Open Access

Fredholm-type theorem for boundary value problems for systems of nonlinear functional differential equations

Boundary Value Problems20142014:113

DOI: 10.1186/1687-2770-2014-113

Received: 30 January 2014

Accepted: 29 April 2014

Published: 13 May 2014

Abstract

A Fredholm-type theorem for boundary value problems for systems of nonlinear functional differential equations is established. The theorem generalizes results known for the systems with linear or homogeneous operators to the case of systems with positively homogeneous operators.

MSC:34K10.

Keywords

functional-differential equations boundary value problems existence of solutions

1 Statement of the problem

Consider the system of functional-differential equations
u i ( t ) = p i ( u 1 , , u n ) ( t ) + f i ( u 1 , , u n ) ( t ) for a.e.  t [ a , b ] ( i = 1 , , n )
(1)
together with the boundary conditions
i ( u 1 , , u n ) = h i ( u 1 , , u n ) ( i = 1 , , n ) .
(2)
Here, p i , f i : C ( [ a , b ] ; R n ) L ( [ a , b ] ; R ) are continuous operators satisfying Carathéodory conditions, i.e. for every r > 0 there exists q r L ( [ a , b ] ; R + ) such that
i = 1 n ( | p i ( u 1 , , u n ) ( t ) | + | f i ( u 1 , , u n ) ( t ) | ) q r ( t ) for a.e.  t [ a , b ] , i = 1 n u i C r ,
and i , h i : C ( [ a , b ] ; R n ) R are continuous functionals which are bounded on every ball by a constant, i.e. for every r > 0 there exists M r > 0 such that
i = 1 n ( | i ( u 1 , , u n ) | + | h i ( u 1 , , u n ) | ) M r whenever  i = 1 n u i C r .
Furthermore, we assume that p i and i satisfy the following condition: there exist positive real numbers λ i j and μ i such that λ i j λ j m = λ i m whenever i , j , m { 1 , , n } , and for every c > 0 and ( u k ) k = 1 n C ( [ a , b ] ; R n ) we have
c p i ( u 1 , , u n ) ( t ) = p i ( c λ i 1 u 1 , , c λ i n u n ) ( t ) for a.e.  t [ a , b ] ,
(3)
c μ i i ( u 1 , , u n ) = i ( c λ i 1 u 1 , , c λ i n u n ) .
(4)

Remark 1 From the above-stated assumptions it follows that λ i i = 1 , λ i j = 1 / λ j i for every i , j { 1 , , n } .

In the case when p i and i are linear bounded operators and f i ( , , ) ( t ) q i ( t ) , h i ( , , ) c i , the relationship between the existence of a solution to problem (1), (2) and the existence of only the trivial solution to its corresponding homogeneous problem, so-called Fredholm alternative, is well known; for more details see e.g. [18] and references therein.

In 1966, Lasota established the Fredholm-type theorem in the case when p i and i are homogeneous operators (see [9]). Recently, Fredholm-type theorems in the case when p i and i are positively homogeneous operators were established by Kiguradze, Půža, Stavroulakis in [10] and also by Kiguradze, Šremr in [11].

In this paper we unify the ideas used in [11] and [9] to obtain a new Fredholm-type theorem for the case when p i and i are positively homogeneous operators. The consequences of the obtained result for particular cases of problem (1), (2) are formulated at the end of the paper.

The following notation is used throughout the paper.

is the set of all natural numbers;

is the set of all real numbers, R + = [ 0 , + ) ;

R n is the linear space of vectors x = ( x i ) i = 1 n with the elements x i R endowed with the norm
x = i = 1 n | x i | ;
C ( [ a , b ] ; R n ) is the Banach space of continuous vector-valued functions u = ( u i ) i = 1 n : [ a , b ] R n with the norm
u C = i = 1 n max { | u i ( t ) | : t [ a , b ] } ;

A C ( [ a , b ] ; R n ) is the set of absolutely continuous vector-valued functions u : [ a , b ] R n ;

L ( [ a , b ] ; R ) is the Banach space of Lebesgue integrable functions p : [ a , b ] R with the norm
p L = a b | p ( s ) | d s ;

L ( [ a , b ] ; R + ) = { p L ( [ a , b ] ; R ) : p ( t ) 0  for a.e.  t [ a , b ] } ;

if Ω is a set then measΩ, intΩ, Ω ¯ , and Ω denotes the measure, interior, closure, and boundary of the set Ω, respectively.

By a solution to (1), (2) we understand a function ( u i ) i = 1 n A C ( [ a , b ] ; R n ) satisfying (1) almost everywhere in [ a , b ] and (2).

