Open Access

On positive solutions for a class of singular nonlinear fractional differential equations

Boundary Value Problems20122012:73

DOI: 10.1186/1687-2770-2012-73

Received: 29 March 2012

Accepted: 18 May 2012

Published: 12 July 2012

Abstract

We study the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem

D 0 + α u ( t ) = f ( t , u ( t ) , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equa_HTML.gif

where 3 < α 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq1_HTML.gif is a real number, D 0 + α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq2_HTML.gif is the Riemann-Liouville fractional derivative and f : ( 0 , 1 ] × [ 0 , + ) × [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq3_HTML.gif is continuous, lim t 0 + f ( t , , ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq4_HTML.gif (f is singular at t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq5_HTML.gif). Our approach is based on a coupled fixed point theorem on ordered metric spaces. An example is given to illustrate our main result.

MSC:34A08, 34B16, 47H10.

Keywords

singular fractional differential equation positive solution coupled fixed point coupled lower and upper solution ordered metric space

1 Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, fluid flows, electrical networks, viscoelasticity, aerodynamics, and many other branches of science. For details, see [111].

In the last few decades, fractional-order models are found to be more adequate than integer-order models for some real world problems. Recently, there have been some papers dealing with the existence and multiplicity of solutions (or positive solutions) of nonlinear initial fractional differential equations by the use of techniques of nonlinear analysis (fixed point theorems, Leray-Schauder theory, etc.); see [2, 4, 5, 11].

Recently, there have been many exciting developments in the field of fixed point theory on partially ordered metric spaces. The first result in this direction was given by Turinici [12]. In [13], Ran and Reurings extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. The obtained result by Ran and Reurings was further extended and refined by many authors; see [1419].

Very recently, Shurong Sun et al. [20] discussed the existence and uniqueness of a positive solution to the singular nonlinear fractional differential equation boundary value problem
D 0 + α u ( t ) = f ( t , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equb_HTML.gif

where 3 < α 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq1_HTML.gif is a real number, D 0 + α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq2_HTML.gif is the Riemann-Liouville fractional derivative and f : ( 0 , 1 ] × [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq6_HTML.gif is continuous, lim t 0 + f ( t , ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq7_HTML.gif (f is singular at t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq5_HTML.gif), f ( t , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq8_HTML.gif is nondecreasing for all t ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq9_HTML.gif.

Motivated by the above mentioned work, in this paper we investigate the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem
D 0 + α u ( t ) = f ( t , u ( t ) , u ( t ) ) , 0 < t < 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ1_HTML.gif
(1)
u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ2_HTML.gif
(2)

where 3 < α 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq1_HTML.gif is a real number, D 0 + α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq2_HTML.gif is the Riemann-Liouville fractional derivative and f : ( 0 , 1 ] × [ 0 , + ) × [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq3_HTML.gif is continuous, lim t 0 + f ( t , , ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq10_HTML.gif (f is singular at t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq5_HTML.gif), for all t ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq9_HTML.gif f ( t , , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq11_HTML.gif is nondecreasing with respect to the first component, and it is decreasing with respect to the second component. Our approach is based on a recent coupled fixed point theorem on ordered metric spaces established by Harjani et al. [17]. We end the paper with an example that illustrates our main result.

2 Preliminaries

In this section, we recall some basic definitions and properties from fractional calculus theory. For more details about fractional calculus, we refer the readers to [1, 3, 10].

Definition 2.1 The Riemann-Liouville fractional derivative of order α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq12_HTML.gif of a continuous function φ : ( 0 , + ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq13_HTML.gif is given by
D 0 + α φ ( t ) = 1 Γ ( n α ) ( d d t ) ( n ) 0 t φ ( s ) ( t s ) α n + 1 d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equc_HTML.gif

where n = [ α ] + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq14_HTML.gif, [ α ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq15_HTML.gif denotes the integer part of number α, provided that the right side is pointwise defined on ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq16_HTML.gif.

Definition 2.2 The Riemann-Liouville fractional integral of order α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq12_HTML.gif of a function φ : ( 0 , + ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq13_HTML.gif is given by
I 0 + α φ ( t ) = 1 Γ ( α ) 0 t ( t s ) α 1 φ ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equd_HTML.gif

provided that the right side is pointwise defined on ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq16_HTML.gif.

