Existence of solutions for impulsive partial neutral functional differential equations

ISSN 1749-3889 (print), 1749-3897 (online)International Journal of Nonlinear Science

Vol.11(2011) No.4,pp.412-426

Existence of Global Solutions for Impulsive Abstract Partial Neutral FunctionalDifferential Equations

S. Sivasankaran1, V. Vijayakumar2 ∗, M. Mallika Arjunan3

1 Department of Mathematics, University College, Sungkyunkwan University, Suwon 440-746, South Korea.2 Department of Mathematics, Info Institute of Engineering, Kovilpalayam, Coimbatore - 641 107, Tamil Nadu, India.

3 Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore- 641 114, Tamil Nadu, India.(Received 20 March 2011, accepted 26 May 2011)

Abstract: In this paper we study the existence of global solutions for a class of impulsive abstract partialneutral functional differential equations. An application is provided to illustrate the theory.

Keywords: impulsive system; neutral differential equations; analytic semigroup theory; mild solutions;global solutions.

1 IntroductionMany evolution processes and phenomena experience abrupt changes of state through short-term perturbations. Since thedurations of the perturbations are negligible in comparison with the duration of each process, it is quite natural to assumethat these perturbations act in terms of impulses; see the monographs of Lakshmikantham et al. [17] and Samoilenkoand Perestyuk [31]. It is to be noted that the recent progress in the development of the qualitative theory of impulsivedifferential equations has been stimulated primarily by a number of interesting applied problems [1–3, 7, 18, 19, 21]. Anatural generalization of impulsive differential equations is abstract impulsive differential equations in Banach space. Forgeneral aspects of impulsive differential equations, monographs [29, 30] are recommended.

Recently in [8], the authors studied the existence of global solutions for an impulsive abstract functional differentialequation of the form ⎧⎨⎩

𝑢′(𝑡) = 𝐴𝑢(𝑡) + 𝑓(𝑡, 𝑢(𝑡), 𝑢(𝜌(𝑡))), 𝑡 ∈ 𝐼 = [0, 𝑎] or [0,∞),

𝑢(0) = 𝑢0,

Δ𝑢(𝑡𝑖) = 𝐽𝑖(𝑢(𝑡𝑖)), 𝑖 ∈ 𝔽 ⊂ ℕ,

by using Leray-Schauder’s alternative theorem and in [4], the existence of solutions for impulsive neutral functionaldifferential equations of the form⎧⎨⎩

𝑑𝑑𝑡 (𝑢(𝑡) + 𝐹 (𝑡, 𝑢𝑡)) = 𝐴(𝑡)𝑢(𝑡) +𝐺(𝑡, 𝑢𝑡), 𝑡 ∈ 𝐼, 𝑡 ∕= 𝑡𝑖,

Δ𝑢(𝑡𝑖) = 𝐼𝑖(𝑢𝑡𝑖),

𝑢0 = 𝜑 ∈ ℬ,by using Leray-Schauder’s alternative theorem.

In this paper, we establish the existence of mild solutions for a class of impulsive neutral functional differentialequations described by

𝑑𝑡(𝑢(𝑡) + 𝑔(𝑡, 𝑢𝑡)) = 𝐴𝑢(𝑡) + 𝑓(𝑡, 𝑢𝑡), 𝑡 ∈ 𝐼, 𝑡 ∕= 𝑡𝑖, (1)

𝑢0 = 𝜑 ∈ ℬ, (2)

Δ𝑢(𝑡𝑖) = 𝐼𝑖(𝑢𝑡𝑖), (3)∗Corresponding author Email address: vijaysarovel@gmail.com

Copyright c⃝World Academic Press, World Academic UnionIJNS.2011.06.30/491

S. Sivasankaran, V. Vijayakumar, M. M. Arjunan: Existence of Global Solutions for Impulsive Abstract Partial Neutral ⋅ ⋅ ⋅ 413

where 𝐴 is the infinitesimal generator of a 𝐶0-semigroup of bounded linear operators ((𝑇 (𝑡))𝑡≥0 defined on a Banachspace (𝕏, ∥ ⋅ ∥); 𝐼 is an interval of the form [0, 𝑇 ); 0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑖 < ⋅ ⋅ ⋅ < 𝑇 are prefixed numbers; the history𝑢𝑡 : (−∞, 0] → 𝕏, 𝑢𝑡(𝜃) = 𝑢(𝑡+ 𝜃), belongs to some abstract phase space ℬ defined axiomatically; 𝑔, 𝑓 : 𝐼 × ℬ → 𝕏,𝐼𝑖 : ℬ → 𝕏 are appropriate functions and the symbol Δ𝜉(𝑡) represents the jump of the function 𝜉 at 𝑡, which is definedby Δ𝜉(𝑡) = 𝜉(𝑡+)− 𝜉(𝑡−).

The existence of global solutions for impulsive ordinary and partial differential system has been considered recentlyin the literature, see among others, [4, 8, 10–15, 23–26, 32–34]. To the best of our knowledge, the study of the existenceof global solutions for the impulsive neutral functional differential equations described in the general “abstract” form(1.1)-(1.3), is a untreated topic in the literature; this will be the main motivation of our paper.

We now turn to a summary of the work. The second section provides the definitions and preliminary results to be usedin the theorems stated and proved in this article; we review some of the standard facts on phase spaces, mild solutions andcertain useful fixed point theorems. In the third section, we focus our attention on the local existence of mild solutions forthe problem (1.1)-(1.3), when 𝐼 = [0, 𝑎]. The fourth section is dedicated to the study of the existence of global solutionsfor the problem (1.1)-(1.3). Finally, in the fifth section, we give an application.

2 PreliminariesWe introduce certain notations which will be used throughout the paper without any further mention. Let (𝕏, ∥ ⋅ ∥𝕏) and(𝕐, ∥ ⋅ ∥𝕐) be Banach spaces, and ℒ(𝕐,𝕏) be the Banach space of bounded linear operators from 𝕐 into 𝕏 equipped withits natural topology; in particular, we use the notation ℒ(𝕏) when 𝕐 = 𝕏.

Throughout this work, 𝐴 is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators(𝑇 (𝑡))𝑡≥0 defined on a Banach space (𝑋, ∥ ⋅ ∥) and �� is a positive constant such that ∥𝑇 (𝑡)∥ ≤ �� for every 𝑡 ∈ 𝐼 . Forthe theory of strongly continuous semigroup, refer the reader to Pazy [27].

In this paper, we employ an axiomatic definition of the phase space ℬ, which is similar to the one introduced in Hinoet al [16] and suitably modified to treat retarded impulsive differential equations. More precisely, ℬ is a linear space offunctions mapping (−∞, 0] into 𝕏 endowed with a seminorm ∥ ⋅ ∥ℬ and we assume that ℬ satisfies the following axioms:

(A) If 𝑥 : (−∞, 𝜎 + 𝑎] → 𝕏, 𝑎 > 0, 𝜎 ∈ ℝ such that 𝑥𝜎 ∈ ℬ, and 𝑥∣[𝜎, 𝜎 + 𝑎] ∈ 𝒫𝒞([𝜎, 𝜎 + 𝑎],𝕏), then for every𝑡 ∈ [𝜎, 𝜎 + 𝑎) the following conditions hold:

(i) 𝑥𝑡 is in ℬ;

(ii) ∥𝑥(𝑡)∥𝕏 ≤ 𝐻∥𝑥𝑡∥ℬ;

(iii) ∥𝑥𝑡∥ℬ ≤ 𝐾(𝑡− 𝜎) sup{∥𝑥(𝑠)∥𝕏 : 𝜎 ≤ 𝑠 ≤ 𝑡}+𝑀(𝑡− 𝜎)∥𝑥𝜎∥ℬ,

where 𝐻 > 0 is a constant; 𝐾,𝑀 : [0,∞) → [1,∞), 𝐾 is continuous, 𝑀 is locally bounded and 𝐻,𝐾,𝑀 areindependent of 𝑥(⋅).

(B) The space ℬ is complete.

Remark 1 In impulsive functional differential systems, the map [𝜎, 𝜎 + 𝑎] → ℬ, 𝑡→ 𝑥𝑡 is in general discontinuous. Forthis reason, this property has been omitted from the description of phase space ℬ.

Example 1 The Phase Space 𝒫𝒞𝑟 × 𝐿2(𝑔,𝕏) .Let 𝑟 > 0 and 𝑔 : (−∞,−𝑟] → ℝ be a non-negative, locally Lebesgue integrable function. Assume that there is a

non-negative measurable, locally bounded function 𝜂(⋅) on (−∞, 0] such that 𝑔(𝜉 + 𝜃) ≤ 𝜂(𝜉)𝑔(𝜃) for all 𝜉 ∈ (−∞, 0]and 𝜃 ∈ (−∞,−𝑟] ∖ 𝑁𝜉, where 𝑁𝜉 ⊂ (−∞,−𝑟] is a set with Lebesgue measure zero. We denote by 𝒫𝒞𝑟 × 𝐿2(𝑔,𝕏)the set of all functions 𝜑 : (−∞, 0] → 𝕏 such that 𝜑∣[−𝑟, 0] ∈ 𝒫𝒞([−𝑟, 0],𝕏) and

∫ −𝑟

−∞ 𝑔(𝜃)∥𝜑(𝜃)∥2𝕏𝑑𝜃 < +∞. In𝒫𝒞𝑟 × 𝐿2(𝑔,𝕏), we consider the seminorm defined by

∥𝜑∥ℬ = sup𝜃∈[−𝑟,0]

∥𝜑(𝜃)∥𝕏 +

(∫ −𝑟

−∞𝑔(𝜃)∥𝜑(𝜃)∥2𝕏𝑑𝜃

From the proceeding conditions, the space 𝒫𝒞𝑟 × 𝐿2(𝑔,𝕏) satisfies the axioms (A) and (B). Moreover, when 𝑟 = 0, we

can take 𝐻 = 1, 𝐾(𝑡) =(1 +

−𝑡𝑔(𝜃)

and 𝑀(𝑡) = 𝜂(−𝑡) for 𝑡 ≥ 0.

