Open Access

Existence of solutions of a system of 3D axisymmetric inviscid stagnation flows

Boundary Value Problems20122012:153

DOI: 10.1186/1687-2770-2012-153

Received: 7 February 2012

Accepted: 6 December 2012

Published: 28 December 2012

Abstract

A system of two integral equations is presented to describe the system of 3D axisymmetric inviscid stagnation flows related to Navier-Stokes equations and existence of its solutions is studied. Utilizing it, we construct analytically the similarity solutions of the 3D system. A nonexistence result is obtained. Previous study was only supported by numerical results.

MSC:34B18.

Keywords

Navier-Stokes equations 3D flows similarity solutions integral systems existence results

1 Introduction

The following system of two differential equations arising in the boundary layer problems in fluid mechanics
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ2_HTML.gif
(1.2)
with boundary conditions
f ( 0 ) = 0 , f ( 0 ) = 0 , f ( ) = 1 , g ( 0 ) = 0 , g ( 0 ) = 0 , g ( ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ3_HTML.gif
(1.3)

has been used to describe the system of 3D axisymmetric inviscid stagnation flow [1, 2], which consists of three partial differential equations [2, 3], where λ is a parameter related to the external flow components.

A solution of (1.1)-(1.3) is called a similarity solution and can be used to express the solutions of the 3D system. Regarding the study of (1.1)-(1.3), Howarth [3] presented a numerical study for the case 0 < λ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq1_HTML.gif which can be applied to the stagnation region of an ellipsoid. Davey [2] investigated numerically the stagnation region near a saddle point ( 1 < λ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq2_HTML.gif). The two-dimensional cases, λ = g = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq3_HTML.gif or λ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq4_HTML.gif and g = f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq5_HTML.gif, and the special cases of the Falkner-Skan equation were solved by Hiemenz [4] and by Homann [5], respectively. Regarding the Falkner-Skan problems, further analytical study can be found in [610]. Also, one may refer to recent review of similarity solutions of the Navier-Stokes equations [11].

However, up to now, there has been very little analytical study on the existence of solutions of (1.1)-(1.3).

The main aim of this paper is to study the existence of solutions of (1.1)-(1.3) analytically for the case of | λ | < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq6_HTML.gif. The method is to present a system of two integral equations and study the existence of its solutions and then use it to construct the solutions of (1.1)-(1.3). Also, a nonexistence result is obtained.

2 A system of two integral equations related to (1.1)-(1.3)

In this section, we present a system of two integral equations to describe a system of (1.1)-(1.3) under suitable conditions, which will be utilized in Section 4.

Let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equa_HTML.gif
and
Γ = { ( f , g ) C 3 [ 0 , ) × C 3 [ 0 , ) : f ( η ) 0 , g ( η ) > 0 , η [ 0 , ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equb_HTML.gif

Lemma 2.1 If ( f , g ) Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq7_HTML.gifis a solution of (1.1)-(1.3), then g ( ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq8_HTML.gif.

Proof Since g ( + ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq9_HTML.gif, we have
lim inf η g ( η ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ4_HTML.gif
(2.1)

Notice that ( f , g ) Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq7_HTML.gif, f ( η ) = 0 η f ( s ) d s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq10_HTML.gif, g ( η ) = 0 η g ( s ) d s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq11_HTML.gif, g ( η ) = 0 η g ( s ) d s > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq12_HTML.gif and 1 > g ( η ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq13_HTML.gif for η ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq14_HTML.gif.

If λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq15_HTML.gif, we know g ( η ) = ( f ( η ) + λ g ( η ) ) g ( η ) λ ( 1 g 2 ( η ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq16_HTML.gif and then g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq17_HTML.gif is decreasing on [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq18_HTML.gif, which implies that lim η g ( η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq19_HTML.gif exists. Hence, g ( ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq20_HTML.gif by (2.1).

If λ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq21_HTML.gif, we have g ( 0 ) = λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq22_HTML.gif by (1.2). By (2.1), there exists η 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq23_HTML.gif such that g ( η 0 ) < g ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq24_HTML.gif and then there exists η https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq25_HTML.gif such that g ( η ) = max { g ( η ) : η [ 0 , η 0 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq26_HTML.gif. Obviously, η ( 0 , η 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq27_HTML.gif by g ( 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq28_HTML.gif. We prove that g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq17_HTML.gif is decreasing on ( η , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq29_HTML.gif.

In fact, if there exist η 1 , η 2 ( η , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq30_HTML.gif with η 1 < η 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq31_HTML.gif such that g ( η 1 ) < g ( η 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq32_HTML.gif. Let η [ η , η 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq33_HTML.gif such that g ( η ) = min { g ( η ) : η [ η , η 2 ] } > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq34_HTML.gif, then g ( η ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq35_HTML.gif and g ( 4 ) ( η ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq36_HTML.gif.

Differentiating (1.2) with η, we have
g ( 4 ) ( η ) = ( λ g ( η ) f ( η ) ) g ( η ) ( f ( η ) + λ g ( η ) ) g ( η ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equc_HTML.gif
then
g ( 4 ) ( η ) = ( λ g ( η ) f ( η ) ) g ( η ) < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equd_HTML.gif

a contradiction. Hence, g ( η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq37_HTML.gif is decreasing on ( η , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq38_HTML.gif and then g ( ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq8_HTML.gif.

