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

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.

Consider the system of functional-differential equations

together with the boundary conditions

Here, p_i,f_i:C([a,b];{\mathbb{R}}^n)\to L([a,b];\mathbb{R}) are continuous operators satisfying Carathéodory conditions, i.e. for every r>0 there exists q_r\in L([a,b];{\mathbb{R}}_{+}) such that

and {\ell }_i,h_i:C([a,b];{\mathbb{R}}^n)\to \mathbb{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

Furthermore, we assume that p_i and {\ell }_i satisfy the following condition: there exist positive real numbers {\lambda }_{ij} and {\mu }_i such that {\lambda }_{ij}{\lambda }_{jm}={\lambda }_{im} whenever i,j,m\in \{1,\dots ,n\}, and for every c>0 and {(u_k)}_{k=1}^n\in C([a,b];{\mathbb{R}}^n) we have

Remark 1 From the above-stated assumptions it follows that {\lambda }_{ii}=1, {\lambda }_{ij}=1/{\lambda }_{ji} for every i,j\in \{1,\dots ,n\}.

In the case when p_i and {\ell }_i are linear bounded operators and f_i(\cdot ,\dots ,\cdot )(t)\equiv q_i(t), h_i(\cdot ,\dots ,\cdot )\equiv 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. [1–8] and references therein.

In 1966, Lasota established the Fredholm-type theorem in the case when p_i and {\ell }_i are homogeneous operators (see [9]). Recently, Fredholm-type theorems in the case when p_i and {\ell }_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 {\ell }_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.

ℕ is the set of all natural numbers;

ℝ is the set of all real numbers, {\mathbb{R}}_{+}=[0,+\mathrm{\infty });

{\mathbb{R}}^n is the linear space of vectors x={(x_i)}_{i=1}^n with the elements x_i\in \mathbb{R} endowed with the norm

C([a,b];{\mathbb{R}}^n) is the Banach space of continuous vector-valued functions u={(u_i)}_{i=1}^n:[a,b]\to {\mathbb{R}}^n with the norm

AC([a,b];{\mathbb{R}}^n) is the set of absolutely continuous vector-valued functions u:[a,b]\to {\mathbb{R}}^n;

L([a,b];\mathbb{R}) is the Banach space of Lebesgue integrable functions p:[a,b]\to \mathbb{R} with the norm

L([a,b];{\mathbb{R}}_{+})=\{p\in L([a,b];\mathbb{R}):p(t)\ge 0\text{ for a.e. }t\in [a,b]\};

if Ω is a set then measΩ, intΩ, \overline{\mathrm{\Omega }}, 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\in AC([a,b];{\mathbb{R}}^n) satisfying (1) almost everywhere in [a,b] and (2).

Notation 1 Define, for every i\in \{1,\dots ,n\}, the following functions:

Theorem 1 Let

If the problem

has only the trivial solution for every \delta \in [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, \mathrm{\Omega }\subseteq X be a symmetric^a bounded domain with 0\in int\mathrm{\Omega }. Let, moreover, A:\overline{\mathrm{\Omega }}\to \overline{\mathrm{\Omega }} be a compact^b continuous operator which has no fixed point on ∂ Ω. If, in addition,

then A has a fixed point in Ω, i.e. there exists x_0\in \mathrm{\Omega } such that x_0=A(x_0).

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

Lemma 1 Let, for every \delta \in [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\in AC([a,b];{\mathbb{R}}^n) and any \delta \in [0,1/2], the a priori estimate

holds, where

Proof Suppose on the contrary that for every m\in \mathbb{N} there exist {(u_{im})}_{i=1}^n\in AC([a,b];{\mathbb{R}}^n) and {\delta }_m\in [0,1/2] such that

where

Put

Then

and from (10) and (11), in view of (3), (4), (12), and (13), we get

On the other hand, from (9) and (12) we have

whence, according to [[13], Corollary IV.8.11] it follows that

Therefore, (14), (15), and (18) imply that the sequences {\{v_{im}\}}_{m=1}^{+\mathrm{\infty }} (i=1,\dots ,n) are uniformly bounded and equicontinuous. Thus, according to Arzelà-Ascoli theorem, without loss of generality we can assume that there exist {(v_{i0})}_{i=1}^n\in C([a,b];{\mathbb{R}}^n) and {\delta }_0\in [0,1/2] such that

