Index at infinity for vector fields with oscillating nonlinearities

Krasnosel’skii A.M.,^{∗}  Mawhin. J.^{†}

March 27, 1998

1 Introduction

This paper concerns an index at infinity calculation for some asymptotically linear completely continuous vector fields x − T (x). If the main linear part x − Ax of the field is non-degenerate, then the index at infinity of x − T (x) is well-defined by the spectrum of this linear part. If this main linear part is degenerate (1 is an eigenvalue of the linear operator A), the index calculation can be done only with the use of some properties of the sublinear part of T (x).

If this sublinear part is asymptotically homogeneous (e.g. satisfies some Landesman – Lazer type conditions), then the calculation of the index can be reduced to the calculation of the degree of some finite dimensional vector field Q (see [1]). If this vector field Q is also degenerate, one can use higher order terms.

Here in this paper we consider the situation where the sublinear terms are not asymptotically homogeneous. We have here the following possibilities: the index is defined and its calculation can be reduced to the index calculation of a finite dimensional field; the index can not be defined (there are some sequence x_{n} of zeros and ∥x_{n}∥ → ∞); known information is insufficient to get any answer and we need to use higher order terms. The last possibility has in some natural sense “zero measure”.

In this paper we consider only the case of one-dimensional degeneration of the linear part: 1 is a simple eigenvalue of the linear operator A. For this case we formulate an abstract theorem and give some general examples for vector fields of Hammerstein type and for a two point boundary value problem. The paper is organized as follows: in Sections 2 and 3 we formulate the main abstract result, in Section 4 we discuss the main condition (1), applications are given in Section 5, the proofs in Section 6 – 8 and some remarks in Section 9.

∗Institute for Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetny lane, 101447 Moscow, Russia; e-mail: amk@ippi.ac.msk.su . This paper was written during the visit of A.M.Krasnosel’skii to Louvain-La-Neuve, Belgium at 1997. A.M.Krasnosel’skii was partially supported by Grants 97-01-00692 and 96-15-96048 of Russian Foundation of Fundamental Researchs

†Institut de Mathématique Pure et Appliquée, Université Catholique de Louvain, B-1348, Louvain-La-Neuve, Belgium; e-mail: mawhin@amm.ucl.ac.be

2 Index at infinity

Consider a vector field Φx = x−T (x) in a Banach space E. We suppose that the operator T (x) is completely continuous, in which case the vector field Φ(x) is also called completely continuous. Suppose that x ≠ T (x) for ∥x∥ ≥ ρ. Then the rotation (see [2]) of this field on the boundary of every ball B(r, 0) = {x ∈ E, ∥x∥ ≤ r} is defined for every r ≥ ρ. This rotation has a common value, called the index at infinity of the field Φ(x) and is denoted as ind_{∞} Φ.

The index at infinity can be used for the study of the equation x = T (x): if ind_{∞} Φ is defined (x ≠ T (x) for ∥x∥ ≥ ρ for some ρ) and ind_{∞} Φ ≠ 0, then there exists at least one solution of the equation x = T (x).

The homotopic invariance of the index makes the index applicable for the study of unbounded branches of solutions or for the study of asymptotic bifurcation points for equations with a parameter ([2]).

Consider the equation x = T (x; λ) with a real parameter λ ∈ Λ = [a, b]. Let the operator T (x; λ) be completely continuous with respect to both variables. If the index at infinity of the field Φ_{λ} = x − T (x; λ) is defined for two distinct values λ_{1} and λ_{2} and ind_{∞} Φ_{λ_{1}} ≠ ind_{∞} Φ_{λ_{2}}, then on the interval [λ_{1}, λ_{2}] ∈ Λ there exists an unbounded branch of solutions: for any R > 0 there exist λ_{R} ∈ (λ_{1}, λ_{2}) and x_{R}, ∥x_{R}∥ ≥ R such that x_{R} = T (x_{R}; λ_{R}). This branch is continuous in a natural sense. In other words, if ind_{∞} Φ_{λ_{1}} ≠ ind_{∞} Φ_{λ_{2}}, then there exists at least one asymptotic bifurcation point on [λ_{1}, λ_{2}], which can be unique or not. In this paper we describe situations, where asymptotic bifurcation points fill some nontrivial interval J ⊂ [λ_{1}, λ_{2}].