This completes the proof. □

Theorem 2.1 If ( f , g ) Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq7_HTML.gifis a solution of (1.1)-(1.2), then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ5_HTML.gif
(2.2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ6_HTML.gif
(2.3)
has a solution ( x , y ) Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq39_HTML.gif, where G 0 , 1 ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq40_HTML.gifdenotes the Green function for u ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq41_HTML.gifwith u ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq42_HTML.gifand u ( b ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq43_HTML.gifdefined by
G 0 , b ( t , s ) = { t ( b s ) / b , 0 t s b , s ( b t ) / b , 0 s t b . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ7_HTML.gif
(2.4)

Proof Assume that ( f , g ) Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq7_HTML.gif. Let η : = η ( t ) = ( g ) 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq44_HTML.gif for t [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq45_HTML.gif be the inverse function to t = g ( η ) : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq46_HTML.gif. It follows that g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq47_HTML.gif is strictly increasing on [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq18_HTML.gif and η ( t ) = ( g ) 1 ( t ) : [ 0 , 1 ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq48_HTML.gif with ( g ) 1 ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq49_HTML.gif, lim t 1 ( g ) 1 ( t ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq50_HTML.gif. Let x ( t ) = g ( η ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq51_HTML.gif for t [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq45_HTML.gif, by Lemma 2.1, x ( 1 ) = lim η g ( η ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq52_HTML.gif. This implies that x ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq53_HTML.gif for t [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq45_HTML.gif and x is continuous on [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq54_HTML.gif. By Lemma 2.1, we see that x is continuous from the left at 1. Hence, we have x ( t ) C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq55_HTML.gif and x ( 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq56_HTML.gif, i.e., x ( t ) Q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq57_HTML.gif.

Using the chain rule to x ( t ) = g ( η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq58_HTML.gif, we obtain g ( η ) d η d t = x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq59_HTML.gif and by the inverse function theorem, we have
d η d t = 1 g ( η ) = 1 x ( t ) for  t [ 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Eque_HTML.gif
This, together with g ( η ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq60_HTML.gif, implies
g ( η ) = x ( t ) x ( t ) , η = 0 t 1 x ( s ) d s and g ( η ) d η d t = t x ( t ) for  t [ 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equf_HTML.gif
Integrating the last equality from 0 to t implies
g ( η ( t ) ) = 0 t s x ( s ) d s for  t [ 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equg_HTML.gif
Let
y ( t ) = f ( η ) = f ( 0 t 1 x ( s ) d s ) for  t [ 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equh_HTML.gif

Then y ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq61_HTML.gif. By f ( ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq62_HTML.gif, we know that y is continuous from the left at 1 and then y ( 1 ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq63_HTML.gif.

Notice that f ( η ) d η d t = y ( t ) x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq64_HTML.gif, t [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq45_HTML.gif, we have f ( η ) = 0 t y ( s ) x ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq65_HTML.gif.

Differentiating y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq66_HTML.gif with t, we have
y ( t ) = f ( η ) d η d t = f ( η ) x ( t ) for  t [ 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equi_HTML.gif

From this, we have f ( η ) = y ( t ) x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq67_HTML.gif for η [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq68_HTML.gif and y Q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq69_HTML.gif.

Differentiating f ( η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq70_HTML.gif with t and utilizing d η d t = 1 x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq71_HTML.gif, we have
f ( η ) x ( t ) = y ( t ) x ( t ) + y ( t ) x ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equj_HTML.gif
Hence,
f ( η ) = y ( t ) x 2 ( t ) + y ( t ) x ( t ) x ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equk_HTML.gif
Substituting g, g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq47_HTML.gif, g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq17_HTML.gif, g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq72_HTML.gif and f into (1.2) implies
x ( t ) = 0 t y ( s ) + λ s x ( s ) d s + λ ( t 2 1 ) x ( t ) , t [ 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ8_HTML.gif
(2.5)
Integrating (2.5) from t to 1, we have
x ( 1 ) x ( t ) = t 1 0 σ y ( s ) + λ s x ( s ) d s d σ + t 1 λ ( s 2 1 ) x ( s ) d s = t 1 λ ( s 2 1 ) x ( s ) d s 0 t ( t 1 y ( s ) + λ s x ( s ) d σ ) d s t 1 ( s 1 y ( s ) + λ s x ( s ) d σ ) d s = t 1 λ ( s 2 1 ) x ( s ) d s 0 t y ( s ) + λ s x ( s ) ( 1 t ) d s t 1 ( y ( s ) + λ s ) ( 1 s ) x ( s ) d s = t 1 λ ( s 2 1 ) ( λ s + y ( s ) ) ( 1 s ) x ( s ) d s ( 1 t ) 0 t λ s + y ( s ) x ( s ) d s = t 1 ( 2 λ s + λ + y ( s ) ) ( s 1 ) x ( s ) d s ( 1 t ) 0 t λ s + y ( s ) x ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equl_HTML.gif
By x ( 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq56_HTML.gif, then
x ( t ) = t 1 ( 2 λ s + λ + y ( s ) ) ( 1 s ) x ( s ) d s + ( 1 t ) 0 t λ s + y ( s ) x ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equm_HTML.gif
Substituting f, f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq73_HTML.gif, f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq74_HTML.gif, f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq75_HTML.gif and g into (1.1) implies
y ( t ) x 2 ( t ) + y ( t ) x ( t ) x ( t ) + y ( t ) x ( t ) 0 t λ s + y ( s ) x ( s ) d s + ( 1 y 2 ( t ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equn_HTML.gif
By 0 t λ s + y ( s ) x ( s ) d s = λ ( t 2 1 ) x ( t ) x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq76_HTML.gif, we have
y ( t ) + λ ( t 2 1 ) y ( t ) + ( 1 y 2 ( t ) ) x 2 ( t ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equo_HTML.gif
Therefore,
y ( t ) = 0 1 G 0 , 1 ( t , s ) λ ( s 2 1 ) y ( s ) + ( 1 y 2 ( s ) ) x 2 ( s ) d s + t , t [ 0 , 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equp_HTML.gif

where G 0 , 1 ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq40_HTML.gif is defined by (2.4). Hence, ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq77_HTML.gif is a solution of (2.2)-(2.3) in Q. □

3 Positive solutions of the system (2.2)-(2.3)

In this section, we will use the fixed point theorem to study the existence of positive solutions of the system (2.2)-(2.3).

