Open Access

Higher order functional boundary value problems without monotone assumptions

Boundary Value Problems20132013:81

DOI: 10.1186/1687-2770-2013-81

Received: 15 December 2012

Accepted: 22 March 2013

Published: 10 April 2013

Abstract

In this paper, given f : [ a , b ] × ( C ( [ a , b ] ) ) n 2 × R 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq1_HTML.gif a L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq2_HTML.gif-Carathéodory function, it is considered the functional higher order equation

u ( n ) ( x ) = f ( x , u , u , , u ( n 2 ) ( x ) , u ( n 1 ) ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equa_HTML.gif

together with the nonlinear functional boundary conditions, for i = 0 , , n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq3_HTML.gif

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equb_HTML.gif

Here, L i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq4_HTML.gif, i = 0 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq5_HTML.gif, are continuous functions. It will be proved an existence and location result in presence of not necessarily ordered lower and upper solutions, without assuming any monotone properties on the boundary conditions and on the nonlinearity f.

1 Introduction

In this paper, it is considered the functional higher order boundary value problem, for n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq6_HTML.gif composed by the equation
u ( n ) ( x ) = f ( x , u , , u ( n 3 ) , u ( n 2 ) ( x ) , u ( n 1 ) ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ1_HTML.gif
(1)
for a.a. x I : = [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq7_HTML.gif, where f : I × ( C ( I ) ) ( n 2 ) × R 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq8_HTML.gif is a L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq2_HTML.gif-Carathéodory function, and the function boundary conditions
L i ( u , u , , u ( n 1 ) , u ( i ) ( a ) ) = 0 , i = 0 , , n 2 , L n 1 ( u , u , , u ( n 1 ) , u ( n 2 ) ( b ) ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ2_HTML.gif
(2)

where L i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq4_HTML.gif, i = 0 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq5_HTML.gif, are continuous functions without assuming monotone conditions or another type of variation.

The functional differential equation (1) can be seen as a generalization of several types of full differential and integro-differential equations and allow to consider delays, maxima or minima arguments, or another kind of global variation on the unknown function or its derivatives until order ( n 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq9_HTML.gif. On the other hand, the functional dependence in (2) makes possible its application to a huge variety of boundary conditions, such as Lidstone, separated, multipoint, nonlocal and impulsive conditions, among others. As example, we mention the problems contained in [115]. A detailed list about the potentialities of functional problems and some applications can be found in [16].

Recently, functional boundary value problems have been studied by several authors following several approaches, as it can be seen, for example, in [1724]. In this work, the lower and upper solutions method is applied together with topological degree theory, according some arguments suggested in [2527].

The novelty of this paper consists in the following items:

  • There is no monotone assumptions on the boundary functions L i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq4_HTML.gif, i = 0 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq5_HTML.gif, by using adequate auxiliary functions and global arguments. This fact with the functional dependence on the unknown function and its derivatives till order ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq10_HTML.gif will allow that problem (1)-(2) can include the periodic and antiperiodic cases, which were not covered by the existent literature on functional boundary value problems. In this sense, the results in this area, as for instance [2832], are improved, even for n = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq11_HTML.gif, where equation (1) loses its functional part.

  • No extra condition on the nonlinear part of (1) is considered, besides a Nagumo-type growth assumption. In fact, as far as we know, it is the first time where lower and upper solutions technique is used without such hypothesis on function f, by the use of stronger definitions for lower and upper solutions.

  • No order between lower and upper solutions is assumed. Putting the ‘well ordered’ case on adequate auxiliary functions, it allows that lower and upper solutions could be well ordered, by reversed order or without a defined order.

The last section contains an example where the potentialities of the functional dependence on the equation and on the boundary conditions are explored.

2 Definitions and auxiliary functions

In this section, it will be introduced the notations and definitions needed forward together with some auxiliary functions useful to construct some ordered functions on the basis of the not necessarily ordered lower and upper solutions of the referred problem.

A Nagumo-type growth condition, assumed on the nonlinear part, will be an important tool to set an a priori bound for the ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq12_HTML.gifth derivative of the corresponding solutions.

In the following, W m , 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq13_HTML.gif denotes the usual Sobolev Spaces in I, that is, the subset of C m 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq14_HTML.gif functions, whose ( m 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq15_HTML.gifth derivative is absolutely continuous in I and the m th derivative belongs to L 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq16_HTML.gif and the usual norms
u p = { ( 0 1 | u ( x ) | p d x ) 1 / p , 1 p < , sup { | u ( x ) | : x I } , p = , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equc_HTML.gif

for spaces L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq17_HTML.gif, 1 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq18_HTML.gif.

