Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 53, 1-10; http://www.math.u-szeged.hu/ejqtde/

Resonant problem for a class of BVPs on the half-line

Toufik Moussaoui, Katarzyna Szymańska-Dȩbowska

Abstract

We provide an existence result for a Neumann nonlinear boundary value problem posed on the half-line. Our main tool is the multivalued version of the Miranda Theorem.

1 Introduction

In this paper we study the following BVP

x′′ = f(t, x, x′), x′(0) = 0, lim_{t→∞} x′(t) = 0,

where f : ℝ_{+} × ℝ × ℝ → ℝ.

In recent years, the existence of solutions for boundary-value problems received wide attention. We can write down almost any nonlinear differential equation in the form Lx = N(x), where L is a linear and N nonlinear operator in appropriate Banach spaces. If kernel L is nontrivial then the equation is called resonant and one can manage the problem by using the coincidence degree in that case. But, if the domain is unbounded the operator is usually non-Fredholm (like in our case). Here, for instance, we can use the perturbation method (see [2]) or we can work in the space H^{2}(ℝ_{+}) (see [7]) to obtain the existence results.

The method presented below enables us to get the existence result under weaker assumptions than those mentioned in the above cited methods. Consequently, we will need some facts from set-valued analysis and multi-valued version of the Miranda Theorem to get the Theorem about the existence of at least one solution to the resonant problem.

Now, we shall set up notation and terminology.

Denote by BC(ℝ_{+}, ℝ) (we write BC) the Banach space of continuous and bounded functions with supremum norm and by BCL(ℝ_{+}, ℝ) (we write BCL) its closed subspace of continuous and bounded functions which have finite limits at +∞.

The following theorem gives a sufficient condition for compactness in the space BC, ([6]).

Theorem 1. If K ⊂ BC satisfies following conditions:

(1) There exists L > 0, that for every x ∈ K and t ∈ [0, ∞) we have |x(t)| ≤ L;

(2) for each t_{0} ≥ 0, the family K is equicontinuous at t_{0};

(3) for each ε > 0 there exists T > 0 and δ > 0 such that if |x(T)−y(T)| ≤ δ, then |x(t) − y(t)| ≤ ε for t ≥ T and all x, y ∈ K, then K is relatively compact in BC.

By a space we mean a metric space. Let X, Y be spaces. A set-valued map Φ : X ⊸ Y is upper semicontinuous (written USC) if, given an open V ⊂ Y, the set {x ∈ X | Φ(x) ⊂ V } is open.

The theorem below is a generalization of the well known Miranda Theorem [5, p. 214] which gives zeros of single-valued continuous maps.

Theorem 2. ([8]) Let g be an USC map from [−M̂, M̂] into convex and compact subsets of ℝ and satisfying the conditions:

if d ∈ g(M̂), then d ≥ 0  (1)

and

if d ∈ g(−M̂), then d ≤ 0.  (2)

Then there exists x̃ ∈ [−M̂, M̂] such that 0 ∈ g(x̃).

The result, we shall present in this paper, is a generalization of the sublinear case considered in [8]. Therefore, the following Lemma will be of crucial importance for our reason.

Lemma 1. ([4])(Bihari’s lemma) Let b : [0, +∞) ⟶ [0, +∞) be a continuous function. Assume that the function y ∈ C([0, +∞), ℝ) is non-negative and satisfies the following inequality

y(t) ≤ C + ∫_{0}^{t} b(s)ω(y(s)) ds, t ≥ 0,

where C ≥ 0 is a real constant and ω(s) is a continuous, positive and nondecreasing function such that W (+∞) = +∞, where

W (t) = ∫_{0}^{t} \frac{ds}{ω(s)}.

Then, one has for all t ≥ 0 that

y(t) ≤ W^{−1}(W (C) + ∫_{0}^{t} b(s)ds).

2 Resonant problem

Let us consider an asymptotic BVP

x′′ = f(t, x, x′), x′(0) = 0, lim_{t→∞} x′(t) = 0,  (3)

where f : ℝ_{+} × ℝ × ℝ → ℝ is continuous.

The following assumption will be needed in this paper:

(i) |f(t, x, y)| ≤ b(t)ω(|y|)+c(t), where b, c ∈ L^{1}(0, ∞), b is continuous and ω : ℝ_{+} ⟶ ℝ_{+} is a nondecreasing function such that W (+∞) = +∞, where

W (t) = ∫_{0}^{t} \frac{ds}{ω(s)}.