From the definition of the Riemann-Liouville derivative, we can obtain the following statement.

Lemma 2.1 (see [10])

Let α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq12_HTML.gif. If we assume u C ( 0 , 1 ) L ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq17_HTML.gif, then the fractional differential equation
D 0 + α u ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Eque_HTML.gif

has u ( 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%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq18_HTML.gif, c i R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq19_HTML.gif, i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq20_HTML.gifas unique solutions, where N is the smallest integer greater than or equal to α.

Lemma 2.2 (see [10])

Assume that u C ( 0 , 1 ) L ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq17_HTML.gifwith a fractional derivative of order α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq12_HTML.gifthat belongs to C ( 0 , 1 ) L ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq21_HTML.gif. Then
I 0 + α D 0 + α u ( t ) = u ( 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%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equf_HTML.gif

for some c i R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq19_HTML.gif, i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq20_HTML.gif, where N is the smallest integer greater than or equal to α.

The Green function of fractional differential equation boundary value problem is given by

Lemma 2.3 (see [10])

Let h C ( [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq22_HTML.gifand 3 < α 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq1_HTML.gif. The unique solution to
D 0 + α u ( t ) = h ( t ) , 0 < t < 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ3_HTML.gif
(3)
u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ4_HTML.gif
(4)
is
u ( t ) = 0 1 G ( t , s ) h ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equg_HTML.gif
where
G ( t , s ) = { ( t s ) α 1 + ( 1 s ) α 2 t α 2 [ ( s t ) + ( α 2 ) ( 1 t ) s ] Γ ( α ) , 0 s t 1 , t α 2 ( 1 s ) α 2 [ ( s t ) + ( α 2 ) ( 1 t ) s ] Γ ( α ) , 0 t s 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equh_HTML.gif

Here G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq23_HTML.gifis called the Green function of boundary value problem (3)-(4).

The following properties of the Green function will be used later.

Lemma 2.4 (see [10])

The following properties hold:
  1. (i)

    G ( t , s ) = G ( 1 s , 1 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq24_HTML.giffor t , s ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq25_HTML.gif;

     
  2. (ii)

    ( α 2 ) t α 2 ( 1 t ) 2 s 2 ( 1 s ) α 2 Γ ( α ) G ( t , s ) M 0 s 2 ( 1 s ) α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq26_HTML.gif, t , s ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq25_HTML.gif;

     
  3. (iii)

    G ( t , s ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq27_HTML.gif, for t , s ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq25_HTML.gif;

     
  4. (iv)

    ( α 2 ) s 2 ( 1 s ) α 2 t α 2 ( 1 t ) 2 Γ ( α ) G ( t , s ) M 0 t α 2 ( 1 t ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq28_HTML.gif, t , s ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq25_HTML.gif, where M 0 = max { α 1 , ( α 2 ) 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq29_HTML.gif.

     

Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq30_HTML.gif be a partially ordered set endowed with a metric d such that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq31_HTML.gif is complete metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq32_HTML.gif be a given mapping.

Definition 2.3 We say that ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq30_HTML.gif is directed if for every ( x , y ) X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq33_HTML.gif there exists z X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq34_HTML.gif such that x z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq35_HTML.gif and y z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq36_HTML.gif.

Definition 2.4 We say that ( X , , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq37_HTML.gif is regular if the following conditions hold: ( c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq38_HTML.gif) = if { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq39_HTML.gif is a nondecreasing sequence in X such that x n x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq40_HTML.gif, then x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq41_HTML.gif for all n;; ( c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq42_HTML.gif) = if { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq43_HTML.gif is a decreasing sequence in X such that y n y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq44_HTML.gif, then y n y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq45_HTML.gif for all n..

