Second-Order Impulsive Differential Equations with Functional Initial Conditions on Unbounded Intervals

Abstract

We present some results on the existence of solutions for second-order impulsive differential equations with deviating argument subject to functional initial conditions. Our results are based on Schaefer's fixed point theorem for completely continuous operators.

1. Introduction

Differential equations with impulses arise quite often in the study of different problems in particular are used as a model for evolutionary processes subject to a sudden rapid change of their state at certain moments. The theory of impulsive differential equations has become recently a quite active area of research. For an introduction to this theory we refer to the books [1–4], which also contain a variety of interesting examples and applications.

In this paper, we establish new results for the existence of solutions for the second-order impulsive differential equation with deviating argument subject to initial conditions: 𝑥^′′(𝑡)(𝑡)((𝑡)(𝛼(𝑡,[)⧵𝑡=𝐴𝑥+𝑓𝑡,𝑥,𝑥)))𝑡∈0,+∞₁,…,𝑡_𝑙𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡_𝑘𝑥,𝑘=1,…,𝑙_′𝑡⁺_𝑘−𝑥^′𝑡⁻_𝑘=𝐽_𝑘𝑥𝑡_𝑘𝑥(𝑡)(𝑡)[],𝑥,𝑘=1,…,𝑙=𝜙,𝑡∈−𝑟,0_′(0)=𝜙^′(0)(P)∶=𝜂, where 𝑥(𝑡) is in the space (ℝ^𝑛,|⋅|) (|⋅| not necessarily the Euclidean norm), Θ∶={0<𝑡₁<⋯<𝑡_𝑙}⊆[0,+∞), 𝐴 is a real 𝑛×𝑛 matrix, 𝑥(𝜏⁻), and 𝑥(𝜏⁺) are the left and right limits of 𝑥 in 𝑡=𝜏.

Such type of problem arises from a few of scientific applications as, for example, the problem of impulsive maneuver of a spacecraft (see [5] in finite-dimentional setting and [6] in infinite-dimentional setting). More recently, impulsive second-order differential equations or inclusions on compact intervals subject to nonhomogeneous conditions have been studied by several authors (see for example [7–14] and the references therein). The study of differential problems on unbounded interval has been done, for example, in [14–18].

Differential equations with deviating argument are investigated, for example, in [6, 7, 9, 12, 14, 16, 17, 19, 20].

Various techniques are utilized in the above papers: Schauder's fixed point Theorem [6, 9, 16, 17, 19, 20], Leray-Schauder's nonlinear alternative [14, 18], Contractions Principle [8, 13], fixed point index theory [12], and Sadovskii's fixed point Theorem [7]. Moreover, in order to prove the compactness of involved operator, the Ascoli-Arzelà Theorem is often used (see [6, 9, 12, 14, 18]).

The methodology here is to write the problem (P) as a perturbed integral equation, and we look for fixed points of an operator 𝑇 in a suitable functions space. For this purpose we want to utilize the Schaefer's fixed point theorem for completely continuous operators. In order to prove that 𝑇 is completely continuous, we make use of a variant of the compactness result in the Banach space 𝐵𝐶(𝑄,ℝ^𝑛) of continuous bounded functions from a topological space 𝑄 into ℝ^𝑛 due to De Pascale, Lewicki and G. Marino. We have used similar techniques and tools in [16, 17, 20] to study first-order, impulsive, or nonimpulsive, differential or integrodifferential equations on unbounded intervals. In this paper we extend these ideas to second-order impulsive equations.

Moreover we prove that our method can be easily used also in the case of functional conditions that, to the best of the our knowledge, are not studied for this class of problems. In fact, we discuss in details the Problem (P) in the case of the initial conditions 𝑥(𝑡)=𝜙(𝑡)𝑡∈[−𝑟,0],𝑥^′(0)=𝜙^′(0)(1) because it sheds light on the techniques used, but the same approach may be applied to impulsive equations subject to a more general functional initial condition that covers a large number of cases, namely 𝐿𝑥=𝐻(𝑥),𝑥^′(0)=𝜂,(2) where 𝐿 is a bounded linear operator and 𝐻 is a completely continuous operator. This is done in the last section.

2. Notations and Preliminaries

Firstly, we recall definitions, notations, and useful facts regarding the cosine families (see [21–23] for the detailed study of cosine and sine families).

Definition 1. A one-parameter family (𝐶(𝑡))_𝑡∈ℝ of bounded linear mappings on ℝ^𝑛 into itself is called a strongly continuous cosine family if and only if (1)𝐶(𝑡+𝑠)+𝐶(𝑡−𝑠)=2𝐶(𝑡)𝐶(𝑠), for all 𝑡,𝑠∈ℝ,(2)𝐶(0)=𝐼, where 𝐼 is the identity map, (3)𝐶(⋅)𝑥∈𝐶(ℝ,ℝ^𝑛), for all 𝑥∈ℝ^𝑛. Moreover, the sine family (𝑆(𝑡))_𝑡∈ℝ is defined as 𝑆(𝑡)∶=^𝑡₀𝐶(𝑠)𝑑𝑠.(3)
By definition one obtains that: (u1)𝑆(𝑡)𝑥 is continuous in 𝑡∈ℝ, 𝑆(0)=0 and 𝑆(−𝑡)=−𝑆(𝑡), for all 𝑡∈ℝ, (u2)𝐶(𝑡)=𝐶(−𝑡), for all 𝑡∈ℝ, (u3)𝐶(𝑡),𝑆(𝑡),𝑆(𝑠),𝐶(𝑠) commute, for all 𝑡,𝑠∈ℝ, (u4)𝑆(𝑡+𝑠)+𝑆(𝑡−𝑠)=2𝑆(𝑡)𝐶(𝑠), for all 𝑡,𝑠∈ℝ, (u5)𝑆(𝑡+𝑠)=𝑆(𝑡)𝐶(𝑠)+𝐶(𝑡)𝑆(𝑠), for all 𝑡,𝑠∈ℝ.