3 Abstract results

Consider a Banach space E and a completely continuous vector field x − T (x). Suppose that T (x) = Ax + F (x), where A is a linear operator and F (x) is bounded nonlinearity: ∥F (x)∥ ≤ r_{1}, x ∈ E.

Suppose 1 is a simple eigenvalue of the linear operator A, let Ae = e, ∥e∥ = 1. There exists a unique projector P on the 1-dimensional subspace E_{1} = {αe, α ∈ IR}, which commutes with A. Denote Q = I − P, E_{2} = QE, the subspace E_{2} has co-dimension 1, it is invariant for A, and in E_{2} the value 1 is regular for the operator A. Denote by p(x) : E → E_{1} the functional defined by the relation P x = p(x)e.

Suppose another Banach space Ẽ ⊂ E is given with a stronger norm: ∥x∥_{E} ≤ const ∥x∥Ẽ , x ∈ E.

Now we can formulate the main assumption for the nonlinearity F (x): for any c > 0

lim_{|ξ|→∞} sup_{∥h∥Ẽ≤c} |p(F (ξe + h)) − p(F (ξe))| = 0.  (1)

This property will be discussed below, and is valid for various classes of concrete nonlinear operators which appear in applications.

Consider the scalar function

Ψ(ξ) = p(F (ξe)),

which is bounded and continuous. Consider the four numbers:

ψ_{±} = lim inf_{ξ→±∞} Ψ(ξ), ψ^{±} = lim sup_{ξ→±∞} Ψ(ξ).  (2)

Theorem 1. Let the operator A act continuously from E to Ẽ. Let the nonlinearity F (x) also act from E to Ẽ and let it be bounded in Ẽ: ∥F (x)∥Ẽ ≤ r_{2}, x ∈ E. Let (1) be valid for any c > 0. Then

a) if either ψ^{+} < 0 < ψ_{−}, or ψ^{−} < 0 < ψ_{+}, then the index at infinity of the field Φ(x) is defined and | ind_{∞} Φ| = 1;

b) if either ψ_{−} > 0 and ψ_{+} > 0, or ψ^{−} < 0 and ψ^{+} < 0, then the index at infinity of the field Φ(x) is defined and ind_{∞} Φ = 0;

c) if either ψ_{−} < 0 < ψ^{−}, or ψ_{+} < 0 < ψ^{+}, or both, then the index at infinity of the field Φ(x) can not be defined: there exists a sequence x_{n} ∈ E such that x_{n} = T (x_{n}) and ∥x_{n}∥_{E} → ∞.

If this theorem is inapplicable, then at least one of the numbers (2) has to be zero. Even if ψ_{−} · ψ^{−} = 0 (ψ_{+} · ψ^{+} = 0) Theorem 1 can be applicable if ψ_{+} < 0 < ψ^{+} (ψ_{−} < 0 < ψ^{−}).

Under the assumptions of Theorem 1, if ψ_{+} < 0 < ψ^{+}, then the sequence x_{n} has the form x_{n} = ξ_{n}e+h_{n}, where h_{n} ∈ E_{2} are uniformly bounded and ξ_{n} → +∞. If ψ_{−} < 0 < ψ^{−}, then ξ_{n} → −∞; if 0 ∈ (ψ_{−}, ψ^{−}) ∩ (ψ_{+}, ψ^{+}), then there are two sequences x_{n}^{±} = ξ_{n}^{±}e + h_{n}^{±}, where ξ_{n}^{±} → ±∞.

If ψ_{−} = ψ^{−} and ψ_{+} = ψ^{+}, then the condition c) of Theorem 1 can not take place, this is the case, when the nonlinearity is asymptotically homogeneous in the sense of [1].

4 Hammerstein type nonlinearities

In this section we formulate some sufficient conditions for the validity of the assumption (1) and the corresponding results for the index at infinity calculation of Hammerstein type vector fields in functional spaces.

Let Ω ⊂ IR^{m} be a domain and let E = L^{2} = L^{2}(Ω, IR^{n}).

Suppose that the nonlinearity F (x) is a Hammerstein type operator generated by the linear operator A (let us recall that we study the vector field Φ(x) = x − T (x) = x−Ax−F (x) where A is the asymptotic derivative of T (x)) and some bounded continuous function f(t, x) : Ω × IR^{n} → IR^{n}:

F (x) = Af(t, x).