IJNS homepage: http://www.nonlinearscience.org.uk/

414 International Journal of Nonlinear Science, Vol.11(2011), No.4, pp. 412-426

Let 0 < 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑛 < 𝑎 be pre-fixed numbers. We introduce the space 𝒫𝒞 = 𝒫𝒞([0, 𝑎];𝕏) formed by all functions𝑢 : [0, 𝑎] → 𝕏 such that are continuous at 𝑡 ∕= 𝑡𝑖, 𝑢(𝑡−𝑖 ) = 𝑢(𝑡𝑖) and 𝑢(𝑡+𝑖 ) exists, for all 𝑖 = 1, ....𝑛. In this paper, weassume that 𝒫𝒞 is endowed with the norm ∥𝑢∥𝒫𝒞 = sup𝑠∈[0,𝑎] ∥𝑢(𝑠)∥𝕏. It is clear that (𝒫𝒞, ∥ ⋅ ∥𝒫𝒞) is a Banach space;see [9] for details.

In what follows, we put 𝑡0 = 0, 𝑡𝑛+1 = 𝑎, and for 𝑢 ∈ 𝒫𝒞, we denote by ��𝑖 ∈ 𝐶([𝑡𝑖, 𝑡𝑖+1],𝕏), 𝑖 = 0, 1, 2, ..., 𝑛, thefunction given by

��𝑖(𝑡) =

{𝑢(𝑡), for 𝑡 ∈ (𝑡, 𝑡𝑖+1],

𝑢(𝑡+𝑖 ), for 𝑡 = 𝑡𝑖.

Moreover, for ℬ ⊂ 𝒫𝒞, we employ the notation ��𝑖, 𝑖 = 0, 1, 2, ..., 𝑛, for the sets ��𝑖 = {��𝑖 : 𝑢 ∈ 𝐵}.

Lemma 2 ([9]). A Set 𝐵 ⊂ 𝒫𝒞 is relatively compact in 𝒫𝒞 if and only if the set 𝐵𝑖 is relatively compact in the space𝐶([𝑡𝑖, 𝑡𝑖+1];𝕏), for every 𝑖 = 0, 1, ..., 𝑛.

In the following definition, we introduce the concept of a mild solution for the problem (1.1)-(1.3).

Definition 1 ([28]). A function 𝑢 : (−∞, 0]∪𝐼 → 𝕏 is called a mild solution of the abstract Cauchy problem (1.1)-(1.3),if 𝑢0 = 𝜙 ∈ ℬ; 𝑢∣𝐼 ∈ 𝒫𝒞(𝐼;𝕏); the function 𝑠→ 𝐴𝑇 (𝑡− 𝑠)𝑔(𝑠, 𝑢𝑠) is integrable in [0, 𝑡) for all 𝑡 ∈ 𝐼 and

𝑢(𝑡) = 𝑇 (𝑡)(𝜙(0) + 𝑔(0, 𝜙)

)− 𝑔(𝑡, 𝑢𝑡)−∫ 𝑡

𝐴𝑇 (𝑡− 𝑠)𝑔(𝑠, 𝑢𝑠)𝑑𝑠+

∫ 𝑡

𝑇 (𝑡− 𝑠)𝑓(𝑠, 𝑢𝑠)𝑑𝑠

+∑𝑡𝑖<𝑡

𝑇 (𝑡− 𝑡𝑖)𝐼𝑖(𝑢𝑡𝑖) +∑𝑡𝑖<𝑡

[𝑔(𝑡, 𝑢𝑡)∣𝑡+𝑖

− 𝑔(𝑡, 𝑢𝑡)∣𝑡−𝑖

], 𝑡 ∈ 𝐼.(4)

Motivated by the previous definition, we introduce the following assumptions. There exists a Banach space (𝕐, ∥ ⋅ ∥𝕐)continuously included in 𝕏 such that

(S1) The function 𝑠→ 𝑇 (𝑡− 𝑠)𝑦 ∈ 𝐶([𝑡,+∞);𝕐), 𝑡 ≥ 0.

(S2) The function 𝑠 → 𝐴𝑇 (𝑡 − 𝑠) defined from (𝑡,+∞), 𝑡 > 0 into 𝐿(𝕐,𝕏) is continuous and there is a functionℋ(⋅) ∈ 𝐿1([0,∞);ℝ+) such that

∥𝐴𝑇 (𝑡− 𝑠)∥ℒ(𝕐,𝕏) ≤ ℋ(𝑡− 𝑠), for all 𝑡 > 𝑠.

Remark 3 Assume that (S1) and (S2) hold and 𝑢 ∈ 𝐶([0, 𝑡];𝕐), then from the Bochner’s criterion for integrable func-tions and the estimate

∥𝐴𝑇 (𝑡− 𝑠)𝑢(𝑠)∥𝕏 ≤ ∥𝐴𝑇 (𝑡− 𝑠)∥ℒ(𝕐,𝕏)∥𝑢(𝑠)∥𝕐 ≤ ℋ(𝑡− 𝑠)∥𝑢(𝑠)∥𝕐,

we have that the function 𝑠 → 𝐴𝑇 (𝑡 − 𝑠)𝑢(𝑠) is integrable over [0, 𝑡) for all 𝑡 > 0. For additional remarks about thesetypes of condition in partial neutral differential equations, see e.g.[20]. In general, we observe that, except in trivialcases, the operator function 𝑠→ 𝐴𝑇 (𝑡− 𝑠) is not integrable over [0,t](see e.g.[5]).

To obtain our results we need the following results.

Theorem 4 ([6, Theorem 6.5.4]) Leray-Schauder’s Alternative Theorem). Let 𝐷 be a closed convex subset of a Banachspace (ℤ, ∥ ⋅ ∥ℤ) and assume that 0 ∈ 𝐷. If 𝐹 : 𝐷 → 𝐷 is a completely continuous map, then the set {𝑥 ∈ 𝐷 : 𝑥 =𝜆𝐹 (𝑥), 0 < 𝜆 < 1} is unbounded or the map 𝐹 has a fixed point in 𝐷.

Theorem 5 ([22, Corollary 4.3.2]). Let 𝐷 be a closed, convex and bounded subset of a Banach space (ℤ, ∥ ⋅ ∥ℤ). If𝐵,𝐶 : 𝐷 → ℤ are continuous functions such that

(a) 𝐵𝑧 + 𝐶𝑧 ∈ 𝐷 for all 𝑧 ∈ ℤ;

(b) 𝐶(𝐷) is compact;

(c) there exists 0 ≤ 𝛾 < 1 such that ∥𝐵𝑧 −𝐵𝑤∥ ≤ 𝛾∥𝑧 − 𝑤∥ for all 𝑧, 𝑤 ∈ 𝐷;

then there exists 𝑥 ∈ 𝐷 such that 𝐵𝑥+ 𝐶𝑥 = 𝑥.

IJNS email for contribution: editor@nonlinearscience.org.uk

3 Local Existence

In this section, we study the existence of mild solutions for the impulsive systems (1.1)-(1.3), when 𝐼 = [0, 𝑎]. To obtainour results, we introduce the following conditions.

(H1) The function 𝑔 : 𝐼 × ℬ → 𝕏 is completely continuous, 𝑔(𝐼 × ℬ) ⊂ 𝑌 , 𝑔 ∈ 𝐶(𝐼 × ℬ,𝕐) and there exist positiveconstants 𝑐1 and 𝑐2 such that

∥𝑔(𝑡, 𝜙)∥𝕐 ≤ 𝑐1∥𝜙∥ℬ + 𝑐2, 𝑡 ∈ 𝐼, 𝜙 ∈ ℬ.

(H2) The function 𝑓 : 𝐼 × ℬ × 𝕏 → 𝕏 satisfies the following conditions.

(a) The function 𝑓(𝑡, ⋅) : ℬ → 𝕏 is continuous for all 𝑡 ∈ 𝐼 .

(b) The function 𝑓(⋅, 𝜙) : 𝐼 → 𝕏 is strongly measurable for all 𝜙 ∈ ℬ.

(c) There exists a continuous function 𝑚 : 𝐼 → [0,∞) and a continuousnon-decreasing function 𝑊 : [0,∞) → (0,∞) such that

∥𝑓(𝑡, 𝜙)∥ ≤ 𝑚(𝑡)𝑊 (∥𝜙∥ℬ), for every (𝑡, 𝜙) ∈ 𝐼 × ℬ.

(H3) The maps 𝐼𝑖 : ℬ → 𝕏 are completely continuous and uniformly bounded, 𝑖 ∈ 𝔽 = {1, 2, ⋅ ⋅ ⋅𝑁}. In what follows,we use the notation: 𝑁𝑖 = sup{∥𝐼𝑖(𝜙)∥ : 𝜙 ∈ ℬ}.

(H4) There are positive constants 𝐿𝑖, such that

∥𝐼𝑖(𝜓1)− 𝐼𝑖(𝜓2)∥ ≤ 𝐿𝑖∥𝜓1 − 𝜓2∥ℬ, 𝜓1, 𝜓2 ∈ ℬ, 𝑖 ∈ 𝔽.

(H5) The function 𝑔 : 𝐼 × ℬ → 𝑌 is Lipschitz constant; that is, there is a constant 𝐿𝑔 > 0 such that

∥𝑔(𝑡, 𝜙1)− 𝑔(𝑡, 𝜙2)∥𝕐 ≤ 𝐿𝑔∥𝜙1 − 𝜙2∥ℬ, 𝑡 ∈ 𝐼, 𝜙𝑖 ∈ ℬ, 𝑖 = 1, 2.

(H6) Let 𝑦 : (−∞, 0] ∪ 𝐼 → 𝕏 be the extension of 𝜙 such that 𝑦(𝑡) = 𝑇 (𝑡)𝜙(0) for all 𝑡 ∈ 𝐼 and 𝑆(𝐼) the space𝑆(𝐼) = {𝑥 : (−∞, 0]∪ 𝐼 → 𝕏 : 𝑥0 = 0, 𝑥∣𝐼 ∈ 𝒫𝒞(𝐼;𝕏)} endowed with uniform convergence topology in 𝐼 . Then

the set of functions{𝑡→ 𝑔(𝑡, 𝑥𝑡 + 𝑦𝑡); 𝑥 ∈ 𝑄

}is equicontinuous in 𝐼 for all bounded sets 𝑄 ⊂ 𝑆(𝐼).

After these preparations, we can formulate the main result of this section.

Theorem 6 Assume that the hypotheses (S1), (S2), (H1), (H2), (H3) and (H6) are fulfilled. Suppose, in addition, thatthe following properties hold.

(a) For all 𝑡, 𝑠 ∈ [0, 𝑎], 𝑡 > 𝑠 and 𝑟 > 0, the set {𝑇 (𝑡− 𝑠)𝑓(𝑠, 𝜓) : 𝑠 ∈ [0, 𝑡], ∥𝜓∥ℬ ≤ 𝑟} is relatively compact in 𝕏.

