Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments

Abstract

In this paper, we propose the study of an integral equation, with deviating arguments, of the type 𝑦(𝑡)=𝜔(𝑡)−∫^∞₀𝑓(𝑡,𝑠,𝑦(𝛾₁(𝑠)),…,𝑦(𝛾_𝑁(𝑠)))𝑑𝑠,𝑡≥0, in the context of Banach spaces, with the intention of giving sufficient conditions that ensure the existence of solutions with the same asymptotic behavior at ∞ as 𝜔(𝑡). A similar equation, but requiring a little less restrictive hypotheses, is 𝑦(𝑡)=𝜔(𝑡)−∫^∞₀𝑞(𝑡,𝑠)𝐹(𝑠,𝑦(𝛾₁(𝑠)),…,𝑦(𝛾_𝑁(𝑠)))𝑑𝑠,𝑡≥0. In the case of 𝑞(𝑡,𝑠)=(𝑡−𝑠)₊, its solutions with asymptotic behavior given by 𝜔(𝑡) yield solutions of the second order nonlinear abstract differential equation 𝑦^′′(𝑡)−𝜔^′′(𝑡)+𝐹(𝑡,𝑦(𝛾₁(𝑡)),…,𝑦(𝛾_𝑁(𝑡)))=0, with the same asymptotic behavior at ∞ as 𝜔(𝑡).

1. Introduction

From the pioneering work of Atkinson [1], and subsequent works found in the literature (see, e.g., [2–10] for recent papers on the subject), we consider the following differential problem, with deviating arguments: 𝑦^′′(𝑡)−𝜔^′′(𝑡)+𝐹𝑡,𝑦𝛾₁(𝑡),…,𝑦𝛾_𝑁(𝑡)=0,𝑡≥0,(1) with the task of finding solutions 𝑦 with the same behavior at ∞ as 𝜔. Solutions with this prescription are given by the solutions of the following integral equation: 𝑦(𝑡)=𝜔(𝑡)−^∞_𝑡(𝑠−𝑡)𝐹𝑠,𝑦𝛾₁(𝑠),…,𝑦𝛾_𝑁(𝑠)𝑑𝑠,𝑡≥0,(2) which, by writing 𝑓(𝑡,𝑠,𝑦(𝛾₁(𝑠)),…,𝑦(𝛾_𝑁(𝑠))) = (𝑠−𝑡)₊𝐹(𝑠,𝑦(𝛾₁(𝑠)),…,𝑦(𝛾_𝑁(𝑠))), is of the type 𝑦(𝑡)=𝜔(𝑡)−^∞₀𝑓𝑡,𝑠,𝑦𝛾₁(𝑠),…,𝑦𝛾_𝑁(𝑠)𝑑𝑠,𝑡≥0.(3)

The purpose of this note is to provide conditions that ensure the existence of solutions to the above integral equation, whose asymptotic behavior at ∞ is the same as that of 𝜔, thus giving a procedure to show existence of solutions with prescribed asymptotic behavior of differential equation of the type (1). Our wish is to also work out this integral equation in the setting of Banach spaces.

Denote by ℝ⁺ the set [0,∞) of nonnegative real numbers. Assume that {𝛾_𝑗}^𝑁_𝑗=1 is a finite set of continuous mappings from ℝ⁺ to ℝ⁺, that (𝑋,‖⋅‖_𝑋) is a Banach space (with norm ‖⋅‖_𝑋), and also that 𝜔 is a continuous mapping from ℝ⁺ to 𝑋. Finally, assume that 𝑓∶ℝ⁺×ℝ⁺×𝑋^𝑁→𝑋 is a given continuous mapping with certain regularity and integrability conditions, to be specified later. In order to give a better aspect to our equation, define, for each continuous 𝑦∶ℝ⁺→𝑋, the mapping Γ(𝑦)∶ℝ⁺→𝑋^𝑁 given by Γ𝑦(𝑡)=𝑦𝛾₁(𝑡),…,𝑦𝛾_𝑁(𝑡),𝑡≥0.(4)

Then, our equation becomes 𝑦(𝑡)=𝜔(𝑡)−^∞₀𝑓𝑡,𝑠,Γ𝑦(𝑠)𝑑𝑠,𝑡≥0,(𝐸) which, by writing 𝑥(𝑡)=𝑦(𝑡)−𝜔(𝑡), 𝑡∈ℝ⁺, is transformed into 𝑥(𝑡)=−^∞₀𝑓(𝑡,𝑠,Γ(𝜔+𝑥)(𝑠))𝑑𝑠,𝑡≥0.𝐸

A bit more of notation and preliminary results are needed. As customary, 𝐵_𝑋(𝑥,𝑟) denotes the open ball in 𝑋 centered at 𝑥 with radius 𝑟. The closure in 𝑋 of any set 𝐴⊆𝑋 is written 𝐴, and its closed convex hull, co(𝐴). The space of continuous 𝑋-valued functions defined on ℝ⁺ is denoted by 𝒞(ℝ⁺,𝑋), while the space of bounded ones is 𝒞_𝑏(ℝ⁺,𝑋). The latter is a Banach space when endowed with the sup norm ‖⋅‖_∞, (i.e., for 𝑥∈𝒞_𝑏(ℝ⁺,𝑋), ‖𝑥‖_∞=sup_𝑡∈ℝ+‖𝑥(𝑡)‖_𝑋).