Definition 2. The infinitesimal generator of a strongly continuous cosine family is the operator 𝐴∶ℝ^𝑛→ℝ^𝑛 defined by 𝑑𝐴∶=²𝑑𝑡²𝐶(𝑡)_𝑡=0(4) and Dom(𝐴)∶={𝑥∈ℝ^𝑛∶𝐶(𝑡)𝑥iscontinuouslydifferentiablein𝑡=0}.
One obtains that, for 𝑥∈ℝ^𝑛 and 𝑡∈ℝ, 𝑑²𝑑𝑡²𝐶(𝑡)𝑥=𝐴𝐶(𝑡)𝑥=𝐶(𝑡)𝐴𝑥(5) and then 𝑑𝐶(𝑡)𝑑𝑡𝑥=^𝑡₀𝐴𝐶(𝑠)𝑥𝑑𝑠=𝐴^𝑡₀𝐶(𝑠)𝑥𝑑𝑠=𝐴𝑆(𝑡)𝑥.(6) So, by (u5) one has (u6)𝐶(𝑡+𝑠)=𝐶(𝑡)𝐶(𝑠)+𝐴𝑆(𝑡)𝑆(𝑠), for all 𝑡,𝑠∈ℝ.

Definition 3. A cosine family (𝐶(𝑡))_𝑡∈ℝ (resp., a sine family (𝑆(𝑡))_𝑡∈ℝ) is uniformly bounded if there exists 𝑀_𝐶>0 (resp., 𝑀_𝑆>0) such that |𝐶(𝑡)|≤𝑀_𝐶resp.|𝑆(𝑡)|≤𝑀_𝑆∀𝑡∈ℝ,(7) where |⋅| denotes the norm in the space of the 𝑛×𝑛 matrices.

Example 4. In ℝ^𝑛, let (𝐶(𝑡))_𝑡∈ℝ be the family of bounded linear operators 𝐶(𝑡)=cos𝑎₁𝑡00cos𝑎₂𝑡𝑎⋮⋮00⋯0⋯0⋮⋮⋯cos_𝑛𝑡.(8) Let us observe that (𝐶(𝑡))_𝑡∈ℝ is a uniformly bounded cosine family and its infinitesimal generator is 𝑑𝐴=²𝑑𝑡²𝐶(𝑡)_𝑡=0−𝑎²₁00−𝑎²₂⋮⋮00…0…0⋮⋮…−𝑎²_𝑛.(9)

We will utilize the Schaefer's fixed point theorem for completely continuous operators.

Theorem 5 (see [24]). Let 𝐸 be a normed linear space. Let 𝑆∶𝐸→𝐸 be a completely continuous operator. Let 𝜁(𝑆)∶=𝑥∈𝐸∶𝑥=𝜆𝑆𝑥𝑓𝑜𝑟𝑠𝑜𝑚𝑒0<𝜆<1.(10) Then either 𝜁(𝑆) in unbounded or 𝑆 has a fixed point.

We use the condition for compactness in the Banach space 𝐵𝐶(𝑄,ℝ^𝑛) of continuous bounded functions 𝑦 from a topological space 𝑄 into ℝ^𝑛, endowed with the norm ‖𝑦‖_∞∶=sup{|𝑦(𝑡)|,𝑡∈𝑄}, due to De Pascale, Lewicki and G. Marino.

Theorem 6 (see [25]). Let 𝑇∶𝐵𝐶(𝑄,ℝ^𝑛)→𝐵𝐶(𝑄,ℝ^𝑛) be a continuous operator. Suppose that, for any bounded set 𝐹⊂𝐵𝐶(𝑄,ℝ^𝑛), 𝑇(𝐹) is a bounded set and there exist 𝜈 bounded functions 𝜑_𝑗∶𝑄→ℝ^𝑛, 𝑗=1,2,…,𝜈, such that, for all 𝑡,𝑠∈𝑄 and for all 𝑦∈𝐹, 𝑇𝑦(𝑡)−𝑇𝑦(𝑠)≤^𝜈_𝑗=1𝜑_𝑗(𝑡)−𝜑_𝑗(𝑠).(11) Then 𝑇 is a compact operator.

3. The Integral Problem

From now on, we assume that the functions 𝑓,𝜙,𝛼,𝐼_𝑘,𝐽_𝑘 and the matrix 𝐴 have the following properties.(ℎ₀)The matrix 𝐴 is the infinitesimal generator of a strongly continuous cosine family (𝐶(𝑡))_𝑡∈ℝ, uniformly bounded by 𝑀_𝐶>0. Suppose moreover that the corresponding sine family (𝑆(𝑡))_𝑡∈ℝ is uniformly bounded by a constant 𝑀_𝑆>0. (ℎ₁)𝑓∶[0,+∞)×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 is a continuous function such that there exist a continuous integrable function 𝑝∶[0,+∞)→[0,+∞) and a continuous nondecreasing function Ψ∶[0,+∞)→[1,+∞) for which 𝑓𝑡,𝑥,𝑦≤𝑝(𝑡)Ψ|𝑥|+𝑦,𝑡≥0,𝑥,𝑦∈ℝ^𝑛,(12)_∞0𝑑𝑠Ψ(𝑠)=+∞.(13)(ℎ₂)𝛼∶[0,+∞)→[−𝑟,+∞) is a continuous increasing function such that 𝛼(𝑡)≤𝑡, for all 𝑡∈[0,+∞). The function 𝜙 belongs to 𝐶¹([−𝑟,0],ℝ^𝑛).(ℎ₃)𝐼_𝑘∶ℝ^𝑛→ℝ^𝑛 and 𝐽_𝑘∶ℝ^𝑛→ℝ^𝑛, 𝑘=1,…,𝑙 are continuous functions such that there are 𝐷_𝐼,𝐷_𝐽>0 for which 𝐼_𝑘(𝑣)≤𝐷_𝐼,𝐽_𝑘(𝑣)≤𝐷_𝐽,∀𝑣∈ℝ^𝑛.(14)

We will work in the Banach space 𝐵𝑃𝐶_Θ[)[)−𝑟,+∞∶=𝑦∶−𝑟,+∞⟶ℝ^𝑛𝑡∣𝑦isboundedandcontinuousin𝑡∉Θ,thereexist𝑦⁻_𝑘=𝑦𝑡_𝑘and𝑦𝑡⁺_𝑘<∞,𝑡_𝑘∈Θ,(15) endowed with the supremum norm ‖𝑦‖_∞=sup{|𝑦(𝑡)|∶𝑡∈[−𝑟,+∞)}.