The function f : I × ( C ( I ) ) ( n 2 ) × R 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq19_HTML.gif is a L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq2_HTML.gif-Carathéodory function, that is, f ( x , , , , , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq20_HTML.gif is a continuous function for a.e. x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq21_HTML.gif; f ( , y 0 , , y n 2 , y n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq22_HTML.gif is measurable for ( y 0 , , y n 2 , y n 1 ) ( C ( I ) ) ( n 2 ) × R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq23_HTML.gif; and for every M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq24_HTML.gif there is a real-valued function ψ M L 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq25_HTML.gif such that
| f ( x , y 0 , y 1 , y 2 , y 3 ) | ψ M ( x ) , for a.e.  x [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equd_HTML.gif

and for every ( y 0 , y 1 , y 2 , y 3 ) ( C ( I ) ) ( n 2 ) × R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq26_HTML.gif with | y i | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq27_HTML.gif, for i = 0 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq5_HTML.gif.

The main tool to obtain the location part is the upper and lower solutions method. However, in this case, they must be defined as a pair, which means that it is not possible to define them independently from each other. Moreover, it is pointed out that lower and upper functions, and the correspondent first derivatives, are not necessarily ordered.

To introduce ‘some order’, some auxiliary functions must be defined.

For any α , β W n 2 , 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq28_HTML.gif define functions α i , β i : I R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq29_HTML.gif, i = 0 , , n 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq30_HTML.gif, as it follows:
α n 3 ( x ) = min { α ( n 3 ) ( a ) , β ( n 3 ) ( a ) } + a x α ( n 2 ) ( s ) d s , β n 3 ( x ) = max { α ( n 3 ) ( a ) , β ( n 3 ) ( a ) } + a x β ( n 2 ) ( s ) d s , α i ( x ) = min { α ( i ) ( a ) , β ( i ) ( a ) } + a x α i + 1 ( s ) d s , β i ( x ) = max { α ( i ) ( a ) , β ( i ) ( a ) } + a x β i + 1 ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ3_HTML.gif
(3)

for i = 0 , , n 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq31_HTML.gif.

The Nagumo-type condition is given by next definition.

Definition 1 Consider Γ i , γ i C ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq32_HTML.gif, i = 0 , , n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq3_HTML.gif, such that Γ i ( x ) γ i ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq33_HTML.gif, x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq34_HTML.gif, and the set
E = { ( x , y 0 , , y n 1 ) I × R n : γ i ( x ) y i Γ i ( x ) , i = 0 , , n 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Eque_HTML.gif
A function f : I × R n R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq35_HTML.gif is said to verify a Nagumo-type condition in E if there exists φ E C ( [ 0 , + ) , ( 0 , + ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq36_HTML.gif such that
| f ( x , y 0 , , y n 1 ) | φ E ( | y n 1 | ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ4_HTML.gif
(4)
for every ( x , y 0 , , y n 1 ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq37_HTML.gif, and
r + t φ E ( t ) d t > max x I Γ n 2 ( x ) min x I γ n 2 ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ5_HTML.gif
(5)
where r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq38_HTML.gif is given by
r : = max { Γ n 2 ( b ) γ n 2 ( a ) b a , Γ n 2 ( a ) γ n 2 ( b ) b a } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equf_HTML.gif

The next result gives an a priori estimate for the ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq12_HTML.gifth derivative of all possible solutions of (1).

Lemma 2 There exists K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq39_HTML.gifsuch that for every L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq2_HTML.gif-Carathéodory function f : I × ( C ( I ) ) n 2 × R 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq40_HTML.gifsatisfying (4) and (5) and every solution u of (1) such that
γ i ( x ) u ( i ) ( x ) Γ i ( x ) , a.e. x I , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ6_HTML.gif
(6)
for i = 0 , , n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq3_HTML.gif, we have
u ( n 1 ) < R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ7_HTML.gif
(7)

Moreover, the constant R depends only on the functions φ and γ i , Γ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq41_HTML.gif ( i = 0 , , n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq42_HTML.gif) and not on the boundary conditions.

Proof The proof is similar to [[19], Lemma 2.1]. □

The upper and lower solution definition is then given by the following.

