EXISTENCE PROBLEMS FOR HOMOCLINIC SOLUTIONS

when a→−∞ and b→ +∞. The boundary value problems on compact intervals have been studied in numerous papers but the boundary value problems on noncompact intervals have been less studied. A first substantial approach of these problems, using functional methods are due to Kartsatos [8]. Last time, this type of results has been published in [2, 3, 4, 5, 6]. For problem (1.3), Mawhin obtained many existence results through topological degree theory; in [9, 10, 11] the reader can find the fundamental ideas of the


Introduction
Let f : R × R n → R n be a continuous function; consider the boundary value problem where x(±∞) := lim t→±∞ x(t) ∈ R n . (1. 2) The solutions of problem (1.1) are often called, by Poincaré, homoclinic solutions.They appear in certain celeste mechanics and cosmogony problems.
The boundary value problems on compact intervals have been studied in numerous papers but the boundary value problems on noncompact intervals have been less studied.A first substantial approach of these problems, using functional methods are due to Kartsatos [8].Last time, this type of results has been published in [2,3,4,5,6].
For problem (1.3), Mawhin obtained many existence results through topological degree theory; in [9,10,11] the reader can find the fundamental ideas of the method developed by Mawhin, the main results, and a rich bibliography in this field.Some approaches of Mawhin dedicated to problem (1.3) can be adjusted for problem (1.1).
The present paper is dedicated to the existence of solutions for problem (1.1); the used method will be the reduction of problem (1.1) to a fixed point problem for a convenient operator defined in a suitable functional space.Such a space is (1.4) Section 2 deals especially to praise the main properties of the space C l .The specified isomorphism between C l and C([a, b], R n ) permits to obtain a compactness criterion in C l (see [1]).We define in C l the notion of an associated operator to problem (1.1) and indicate the construction method of such operator together with its main properties.An associated operator for problem (1.1) is an operator whose fixed points are solutions for (1.1).
In Section 3, assuming the existence and uniqueness on R of the solutions for the problem ẋ = f (t, x), x(0) = y, ( one builds up associated operators mapping in R n ; consequently, their topological degree will be a Brouwer one.In Section 4, the continuation method is presented (see Proposition 4.1).Through this method we obtain existence results for perturbed equations.The starting equation is chosen such that the topological degree of its associated operator is easy to be evaluated, and the perturbation is done through homogeneous or "small" functions.
For further details about the construction of the associated operators, the reader can consult [12].For the topological degree theory we recommend the delightful book [13].

Introduction.
Let f : R × R n → R n be a continuous mapping; consider problem (1.1) where x(±∞) := lim t→±∞ x(t) ∈ R n (notation used throughout this paper).
It is clear from the introduction that the aim of this paper is to find sufficient conditions to assure the existence of solutions for problem (1.1).The method will be the reduction of the existence solutions for problem (1.1) to the existence of fixed points for an adequate operator which maps in an adequate functional space.
In this section, we present the principal function spaces, their main properties, the notations, and the principal theoretical results needed in what follows.

Function spaces. Denote by
As is well known, C c is a Fréchet space endowed with the uniform convergence on compact subsets of R with the usual topology.Let C 1 c denote the linear subspace of C 1 functions in C c .
The principal function spaces are where C l and C ll are Banach spaces with respect to the norm where R n will be identified naturally with the constant functions subspace.Consider Another function space, interesting only as linear space, is the space of all Riemann integrable functions on R, where Finally, we use the spaces endowed with the usual norm x(t) . (2.7) In the case of a Banach space X, where X = C l or X = C ll , set Proof.Indeed, consider ϕ : (a, b) → R a continuous and bijective mapping; define the mapping Φ : C l → C (a,b) by the equality (2.9) It is clear that Φ is an isometric isomorphism and the proof ends.The property in Proposition 2.2 allows us to obtain a compactness criterion in C l ; obviously, it will work in C ll too, since C ll is a closed subspace of C l .

Definition 2.4. A family
(2.10) Proposition 2.5.A family A ⊂ C l is relatively compact if and only if the following three conditions are fulfilled: (ii) A is equicontinuous on every compact interval of R; (iii) A is equiconvergent.
Proposition 2.5 results immediately from the fact that the isomorphism Φ given by (2.9) transforms a set A, satisfying conditions (i), (ii), and (iii), into an equicontinuous and uniformly bounded set in C (a,b) .

Definition 2.6. A family
(2.11) , uniformly bounded on R having the family of derivatives C R -bounded, is relatively compact in C l .