We define for 𝑥∈𝐵𝑃𝐶_Θ[−𝑟,+∞): 𝑤_𝑥(𝑡)∫∶=^𝑡0𝑆()𝑓((𝑠)(𝛼(𝑠(𝑡)∑𝑡−𝑠𝑠,𝑥,𝑥)))𝑑𝑠,𝑡≥0,0,−𝑟≤𝑡≤0𝐼𝑥)(∶=^{0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘,𝑡>𝑡₁,0,−𝑟≤𝑡≤𝑡₁(𝑡)∑𝐽𝑥)(∶=^{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐽_𝑘𝑥𝑡_𝑘,𝑡>𝑡₁,0,−𝑟≤𝑡≤𝑡₁.(16) For any 𝑥∈𝐵𝑃𝐶_Θ[−𝑟,+∞) let 𝑇𝑥 be the function defined on [−𝑟,+∞) by (𝑡)𝐶(𝑡)𝜙(0)(𝑡)𝑇𝑥)(∶=+𝑆𝜂+𝑤_𝑥(𝑡)+(𝑡)+(𝑡)𝜙(𝑡)𝐼𝑥)(𝐽𝑥)(,𝑡≥0,−𝑟≤𝑡≤0.(17) In the next propositions we show the properties of the operator 𝑇 useful for us.

Proposition 7. The operator 𝑇 maps 𝐵𝑃𝐶_Θ[−𝑟,+∞) into 𝐵𝑃𝐶_Θ[−𝑟,+∞) and (𝑇𝑥)(𝑡⁺_𝑘)−(𝑇𝑥)(𝑡⁻_𝑘)=𝐼_𝑘(𝑥(𝑡_𝑘)). Moreover Tx has derivative in 𝑡∈[−𝑟,+∞)⧵Θ, (𝑇𝑥)^′(𝑡⁻_𝑘)=(𝑇𝑥)^′(𝑡_𝑘), (𝑇𝑥)^′(𝑡⁺_𝑘)<∞ and (𝑇𝑥)^′(𝑡⁺_𝑘)−(𝑇𝑥)^′(𝑡⁻_𝑘)=𝐽_𝑘(𝑥(𝑡_𝑘)).