(ii) there exists M > 0 such that xf(t, x, y) ≥ 0 for t ≥ 0, y ∈ ℝ and |x| ≥ M.

Now, we can formulate our main result.

Theorem 3. Under assumption (i) and (ii) problem (3) has at least one solution.

The proof will be divided into a sequence of Lemmas.

First, we consider problem

y′ = f(t, c + ∫_{0}^{t} y, y), y(0) = 0,  (4)

for fixed c ∈ ℝ. Observe that (4) is equivalent to an initial value problem

x′′ = f(t, x, x′), x(0) = c, x′(0) = 0.  (5)

Since f is continuous, then by the Local Existence Theorem we get that problem (5) has at least one local solution. We can write (4) as

y_{c}(t) = ∫_{0}^{t} f(s, c + ∫_{0}^{s} y_{c}(u)du, y_{c}(s))ds  (6)

Set

B := ∫_{0}^{∞} b(s)ds, C := ∫_{0}^{∞} c(s)ds.  (7)

By (i) and (7) we get

|y_{c}(t)| ≤ ∫_{0}^{t} (b(s)ω(|y_{c}(s)|) + c(s))ds ≤ C + ∫_{0}^{t} b(s)ω(|y_{c}(s)|)ds. (8)

Now, due to Bihari’s Lemma (see Lemma 1), we have

|y_{c}(t)| ≤ W^{−1}(W (C) + ∫_{0}^{t} b(s)ds).

Hence, by the Theorem on a Priori Bounds [5, p. 146], (5) has a global solution for t ≥ 0. We obtain that (4) has a global solution for t ≥ 0. Moreover, by assumption (i) and (7), we have

|y_{c}(t)| ≤ W^{−1}(W (C) + ∫_{0}^{t} b(s)ds) ≤ W^{−1}(W (C) + ∫_{0}^{∞} b(s)ds)
= W^{−1}(W (C) + B) < ∞  (9)

Hence all global solutions are bounded for t ≥ 0.

The function t ↦ f(t, c + ∫_{0}^{t} y_{c}(u)du, y_{c}(t)) is absolutely integrable; i.e.,

∀_{ε>0} ∃_{M>0} | ∫_{M}^{∞} f(t, c + ∫_{0}^{t} y_{c}(u)du, y_{c}(t))dt| < ε.

In particular, there exists a limit lim_{t→∞} y_{c}(t), for every c. Thus all solutions of (4) have finite limits at +∞.

Let us consider the nonlinear operator A, ℝ × BCL ∋ (c, x) → A_{c}(x) ∈ BCL, given by

A_{c}(x)(t) = ∫_{0}^{t} f(s, c + ∫_{0}^{s} x(u)du, x(s))ds.  (10)

It is easy to see that A is well defined. By using the Lebesgue Dominated Convergence Theorem one can prove the continuity of A.

Lemma 2. Under assumption (i) operator A is completely continuous.

Proof. We shall show that the image of K := {(c, x) ∈ ℝ × BCL | ∥(c, x)∥_{ℝ×BC} ≤ R} under A is relatively compact. Condition (1) of Theorem 1 is satisfied, since |A_{c}(x)(t)| ≤ C + ω(R)B.

Now, we prove condition (2). We show that for any t_{0} ≥ 0 and ε > 0 there exists δ > 0 that for each x ∈ K if |t−t_{0}| < δ, then |A_{c}(x)(t) − A_{c}(x)(t_{0})| < ε. Let us choose an arbitrary ε > 0. By (i) there exist δ_{1}, δ_{2} > 0 such that

if |t − t_{0}| < δ_{1}, then ∫_{min\{t_{0},t\}}^{max\{t_{0},t\}} b(s)ds < \frac{ε}{2ω(R)},

if |t − t_{0}| < δ_{2}, then ∫_{min\{t_{0},t\}}^{max\{t_{0},t\}} c(s)ds < \frac{ε}{2}.

Let δ = min{δ_{1}, δ_{2}}. Then, for |t − t_{0}| < δ, we get

|A_{c}(x)(t) − A_{c}(x)(t_{0})| ≤ ∫_{min{t_{0},t}}^{max{t_{0},t}} |f(s, c + ∫_{0}^{s} x(u)du, x(s))|ds

≤ ω(R) ∫_{min{t_{0},t}}^{max{t_{0},t}} b(s)ds + ∫_{min{t_{0},t}}^{max{t_{0},t}} c(s)ds

< ω(R)\frac{ε}{2ω(R)} + \frac{ε}{2} = ε.

