Asymptotic behavior of solutions to an integral equation underlying a second-order differential equation

Nonlinear Analysis 70 (2009) 822–829 www.elsevier.com/locate/na

Asymptotic behavior of solutions to an integral equation underlying a second-order differential equationI Crist´obal Gonz´alez ∗ , Antonio Jim´enez-Melado Dept. An´alisis Matem´atico, Fac. Ciencias, Univ. M´alaga, 29071 M´alaga, Spain Received 7 November 2007; accepted 9 January 2008

Abstract In this paper we propose the study of an integral equation of the type Z ∞ y(t) = ω(t) − f (t, s, y(s))ds, t ≥ 0. 0

We investigate which conditions give existence, and which ones uniqueness, of solutions behaving like the function ω(t) at ∞. In applying our results to second-order nonlinear differential equations, we are able to recover the previous results and some generalizations. c 2008 Elsevier Ltd. All rights reserved.

MSC: 45G10; 34A34; 45M05; 47J05 Keywords: Nonlinear integral equation; Asymptotic behavior; Schauder fixed point theorem

1. Introduction In recent papers, much interest has been given to studying conditions that ensure the existence (and uniqueness) of asymptotically constant solutions to the equation: y 00 (t) + F(t, y(t)) = 0,

t ≥ 0.

(1)

We have to mention that the pioneering work for this type of research comes from the hands of Atkinson [1], when investigating the existence of non-oscillatory solutions for differential equations of the above type. For just a few references on the subject we indicate the papers [1–10] and the references therein. This paper is motivated by recent works of Dub´e and Mingarelli [3], Wahl´en [9], and Ehrnstr¨om [4,5]. In each of these works, they look for solutions of (1) asymptotically equal to a real number ω (Ehrnstr¨om also considers I Research partially supported by the Spanish (Grants MTM2007-60854 and MTM2006-26627) and regional Andalusian (Grants FQM210 and P06-FQM01504) Governments. ∗ Corresponding author. E-mail addresses: [email protected] (C. Gonz´alez), [email protected] (A. Jim´enez-Melado).

c 2008 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2008.01.012

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dependence on the y 0 variable), and obtained their results through the study of the following integral equation, Z ∞ y(t) = ω − (s − t)F(s, y(s))ds, t ≥ 0. t

Observe that this integral equation fits into the following type: Z ∞ y(t) = ω(t) − f (t, s, y(s))ds, t ≥ 0, 0

in which, f (t, s, y) = (s − t)+ F(s, y), and ω(t) ≡ ω. Here, x+ = x if x ≥ 0, and x+ = 0, otherwise. Our intention with this note is to consider this more general integral equation by itself. In our study, we shall stress which conditions give rise to uniqueness, as well as which ones yield existence, of solutions that behave like the function ω(t) at ∞. In contrast with the works mentioned above, the existence results shall be obtained with conditions that are not of Lipschitz type, while a similar type of contractivity condition will suffice for the uniqueness result. One point we should mention is that the function ω(t) will be continuous but not necessarily bounded. As a consequence of this, our results also contain a result by Lipovan [7], about the existence of solutions asymptotically equal to a given polynomial of degree 1. A great number of authors also include dependence on the y 0 (t) ‘variable’, something that we do not deal with in this paper, although some of our results might be adapted to include this situation, particularly Theorem 1 below. The notation used throughout the paper is as follows. We usually write C(X, Y ) to denote the space of continuous functions from X to Y . R+ = [0, ∞). C(R+ ) is the vector space of real-valued continuous functions defined on R+ . Cb (R+ ) is the subspace of bounded continuous real-valued functions defined on R+ . It is a Banach space when endowed with the sup norm k · k∞ . Some more terminologies will be used. By a compact operator we mean a continuous operator which maps bounded sets onto relatively compact sets. Under this definition the Schauder Fixed Point Theorem asserts that any compact operator defined on a nonempty, bounded, closed and convex subset of a Banach space, which remains invariant by the operator, has a fixed point. We will also use a well-known version of the Arzel`a–Ascoli Theorem that asserts that if a family F of real-valued functions defined on R+ is equicontinuous at each t ∈ R+ and is pointwise relatively compact in R, i.e., for each t ∈ R+ the set F(t) = {u(t) : u ∈ F} is relatively compact in R, then each sequence {u n } ⊂ F contains a subsequence that converges to a given real-valued function u defined on R+ , this convergence being uniform in each compact subset of R+ . 2. On the existence of solutions As we said before, we propose the study of the following integral equation, Z ∞ y(t) = ω(t) − f (t, s, y(s))ds, t ≥ 0.