Theorem 2. Let Ẽ = E and

mes{t ∈ Ω : e(t) = 0} = 0.  (3)

Let the function f(t, x) satisfy the Lipschitz condition

∥f(t, x) − f(t, y)∥_{I}_{R}_{n} ≤ L(r) ∥x − y∥_{I}_{R}_{n} ,  ∥x∥_{I}_{R}_{n} , ∥y∥_{I}_{R}_{n} > r  (4)

where

lim_{r→∞} L(r) = 0.  (5)

Then condition (1) is valid for any c > 0, and moreover

lim_{|ξ|→∞} sup_{∥h∥_{L^{2}}≤c} ∥f(ξe(t) + h(t)) − f(ξe(t))∥_{L^{2}} = 0.  (6)

For scalar functions e(t), defined for Ω = [a, b] ⊂ IR it is possible to obtain a less restrictive result.

Suppose the linear projector P x is generated by the function g(t) ∈ L^{2}:

P x = e(t) ∫_{a}^{b} g(s)x(s) ds.

Theorem 3. Let E = L^{2} = L^{2}([a, b], IR) and Ẽ = C^{1} = C^{1}([a, b], IR). Let e(t) ∈ C^{1} and

mes{t ∈ [a, b] : e′(t) = 0} = 0.  (7)

Let the function f(t, x) be uniformly continuous in t with respect to x ∈ IR. Then condition (1) is valid for any c > 0:

lim_{|ξ|→∞} sup_{∥h(t)∥_{C^{1}}≤c} |∫_{a}^{b} g(t)[f(t, ξe(t) + h(t)) − f(t, ξe(t))] dt| = 0.  (8)

Theorem 3 does not contain any Lipschitz conditions like (4) – (5), which, of course, are very restrictive. The bounded function f(t, x) is uniformly continuous in t with respect to x for example if f(t, x) = a(t) + b(t)f(x).

From the proof given below one can see that Theorem 3 can be generalized for the case E = L^{2}(Ω, IR^{n}), Ω ⊂ IR^{m} where m ≥ n. It seems that for m < n we have essential difficulties and need some additional hypotheses for f(t, x).

Theorem 1 and Theorems 2 and 3 allow us to formulate statements for the index at infinity calculation of Hammerstein type vector fields in functional spaces. Corollary 1. Let E = L^{2}(Ω, IR^{n}) where Ω is a domain in IR^{m}. Let the linear completely continuous operator A act in E, let 1 be a simple eigenvalue for A, let Ae(t) = e(t) and let condition (3) be valid. Let a bounded continuous function f(t, x) : Ω × IR^{n} → IR^{n} satisfy (4) and (5). Then for the vector field Φ(x) = x − A(x + f(t, x(t))) all conclusions of Theorem 1 are valid.

Corollary 2. Let E = L^{2}([0, π], IR). Let the linear completely continuous operator A : E → E act continuously from E to the space C^{1}(0, π). Let 1 be a simple eigenvalue for A, let Ae(t) = e(t) and let condition (7) be valid. Let a bounded continuous function f(t, x) : Ω × IR → IR be uniformly continuous in t with respect to x. Then for the vector field Φ(x) = x − A(x + f(t, x(t))) all conclusions of Theorem 1 are valid.

Corollary 1 is a direct consequence of Theorems 1 and 2, corollary 2 is a direct consequence of Theorems 1 and 3.

5 Two point boundary value problem

In this section we give some results for two point BVP at resonance. Consider the problem

x′′ + n^{2}x + f(t, x) = 0, x(0) = x(π) = 0  (9)

for some positive integer n and for some bounded continuous function f(t, x) : [0, π]×IR → IR. Suppose that f(t, x) is uniformly continuous in t with respect to x. Put

e(t) = \sqrt{\frac{2}{π}} sin nt, Ψ(ξ) = ∫_{0}^{π} e(t) f(t, ξe(t)) dt

and define numbers (2). Theorem 4. If either ψ^{+} < 0 < ψ_{−}, or ψ^{−} < 0 < ψ_{+}, then problem (9) has at least one solution, the set K of all solutions of (9) is bounded. If either ψ_{−} < 0 < ψ^{−}, or ψ_{+} < 0 < ψ^{+}, or both, then problem (9) has an infinite number of solutions, the set K is unbounded.

This theorem follows directly from Corollary 2.