Proof. The continuity of 𝑓, 𝑆, and 𝐶 guarantees that 𝑇𝑥 is continuous in 𝑡∈[−𝑟,+∞)⧵Θ, and there exists (𝑇𝑥)^′(𝑡) for 𝑡∈[−𝑟,+∞)⧵Θ. Moreover, since 𝑝 is integrable, for 𝑡∈[−𝑟,+∞)⧵Θ, one has |(𝑇𝑥)(𝑡)|≤𝑀_𝐶𝜙(0)+𝑀_𝑆𝜂+𝑀_𝑆Ψ2‖𝑥‖_∞×^∞₀𝑝(𝑠)𝑑𝑠+𝑙𝑀_𝐶𝐷_𝐼+𝑙𝑀_𝑆𝐷_𝐽,(18) so ‖𝑇𝑥‖_∞<∞. We need to show that, for 𝑘=1,…,𝑙, there exist (𝑇𝑥)(𝑡⁺_𝑘) and (𝑇𝑥)(𝑡⁻_𝑘)=(𝑇𝑥)(𝑡_𝑘). Note that(i)lim_{𝑡→𝑡𝑘}𝑤_𝑥(𝑡)=𝑤_𝑥(𝑡_𝑘). Indeed, using (u5), for 𝑡→𝑡_𝑘𝑤_𝑥(𝑡)−𝑤_𝑥𝑡_𝑘≤^𝑡₀𝑆(𝑡−𝑠)−𝑆𝑡_𝑘⋅𝑓((𝑠)(𝛼(𝑠+−𝑠𝑠,𝑥,𝑥)))𝑑𝑠_𝑡𝑘𝑡𝑆𝑡_𝑘⋅𝑓((𝑠)(𝛼(𝑠ℎ−𝑠𝑠,𝑥,𝑥)))𝑑𝑠by₁≤^𝑡₀𝑆(𝑡−𝑠)−𝑆𝑡_𝑘−𝑠Ψ2‖𝑥‖_∞𝑝(𝑠)𝑑𝑠+𝑀_𝑆Ψ2‖𝑥‖_∞^𝑡𝑘_𝑡𝑝(𝑠)≤𝑆(𝑡)𝑡𝑑𝑠−𝑆_𝑘+𝐶(𝑡)−𝐶𝑡_𝑘×max𝑀_𝑆,𝑀_𝐶Ψ2‖𝑥‖_∞^𝑡₀𝑝(𝑠)𝑑𝑠+𝑀_𝑆Ψ2‖𝑥‖_∞^𝑡𝑘_𝑡𝑝(𝑠)𝑑𝑠⟶0.(19)(ii)It results that (𝐼𝑥)(𝑡⁻_𝑘)=(𝐼𝑥)(𝑡_𝑘)=∑^𝑘−1_𝑗=1𝐶(𝑡_𝑘−𝑡_𝑗)𝐼_𝑗(𝑥(𝑡_𝑗)). Indeed let 𝑡_𝑘−1<𝑡<𝑡_𝑘. Then (𝐼𝑥)(𝑡)−(𝐼𝑥)𝑡_𝑘=_{0<𝑡𝑗<𝑡}𝐶𝑡−𝑡_𝑗𝐼_𝑗𝑥𝑡_𝑗−_{0<𝑡𝑗<𝑡𝑘}𝐶𝑡_𝑘−𝑡_𝑗𝐼_𝑗𝑥𝑡_𝑗,byℎ₃≤𝐷_𝐼^𝑘−1_𝑗=1𝐶𝑡−𝑡_𝑗−𝐶𝑡_𝑘−𝑡_𝑗⟶0.(20) Analogously one can see that (𝐼𝑥)𝑡⁺_𝑘=^𝑘_𝑗=1𝐶𝑡_𝑘−𝑡_𝑗𝐼_𝑗𝑥𝑡_𝑗,(21) in such a way that (𝐼𝑥)(𝑡⁺_𝑘)−(𝐼𝑥)(𝑡⁻_𝑘)=𝐼_𝑘(𝑥(𝑡_𝑘)). Since, by (u1), 𝑆(𝑡_𝑘−𝑡_𝑘)=0, a similar proof permits to verify that (𝐽𝑥)𝑡⁺_𝑘−(𝐽𝑥)𝑡⁻_𝑘=^𝑘_𝑗=1𝑆𝑡_𝑘−𝑡_𝑗𝐽_𝑗𝑥𝑡_𝑗+^𝑘−1_𝑗=1𝑆𝑡_𝑘−𝑡_𝑗𝐽_𝑗𝑥𝑡_𝑗=0.(22)By the above steps, it follows that (𝑇𝑥)(𝑡⁺_𝑘)−(𝑇𝑥)(𝑡⁻_𝑘)=𝐼_𝑘(𝑥(𝑡_𝑘)).
Since there exists (𝑇𝑥)^′(𝑡) if 𝑡∈[0,+∞)⧵Θ, to conclude our proof we show that there exists (𝑇𝑥)^′(𝑡⁻_𝑘)=(𝑇𝑥)^′(𝑡_𝑘), (𝑇𝑥)^′(𝑡⁺_𝑘) is finite, and (𝑇𝑥)^′(𝑡⁺_𝑘)−(𝑇𝑥)^′(𝑡⁻_𝑘)=𝐽_𝑘(𝑥(𝑡_𝑘)). We also observe that, for 𝑡≥0 and 𝑡∉Θ, by (u6), (𝑇𝑥)^′(𝑡)(𝑡)𝜙(0)(𝑡)𝜂+=𝐴𝑆+𝐶_𝑡0𝐶(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),𝑥(𝛼(𝑠)))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐴𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐽_𝑘𝑥𝑡_𝑘.(23) Thus, following the same idea, when 𝑡→𝑡_𝑘, we have ^𝑡₀𝐶()𝑓((𝑠)(𝛼(𝑠−𝑡−𝑠𝑠,𝑥,𝑥)))𝑑𝑠_𝑡𝑘0𝐶𝑡_𝑘𝑓((𝑠)(𝛼(𝑠≤−𝑠𝑠,𝑥,𝑥)))𝑑𝑠_𝑡0𝐶(𝑡−𝑠)−𝐶𝑡_𝑘−𝑠𝑝(𝑠)Ψ2‖𝑥‖_∞𝑑𝑠+𝑀_𝐶Ψ2‖𝑥‖_∞^𝑡𝑘_𝑡𝑝(𝑠)𝑑𝑠⟶0.(24) Moreover, for 𝑡→𝑡⁻_{𝑘0<𝑡𝑗<𝑡}𝐴𝑆𝑡−𝑡_𝑗𝐼_𝑗𝑥𝑡_𝑗−^𝑘−1_𝑗=1𝐴𝑆𝑡_𝑘−𝑡_𝑗𝐼_𝑗𝑥𝑡_𝑗≤𝐷_𝐼|𝐴|^𝑘−1_𝑗=1𝑆𝑡−𝑡_𝑗−𝑆𝑡_𝑘−𝑡_𝑗⟶0,lim_{𝑡→𝑡+𝑘0<𝑡𝑘<𝑡}𝐴𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘=^𝑘−1_𝑗=1𝐴𝑆𝑡−𝑡_𝑘𝐼_𝑗𝑥𝑡_𝑗.(25) Finally, for 𝑡→𝑡⁻_{𝑘0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐽_𝑘𝑥𝑡_𝑘−^𝑘−1_𝑗=1𝐶𝑡_𝑘−𝑡_𝑗𝐽_𝑗𝑥𝑡_𝑗≤𝐷_𝐽^𝑘−1_𝑗=1𝑆𝑡−𝑡_𝑗−𝑆𝑡_𝑘−𝑡_𝑗⟶0,(26) while lim_{𝑡→𝑡+𝑘0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐽_𝑘𝑥𝑡_𝑘=^𝑘_𝑗=1𝐶𝑡−𝑡_𝑘𝐽_𝑗𝑥𝑡_𝑗,(27) and these permit to obtain that there exist (𝑇𝑥)^′(𝑡⁺_𝑘) and (𝑇𝑥)^′(𝑡⁻_𝑘) and that one has (𝑇𝑥)^′(𝑡⁺_𝑘)−(𝑇𝑥)^′(𝑡⁻_𝑘)=𝐽_𝑘(𝑥(𝑡_𝑘)).

Proposition 8. The fixed points of 𝑇 are solutions of the problem (P).

Proof. In fact, 𝑇𝑥|_[−𝑟,0]=𝜙 and (𝑇𝑥)^′(0)=𝜂. Moreover, by the hypotheses on 𝑓 and on the sine and cosine families, one obtains that (𝑇𝑥)^′ is derivable in 𝑡∈[−𝑟,+∞)⧵Θ, and it results in the following: (𝑇𝑥)^′′(𝑡)(𝑡)𝜙(0)(𝑡)𝜂+=𝐴𝐶+𝐴𝑆_𝑡0()𝑓((𝑠)(𝛼(𝑠((𝑡)(𝛼(𝑡𝐴𝑆𝑡−𝑠𝑠,𝑥,𝑥)))𝑑𝑠+𝑓𝑡,𝑥,𝑥)))+_{0<𝑡𝑘<𝑡}𝐴𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐴𝑆𝑡−𝑡_𝑘𝐽_𝑘𝑥𝑡_𝑘.(28) So one can conclude that (𝑇𝑥)^′′(𝑡)=𝐴𝑇𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡),𝑥(𝛼(𝑡))),(29) that is, 𝑇𝑥 satisfies the differential equation in problem (P). This combined with the results in Proposition 7 yields the thesis.