(E)

0

Let us somehow motivate the conditions that ensure the existence of solutions. We start imposing a natural one, which is the continuity of the elements that appear in the equation, f ∈ C(R+ × R+ × R, R),

and

ω ∈ C(R+ ).

(H0)

Next, as we want the solution y(t) to satisfy y(t) − ω(t) → 0 as t → ∞, it is natural to ask that the integral in (E) approaches 0 as t → ∞, and we do so, but uniformly in a set C of continuous functions that are likely to be solutions with the desired asymptotic behavior,  t→∞ +  R+ bounded, with g(t) −−−→ 0, such that,  There exists g : R → for C = {y ∈ C(R+ ) : |y(t) − ω(t)| ≤ g(t), t ∈ R+ }, (H1) Z ∞    | f (t, s, y(s))|ds ≤ g(t), t ∈ R+ , y ∈ C. 0

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One more condition is needed, and its motivation comes from taking a closer look at what happens when f (t, s, y) = (s − t)+ F(s, y) and F ≥ 0. In this case, under (H0) and (H1), we have for y ∈ C and 0 ≤ τ ≤ t, Z ∞ Z ∞ τ →∞ (s − t)+ F(s, y(s))ds ≤ (s − τ )+ F(s, y(s))ds ≤ g(τ ) −−−→ 0, τ

τ

and for y ∈ C and 0 ≤ t ≤ τ , Z ∞ Z ∞ (s − t)+ F(s, y(s))ds = (s − t)F(s, y(s))ds τ τ Z ∞ Z ∞ ≤ (s − τ )F(s, y(s))ds + τ F(s, y(s))ds τ τ Z ∞ ≤ g(τ ) + 2 (s − τ/2)F(s, y(s))ds τ

τ →∞

≤ g(τ ) + 2g(τ/2) −−−→ 0. R∞ That is, the integrals τ (s − t)+ F(s, y(s))ds are bounded above by a function of τ which approaches 0 at ∞, and it is done uniformly in the variables t ∈ R+ and y ∈ C. This fact provides us the following condition to impose in the general case.  τ →∞ There exists h : R+ → R+ , bounded, with h(τ ) −−−→ 0, such that, Z ∞ (H2)  | f (t, s, y(s))|ds ≤ h(τ ), t, τ ∈ R+ , y ∈ C. τ

These conditions provide an existence result. Theorem 1. Under hypotheses (H0) , (H1) and (H2) , Eq. (E) has a solution y(t) that satisfies y(t) − ω(t) → 0 as t → ∞. Proof. First observe that, by replacing y − ω with x, it suffices to find a solution to the integral equation Z ∞ x(t) = − f (t, s, ω(s) + x(s))ds,

˜ (E)

0

in the set C˜ = {x ∈ Cb (R+ ) : |x(t)| ≤ g(t)}, which in turn, is equivalent to find a fixed point in C˜ to the operator Z ∞ ˜ T x(t) = − f (t, s, ω(s) + x(s))ds, t ≥ 0, x ∈ C. 0

The idea then is to use the Schauder Fixed Point Theorem. Note that C˜ is a nonempty, bounded, closed and ˜ so convex subset of the Banach space Cb (R+ ). Also, by (H1), |T x(t)| ≤ g(t) for all t ∈ R+ and all x ∈ C, ˜ ˜ is relatively compact in R for each t ∈ R+ , even more, T C˜ ⊆ C˜ provided T x ∈ C(R+ ) T C(t) = {T x(t) : x ∈ C} ˜ for all x ∈ C. We will in fact show that T C˜ is equicontinuous at each t ∈ R+ , giving us the opportunity to use the above ˜ Fix t0 ∈ R+ and ε > 0. By (H2), find τε > 0 such that mentioned Arzel`a–Ascoli Theorem on the family T C. h(τ ) < ε/4 for all τ ≥ τε . Next observe that the continuity of ω and the uniform bound for functions in C˜ (given by ˜ such the bound of g), imply that there exists a finite interval J ⊂⊂ R, depending on τε , ω, and g, but not on x ∈ C, that, ω(s) + x(s) ∈ J,

for all x ∈ C˜ and all s ∈ [0, τε ].