The Schauder fixed point theorem states that any continuous operator 𝑇 defined on a nonempty, bounded, closed and convex subset 𝐶 of a Banach space has necessarily a fixed point, provided that 𝑇(𝐶) is a relatively compact subset of 𝐶. We will also be needing a well-known version of the Arzelà-Ascoli theorem which, in the case that occupies us, is as follows: if a family ℱ⊆𝒞(ℝ⁺,𝑋) is equicontinuous at each 𝑡∈ℝ⁺, and each section ℱ(𝑡)∶={𝑢(𝑡)∶𝑢∈ℱ} is relatively compact in 𝑋, then each sequence {𝑢_𝑛}⊆ℱ contains a subsequence that converges uniformly on compact subsets of ℝ⁺ to a given 𝑋-valued function 𝑢.

Let us also say a word about vector integrals. For a brief introduction see, for example, [11] or [12]. If 𝑓∶[𝑎,𝑏]→𝑋 is a bounded vector function, the Riemann sum of 𝑓 associated to a finite partition of [𝑎,𝑏], ℘={𝑎=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝑏} (with norm ‖℘‖=max_𝑗|𝑡_𝑗−𝑡_𝑗−1|), and to a selection {𝑡^∗_𝑗}^𝑛_𝑗=1, 𝑡^∗_𝑗∈[𝑡_𝑗−1,𝑡_𝑗], is ∑^𝑛_𝑗=1𝑓(𝑡^∗_𝑗)(𝑡_𝑗−𝑡_𝑗−1). Without specifying the type of limit alluded, we say that 𝑓 is Riemann integrable on [𝑎,𝑏] if there exists the limit of its Riemann sums as the norm of the partitions of [𝑎,𝑏] tends to 0, in which case, this limit (which is unique) is called the integral of 𝑓 over [𝑎,𝑏] and is denoted by ∫^𝑏_𝑎𝑓(𝑡)𝑑𝑡. That is, being a little bit sloppy with the notation, we have ^𝑏_𝑎𝑓(𝑡)𝑑𝑡=lim_℘→0𝑗𝑓𝑡^∗_𝑗𝑡_𝑗−𝑡_𝑗−1.(5)

Observe that whenever 𝑓 is integrable on [𝑎,𝑏], then each Riemann sum associated to 𝑓 is (𝑏−𝑎) times a linear convex combination of elements of 𝑓([𝑎,𝑏]). Therefore, the integral of 𝑓 over [𝑎,𝑏], being a limit of Riemann sums, is a multiple of an element of the closed convex hull of 𝑓([𝑎,𝑏]), that is, ^𝑏_𝑎𝑓(𝑡)𝑑𝑡=lim_℘→0(𝑏−𝑎)×_𝑗𝑓𝑡^∗_𝑗𝑡_𝑗−𝑡_𝑗−1∈()𝑏−𝑎𝑏−𝑎co𝑓([𝑎,𝑏]).(6)

2. Existence of Solutions

We begin this section enumerating the conditions that will be basic for our results on existence of solutions of  (𝐸). Take 𝑓, 𝛾_𝑗, and 𝜔 as above and, in analogy with what would happen with continuous mappings on finite dimensional spaces, assume that 𝑓isuniformlycontinuousonboundedsetsofℝ⁺×ℝ⁺×𝑋^𝑁,andmapsbounded()setsintorelativelycompactsetsof𝑋.H0

The following conditions have already been motivated in previous work [7]. Recall that 𝜔∶ℝ⁺→𝑋 is continuous, but not necessarily bounded. Thereexists𝑔∶ℝ⁺⟶ℝ⁺bounded,with𝑔(𝑡)^𝑡→∞ℝ−−−−−→0,suchthatfor𝐶=𝑦∈𝒞⁺∶𝑦(𝑡)(𝑡,𝑋−𝜔)_𝑋≤𝑔(𝑡),𝑡∈ℝ⁺,_∞0𝑓𝑡,𝑠,Γ𝑦(𝑠)_𝑋𝑑𝑠≤𝑔(𝑡),∀𝑡∈ℝ⁺(),∀𝑦∈𝐶.H1Thereexistsℎ∶ℝ⁺⟶ℝ⁺withℎ(𝜏)−−−−−−^𝜏⟶∞→0,suchthat_∞𝜏𝑓𝑡,𝑠,Γ𝑦(𝑠)_𝑋𝑑𝑠≤ℎ(𝜏),∀𝑡,𝜏∈ℝ⁺()∀𝑦∈𝐶.H2

The result on existence of solutions to the integral equation is the following.