We illustrate this theorem with the description of the set of solutions of the problem

x′′ + n^{2}x + f(t, x) − λe(t) = 0, x(0) = x(π) = 0  (10)

with a scalar parameter λ. Define the numbers ψ_{−}, ψ^{−}, ψ_{+}, ψ^{+} as in Theorem 4 and let ψ_{−} < ψ^{−} < 0 < ψ_{+} < ψ^{+}. For every ξ ∈ IR we can find at least one λ and a nonempty set H(ξ) of functions h(t) such that x(t) = ξe(t) + h(t) is a solution of problem (10).

On the plane {λ, ξ} the graph of the mapping ξ ↦ λ can be found at Fig.1.

[figure: λ; ξ; ψ_{−}; ψ^{−}; ψ_{+}; ψ^{+}; Fig.1]

Here the set [ψ_{−}, ψ^{−}] ∪ [ψ_{+}, ψ^{+}] is the set of asymptotic bifurcation points for the problem (10). Of course, the behaviour of the graph for small values of ξ can be much more cumbersome, as we had no assumptions concerning the behaviour of the function f(t, x) for small x. If f′_{x}(t, x) is uniformly small enough (the constant depends only on n), then for any ξ there exist unique h(t) and unique λ such that x(t) = ξe(t) + h(t) is a solution of problem (10).

The natural question arises: how to compute the numbers (2)? We want to mention, that if f(t, x) = f_{1}(t, x) + f_{2}(t, x) and

lim_{|ξ|→∞} ∫_{Ω} g(t) f_{2}(t, ξe(t)) dt = 0,  (11)

then numbers (2) computed for the functions f(t, x) and f_{1}(t, x) coincide. If the limits

lim_{ξ→±∞} ∫_{Ω} g(t) f_{2}(t, ξe(t)) dt = ψ̃(±)  (12)

exist, then they are the difference between numbers (2) computed for the functions f(t, x) and the same numbers, computed for f_{1}(t, x).

This simple idea allows us to split the initial function f(t, x) into a sum of functions such that it is possible either to see that (11) is valid or limits (12) exist and for one last function we need to do computations of different upper and lower limits.

Relation (11) is valid for various functions f_{2}(t, x). Let us give some examples.

1. Let f_{2}(t, x) → 0 as |x| → ∞ uniformly with respect to t. Let e(t) satisfy (3). Then (11) is valid.

2. Let f_{2}(t, x) satisfy some Landesman–Lazer type conditions f_{2}(t, x) → φ^{±}(t) as x → ∞. Let e(t) satisfy (3). Then limits (12) exist and

ψ̃(±) = ∫_{{±e(t)≥0}} g(t) φ^{±}(t) dt + ∫_{{±e(t)≤0}} g(t) φ^{∓}(t) dt.

3. Let f_{2}(t, x) = f(x) be an even function, t ∈ [0, π], e(t) = \sqrt{2/π} sin nt and n be even. Then for any ξ

∫_{0}^{π} e(t) f(ξe(t)) dt = 0.

Of course (11) is valid for this case.

4. Let e(t) ∈ C^{1} and e′(t) satisfy (7). Let again f_{2}(t, x) = f(x) and let its primitive F(x) be sublinear:

lim_{x→∞} \frac{1}{x} ∫_{0}^{x} f(u) du = 0.  (13)

Then (11) is valid.

Some analog of this statement is formulated in [3] in the context of unbounded solution existence. It can be proved in the following way. The set where e′(t) is small has a small measure due to (7). This means that without loss of generality we can consider only the case, where the function e(t) is monotone (for example, it increases), e′(t) > ε and g(t) ∈ C^{1}. Then

lim_{ξ→∞} ∫_{a}^{b} g(t) f_{2}(t, ξe(t)) dt = lim_{ξ→∞} ∫_{a}^{b} g(t) f(ξe(t)) dt = lim_{ξ→∞} ∫_{a}^{b} \frac{g(t)}{ξe′(t)} dF(ξe(t)) =

= lim_{ξ→∞} ( \frac{g(t)}{e′(t)} \frac{F(ξe(t))}{ξ} |_{t=a}^{t=b}) − lim_{ξ→∞} ∫_{a}^{b} [\frac{d}{dt} (\frac{g(t)}{e′(t)})] \frac{F(ξe(t))}{ξ} dt → 0.