Let
δ = δ ( λ ) = λ 2 λ + 1 , λ ( 1 3 , 0 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equq_HTML.gif
It is easy to verify
0 < δ < 1 if and only if 1 3 < λ < 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equr_HTML.gif
We define some functions
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equs_HTML.gif

By computation, ω ( 0 ) = 1 36 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq78_HTML.gif, ω ( 1 3 ) = 4 9 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq79_HTML.gif, there exists λ 0 ( 1 3 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq80_HTML.gif such that ω ( λ ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq81_HTML.gif for λ ( λ 0 , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq82_HTML.gif and ω ( λ 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq83_HTML.gif.

In order to study the existence of solutions of (2.2)-(2.3) in Q for λ ( λ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq84_HTML.gif, we denote the norm of the Banach space C [ 0 , 1 ] × C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq85_HTML.gif by
( x , y ) = x + y + y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equt_HTML.gif

where x = max { | x ( t ) | : t [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq86_HTML.gif.

Let ( x , y ) C [ 0 , 1 ] × C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq87_HTML.gif and n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq88_HTML.gif be a natural number, we define
φ x ( t ) = max { x ( t ) , c ( t ) } , φ n x ( t ) = max { x ( t ) , c ( t ) , 1 n } , θ y ( t ) = max { y ( t ) , t } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equu_HTML.gif
where c ( t ) = c λ ( 1 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq89_HTML.gif, t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif,
c λ = { 1 n , λ 0 , min { h ( λ ) , ω ( λ ) , ( 1 + λ ) ( 1 δ ) δ 2 4 σ ( λ ) } , λ 0 < λ < 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ9_HTML.gif
(3.1)
Notation
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equv_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equw_HTML.gif

where G 0 , 1 ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq40_HTML.gif is defined by (2.4).

Let ( x , y ) C [ 0 , 1 ] × C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq87_HTML.gif, we define an operator F as follows:
F n ( x , y ) ( t ) = ( A n ( x , y ) ( t ) , B n ( x , y ) ( t ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equx_HTML.gif
where
A n ( x , y ) ( t ) = S n ( x , y ) ( t ) + ( 1 t ) T n ( x , y ) + 1 n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equy_HTML.gif

It is easy to verify that φ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq91_HTML.gif, θ are continuous operators from C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq92_HTML.gif into C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq92_HTML.gif and φ n x ( t ) 1 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq93_HTML.gif, t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif, we know the following proposition holds:

Lemma 3.1 F n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq94_HTML.gifis a continuous and compact operator from C [ 0 , 1 ] × C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq85_HTML.gifto C [ 0 , 1 ] × C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq85_HTML.gif.

Lemma 3.2 Let ( λ , z , w ) ( 1 , 1 ) × C [ 0 , 1 ] × C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq95_HTML.gif and 0 < μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq96_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ10_HTML.gif
(3.2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ11_HTML.gif
(3.3)
Then the following assertions hold:
  1. (i)

    μ t y ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq97_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif.

     
  2. (ii)

    0 1 | y ( s ) | d s 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq98_HTML.gif and V 0 1 ( y ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq99_HTML.gif, where V 0 1 ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq100_HTML.gif is a total variation of y on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq101_HTML.gif.

     
  3. (iii)

    If μ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq102_HTML.gif, then y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq66_HTML.gif is increasing on ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq103_HTML.gif and then θ y ( t ) = y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq104_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif.

     
Proof We shall use the basic fact: let u ( t ) C [ a , b ] × C 2 ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq105_HTML.gif and u ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq106_HTML.gif ( ξ ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq107_HTML.gif) be local minimum (maximum), then u ( ξ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq108_HTML.gif (≤0).
  1. (i)
    If there exists t 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq109_HTML.gif such that y ( t 0 ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq110_HTML.gif, by y ( 0 ) = 0 < μ = y ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq111_HTML.gif, we know that there exists t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq112_HTML.gif such that y ( t ) = max { y ( t ) : t [ 0 , 1 ] } > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq113_HTML.gif. Differentiating (3.3) with t twice, we have
    y ( t ) = μ h ( y ) ( t ) ( φ n x ( t ) ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ12_HTML.gif
    (3.4)
     
By y ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq114_HTML.gif and (3.4), we have
y ( t ) = μ ( 1 y 2 ( t ) ) ( φ n x ( t ) ) 2 > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equz_HTML.gif

a contradiction. Hence, y ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq115_HTML.gif for t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq116_HTML.gif.