In order to prove that 𝑇 is a completely continuous operator, we make use of the following compactness criterion in 𝐵𝑃𝐶_Θ[−𝑟,+∞).

Proposition 9 (see [16]). Let 𝑇∶𝐵𝑃𝐶_Θ[−𝑟,+∞)→𝐵𝑃𝐶_Θ[−𝑟,+∞) be a continuous operator. Suppose that, for any bounded set 𝐹⊂𝐵𝑃𝐶_Θ[−𝑟,+∞), 𝑇(𝐹) is a bounded set, and there exist 𝜈 bounded functions 𝜑_𝑗∶[−𝑟,+∞)→ℝ^𝑛, 𝑗=1,2,…,𝜈, such that, for all 𝑡,𝑠∈[−𝑟,+∞) and, for all 𝑦∈𝐹𝑇𝑦(𝑡)−𝑇𝑦(𝑠)≤^𝜈_𝑗=1𝜑_𝑗(𝑡)−𝜑_𝑗(𝑠).(30) Then 𝑇 is a compact operator.

Proof. The Banach space 𝐵𝑃𝐶_Θ[−𝑟,+∞) is isometric to the Banach space: Ω=𝑦∈𝐵𝐶𝑄,ℝ^𝑛∶thereexist𝑦𝑡⁺_𝑘and𝑦𝑡⁻_𝑘,for𝑘=1,…,𝑙,(31) where 𝑄=[−𝑟,+∞)⧵Θ. Of course, Ω is closed in 𝐵𝐶(𝑄,ℝ^𝑛). So the thesis follows from Theorem 6.

Proposition 10. 𝑇 is a completely continuous operator.