Notation 1 Define, for every i { 1 , , n } , the following functions:
q i ( t , ρ ) = def sup { | f i ( u 1 , , u n ) ( t ) | : u k C ρ λ i k , k = 1 , , n } for a.e.  t [ a , b ] , η i ( ρ ) = def sup { | h i ( u 1 , , u n ) | : u k C ρ λ i k μ i , k = 1 , , n } .

2 Main result

Theorem 1 Let
lim ρ + a b q i ( s , ρ ) ρ d s = 0 , lim ρ + η i ( ρ ) ρ = 0 ( i = 1 , , n ) .
(5)
If the problem
u i ( t ) = ( 1 δ ) p i ( u 1 , , u n ) ( t ) δ p i ( u 1 , , u n ) ( t ) for a.e.  t [ a , b ] ( i = 1 , , n ) ,
(6)
( 1 δ ) i ( u 1 , , u n ) δ i ( u 1 , , u n ) = 0 ( i = 1 , , n )
(7)

has only the trivial solution for every δ [ 0 , 1 / 2 ] , then problem (1), (2) has at least one solution.

The proof of Theorem 1 is based on the following result by Krasnosel’skii (see [[12], Theorem 41.3, p.325]). We will formulate it in a form suitable for us.

Theorem 2 Let X be a Banach space, Ω X be a symmetrica bounded domain with 0 int Ω . Let, moreover, A : Ω ¯ Ω ¯ be a compactb continuous operator which has no fixed point on Ω. If, in addition,
A ( x ) x A ( x ) x A ( x ) + x A ( x ) + x for  x Ω

then A has a fixed point in Ω, i.e. there exists x 0 Ω such that x 0 = A ( x 0 ) .

Furthermore, to prove Theorem 1 we will need the following lemma.

Lemma 1 Let, for every δ [ 0 , 1 / 2 ] , problem (6), (7) has only the trivial solution. Then there exists r > 0 such that for any ( u i ) i = 1 n A C ( [ a , b ] ; R n ) and any δ [ 0 , 1 / 2 ] , the a priori estimate
k = 1 n u k C λ k 1 r i = 1 n ( f ˜ i L λ i 1 + | h ˜ i | λ i 1 μ i )
(8)
holds, where
f ˜ i ( t ) = def u i ( t ) ( 1 δ ) p i ( u 1 , , u n ) ( t ) + δ p i ( u 1 , , u n ) ( t ) for a.e.  t [ a , b ] ( i = 1 , , n ) , h ˜ i = def ( 1 δ ) i ( u 1 , , u n ) δ i ( u 1 , , u n ) ( i = 1 , , n ) .
Proof Suppose on the contrary that for every m N there exist ( u i m ) i = 1 n A C ( [ a , b ] ; R n ) and δ m [ 0 , 1 / 2 ] such that
k = 1 n u k m C λ k 1 > m i = 1 n ( f ˜ i m L λ i 1 + | h ˜ i m | λ i 1 μ i ) ,
(9)
where
f ˜ i m ( t ) = def u i m ( t ) ( 1 δ m ) p i ( u 1 m , , u n m ) ( t ) + δ m p i ( u 1 m , , u n m ) ( t ) for a.e.  t [ a , b ] ( i = 1 , , n ) ,
(10)
h ˜ i m = def ( 1 δ m ) i ( u 1 m , , u n m ) δ m i ( u 1 m , , u n m ) ( i = 1 , , n ) .
(11)
Put
ρ m = k = 1 n u k m C λ k 1 for  m N ,
(12)
v i m ( t ) = u i m ( t ) ρ m λ 1 i for  t [ a , b ] , m N .
(13)
Then
i = 1 n v i m C λ i 1 = 1 for  m N
(14)
and from (10) and (11), in view of (3), (4), (12), and (13), we get
f ˜ i m ( t ) ρ m λ 1 i = v i m ( t ) ( 1 δ m ) p i ( v 1 m , , v n m ) ( t ) + δ m p i ( v 1 m , , v n m ) ( t ) for a.e.  t [ a , b ] ( i = 1 , , n ; m N ) ,
(15)
h ˜ i m ρ m λ 1 i μ i = ( 1 δ m ) i ( v 1 m , , v n m ) δ m i ( v 1 m , , v n m ) ( i = 1 , , n ; m N ) .
(16)
On the other hand, from (9) and (12) we have
i = 1 n ( f ˜ i m ρ m λ 1 i L λ i 1 + | h ˜ i m ρ m λ 1 i μ i | λ i 1 μ i ) < 1 m for  m N ,
(17)
whence, according to [[13], Corollary IV.8.11] it follows that
lim meas E 0 E f ˜ i m ( s ) ρ m λ 1 i d s = 0 uniformly for  m N ( i = 1 , , n ) .
(18)
Therefore, (14), (15), and (18) imply that the sequences { v i m } m = 1 + ( i = 1 , , n ) are uniformly bounded and equicontinuous. Thus, according to Arzelà-Ascoli theorem, without loss of generality we can assume that there exist ( v i 0 ) i = 1 n C ( [ a , b ] ; R n ) and δ 0 [ 0 , 1 / 2 ] such that
lim m + δ m = δ 0 , lim m + v i m v i 0 C = 0 ( i = 1 , , n ) .
(19)
Furthermore, (15)-(17) yield ( v i 0 ) i = 1 n A C ( [ a , b ] ; R n ) and show that it is a solution to (6), (7). However, (14) and (19) result in
i = 1 n v i 0 C λ i 1 = 1 ,