Theorem 1. Under hypotheses (𝐻0), (𝐻1), and (𝐻2), the integral equation (𝐸) has a solution 𝑦(𝑡) asymptotically equal to 𝜔(𝑡) as 𝑡→∞.

Remark 2. This theorem represents a generalization of the one presented in the work [7] in two aspects. First, we have made the jump to deal with integral equations in the setting of infinite dimensional spaces. And second, we have included deviating arguments in the equation.

Proof of Theorem 1. First observe that it suffices to find a solution to the integral equation 𝐸 in 𝐶={𝑥∈𝒞_𝑏(ℝ⁺,𝑋)∶‖𝑥(𝑡)‖_𝑋≤𝑔(𝑡),𝑡∈ℝ⁺}, and this will be achieved by proving the existence of a fixed point in 𝐶 of the operator: (𝑡)𝑇∶𝑥∈𝐶↦𝑇𝑥,𝑇𝑥=−^∞₀𝑓(𝑡,𝑠,Γ(𝜔+𝑥)(𝑠))𝑑𝑠,𝑡≥0.(7)
We proceed to check that the conditions of the Schauder fixed point Theorem are fulfilled. First observe that 𝐶 is a nonempty, bounded, closed, and convex subset of the Banach space 𝒞_𝑏(ℝ⁺,𝑋). Also, by (H1), ‖𝑇𝑥(𝑡)‖_𝑋≤𝑔(𝑡) for all 𝑡∈ℝ⁺ and all 𝑥∈𝐶. So 𝑇𝐶⊆𝐶 provided 𝑇𝑥∈𝒞(ℝ⁺,𝑋) for all 𝑥∈𝐶.
In fact, we will prove four assertions. (a) 𝑇𝐶 is uniformly equicontinuous on ℝ⁺. This will give the desired continuity of 𝑇𝑥 for each 𝑥∈𝐶. (b) 𝑇 is uniformly continuous on 𝐶, hence it will be continuous on 𝐶. (c) 𝑇𝐶(ℝ⁺) is relatively compact in 𝑋. Thus, (a) and (c) will imply, by the Arzelà-Ascoli theorem, that each sequence in 𝑇𝐶 has a subsequence which converges uniformly on each compact subset of ℝ⁺ to a given function in 𝒞(ℝ⁺,𝑋) (actually in 𝐶). Finally, in order to obtain that 𝑇𝐶 is relatively compact in 𝒞_𝑏(ℝ⁺,𝑋), we will use the “funnel” structure of 𝐶 to prove (d), that any sequence in 𝐶 which converges uniformly on each compact subset of ℝ⁺ to a given function in 𝐶 must indeed converge uniformly to that function in all of ℝ⁺. With all these assertions, the Schauder fixed point theorem can be applied to conclude the existence of a fixed point of 𝑇, as we want.
Start fixing an arbitrary 𝜀>0 once for all. In what follows, we will build up different objects indexed by this 𝜀, (𝑡_𝜀, 𝜏_𝜀, 𝐷_𝜀, 𝐵_𝜀), knowing that even if for each assertion we have to start taking an arbitrary 𝜀>0, the objects will vary accordingly but not the way to obtain them.
Since 𝑔(𝑡)→0 as 𝑡→∞, there exists 𝑡_𝜀>0 such that 𝑔(𝑡)<𝜀2,∀𝑡≥𝑡_𝜀.(8)
We start proving (d), as it is quite independent of the rest of assertions. Assume that a sequence {𝑢_𝑛}⊆𝐶 converges uniformly on each compact subset of ℝ⁺ to a function 𝑢∈𝐶, and let us show that indeed converges uniformly to 𝑢 in all of ℝ⁺. For the 𝜀>0 above (so for any 𝜀>0), find the corresponding 𝑡_𝜀>0 to satisfy (8). Since 𝑢_𝑛, 𝑛∈ℕ, and 𝑢 all belong to 𝐶, then 𝑢_𝑛(𝑡)−𝑢(𝑡)_𝑋≤2𝑔(𝑡),∀𝑛∈ℕ,∀𝑡∈ℝ⁺.(9)
In particular, 𝑢_𝑛(𝑡)−𝑢(𝑡)_𝑋≤2𝑔(𝑡)<𝜀,∀𝑛∈ℕ,∀𝑡≥𝑡_𝜀.(10)
Now, since {𝑢_𝑛} converges uniformly to 𝑢 in [0,𝑡_𝜀], there exists 𝑛_𝜀∈ℕ such that 𝑢_𝑛(𝑡)−𝑢(𝑡)_𝑋<𝜀,∀𝑛≥𝑛_𝜀,∀𝑡∈0,𝑡_𝜀.(11)
This tells us that if 𝑛≥𝑛_𝜀, then ‖𝑢_𝑛−𝑢‖_∞≤𝜀, proving that the convergence of {𝑢_𝑛} to 𝑢 is uniform on ℝ⁺, and thus (d) is proven.
Next, continue building up other objects associated with the arbitrary 𝜀 fixed above. Observe that, by (8) and (H1), ‖𝑇𝑥(𝑡)‖_𝑋≤^∞₀𝑓(𝑡,𝑠,Γ(𝜔+𝑥)(𝑠))_𝑋(𝑡)<𝜀𝑑𝑠≤𝑔2,∀𝑡≥𝑡_𝜀∀𝑥∈𝐶.(12)
Also, since ℎ(𝜏)→0 as 𝜏→∞, there exists 𝜏_𝜀>0 such that ℎ(𝜏)<𝜀4,∀𝜏≥𝜏_𝜀,(13) so, by (H2), ^∞_𝜏𝜀𝑓(𝑡,𝑠,Γ(𝜔+𝑥)(𝑠))_𝑋𝜏𝑑𝑠≤ℎ_𝜀<𝜀4,∀𝑡∈ℝ⁺,∀𝑥∈𝐶.(14)