Definition 3 The functions α , β W n , 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq43_HTML.gif are a pair of lower and upper solutions for problem (1)-(2) if α ( n 2 ) ( x ) β ( n 2 ) ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq44_HTML.gif, on I, for all ( v 0 , , v n 3 ) A : = [ α 0 , β 0 ] × × [ α n 3 , β n 3 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq45_HTML.gif, and for every ( w 1 , w 2 ) B : = [ α ( n 2 ) , β ( n 2 ) ] × [ K , K ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq46_HTML.gif, for some K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq39_HTML.gif, the following inequalities hold for a.e. x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq47_HTML.gif,
α ( n ) ( x ) f ( x , v 0 , , v n 3 , α ( n 2 ) ( x ) , α ( n 1 ) ( x ) ) , β ( n ) ( x ) f ( x , v 0 , , v n 3 , β ( n 2 ) ( x ) , β ( n 1 ) ( x ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ8_HTML.gif
(8)
and for j = 0 , , n 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq48_HTML.gif,
L j ( v 0 , , v n 3 , w 1 , w 2 , α j ( a ) ) 0 , L n 2 ( v 0 , , v n 3 , w 1 , w 2 , α ( n 2 ) ( a ) ) 0 , L n 1 ( v 0 , , v n 3 , w 1 , w 2 , α ( n 2 ) ( b ) ) 0 , L j ( v 0 , , v n 3 , w 1 , w 2 , β j ( a ) ) 0 , L n 2 ( v 0 , , v n 3 , w 1 , w 2 , β ( n 2 ) ( a ) ) 0 , L n 1 ( v 0 , , v n 3 , w 1 , w 2 , β ( n 2 ) ( b ) ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ9_HTML.gif
(9)

3 Existence and location result

In this section, it is provided an existence and location theorem for the problem (1)-(2). More precisely, sufficient conditions are given for, not only the existence of a solution u, but also to have information about the location of u, and all its derivatives up to the ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq10_HTML.gif order.

The arguments of the proof require the following lemma, given on [29].

Lemma 4 For v , w C ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq49_HTML.gifsuch that v ( x ) w ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq50_HTML.gif, for every x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq21_HTML.gif, define
q ( x , u ) = max { v , min { u , w } } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equg_HTML.gif
Then, for each u C 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq51_HTML.gifthe next two properties hold:
  1. (a)

    d d x q ( x , u ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq52_HTML.gif exists for a.e. x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq21_HTML.gif.

     
  2. (b)
    If u , u m C 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq53_HTML.gif and u m u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq54_HTML.gif in C 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq55_HTML.gif then
    d d x q ( x , u m ( x ) ) d d x q ( x , u ( x ) ) for a.e. x I . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equh_HTML.gif
     

Now, we are in a position to prove the main result of this paper.

Theorem 5 Assume that there exists a pair of lower and upper solutions ( α , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq56_HTML.gifof problem (1)-(2).

If f : I × ( C ( I ) ) n 2 × R 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq57_HTML.gifis a L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq2_HTML.gif-Carathéodory function, satisfying a Nagumo-type condition in
E = { ( x , y 0 , , y n 1 ) I × R n 1 : α i ( x ) y i β i ( x ) , i = 0 , , n 3 , α ( n 2 ) ( x ) y n 2 β ( n 2 ) ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equi_HTML.gif
then problem (1)-(2) has at least one solution u such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equj_HTML.gif
for every x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq21_HTML.gif, and | u ( n 1 ) ( x ) | K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq58_HTML.gif, x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq34_HTML.gif, where
K = max { R , | α ( n 1 ) ( x ) | , | β ( n 1 ) ( x ) | } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ10_HTML.gif
(10)

and R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq59_HTML.gifis given by (7).

Proof Define the continuous functions, for i = 0 , , n 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq30_HTML.gif,
δ i ( x , y i ) = max { α i ( x ) , min { y i , β i ( x ) } } , δ n 2 ( x , y n 2 ) = max { α ( n 2 ) ( x ) , min { y n 2 , β ( n 2 ) ( x ) } } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ11_HTML.gif
(11)
and the truncation, not necessarily continuous,
ξ ( z ) = max { K , min { z , K } } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equk_HTML.gif

with K given by (10).