If there exists t 0 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq117_HTML.gif such that μ t 0 > y ( t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq118_HTML.gif, let τ ( t ) = μ t y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq119_HTML.gif, by τ ( 0 ) = 0 = τ ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq120_HTML.gif and τ ( t 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq121_HTML.gif, we may assume t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq112_HTML.gif such that τ ( t ) = max { τ ( t ) : t [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq122_HTML.gif. This implies τ ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq123_HTML.gif, i.e., y ( t ) = μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq124_HTML.gif, and τ ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq125_HTML.gif. By (3.4) and θ y ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq126_HTML.gif, we know
h ( y ) ( t ) = λ ( t 2 1 ) μ + ( 1 t 2 ) = ( 1 λ μ ) ( 1 t 2 ) > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equaa_HTML.gif
then
τ ( t ) = y ( t ) = μ h ( y ) ( t ) ( φ n x ( t ) ) 2 > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equab_HTML.gif
a contradiction. Hence, (i) holds.
  1. (ii)

    Let t ˜ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq127_HTML.gif such that y ( t ˜ ) = max { y ( t ) : t [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq128_HTML.gif and γ = sup { t ˜ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq129_HTML.gif. If γ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq130_HTML.gif, we prove that y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq66_HTML.gif is increasing on ( 0 , γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq131_HTML.gif and decreasing on ( γ , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq132_HTML.gif.

     
Since y ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq61_HTML.gif and y ( 1 ) = μ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq133_HTML.gif, then γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq134_HTML.gif. Let γ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq130_HTML.gif. If there exist t 1 , t 2 ( 0 , γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq135_HTML.gif with t 1 < t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq136_HTML.gif such that y ( t 1 ) > y ( t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq137_HTML.gif, let t ( t 1 , γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq138_HTML.gif such that y ( t ) = min { y ( t ) : t [ t 1 , γ ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq139_HTML.gif, then y ( t ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq140_HTML.gif by (i). From y ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq114_HTML.gif, t θ y ( t ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq141_HTML.gif and (3.4), we know
y ( t ) = μ ( 1 ( θ y ( t ) ) 2 ) ( φ n x ( t ) ) 2 < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equac_HTML.gif

a contradiction.

If there exist t 1 , t 2 ( γ , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq142_HTML.gif with t 1 < t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq136_HTML.gif such that y ( t 1 ) < y ( t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq143_HTML.gif, let t ( γ , t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq144_HTML.gif such that y ( t ) = min { y ( t ) : t [ γ , t 2 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq145_HTML.gif, then y ( t ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq140_HTML.gif by (i). Analogously, we know easily
y ( t ) = μ ( 1 ( θ y ( t ) ) 2 ) ( φ n x ( t ) ) 2 < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equad_HTML.gif
a contradiction. Hence,
0 1 | y ( s ) | d s = 0 γ y ( s ) d s γ 1 y ( s ) d s = 2 y ( γ ) μ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equae_HTML.gif
and V 0 1 ( y ) = 0 1 | y ( s ) | d s 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq146_HTML.gif, i.e., (ii) holds.
  1. (iii)

    Let μ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq102_HTML.gif. By (i) and y ( 1 ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq63_HTML.gif, we know γ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq147_HTML.gif and then y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq66_HTML.gif is increasing on ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq103_HTML.gif and then θ y ( t ) = y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq104_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif. Hence, (iii) holds. □

     

Lemma 3.3[12]

Let E be a Banach space, D be a bounded open set of E and θ D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq148_HTML.gif, F : D ¯ E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq149_HTML.gifis compact. If x μ F x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq150_HTML.giffor any 0 < μ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq151_HTML.gifand x D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq152_HTML.gif, then F has a fixed point in D ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq153_HTML.gif.

Lemma 3.4 Let λ ( 1 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq154_HTML.gif, then F has a fixed point ( x n , y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq155_HTML.gifin C [ 0 , 1 ] × C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq85_HTML.gif, i.e., there exists ( x n , y n ) C [ 0 , 1 ] × C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq156_HTML.gifsuch that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ13_HTML.gif
(3.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ14_HTML.gif
(3.6)

hold.

Proof Let
Ω = { ( x , y ) : ( x , y ) C [ 0 , 1 ] × C 1 [ 0 , 1 ] , ( x , y ) < R } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equaf_HTML.gif

where R = 16 n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq157_HTML.gif. We prove ( x , y ) μ F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq158_HTML.gif for 0 < μ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq151_HTML.gif and with ( x , y ) = R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq159_HTML.gif.

In fact, if there exist ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq77_HTML.gif and μ with ( x , y ) = R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq159_HTML.gif and 0 < μ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq160_HTML.gif such that ( x , y ) μ F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq158_HTML.gif, by Lemma 3.2(i) and (iii), we have y 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq161_HTML.gif.

Since | α ( y ) ( s ) | ( 2 | λ | + | λ | + 1 ) = 3 | λ | + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq162_HTML.gif and | β ( y ) ( s ) | | λ | + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq163_HTML.gif for s [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq164_HTML.gif, this, together with 1 t 1 s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq165_HTML.gif for s t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq166_HTML.gif and φ n x ( t ) 1 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq93_HTML.gif, implies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equag_HTML.gif

And then | x ( t ) | | S n ( x , y ) ( t ) | + ( 1 t ) | T n ( x , y ) ( t ) | + 1 2 ( 2 | λ | + 1 ) n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq167_HTML.gif, i.e., x 2 ( 2 | λ | + 1 ) n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq168_HTML.gif.

By (3.3), we have
y ( t ) = 0 t s h ( y ) ( s ) ( φ n x ( s ) ) 2 d s + t 1 ( 1 s ) h ( y ) ( s ) ( φ n x ( s ) ) 2 d s + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ15_HTML.gif
(3.7)
Noticing that | h ( y ) ( s ) | | λ | | y ( s ) | + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq169_HTML.gif and φ n x ( s ) 1 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq170_HTML.gif for s [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq164_HTML.gif, we obtain | h ( y ) ( s ) ( φ n x ( s ) ) 2 | n 2 ( | λ | | y ( s ) | + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq171_HTML.gif for s [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq172_HTML.gif. This, together with (3.7) and Lemma 3.2(ii), implies
| y ( t ) | 0 1 | h ( y ) ( s ) ( φ n x ( s ) ) 2 | d s + 0 1 | h ( y ) ( s ) ( φ n x ( s ) ) 2 | d s + 1 2 ( 2 | λ | + 1 ) n 2 + 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equah_HTML.gif
i.e., y 2 ( 2 | λ | + 1 ) n 2 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq173_HTML.gif. Hence,
( x , y ) = x + y + y 2 ( 2 | λ | + 1 ) n + 1 + 2 ( 2 | λ | + 1 ) n 2 + 1 < R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equai_HTML.gif

a contradiction.