Example 2.1 Let X = C ( [ 0 , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq46_HTML.gif, T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq47_HTML.gif, be the set of real continuous functions on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq48_HTML.gif. We endow X with the standard metric d given by
d ( u , v ) = max 0 t T | u ( t ) v ( t ) | , u , v X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equi_HTML.gif
We define the partial order on X by
u , v X , u v u ( t ) v ( t ) for all t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equj_HTML.gif

Let u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq49_HTML.gif. For w = max { u , v } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq50_HTML.gif, that is, w ( t ) = max { u ( t ) , v ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq51_HTML.gif for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq52_HTML.gif, we have u w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq53_HTML.gif and v w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq54_HTML.gif. This implies that ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq30_HTML.gif is directed. Now, let { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq39_HTML.gif be a nondecreasing sequence in X such that d ( x n , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq55_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq56_HTML.gif, for some x X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq57_HTML.gif. Then, for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq52_HTML.gif, { x n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq58_HTML.gif is a nondecreasing sequence of real numbers converging to x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq59_HTML.gif. Thus we have x n ( t ) x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq60_HTML.gif for all n, that is, x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq41_HTML.gif for all n. Similarly, if { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq43_HTML.gif is a decreasing sequence in X such that d ( y n , y ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq61_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq56_HTML.gif, for some y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq62_HTML.gif, we get that y n y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq45_HTML.gif for all n. Then we proved that ( X , , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq63_HTML.gif is regular.

Definition 2.5 (see [15])

An element ( x , y ) X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq33_HTML.gif is called a coupled fixed point of F if F ( x , y ) = x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq64_HTML.gif and F ( y , x ) = y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq65_HTML.gif.

Definition 2.6 (see [15])

We say that F has the mixed monotone property if for all ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq66_HTML.gif, ( u , v ) X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq67_HTML.gif, we have
x u , y v F ( x , y ) F ( u , v ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equk_HTML.gif

Denote by Φ the set of functions φ : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq68_HTML.gif satisfying: ( φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq69_HTML.gif) = φ is continuous;; ( φ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq70_HTML.gif) = φ is nondecreasing;; ( φ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq71_HTML.gif) = φ 1 ( { 0 } ) = { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq72_HTML.gif..

The following two lemmas are fundamental in the proofs of our main results.

Lemma 2.5 (see [17])

Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq30_HTML.gifbe a partially ordered set and suppose that there exists a metric d on X such that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq31_HTML.gifis a complete metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq32_HTML.gifbe a mapping having the mixed monotone property on X such that
ψ ( d ( F ( x , y ) , F ( u , v ) ) ) ψ ( max { d ( x , u ) , d ( y , v ) } ) φ ( max { d ( x , u ) , d ( y , v ) } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ5_HTML.gif
(5)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq73_HTML.gifwith x u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq74_HTML.gifand y v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq75_HTML.gif, where ψ , φ Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq76_HTML.gif. Suppose also that ( X , , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq37_HTML.gifis regular and there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq77_HTML.gifsuch that
x 0 F ( x 0 , y 0 ) , y 0 F ( y 0 , x 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equl_HTML.gif

Then F has a coupled fixed point ( x , y ) X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq78_HTML.gif. Moreover, if { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq79_HTML.gifand { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq43_HTML.gifare the sequences in X defined by

x n + 1 = F ( x n , y n ) , y n + 1 = F ( y n , x n ) , n = 0 , 1 , , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equm_HTML.gif
then
lim n d ( x n , x ) = lim n d ( y n , y ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ6_HTML.gif
(6)

Lemma 2.6 (see [17])

Adding to the hypotheses of Lemma 2.5 the condition ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq30_HTML.gifis regular, we obtain the uniqueness of the coupled fixed point. Moreover, we have the equality x = y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq80_HTML.gif.

3 Main result

Let Banach space E = C ( [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq81_HTML.gif be endowed with the norm u = max 0 t 1 | u ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq82_HTML.gif. We define the partial order on E by
u , v E , u v u ( t ) v ( t ) for all t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ7_HTML.gif
(7)
In Example 2.1, we proved that ( E , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq83_HTML.gif with the classic metric given by
d ( u , v ) = max 0 t 1 | u ( t ) v ( t ) | , u , v E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equn_HTML.gif

satisfies the following properties: ( E , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq83_HTML.gif is directed and ( E , , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq84_HTML.gif is regular.

Define the closed cone P E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq85_HTML.gif by
P = { u E : u 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equo_HTML.gif

where 0 denotes the zero function.