Consider the modified problem composed by the equation
u ( n ) ( x ) = f ( x , δ 0 ( , u ) , , δ n 3 ( , u ( n 3 ) ) , δ n 2 ( x , u ( n 2 ) ( x ) ) , ξ ( d d x ( δ n 2 ( x , u ( n 2 ) ( x ) ) ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ12_HTML.gif
(12)
and the boundary conditions, for i = 0 , , n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq3_HTML.gif,
u ( i ) ( a ) = δ i ( a , u ( i ) ( a ) + L i ( δ 0 ( , u ) , , δ n 2 ( , u ( n 2 ) ) , ξ ( d d x ( δ n 2 ( , u ( n 2 ) ) ) ) , u ( i ) ( a ) ) ) , u ( n 2 ) ( b ) = δ n 2 ( b , u ( n 2 ) ( b ) + L n 1 ( δ 0 ( , u ) , , δ n 2 ( , u ( n 2 ) ) , ξ ( d d x ( δ n 2 ( , u ( n 2 ) ) ) ) , u ( n 2 ) ( b ) ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ13_HTML.gif
(13)

The proof will follow the next steps:

Step 1. Every solution u of problem (12)-(13), satisfies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equl_HTML.gif

and | u ( n 1 ) ( x ) | < K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq60_HTML.gif, for every x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq21_HTML.gif, with K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq39_HTML.gif given in (10).

Let u be a solution of the modified problem (12)-(13). Assume, by contradiction, that there exists x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq21_HTML.gif such that α ( n 2 ) ( x ) > u ( n 2 ) ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq61_HTML.gif and let x 0 I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq62_HTML.gif be such that
min x I ( u α ) ( n 2 ) ( x ) : = ( u α ) ( n 2 ) ( x 0 ) < 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equm_HTML.gif
As, by (13), u ( n 2 ) ( a ) α ( n 2 ) ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq63_HTML.gif and u ( n 2 ) ( b ) α ( n 2 ) ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq64_HTML.gif, then x 0 ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq65_HTML.gif. So, there is ( x 1 , x 2 ) ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq66_HTML.gif such that
u ( n 2 ) ( x ) < α ( n 2 ) ( x ) , x ( x 1 , x 2 ) , ( u α ) ( n 2 ) ( x 1 ) = ( u α ) ( n 2 ) ( x 2 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ14_HTML.gif
(14)
Therefore,
δ n 2 ( x , u ( n 2 ) ( x ) ) = α ( n 2 ) ( x ) , x ( x 1 , x 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equn_HTML.gif
and
d d x δ n 2 ( x , u ( n 2 ) ( x ) ) = α ( n 1 ) ( x ) , a.e.  x ( x 1 , x 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equo_HTML.gif
Now, since for all u C n 2 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq67_HTML.gif it is satisfied that ( δ 0 ( , u ) , , δ n 3 ( , u ) ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq68_HTML.gif, we deduce that
u ( n ) ( x ) = f ( x , δ 0 ( , u ) , , δ n 3 ( , u ( n 3 ) ) , δ n 2 ( x , u ( n 2 ) ( x ) ) , ξ ( d d x ( δ n 2 ( x , u ( n 2 ) ( x ) ) ) ) ) = f ( x , δ 0 ( , u ) , , δ n 3 ( , u ( n 3 ) ) , α ( n 2 ) ( x ) , α ( n 1 ) ( x ) ) α ( n ) ( x ) for a.e.  x ( x 1 , x 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equp_HTML.gif

As ( u α ) ( n 1 ) ( x 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq69_HTML.gif and ( u α ) ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq70_HTML.gif is nonincreasing in ( x 1 , x 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq71_HTML.gif, this contradicts the definitions of x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq72_HTML.gif and x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq73_HTML.gif.

The inequality u ( n 2 ) ( x ) β ( n 2 ) ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq74_HTML.gif, in I, can be proved in same way and so,
α ( n 2 ) ( x ) u ( n 2 ) ( x ) β ( n 2 ) ( x ) , x I . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ15_HTML.gif
(15)
By (13) and (3), the following inequalities hold for every x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq21_HTML.gif:
u ( n 3 ) ( x ) = u ( n 3 ) ( a ) + a x u ( n 2 ) ( s ) d s α n 3 ( a ) + a x α ( n 2 ) ( s ) d s min { α ( n 3 ) ( a ) , β ( n 3 ) ( a ) } + a x α ( n 2 ) ( s ) d s = α n 3 ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equq_HTML.gif

Analogously, it can be obtained u ( n 3 ) ( x ) β n 3 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq75_HTML.gif, for x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq21_HTML.gif.

The remaining inequalities are obtained by the same integration process.

Applying previous bounds in Lemma 2, and remarking that
r K s φ ( s ) d s r R s φ ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equr_HTML.gif

for K given by (10), it is obtained, by Lemma 2, the a priori bound | u ( n 1 ) ( x ) | < K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq76_HTML.gif, for x I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq21_HTML.gif. For details, see [[33], Lemma 2].