For example f(x) has a sublinear primitive if f(x) is periodic or almost periodic with zero average. Other examples: the functions sin \sqrt{|x|} and sin(x^{2}) also have sublinear primitives. The function sin(ln(ln(3 + |x|))) has not sublinear primitive and the limit in (11) can be non-zero. If a primitive of a function g(x) is bounded and

lim_{x→∞} \frac{f′′(x)}{(f′(x))^{2}} = 0,

then g(f(x)) has a sublinear primitive.

As a last example consider the problem

x′′ + x + sin(ln(ln(3 + |x|))) + \frac{3}{π} arctan x + 100 cos x = λ sin t, x(0) = x(π) = 0.

Here p(x) = ∫_{0}^{π} e(t) x(t) dt. The function 100 cos x has a bounded primitive and satisfies (11), for the function (3/π) arctan x limits (12) exist and ψ̃(±) = ±3\sqrt{2/π}. And for the function sin(ln(ln(3 + |x|))) the numbers (2) take the following values: ψ_{±} = −2\sqrt{2/π}, ψ^{±} = 2\sqrt{2/π}. This means that for the complete nonlinearity sin(ln(ln(3 +

|x|))) + 3/π arctan x + 100 cos x our numbers take four different values ψ_{−} = −5\sqrt{2/π},

ψ^{−} = −\sqrt{2/π}, ψ_{+} = \sqrt{2/π} and ψ^{+} = 5\sqrt{2/π}.

If 2 < |λπ| < 10, then the set of solutions is unbounded; if |λπ| < 2, then the set of solutions is nonempty and bounded; if |λπ| > 10, then the set of solutions is bounded (may be empty).

6 Proof of Theorem 1

We prove the different conclusions of Theorem 1 separately.

a) Let ψ^{+} < 0 < ψ_{−} (the case ψ^{−} < 0 < ψ_{+} can be considered in a similar way). Put

s(ξ) = \{ \begin{matrix}1, & if & ξ ≥ 1, \\ ξ, & if & |ξ| < 1, \\ −1, & if & ξ ≤ −1. \end{matrix}  (14)

Consider for λ ∈ [0, 1] the homotopy

Ξ_{1}(λ, x) = Qx − AQx − λF (x) + (1 − λ)s(p(x))e.

Obviously, Ξ_{1}(1, x) = Φ(x) and Ξ_{1}(0, x) = Qx − AQx + s(p(x))e. To prove the conclusion a) of Theorem 1 it is sufficient to see that | ind_{∞} Ξ_{1}(0, ·)| = 1 and to state an a priori estimate ∥x∥ ≤ const < ∞ for all the zeros x of the homotopy Ξ_{1}(λ, x) for all λ’s, 0 ≤ λ ≤ 1.

The first step is rather simple: the equation Ξ_{1}(0, x) has a unique solution x = 0, the index (in any sense) of this solution equals ±1. This follows from the formula of the index product: the index of the origin in E_{2} of the vector field Qx − AQx is equal to (−1)^{β}, where β is a sum of multiplicities of all real eigenvalues of A which are strictly greater than 1 and the index of the origin in E_{1} of s(ξ) equals 1.

Now let us prove an a priori estimate. Let x = ξe + h, h ∈ E_{2} and Ξ_{1}(λ, x) = 0 for some λ ∈ [0, 1]. This implies two equalities

Qx = AQx + λQF (x)  (15)

and

p(Ξ_{1}(λ, x)) = −λP F (x) + (1 − λ)s(ξ)e = 0.  (16)

Equality (15), rewritten as h = (I − AQ)^{−1}QF (x), implies the estimate

∥h∥Ẽ ≤ ∥(I − AQ)^{−1}∥_{L(Ẽ∩E_{2})} sup_{x∈E} ∥QF (x)∥Ẽ Δ = c < ∞.  (17)

Consider equality (16). If |ξ| is sufficiently large, then s(ξ) = sign ξ. We consider only the case ξ > 1, the case ξ < −1 is analogous. Since

p(Ξ_{1}(λ, x)) = −λp(F (x)) + (1 − λ) =
= −λ(p(F (ξe + h)) − p(F (ξe))) − λΨ(ξ) + (1 − λ) ≥ o(1) − λψ^{+} + 1 − λ

and −λψ^{+} + 1 − λ ≥ max{1, −ψ^{+}} > 0 then for ξ → +∞ we have p(Ξ_{1}(λ, x)) > 0. This gives the a priori estimate for ξ, together with (17) it proves the conclusion a).

b) Let ψ_{−} ≥ ψ_{+} > 0. The case ψ_{+} > ψ_{−} > 0 and the case of negative ψ^{−} and ψ^{+} are analogous to the case considered. Consider for λ ∈ [0, 1] the homotopy

Ξ_{2}(λ, x) = Qx − AQx − λF (x) − (1 − λ)ψ_{+}e.