(b) 𝜇 = 𝑐1𝐾𝑎

(∥𝑖𝑐∥ℒ(𝕐,𝕏) +

∫ 𝑎

0ℋ(𝑠)𝑑𝑠

)< 1, ��𝐾𝑎

1−𝜇

∫ 𝑎

0𝑚(𝑠)𝑑𝑠 <

∫∞𝑐

𝑑𝑠𝑊 (𝑠) , where 𝑖𝑐 : 𝕐 → 𝕏 is the inclusion

operator and

𝑐 =𝐾𝑎

1− 𝜇

[(��𝐻 + 𝑐1��∥𝑖𝑐∥ℒ(𝕐,𝕏) +𝐾−1

𝑎 𝑀𝑎

)∥𝜙∥ℬ + 𝑐2∥𝑖𝑐∥ℒ(𝕐,𝕏)(�� + 1) + 𝑐2

∫ 𝑎

ℋ(𝑠)𝑑𝑠+ ��𝑁∑𝑖=1

𝑁𝑖

where𝐻 is a constant given by Axiom (𝐴),𝐾𝑎 and𝑀𝑎 are given by𝐾𝑎 = sup0≤𝑡≤𝑎𝐾(𝑡) and𝑀𝑎 = sup0≤𝑡≤𝑎𝑀(𝑡),respectively.

Then there exists a mild solution of the initial value problem (1.1)-(1.3).

Proof.Let 𝑆(𝐼) and 𝑦 : (−∞, 𝑎] → 𝕏 be introduced in (H6) and Γ : 𝑆(𝐼) → 𝑆(𝐼) be the operator defined by (Γ𝑢)0 = 0

Γ𝑢(𝑡) = 𝑇 (𝑡)𝑔(0, 𝜙)−𝑔(𝑡, 𝑢𝑡+𝑦𝑡)−∫ 𝑡

𝐴𝑇 (𝑡−𝑠)𝑔(𝑠, 𝑢𝑠+𝑦𝑠)𝑑𝑠+∫ 𝑡

𝑇 (𝑡−𝑠)𝑓(𝑠, 𝑢𝑠+𝑦𝑠)𝑑𝑠+∑

0<𝑡𝑖<𝑡

𝑇 (𝑡−𝑡𝑖)𝐼𝑖(𝑢𝑡𝑖+𝑦𝑡𝑖),(6)

To prove that the function Γ𝑢 ∈ 𝑆(𝐼), we need to show that the following properties are satisfied:(i) The function Γ𝑢 is continuous in 𝑡 ∕= 𝑡𝑖.(ii) The limit lim𝑡→𝑡−𝑖

Γ𝑢(𝑡) = Γ𝑢(𝑡𝑖), for all 𝑖 = 1, ..., 𝑁 .(iii) The limit lim𝑡→𝑡+𝑖

Γ𝑢(𝑡), exists, for all 𝑖 = 1, ..., 𝑁 .Taking into the account that 𝑔 and 𝑡 → 𝑇 (𝑡)𝑔(0, 𝜙) are continuous, it follows from (S1), (H2) and the Remark 2

that the function Γ𝑢(𝑡) is continuous in 𝑡 ∕= 𝑡𝑖. On the other hand, from the definition of Γ, the Dominated ConvergenceTheorem and the condition (S2), we have that lim𝑡→𝑡−𝑖

Γ𝑢(𝑡) = Γ𝑢(𝑡𝑖) and that lim𝑡→𝑡+𝑖Γ𝑢(𝑡) = Γ𝑢(𝑡𝑖) + 𝐼𝑖(𝑢𝑡𝑖).

Next, we show that Γ is a continuous operator. Let (𝑢𝑛)𝑛∈ℕ be a sequence in 𝑆(𝐼) and 𝑢 ∈ 𝑆(𝐼) such that 𝑢𝑛 → 𝑢in 𝑆(𝐼). By using the continuity of 𝑔 and conditions ((H2)− (a)), (H3) and (H6), we prove that 𝑔(𝑡, 𝑢𝑛𝑠 + 𝑦𝑛𝑠 ) →𝑔(𝑡, 𝑢𝑠 + 𝑦𝑠) uniformly in 𝐼 , 𝑓(𝑠, 𝑢𝑛𝑠 + 𝑦𝑠) → 𝑓(𝑠, 𝑢𝑠 + 𝑦𝑠) for all 𝑠 ∈ 𝐼 , and 𝐼𝑖(𝑢𝑛𝑡 + 𝑦𝑡) → 𝐼(𝑢𝑡 + 𝑦𝑡) uniformly on 𝐼 .Using conditions (S2), ((H2)− (b, c)) and Lebesgue’s Dominated Convergence Theorem, we conclude that∫ 𝑡

𝐴𝑇 (𝑡−𝑠)𝑔(𝑠, 𝑢𝑛𝑠+𝑦𝑛𝑠 )𝑑𝑠→∫ 𝑡

𝐴𝑇 (𝑡−𝑠)𝑔(𝑠, 𝑢𝑠+𝑦𝑠)𝑑𝑠,∫ 𝑡

𝑇 (𝑡−𝑠)𝑓(𝑠, 𝑢𝑛𝑠+𝑦𝑛𝑠 )𝑑𝑠→∫ 𝑡

𝑇 (𝑡−𝑠)𝑓(𝑠, 𝑢𝑠+𝑦𝑠)𝑑𝑠,

as 𝑛→ ∞, which clearly implies that Γ is a continuous operator.In order to use Theorem 2, we need to obtain a 𝑝𝑟𝑖𝑜𝑟𝑖 bound for the solutions of the integral equations 𝜆Γ𝑢 = 𝑢,

𝜆 ∈ (0, 1). Let 𝜆 ∈ (0, 1) and let 𝑥𝜆 be a solution of 𝜆Γ𝑢 = 𝑢, 0 < 𝜆 < 1; taking into account that ∥𝑦𝑡∥ℬ ≤(𝐾𝑎��𝐻 +𝑀𝑎)∥𝜙∥ℬ and using Remark 2, (H2) and (H4), we find that

∥𝑥𝜆(𝑡)∥ ≤��(∥𝑖𝑐∥ℒ(𝕐,𝕏)(𝑐1∥𝜙∥ℬ + 𝑐2)) + ∥𝑖𝑐∥ℒ(𝕐,𝕏)

(𝑐1(𝐾𝑎∥𝑥𝜆∥𝑡 + (𝐾𝑎��𝐻 +𝑀𝑎)∥𝜙∥ℬ) + 𝑐2

∫ 𝑡

ℋ(𝑡− 𝑠)(𝑐1(𝐾𝑎∥𝑥𝜆∥𝑠 + (𝐾𝑎��𝐻 +𝑀𝑎)∥𝜙∥ℬ) + 𝑐2

)𝑑𝑠+ ��

∫ 𝑡

𝑚(𝑠)𝑊 (𝐾𝑎∥𝑥𝜆∥𝑠

+(𝐾𝑎��𝐻 +𝑀𝑎)∥𝜙∥ℬ)𝑑𝑠+𝑁∑𝑖=1

��𝑁𝑖,

where we are using the following notation: ∥𝑓∥𝑡 = sup0≤𝑠≤𝑡 ∥𝑓(𝑠)∥𝕏. Next, putting

𝛿𝜆(𝑡) = 𝐾𝑎∥𝑥𝜆∥𝑡 + (𝐾𝑎��𝐻 +𝑀𝑎)∥𝜙∥ℬ,it follows that

∥𝑥𝜆(𝑡)∥ ≤ ∥𝑖𝑐∥ℒ(𝕐,𝕏)(��(𝑐1∥𝜙∥ℬ + 𝑐2)) + ∥𝑖𝑐∥ℒ(𝕐,𝕏)(𝑐1𝛿𝜆(𝑡) + 𝑐2) +

∫ 𝑡

ℋ(𝑡− 𝑠)(𝑐1𝛿𝜆(𝑠) + 𝑐2)𝑑𝑠

+��

∫ 𝑡

𝑚(𝑠)𝑊 (𝛿𝜆(𝑠))𝑑𝑠+

𝑁∑𝑖=1

��𝑁𝑖.

We obtain after some simplification and a rearrangement of terms

𝛿𝜆(𝑡) ≤[𝐾𝑎��𝐻 + 𝑐1𝐾𝑎��∥𝑖𝑐∥ℒ(𝕐,𝕏) +𝑀𝑎

]∥𝜙∥ℬ + 𝑐2𝐾𝑎∥𝑖𝑐∥ℒ(𝕐,𝕏)(�� + 1) +𝐾𝑎𝑐2

∫ 𝑎

ℋ(𝑠)𝑑𝑠

+𝑐1𝐾𝑎

(∥𝑖𝑐∥ℒ(𝕐,𝕏) +

∫ 𝑎

ℋ(𝑠)𝑑𝑠)𝛿𝜆(𝑡) +𝐾𝑎��

∫ 𝑡

𝑚(𝑠)𝑊 (𝛿𝜆(𝑠))𝑑𝑠+𝐾𝑎��

𝑁∑𝑖=1

𝑁𝑖.

From the hypothesis (b), we have that

𝛿𝜆(𝑡) ≤ 𝑐+��𝐾𝑎

1− 𝜇

∫ 𝑡

𝑚(𝑠)𝑊 (𝛿𝜆(𝑠))𝑑𝑠.

Denote by 𝛽𝜆(𝑡) the right-hand side of the previous inequality. Computing 𝛽′𝜆(𝑡) and observing that 𝛿𝜆(𝑡) ≤ 𝛽𝜆(𝑡)

for 𝑡 ∈ 𝐼 , we arrive at

𝛽′𝜆(𝑡) ≤

��𝐾𝑎

1− 𝜇𝑚(𝑡)𝑊 (𝛽𝜆(𝑡)),

hence ∫ 𝛽𝜆(𝑡)

𝛽𝜆(0)

𝑑𝑠

𝑊 (𝑠)≤ ��𝐾𝑎

1− 𝜇

∫ 𝑎

𝑚(𝑠)𝑑𝑠 ≤∫ ∞

𝑑𝑠

𝑊 (𝑠).

This allow us to conclude that the set of functions {𝛽𝜆 : 𝜆 ∈ (0, 1)} is bounded. This implies that {𝑢𝜆 : 𝑢𝜆 = Γ𝑢𝜆, 𝜆 ∈(0, 1)} is bounded in 𝑆(𝐼).