It remains to prove condition (3). By assumption (i) for every ε > 0 there exist t_{1}, t_{2} large enough that

∫_{t_{1}}^{∞} b(s)ds < \frac{ε}{6ω(R)}, ∫_{t_{2}}^{∞} c(s)ds < \frac{ε}{6}.

Let T = max{t_{1}, t_{2}} and δ := ε/3. If |A_{c}(x)(T) − A_{c}(y)(T)| ≤ δ, then for t ≥ T we get

|A_{c}(x)(t) − A_{c}(y)(t)| ≤

≤ |A_{c}(x)(T) − A_{c}(y)(T)| + ∫_{T}^{∞} |f(s, c + ∫_{0}^{s} x(u)du, x(s))|ds

+ ∫_{T}^{∞} |f(s, c + ∫_{0}^{s} y(u)du, y(s))|ds

≤ |A_{c}(x)(T) − A_{c}(y)(T)| + 2 ∫_{T}^{∞} ω(R)b(s)ds + 2 ∫_{T}^{∞} c(s)ds

≤ \frac{ε}{3} + 2ω(R)\frac{ε}{6ω(R)} + 2\frac{ε}{6} = ε.

The proof is complete.

Note that the solutions of (4) are fixed points of operator A defined by (10). Let fix A_{c}(·) denote the set of fixed points of operator A_{c}, where c is given. Let us consider the multifunction g : ℝ ⊸ ℝ given by

g(c) := { lim_{t→∞} y_{c}(t) : y_{c} ∈ fix A_{c}(·)}.  (11)

Now, let φ : BCL → ℝ be a single-valued function such that

φ(y_{c}) = lim_{t→∞} y_{c}(t).  (12)

It is easily seen that function φ is continuous.

Set

Φ : ℝ ∋ c ⊸ FixA_{c}(·) ⊂ BCL  (13)

and notice that g = φ ∘ Φ.

Lemma 3. Let assumption (i) holds. Then the set-valued map Φ is USC with compact values.

Proof. The set-valued map Φ is upper semicontinuous with compact values if given a sequence (c_{n}) in ℝ^{k}, c_{n} → c_{0} and (x_{n}) ∈ Φ(c_{n}), (x_{n}) has a convergent subsequence to some x_{0} ∈ Φ(x_{0}). Taking any sequence (c_{n}), c_{n} → c_{0} and (x_{n}) ∈ Φ(c_{n}) we have

x_{n} = A_{c_{n}}(x_{n}).  (14)

By (9), we get that the solutions of (4) are equibounded for any c. Hence both sequences (x_{n}) and (c_{n}) are bounded. Proposition 2 yields that operator A is completely continuous. Then, by (14), (x_{n}) is relatively compact. Hence, passing to a subsequence if necessary, we may assume that x_{n} → x_{0} in BCL. The continuity of A implies that

x_{0} = A_{c_{0}}(x_{0}).

Hence, x_{0} ∈ Φ(c_{0}) and the proof is complete.

Lemma 4. Let assumption (i) hold. Then the set-valued map Φ has connected values.

Proof. We shall show that fix A_{c}(·) is connected in BCL. On the contrary, suppose that the set is not connected. Since fix A_{c}(·) is compact, there exist compact sets A and B such that A, B ≠ ∅, A∩B = ∅ and A∪B = fix F (c, ·). Let ε := dist(A, B), ε > 0. Then

∀_{y∈A,z∈B} ∥y − z∥ ≥ ε.  (15)

By (9) there exists T > 0 such that for any y ∈ fix A_{c}(·) we get

∫_{T}^{∞} |f(t, c + ∫_{0}^{t} y(u)du, y(t))|dt < \frac{1}{3}ε.  (16)

Let y ∈ A, i z ∈ B. Now, consider the functions y and z cut to the compact set [0, T] and set y|_{[0,T]} and z|_{[0,T]}. By Kneser’s Theorem [3, p. 413], the set fix A_{c}(·)|_{[0,T]} is connected in C([0, T], ℝ). From this, there exist x_{1}, …, x_{k} in fix A_{c}(·)|_{[0,T]} such that x_{1} = y |_{[0,T]}, x_{k} = z |_{[0,T]} and

∥x_{i} − x_{i+1}∥_{C([0,T],ℝ)} < \frac{1}{3}ε.  (17)

