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

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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.

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 ] } ;

AC([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)|ds;

L([a,b]; R + )={pL([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 AC([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)
(1δ) 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, ΩX be a symmetrica bounded domain with 0intΩ. 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 xΩ

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 AC([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 mN there exist ( u i m ) i = 1 n AC([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 mN,
(12)
v i m (t)= u i m ( t ) ρ m λ 1 i for t[a,b],mN.
(13)

Then

i = 1 n v i m C λ i 1 =1for mN
(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;mN).
(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 mN,
(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 ds=0uniformly for mN(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 AC([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 xX, 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:XX 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)xν ( A ( x ) + x ) for xΩ,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 AC([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 pL([a,b];R), Φ λ (x)= | x | λ sgnx, 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 fL([a,b];R) and h 1 , h 2 R.

Endnotes

aIf xΩ then xΩ.

bIt transforms bounded sets into relatively compact sets.

References

  1. 1.

    Azbelev NV, Maksimov VP, Rakhmatullina LF: Introduction to the Theory of Functional Differential Equations. Nauka, Moscow; 1991.

  2. 2.

    Azbelev NV, Maksimov VP, Rakhmatullina LF: Methods of Modern Theory of Linear Functional Differential Equations. R&C Dynamics, Moscow; 2000.

  3. 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.

  4. 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.

  5. 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.

  6. 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:1022829931363

  7. 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.

  8. 8.

    Schwabik Š, Tvrdý M, Vejvoda O: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia, Praha; 1979.

  9. 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.

  10. 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.

  11. 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.2011.06.038

  12. 12.

    Krasnosel’skii MA, Zabreiko PP: Geometricheskie metody nelineinogo analiza. Nauka, Moscow; 1975.

  13. 13.

    Dunford N, Schwartz JT: Linear Operators: General Theory. Wiley-Interscience, New York; 1961.

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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.

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Correspondence to Robert Hakl.

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Dedicated to Professor Ivan Kiguradze.

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The authors declare that they have no competing interests.

Authors’ contributions

RH and MZ obtained the results in a joint research. Both authors read and approved the final manuscript.

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Keywords

  • functional-differential equations
  • boundary value problems
  • existence of solutions