Saying it otherwise, we have, y(s) ∈ J,

for all y ∈ C and all s ∈ [0, τε ].

We use this to say that f is uniformly continuous in [0, t0 + 1] × [0, τε ] × J , so there exists δ ∈ (0, 1) such that, for all t1 , t2 ∈ [0, t0 + 1], s1 , s2 ∈ [0, τε ], y1 , y2 ∈ J , with |t1 − t2 | ≤ δ, |s1 − s2 | ≤ δ, and |y1 − y2 | ≤ δ, ε | f (t1 , s1 , y1 ) − f (t2 , s2 , y2 )| < . 4τε

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˜ or y ≡ ω + x ∈ C, we have, Hence, if t ∈ [0, t0 + 1] ∩ [t0 − δ, t0 + δ], and if x ∈ C, Z τε Z ∞ Z | f (t, s, y(s)) − f (t0 , s, y(s))| ds + |T x(t) − T x(t0 )| ≤ | f (t, s, y(s))|ds + 0

< τε

ε + 2h(τε ) < ε. 4τε

τε

∞

τε

| f (t0 , s, y(s))|ds

This proves the equicontinuity of T C˜ at t0 ∈ R+ . Therefore, by the Arzel`a–Ascoli Theorem, each sequence in T C˜ has a subsequence which converges uniformly on ˜ each compact subset of R+ to a given function in Cb (R+ ), that also belongs to C. ˜ But given the structure of ‘funnel’ that presents the set C, a sequence {u n } ⊂ C which converges uniformly on ˜ must, in fact, converge uniformly to u in R+ . To see it, take ε > 0 each compact subset of R+ to a function u ∈ C, and find t0 > 0 such that g(t) < ε/2 for all t ≥ t0 . Since u n , n ∈ N, and u, all belong to C, then |u n (t) − u(t)| ≤ 2g(t),

all n ∈ N, all t ∈ R+ .

In particular, |u n (t) − u(t)| ≤ 2g(t) < ε. all n ∈ N, all t ≥ t0 . Now, since {u n } converges uniformly to u in [0, t0 ], there exists n 0 ∈ N such that |u n (t) − u(t)| < ε. all n ≥ n 0 , all t ∈ [0, t0 ]. This tells us that if n ≥ n 0 , then ku n − uk∞ ≤ ε, proving the uniform convergence of {u n } to u in R+ . With this observation, we have proved that T C˜ is relatively compact in Cb (R+ ). It only remains to prove that T is ˜ Fix x0 ∈ C˜ (y0 ≡ ω + x0 ∈ C) and fix ε > 0. By (H1), find tε > 0 such that g(t) < ε/4 continuous at each x ∈ C. for all t ≥ tε ; and by (H2), find τε > 0 such that h(τ ) < ε/4 for all τ ≥ τε . Again, we realize that there exists a finite interval J , depending on τε , ω and g, but not on x ∈ C˜ (neither on y ∈ C), such that y(s) ∈ J for all y ∈ C and all s ∈ [0, τε ]. And again, noticing that f is uniformly continuous in [0, tε ] × [0, τε ] × J , there exists δ > 0, such that, for t1 , t2 ∈ [0, tε ], s1 , s2 ∈ [0, τε ], and y1 , y2 ∈ J , with |t1 − t2 | < δ, |s1 − s2 | ≤ δ, and |y1 − y2 | ≤ δ, ε | f (t1 , s1 , y1 ) − f (t2 , s2 , y2 )| < . 4τε Now take x ∈ C˜ (i.e., y ≡ ω + x ∈ C) with kx − x0 k∞ = ky − y0 k∞ < δ. Then we have, for t ≤ tε , Z τε Z ∞ Z ∞ | f (t, s, y(s)) − f (t, s, y0 (s))| ds + |T x(t) − T x0 (t)| ≤ | f (t, s, y(s))|ds + | f (t, s, y0 (s))|ds 0

ε < τε + 2h(τε ) < ε, 4τε

τε

τε

and for t ≥ tε , the quantity |T x(t) − T x0 (t)| is less than or equal to Z ∞ | f (t, s, y(s))| + | f (t, s, y0 (s))| ds ≤ 2g(t) < ε. 0