By Lemmas 3.1 and 3.3, F has a fixed point ( x n , y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq155_HTML.gif in C [ 0 , 1 ] × C 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq85_HTML.gif. □

Lemma 3.5 Let ( x n , y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq155_HTML.gifbe in Lemma  3.4, then
  1. (i)

    { x n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq174_HTML.gif is bounded on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq101_HTML.gif.

     
  2. (ii)

    { x n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq175_HTML.gif is bounded on [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq176_HTML.gif for any b ( 1 2 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq177_HTML.gif.

     
Proof By Lemma 3.3(i), we know 0 y n ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq178_HTML.gif. By (3.5), we have
x n ( t ) = λ ( 1 t 2 ) φ x n ( t ) 0 t y n ( s ) + λ s φ x n ( s ) d s , t [ 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ16_HTML.gif
(3.8)
  1. (i)
    For λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq15_HTML.gif, we know x n ( t ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq179_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif, i.e., x n ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq180_HTML.gif is decreasing in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq101_HTML.gif, by x n ( 1 ) = 1 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq181_HTML.gif, φ x n ( t ) = x n ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq182_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif. By α ( y n ) ( t ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq183_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif and (3.5), we have
    x n ( t ) S n ( x n , y n ) ( t ) t 1 s φ n x ( s ) d s t x n ( t ) t 1 ( 1 s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ17_HTML.gif
    (3.9)
     
And then x n ( t ) ( 1 t ) 2 t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq184_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif. Obviously, x n ( t ) 1 t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq185_HTML.gif for t [ 1 2 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq186_HTML.gif. This, together with the decrease in x n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq187_HTML.gif, implies
x n ( t ) 1 t 4 for  t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ18_HTML.gif
(3.10)
Let c ( t ) = μ ( 1 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq188_HTML.gif, μ defined by
μ = { 1 4 if  λ 0 , c λ if  λ < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equaj_HTML.gif

where c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq189_HTML.gif defined in (3.1).

It is easy to verify φ n x n ( t ) c ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq190_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif. And then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equak_HTML.gif
The last two inequalities imply that { x n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq174_HTML.gif is bounded on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq101_HTML.gif.
  1. (ii)
    By (3.8),
    | x n ( t ) | | λ | ( 1 + t ) μ + 0 t 2 c ( s ) d s , t [ 0 , 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equal_HTML.gif
     

we know that { x n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq175_HTML.gif is bounded on [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq176_HTML.gif for any b ( 1 2 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq191_HTML.gif. □

Lemma 3.6 Let ( x n , y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq155_HTML.gifbe in Lemma  3.4, then
  1. (i)

    t y n ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq192_HTML.gif and y n ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq193_HTML.gif is increasing in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq101_HTML.gif.

     
  2. (ii)

    { y n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq194_HTML.gif is bounded and equicontinuous in [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq176_HTML.gif for any b ( 1 2 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq195_HTML.gif.

     
Proof
  1. (i)

    Lemma 3.2(i) and (iii) imply the desired results.

     
  2. (ii)

    For b ( 1 2 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq177_HTML.gif, let t b [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq196_HTML.gif such that y n ( t b ) = min { y n ( t ) : t [ 0 , b ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq197_HTML.gif. Since y n ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq198_HTML.gif on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq101_HTML.gif, by Lemma 3.2(ii), y n ( t b ) b 0 b y n ( s ) d s 0 1 y n ( s ) d s 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq199_HTML.gif, we obtain y n ( t b ) 2 b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq200_HTML.gif.

     
Differentiating (3.6) with t twice, we have y n ( t ) = h ( y n ) ( t ) ( φ n x n ( t ) ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq201_HTML.gif. Integrating this equality from 0 to t b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq202_HTML.gif, we have
y n ( t ) y n ( t b ) = t b t h ( y n ) ( s ) ( φ n x n ( s ) ) 2 d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equam_HTML.gif
Noticing that | h ( y n ) ( t ) | | λ | y n ( t ) + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq203_HTML.gif and c ( t ) c ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq204_HTML.gif for t [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq205_HTML.gif and Lemma 3.2(ii), we know
| y n ( t ) | 2 | λ | + 1 ( c ( b ) ) 2 + | y n ( t b ) | 2 | λ | + 1 ( c ( b ) ) 2 + 2 b , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equan_HTML.gif
i.e., { y n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq194_HTML.gif is bounded on [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq176_HTML.gif. Let M b = sup { M n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq206_HTML.gif (where M n = max { y n ( t ) : t [ 0 , b ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq207_HTML.gif), we know
| y n ( t ) | = | h ( y n ) ( t ) | ( φ x n ( t ) ) 2 | λ | M b + 1 ( c ( b ) ) 2 < + for  0 t b . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equao_HTML.gif

This implies that { y n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq194_HTML.gif is equicontinuous on [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq176_HTML.gif. □