The continuity of 𝜔 and the 𝛾_𝑗's, 𝑗=1,…,𝑁, and the uniform bound for functions in 𝐶 (given by the bound of 𝑔) imply that there exists a bounded set 𝐷_𝜀⊆𝑋^𝑁, depending on 𝜏_𝜀, 𝜔, the 𝛾_𝑗's, and 𝑔, but not on 𝑥∈𝐶, such that Γ(𝑠)=𝜔𝛾𝜔+𝑥)(₁(𝑠)+𝑥𝛾₁(𝑠),…,𝜔𝛾_𝑁(𝑠)+𝑥𝛾_𝑁(𝑠)∈𝐷_𝜀,∀𝑥∈𝐶,∀𝑠∈0,𝜏_𝜀.(15)
Now, observe that 𝐵_𝜀∶=[0,𝑡_𝜀+1]×[0,𝜏_𝜀]×𝐷_𝜀 is bounded in ℝ⁺×ℝ⁺×𝑋^𝑁, so by (H0), 𝑓isuniformlycontinuouson𝐵_𝜀,and𝑓𝐵_𝜀isrelativelycompactin𝑋.(16)
By the uniform continuity of 𝑓 on 𝐵_𝜀, there exists 𝛿_𝜀∈(0,1) such that, whenever (𝑡₁,𝑠₁,𝑦_1,1,…,𝑦_𝑁,1)∈𝐵_𝜀 and (𝑡₂,𝑠₂,𝑦_1,2,…,𝑦_𝑁,2)∈𝐵_𝜀 with |𝑡₁−𝑡₂|<𝛿_𝜀, |𝑠₁−𝑠₂|<𝛿_𝜀, and ‖𝑦_𝑗,1−𝑦_𝑗,2‖_𝑋<𝛿_𝜀, 𝑗=1,…,𝑁, then 𝑓𝑡₁,𝑠₁,𝑦_1,1,…,𝑦_𝑁,1−𝑓𝑡₂,𝑠₂,𝑦_1,2,…,𝑦_𝑁,2𝑋<𝜀4𝜏_𝜀.(17)
Notice that if 𝑡₁,𝑡₂∈ℝ⁺ with |𝑡₁−𝑡₂|<𝛿_𝜀 and, without loss of generality, 𝑡₁≤𝑡₂, then several cases are possible. If 𝑡₂≥𝑡_𝜀+1, then, as 𝛿_𝜀<1, 𝑡₂≥𝑡₁>𝑡₂−𝛿_𝜀>𝑡_𝜀+1−1=𝑡_𝜀, and so, by (12), 𝑇𝑥𝑡₁−𝑇𝑥𝑡_2𝑋≤𝑇𝑥𝑡_1𝑋+𝑇𝑥𝑡_2𝑋<𝜀,∀𝑥∈𝐶.(18)
If, on the other hand, 𝑡₂≤𝑡_𝜀+1, then, by (15), Γ(𝜔+𝑥)([0,𝜏_𝜀])⊆𝐷_𝜀 for all 𝑥∈𝐶, and, by (17) and (14), we have, for any 𝑥∈𝐶, 𝑇𝑥𝑡₁−𝑇𝑥𝑡_2𝑋≤^𝜏𝜀₀𝑓𝑡₂(𝑠)𝑡,𝑠,Γ𝜔+𝑥)(−𝑓₁,𝑠,Γ(𝜔+𝑥)(𝑠)_𝑋+𝑑𝑠_∞𝜏𝜀𝑓𝑡₁,𝑠,Γ(𝜔+𝑥)(𝑠)_𝑋+𝑑𝑠_∞𝜏𝜀𝑓𝑡₂,𝑠,Γ(𝜔+𝑥)(𝑠)_𝑋𝑑𝑠<𝜏_𝜀𝜀4𝜏_𝜀𝜀+24<𝜀.(19) This proves (a), the uniform equicontinuity of 𝑇𝐶 over ℝ⁺.
Now notice that if 𝑥₁,𝑥₂∈𝐶 with ‖𝑥₁−𝑥₂‖_∞<𝛿_𝜀 and 𝑡∈ℝ⁺, then, again, two cases are possible. If 𝑡≥𝑡_𝜀, then, by (12), 𝑇𝑥₁(𝑡)−𝑇𝑥₂(𝑡)_𝑋≤𝑇𝑥₁(𝑡)_𝑋+𝑇𝑥₂(𝑡)_𝑋≤2𝑔(𝑡)<𝜀,(20) while for 𝑡≤𝑡_𝜀, using (15), (17), (14), and (13), 𝑇𝑥₁(𝑡)−𝑇𝑥₂(𝑡)_𝑋≤^𝜏𝜀₀𝑓𝑡,𝑠,Γ𝜔+𝑥₂(𝑠)−𝑓𝑡,𝑠,Γ𝜔+𝑥₁(𝑠)_𝑋+𝑑𝑠_∞𝜏𝜀𝑓𝑡,𝑠,Γ𝜔+𝑥₁(𝑠)_𝑋+𝑑𝑠_∞𝜏𝜀𝑓𝑡,𝑠,Γ𝜔+𝑥₂(𝑠)_𝑋𝑑𝑠<𝜏_𝜀𝜀4𝜏_𝜀+2ℎ𝜏_𝜀<𝜀,(21) This proves that ‖𝑇𝑥₁−𝑇𝑥₂‖_∞<𝜀, showing (b), that 𝑇 is uniformly continuous on 𝐶.
For the compactness of 𝑇𝐶(ℝ⁺) in 𝑋, it suffices to show that 𝑇𝐶(ℝ⁺) is totally bounded [13, page 298]; that is, for the given 𝜀>0 (so for any 𝜀>0) there exists a finite covering of 𝑇𝐶(ℝ⁺) with balls of radii not bigger than 𝜀. Observe first that, by (12), ‖𝑇𝑥(𝑡)‖_𝑋<𝜀/2 for all 𝑥∈𝐶 and all 𝑡≥𝑡_𝜀, that is, 𝑇𝐶𝑡_𝜀,∞⊆𝐵_𝑋𝜀0,2.(22)
Now, in order to control the elements of 𝑇𝐶([0,𝑡_𝜀]), observe that each of these can be decomposed as the sum of a “head” and a “tail,” 𝑇𝑥(𝑡)=−^𝜏𝜀₀𝑓(𝑡,𝑠,Γ(𝜔+𝑥)(𝑠))𝑑𝑠−^∞_𝜏𝜀𝑓(𝑡,𝑠,Γ(𝜔+𝑥)(𝑠))𝑑𝑠,∀𝑥∈𝐶andall𝑡∈0,𝑡_𝜀.(23)
The “head” can be approximated by Riemann sums, which, in turn, are nothing else but 𝜏_𝜀 times a convex linear combination of elements of −𝑓(𝐵_𝜀), that is, the “head” is an element of 𝜏_𝜀co(−𝑓(𝐵_𝜀)). By (16), 𝑓(𝐵_𝜀) has compact closure in 𝑋, so by Mazur's theorem [14], co(−𝑓(𝐵_𝜀)) is compact, and therefore it can be covered with a finite number of balls, say 𝐵₁,…,𝐵_ℓ, of radii not bigger than 𝜀/(2𝜏_𝜀). This yields a finite covering of 𝜏_𝜀co(−𝑓(𝐵_𝜀)) with balls of radii not bigger than 𝜀/2, precisely the collection {𝜏_𝜀𝐵_𝑗}^ℓ_𝑗=1. On the other hand, by (14), the “tail” of each of the above integrals is bounded by ℎ(𝜏_𝜀)<𝜀/4, so they are elements of 𝐵_𝑋(0,𝜀/4). All this can be summarized as follows: 𝑇𝐶0,𝑡_𝜀⊆𝜏_𝜀co−𝑓𝐵_𝜀+𝐵_𝑋𝜀0,4⊆^ℓ_𝑗=1𝜏_𝜀𝐵_𝑗+𝐵_𝑋𝜀0,4=^ℓ_𝑗=1𝜏_𝜀𝐵_𝑗+𝐵_𝑋𝜀0,4;(24) that is, 𝑇𝐶([0,𝑡_𝜀]) can be covered with a finite collection of balls of radii smaller than 𝜀, because each 𝜏_𝜀𝐵_𝑗+𝐵_𝑋(0,𝜀/4) is readily seen to be a ball of radius not bigger than 3𝜀/4.
At the end, by (22) and (24), we have 𝑇𝐶ℝ⁺⊆𝑇𝐶0,𝑡_𝜀∪𝑇𝐶𝑡_𝜀⊆,∞_ℓ𝑗=1𝜏_𝜀𝐵_𝑗+𝐵_𝑋𝜀0,4∪𝐵_𝑋𝜀0,2;(25) that is, we have given a finite covering of 𝑇𝐶(ℝ⁺) with balls of radii not bigger than 𝜀. With this we conclude (c) and, with all four assertions proved, the theorem too.