Proof. First we prove that 𝑇 is a continuous operator. Let (𝑦_𝑚)_𝑚∈ℕ a sequence in 𝐵𝑃𝐶_Θ[−𝑟,+∞) such that 𝑦_𝑚→𝑦 in 𝐵𝑃𝐶_Θ[−𝑟,+∞). We prove that 𝑇𝑦_𝑚→𝑇𝑦. By the continuity of 𝑓 it results that, for fixed 𝑠∈[0,+∞), 𝜎_𝑚(𝑠)∶=𝑓𝑠,𝑦_𝑚(𝑠),𝑦_𝑚(𝛼(𝑠(𝑠)(𝛼(𝑠))−𝑓𝑠,𝑦,𝑦))⟶0.(32) Moreover, by the monotonicity of Ψ, we have 𝜎_𝑚(𝑠)≤𝑓𝑠,𝑦_𝑚(𝑠),𝑦_𝑚(𝛼(𝑠+𝑓(𝑠)(𝛼(𝑠(𝑠Ψ2𝑦))𝑠,𝑦,𝑦))≤𝑝)_𝑚∞+Ψ2𝑦_∞≤2𝑝(𝑠)Ψ(2Λ),(33) where Λ=max{sup_𝑚∈ℕ‖𝑦_𝑚‖_∞,‖𝑦‖_∞}. Therefore |𝜎_𝑚(𝑠)| is dominated by an integrable function that does not depend on 𝑚∈ℕ. Then, from the uniform boundedness of the sine family and by the dominated convergence theorem, we have 𝑤_𝑦𝑚−𝑤_𝑦∞=sup_{0≤𝑡<+∞}𝑤_𝑦𝑚(𝑡)−𝑤_𝑦(𝑡)=sup_{0≤𝑡<+∞}^𝑡₀𝑆(𝑡−𝑠)𝜎_𝑚(𝑠)𝑑𝑠≤𝑀_𝑆^∞₀𝜎_𝑚(𝑠)𝑑𝑠⟶0.(34) The uniform boundedness of the cosine family and the continuity of 𝐼_𝑘 and 𝐽_𝑘 imply that, as 𝑚→∞, 𝐼𝑦_𝑚−𝐼𝑦_∞≤sup_{𝑡∈[𝑡1,+∞)0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑦_𝑚𝑡_𝑘−𝐼_𝑘𝑦𝑡_𝑘≤𝑀_𝐶^𝑙_𝑘=1𝐼_𝑘𝑦_𝑚𝑡_𝑘−𝐼_𝑘𝑦𝑡_𝑘⟶0,𝐽𝑦_𝑚−𝐽𝑦_∞⟶0.(35) Thus 𝑇𝑦_𝑚−𝑇𝑦_∞≤𝑤_𝑦𝑚−𝑤_𝑦∞+𝐽𝑦_𝑚−𝐽𝑦_∞+𝐼𝑦_𝑚−𝐼𝑦_∞⟶0.(36) Now, let 𝐵_𝜌∶=𝑢∈𝐵𝑃𝐶_Θ[−𝑟,+∞)∶‖𝑢‖_∞≤𝜌.(37) Thanks to Proposition 9, it is enough to show that 𝑇(𝐵_𝜌) is a bounded set and that it is possible to control the oscillations of each function in 𝑇(𝐵_𝜌) by means of a finite number of bounded functions. The boundedness of 𝑇(𝐵_𝜌) follows by the inequalities: 𝑤_𝑥∞≤𝑀_𝑆Ψ2𝜌^∞₀𝑝(𝑠)‖‖𝑑𝑠,𝐼𝑥_∞≤𝑀_𝐶^𝑙_𝑘=1𝐼_𝑘𝑥𝑡_𝑘≤𝑙𝑀_𝐶𝐷_𝐼,‖𝐽𝑥‖_∞≤𝑀_𝑆^𝑙_𝑘=1𝐽_𝑘𝑥𝑡_𝑘≤𝑙𝑀_𝑆𝐷_𝐽.(38) To control the oscillations of 𝑇(𝐵_𝜌) we should need to distinguish three cases: 0≤𝜏₁<𝜏₂, 𝜏₁<0<𝜏₂, and 𝜏₁<𝜏₂≤0.
Here we study the case 0≤𝜏₁<𝜏₂ only because the proofs of the other cases are similar.
Let 0≤𝜏₁<𝜏₂ and 𝑥∈𝐵_𝜌. Then (𝑇𝑥)𝜏₁−(𝑇𝑥)𝜏₂≤𝐶𝜏₁−𝐶𝜏₂𝜙(0)+𝑆𝜏₁−𝑆𝜏₂𝜂+^𝜏1₀𝑆𝜏₁−𝑠−𝑆𝜏₂𝑓((𝑠)(𝛼(𝑠+−𝑠𝑠,𝑥,𝑥)))𝑑𝑠_𝜏2𝜏1𝑆𝜏₂𝑓((𝑠)(𝛼(𝑠+)𝜏−𝑠𝑠,𝑥,𝑥)))𝑑𝑠(𝐼𝑥₁−(𝐼𝑥)𝜏₂+(𝐽𝑥)𝜏₁−(𝐽𝑥)𝜏₂.(39) Now, by property (u5), one obtains that 𝑆𝜏₂−𝑠−𝑆𝜏₁=𝑆𝜏−𝑠₂−𝑆𝜏₁𝐶(𝑠)−𝐶𝜏₂−𝐶𝜏₁𝑆(𝑠),(40) so we have ^𝜏1₀𝑆𝜏₁−𝑠−𝑆𝜏₂𝑓((𝑠)(𝛼(𝑠≤𝑆𝜏−𝑠𝑠,𝑥,𝑥)))𝑑𝑠₂−𝑆𝜏₁×^𝜏1₀|𝐶(𝑠𝑓((𝑠)(𝛼(𝑠+𝐶𝜏)|𝑠,𝑥,𝑥)))𝑑𝑠₂−𝐶𝜏₁×^𝜏1₀|𝑆(𝑠𝑓((𝑠)(𝛼(𝑠≤𝑆𝜏)|𝑠,𝑥,𝑥)))𝑑𝑠₂−𝑆𝜏₁𝑀_𝐶Ψ2𝜌^∞₀𝑝(𝑠)+𝐶𝜏𝑑𝑠₂−𝐶𝜏₁𝑀_𝑆Ψ2𝜌^∞₀𝑝(𝑠)𝑑𝑠.(41) Using (ℎ₃) we obtain also that ^𝜏2_𝜏1𝑆𝜏₂−𝑠𝑓(𝑠,𝑥(𝑠),𝑥(𝛼(𝑠)))𝑑𝑠≤𝑀_𝑆Ψ2𝜌^𝜏2₀𝑝(𝑠)𝑑𝑠−^𝜏1₀𝑝(𝑠)𝑑𝑠.(42) To control the oscillations of the operators 𝐼 and 𝐽, first we note that if 𝜏₁<𝑡₁, and 𝜏₂<𝑡₁ we have nothing to prove because (𝐼𝑥)(𝜏₁)=(𝐼𝑥)(𝜏₂)=(𝐽𝑥)(𝜏₁)=(𝐽𝑥)(𝜏₂)=0. The boundedness of 𝐼 and 𝐽 solves also the case 𝜏₁<𝑡₁<𝜏₂ since we have (𝐼𝑥)𝜏₂−(𝐼𝑥)𝜏₁=(𝐼𝑥)𝜏₂≤𝐷_{𝐼0<𝑡𝑘<𝜏2}𝑀_𝐶,(𝐽𝑥)𝜏₂−(𝐽𝑥)𝜏₁=(𝐽𝑥)𝜏₂≤𝐷_{𝐽0<𝑡𝑘<𝜏2}𝑀_𝑆.(43) Thus it remains to prove only the case 𝑡₁<𝜏₁<𝜏₂. We have that (𝐼𝑥)𝜏₁−(𝐼𝑥)𝜏₂=_{0<𝑡𝑘<𝜏1}𝐶𝜏₁−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘−_{0<𝑡𝑘<𝜏2}𝐶𝜏₂−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘≤_{0<𝑡𝑘<𝜏1}𝐶𝜏₁−𝑡_𝑘−𝐶𝜏₂−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+_{𝜏1≤𝑡𝑘<𝜏2}𝐶𝜏₂−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘≤𝐷_𝐼^𝑙_𝑘=1𝐶𝜏₁−𝑡_𝑘−𝐶𝜏₂−𝑡_𝑘+𝐷_{𝐼0<𝑡𝑘<𝜏2}𝑀_𝐶−_{0<𝑡𝑘<𝜏1}𝑀_𝐶.(44) In a similar way we can show that (𝐽𝑥)𝜏₁−(𝐽𝑥)𝜏₂≤𝐷_𝐽^𝑙_𝑘=1𝑆𝜏₁−𝑡_𝑘−𝑆𝜏₂−𝑡_𝑘+𝐷_{𝐽0<𝑡𝑘<𝜏2}𝑀_𝑆−_{0<𝑡𝑘<𝜏1}𝑀_𝑆.(45) So, defining 𝜓₁(𝑡)=𝐶(𝑡)𝜙(0),𝜓₂(𝑡)(𝑡)𝜓=𝑆𝜂,₃(𝑡)=𝑀_𝐶𝐾𝑆(𝑡),𝜓₄(𝑡)=𝑀_𝑆(𝑡),𝜓𝐾𝐶₅(𝑡)=𝑀_𝑆Ψ2𝜌^𝑡₀𝑝(𝑠)𝜓𝑑𝑠,₆(𝑡)=𝐷_𝐼𝐶𝑡−𝑡₁,𝜓₇(𝑡)=𝐷_𝐼𝐶𝑡−𝑡₂,…,𝜓_𝑙+5(𝑡)=𝐷_𝐼𝐶𝑡−𝑡_𝑙,𝜓_𝑙+6(𝑡)=𝐷_𝐼𝑀_{𝐶0<𝑡𝑘<𝑡}𝜓1,_𝑙+7(𝑡)=𝐷_𝐽𝑆𝑡−𝑡₁,𝜓_𝑙+8(𝑡)=𝐷_𝐽𝑆𝑡−𝑡₂,…,𝜓_2𝑙+6(𝑡)=𝐷_𝐽𝑆𝑡−𝑡_𝑙,𝜓_2𝑙+7(𝑡)=𝐷_𝐽𝑀_{𝑆0<𝑡𝑘<𝑡}1,(46) where 𝐾=Ψ(2𝜌)∫^∞₀𝑝(𝑠)𝑑𝑠, and we obtain that for all 𝑥∈𝐵_𝜌(𝑇𝑥)𝜏₁−(𝑇𝑥)𝜏₂≤^2𝑙+7_𝑘=1𝜓_𝑘𝜏₁−𝜓_𝑘𝜏₂.(47)

4. Main Result

Theorem 11. Assume that the hypotheses (ℎ₀)−(ℎ₃) hold. Then the problem (P) has at least one solution.