˜ This proves that kT x − T x0 k∞ < ε whenever x ∈ C˜ and kx − x0 k∞ < δ, showing that T is continuous at x0 ∈ C. ˜ Therefore by the Schauder Fixed Point Theorem, the operator All this shows that T is a compact operator on C. ˜ i.e., the integral equation (E) ˜ which yields y0 = ω + x0 ∈ C as a ˜ has a solution x0 ∈ C, T has a fixed point in C, solution to Eq. (E).  Remark 1. The fact to have located our equation in R+ = [0, ∞) is of no importance. If the conditions and the hypotheses are shifted to some interval of the type [t0 , ∞) ⊂ R, then the result remains unchanged. Remark 2. The motivations that initiated this section serve now to say that some of the results in [3,9,6] are generalized with Theorem 1, at least in the matter of existence of solutions. The following result might be considered as an application of Theorem 1 to the existence of solutions of a secondorder nonlinear differential equation with a prescribed asymptotic behavior.

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Theorem 2. Let F : [α, ∞) × R → R be a continuous function such that for t ∈ [α, ∞), y ∈ R,   y + ψ2 (t), |F(t, y)| ≤ ψ1 (t) ϕ ω0 (t)

(2)

where ϕ ∈ C(R, R+ ),

(3)

ω0 ∈ C([α, ∞), (0, ∞)),

t→∞

ω0 (t) −−−→ ∞, Z ∞ + ψi : [α, ∞) → R , i ∈ {1, 2}, satisfy sψi (s) < ∞. and

α

(4) (5)

Then for any ω ∈ C 2 ([α, ∞), R) with ω(t)/ω0 (t) → λ ∈ R as t → ∞, there exists t0 ≥ α such that the second-order differential equation y 00 (t) = ω00 (t) − F(t, y(t)),

t ≥ t0 ,

(6)

has a solution y ∈ C 2 ([t0 , ∞), R) with y(t) − ω(t) → 0 as t → ∞. This result is in fact a generalized version of Theorem 1 in Lipovan’s paper [7]: Corollary 1 (Lipovan). Under the same hypotheses as in Theorem 2, where in this case α = 1 and ω0 (t) = t, t ≥ 1, we have that for any a, b ∈ R, there exists t0 ≥ 1 for which the equation y 00 (t) = −F(t, y(t)),

t ≥ t0 , t→∞

has a solution y(t) ∈ C 2 ([t0 , ∞), R) with y(t) − at − b −−−→ 0. Proof of Theorem 2. The idea is to apply Theorem 1 to an appropriate integral equation, and for that, the main task is to find what will be the ‘invariant’ test set C. Since ϕ is continuous at λ, it is bounded in a neighborhood of it, so there exist δ > 0 and M > 0 such that 0 ≤ ϕ(x) ≤ M,

for all x ∈ [λ − δ, λ + δ].

(7)

Now, (ω(t) − 1) /ω0 (t) and (ω(t) + 1) /ω0 (t), both have limit λ at ∞. So there exists t1 ≥ α such that, for all t ≥ t1 , ω(t) 1 ω(t) 1 − < + < λ + δ. (8) ω0 (t) ω0 (t) ω0 (t) ω0 (t) R∞ Also, by the hypotheses, the function g(t) = t s (ψ1 (s)M + ψ2 (s)) ds, t ≥ α, is non-negative and has limit equal to 0 at ∞, so there exists t0 ≥ t1 such that λ−δ <

0 ≤ g(t) < 1,

for all t ≥ t0 .

With all these settings we consider the following integral equation in the interval [t0 , ∞), Z ∞ ω(t) 1 − (s − t)F(s, ω0 (s)x(s))ds. x(t) = ω0 (t) ω0 (t) t

(9)

We claim that this integral equation has a solution in the set   ω(t) g(t) C = x ∈ C([t0 , ∞), R) : x(t) − . ≤ ω0 (t) ω0 (t) And this is readily true if we just verify that for all t ≥ t0 and all x ∈ C, Z ∞ 1 g(t) (s − t)|F(s, ω0 (s)x(s))|ds ≤ , ω0 (t) t ω0 (t) for the rest of the conditions to apply Theorem 1 are easily satisfied.