Coming back to the integral equation (2) underlying the differential equation (1), we just need to adapt the hypotheses presented above to the function 𝑓(𝑡,𝑠,𝑦₁,…,𝑦_𝑁)=(𝑠−𝑡)₊𝐹(𝑠,𝑦₁,…,𝑦_𝑁), to obtain a corresponding result on existence of solutions to (2) with asymptotic behavior given by 𝜔(𝑡). Notice that these hypotheses, (H0), (H1), and (H2), are natural generalizations from the 1-dimensional case. However, the proof in the abstract setting has shown many more properties for the operator 𝑇 that needed in order to show the existence of a fixed point. This, somehow, is telling us that the hypotheses could be weakened. For the moment, we content ourselves noticing that, in (2), the corresponding hypothesis of uniform continuity on bounded sets is “not needed,” because it will be “consequence” of the other hypotheses, and a little trick of changing the domain of definition of the operator. Thus, for our next result, we will be using the following hypothesis: 𝐹∶ℝ⁺×𝑋^𝑁⟶𝑋iscontinuousandmapsboundedsetsintorelativelycompactones.H0^′ Also, instead of considering just the integral equation (2), we generalize a little bit to a convolution type integral equation, 𝑦(𝑡)=𝜔(𝑡)−^∞₀𝑞(𝑡,𝑠)𝐹𝑠,Γ𝑦(𝑠)𝑑𝑠,𝑡≥0,𝐸₁ with a kernel 𝑞(𝑡,𝑠) satisfying the right properties for us. Observe that, when 𝑞(𝑡,𝑠)=(𝑠−𝑡)₊, and 0≤𝑡₁≤𝑡₂, 𝑞𝑡₁,𝑠−𝑞𝑡₂,𝑠=𝑠−𝑡₁₊−𝑠−𝑡₂₊=0if0≤𝑠≤𝑡₁𝑠−𝑡₁if𝑡₁≤𝑠≤𝑡₂𝑡₂−𝑡₁if𝑡₂≤𝑠≤𝑡₂−𝑡₁.(26)