Now Ξ_{2}(1, x) = Φ(x) and Ξ_{2}(0, x) = Qx − AQx − ψ_{+}e, the equation Ξ_{2}(0, x) = 0 has no solutions and ind_{∞} Ξ_{2}(0, ·) = 0. To prove the conclusion b) of Theorem 1 it is sufficient to prove an a priori estimate ∥x∥ ≤ const < ∞ for all the zeros x of the homotopy Ξ_{2}(λ, x) for 0 ≤ λ ≤ 1. This a priori estimate follows from the estimate (17) (it is valid here too) and the relations

p(Ξ_{2}(λ, x)) = −λp(F (x))−(1−λ)ψ_{+} = −λ(p(F (ξe+h))−p(F (ξe)))−λΨ(ξ)−(1−λ)ψ_{+}

≤ o(1) − λψ_{+} − ψ_{+} + λψ_{+} = o(1) − ψ_{+} < 0.

c) Let ψ_{+} < 0 < ψ^{+}. Again, the case ψ_{−} < 0 < ψ^{−} can be considered in a similar way. We prove that there exists a sequence of zeros x_{n} = ξ_{n}e+h_{n} of the vector field Φ(x), satisfying ∥x_{n}∥_{E} → ∞.

Put

ε = \frac{1}{3} min\{ψ^{+}, −ψ_{+}\}, c = ∥(I − A)^{−1}∥_{L(Ẽ∩E_{2})} sup_{x∈E} ∥QF (x)∥Ẽ ,

c_{1} = ∥(I − A)^{−1}∥_{L(E_{2})} sup_{x∈E} ∥QF (x)∥_{E} .

According to (1) choose a ξ_{0} > 0 such that for ξ ≥ ξ_{0}

|p(F (ξe + h)) − Ψ(ξ)| < ε, ∥h∥Ẽ ≤ c.  (18)

According to the definition of the numbers ψ_{+} and ψ^{+} there exist numbers ξ_{∗} > ξ_{0} and ξ^{∗} > ξ_{∗} such that

Ψ(ξ_{∗}) < −2ε, Ψ(ξ^{∗}) > 2ε.  (19)

Consider the cylinder Ω = {∥Qx∥_{E} ≤ c_{1} + 1, p(x) ∈ [ξ_{∗}, ξ^{∗}]}. Let us prove that the vector field x − T (x) is non-zero on ∂Ω. The boundary ∂Ω consists from 3 parts:

G_{∗} = {∥Qx∥_{E} ≤ c_{1} + 1, p(x) = ξ_{∗}}, G^{∗} = {∥Qx∥_{E} ≤ c_{1} + 1, p(x) = ξ^{∗}},
G = {∥Qx∥_{E} = c_{1} + 1, p(x) ∈ [ξ_{∗}, ξ^{∗}]}.

The vector field Φ(x) is non-zero on G because of the estimate ∥Qx∥_{E} ≤ c_{1} of all solutions x of QΦ(x) = 0. The vector field Φ(x) is non-zero on G_{∗} due to relations

p(Φ(x)) = p(F (x)) − Ψ(p(x)) + Ψ(p(x)) < ε − 2ε = −ε < 0,

The vector field Φ(x) is non-zero on G^{∗} due to relations

p(Φ(x)) = p(F (x)) − Ψ(p(x)) + Ψ(p(x)) > −ε + 2ε = ε > 0.

In the last two formulas we used the following arguments: if Φ(x) = 0, then ∥Qx∥Ẽ ≤ c and therefore

|p(F (x)) − Ψ(p(x))| < ε, Ψ(p(x)) < −2ε, x ∈ G_{∗}

and

|p(F (x)) − Ψ(p(x))| < ε, Ψ(p(x)) > 2ε, x ∈ G^{∗}.

We additionally proved that p(Φ(x)) < 0 if QΦ(x) = 0 and p(x) = ξ_{∗} and p(Φ(x)) > 0 if QΦ(x) = 0 and p(x) = ξ^{∗}.