(10)

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To see (10), take t ≥ t0 and x ∈ C. Then from the same definition of the set C, and the fact that t ≥ t0 ≥ t1 , we have, ω(t) 1 ω(t) g(t) λ−δ ≤ − ≤ − ω0 (t) ω0 (t) ω0 (t) ω0 (t) g(t) ω(t) 1 ω(t) + ≤ + ≤ x(t) ≤ ω0 (t) ω0 (t) ω0 (t) ω0 (t) ≤ λ + δ. In consequence, by (7), 0 ≤ ϕ(x(t)) ≤ M for all t ≥ t0 and all x ∈ C. Therefore, Z ∞ Z ∞ (s − t)|F(s, ω0 (s)x(s))|ds ≤ s [ψ1 (s)ϕ(x(s)) + ψ2 (s)] ds t Zt ∞ ≤ s [ψ1 (s)M + ψ2 (s)] ds = g(t), t

and (10) follows. Once we have a solution x ∈ C to the integral equation (9), we obtain that y(t) = ω0 (t) x(t), t ≥ t0 , satisfies the following integral equation, Z ∞ y(t) = ω(t) − (s − t)F(s, y(s))ds, t ≥ t0 , (11) t

C 2 ([t

2 implying that y ∈ 0 , ∞), R), for both defining summands are in C ([α, ∞), R). This tells us that y is a solution to the differential equation (6). It only remains to show the asymptotic behavior of y: for t ≥ t0 , and having in mind that x = y/ω ∈ C,   ω(t) t→∞ |y(t) − ω(t)| = |ω0 (t)x(t) − ω(t)| = x(t) − ω0 (t) ≤ g(t) −−−→ 0. ω (t) 0

With this, we conclude the proof of the theorem.



3. Some qualitative properties of the solutions This is a short section in which we study the effect that additional properties on f might have on the solutions to the integral equation (E), and whether Theorem 1 can be improved upon these additional properties. The set C appearing in hypothesis (H1) can be considered as a test set where an integral estimate must be verified. On one side, it has to be large enough so to contain the sought after solution. On the other side, from the practical point of view, it is desirable that its size be reduced in order not to cumber the verifications of the integral estimates. Also, as the purpose is to apply the Schauder Fixed Point Theorem to an operator defined on a translate C˜ ≡ C − ω of C, the idea is to keep this translate C˜ nonempty, bounded, closed and convex. With this in mind, additional properties on the function f = f (t, s, y) might have a reducing effect on the test set C. For instance, if f ≥ 0, then any solution y to the integral equation (E) must satisfy Z ∞ y(t) = ω(t) − f (t, s, y(s))ds ≤ ω(t), t ≥ 0, 0

i.e., the solutions must remain below ω. Knowing this, it gives us the hint to consider the following replacement condition for (H1), in which the test set C is smaller,  t→∞ + +  There exists g : R → R bounded, with g(t) −−−→ 0, such that,  + forZ C = {y ∈ C(R ) : 0 ≤ ω(t) − y(t) ≤ g(t), t ∈ R+ }, (H1’) ∞   + 0 ≤ f (t, s, y(s))ds ≤ g(t), t ∈ R , y ∈ C. 0

There is no problem in checking that the translate C˜ ≡ C − ω of the set C is nonempty, bounded, closed and convex in Cb (R+ ), and remains invariant under the corresponding operator T . Then going through the same steps as in the proof of Theorem 1, we conclude that a solution to (E) exists in the new test set C.

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Some other additional properties, should give more qualitative properties on the solutions, likely providing us with a smaller test set C. Here we enumerate just a few easy situations, with no other purpose than to illustrate the mechanism, leaving the details of its correctness to the interested reader. (1) As previously discussed, if f ≥ 0 then the sought after solutions remain below ω, and the test set to be considered can be reduced to C = {y ∈ C(R+ ) : − g(t) ≤ y(t) − ω(t) ≤ 0, t ∈ R+ }. (2) Similarly for non-positive f . In this case, the sought after solutions must be above ω, and the test set could be C = {y ∈ C(R+ ) : 0 ≤ y(t) − ω(t) ≤ g(t), t ∈ R+ }. (3) If f = f (t, s, y) is non-increasing in t ∈ R+ , then any solution y to the integral equation (E) satisfies that y − ω is non-decreasing, giving the opportunity to consider C with translate (which is where the Schauder Fixed Point Theorem is applied), C˜ = C − ω, given by C˜ = {non-decreasing x ∈ Cb (R+ ) : |x(t)| ≤ g(t), t ∈ R+ }. (4) More generally, if f presents a certain monotonicity behavior in the variable t along a given interval I ⊆ R+ , then the translate of any solution, y − ω, must present the opposite monotonicity behavior in the same interval. This yields the possibility of considering a test set C, whose translate, C˜ = C − ω, presents the corresponding monotonicity behavior in its elements. (5) Likewise, if f = f (t, s, y) is convex in the variable t along an interval I ⊆ R+ , then the translate of any solution, y − ω, must be concave in I , yielding a test set C with C˜ = {x ∈ Cb (R+ ) concave in I : |x(t)| ≤ g(t), t ∈ R+ }. (Note that the function (s − t)+ F(s, y) is non-increasing and convex in the t variable.) (6) Finally, we mention that combinations of some of the above items (or other ones) are possible, then providing a better behavior on the solutions and a smaller test set. 4. On the uniqueness of solutions In this section we look for conditions that ensure the uniqueness of solutions. Our study is inspired on the works by Dub´e and Mingarelli [3], Wahl´en [9], and Ehrnstr¨om [4,5], in which uniqueness of solutions is achieved by imposing a certain Lipschitz condition on the y variable, and then using the Banach contraction principle on an appropriate complete metric space of functions. To motivate our condition, let us study the condition imposed inRthe works mentioned above. The basic one is that, ∞ for f (t, s, y) = (s − t)+ F(s, y), there exists a function k(s) with 0 sk(s)ds < ∞, such that |F(s, y1 ) − F(s, y2 )| ≤ k(s)|y1 − y2 |. To see what this condition reveals, observe that Z ∞ Z ∞ t→∞ (s − t)+ k(s)ds ≤ sk(s)ds −−−→ 0. 0