It remains to show that Γ is completely continuous. To this end, we introduce the decomposition Γ = Γ1 +Γ2, where(Γ𝑖𝑢)0 = 0, 𝑖 = 1, 2, and

Γ1𝑢(𝑡) = 𝑇 (𝑡)𝑔(0, 𝜙)− 𝑔(𝑡, 𝑢𝑡 + 𝑦𝑡)−∫ 𝑡

𝐴𝑇 (𝑡− 𝑠)𝑔(𝑠, 𝑢𝑠 + 𝑦𝑠)𝑑𝑠+

∫ 𝑡

𝑇 (𝑡− 𝑠)𝑓(𝑠, 𝑢𝑠 + 𝑦𝑠)𝑑𝑠, 𝑡 ∈ [0, 𝑎],

Γ2𝑢(𝑡) =∑

0<𝑡𝑖<𝑡

𝑇 (𝑡− 𝑡𝑖)𝐼𝑖(𝑢𝑡𝑖 + 𝑦𝑡𝑖), 𝑡 ∈ [0, 𝑎].

We now prove that Γ1 is a completely continuous operator. We consider 𝑢 in 𝐵𝑟(0, 𝑆(𝐼)), an open ball of radius 𝑟.Applying now the Mean Value Theorem for the Bochner integral (see [22, Lemma 2.1.3]), we infer that

(Γ1𝑢)(𝑡) ⊂ 𝑇 (𝑡)𝑔(0, 𝜙)− 𝑔(𝑡, 𝑢𝑡 + 𝑦𝑡)− 𝑡𝑐𝑜{𝐴𝑇 (𝑡− 𝜃)𝑔(𝜃, 𝑢𝜃 + 𝑦𝜃), 𝜃 ∈ [0, 𝑡], 𝑢 ∈ 𝐵𝑟(0, 𝑆(𝐼))}𝕐

+𝑡𝑐𝑜{𝑇 (𝑡− 𝜃)𝑓(𝜃, 𝑢𝜃 + 𝑦𝜃), 𝜃 ∈ [0, 𝑡], 𝑢 ∈ 𝐵𝑟(0, 𝑆(𝐼))}.Taking the advantage of assumptions (a) and (H1), we conclude that

{(Γ1𝑢)(𝑡) : 𝑢 ∈ 𝐵𝑟(0, 𝑆(𝐼))} is a compact subset of 𝕏.Now, we prove that the set of functions {(Γ1(𝑢);𝑢 ∈ 𝐵𝑟(0, 𝑆(𝐼))} is equicontinuous in 𝐼 . To this end, consider

0 < 𝜀 < 𝑡 < 𝑡′ for 𝑡, 𝑡′ ∈ [0, 𝑎]; then there is 0 < 𝛿 < 𝜀 such that

∥𝑇 (𝑡)𝑔(0, 𝜙)− 𝑇 (𝑡′)𝑔(0, 𝜙)∥ < 𝜀,

∥𝑔(𝑡, 𝑢𝑡 + 𝑦𝑡)− 𝑔(𝑡′, 𝑢𝑡′ + 𝑦𝑡′)𝑑𝑠)∥ < 𝜀,

∥𝑇 (𝑡− 𝑠)𝑓(𝑢𝑠 + 𝑦𝑠)− 𝑇 (𝑡′ − 𝑠)𝐺(𝑢𝑠 + 𝑦𝑠, )∥ < 𝜀,

for all 𝑠 ∈ [0, 𝑡], ∣𝑡− 𝑡′∣ < 𝛿 and 𝑢 ∈ 𝐵𝑟(0, 𝑆(𝐼)). On the other-hand, using the continuity of the map (𝑡, 𝑠) → 𝐴𝑇 (𝑡− 𝑠)for 𝑡 > 𝑠, we arrive at

∥𝐴𝑇 (𝑡− 𝑠)𝑔(𝑠, 𝑢𝑠 + 𝑦𝑠)−𝐴𝑇 (𝑡′ − 𝑠)𝑔(𝑠, 𝑢𝑠 + 𝑦𝑠)∥ < 𝜀,

for all 𝑠 ∈ [0, 𝑡− 𝜀], ∣𝑡− 𝑡′∣ < 𝛿 and 𝑢 ∈ 𝐵𝑟(0, 𝑆(𝐼)). Under these conditions, for 𝑢 ∈ 𝐵𝑟(0, 𝑆(𝐼)), ∣𝑡− 𝑡′∣ < 𝛿, we inferthat

∥Γ1𝑢(𝑡)− Γ1𝑢(𝑡′)∥ ≤ ∥(𝑇 (𝑡)− 𝑇 (𝑡′))𝑔(0, 𝜙)∥+ ∥𝑔(𝑡, 𝑢𝑡 + 𝑦𝑡)− 𝑔(𝑡′, 𝑢𝑡′ + 𝑦𝑡′)∥+

∫ 𝑡′

∥𝐴𝑇 (𝑡′ − 𝑠)𝑔(𝑠, 𝑢𝑠 + 𝑦𝑠)∥𝑑𝑠

∫ 𝑡−𝜀

∥(𝐴𝑇 (𝑡− 𝑠)−𝐴𝑇 (𝑡′ − 𝑠))𝑔(𝑠, 𝑢𝑠 + 𝑦𝑠)∥𝑑𝑠+∫ 𝑡

𝑡−𝜀

∥(𝐴𝑇 (𝑡− 𝑠)−𝐴𝑇 (𝑡′ − 𝑠))𝑔(𝑠, 𝑢𝑠 + 𝑦𝑠)∥𝑑𝑠

∫ 𝑡

∥(𝑇 (𝑡− 𝑠)− 𝑇 (𝑡′ − 𝑠))𝑓(𝑠, 𝑢𝑠 + 𝑦𝑠)∥𝑑𝑠+∫ 𝑡′

∥𝑇 (𝑡′ − 𝑠)𝑓(𝑠, 𝑢𝑠 + 𝑦𝑠)∥𝑑𝑠.

From the above estimate, one can deduce the following inequality:

∥Γ1𝑢(𝑡)− Γ1𝑢(𝑡′)∥ ≤ 2𝜀+ 𝜀(𝑡− 𝜀) + 𝜀𝑡+ (𝑐1(𝐾𝑎𝑟 +𝐾𝑎��𝐻 +𝑀𝑎))

∫ 𝑡

𝑡−𝜀

ℋ(𝑡− 𝑠)𝑑𝑠+

∫ 𝑡′

ℋ(𝑡− 𝑠)𝑑𝑠

+��𝑊 (𝑐1(𝐾𝑎𝑟 +𝐾𝑎��𝐻 +𝑀𝑎))

∫ 𝑡′

𝑚(𝑠)𝑑𝑠,

which shows the equicontinuity at 𝑡 ∈ 𝐼 . So, to conclude the proof, we show that Γ2 is completely continuous. We observethat the continuity of Γ2 is obvious. On the other hand, for 𝑟 > 0, 𝑡 ∈ [𝑡𝑖, 𝑡𝑖+1], 𝑖 ≥ 1, and 𝑢 ∈ 𝐵𝑟 = 𝐵𝑟(0,𝒫𝒞([0, 𝑎];𝕏)),we have that there exists 𝑟 > 0 such that

[ ˜Γ2𝑢]𝑖(𝑡) ∈

⎧⎨⎩∑𝑖

𝑗=1 𝑇 (𝑡− 𝑡𝑗)𝐼𝑗(𝐵𝑟(0;ℬ)), 𝑡 ∈ (𝑡𝑖, 𝑡𝑖+1),∑𝑖𝑗=1 𝑇 (𝑡𝑖+1 − 𝑡𝑗)𝐼𝑗(𝐵𝑟(0;ℬ)), 𝑡 = 𝑡𝑖+1,∑𝑖−1𝑗=1 𝑇 (𝑡𝑖 − 𝑡𝑗)𝐼𝑗(𝐵𝑟(0;ℬ)) + 𝐼𝑖(𝐵𝑟(0;ℬ)), 𝑡 = 𝑡𝑖,

where 𝐵𝑟(0;ℬ) is an open ball of radius 𝑟. From condition (H4) it follows that [ ˜Γ2(𝐵𝑟)]𝑖(𝑡) is relatively compact in 𝕏,for all 𝑡 ∈ [𝑡𝑖, 𝑡𝑖+1], 𝑖 ≥ 1. Moreover, using the fact that the operators {𝐼𝑖}𝑖 are compact and the strong continuity of(𝑇 (𝑡))𝑡≥0, we conclude that for 𝜀 > 0, there exists 𝛿 > 0 such that

∥𝑇 (𝑡− 𝑡𝑖)𝑧 − 𝑇 (𝑠− 𝑡𝑖)𝑧∥ ≤ 𝜀, 𝑧 ∈𝑁∪𝑖=1

𝐼𝑖(𝐵𝑟(0,ℬ)), (3.3)

for all 𝑡𝑖, 𝑖 = 1, ..., 𝑁 , 𝑡, 𝑠 ∈ (𝑡𝑖, 𝑡𝑖+1] with ∣𝑡 − 𝑠∣ < 𝛿. Thus for 𝑢 ∈ 𝐵𝑟, 𝑡 ∈ [𝑡𝑖, 𝑡𝑖+1], 𝑖 ≥ 0, and 0 < ∣ℎ∣ < 𝛿 with𝑡+ ℎ ∈ [𝑡𝑖, 𝑡𝑖+1], we have that

∥[ ˜Γ2𝑢]𝑖(𝑡+ ℎ)− [ ˜Γ2𝑢]𝑖(𝑡)∥ ≤ 𝑁𝜀, 𝑖 = 1, ..., 𝑁. (3.4)

We have shown that the set ˜[Γ2(𝐵𝑟)]𝑖 is uniformly equicontinuous for 𝑖 ≥ 0. Now, from Lemma 1, we conclude that Γ2

is completely continuous.We have proven that Γ satisfies the conditions of Theorem 2, which allows us infer the existence of a mild solution of

the problem (1.1)-(1.3). This completes the proof of the theorem.If the map 𝐹 and 𝐼𝑖, 𝑖 = 1, ..., 𝑁 fulfill some Lipschitz condition instead of the compactness properties considered in

Theorem 4, we also can establish an existence result.