Theorem 3.1 There exists ( x , y ) C [ 0 , 1 ] × ( C [ 0 , 1 ] C 1 [ 0 , 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq208_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ19_HTML.gif
(3.11)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ20_HTML.gif
(3.12)
hold, where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equap_HTML.gif

Proof Let ( x n , y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq155_HTML.gif be in Lemma 3.4, by Lemma 3.5(ii) and (iii), we know that { x n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq174_HTML.gif is bounded and equicontinuous on [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq176_HTML.gif for any b ( 1 2 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq177_HTML.gif. Letting b = 1 1 k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq209_HTML.gif ( k = 3 , 4 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq210_HTML.gif), utilizing the diagonal principle and the Arzela-Ascoli theorem, we know that there exists a subsequence { x n k ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq211_HTML.gif of { x n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq174_HTML.gif and x ( t ) C [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq212_HTML.gif such that x n k ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq213_HTML.gif converges to x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq214_HTML.gif for t [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq45_HTML.gif. Without loss of generality, we assume that { x n k ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq211_HTML.gif is itself of { x n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq174_HTML.gif.

By Lemma 3.6, we know that { y n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq194_HTML.gif is bounded and equicontinuous on [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq176_HTML.gif for any b ( 1 2 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq177_HTML.gif and then { y n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq215_HTML.gif is bounded and equicontinuous on [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq176_HTML.gif. Let b = 1 1 k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq216_HTML.gif ( k = 3 , 4 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq210_HTML.gif), the diagonal principle and the Arzela-Ascoli theorem imply that there exist y and y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq217_HTML.gif in C [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq218_HTML.gif and two subsequences { y n k ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq219_HTML.gif and { y n i ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq220_HTML.gif with { y n i ( t ) } { y n k ( t ) } { y n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq221_HTML.gif such that y n k ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq222_HTML.gif converges to y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq66_HTML.gif for t [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq45_HTML.gif with y ( 1 ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq63_HTML.gif and y n i ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq223_HTML.gif converges to y 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq224_HTML.gif for each t [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq45_HTML.gif. For the sake of convenience, we assume that { y n i ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq225_HTML.gif and { y n k ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq226_HTML.gif are itself of { y n ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq215_HTML.gif. By y n ( t ) = 0 t y n ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq227_HTML.gif, we obtain y ( t ) = 0 t y 0 ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq228_HTML.gif and then y 0 ( t ) = y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq229_HTML.gif for t [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq45_HTML.gif.

Since
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equaq_HTML.gif

α ( y n ) ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq230_HTML.gif converges to α ( y ) ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq231_HTML.gif and β ( y n ) ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq232_HTML.gif converges to β ( y ) ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq233_HTML.gif for t [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq45_HTML.gif, by the Lebesgue dominated theorem (the dominated function F ( s ) = 3 | λ | + 1 c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq234_HTML.gif, s [ 0 , 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq235_HTML.gif, we have that ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq77_HTML.gif satisfies (3.11) and x Q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq236_HTML.gif.

Fix t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq116_HTML.gif and choose b ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq237_HTML.gif such that t b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq202_HTML.gif, then
y n ( t ) = 0 b G 0 , b h ( y n ) ( s ) ( φ x n ( s ) ) 2 d s + t b y n ( b ) for  t [ 0 , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equar_HTML.gif
Noticing that | h ( y n ) ( s ) | | λ | | y n ( s ) | + 1 | λ | M b + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq238_HTML.gif and h ( y n ) ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq239_HTML.gif converges to h ( y ) ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq240_HTML.gif for s [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq241_HTML.gif, by the Lebesgue dominated theorem (the dominated function F ( s ) = M b + 1 ( c ( b ) ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq242_HTML.gif on [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq243_HTML.gif), we have
y ( t ) = 0 b G 0 , b h ( y ) ( s ) ( φ x ( s ) ) 2 d s + t b y ( b ) for  t [ 0 , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equas_HTML.gif
Differentiating the last equality twice, we know
y ( t ) = h ( y ) ( t ) ( φ x ( t ) ) 2 for  t [ 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equat_HTML.gif

By (i), we know t y ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq244_HTML.gif and lim t 1 y ( t ) = 1 = y ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq245_HTML.gif and then y C [ 0 , 1 ] C 1 [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq246_HTML.gif. This, together with (2.4), implies that y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq66_HTML.gif satisfies (3.12). Clearly, ( x , y ) Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq247_HTML.gif. □

Theorem 3.2 For λ ( λ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq84_HTML.gif, the system (2.2)-(2.3) has at least a solution ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq77_HTML.gifin Q.

Proof Let ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq77_HTML.gif in Theorem 3.1. It is clear that we only prove φ x ( t ) = x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq248_HTML.gif. If λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq15_HTML.gif, by (3.10), we obtain x ( t ) 1 t 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq249_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif and then φ x ( t ) = x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq248_HTML.gif. Next, we prove x ( t ) c λ ( 1 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq250_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif for λ 0 < λ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq251_HTML.gif.

Let γ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq252_HTML.gif such that M = φ x ( γ ) = max { φ x ( t ) : t [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq253_HTML.gif, then
M x ( δ ) = δ 1 ( 2 λ s + λ + y ( s ) ) ( 1 s ) φ x ( s ) + ( 1 δ ) 0 δ λ s + y ( s ) φ x ( s ) d s δ 1 ( 2 λ s + λ + s ) ( 1 s ) max { M , c λ } d s + ( 1 δ ) 0 δ λ s + s max { M , c λ } d s = 1 max { M , c λ } δ 1 ( 2 λ s + λ + s ) ( 1 s ) d s + ( 1 δ ) 0 δ ( λ s + s ) d s = h ( λ ) max { M , c λ } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equau_HTML.gif

From this and c λ h ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq254_HTML.gif, we obtain M h ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq255_HTML.gif and x ( γ ) = φ x ( γ ) = M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq256_HTML.gif.