That is, 𝑞(𝑡,𝑠) satisfies a Lipschitz condition on the first variable, independent of the second. Actually, much less is needed, just continuity of 𝑞(𝑡,𝑠) suffices.

Theorem 3. Let 𝜔, 𝛾_𝑗, 𝑗=1,…,𝑁, Γ be as for Theorem 1, and let 𝑞∶ℝ⁺×ℝ⁺→ℝ  be continuous. Under hypotheses (𝐻0^′), (𝐻1), and (𝐻2)  (where 𝑓(𝑡,𝑠,𝑦₁,…,𝑦_𝑁)=𝑞(𝑡,𝑠)𝐹(𝑠,𝑦₁,…,𝑦_𝑁)), the integral equation (E₁) has a solution 𝑦(𝑡) asymptotically equal to 𝜔(𝑡) as 𝑡→∞.

Remark 4. Observe, as in [7], that if the kernel 𝑞 is the one we started with, 𝑞(𝑡,𝑠)=(𝑠−𝑡)₊, then hypothesis (H2) is redundant, because in that case, if 𝑠≥𝜏, we always have, for 𝜏≤𝑡, (𝑠−𝑡)₊≤(𝑠−𝜏)₊≤2(𝑠−𝜏/2)₊, and for 𝜏>𝑡, (𝑠−𝑡)₊𝜏=𝑠−𝑡≤𝑠≤2𝑠−2𝜏=2𝑠−2₊.(27) Consequently, from hypothesis (H1), taking ℎ(𝜏)=3𝑔(𝜏/2), we have ℎ(𝜏)→0 as 𝜏→∞, and ^∞_𝜏(𝑠−𝑡)₊𝐹𝑠,Γ𝑦(𝑠)_𝑋𝜏𝑑𝑠≤3𝑔2=ℎ(𝜏),∀𝜏,𝑡∈ℝ⁺,∀𝑦∈𝐶.(28)
In general, hypothesis (H2) is superfluous whenever there exist constants, 𝑎∈(0,1) and 𝐴>1, such that 𝑞(𝑡,𝑠)≤𝐴𝑞(𝑎𝜏,𝑠),∀𝑡,𝑠,𝜏∈ℝ⁺,with𝑠≥𝜏.(29)