which contradicts our assumptions. □

Proof of Theorem 1 Let X = C ( [ a , b ] ; R n ) × R n and for x X , i.e. x = ( u , α ) = ( ( u i ) i = 1 n , ( α i ) i = 1 n ) , define the norm
x = u C + α .
Then ( X , ) is a Banach space. Let the operators T , F , A : X X be defined as follows:
T ( x ) = def ( ( u i ( a ) + α i + a t p i ( u 1 , , u n ) ( s ) d s ) i = 1 n , ( α i + i ( u 1 , , u n ) ) i = 1 n ) ,
(20)
F ( x ) = def ( ( a t f i ( u 1 , , u n ) ( s ) d s ) i = 1 n , ( h i ( u 1 , , u n ) ) i = 1 n ) ,
(21)
A ( x ) = def T ( x ) + F ( x ) ,
(22)
and consider the operator equation
x = A ( x ) .
(23)

It can easily be seen that problem (1), (2), and (23) are equivalent in the following sense: if x = ( u , α ) is a solution to (23), then α i = 0 ( i = 1 , , n ) and ( u i ) i = 1 n is a solution to (1), (2); and vice versa if ( u i ) i = 1 n is a solution to (1), (2), then x = ( u , 0 ) is a solution to (23).

Let r > 0 be such that the conclusion of Lemma 1 is valid. According to (5) we can choose ρ 0 > 0 such that
1 ρ 0 i = 1 n ( q i ( , ρ 0 λ 1 i ) L λ i 1 + | η i ( ρ 0 λ 1 i μ i ) | λ i 1 μ i ) < 1 r .
(24)
Let, moreover,
Ω = { x X : k = 1 n ( u k C λ k 1 + | α k | ) < ρ 0 } .
(25)
Now we will show that the operator A has a fixed point in Ω. According to Theorem 2 it is sufficient to show that
A ( x ) x ν ( A ( x ) + x ) for  x Ω , ν ( 0 , 1 ] .
Assume on the contrary that there exist x 0 = ( ( u i 0 ) i = 1 n , ( α i 0 ) i = 1 n ) Ω and ν 0 ( 0 , 1 ] such that
A ( x 0 ) x 0 = ν 0 ( A ( x 0 ) + x 0 ) .
(26)
Then from (26), in view of (20)-(22) we obtain
x 0 = ( 1 δ 0 ) T ( x 0 ) δ 0 T ( x 0 ) + ( 1 δ 0 ) F ( x 0 ) δ 0 F ( x 0 ) ,
where δ 0 = ν 0 / ( 1 + ν 0 ) ( 0 , 1 / 2 ] , i.e.
u i 0 ( t ) = u i 0 ( a ) + α i 0 + ( 1 δ 0 ) a t p i ( u 10 , , u n 0 ) ( s ) d s u i 0 ( t ) = δ 0 a t p i ( u 10 , , u n 0 ) ( s ) d s + ( 1 δ 0 ) a t f i ( u 10 , , u n 0 ) ( s ) d s u i 0 ( t ) = δ 0 a t f i ( u 10 , , u n 0 ) ( s ) d s for  t [ a , b ] ( i = 1 , , n ) ,
(27)
α i 0 = α i 0 + ( 1 δ 0 ) i ( u 10 , , u n 0 ) δ 0 i ( u 10 , , u n 0 ) α i 0 = ( 1 δ 0 ) h i ( u 10 , , u n 0 ) + δ 0 h i ( u 10 , , u n 0 ) ( i = 1 , , n ) .
(28)
Now from (27) and (28) it follows that ( u i ) i = 1 n A C ( [ a , b ] ; R n ) ,
α i 0 = 0 ( i = 1 , , n ) ,
(29)
u i 0 ( t ) = ( 1 δ 0 ) p i ( u 10 , , u n 0 ) ( t ) δ 0 p i ( u 10 , , u n 0 ) ( t ) u i 0 ( t ) = + ( 1 δ 0 ) f i ( u 10 , , u n 0 ) ( t ) δ 0 f i ( u 10 , , u n 0 ) ( t ) u i 0 ( t ) = for a.e.  t [ a , b ] ( i = 1 , , n ) ,
(30)
( 1 δ 0 ) i ( u 10 , , u n 0 ) δ 0 i ( u 10 , , u n 0 ) = ( 1 δ 0 ) h i ( u 10 , , u n 0 ) δ 0 h i ( u 10 , , u n 0 ) ( i = 1 , , n ) .
(31)
Moreover, since x 0 Ω , on account of (25) and (29) we have
ρ 0 = k = 1 n u k 0 C λ k 1 .
(32)
Now the equality (32), according to Notation 1, implies
| ( 1 δ 0 ) f i ( u 10 , , u n 0 ) ( t ) δ 0 f i ( u 10 , , u n 0 ) ( t ) | q i ( t , ρ 0 λ 1 i ) for a.e.  t [ a , b ] ( i = 1 , , n ) ,
(33)
| ( 1 δ 0 ) h i ( u 10 , , u n 0 ) δ 0 h i ( u 10 , , u n 0 ) | η i ( ρ 0 λ 1 i μ i ) ( i = 1 , , n ) .
(34)
Therefore, in view of Lemma 1, with respect to (30)-(34) we obtain
ρ 0 r i = 1 n ( q i ( , ρ 0 λ 1 i ) L λ i 1 + | η i ( ρ 0 λ 1 i μ i ) | λ i 1 μ i ) .