Theorem 7 Suppose that the assumptions (S1), (S2), (H2), (H4) and (H5) are satisfied and that the condition (a) ofTheorem 4 is fulfilled. If

[𝐾𝑎𝐿𝑔

(∥𝑖𝑐∥ℒ(𝕐,𝕏) +

∫ 𝑎

ℋ(𝑠)𝑑𝑠)+ ��𝐾𝑎

𝑁∑𝑖=1

𝐿𝑖 + ��𝐾𝑎 lim𝑟→∞ inf

𝑊 (𝑟)

∫ 𝑎

𝑚(𝑠)𝑑𝑠]< 1,

where 𝑖𝑐 denotes the inclusion operator from 𝕐 into 𝕏, then there exists a mild solution of the impulsive problem (1.1)-(1.3).

Proof. Let Γ be the function given in the proof of Theorem 4 and consider the decomposition Γ = Γ1 + Γ2, where(Γ𝑖𝑢) = 0 on (−∞, 0], 𝑖 = 1, 2, and

Γ1𝑢(𝑡) = 𝑇 (𝑡)𝑔(0, 𝜙)− 𝑔(𝑡, 𝑢𝑡 + 𝑦𝑡)−∫ 𝑡

𝐴𝑇 (𝑡− 𝑠)𝑔(𝑠, 𝑢𝑠 + 𝑦𝑠)𝑑𝑠+∑

0<𝑡𝑖<𝑡

𝑇 (𝑡− 𝑡𝑖)𝐼𝑖(𝑢𝑡𝑖 + 𝑦𝑡𝑖), 𝑡 ∈ [0, 𝑎],

Γ2𝑢(𝑡) =

∫ 𝑡

𝑇 (𝑡− 𝑠)𝑓(𝑠, 𝑢𝑠 + 𝑦𝑠)𝑑𝑠, 𝑡 ∈ [0, 𝑎].

We claim that there exists 𝑟 > 0 such that Γ(𝐵𝑟(0, 𝑆(𝐼))) ⊂ 𝐵𝑟(0, 𝑆(𝐼)). In fact, if we assume that this assertion is false,then for all 𝑟 > 0 we can choose 𝑥𝑟 ∈ 𝐵𝑟 = 𝐵𝑟(0, 𝑆(𝐼)) and 𝑡𝑟 ∈ [0, 𝑎] such that ∥Γ𝑥𝑟(𝑡𝑟)∥ > 𝑟. Observe that standardcomputations involving the phase space axioms yield

𝑟 ≤ ∥Γ𝑥𝑟(𝑡𝑟)∥ ≤ ��∥𝑔(0, 𝜙)∥𝕐 + ∥𝑖𝑐∥𝐿(𝕐,𝕏) sup0≤𝑡≤𝑎

∥𝑔(𝑡, 0)∥𝕐 + 𝐿𝑔∥𝑖𝑐∥ℒ(𝕐,𝕏)(𝐾𝑎𝑟 + (𝐾𝑎��𝐻 +𝑀𝑎)∥𝜙∥ℬ)

∫ 𝑎

ℋ(𝑠)𝑑𝑠(𝐿𝑔(𝐾𝑎𝑟 + (𝐾𝑎��𝐻 +𝑀𝑎)∥𝜙∥ℬ) + sup

0≤𝜏≤𝑎∥𝑔(𝜏, 0)∥𝕐

)+��

∫ 𝑎

𝑚(𝑠)𝑊 ((𝐾𝑎𝑟 +𝐾𝑎��𝐻 +𝑀𝑎)∥𝜙∥ℬ)𝑑𝑠+ ��𝑁∑𝑖=1

(𝐿𝑖(𝐾𝑎𝑟 + (𝐾𝑎��𝐻 +𝑀𝑎)∥𝜙∥ℬ) + ∥𝐼𝑖(0)∥).

1 ≤[𝐾𝑎𝐿𝑔

(∥𝑖𝑐∥ℒ(𝕐,𝕏) +

∫ 𝑎

𝑁∑𝑖=1

𝐿𝑖 + ��𝐾𝑎 lim𝑟→∞ inf

𝑊 (𝑟)

∫ 𝑎

𝑚(𝑠)𝑑𝑠],

which is contrary to our assumptions.Let 𝑟 > 0 such that Γ(𝐵𝑟(0, 𝑆(𝐼))) ⊂ 𝐵𝑟(0, 𝑆(𝐼)). It follows from the proof of Lemma 3.1 in [10] that Γ2 is

completely continuous in 𝐵𝑟(0, 𝑆(𝐼)). Moreover, the estimate

∥Γ1𝑢− Γ1𝑣∥∞ ≤[𝐾𝑎𝐿𝐹

(∥𝑖𝑐∥ℒ(𝕐,𝕏) +

∫ 𝑎

𝑁∑𝑖=1

𝐿𝑖

]∥𝑢− 𝑣∥∞,

𝑢, 𝑣 ∈ 𝐵𝑟, shows that Γ1 is a contraction on 𝐵𝑟. Consequently, Γ is a condensing operator from 𝐵𝑟 into 𝐵𝑟. Then theexistence of a mild solution of (1.1)-(1.3) is a consequence of Theorem 3. This completes the proof.

4 Global solutionsIn this section, we study the existence of mild solutions for the impulsive problem

𝑑𝑡(𝑢(𝑡) + 𝑔(𝑡, 𝑢𝑡)) = 𝐴𝑢(𝑡) + 𝑓(𝑡, 𝑢𝑡), 𝑡 ∈ 𝐼 = [0,∞), (7)

𝑢0 = 𝜑 ∈ ℬ (8)

Δ𝑢(𝑡𝑖) = 𝐼𝑖(𝑢𝑡𝑖), 0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑛 < ⋅ ⋅ ⋅ (9)

where (𝑡𝑖)𝑖∈ℕ is an increasing sequence of positive numbers.Let ℎ : [0,∞) → [0,∞) be a positive, non-decreasing, continuous function such that ℎ(0) = 1 and lim𝑡→∞ ℎ(𝑡) =

+∞. In addition, suppose that the map (𝑡, 𝑠) → 𝑇 (𝑡−𝑠) is uniformly bounded. Moreover, 𝒫𝒞([0,∞);𝕏) and (𝒫𝒞ℎ)0(𝕏)

denote the spaces

𝒫𝒞([0,∞);𝑋) =

⎧⎨⎩𝑥 : [0,∞) → 𝑋 : 𝑥∣[0,𝑎]

∈ 𝒫𝒞, ∀𝑎 ∈ (0,∞) ∖ {𝑡𝑖 : 𝑖 ∈ ℕ},𝑥0 = 0,

∥𝑥∥∞ = sup𝑡≥0 ∥𝑥(𝑡)∥ <∞

⎫⎬⎭(𝒫𝒞)0ℎ(𝕏) =

{𝑥 ∈ 𝒫𝒞([0,∞);𝕏) : lim

𝑡→∞∥𝑥(𝑡)∥ℎ(𝑡)

endowed with the norms ∥𝑥∥∞ = sup𝑡≥0 ∥𝑥(𝑡)∥ and ∥𝑥∥𝒫𝒞ℎ= sup𝑡≥0

∥𝑥(𝑡)∥ℎ(𝑡) , respectively.

To get the next results, we need a very detailed knowledge of the relativity compact sets of the space (𝒫𝒞)0ℎ(𝕏). Wewill use the following result.

Lemma 8 A bounded set ℬ ⊂ (𝒫𝒞)0ℎ(𝕏) is relatively compact in (𝒫𝒞)0ℎ(𝕏) if, and only if:

(a) The set 𝐵𝑎 = {𝑢∣[0,𝑎]: 𝑢 ∈ 𝐵} is relatively compact in 𝒫𝒞([0, 𝑎];𝕏), for all 𝑎 ∈ (0,∞) ∖ {𝑡𝑖; 𝑖 ∈ ℕ}.

(b) lim𝑡→∞∥𝑥(𝑡)∥ℎ(𝑡) = 0, uniformly for 𝑥 ∈ 𝐵.

We introduce the following concept of global solution of system (4.1)-(4.3).

Definition 2 A function 𝑢 : ℝ → 𝕏 is called a global solution of the problem (4.1)-(4.3), if the conditions (4.2) and (4.3)are verified; 𝑢∣[0,𝑎]

∈ 𝒫𝒞([0, 𝑎];𝕏) for all 𝑎 ∈ (0,+∞) ∖ {𝑡𝑖 : 𝑖 ∈ ℕ} and

𝑢(𝑡) = 𝑇 (𝑡)(𝜙(0)+𝑔(0, 𝜙)

)−𝑔(𝑡, 𝑢𝑡)−∫ 𝑡

𝐴𝑇 (𝑡−𝑠)𝑔(𝑠, 𝑢𝑠)𝑑𝑠+∫ 𝑡

𝑇 (𝑡−𝑠)𝑓(𝑠, 𝑢𝑠)𝑑𝑠+∑𝑡𝑖<𝑡

𝑇 (𝑡−𝑡𝑖)𝐼𝑖(𝑢𝑡𝑖), 𝑡 ∈ 𝐼 = [0,∞).

We have to prove the following theorem.

Theorem 9 Assume that the hypotheses (S1), (S2), (H2), (H3), (H6) and the condition (a) of Theorem 4 are valid in𝐼 = [0, 𝑎] for all 𝑎 > 0. Suppose, in addition, that functions 𝑀(⋅) and 𝐾(⋅) given by Axiom (A) are bounded and thatthe following conditions hold.

(a) For all 𝐽 > 0, lim𝑡→∞ 1ℎ(𝑡)

∫ 𝑡

0𝑚(𝑠)𝑊 (𝐽ℎ(𝑠))𝑑𝑠 = 0.

(b) The function 𝑔 : 𝐼 ×ℬ → 𝕐 is completely continuous and there exists a positive continuous function 𝑐1 : [0,∞) →[0,∞) with 𝑐1(𝑡) → 0 when 𝑡 → ∞ and a positive constant 𝑑2 such that ∥𝑔(𝑡, 𝜓)∥𝕐 ≤ 𝑐1(𝑡)∥𝜓∥ℬ + 𝑑2 for all𝑡 > 0, 𝜓 ∈ ℬ and

lim𝑡→∞

ℎ(𝑡)

∫ 𝑡

ℋ(𝑡− 𝑠)ℎ(𝑠)𝑑𝑠 = 0.