Proof of Theorem 3. We proceed as before. Consider initially the same nonempty, bounded, closed, and convex subset 𝐶 as before, as well as the same operator 𝑇. The idea is to find a fixed point for 𝑇 using the Schauder fixed point theorem. For that, we start repeating the same scheme of four assertions established in the previous theorem. The two assertions that do not depend on the uniform continuity of 𝑓 on bounded sets are done the same way as before; hence we omit their proofs. These are (c), that 𝑇𝐶(ℝ⁺) is compact in 𝑋, and (d), that uniform convergence on compact subsets of ℝ⁺ of a sequence in 𝐶 turns into actual uniform convergence on ℝ⁺.
Let us now prove (a), that 𝑇𝐶 is uniformly equicontinuous on ℝ⁺. This will finish showing that 𝑇𝐶⊆𝐶, and, by (c), (d), and the Arzelà-Ascoli theorem, that 𝑇𝐶 has compact closure in 𝒞_𝑏(𝑅⁺,𝑋).
Let 𝜀>0 be fixed. Find, as before, 𝑡_𝜀>0, 𝜏_𝜀>0, and 𝐷_𝜀 bounded subset of 𝑋^𝑁, so as to satisfy (8), (12), (13), (14), and (15). Now, observe that [0,𝜏_𝜀]×𝐷_𝜀 is bounded in ℝ⁺×𝑋^𝑁, so by H0^′, 𝐹([0,𝜏_𝜀]×𝐷_𝜀) is relatively compact in 𝑋; hence there exists 𝑀_𝜀>0 such that ‖𝐹(𝑠,Γ(𝜔+𝑥)(𝑠))‖_𝑋≤𝑀_𝜀,∀𝑠∈0,𝜏_𝜀,∀𝑥∈𝐶.(30)
Notice now that 𝑞 is uniformly continuous on 𝑅_𝜀=[0,𝑡_𝜀+1]×[0,𝜏_𝜀]. So there is 𝛿_𝜀∈(0,1) such that, whenever (𝑡₁,𝑠₁) and (𝑡₂,𝑠₂) are in 𝑅_𝜀 with |𝑡₁−𝑡₂|<𝛿 and |𝑠₁−𝑠₂|<𝛿, then 𝑞𝑡₁,𝑠₁−𝑞𝑡₂,𝑠₂<𝜀4𝜏_𝜀𝑀_𝜀.(31)
Now take 𝑡₁,𝑡₂∈ℝ⁺ with |𝑡₁−𝑡₂|<𝛿_𝜀 and, without loss of generality, assume that 𝑡₁≤𝑡₂. Then again, several cases are possible. If 𝑡₂≥𝑡_𝜀+1, then, as 𝛿_𝜀<1, 𝑡₂≥𝑡₁>𝑡₂−𝛿_𝜀>𝑡_𝜀+1−1=𝑡_𝜀, and so, by (12), 𝑇𝑥𝑡₁−𝑇𝑥𝑡_2𝑋≤𝑇𝑥𝑡_1𝑋+𝑇𝑥𝑡_2𝑋<𝜀,∀𝑥∈𝐶.(32)
If, on the other hand, 𝑡₂≤𝑡_𝜀+1, then, by (31), (30), and (14), we have, for any 𝑥∈𝐶, 𝑇𝑥𝑡₁−𝑇𝑥𝑡_2𝑋≤^𝜏𝜀₀𝑞𝑡₂,𝑠−𝑞𝑡₁×‖𝐹((𝑠,𝑠𝑠,Γ𝜔+𝑥)())‖_𝑋+𝑑𝑠_∞𝜏𝜀𝑞𝑡₁,𝑠‖𝐹(𝑠,Γ(𝜔+𝑥)(𝑠))‖_𝑋+𝑑𝑠_∞𝜏𝜀𝑞𝑡₂,𝑠‖𝐹(𝑠,Γ(𝜔+𝑥)(𝑠))‖_𝑋<𝜀𝑑𝑠4𝜏_𝜀𝑀_𝜀𝑀_𝜀𝜏_𝜀𝜀+24<𝜀.(33) This proves the uniform equicontinuity of 𝑇𝐶 over ℝ⁺.
To finish the proof, we have to prove (b), that 𝑇 is uniformly continuous. It is here that we restrict the domain of definition of 𝑇. Observe that co(𝑇𝐶) is nonempty, closed and convex. Also, co(𝑇𝐶)⊆𝐶 because 𝑇𝐶⊆𝐶 and 𝐶 is closed and convex. More is true, since 𝑇𝐶 has compact closure, then, by Mazur's Theorem, co(𝑇𝐶) is compact too. One more thing, 𝑇 leaves invariant co(𝑇𝐶): co𝑇𝐶⊆𝐶⟹𝑇co𝑇𝐶⊆𝑇𝐶⊆co𝑇𝐶.(34)
With all of this, it suffices to prove that 𝑇 is continuous on this new 𝑇-invariant set. Let us prove that, indeed, 𝑇 is uniformly continuous on co(𝑇𝐶). Let 𝜀>0 be given. Find that 𝑡_𝜀>0 and 𝜏_𝜀>0 as before, so as to satisfy (8), (12), (13), and (14). The continuity of 𝜔 and the 𝛾_𝑗's, 𝑗=1,…,𝑁, and the compactness of co(𝑇𝐶) tells us that 𝐷_𝜀={Γ(𝜔+𝑥)(𝑠)∶𝑠∈[0,𝜏_𝜀],𝑥∈co(𝑇𝐶)} is a compact subset of 𝑋^𝑁, giving us the opportunity to say that 𝐹 is uniformly continuous on [0,𝜏_𝜀]×𝐷_𝜀 and, consequently, that 𝑓(𝑡,𝑠,𝑦₁,…,𝑦_𝑁)=𝑞(𝑡,𝑠)𝐹(𝑠,𝑦₁,…,𝑦_𝑁) is uniformly continuous on 𝐵_𝜀∶=[0,𝑡_𝜀+1]×[0,𝜏_𝜀]×𝐷_𝜀. So there exists 𝛿_𝜀∈(0,1) such that, for (𝑡₁,𝑠₁,𝑦_1,1,…,𝑦_𝑁,1)∈𝐵_𝜀 and (𝑡₂,𝑠₂,𝑦_1,2,…,𝑦_𝑁,2)∈𝐵_𝜀 with |𝑡₁−𝑡₂|<𝛿_𝜀, |𝑠₁−𝑠₂|<𝛿_𝜀, and ‖𝑦_𝑗,1−𝑦_𝑗,2‖_𝑋<𝛿_𝜀, 𝑗=1,…,𝑁, then 𝑓𝑡₁,𝑠₁,𝑦_1,1,…,𝑦_𝑁,1−𝑓𝑡₂,𝑠₂,𝑦_1,2,…,𝑦_𝑁,2𝑋<𝜀4𝜏_𝜀.(35) Now, it is a matter of repeating the same steps as was done in Theorem 1 to prove the uniform continuity of 𝑇.
This concludes the proof of the theorem.