Let S ( t ) = S ( x , y ) ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq257_HTML.gif and S = max { S ( t ) : t [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq258_HTML.gif, we prove
S 3 + 5 λ 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ21_HTML.gif
(3.13)

By α ( y ) ( 0 ) = λ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq259_HTML.gif and α ( y ) ( 1 ) = 3 λ + 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq260_HTML.gif, there exists t 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq261_HTML.gif such that α ( y ) ( t 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq262_HTML.gif. Since α ( y ) ( t ) = y ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq263_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq90_HTML.gif, i.e., α ( y ) ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq231_HTML.gif is concave down on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq101_HTML.gif, then α ( y ) ( s ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq264_HTML.gif for s [ 0 , t 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq265_HTML.gif and α ( y ) ( s ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq266_HTML.gif for s [ t 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq267_HTML.gif. Hence, S = S ( t 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq268_HTML.gif.

By (3.11), we have
φ x ( t ) x ( t ) S ( t ) for  t [ t 0 , 1 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equav_HTML.gif
we know
S ( t ) ( S ( t ) ) = S ( t ) α ( y ) ( t ) φ x ( t ) 2 λ t + λ + 1 for  t [ t 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equaw_HTML.gif
Integrating the last inequality from t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq269_HTML.gif to 1 and utilizing S ( 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq270_HTML.gif, we have
S 2 ( t 0 ) 2 t 0 1 ( 2 λ s + λ + 1 ) ( 1 s ) d s 0 1 ( 2 λ s + λ + 1 ) ( 1 s ) d s = 3 + 5 λ 6 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equax_HTML.gif

Hence, (3.13) holds.

By x ( 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq271_HTML.gif, x ( δ ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq272_HTML.gif and x ( 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq56_HTML.gif, we have 0 < γ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq273_HTML.gif and x ( γ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq274_HTML.gif, then
0 = x ( γ ) = λ ( 1 γ 2 ) φ x ( γ ) 0 γ λ s + y ( s ) φ x ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equay_HTML.gif
i.e.,
0 γ λ s + y ( s ) φ x ( s ) d s = λ ( 1 γ 2 ) φ x ( γ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equaz_HTML.gif
Hence,
( 1 γ ) T ( x , y ) ( γ ) = λ ( 1 γ ) ( 1 γ 2 ) φ x ( γ ) λ M . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equba_HTML.gif
This, together with (3.13), implies
M = x ( γ ) = S ( x , y ) ( γ ) + ( 1 γ ) T ( x , y ) ( γ ) 3 + 5 λ 3 λ M , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbb_HTML.gif
i.e.,
M 3 + 5 λ 3 + 3 7 λ 3 2 = σ ( λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbc_HTML.gif
Since α ( y ) ( t ) 2 λ t + λ + t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq275_HTML.gif for t [ δ , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq276_HTML.gif, we have
x ( t ) ( 1 t ) T ( x , y ) ( t ) ( 1 t ) T ( x , y ) ( δ ) ( 1 t ) 0 δ λ s + s σ ( λ ) d s ( λ + 1 ) δ 2 2 σ ( λ ) ( 1 t ) , t [ δ , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbd_HTML.gif

And then x ( t ) c λ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq277_HTML.gif for t [ δ , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq278_HTML.gif.

Finally, we prove x ( t ) c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq279_HTML.gif for t [ 0 , δ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq280_HTML.gif.

In fact, if there exists t [ 0 , δ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq281_HTML.gif such that x ( t ) < c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq282_HTML.gif, by x ( δ ) > c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq283_HTML.gif, there exists t ( 0 , δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq284_HTML.gif such that x ( t ) > c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq285_HTML.gif for t ( t , δ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq286_HTML.gif and x ( t ) = c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq287_HTML.gif.

From
x ( δ ) = S ( x , y ) ( δ ) + ( 1 δ ) T ( x , y ) ( δ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Eqube_HTML.gif
S ( x , y ) ( δ ) δ 1 2 λ s + λ + s σ ( λ ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq288_HTML.gif and T ( x , y ) ( δ ) 0 δ λ s + s σ ( λ ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq289_HTML.gif, we obtain
x ( δ ) h ( λ ) σ ( λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbf_HTML.gif
By (3.11), we have
x ( t ) = λ ( 1 t 2 ) x ( t ) 0 t λ s + y ( s ) φ x ( s ) d s λ ( 1 t 2 ) x ( t ) , t [ t , δ ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbg_HTML.gif
i.e., x ( t ) x ( t ) λ ( 1 t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq290_HTML.gif, t [ t , δ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq291_HTML.gif. Integrating this inequality from t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq292_HTML.gif to δ, we have
x 2 ( δ ) c λ 2 2 t δ λ ( 1 s 2 ) d s < 0 δ λ ( 1 s 2 ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbh_HTML.gif

and then c λ 2 > x 2 ( δ ) + 2 0 δ λ ( 1 s 2 ) d s h 2 ( λ ) σ 2 ( λ ) 2 l ( λ ) = ω ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq293_HTML.gif, a contradiction.

This completes the proof. □

4 Existence of solutions of (1.1)-(1.3)

In this section, we use positive solutions obtained in Theorem 3.2 to construct the solutions of (1.1)-(1.3) in Γ.

Theorem 4.1 For λ ( λ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq84_HTML.gif, the system (1.1)-(1.3) has at least a solution ( f , g ) Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq7_HTML.gif.