However, the latter inequality contradicts (24). □

3 Corollaries

If the operators p i and i are homogeneous, i.e. if moreover
p i ( u 1 , , u n ) ( t ) = p i ( u 1 , , u n ) ( t ) for a.e.  t [ a , b ] , ( u k ) k = 1 n C ( [ a , b ] ; R n ) ( i = 1 , , n ) ,
(35)
i ( u 1 , , u n ) = i ( u 1 , , u n ) , ( u k ) k = 1 n C ( [ a , b ] ; R n ) ( i = 1 , , n ) ,
(36)

then from Theorem 1 we obtain the following assertion.

Corollary 1 Let (5), (35), and (36) be fulfilled. If the problem
u i ( t ) = p i ( u 1 , , u n ) ( t ) for a.e.  t [ a , b ] ( i = 1 , , n ) ,
(37)
i ( u 1 , , u n ) = 0 ( i = 1 , , n )
(38)

has only the trivial solution then problem (1), (2) has at least one solution.

For a particular case when p i are defined by
p i ( u 1 , , u n ) ( t ) = def p ˜ i ( t ) | u i + 1 ( τ i ( t ) ) | λ i sgn u i + 1 ( τ i ( t ) ) for a.e.  t [ a , b ] ( i = 1 , , n 1 ) ,
(39)
p n ( u 1 , , u n ) ( t ) = def p ˜ n ( t ) | u 1 ( τ n ( t ) ) | λ n sgn u 1 ( τ n ( t ) ) for a.e.  t [ a , b ] ,
(40)

where p ˜ i L ( [ a , b ] ; R ) and τ i : [ a , b ] [ a , b ] are measurable functions, we have the following assertion.

Corollary 2 Let (5), (36), (39), and (40) be fulfilled. Let, moreover,
i = 1 n λ i = 1 ,

and let problem (37), (38) have only the trivial solution. Then problem (1), (2) has at least one solution.

Namely, for a two-dimensional system of ordinary equations and a particular case of boundary conditions we get the following.

Corollary 3 Let λ 1 λ 2 = 1 , and let
u 1 = p ˜ 1 ( t ) | u 2 | λ 1 sgn u 2 , u 2 = p ˜ 2 ( t ) | u 1 | λ 2 sgn u 1 , u 1 ( a ) c 1 u 1 ( b ) = 0 , u 2 ( a ) c 2 u 2 ( b ) = 0
with p ˜ 1 , p ˜ 2 L ( [ a , b ] ; R ) , c 1 , c 2 R have only the trivial solution. Then the problem
u 1 = p ˜ 1 ( t ) | u 2 | λ 1 sgn u 2 + f 1 ( t ) , u 2 = p ˜ 2 ( t ) | u 1 | λ 2 sgn u 1 + f 2 ( t ) , u 1 ( a ) c 1 u 1 ( b ) = h 1 , u 2 ( a ) c 2 u 2 ( b ) = h 2

has at least one solution for every f 1 , f 2 L ( [ a , b ] ; R ) and h 1 , h 2 R .