Remark 5. Instead of working with functions defined on ℝ⁺, we could have worked with functions defined on any interval of the type [𝑡₀,∞), obtaining a result completely similar. Also, many times, one is just interested in giving partial solutions; that is, solutions defined not on the whole interval [𝑡₀,∞), but on some other interval [𝑡₁,∞) with 𝑡₁≥𝑡₀.

Next, we just mention an easy result on the existence of solutions of the underlying differential equation, just to illustrate the type of functions 𝐹 that could generate a condition like (H1). This result is inspired from [7, Thm. 2] and [8, Thm. 1].

Theorem 6. Let 𝐹∶ℝ⁺×𝑋^𝑁→𝑋 be a continuous function mapping bounded sets into relatively compact ones, such that, 𝐹𝑡,𝑦₁,…,𝑦_𝑁𝑋≤𝜓(𝑡),∀𝑡∈ℝ⁺,∀𝑦₁,…,𝑦_𝑁∈𝑋^𝑁,(36) where 𝜓∶ℝ⁺⟶ℝ⁺,𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠^∞₀𝑠𝜓(𝑠)𝑑𝑠<∞.(37) Let also {𝛾_𝑗}^𝑁_𝑗=1 be a set of continuous functions from ℝ⁺ to ℝ⁺.
Then, for any 𝜔∈𝒞²(ℝ⁺,𝑋), (1) has a solution 𝑦∈𝒞²(ℝ⁺,𝑋) with 𝑦(𝑡)−𝜔(𝑡)→0 as 𝑡→∞.

Proof. Take 𝜔∈𝒞²(ℝ⁺,𝑋), consider the corresponding integral equation (2), and define 𝑔(𝑡)=^∞_𝑡𝑠𝜓(𝑠)𝑑𝑠,𝑡≥0.(38)
Observe that, by (37), 𝑔(𝑡)→0 as 𝑡→∞. This allows us to consider the set 𝐶=𝑦∈𝒞ℝ⁺,𝑋∶𝑦(𝑡)−𝜔(𝑡)_𝑋≤𝑔(𝑡),𝑡≥0,(39) which, adopting the notation used throughout the paper, gives by (36), for 𝑦∈𝐶 and 𝑡≥0, (𝑠−𝑡)^∞_𝑡𝐹𝑠,Γ𝑦(𝑠)_𝑋𝑑𝑠≤^∞_𝑡(𝑠−𝑡)𝜓(𝑠)𝑑𝑠≤𝑔(𝑡).(40)
With this, Theorem 3 applies (hypothesis (H2) need not be verified by Remark 4) and then there exists 𝑦∈𝒞(ℝ⁺,𝑋), solution of (2) with 𝑦(𝑡)−𝜔(𝑡)→0 as 𝑡→∞. Finally, it is just an exercise to check that 𝑦 is twice continuously differentiable and that satisfies the differential equation (1).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Research partially supported by the Spanish (Grant MTM2011-25502) and regional Andalusian (Grants FQM210 and P09-FQM4468) Governments.