Now let us calculate the rotation γ(Φ, ∂Ω) of non-degenerate vector field Φ(x) on ∂Ω. This can be done with the use of Theorem 3 from [1]. Let us formulate this theorem in a form which is convenient for our very particular case.

Let X be a Banach space. Let some completely continuous operator A = {A_{1}, A_{2}} be defined on the set Ω = Ω_{1} × Ω_{2} where Ω_{1} = B(0, r) ⊂ X and Ω_{2} = [ξ_{∗}, ξ^{∗}]. Let for any ξ ∈ Ω_{2} the vector field x_{1} −A_{1}(x_{1}, ξ) be non-degenerate on ∂Ω_{1} = {∥x∥_{X} = r} and let the rotation γ_{1} = γ(x_{1} − A_{1}(x_{1}, ξ), ∂Ω_{1}) be non-zero. This γ_{1} does not depend on ξ. Denote for any ξ ∈ Ω_{2} by K(ξ) the non-empty set of solutions x_{1} of the equation x_{1} = A_{1}(x_{1}, ξ). Theorem [1]. If ξ_{∗} − A_{2}(x_{1}, ξ_{∗}) < 0 for x_{1} ∈ K(ξ_{∗}) and ξ^{∗} − A_{2}(x_{1}, ξ^{∗}) > 0 for x_{1} ∈ K(ξ^{∗}), then γ(A, ∂Ω) = γ_{1}.

This theorem implies for our situation the relations γ(Φ, ∂Ω) = γ(QΦ, {∥Qx∥_{E} = c_{1} + 1}) = ± − 1 ≠ 0.

We proved that in the set Ω there is at least one zero of the vector field Φ. Denote it as x_{1} and choose other numbers ξ_{∗} and ξ^{∗}, both greater than both old ones. Again, in the corresponding cylinder there exists at least one zero of Φ. Since we can choose arbitrary large numbers ξ_{∗} and ξ^{∗}, satisfying (18), there exists a sequence x_{n} of zeros with ∥x_{n}∥_{E} → ∞.

7 Proof of Theorem 2

Let us choose the function μ : [ξ_{0}, ∞) → IR satisfying the following conditions

μ(ξ) → ∞, ξ → ∞,  μ(ξ) ≤ \sqrt{ξ},  μ(ξ) ≤ \frac{1}{\sqrt{L(\sqrt{ξ})}},

where L(r) is the function from (4), satisfying (5). Since

∥f(t, ξe(t) + h(t)) − f(t, ξe(t))∥_{L^{2}}^{2} =

= ∫_{Ω} ∥f(t, ξe(t) + h(t)) − f(t, ξe(t))∥_{IR^{n}}^{2} dt ≤ ∫_{Ω_{1}} … + ∫_{Ω_{2}} … + ∫_{Ω_{3}} …,

where

Ω_{1} = \{t ∈ Ω : ∥e(t)∥_{I}_{R}_{n} ≤ \frac{2}{\sqrt{|ξ|}}\}, Ω_{2} = \{t ∈ Ω : ∥h(t)∥_{I}_{R}_{n} ≥ μ(|ξ|)\},

Ω_{3} = \{t ∈ Ω : ∥e(t)∥_{I}_{R}_{n} ≥ \frac{2}{\sqrt{|ξ|}}, ∥h(t)∥_{I}_{R}_{n} ≤ μ(|ξ|)\},

then it is necessary to estimate the last three integrals. According to (3)

|∫_{Ω_{1}} …| ≤ 4 sup |f^{2}(t, x)| · mes Ω_{1} → 0,

according to Chebyshev inequality

|∫_{Ω_{2}} …| ≤ 4 sup |f^{2}(t, x)| · mes Ω_{2} ≤ 4 sup |f^{2}(t, x)| · \frac{∥h(t)∥_{L^{2}}^{2}}{μ^{2}(|ξ|)} → 0.

The estimate of the third integral follows from the inequality

∥ξe(t) + h(t)∥_{I}_{R}_{n} ≥ |ξ| ∥e(t)∥_{I}_{R}_{n} − ∥h(t)∥_{I}_{R}_{n} ≥ 2\sqrt{|ξ|} − μ(|ξ|) ≥ \sqrt{|ξ|},

which is valid for t ∈ Ω_{3}. Now according to (4)

|∫_{Ω_{3}} …| ≤ ∫_{Ω_{3}} L^{2}(\sqrt{|ξ|}) ∥h(t)∥_{IR^{n}}^{2} dt ≤ mes Ω L^{2}(\sqrt{|ξ|}) μ^{2}(|ξ|) ≤ mes Ω L(\sqrt{|ξ|}) → 0. Theorem 2 is proved.