Furthermore, (15)-(17) yield {(v_{i0})}_{i=1}^n\in AC([a,b];{\mathbb{R}}^n) and show that it is a solution to (6), (7). However, (14) and (19) result in

which contradicts our assumptions. □

Proof of Theorem 1 Let X=C([a,b];{\mathbb{R}}^n)\times {\mathbb{R}}^n and for x\in X, i.e. x=(u,\alpha )=({(u_i)}_{i=1}^n,{({\alpha }_i)}_{i=1}^n), define the norm

Then (X,\parallel \cdot \parallel ) is a Banach space. Let the operators T,F,A:X\to X be defined as follows:

and consider the operator equation

It can easily be seen that problem (1), (2), and (23) are equivalent in the following sense: if x=(u,\alpha ) is a solution to (23), then {\alpha }_i=0 (i=1,\dots ,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 {\rho }_0>0 such that

Let, moreover,

Now we will show that the operator A has a fixed point in Ω. According to Theorem 2 it is sufficient to show that

Assume on the contrary that there exist x_0=({(u_{i0})}_{i=1}^n,{({\alpha }_{i0})}_{i=1}^n)\in \partial \mathrm{\Omega } and {\nu }_0\in (0,1] such that

Then from (26), in view of (20)-(22) we obtain

where {\delta }_0={\nu }_0/(1+{\nu }_0)\in (0,1/2], i.e.

Now from (27) and (28) it follows that {(u_i)}_{i=1}^n\in AC([a,b];{\mathbb{R}}^n),

Moreover, since x_0\in \partial \mathrm{\Omega }, on account of (25) and (29) we have

Now the equality (32), according to Notation 1, implies

Therefore, in view of Lemma 1, with respect to (30)-(34) we obtain

However, the latter inequality contradicts (24). □

If the operators p_i and {\ell }_i are homogeneous, i.e. if moreover

then from Theorem 1 we obtain the following assertion.

Corollary 1 Let (5), (35), and (36) be fulfilled. If the problem

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

For a particular case when p_i are defined by

where {\tilde{p}}_i\in L([a,b];\mathbb{R}) and {\tau }_i:[a,b]\to [a,b] are measurable functions, we have the following assertion.

Corollary 2 Let (5), (36), (39), and (40) be fulfilled. Let, moreover,

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 {\lambda }_1{\lambda }_2=1, and let

with {\tilde{p}}_1,{\tilde{p}}_2\in L([a,b];\mathbb{R}), c_1,c_2\in \mathbb{R} have only the trivial solution. Then the problem

has at least one solution for every f_1,f_2\in L([a,b];\mathbb{R}) and h_1,h_2\in \mathbb{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 {\tilde{p}}_1\equiv 1, Corollary 3 yields the following.

Corollary 4 Let the problem

with p\in L([a,b];\mathbb{R}), {\mathrm{\Phi }}_{\lambda }(x)={|x|}^{\lambda }sgnx, c_1,c_2\in \mathbb{R} have only the trivial solution. Then the problem

has at least one solution for every f\in L([a,b];\mathbb{R}) and h_1,h_2\in \mathbb{R}.

^aIf x\in \mathrm{\Omega } then -x\in \mathrm{\Omega }.

^bIt transforms bounded sets into relatively compact sets.

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.

Affiliations

1. Institute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Žižkova 22, Brno, 61662, Czech Republic

   • Robert Hakl

2. Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, 4051381, Chile

   • Manuel Zamora

Corresponding author

Correspondence to Robert Hakl.

Dedicated to Professor Ivan Kiguradze.

Competing interests

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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