Hence, at least two sequel x_{i}, x_{i+1} in fix A_{c}(·)|_{[0,T]} are such that x_{i} ∈ A i x_{i+1} ∈ B. Moreover, x_{i}, x_{i+1} ∈ fix A_{c}(·). By (15), (16) and (17) we get a contradiction. Indeed

ε ≤ ∥x_{i} − x_{i+1}∥_{BCL}

≤ max{∥x_{i} − x_{i+1}∥_{C([0,T],ℝ)}, sup_{t≥T} |x_{i}(t) − x_{i+1}(t)|}

≤ ∥x_{i} − x_{i+1}∥_{C([0,T],ℝ)} + ∑_{j=i}^{i+1} ∫_{T}^{∞} |f(t, c + ∫_{0}^{t} x_{j}(u)du, x_{j}(t))|dt

< \frac{1}{3}ε + \frac{1}{3}ε + \frac{1}{3}ε = ε.

Hence, fix A_{c}(·) is connected in BCL, which proves the Lemma.

Lemma 5. Let assumption (i) hold. Then the multifunction g is USC map from ℝ into its compact intervals.

Proof. By Proposition 4 we know that Φ has compact and connected values. Now, by continuity of φ, the set

{ lim_{t→∞} y_{c}(t) | y_{c} ∈ fix A_{c}(·)}

is compact and connected subset of ℝ for every c. From this, we conclude that multifunction g(c) = (φ∘Φ)(c) maps ℝ into its compact intervals. Moreover, g is USC as a superposition of set-valued map with compact values and continuous function [1, p. 47].

By the above Lemmas, we get the proof of Theorem 3.

The proof of Theorem 3. Let y_{c} ∈ fix A_{c}(·) be the bounded global solution of (4) and g be given by (11).

Observe that x(t) = c + ∫_{0}^{t} y_{c}(s)ds is a solution of (3) if there exists an c ∈ ℝ such that 0 ∈ g(c).

We shall show that g satisfies assumptions of the multi-valued version of Miranda Theorem (see Theorem 2).

By Lemma 5 we get that g is USC map from ℝ into its compact (so convex) intervals.

Let c = M + 1, where M is as in (ii). We will show that y_{c}(t) ≥ 0 for t ≥ 0 and all y_{c} ∈ fix A_{c}(·).

By (4) we have y_{c}(0) = 0. Assume that for some t and y_{c} ∈ fix A_{c}(·) we have y_{c}(t) < 0. Then there exist t_{∗} := inf{t | y_{c}(t) < 0} such that y_{c}(t_{∗}) = 0 and y_{c}(t) ≥ 0 for t < t_{∗} (if t_{∗} ≠ 0). By continuity of y_{c}(t) there exists t_{1} > t_{∗} such that ∫_{t_{∗}}^{t_{1}} |y_{c}(t)|dt ≤ 1. Hence, we get

x(t) = c + ∫_{t_{∗}}^{t} y_{c}(s)ds = M + 1 + ∫_{t_{∗}}^{t} y_{c}(s)ds ≥ M for t ∈ [t_{∗}, t_{1}] .

Now, by (ii) we have

x(t)f(t, x(t), y(t)) = x(t)y′_{c}(t) ≥ 0.

Hence y′_{c}(t) ≥ 0 for t ∈ [t_{∗}, t_{1}]. It means that y_{c}(t) is nondecreasing on [t_{∗}, t_{1}]. Since y_{c}(t_{∗}) = 0 we get a contradiction. Hence y_{c}(t) ≥ 0 for t ≥ 0. In consequence, if d ∈ g(c) = lim_{t→∞} y_{c}(t), then d ≥ 0. Hence, the condition (1) is satisfied for M̂:= M + 1

To prove that if d ∈ g(−M̂), then d ≤ 0 we proceed analogously. Hence, by multi-valued version of Miranda Theorem, there exists an c̃ ∈ [−M̂, M̂] such that, 0 ∈ g(c̃). This completes the proof.

References

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Toufik Moussaoui (e-mail: moussaoui@ens-kouba.dz) Department of Mathematics, Lab. EDP & HM, E.N.S. Po Box 92 Kouba, 16050. Algiers, Algeria Katarzyna Szymańska-Dȩbowska (e-mail: grampa@zb.net.lodz.pl ) Institute of Mathematics Technical University of Lodz 90-924 Lodz, ul. Wolczanska 215, Poland

(Received March 5, 2009)