Proof. Our problem (P) can be reduced, by Propositions 7 and 8, to find a fixed point for the operator 𝑇. Proposition 10 assures that 𝑇 is a completely continuous operator.
To apply the Schaefer's fixed point theorem, it remains to prove that the set 𝜁(𝑇)∶=𝑥∈ℝ^𝑛∶𝑥=𝜆𝑇𝑥,forsome0<𝜆<1(48) is bounded. The proof is based on an idea in [19].
Let 𝑥=𝜆(𝑇𝑥) with 𝜆∈(0,1). First of all, since {𝑇𝑥|_[−𝑟,0]∶𝑥∈𝐵𝑃𝐶_Θ[−𝑟,+∞)}={𝜙}, for 𝑡∈[−𝑟,0] we have |𝑥(𝑡)|=𝜆|(𝑇𝑥)(𝑡)|<|(𝑇𝑥)(𝑡)|=𝜙(𝑡)≤𝜙_∞.(49) We consider now that 𝑡≥0. Thus we have |𝑥(𝑡)|=𝜆|(𝑇𝑥)(𝑡)|<𝑀_𝐶𝜙_∞+𝑀_𝑆𝜂+𝑀_𝑆^𝑡₀𝑝(𝑠)Ψ(|𝑥(𝑠)|+|𝑥(𝛼(𝑠))|)𝑑𝑠+𝑙𝑀_𝐶𝐷_𝐼+𝑙𝑀_𝑆𝐷_𝐽.(50) Consider the function 𝜇_𝑥∶[0,+∞)→[0,+∞) defined by 𝜇_𝑥(𝑡)∶=sup𝑥(𝜉)∶0≤𝜉≤𝑡.(51) Observe that 𝜇_𝑥 is not necessarily continuous in Θ but 𝜇_𝑥(𝑡⁺_𝑘) and 𝜇_𝑥(𝑡⁻_𝑘)=𝜇_𝑥(𝑡_𝑘) exist, for all 𝑡_𝑘∈Θ.
For 𝜉∈[0,𝑡], we have 𝑥(𝜉)≤𝑀_𝐶𝜙_∞+𝑀_𝑆𝜂+𝑀_𝑆^𝑡₀𝑝(𝑠)Ψ(|𝑥(𝑠)|+|𝑥(𝛼(𝑠))|)𝑑𝑠+𝑙𝑀_𝑆𝐷_𝐼+𝑙𝑀_𝐶𝐷_𝐽.(52) Moreover, being 𝛼 increasing and 𝛼(𝑡)≤𝑡 for 𝑡≥0, one has sup_{0≤𝑠≤𝑡}|𝑥(𝛼(𝑠))|=sup_{𝛼(0)≤𝑠≤𝛼(𝑡)}|𝑥(𝑠)|≤sup_{0≤𝑠≤𝑡}|𝑥(𝑠)|=𝜇_𝑥(𝑡)(53) so, taking the supremum over [0,𝑡] in the inequality (52), we obtain that 𝜇_𝑥(𝑡)≤𝑀_𝐶𝜙_∞+𝑀_𝑆𝜂+𝑀_𝑆^𝑡₀𝑝(𝑠)Ψ2𝜇_𝑥(𝑡)𝑑𝑠+𝑙𝑀_𝑆𝐷_𝐼+l𝑀_𝐶𝐷_𝐽.(54) Denoting by 𝑣_𝑥(𝑡) the right-hand side of the last inequality, we have that the function 𝑣_𝑥 is continuous: 𝑐∶=𝑣_𝑥(0)=1+𝑀_𝐶𝜙_∞+𝑀_𝑆𝜂+𝑙𝑀_𝑆𝐷_𝐼+𝑙𝑀_𝐶𝐷_𝐽(55) and 𝜇_𝑥(𝑡)≤𝑣_𝑥(𝑡) for 𝑡≥0.
Moreover, since Ψ is nondecreasing, for 𝑡∉Θ, 𝑣^′_𝑥(𝑡)≤𝑀_𝑆𝑝(𝑡)Ψ2𝑣_𝑥(𝑡).(56) This implies that 𝑣^′_𝑥(𝑡)Ψ2𝑣_𝑥(𝑡)≤𝑀_𝑆𝑝(𝑡),𝑡∉Θ(57) and so, for any 𝑏>0, ^𝑏₀𝑣^′_𝑥(𝑡)Ψ2𝑣_𝑥(𝑡)𝑑𝑡≤𝑀_𝑆^𝑏₀𝑝(𝑡)∶=Γ_𝑏<∞.(58) Since 𝑣^′_𝑥 is a continuous function for all 𝑡∉Θ, we have ^{2𝑣𝑥(𝑏)}_2𝑐𝑑𝑠2Ψ(𝑠)≤Γ_𝑏.(59) This, together with condition (13), permits us to conclude that 𝑣_𝑥 is bounded by a constant Δ depending on the functions Ψ and 𝑝 only.
Summarizing, 𝑥∈𝜁(𝑇) implies that ‖𝑥‖_∞≤max{‖𝜙‖_∞,Δ}.