Definition 3.1 (see [15])

We say that ( u , u + ) C ( [ 0 , 1 ] ) × C ( [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq86_HTML.gif is a coupled lower and upper solution to (1)-(2) if
u ( t ) 0 1 G ( t , s ) f ( s , u ( s ) , u + ( s ) ) d s , for all 0 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equp_HTML.gif
and
u + ( t ) 0 1 G ( t , s ) f ( s , u + ( s ) , u ( s ) ) d s , for all 0 t 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equq_HTML.gif

Our main result is the following.

Theorem 3.1 Let 0 < σ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq87_HTML.gif, 3 < α 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq1_HTML.gif, f : ( 0 , 1 ) × [ 0 , + ) × [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq88_HTML.gifis continuous, lim t 0 + f ( t , , ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq89_HTML.gifand t t σ f ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq90_HTML.gifis continuous on [ 0 , 1 ] × [ 0 , + ) × [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq91_HTML.gif. Assume that there exists 0 < λ ( 1 σ ) Γ ( α σ + 1 ) / ( 2 Γ ( 3 σ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq92_HTML.gifsuch that for x , y , z , w [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq93_HTML.gifwith x z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq94_HTML.gif, y w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq95_HTML.gifand t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq96_HTML.gif,
0 t σ ( f ( t , x , y ) f ( t , z , w ) ) λ η ( max { ( x z ) , ( w y ) } ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ8_HTML.gif
(8)
where η : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq97_HTML.gif, β : u u η ( u ) Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq98_HTML.gif. Suppose also that (1)-(2) has a coupled lower and upper solution ( u , u + ) P × P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq99_HTML.gif. Then the boundary value problem (1)-(2) has a unique positive solution u C ( [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq100_HTML.gif. The sequences { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq101_HTML.gifand { v n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq102_HTML.gifdefined by
u 0 = u , u n + 1 = 0 1 G ( t , s ) f ( s , u n ( s ) , v n ( s ) ) d s , n = 0 , 1 , , v 0 = u + , v n + 1 = 0 1 G ( t , s ) f ( s , v n ( s ) , u n ( s ) ) d s , n = 0 , 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equr_HTML.gif

converge uniformly to u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq103_HTML.gif.

Proof Suppose that u is a solution of boundary value problem (1)-(2). Then
u ( t ) = 0 1 G ( t , s ) f ( s , u ( s ) , u ( s ) ) d s , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ9_HTML.gif
(9)
We define the operator F : P × P E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq104_HTML.gif by
F ( u , v ) ( t ) = 0 1 G ( t , s ) f ( s , u ( s ) , v ( s ) ) d s , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equs_HTML.gif
• Step 1. We shall prove that
F ( P × P ) P . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ10_HTML.gif
(10)
Let u , v P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq105_HTML.gif. Let us prove that F ( u , v ) C ( [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq106_HTML.gif. We have
F ( u , v ) ( t ) = 0 1 G ( t , s ) s σ s σ f ( s , u ( s ) , v ( s ) ) d s , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equt_HTML.gif
By the continuity of s σ f ( s , u ( s ) , v ( s ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq107_HTML.gif in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq108_HTML.gif, it is easy to check that F ( u , v ) ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq109_HTML.gif. Now, let t 0 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq110_HTML.gif. We have to prove that
| F ( u , v ) ( t ) F ( u , v ) ( t 0 ) | 0 as t t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equu_HTML.gif
We distinguish three cases:
t 0 = 0 , t 0 ( 0 , 1 ] and t ( t 0 , 1 ] , t 0 ( 0 , 1 ] and t [ 0 , t 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equv_HTML.gif
Case 1. t 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq111_HTML.gif. Since s s σ f ( s , u ( s ) , v ( s ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq112_HTML.gif is continuous on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq108_HTML.gif, there exists a constant M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq113_HTML.gif such that | s σ f ( s , u ( s ) , v ( s ) ) | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq114_HTML.gif for all s [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq115_HTML.gif. We have
| F ( u , v ) ( t ) F ( u , v ) ( t 0 ) | = | 0 1 G ( t , s ) s σ s σ f ( s , u ( s ) , v ( s ) ) d s | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equw_HTML.gif
Using Lemma 2.3, we have
| F ( u , v ) ( t ) F ( u , v ) ( t 0 ) | = | 0 1 t α 2 ( 1 s ) α 2 [ ( s t ) + ( α 2 ) ( 1 t ) s ] Γ ( α ) s σ s σ f ( s , u ( s ) , v ( s ) ) d s + 0 t ( t s ) α 1 Γ ( α ) s σ s σ f ( s , u ( s ) , v ( s ) ) d s | | 0 1 t α 2 ( 1 s ) α 2 [ ( s t ) + ( α 2 ) ( 1 t ) s ] Γ ( α ) s σ s σ f ( s , u ( s ) , v ( s ) ) d s | + | 0 t ( t s ) α 1 Γ ( α ) s σ s σ f ( s , u ( s ) , v ( s ) ) d s | M 0 1 ( α 1 ) t α 2 ( 1 s ) α 2 Γ ( α ) s 1 σ d s + M 0 1 t α 1 ( 1 s ) α 2 Γ ( α ) s σ d s + M 0 1 ( α 2 ) t α 1 ( 1 s ) α 2 Γ ( α ) s 1 σ d s + M 0 t ( t s ) α 1 Γ ( α ) s σ d s = M t α 2 Γ ( α 1 ) B ( 2 σ , α 1 ) + M t α 1 Γ ( α ) B ( 1 σ , α 1 ) + M ( α 2 ) t α 1 Γ ( α ) B ( 2 σ , α 1 ) + M t α σ Γ ( α ) B ( 1 σ , α ) 0 as t 0 + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equx_HTML.gif