Step 2. Problem (12)-(13) has at least one solution.

For λ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq77_HTML.gif let us consider the homotopic problem given by
u ( n ) ( x ) = λ f ( x , δ 0 ( , u ) , , δ n 3 ( , u ( n 3 ) ) , δ n 2 ( x , u ( n 2 ) ( x ) ) , ξ ( d d x ( δ n 2 ( x , u ( n 2 ) ( x ) ) ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ16_HTML.gif
(16)
and the boundary conditions, for i = 0 , , n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq3_HTML.gif,
u ( i ) ( a ) = λ δ i ( a , u ( i ) ( a ) + L i ( δ 0 ( , u ) , , δ n 2 ( , u ( n 2 ) ) , ξ ( d d x ( δ n 2 ( , u ( n 2 ) ) ) ) , u ( i ) ( a ) ) ) : = λ L A i , u ( n 2 ) ( b ) = λ δ n 2 ( b , u ( n 2 ) ( b ) + L n 1 ( δ 0 ( , u ) , , δ n 2 ( , u ( n 2 ) ) , ξ ( d d x ( δ n 2 ( , u ( n 2 ) ) ) ) , u ( n 2 ) ( b ) ) ) u ( n 2 ) ( b ) : = λ L B . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ17_HTML.gif
(17)
Let us consider the norms in C n 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq78_HTML.gif and in L 1 ( I ) × R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq79_HTML.gif, respectively,
v C n 1 = max { v , , v ( n 1 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equs_HTML.gif
and
| ( h , h 1 , , h n ) | = max { h L 1 , max { | h 1 | , , | h n | } } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equt_HTML.gif
Define the operators L : W n , 1 ( I ) C n 1 ( I ) L 1 ( I ) × R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq80_HTML.gif by L u = ( u ( n ) , u ( a ) , , u ( n 2 ) ( a ) , u ( n 2 ) ( b ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq81_HTML.gif and, for λ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq82_HTML.gif, i = 0 , , n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq3_HTML.gif, N λ : C n 1 ( I ) L 1 ( I ) × R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq83_HTML.gif by
N λ u = ( λ f ( x , δ 0 ( , u ) , , δ n 3 ( , u ( n 3 ) ) , δ n 2 ( x , u ( n 2 ) ( x ) ) , ξ ( d d x ( δ n 2 ( x , u ( n 2 ) ( x ) ) ) ) ) , λ L A 1 , , λ L A n 2 , λ L B ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equu_HTML.gif

Since L 0 , , L n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq84_HTML.gif are continuous and f is a L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq2_HTML.gif-Carathéodory function, then, from Lemma 4, N λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq85_HTML.gif is continuous. Moreover, as L 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq86_HTML.gif is compact, it can be defined the completely continuous operator T λ : C n 1 ( I ) C n 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq87_HTML.gif by T λ u = L 1 N λ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq88_HTML.gif.

It is obvious that the fixed points of operator T λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq89_HTML.gif coincide with the solutions of problem (16)-(17).

As N λ u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq90_HTML.gif is bounded in L 1 ( I ) × R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq91_HTML.gif and uniformly bounded in C n 1 ( I ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq78_HTML.gif, we have that any solution of the problem (16)-(17), verifies the following a priori bound
u C n 1 L 1 C n 1 | N λ ( u ) | K ¯ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equv_HTML.gif

for some K ¯ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq92_HTML.gif independent of λ.

In the set Ω = { u C n 1 ( I ) : u C n 1 < K ¯ + 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq93_HTML.gif, the degree d ( I T λ , Ω , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq94_HTML.gif is well defined for every λ [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq95_HTML.gif and, by the invariance under homotopy, d ( I T 0 , Ω , 0 ) = d ( I T 1 , Ω , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq96_HTML.gif.

As the equation x = T 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq97_HTML.gif is equivalent to the problem
{ u ( n ) ( x ) = 0 , u ( i ) ( a ) = u ( n 2 ) ( b ) = 0 , i = 0 , , n 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equw_HTML.gif

which has only the trivial solution, then d ( I T 0 , Ω , 0 ) = ± 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq98_HTML.gif. So, by degree theory, the equation x = T 1 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq99_HTML.gif has at least one solution, that is, the problem (12)-(13) has at least a solution in Ω.