Operators.
The first operator is the Nemitzky operator, F : C c → C c generated by the continuous function f : R × R n → R n and defined by (Fx)(t) := f t, x(t) . (2.12) Taking into account Remark 2.1, it results that for every solution x of (1.1) it holds (2.13) Similarly, for every solution x of (1.1), (2.17) Examples of such operators T are Tx = θ(•)x(0) or T = θ(•)x, where θ : R → R is a continuous and strictly positive mapping with +∞ −∞ θ(t) dt = 1.There exist general procedures to build up the associated operators, like the one from below having a pure algebraic character.
Let X and Z be two linear spaces, L : D(L) ⊂ X → Z a linear operator, and if and only if x is a fixed point for the operator where I is the identity operator in Z, a ∈ R, a = 0, and K is the right inverse of L (more precisely K = (L | D(L)∩N(P) ) −1 ).This result has been successfully used by Mawhin for the building of associated operators to the boundary value problem (closely related to the periodic solutions problem [9,10,11]) (2.20) By this model in the next subsection, we briefly describe how can we construct associated operators to problem (1.1) and their properties.

Construction of associated operators.
The form of associated operators depends firstly on the fundamental space X and next on the space Z and on the choice of the operators L, N and the choice of the projectors P, Q; only after this K can be determined and also the final form of the operator U. Having so many arbitrary elements we can find many associated operators.
In what follows, we sketch the building of associated operators in two important cases: X = C l and X = C ll ; further details about the construction can be found in [4,12].
In the case X = C l we distinguish three subcases related to L and N; this choice must be made such that the equation (L, N) does contain (1.1).The expression of projectors P and Q depends on the considered case.
In all three cases, we have In this case P = Q = 0, so the operator L is invertible and therefore U = L −1 N.This case gives us the easiest associated operators, and the symmetric form (2.25) For P, we take or (2.28) For U, we can construct (2.30) In this case, and hence the projector Q must be changed; we can take for example and therefore, and other more complicated forms.
In the case We can take, for example, In general, in this case the expression of U is more complicated since all its values must be in C ll .For example, for (2.26) we get and with (2.27), where e(t) (2.38)

Admissible operators.
It is obvious that this construction of the associated operators has an algebraic character; the condition is sufficient for the existence of these operators, but it is not sufficient to confer their important topological properties.
Definition 2.9.An associated operator on the set D ⊂ X to problem (1.1), constructed as in Section 2.5, is called admissible if and only if Proposition 2.10.Let X be a subspace of C l and D ⊂ C l be a bounded subset.If FD is C R -bounded, then every associated operator constructed as in Section 2.5 is compact.
The proof of this proposition is complicated in calculus, but it is basically an easy application of the elementary known properties of uniform convergence, which allows to establish immediately the continuity of the operator U which contains finite rank projectors and application of type The compactness of the operator U is an immediate consequence of Corollary 2.7.At least for the operators U given by (2.22), (2.23), (2.29), (2.30), (2.34), (2.37), and (2.38) the verification of compactity is immediate.
Remark that if f satisfies the condition where where The situation is more complicated in the case when (1.1) proceeds from a second-order equation where h : , where Therefore, the boundary value problem defining the homoclinic solutions for (2.34) has the form (2.50) We give an example to obtain the C R -boundedness of F(D) in this case.
Let α 1 ,α 2 ∈ C R , α 1 ,α 2 be positive; in addition, suppose that α 2 (±∞) = 0. Let γ, β : R → R be two continuous and positive functions.We take as fundamental space X = C l × C Rl , where Let D 1 be a bounded set in C R and let where It is easy to check that if The case of second-order equation is different from the first-order equation; this is why it will not be treated here, but it will make the object of a future note.

Remarks on the topological degree of the admissible operators.
Let Ω ⊂ X be an open and bounded set, where X is C l or C ll .
Suppose that F(Ω) is C R -bounded, for an admissible operator U, if where ∂Ω is the boundary of Ω, we can consider its topological degree If this degree is nonzero, then U admits fixed points and so problem (1.1) has solutions.
As we said, the results contained in this section are based on the ones by Mawhin related to the boundary value problem This author proves that the associated operators to problem (2.55) in the space C (0,T) or C [0,T] are compact on the bounded sets without supplementary conditions on the mapping f as it was to be expected.Moreover, these operators have the same topological degree which does not depend on the choice of L, N, P, Q.In addition, if in particular f (t, x) = g(x), then for each associated operator U to problem (2.55) on the bounded and open set Ω from C (0,T) (or C [0,T] ), we have where deg B denotes the Brouwer degree.
The associated operators to problem (1.1) on Ω from C l or C ll have the degrees invariant with respect to L, N, P, Q; the proof, based on the invariance of topological degree to homeomorphisms, is essentially simple but complicated to achieve.As we do not use this property in the present paper, we renounce to its proof.
Cezar Avramescu 11 Finally we make only a remark on the isomorphism Φ given by (2.9).Let Ω ⊂ X be an open and bounded set in X (X = C l or C ll ) and let U be an admissible operator on Ω for problem (1.1) fulfilling (2.56). Set where Φ is given by (2.9) with a = 0, b = T.
Then Ω Φ is open and bounded, Φ(∂Ω) = ∂Ω Φ and Ω Φ ⊂ C (0,T) (resp., Furthermore, U Φ is compact and since ∂Ω Φ = Φ(∂Ω), we have has a unique solution defined on the whole real axis R, for every G a bounded set in R n and for every y ∈ G; denote the solution of (3.1) by The uniqueness condition is fulfilled in particular if f (t, x) is locally Lipschitz with respect to x. Condition (2.41) is sufficient to assure the existence on R of the solution (3.2), it is in particular fulfilled in conditions of type (2.41) and even more general.
It is known that the uniqueness condition assures the continuous dependence of the function x(t; •); this property would be stated as: for every [a, b] ⊂ R and for every y n ∈ G, y n → y ∈ G, the sequence x(t; y n ) converges uniformly on [a, b] to x(t; y).
In this section, we present certain existence results for problem (1.1), exploiting this continuous dependence with respect to initial data.