where B ( , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq116_HTML.gif denotes the beta function.

Case 2. t 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq117_HTML.gif and t ( t 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq118_HTML.gif. In this case,
| F ( u , v ) ( t ) F ( u , v ) ( t 0 ) | = | 0 1 ( α 1 ) ( t α 2 t 0 α 2 ) ( 1 s ) α 2 s 1 σ Γ ( α ) s σ f ( s , u ( s ) , v ( s ) ) d s + 0 1 ( t 0 α 1 t α 1 ) ( 1 s ) α 2 s σ Γ ( α ) s σ f ( s , u ( s ) , v ( s ) ) d s + 0 1 ( α 2 ) ( t 0 α 1 t α 1 ) ( 1 s ) α 2 s 1 σ Γ ( α ) s σ f ( s , u ( s ) , v ( s ) ) d s + 0 t 0 [ ( t s ) α 1 ( t 0 s ) α 1 ] s σ Γ ( α ) s σ f ( s , u ( s ) , v ( s ) ) d s + t 0 t ( t s ) α 1 s σ Γ ( α ) s σ f ( s , u ( s ) , v ( s ) ) d s | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equy_HTML.gif
Now, we have
| F ( u , v ) ( t ) F ( u , v ) ( t 0 ) | M ( α 1 ) ( t α 2 t 0 α 2 ) Γ ( α ) 0 1 ( 1 s ) α 2 s 1 σ d s + M ( t α 1 t 0 α 1 ) Γ ( α ) 0 1 ( 1 s ) α 2 s σ d s + M ( α 2 ) ( t α 1 t 0 α 1 ) Γ ( α ) 0 1 ( 1 s ) α 2 s 1 σ d s + M Γ ( α ) 0 t 0 [ ( t s ) α 1 ( t 0 s ) α 1 ] s σ d s + M Γ ( α ) t 0 t ( t s ) α 1 s σ d s = M ( t α 2 t 0 α 2 ) Γ ( α 1 ) B ( 2 σ , α 1 ) + M ( t α 1 t 0 α 1 ) Γ ( α ) [ B ( 1 σ , α 1 ) + ( α 2 ) B ( 2 σ , α 1 ) ] + M t α σ Γ ( α ) B ( 1 σ , α ) M t 0 α σ Γ ( α ) B ( 1 σ , α ) 0 as t t 0 + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equz_HTML.gif

Case 3. t 0 ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq119_HTML.gif and t [ 0 , t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq120_HTML.gif. The proof is similar to that of Case 2, so we omit it.