t

But more is true. Defining A1 (t) = can easily check as an exercise that 1 + A1 (t) ≤ e A1 (t) ,

R∞ t

sk(s)ds, and then, recursively, An+1 (t) =

R∞ t

An (s)sk(s)ds, for n ∈ N, we

t ∈ R+ ,

and by induction on n, that 1 + A1 (t) + A2 (t) + · · · + An (t) ≤ e A1 (t) , n→∞

t ∈ R+ ,

showing in particular that An (t) −−−→ 0 uniformly in t ∈ R+ . This is essentially what will be needed. We start introducing a new class of functions. The class A consists of those functions a : R+ × R+ → R+ , for which the associated recursive sequence of functions given by Z ∞ A1 (t) = a(t, s)ds, 0 Z ∞ An+1 (t) = An (s)a(t, s)ds, n ∈ N, 0

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is well defined in R+ and satisfies further the following properties: t→∞

A1 (t) −−−→ 0,

and

(A1)

There exist n 0 ∈ N, and c ∈ (0, 1), such that An 0 (t) ≤ c, for all t ∈ R . (An) R∞ Notice that, when 0 sk(s)ds < ∞, the function (s − t)+ k(s) belongs to the class A. With this definition, we introduce our condition,  There exists a ∈ A, such that for t, s ∈ R+ , and y1 , y2 ∈ R, (H3) | f (t, s, y1 ) − f (t, s, y2 )| ≤ a(t, s)|y1 − y2 |. +

We are ready to state our result. Theorem 3. Under (H0) and (H3) , (E) has at most one solution y(t) ∈ C(R+ ) that satisfies [y(t) − ω(t)] → 0 as t → ∞. t→∞

Proof. The set X = {y ∈ C(R+ ) : [y(t) − ω(t)] −−−→ 0} becomes a complete metric space when endowed with the distance function given by d(y1 , y2 ) = sup{|y1 (t) − y2 (t)| : t ∈ R+ }. t→∞ Now assume that y1 and y2 are two solutions of (E) satisfying yi (t) − ω(t) −−−→ 0 for i ∈ {1, 2}. Then they belong to X and, by virtue of (H3), we have, for t ∈ R+ , Z ∞ |y1 (t) − y2 (t)| ≤ | f (t, s, y1 (s)) − f (t, s, y2 (s))|ds Z0 ∞ ≤ a(t, s)|y1 (s) − y2 (s)|ds 0

≤ A1 (t) d(y1 , y2 ), and using this estimate in the same situation, we have for t ∈ R+ , Z ∞ |y1 (t) − y2 (t)| ≤ a(t, s)|y1 (s) − y2 (s)|ds Z0 ∞ ≤ a(t, s)A1 (s) d(y1 , y2 )ds 0

≤ A2 (t) d(y1 , y2 ), so by induction on n we obtain |y1 (t) − y2 (t)| ≤ An (t) d(y1 , y2 ),

n ∈ N, t ∈ R+ .

Now condition (An) tells us that this is a contradiction unless y1 ≡ y2 .



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