Step 3. Every solution u of problem (12)-(13) is a solution of (1)-(2).

Let u be a solution of the modified problem (12)-(13). By previous steps, function u fulfills equation (1). So, it will be enough to prove the following inequalities, for i = 0 , , n 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq30_HTML.gif:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equx_HTML.gif
and
α ( n 2 ) ( b ) u ( n 2 ) ( b ) + L n 1 ( δ 0 ( , u ) , , δ n 2 ( x , u ( n 2 ) ( x ) ) , ξ ( d d x ( δ n 2 ( x , u ( n 2 ) ( x ) ) ) ) , u ( n 2 ) ( b ) ) β ( n 2 ) ( b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equy_HTML.gif
Assume that
u ( a ) + L 0 ( δ 0 ( , u ) , , δ n 2 ( x , u ( n 2 ) ( x ) ) , ξ ( d d x ( δ n 2 ( x , u ( n 2 ) ( x ) ) ) ) , u ( a ) ) > β 0 ( a ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ18_HTML.gif
(18)
Then, by (13), u ( a ) = β 0 ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq100_HTML.gif. By previous steps, it is obtained the following contradiction with (18):
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equz_HTML.gif
Applying similar arguments, it can be proved that
α 0 ( a ) u ( a ) + L 0 ( δ 0 ( , u ) , , δ n 2 ( x , u ( n 2 ) ( x ) ) , ξ ( d d x ( δ n 2 ( x , u ( n 2 ) ( x ) ) ) ) , u ( a ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equaa_HTML.gif
and analogously, for j = 1 , , n 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq101_HTML.gif,
α j ( a ) u ( j ) ( a ) + L j ( δ 0 ( , u ) , , δ n 2 ( x , u ( n 2 ) ( x ) ) , ξ ( d d x ( δ n 2 ( x , u ( n 2 ) ( x ) ) ) ) , u ( j ) ( a ) ) β j ( a ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equab_HTML.gif
Also, using the same arguments and the same techniques, it can be proved that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equac_HTML.gif

 □

4 Example

This section contains a problem composed by an integro-differential equation with some functional boundary conditions, whose solvability is proved in presence of nonordered lower and upper solutions. We remark that such fact was not possible with the results in the current literature. This example does not model any particular problem arising in real phenomena. Our purpose consists on emphasizing the powerful of the developed theory in this paper by showing what kind of problems we can deal with.

Consider, for x [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq102_HTML.gif, the fourth-order equation
u ( i v ) ( x ) = 0 x u ( s ) d s + max x [ 0 , 1 ] { u ( x ) } + ( u ( x ) ) 3 ( u ( x ) + 1 ) 2 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ19_HTML.gif
(19)
coupled with the boundary value conditions
min x [ 0 , 1 ] u ( x ) 26 u ( 0 ) = 0 , u ( s ) ( u ( 0 ) ) 3 + 14 = 0 , max x [ 0 , 1 ] u ( x ) 2 u ( 0 ) = 0 , u ( 1 ) 3 = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equ20_HTML.gif
(20)
One can verify that functions
α ( x ) = x 6 3 12 x 2 + 20 x 1 and β ( x ) = x 3 3 + 12 x 2 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equad_HTML.gif
are, respectively, lower and upper solutions for the problem (19)-(20). Moreover, we deduce that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equae_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equaf_HTML.gif
As the continuous function f verifies (4) and (5) for φ E ( y 3 ) = 1 , 847 12 + ( y 3 + 1 ) 2 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq103_HTML.gif in
E = { ( x , y 0 , y 1 , y 2 , y 3 ) [ 0 , 1 ] × R 4 : x 6 3 12 x 2 9 2 x 1 y 0 x 3 3 + 12 x 2 + 125 x + 40 3 x 2 2 24 x 9 2 y 1 x 2 + 24 x + 25 x 24 y 2 2 x + 24 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equag_HTML.gif
then, by Theorem 5, there is a nontrivial solution u for problem (19)-(20) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_Equah_HTML.gif

for all x [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-81/MediaObjects/13661_2012_Article_336_IEq102_HTML.gif.

Declarations

Acknowledgements

Dedicated to Professor Jean Mawhin on the occasion of his 70th anniversary.

Authors’ Affiliations

(1)
School of Mathematics, Physics and Technology, College of the Bahamas
(2)
Department of Mathematics, School of Sciences and Technology, University of Évora
(3)
Research Centre on Mathematics and Applications, University of Évora (CIMA-UE)

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© Fialho and Minhós; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.