Thus we proved that F ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq121_HTML.gif is continuous on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq108_HTML.gif for all u , v C ( [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq122_HTML.gif. Moreover, taking into account Lemma 2.4 and as t σ f ( t , x , y ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq123_HTML.gif for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq96_HTML.gif, x , y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq124_HTML.gif, our claim (10) is proved. Now the mapping
F : P × P P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equaa_HTML.gif

is well defined.

  • Step 2. We shall prove that F has the mixed monotone property with respect to the partial order given by (7).

Let ( x , y ) , ( u , v ) P × P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq125_HTML.gif such that x u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq126_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq127_HTML.gif. From (8), we have
s σ f ( s , x ( s ) , y ( s ) ) s σ f ( s , u ( s ) , v ( s ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equab_HTML.gif
for all s [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq115_HTML.gif. This implies that
0 1 G ( t , s ) s σ s σ f ( s , x ( s ) , y ( s ) ) d s 0 1 G ( t , s ) s σ s σ f ( s , u ( s ) , v ( s ) ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equac_HTML.gif
that is,
0 1 G ( t , s ) f ( s , x ( s ) , y ( s ) ) d s 0 1 G ( t , s ) f ( s , u ( s ) , v ( s ) ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equad_HTML.gif
which gives us that
F ( x , y ) ( t ) F ( u , v ) ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equae_HTML.gif
for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq96_HTML.gif, and then we have
F ( x , y ) F ( u , v ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equaf_HTML.gif

Then F has the mixed monotone property.

  • Step 3. We shall prove that F satisfies the contractive condition (5) for some ψ , φ Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq128_HTML.gif.

Let ( x , y ) , ( u , v ) P × P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq129_HTML.gif such that x u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq130_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq131_HTML.gif. For all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq96_HTML.gif, using (8), we have
| F ( x , y ) ( t ) F ( u , v ) ( t ) | = 0 1 G ( t , s ) [ f ( s , x ( s ) , y ( s ) ) f ( s , u ( s ) , v ( s ) ) ] d s = 0 1 G ( t , s ) s σ s σ [ f ( s , x ( s ) , y ( s ) ) f ( s , u ( s ) , v ( s ) ) ] d s 0 1 G ( t , s ) s σ λ η ( max { x ( s ) u ( s ) , v ( s ) y ( s ) } ) d s λ η ( max { d ( x , u ) , d ( y , v ) } ) 0 1 G ( t , s ) s σ d s λ η ( max { d ( x , u ) , d ( y , v ) } ) max z [ 0 , 1 ] 0 1 G ( z , s ) s σ d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equag_HTML.gif
Thus we have
d ( F ( x , y ) , F ( u , v ) ) λ η ( max { d ( x , u ) , d ( y , v ) } ) max z [ 0 , 1 ] 0 1 G ( z , s ) s σ d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ11_HTML.gif
(11)
Now, let z [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq132_HTML.gif. We have
0 1 G ( z , s ) s σ d s = 0 1 z α 2 ( 1 s ) α 2 [ ( s z ) + ( α 2 ) ( 1 z ) s ] Γ ( α ) s σ d s + 0 z ( z s ) α 1 Γ ( α ) s σ d s 0 1 ( α 1 ) z α 2 ( 1 s ) α 2 Γ ( α ) s 1 σ d s + 0 1 z α 1 ( 1 s ) α 2 Γ ( α ) s σ d s + 0 1 ( α 2 ) z α 1 ( 1 s ) α 2 Γ ( α ) s 1 σ d s + 0 z ( z s ) α 1 Γ ( α ) s σ d s z α 2 Γ ( α 1 ) B ( 2 σ , α 1 ) + z α 1 Γ ( α ) B ( 1 σ , α 1 ) + ( α 2 ) z α 1 Γ ( α ) B ( 2 σ , α 1 ) + z α σ Γ ( α ) B ( 1 σ , α ) 1 Γ ( α 1 ) B ( 2 σ , α 1 ) + 1 Γ ( α ) B ( 1 σ , α 1 ) + ( α 2 ) Γ ( α ) B ( 2 σ , α 1 ) + 1 Γ ( α ) B ( 1 σ , α ) = 2 Γ ( 3 σ ) ( 1 σ ) Γ ( α σ + 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equah_HTML.gif
which implies that
max z [ 0 , 1 ] 0 1 G ( z , s ) s σ d s 2 Γ ( 3 σ ) ( 1 σ ) Γ ( α σ + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equai_HTML.gif
Now, using the above inequality, (11) and the fact that λ < ( 1 σ ) Γ ( α σ + 1 ) 2 Γ ( 3 σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq133_HTML.gif, we get
d ( F ( x , y ) , F ( u , v ) ) η ( max { d ( x , u ) , d ( y , v ) } ) = max { d ( x , u ) , d ( y , v ) } β ( max { d ( x , u ) , d ( y , v ) } ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equaj_HTML.gif
Thus we proved that for all ( x , y ) , ( u , v ) P × P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq129_HTML.gif such that x u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq130_HTML.gif and y v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq131_HTML.gif, we have
ψ ( d ( F ( x , y ) , F ( u , v ) ) ) ψ ( max { d ( x , u ) , d ( y , v ) } ) φ ( max { d ( x , u ) , d ( y , v ) } ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equak_HTML.gif

