/her valuable suggestions. The first author was supported financially by the Youth Science Foundations of China (11201272) and Shanxi Province (2010021002-1).

      Authors’ Affiliations

      (1)
      School of Mathematical Sciences, Shanxi University

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      Copyright

      © Zhai and Hao; licensee Springer. 2013

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