(c) 𝜂 =(∥𝑖𝑐∥ℒ(𝕐,𝕏)𝐾∞𝑐1,∞ + sup𝑡≥0𝐾∞

∫ 𝑡

0ℋ(𝑡− 𝑠)𝑐1(𝑠)𝑑𝑠

(d) ��𝐾∞1−𝜂

∫∞0𝑚(𝑠)𝑑𝑠 <

∫∞𝑐

1𝑊 (𝑠)𝑑𝑠, with

𝑐 = (1− 𝜂)−1((��𝐾∞𝐻 +𝑀∞ + ��𝐾∞∥𝑖𝑐∥ℒ(𝕐,𝕏)𝑐1,∞)∥𝜙∥ℬ + ∥𝑖𝑐∥ℒ(𝕐,𝕏)𝐾∞(�� + 1)𝑑2 + ��𝐾∞

∞∑𝑖=1

𝑁𝑖

+𝑑2𝐾∞∫ ∞

ℋ(𝑠)𝑑𝑠)<∞,

where 𝑐1,∞ = sup𝑡≥0 𝑐1(𝑡), 𝐾∞ = sup𝑡≥0𝐾(𝑡), 𝑀∞ = sup𝑡≥0𝑀(𝑡).

Then there exists a global solution of (4.1)-(4.3).

Proof. Let 𝑦 : ℝ → 𝕏 be the extension of 𝜙 such that 𝑦(𝑡) = 𝑇 (𝑡)𝜙(0), 𝑡 ≥ 0 and 𝑆(∞) be the set defined by

𝑆(∞) ={𝑥 : ℝ → 𝕏 : 𝑥0 = 0 𝑎𝑛𝑑 𝑥∣[0,∞)

∈ 𝒫𝒞0ℎ(𝕏)

endowed with the norm ∥𝑥∥𝒫𝒞ℎ= sup𝑡≥0

∥𝑥(𝑡)∥ℎ(𝑡) and Γ : 𝑆(∞) → 𝑆(∞) be the operator defined by (Γ𝑢)0 = 0 and

Γ𝑢(𝑡) = 𝑇 (𝑡)𝑔(0, 𝜙)− 𝑔(𝑡, 𝑦𝑡 + 𝑢𝑡)−∫ 𝑡

𝐴𝑇 (𝑡− 𝑠)𝑔(𝑠, 𝑦𝑠 + 𝑢𝑠)𝑑𝑠+

∫ 𝑡

𝑇 (𝑡− 𝑠)𝑓(𝑠, 𝑦𝑠 + 𝑢𝑠)𝑑𝑠

+∑𝑡𝑖<𝑡

𝑇 (𝑡− 𝑡𝑖)𝐼𝑖(𝑦𝑡𝑖 + 𝑢𝑡𝑖), 𝑡 ≥ 0.

Since ∥𝑢(𝑡)∥ ≤ ∥𝑢∥𝒫𝒞ℎℎ(𝑡) for all 𝑡 ≥ 0, we have that

∥Γ𝑢(𝑡)∥ℎ(𝑡)

≤ ��∥𝑔(0, 𝜙)∥ℎ(𝑡)

+∥𝑖𝑐∥ℒ(𝕐,𝕏)

[𝑐1(𝑡)𝐾∞∥𝑢∥𝒫𝒞ℎ

ℎ(𝑡)]

ℎ(𝑡)+

∥𝑖𝑐∥ℒ(𝕐,𝕏)[𝑐1(𝑡)(��𝐾∞𝐻 +𝑀∞)∥𝜙∥ℬ + 𝑑2]

ℎ(𝑡)+

��

ℎ(𝑡)

∞∑𝑖=1

𝑁𝑖

ℎ(𝑡)

∫ 𝑡

ℋ(𝑡− 𝑠)(𝑐1(𝑠)(𝐾∞∥𝑢∥𝒫𝒞ℎ

+ (��𝐾∞𝐻 +𝑀∞)∥𝜙∥ℬ) + 𝑑2

)ℎ(𝑠)𝑑𝑠

+��

ℎ(𝑡)

∫ 𝑡

𝑚(𝑠)𝑊 ((𝐾∞∥𝑢∥𝒫𝒞ℎ+ (��𝐾∞𝐻 +𝑀∞)∥𝜙∥ℬ)ℎ(𝑠))𝑑𝑠,

so by (a) and (b), we have that Γ𝑢 ∈ 𝑆(∞).Next, we show that Γ(𝐵𝑟), where 𝐵𝑟 = 𝐵𝑟(0, (𝒫𝒞)0ℎ(𝕏)), satisfies the conditions of Lemma 2. To do this, we show

the continuity of the operator Γ. Let (𝑢𝑛)𝑛∈ℕ be a sequence in 𝑆(∞) and 𝑢 ∈ 𝑆(∞) such that 𝑢𝑛 → 𝑢 in 𝑆(∞). Take𝐶 = sup{∥𝑢𝑛∥𝒫𝒞ℎ

, ∥𝑢∥𝒫𝒞ℎ; 𝑛 ∈ ℕ}, 𝐽 = (��𝐾∞𝐻 +𝑀∞)∥𝜙∥ℬ + 𝐶𝐾∞ and the function 𝜇 : [0,∞) → ℝ defined

by 𝜇(𝑡) = 𝑐1(𝑡)𝐽 + 𝑑2. From the conditions (a), (b) and (c), there exists 𝐿1 > 0 such that

ℎ(𝑡)

∫ 𝑡

ℋ(𝑡− 𝑠)𝜇(𝑠)ℎ(𝑠)𝑑𝑠+2��

ℎ(𝑡)

∫ 𝑡

𝑚(𝑠)𝑊 (𝐽ℎ(𝑠))𝑑𝑠+2��

ℎ(𝑡)

∞∑𝑖=1

𝑁𝑖 <𝜀

for 𝑡 ≥ 𝐿1. From the proof of Theorem 4, we have that 𝑔(𝑡, 𝑢𝑛𝑡 + 𝑦𝑡) → 𝑔(𝑡, 𝑢𝑡 + 𝑦𝑡) uniformly for 𝑡 ∈ [0, 𝐿1], when𝑛→ ∞. From Lebesgue’s Dominated Convergence Theorem, we can fix a positive number 𝑁𝜀 > 0 such that

ℎ(𝑡)∥𝑔(𝑡, 𝑢𝑛𝑡 + 𝑦𝑡)− 𝑔(𝑡, 𝑢𝑡 + 𝑦𝑡)∥+ ��

ℎ(𝑡)

∫ 𝐿1

∥𝑓(𝑠, 𝑢𝑛𝑠 + 𝑦𝑠)− 𝑓(𝑠, 𝑢𝑠 + 𝑦𝑠)∥𝑑𝑠

ℎ(𝑡)

∫ 𝑡

ℋ(𝑡− 𝑠)∥𝑔(𝑠, 𝑢𝑛𝑠 + 𝑦𝑠)− 𝑔(𝑠, 𝑢𝑠 + 𝑦𝑠)∥𝑑𝑠

+��

ℎ(𝑡)

∑𝑡𝑖≤𝐿1

∥𝐼𝑖(𝑢𝑛𝑡𝑖 + 𝑦𝑡𝑖)− 𝐼𝑖(𝑢𝑡𝑖 + 𝑦𝑡𝑖)∥ <𝜀

for all 𝑡 ∈ [0, 𝐿1] and 𝑛 ≥ 𝑁𝜀. Using the above inequality, for 𝑡 ∈ [0, 𝐿1] and 𝑛 ≥ 𝑁𝜀, we have that

{∥Γ𝑢𝑛(𝑡)− Γ𝑢(𝑡)∥ℎ(𝑡)

: 𝑡 ∈ [0, 𝐿1], 𝑛 ≥ 𝑁𝜀

}≤ 𝜀 (11)

On the other hand, for 𝑡 ≥ 𝐿1 and 𝑛 ≥ 𝑁𝜀, we get

∥Γ𝑢𝑛(𝑡)− Γ𝑢(𝑡)∥ℎ(𝑡)

≤ ∥𝑔(𝑡, 𝑢𝑛𝑡 + 𝑦𝑡)− 𝑔(𝑡, 𝑢𝑡 + 𝑦𝑡)∥ℎ(𝑡)

ℎ(𝑡)

∫ 𝐿1

∥𝐴𝑇 (𝑡− 𝑠)(𝑔(𝑠, 𝑢𝑛𝑠 + 𝑦𝑠)− 𝑔(𝑠, 𝑢𝑠 + 𝑦𝑠))∥𝑑𝑠

ℎ(𝑡)

∫ 𝑡

∥𝐴𝑇 (𝑡− 𝑠)(𝑔(𝑠, 𝑢𝑛𝑠 + 𝑦𝑠)− 𝑔(𝑠, 𝑢𝑠 + 𝑦𝑠))∥𝑑𝑠

ℎ(𝑡)

∫ 𝐿1

∥𝑇 (𝑡− 𝑠)(𝑓(𝑠, 𝑢𝑛𝑠 + 𝑦𝑠)− 𝑓(𝑠, 𝑢𝑠 + 𝑦𝑠))∥𝑑𝑠+ 1

ℎ(𝑡)

∫ 𝑡

∥𝑇 (𝑡− 𝑠)(𝑓(𝑠, 𝑢𝑛𝑠 + 𝑦𝑠)− 𝑓(𝑠, 𝑢𝑠 + 𝑦𝑠))∥𝑑𝑠

+��

ℎ(𝑡)

∑𝑡𝑖≤𝐿1

∥𝐼𝑖(𝑢𝑛𝑡𝑖 + 𝑦𝑡𝑖)− 𝐼𝑖(𝑢𝑡𝑖 + 𝑦𝑡𝑖)∥+2��

ℎ(𝑡)

∑𝑡𝑖≥𝐿1

𝑁𝑖

ℎ(𝑡)

∫ 𝑡

ℋ(𝑡− 𝑠)𝜇(𝑠)ℎ(𝑠)𝑑𝑠+2��

ℎ(𝑡)

∫ 𝑡

𝑚(𝑠)𝑊 (𝐽ℎ(𝑠))𝑑𝑠+2��

ℎ(𝑡)

∞∑𝑖=1

𝑁𝑖 +𝜀

and thus

sup{∥Γ𝑢𝑛(𝑡)− Γ𝑢(𝑡)∥

ℎ(𝑡): 𝑡 ≥ 𝐿1, 𝑛 ≥ 𝑁𝜀

}≤ 𝜀. (12)

Using inequalities (4.5) and (4.6), we conclude that Γ is continuous.Next, we prove that the set Γ(𝐵𝑟) is relatively compact. Let 𝑟 > 0 be a positive real number, 𝐽 = (��𝐾∞𝐻 +