5. Functional Initial Conditions

Let us consider the problem 𝑥^′′(𝑡)(𝑡)((𝑡)(𝛼(𝑡,[)⧵𝑡=𝐴𝑥+𝑓𝑡,𝑥,𝑥)))𝑡∈0,+∞₁,…,𝑡_𝑙𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡_𝑘𝑥,𝑘=1,…,𝑙_′𝑡⁺_𝑘−𝑥^′𝑡⁻_𝑘=𝐽_𝑘𝑥𝑡_𝑘(𝑥),𝑘=1,…,𝑙𝐿𝑥=𝐻,𝑥^′(0)P=𝜂.^F The modularity of the operator 𝑇 used in the proof of our Theorem 11 permits to prove a result on the existence of solutions for Problem P_F modifying few parts of the above proof.

For 𝑢∈𝐵𝑃𝐶_Θ[−𝑟,+∞), we define the function 𝐶𝑢∶[−𝑟,+∞)→ℝ^𝑛 as 𝑡)𝑢(𝑡)𝐶(𝑡)𝑢(0)𝐶𝑢(∶=,−𝑟≤𝑡≤0,𝑡≥0.(60) We suppose that (ℎ₄)𝐿∶𝐵𝑃𝐶_Θ[−𝑟,+∞)→𝐶([−𝑟,0],ℝ^𝑛) is a bounded linear operator for which if 𝑢,𝑣∈𝐵𝑃𝐶_Θ[−𝑟,+∞) are such that 𝑢|_[−𝑟,0]=𝑣|_[−𝑟,0], then 𝐿𝑢=𝐿𝑣. 𝐻∶𝐵𝑃𝐶_Θ[−𝑟,+∞)→𝐶([−𝑟,0];ℝ^𝑛) is a completely continuous operator such that ‖𝐻(𝑢)‖_∞≤𝑀₁ for a certain constant 𝑀₁. (ℎ₅) For 𝑢∈𝐵𝑃𝐶_Θ[−𝑟,+∞), 𝐻(𝑢)=(𝐿𝐶𝐻)(𝑢).

Theorem 12. Assume that the hypotheses (ℎ₀)−(ℎ₅) hold. Then the problem (𝑃_𝐹) has at least one solution.

Proof. Let us consider the operator 𝑊 defined on 𝐵𝑃𝐶_Θ[−𝑟,+∞) as (𝑊𝑥)(𝑡)∶=𝐶𝐻(𝑥)(𝑡)+𝑆(𝑡)𝜂+𝑤_𝑥(𝑡)+(𝐼𝑥)(𝑡)+(𝐽𝑥)(𝑡).(61) Following the proof of Theorem 11, it is necessary to study only the part of the operator arising from the functional initial condition. First, for 𝑡≥0, one has 𝐶𝐻(𝑥)(𝑡)≤𝑀_𝐶|(𝐻(𝑢))(𝑡)|≤𝑀_𝐶𝑀₁⇒𝐶𝐻(𝑥)_∞≤𝑀_𝐶𝑀₁(62) and, by the continuity of 𝐶(𝑡), as 𝑡→𝑡_𝑘𝐶𝐻(𝑥)(𝑡)−𝐶𝐻(𝑥)𝑡_𝑘≤𝑀₁𝐶(𝑡)−𝐶𝑡_𝑘⟶0.(63) These, following Proposition 7, are enough to prove that 𝑊𝑥∈𝐵𝑃𝐶_Θ[−𝑟,+∞) and (𝑊𝑥)(𝑡⁺_𝑘)−(𝑊𝑥)(𝑡⁻_𝑘) = 𝐼_𝑘(𝑥(𝑡_𝑘)).
Moreover, it is not difficult to verify that (𝐶𝐻(𝑥))^′(𝑡)=𝐴𝑆(𝑡)(𝐻(𝑢)(𝑡)) for 𝑡≥0, and this is enought to prove that (𝐶𝐻(𝑥))^′(𝑡⁺_𝑘)=(𝐶𝐻(𝑥))^′(𝑡⁻_𝑘) so that one has (𝑊𝑥)^′𝑡⁺_𝑘−(𝑊𝑥)^′𝑡⁻_𝑘=𝐽_𝑘𝑥𝑡_𝑘.(64) From the hypotheses on 𝐿 one has 𝐿𝑆(𝑡)𝜂=𝐿𝑤_𝑥(𝑡)=𝐿(𝐼𝑥)(𝑡)=𝐿(𝐽𝑥)(𝑡)=0,(65) so, it follows from (ℎ₅) that 𝐿(𝑊𝑥)(𝑡)=𝐿𝐶𝐻(𝑥)(𝑡)+𝐿𝑆(𝑡)𝜂+𝐿𝑤_𝑥(𝑡)(𝑡)(𝑡)(𝑥)(𝑡)=(𝐻(𝑥𝑡).+𝐿𝐼𝑥)(+𝐿𝐽𝑥)(=𝐿𝐶𝐻))((66)
Moreover, (𝐶𝐻(𝑥))^′′(𝑡)=𝐴(𝐶𝐻(𝑥))(𝑡) and so, using Proposition 8, the fixed points of 𝑊 are solutions of P_F. It remains to prove that we can control the oscillations of 𝑊𝑥 by a finite number of bounded functions.
We need to control only the oscillations of 𝐶𝐻(𝑥). If 0<𝜏₁<𝜏₂ (the other case are similar), we have 𝐶𝐻(𝑥)𝜏₁−𝐶𝐻(𝑥)𝜏₂≤𝐶𝜏₁−𝐶𝜏₂𝑀₁.(67) So, by defining 𝛾(𝑡)∶=𝑀₁𝐶(𝑡) we have 𝐶𝐻(𝑥)𝜏₁−𝐶𝐻(𝑥)𝜏₂≤𝛾𝜏₁−𝛾𝜏₂.(68)
Hence 𝑊 is a completely continuous operator. Following the same proof of Theorem 11 one can see that the set 𝜁(𝑊)∶=𝑥∈ℝ^𝑛∶𝑥=𝜆𝑊𝑥,forsome0<𝜆<1(69) is bounded, so 𝑊 has fixed points that are solutions of P_F. This complete the proof.

Remark 13. The reader is referred to [5, 9, 13, 19] for some examples and applications.