Generalized Poincaré operator.
Let Ω ⊂ C l be a bounded and open set; let By hypothesis (i) it results that the integral in (3.8) is uniformly convergent with respect to y ∈ Ḡ; on the other hand, since the mapping y → x(•; y) is continuous (as mentioned in the previous paragraph) we conclude the continuity of the mapping (t, y) → f (t, x(t; y)) on every set of type [−A, A] × Ḡ. Hence the mapping y → P y is continuous on Ḡ.
Define the application h : Ḡ Cezar Avramescu 13 it follows that h is continuous.
If for y ∈ ∂G we have P y = 0, then x(•; y) is a solution for (1.1).Suppose then P y = 0, for every y ∈ ∂G; by hypotheses (ii) and (iii) it results that h(y, λ) = 0, ∀λ ∈ [0, 1], ∀y ∈ ∂G. (3.12) By homotopic invariance property of the topological degree it results that deg and hence, by (3.6) which assures the existence of y ∈ G with P y = 0.The theorem is proved.

3.3.
The case Ω connected.The advantage of the previous result is that the topological degrees appearing are Brouwer degrees; the drawback is that condition (3.4) is not easy to be checked.We state now another existence result.
As usual, suppose that Ω ⊂ C l is a bounded and open set; define on Ω the operators (3.17)If x = z, then there exists t 0 ∈ R such that x(t 0 ) = z(t 0 ); we can assume that t 0 > 0. Let A > 0 be such that t 0 ∈ [0,A] and r = max{ x ∞ , y ∞ }. Since we obtain by using Gronwall's lemma Remark 3.3.The mapping S : Ω → S( Ω) is a homeomorphism.In addition, since H is a compact operator, S −1 is a compact perturbation of identity, too.Observe that the operator is just the admissible operator (2.22), where b = 0 and a = 1/2.
Remark 3.5.The following identity holds: (iv) the following relations hold: In addition, if

Existence results using Miranda
then the equation admits solutions in K. Suppose that f satisfies the following hypotheses: (H 1 ) for every l > 0, problem (3.1) has a unique solution defined on the whole R, for every y ∈ K; ) there exists a constant c > 0 such that for every i ∈ 1,n and for every (t, y) ∈ R × R n with |y i | > c, we have Theorem 3.9.Assume that the hypotheses (H 1 ), (H 2 ), and (H 3 ) are fulfilled.Then problem (1.1) admits solutions.
Proof.Consider the operator P on K given by (3.8), that is, The operator P is well defined since hypotheses (H 1 ), (H 2 ), and (H 3 ) are assumed; in addition, as remarked, it is continuous on R n .Set (3.39) Then we have for every solution x(t; y) = (x i (t; y)) i∈1,n , Considering l ≥ 0 such that we obtain If relation (3.41) is fulfilled, it follows from (H 3 ), where P = (P i ) i∈1,n .

Cezar Avramescu 17
Applying Miranda's theorem, it results that P has a zero in K.The proof is now complete.