where ψ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq134_HTML.gif and φ ( t ) = β ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq135_HTML.gif.

  • Step 4. Existence of ( x 0 , y 0 ) P × P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq136_HTML.gif such that x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq137_HTML.gif and y 0 F ( y 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq138_HTML.gif.

We take ( x 0 , y 0 ) = ( u , u + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq139_HTML.gif, the coupled lower and upper solution to (1)-(2).

Now, from Lemmas 2.5 and 2.6, there exists a unique u P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq140_HTML.gif such that u = F ( u , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq141_HTML.gif, that is u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq103_HTML.gif is the unique positive solution to (1)-(2). The convergence of the sequences { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq101_HTML.gif and { v n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq142_HTML.gif to u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq103_HTML.gif follows immediately from (6). □

Now, we end this paper with the following example.

Example 3.1 Consider the boundary value problem
D 0 + 7 / 2 u ( t ) = ( t 1 / 2 ) 2 2 t ( u ( t ) + 1 u ( t ) + 1 ) , 0 < t < 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ12_HTML.gif
(12)
u ( 0 ) = u ( 1 ) = u ( 0 ) = u ( 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equ13_HTML.gif
(13)
In this case, f ( t , x , y ) = ( t 1 / 2 ) 2 2 t ( x + 1 y + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq143_HTML.gif, for ( t , x , y ) ( 0 , 1 ] × [ 0 , + ) × [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq144_HTML.gif. Note that f is continuous on ( 0 , 1 ] × [ 0 , + ) × [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq145_HTML.gif and lim t 0 + f ( t , , ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq146_HTML.gif. Let σ = λ = 1 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq147_HTML.gif and η ( t ) = ( 1 / 2 ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq148_HTML.gif. For all x , y , z , w [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq93_HTML.gif with x z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq94_HTML.gif, y w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq95_HTML.gif and t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq96_HTML.gif, we have
0 t 1 / 2 ( f ( t , x , y ) f ( t , z , w ) ) = ( t 1 / 2 ) 2 2 [ ( x z ) + ( w y ) ] ( t 1 / 2 ) 2 max { x z , w y } 1 4 max { x z , w y } = λ η ( max { x z , w y } ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equal_HTML.gif
On the other hand,
( 1 σ ) Γ ( α σ + 1 ) 2 Γ ( 3 σ ) = 3 2 Γ ( 5 / 2 ) = 2 π > λ = 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_Equam_HTML.gif

Consider now, the pair ( u , u + ) P × P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq99_HTML.gif defined by u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq149_HTML.gif and u + 7 / 17 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq150_HTML.gif. Using Lemma 2.4(iv), one can show easily that ( u , u + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq151_HTML.gif is a coupled lower and upper solution to (12)-(13).

Finally, applying Theorem 3.1, we deduce that (12)-(13) has one and only one positive solution u C ( [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-73/MediaObjects/13661_2012_Article_170_IEq100_HTML.gif.

Declarations

Acknowledgement

This work was supported by the Research Center, College of Science, King Saud University.

Authors’ Affiliations

(1)
Department of Mathematics, King Saud University

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