𝑀∞)∥𝜙∥ℬ + 𝑟𝐾∞. From the proof of Theorem 4, it follows that the set Γ(𝐵𝑟)∣[0,𝑎] = {Γ𝑢∣[0,𝑎] : 𝑢 ∈ 𝐵𝑟} is relativelycompact in 𝒫𝒞([0, 𝑎];𝕏) for all 𝑎 ∈ (0,∞) ∖ {𝑡𝑖, 𝑖 ∈ ℕ}. Moreover, for 𝑥 ∈ 𝐵𝑟, we have that

∥Γ𝑢(𝑡)∥ℎ(𝑡)

ℎ(𝑡)[��∥𝑖𝑐∥ℒ(𝕐,𝕏)(𝑐1(0)∥𝜙∥ℬ + 𝑑2)] +

ℎ(𝑡)

(∥𝑖𝑐∥ℒ(𝕐,𝕏)𝑐1(𝑡)𝐽ℎ(𝑡) + 𝑑2)

+��

ℎ(𝑡)

∫ 𝑡

𝑚(𝑠)𝑊 (𝐽ℎ(𝑠))𝑑+��

ℎ(𝑡)

∞∑𝑖=1

𝑁𝑖 +1

ℎ(𝑡)

∫ 𝑡

ℋ(𝑡− 𝑠)(𝑐1(𝑠)𝐽ℎ(𝑠) + 𝑑2

)𝑑𝑠

ℎ(𝑡)[��∥𝑖𝑐∥ℒ(𝕐,𝕏)(𝑐1(0)∥𝜙∥ℬ + 𝑑2)] +

𝐽𝑐1,∞ℎ(𝑡)

∫ 𝑡

ℋ(𝑡− 𝑠)ℎ(𝑠)𝑑𝑠+𝑑2ℎ(𝑡)

∫ 𝑡

ℋ(𝑡− 𝑠)𝑑𝑠+ ∥𝑖𝑐∥ℒ(𝕐,𝕏)(𝑐1(𝑡)𝐽

+𝑑2ℎ(𝑡)

��

ℎ(𝑡)

∫ 𝑡

𝑚(𝑠)𝑊 (𝐽ℎ(𝑠))𝑑𝑠+��

ℎ(𝑡)

∞∑𝑖=1

𝑁𝑖,

which enables us to conclude that ∥Γ𝑢(𝑡)∥ℎ(𝑡) → 0, when 𝑡→ ∞, uniformly for 𝑥 ∈ 𝐵𝑟.

By Lemma 4.1, we infer that Γ(𝐵𝑟) is relatively compact in (𝒫𝒞)0ℎ(𝕏). Thus, Γ is completely continuous. To finishthe proof, we need to obtain an 𝑎 𝑝𝑟𝑖𝑜𝑟𝑖 estimate for the solutions of the integral equations 𝜆Γ𝑢 = 𝑢, 𝜆 ∈ (0, 1). To this

end, let 𝑢𝜆 be the solution of the equation 𝜆Γ𝑢𝜆 = 𝑢𝜆. For 𝑡 ≥ 0, we have that

∥𝑢𝜆(𝑡)∥ ≤ ∥𝑖𝑐∥ℒ(𝕐,𝕏)[𝑐1,∞

((��(1 +𝐾∞𝐻) +𝑀∞)∥𝜙∥ℬ +𝐾∞∥𝑢𝜆∥𝑡

)]+ ∥𝑖𝑐∥ℒ(𝕐,𝕏)(�� + 1)𝑑2

∫ 𝑡

ℋ(𝑡− 𝑠)(𝑐1(𝑠)((��𝐾∞𝐻 +𝑀∞)∥𝜙∥ℬ +𝐾∞∥𝑢𝜆∥𝑠) + 𝑑2

)𝑑𝑠

+��

∫ 𝑡

𝑚(𝑠)𝑊 ((��𝐾∞𝐻 +𝑀∞)∥𝜙∥ℬ +𝐾∞∥𝑢𝜆∥𝑠)𝑑𝑠+ ��∞∑𝑖=1

𝑁𝑖.

Putting𝛿𝜆(𝑡) = (��𝐾∞𝐻 +𝑀∞)∥𝜙∥ℬ +𝐾∞∥𝑢𝜆∥𝑡,

we have that

𝛿𝜆(𝑡) ≤ (1− 𝜂)−1((��𝐾∞𝐻 +𝑀∞ + ��𝐾∞∥𝑖𝑐∥ℒ(𝕐,𝕏)𝑐1,∞)∥𝜙∥ℬ∥𝑖𝑐∥ℒ(𝕐,𝕏)𝐾∞(�� + 1)𝑑2 + 𝑑2𝐾∞

∫ ∞

ℋ(𝑠)𝑑𝑠

+��𝐾∞∫ 𝑡

𝑚(𝑠)𝑊 (𝛿𝜆(𝑠))𝑑𝑠+ ��𝐾∞∞∑𝑖=1

𝑁𝑖

where 𝜂 =(∥𝑖𝑐∥ℒ(𝕐,𝕏)𝐾∞𝑐1,∞ + sup𝑡≥0𝐾∞

∫ 𝑡

0ℋ(𝑡 − 𝑠)𝑐1(𝑠)𝑑𝑠

). Denoting the right-hand side of the previous

expression as 𝛽𝜆(𝑡), we see that

𝛽′𝜆(𝑡) ≤

��𝐾∞1− 𝜂

𝑚(𝑡)𝑊 (𝛽𝜆(𝑡)),

and subsequently, upon integrating over [0, 𝑡], we obtain∫ 𝛽𝜆(𝑡)

𝛽𝜆(0)

𝑑𝑠

𝑊 (𝑠)≤ ��𝐾∞

1− 𝜇

∫ 𝑎

𝑚(𝑠)𝑑𝑠 ≤∫ ∞

𝑑𝑠

𝑊 (𝑠)

The above inequality together with condition (d) enables us to conclude that the set of functions {𝑢𝜆 : 𝑢𝜆 = 𝜆Γ𝑢𝜆} isbounded in 𝒫𝒞([0,∞);𝕏). Note that, if 𝑥 ∈ 𝒫𝒞([0,∞);𝕏), then ∥𝑥∥𝒫𝒞ℎ

≤ ∥𝑥∥∞. This allows us to conclude that theset {𝑢𝜆 : 𝑢𝜆 = 𝜆Γ𝑢𝜆, 𝜆 ∈ (0, 1)} is bounded in 𝑆(∞). Finally, from Theorem 2 we infer the existence of a fixed pointof Γ, and consequently, the existence of a global solution of (4.1)-(4.3). This completes the proof.

If we suppose that the functions 𝐼𝑖, 𝑖 ∈ ℕ are Lipschitz continuous, then we have the following result.

Theorem 10 Suppose that the hypotheses (S1), (S2), (H2), (H4) and (H6) are valid for all 𝑎 > 0. Assume also thatcondition (a) of Theorem 4 and condition (b) of Theorem 6 are satisfied and that

∑∞𝑖=1 ∥𝐼𝑖(0)∥ < +∞. If

𝑐1,∞𝐾∞(∥𝑖𝑐∥ℒ(𝕐,𝕏) +

∫ ∞

ℋ(𝑠)𝑑𝑠)+𝐾∞��

∞∑𝑖=1

𝐿𝑖 + lim inf𝑟→∞ ��

∫ ∞

𝑚(𝑠)𝑊 (𝑇 (𝑟, 𝑠))

𝑟𝑑𝑠 < 1 (14)

where 𝑇 (𝑟, 𝑠) = (��𝐾∞𝐻 +𝑀∞)∥𝜙∥𝐵 +𝐾∞𝑟ℎ(𝑠), then there exists a global solution of the problem (4.1)-(4.3).

Proof. Let Γ be the operator introduced in the Theorem 6 and consider the decomposition Γ = Γ1+Γ2, where (Γ𝑖𝑢)0 = 0,𝑖 = 1, 2, and

Γ1𝑢(𝑡) = 𝑇 (𝑡)𝑔(0, 𝜙)− 𝑔(𝑡, 𝑦𝑡 + 𝑢𝑡)−∫ 𝑡

𝐴𝑇 (𝑡− 𝑠)𝑔(𝑠, 𝑦𝑠 + 𝑢𝑠)𝑑𝑠+

∫ 𝑡

𝑇 (𝑡− 𝑠)𝑓(𝑠, 𝑦𝑠 + 𝑢𝑠)𝑑𝑠, 𝑡 ≥ 0,

Γ2𝑢(𝑡) =∑𝑡𝑖<𝑡

𝑇 (𝑡− 𝑡𝑖)𝐼𝑖(𝑦𝑡𝑖 + 𝑢𝑡𝑖), 𝑡 ≥ 0.

Proceeding as in the proof of the Theorem 6, we infer that the map Γ1 is completely continuous. Moreover, it is easy tosee that

∥Γ2𝑢− Γ2𝑣∥𝒫𝒞ℎ≤ ��𝐾∞

∞∑𝑖=1

𝐿𝑖∥𝑢− 𝑣∥𝒫𝒞ℎ, 𝑢, 𝑣 ∈ 𝑆(∞),

which implies that Γ2 is a contraction in 𝑆(∞). We prove that there is 𝑟 > 0 such that Γ𝐵𝑟 ⊂ 𝐵𝑟, where 𝐵𝑟 =𝐵𝑟(0, 𝑆(∞)). In fact, if we assume that the assertion is false, then, for every 𝑟 > 0, there exists 𝑢𝑟 ∈ 𝐵𝑟 and 𝑡𝑟 ≥ 0 suchthat ∥Γ𝑢𝑟(𝑡

𝑟)ℎ(𝑡𝑟) ∥ > 𝑟. This yields that

𝑟 ≤ 1

ℎ(𝑡𝑟)��∥𝑔(0, 𝜙)∥+ ��

ℎ(𝑡𝑟)

∑𝑡𝑖<𝑡𝑟

∥𝐼𝑖(𝑢𝑟𝑡𝑖 + 𝑦𝑡𝑖)∥+∥𝑔(𝑡𝑟, 𝑢𝑟𝑡𝑟 + 𝑦𝑡𝑟 )∥

ℎ(𝑡𝑟)

∫ 𝑡𝑟

∥𝐴𝑇 (𝑡𝑟 − 𝑠)∥ℒ(𝕐,𝕏)∥𝑔(𝑠, 𝑢𝑟𝑠 + 𝑦𝑠)∥𝕐𝑑𝑠+ 1

ℎ(𝑡𝑟)