Continuation method
4.1.Introduction.In this section f : R × R n → R is a continuous function, X is the space C l or C ll and Ω ⊂ X is an open and bounded set.If F Ω is C R -bounded, as remarked in Section 2, one can associate to problem (1.1) operators U : Ω → X which are compact and whose fixed points coincide with the solutions of (1.1).
In particular, if then we can define the topological degree of U and if then U admits fixed points in Ω.
However, when we face to check condition (4.2), then we can use the so-called continuation method, which is based on the well-known homotopic invariance property of the topological degree (used in Section 3).
One of the most used forms of this method is the following.Let h : R × R n × [0, 1] → R n be a continuous and C R -bounded on Ω function in the sense that there exists θ ∈ C R , θ > 0, such that for every x ∈ Ω and for every λ ∈ [0, 1] we have |h(t, x(t),λ)| ≤ θ(t), t ∈ R.
Consider the problem We can associate to problem (4.3) an operator U λ which in addition is compact for every λ.
If the condition is fulfilled, then we can define the degree deg(I − U λ , Ω, 0); but a homotopic invariance property tells us that this degree is constant with respect to λ.In particular, Equality (4.5) is useful if U 0 is an associated operator to problem (1.1) (h(t, x, 0) = f (t, x)) and the degree of I − U 1 is easier to be computed, for example, when it is a Brouwer degree.
Condition (4.4) can be formulated under the following form: for every λ ∈ [0, 1] problem (4.3) has no solutions x(•; λ) with x ∈ ∂Ω.If this condition is fulfilled, every associated operator U λ satisfies (4.4) because the fixed points of an associated operator coincide with the set of solutions for the problem whose it is associated.
We get therefore the following proposition.
The question that problem (4.3) has no solutions in ∂Ω can be formulated under the following form.
"A priori estimates": for every possible solution x(•) of problem (4.3) with x ∈ Ω we have x ∈ Ω.
Another form of the same condition is the next."A priori bound": there exists a number r > 0 such that problem (4.3) does not admit solutions x(•) with x ∞ = r.
In this case we set Ω := {x ∈ X, x < r}.
Another variant of the same condition is the following."Bounded set condition": for every λ ∈ [0, 1] for which problem (4.3) has solutions x(•) with x(t) ∈ D, t ∈ R, we have x(t) ∈ D, for every t ∈ R.
In this case when D ⊂ R n is an open and bounded set we take Ω :={x ∈ X, x(t) ∈ D}.
In this section, we indicate certain simple functions candidates to be homotopic linked through h with f , functions for which the computation of their topological degree is more advantageously.
The most difficult problem remains to establish the fact that problem (4.3) has no solutions in ∂Ω; in what follows we consider certain cases when this thing is easy to be checked.

Homotopy with a linear equation. In this paragraph consider X
Let A : R → M n (R) be a continuous quadratic matrix; denote by | • | an arbitrary norm for the constant matrices.
Consider the system and denote by X = X(t) its fundamental matrix with X(0) = I.In [5], the following result is proved.

Auxiliary results.
In Section 4.2, the homotopy has been achieved through a linear mapping for which it was easy to evaluate its topological degree.We give rise to another case when the topological degree computation is not too difficult in the sense that it becomes a Brouwer degree.This result will be a consequence of a more general result which links the existence of solutions for problem (1.1) to the existence of solutions for the problems of the type ẏ = g(t, y), y(0) = y(T), 0 < T < ∞. (4.28) Let θ : R → R, θ ∈ C R , θ(t) > 0, for every t ∈ R; set Obviously, through (2.9), ϕ : (0,T) → R determines by (2.9) an isomorphism between C l and C (0,T) (or between C ll and C [0,T] ).Let x(t) be a solution for (1.1); then y = Φ(x) is a solution for the differential equation appearing in (4.28) on the interval (0,T).Since y(t) has limits in 0 and T, it can be prolonged as solution on [0,T]; but by definition of y(t) it follows that y(0) = ϕ x(−∞) = ϕ x(+∞) = y(T).But the operator U Φ is associated to problem (4.28).By using the remarks from 2.4 we obtain the following result.(Obviously, the first degree is computed in C l , the second in C (0,T) .)

Cezar Avramescu 23
An important particular case is f (t, x) = θ(t) • g(x).As it is proved in [7], for every associated operator to problem (4.41) in C (0,T) or C [0,T] (so for U Φ , too) we have deg

Remark 2 . 3 .
The same mapping Φ is an isomorphism between C ll and C [a,b] .

Corollary 4 . 6 .
If x = Ux, for every x ∈ ∂Ω, then deg(I − U, Ω, 0) = deg I − U Φ , Ω Φ , 0 .(4.39) this case, when (4.30) is fulfilled, (4.24) becomes ẏ = g(y), y(0) = y(T).(4.41) | • | an arbitrary norm in R n and .14) In what follows, X ⊆ C l denotes a closed subspace of C l and D ⊂ X is a void set.Define on D an important category of operators called associated.Definition 2.8.The operator U : D ⊂ X → X is associated to problem (1.1) on the set D if and only if every fixed point of U is a solution for problem (1.1).
R × R n → R n and p : R n → R n are continuous functions.
.26) By Corollary 2.7, we get the compactness of the sequence(u k ) k in C ll .Let u ∈ (u k ) k , λ ∈ (λ k ) k; by using the classical properties of uniform convergence we obtain, after computations, .30) the convergence being uniform with respect to y on every compact subset ofR n .Remark that if D ⊂ C l is a bounded set, then FD ⊂ C R ; indeed, for |t| ≥ A, + (u), γ − (u), u ∈ D ∩ R n .