The particular case of the system discussed in Corollary 3 is so-called second-order differential equation with λ-Laplacian. Therefore, in the case when p ˜ 1 1 , Corollary 3 yields the following.

Corollary 4 Let the problem
( Φ λ ( u ( t ) ) ) = p ( t ) Φ λ ( u ( t ) ) , u ( a ) c 1 u ( b ) = 0 , u ( a ) c 2 u ( b ) = 0
with p L ( [ a , b ] ; R ) , Φ λ ( x ) = | x | λ sgn x , c 1 , c 2 R have only the trivial solution. Then the problem
( Φ λ ( u ( t ) ) ) = p ( t ) Φ λ ( u ( t ) ) + f ( t ) , u ( a ) c 1 u ( b ) = h 1 , u ( a ) c 2 u ( b ) = h 2

has at least one solution for every f L ( [ a , b ] ; R ) and h 1 , h 2 R .

Endnotes

aIf x Ω then x Ω .

bIt transforms bounded sets into relatively compact sets.

Notes

Declarations

Acknowledgements

Robert Hakl has been supported by RVO: 67985840. Manuel Zamora has been supported by Ministerio de Educación y Ciencia, Spain, project MTM2011-23652, and by a postdoctoral grant from University of Granada.

Authors’ Affiliations

(1)
Institute of Mathematics, Academy of Sciences of the Czech Republic
(2)
Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío

References

  1. Azbelev NV, Maksimov VP, Rakhmatullina LF: Introduction to the Theory of Functional Differential Equations. Nauka, Moscow; 1991.Google Scholar
  2. Azbelev NV, Maksimov VP, Rakhmatullina LF: Methods of Modern Theory of Linear Functional Differential Equations. R&C Dynamics, Moscow; 2000.Google Scholar
  3. Bravyi E: A note on the Fredholm property of boundary value problems for linear functional differential equations. Mem. Differ. Equ. Math. Phys. 2000, 20: 133-135.MathSciNetGoogle Scholar
  4. Hakl R, Lomtatidze A, Stavroulakis IP: On a boundary value problem for scalar linear functional differential equations. Abstr. Appl. Anal. 2004, 9(1):45-67.MathSciNetView ArticleGoogle Scholar
  5. Hakl R, Mukhigulashvili S: On a boundary value problem for n -th order linear functional differential systems. Georgian Math. J. 2005, 12(2):229-236.MathSciNetGoogle Scholar
  6. Kiguradze I, Půža B: On boundary value problems for systems of linear functional differential equations. Czechoslov. Math. J. 1997, 47(2):341-373. 10.1023/A:1022829931363View ArticleGoogle Scholar
  7. Kiguradze I, Půža B Folia Facult. Scien. Natur. Univ. Masarykiana Brunensis. Mathematica 12. In Boundary Value Problems for Systems of Linear Functional Differential Equations. Masaryk University, Brno; 2003.Google Scholar
  8. Schwabik Š, Tvrdý M, Vejvoda O: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia, Praha; 1979.Google Scholar
  9. Lasota A: Une généralisation du premier théorème de Fredholm et ses applications à la théorie des équations différentielles ordinaires. Ann. Pol. Math. 1966, 18: 65-77.MathSciNetGoogle Scholar
  10. Kiguradze I, Půža B, Stavroulakis IP: On singular boundary value problems for functional differential equations of higher order. Georgian Math. J. 2001, 8(4):791-814.MathSciNetGoogle Scholar
  11. Kiguradze I, Šremr J: Solvability conditions for non-local boundary value problems for two-dimensional half-linear differential systems. Nonlinear Anal. TMA 2011, 74(17):6537-6552. 10.1016/j.na.2011.06.038View ArticleGoogle Scholar
  12. Krasnosel’skii MA, Zabreiko PP: Geometricheskie metody nelineinogo analiza. Nauka, Moscow; 1975.Google Scholar
  13. Dunford N, Schwartz JT: Linear Operators: General Theory. Wiley-Interscience, New York; 1961.Google Scholar

Copyright

© Hakl and Zamora; licensee Springer. 2014

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 credited.