∫ 𝑡𝑟

��∥𝑓(𝑠, 𝑢𝑟𝑠 + 𝑦𝑠)∥𝑑𝑠

ℎ(𝑡𝑟)��∥𝑔(0, 𝜙)∥+ ∥𝑖𝑐∥ℒ(𝕐,𝕏)𝑐1,∞((��𝐾∞𝐻 +𝑀∞)∥𝜙∥ℬ +𝐾∞𝑟ℎ(𝑡𝑟))

ℎ(𝑡𝑟)

∫ 𝑡𝑟

ℋ(𝑡𝑟 − 𝑠)[𝑐1,∞((��𝐾∞𝐻 +𝑀∞)∥𝜙∥ℬ +𝐾∞𝑟ℎ(𝑠)) + 𝑑2

]𝑑𝑠

ℎ(𝑡𝑟)

∫ 𝑡𝑟

��𝑚(𝑠)𝑊 ((��𝐾∞𝐻 +𝑀∞)∥𝜙∥ℬ +𝐾∞𝑟ℎ(𝑠))𝑑𝑠

+��

ℎ(𝑡𝑟)

∞∑𝑖=1

𝐿𝑖((��𝐾∞𝐻)∥𝜙∥+𝐾∞𝑟ℎ(𝑡𝑖)) +∥𝑖𝑐∥ℒ(𝕐,𝕏)𝑑2

ℎ(𝑡𝑟)+

��

ℎ(𝑡𝑟)

∞∑𝑖=1

∥𝐼𝑖(0)∥,

which implies that

1 ≤ 𝑐1,∞𝐾∞(∥𝑖𝑐∥ℒ(𝕐,𝕏) +

∫ ∞

ℋ(𝑠)𝑑𝑠)+𝐾∞��

∞∑𝑖=1

𝐿𝑖 + lim inf𝑟→∞ ��

∫ ∞

𝑚(𝑠)𝑊 (𝑇 (𝑟, 𝑠))

𝑟𝑑𝑠,

which is contrary to our assumptions. This prove that there exists 𝑟 > 0 such that Γ is a condensing operator from 𝐵𝑟 into𝐵𝑟. This completes the proof.

5 An applicationLet 𝕏 = 𝐿2([0, 𝜋]) be the space of functions which are square integrable and ℬ = 𝒫𝒞𝑟 × 𝐿2(𝑔,𝕏). Now, we considerthe operator 𝐴 : 𝐷(𝐴) ⊂ 𝕏 → 𝕏 defined by 𝐴𝑥 = 𝑥′′, where

𝐷(𝐴) = {𝑥 ∈ 𝕏 : 𝑥′′ ∈ 𝕏, 𝑥(0) = 𝑥(𝜋) = 0}.Is well known that 𝐴 is the infinitesimal generator of an analytic and compact semigroup (𝑇 (𝑡))𝑡≥0 on 𝕏. Furthermore,𝐴 has a discrete spectrum, the eigenvalues are −𝑛2, 𝑛 ∈ ℕ, with corresponding normalized eigenfunctions 𝑧𝑛(𝜉) =( 2𝜋 )

1/2𝑠𝑖𝑛(𝑛𝜉), and the following properties hold.

(a) {𝑧𝑛 : 𝑛 ∈ ℕ} is an orthonormal basis for 𝕏.

(b) If 𝑥 ∈ 𝐷(𝐴) then 𝐴𝑥 = −∑∞𝑛=1 𝑛

2⟨𝑥, 𝑧𝑛⟩𝑧𝑛.

(c) For 𝑥 ∈ 𝕏, 𝑇 (𝑡)𝑥 =∑∞

𝑛=1 𝑒−𝑛2𝑡⟨𝑥, 𝑧𝑛⟩𝑧𝑛. In particular, we have that (𝑇 (𝑡))𝑡≥0 is a uniformly stable semigroup

with ∥𝑇 (𝑡)∥ ≤ 𝑒−𝑡 for all 𝑡 ≥ 0. Moreover, it is possible to define the fractional power of 𝐴; see e.g. [5, 15, 20].

(d) For each 𝑥 ∈ 𝕏 and each 𝛽 ∈ (0, 1), (−𝐴)−𝛽𝑥 =∑𝑛

𝑖=11

𝑛2𝛽 ⟨𝑥, 𝑧𝑛⟩𝑧𝑛. In particular, ∥(−𝐴)−1/2∥ = 1.

(e) For each 𝑥 ∈ 𝕏 and 𝛽 ∈ (0, 1), (−𝐴)𝛽𝑥 =∑𝑛

𝑖=1 𝑛2𝛽⟨𝑥, 𝑧𝑛⟩𝑧𝑛. Moreover,

𝐷((−𝐴)𝛽) ={𝑥 : 𝑥 ∈ 𝕏,

𝑛∑𝑖=1

𝑛2𝛽⟨𝑥, 𝑧𝑛𝑧𝑛 ∈ 𝕏

Consider the impulsive partial neutral differential system,

∂𝑡

[𝑤(𝑡, 𝜉) +

∫ 𝑡

−∞

∫ 𝜋

𝑎1(𝑡)𝑏(𝑠− 𝑡, 𝜂, 𝜉)𝑤(𝜉, 𝑠)𝑑𝜂𝑑𝑠]=

∂𝜉2𝑤(𝑡, 𝜉) + 𝑣(𝑡, 𝑤(𝑡, 𝜉))𝑡 ≥ 0, 0 ≤ 𝜉 ≤ 𝜋, (15)

𝑤(𝑡, 0) = 𝑤(𝑡, 𝜋) = 0, 𝑡 ≥ 0, (16)

𝑤(𝜏, 𝜉) = 𝜑(𝜏, 𝜋), 𝜏 ≤ 0, 0 ≤ 𝜉 ≤ 𝜋, (17)

Δ𝑤(𝑡𝑖, ⋅) = 𝑤(𝑡+𝑖 , ⋅)− 𝑤(⋅, 𝑡−𝑖 ) =∫ 𝜋

𝑝𝑖(𝜉, 𝑤(𝑡𝑖, 𝑠))𝑑𝑠, (18)

where (𝑡𝑖)𝑖∈ℕ is a strictly increasing sequence of positive real numbers. To treat this system, we assume the followingconditions.

(a)∗ The functions 𝑣 : ℝ × ℝ → ℝ, is continuous and there is a continuous and integrable function 𝑚 : ℝ → [0,∞)such that

∣𝑣(𝑡, 𝑥)∣ ≤ 𝑚(𝑡)(∣𝑥∣), 𝑡 ≥ 0, 𝑥 ∈ ℝ.

(b)∗ The function 𝑎1 : ℝ → [0,∞) is continuous and lim𝑡→+∞ 𝑎1(𝑡) = 0.

(c)∗ The functions 𝑏(𝑠, 𝜂, 𝜉), ∂∂𝜉 𝑏(𝑠, 𝜂, 𝜉) are integrable, 𝑏(𝑠, 𝜂, 𝜋) = 𝑏(𝑠, 𝜂, 0) = 0 and

𝐿𝐹 = sup

{∫ 𝜋

−∞

∫ 𝜋

𝑔(𝑠)

∂𝑖𝑏(𝑠, 𝜂, 𝜉)

∂𝜉𝑖𝑑𝜂𝑑𝑠𝑑𝜉

: 𝑖 = 0, 1

}< +∞.

(d)∗ The functions 𝑝𝑖 : [0, 𝜋]× ℝ → ℝ, 𝑖 ∈ ℕ, are continuous and there are positive constants 𝐿𝑖 such that

∣𝑝𝑖(𝜉, 𝑠)− 𝑝𝑖(𝜉, 𝑠)∣ ≤ 𝐿𝑖∣𝑠− 𝑠∣, 𝜉 ∈ [0, 𝜋], 𝑠, 𝑠 ∈ ℝ.

We now define the functions 𝑔, 𝑓 : [0,∞)× ℬ → 𝕏, 𝐼𝑖 : 𝕏 → 𝕏 by

𝑔(𝑡, 𝜙)(𝜉) =

−∞

∫ 𝜋

𝑎1(𝑡)𝑏(𝑠, 𝜂, 𝜉)𝜙(𝑠, 𝜂)𝑑𝜂𝑑𝑠, 𝑡 ≥ 0, 𝜉 ∈ [0, 𝜋],

𝑓(𝑡, 𝜙)(𝜉) = 𝑣(𝑡, 𝜙(0, 𝜉)) 𝑡 ≥ 0, 𝜉 ∈ [0, 𝜋],

𝐼𝑖(𝜙)(𝜉) =

∫ 𝜋

𝑝𝑖(𝜉, 𝜙(0, 𝑠))𝑑𝑠, 𝑖 ∈ ℕ, 𝜉 ∈ [0, 𝜋],

𝕐 = 𝐷(𝐴1/2) =

{𝑥(⋅) ∈ 𝕏 :

∞∑𝑖=1

𝑛 < 𝑥, 𝑧𝑛 > 𝑧𝑛 ∈ 𝕏

Using the properties of semigroup (𝑇 (𝑡))𝑡≥0, we have that 𝕐 verifies the conditions (H1) (H2) and (H3). Moreover, inthis case

∥(−𝐴) 12𝑇 (𝑡)∥𝕏 ≤ 𝑒

−𝑡2 𝑡

−12√

2, 𝑓𝑜𝑟 𝑡 > 0.

It easy to see that problem (5.1)-(5.4) can be modeled as the abstract impulsive Cauchy problem (4.1)-(4.3).The next result follows directly from Theorem 7.

Proposition 11 Assume that the previous conditions hold. Moreover, suppose that∑∞

𝑖=1[∫ 𝜋

0∣𝑝𝑖(𝜉, 0)∣2𝑑𝜉]1/2 <∞. If

𝐾∞[𝐿𝑔

(1 + Γ(1/2)

)+ ��

∫ ∞

𝑚(𝑠)𝑑𝑠+

∞∑𝑖=1

𝐿𝑖

where Γ(⋅) : (0,∞) → ℝ denotes the Gamma function and 𝑐1,∞ = 𝐿1/2 sup𝑡≥0 𝑎1(𝑡), then there is a global solution ofthe impulsive system (5.1)-(5.4).

AcknowledgmentsThe author V. Vijayakumar would like to thank Dr. V. Palanisamy, Principal and Dr. Chitra Manohar, Secretary, InfoInstitute of Engineering, Coimbatore, for their constant encouragements and support rendered to him during his researchwork.

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Existence of solutions for impulsive partial neutral functional differential equations

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