8 Proof of Theorem 3

We have to prove that for any ε > 0 the supremum in (8) is less than ε for sufficiently large |ξ|. First let us choose a continuous g̃(t) such that

2 sup |f(t, x)| ∫_{a}^{b} |g(t) − g̃(t)|dt < ε/3.

We need to prove that

sup_{∥h(t)∥_{C^{1}}≤c} |∫_{a}^{b} g̃(t)[f(t, ξe(t) + h(t)) − f(t, ξe(t))] dt| < \frac{2}{3}ε.  (20)

Now let us split the interval [a, b] into a finite number of subintervals [a_{i}, b_{i}] and [b_{i}, a_{i+1}] in the following way. The intervals [b_{i}, a_{i+1}] contain the set {t ∈ [a, b] : e′(t) = 0}, according to (7) the union of these intervals can have arbitrary small measure. Let

2 sup |f(t, x)| ∫_{∪[b_{i},a_{i+1}]} |g̃(t)| dt < ε/3.  (21)

Suppose that the points a_{i} and b_{i} are fixed up to the end of the proof. For any [a_{i}, b_{i}] the estimate

inf_{t∈[a_{i},b_{i}]} |e′(t)| ≥ δ > 0

holds. This means that the function e(t) is strictly monotone on every [a_{i}, b_{i}], for sufficiently large |ξ| (|ξ| > 2cδ^{−1}) the function ξe(t)+h(t) is also strictly monotone, |ξe′(t)+h′(t)| > 1/2 |ξ|δ. Consider the integrals

J_{i} = ∫_{a_{i}}^{b_{i}} g̃(t)f(t, ξe(t) + h(t)) dt.

Fix any of them and do the change of variables t = t_{ξ}(τ) in this integral, defined by the formula ξe(τ) = ξe(t) + h(t):

J_{i} = ∫_{t_{ξ}^{−1}(a_{i})}^{t_{ξ}^{−1}(b_{i})} \frac{g̃(t_{ξ}(τ)) f(t_{ξ}(τ), ξe(τ))}{t′_{ξ}(τ)}dτ.

The function t_{ξ}(τ) is one-to-one, t_{ξ}(τ) → τ and t′_{ξ}(τ) → 1 uniformly in τ as |ξ| → ∞. Now

t_{ξ}^{−1}(a_{i}) → a_{i}, t_{ξ}^{−1}(b_{i}) → b_{i},

g̃(t_{ξ}(τ)) → g̃(τ) due to the continuity of g̃(·), and

f(t_{ξ}(τ), ξe(τ)) → f(τ, ξe(τ))

due to the assumption of the uniform continuity of f(t, x) in t. Consequently,

J_{i} − ∫_{a_{i}}^{b_{i}} g̃(τ)f(τ, ξe(τ) + h(τ)) dτ → 0

for every i. This together with (20) and (21) proves the theorem.

9 Remarks

1. The conclusions of Theorem 1 can be obtained for asymptotically linear degenerate vector fields with unbounded nonlinearities.

2. If we cannot calculate the exact values of the numbers (2) we can use some estimates of these numbers.

3. The main condition (1) is valid if

|p(F (x)) − p(F (y))| ≤ L(r) ∥x − y∥_{E} , ∥x∥_{E} , ∥y∥_{E} ≥ r  (22)

and (5) holds. Unfortunately we do not know any natural examples of nonlinearities, satisfying (22).

4. Analogs of Theorem 1 can be formulated for vector fields with non-compact operators (e.g. with condensing ones, see [2]).

5. The conditions for conclusion c) of Theorem 1 give us the information about the localization of solutions x_{n} .

References

[1] Krasnosel’skii A.M., Krasnosel’skii M.A. Vector fields in a product of spaces and applications to differential equations, Differencialnyje Uravnenija, #1, 33, 1997, 60 – 67 [Russian]

[2] Krasnosel’skii M.A., Zabreiko P.P. Geometrical methods of nonlinear analysis, Springer – Verlag, Berlin, Heidelberg, 1984

[3] Alonso J.M., Ortega R. Unbounded solutions of semilinear equations at resonance, Nonlinearity, 9, 1996, 1099 – 1111