Proof Let λ ( λ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq84_HTML.gif, by Theorem 3.2, the system (2.2)-(2.3) has at least a solution ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq77_HTML.gif in Q. By x ( t ) c ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq294_HTML.gif and (2.2), we know
x ( t ) t 1 ( 1 s ) ( 3 | λ | + 1 ) c ( s ) d s + ( 1 t ) 0 t | λ | + 1 c ( s ) d s 1 c ( t 1 ( 3 | λ | + 1 ) d s + ( 1 t ) 0 t 1 + | λ | 1 s d s ) 1 c ( 3 | λ | + 1 ( 1 + | λ | ) ln ( 1 t ) ) ( 1 t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbi_HTML.gif
Let u ( t ) = 1 c ( 3 | λ | + 1 ( 1 + | λ | ) ln ( 1 t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq295_HTML.gif, d u = 1 + | λ | c ( 1 t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq296_HTML.gif and then
0 1 1 z ( s ) d s 0 1 1 u ( s ) ( 1 s ) d s = c 1 + | λ | 0 d u u = , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbj_HTML.gif

we have 0 1 1 x ( s ) d s = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq297_HTML.gif.

Let
η : = η ( t ) = 0 t 1 x ( s ) d s , 0 t < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ22_HTML.gif
(4.1)
Then η ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq298_HTML.gif is strictly increasing on [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq54_HTML.gif and
η ( 0 ) = 0 , η ( 1 0 ) = 0 1 1 x ( s ) d s = + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbk_HTML.gif
Let t = h ( η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq299_HTML.gif be the inverse function to η = η ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq300_HTML.gif, we define the function
g ( η ) = 0 η h ( s ) d s , f ( η ) = 0 η y ( h ( s ) ) d s , 0 η < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbl_HTML.gif
Then
g ( η ) = h ( η ) , g ( 0 ) = 0 , g ( 0 ) = 0 , g ( ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbm_HTML.gif
and
f ( η ) = y ( h ( η ) ) , f ( 0 ) = 0 , f ( 0 ) = 0 , f ( ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbn_HTML.gif
From (4.1), we have
η = η ( g ( η ) ) = 0 g ( η ) 1 x ( s ) d s , 0 η < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ23_HTML.gif
(4.2)
Differentiating (4.2) with respect to η, we have
g ( η ) = x ( g ( η ) ) = x ( t ) , 0 η < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ24_HTML.gif
(4.3)

Then g ( η ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq301_HTML.gif for 0 η < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq302_HTML.gif.

Differentiating (4.3) with respect to η, we have
g ( η ) = x ( g ( η ) ) , g ( η ) = x ( t ) x ( t ) , 0 t < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ25_HTML.gif
(4.4)
Differentiating (2.2) with respect to t, we have
x ( t ) = 0 t λ s + y ( s ) x ( s ) d s + λ ( 1 t 2 ) x ( t ) , 0 t < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ26_HTML.gif
(4.5)
By setting s = g ( σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq303_HTML.gif and utilizing t = g ( η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq304_HTML.gif and (4.3), we have
0 t λ s + y ( s ) x ( s ) d s = 0 g ( η ) λ s + y ( s ) x ( s ) d s = 0 η ( f ( σ ) + λ g ( σ ) ) d σ = f ( η ) + λ g ( η ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equ27_HTML.gif
(4.6)
By (4.3), (4.4), (4.5) and (4.6), we have
g = ( f + λ g ) g + λ ( g 2 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbo_HTML.gif
By (4.1), we have d t d η = x ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq305_HTML.gif. Differentiating f ( η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq306_HTML.gif with respect to η, we have
f ( η ) = y ( t ) d t d η = y ( t ) x ( t ) , f ( η ) = y ( t ) x 2 ( t ) + y ( t ) x ( t ) x ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbp_HTML.gif
Differentiating (2.3) with t twice and combining (4.5) and (4.6), we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbq_HTML.gif

This completes the proof. □

Remark 4.1 For λ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq307_HTML.gif, by Theorem 1 [2], (1.1)-(1.3) has no solution such that lim η g ( η ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq308_HTML.gif with | g ( η ) | < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq309_HTML.gif for η η 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq310_HTML.gif, η 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq311_HTML.gif is a constant.

Utilizing the system (2.2)-(2.3), we know easily that (1.1)-(1.3) has no solution in Γ for λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq312_HTML.gif.

In fact, if (1.1)-(1.3) has a solution ( f , g ) Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq313_HTML.gif for some λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq312_HTML.gif, by Theorem 2.1, then (1.1)-(1.3) has a solution in ( x , y ) Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_IEq247_HTML.gif. Noticing that
α ( y ) ( t ) = 2 λ t + λ + y ( t ) 2 λ t + λ + 1 < 0 for  t ( 0 , 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbr_HTML.gif
we know
g ( 0 ) = x ( 0 ) = 0 1 α ( y ) ( s ) ( 1 s ) φ x ( s ) d s < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-153/MediaObjects/13661_2012_Article_258_Equbs_HTML.gif

a contradiction.

This research uses integrals of equations to investigate the existence of solutions of the 3D axisymmetric inviscid stagnation flows related to Navier-Stokes equations and supplies a gap of analytical study in this field.

Declarations

Acknowledgements

The authors wish to thank the anonymous referees for their valuable comments. This research was supported by the National Natural Science Foundation of China (Grant No. 11171046) and the Scientific Research Foundation of the Education Department of Sichuan Province, China.

Authors’ Affiliations

(1)
College of Mathematics, Chengdu University of Information Technology

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