Existence of solutions for impulsive partial neutral functional differential equations
Transcript of 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 β, 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 [1β3, 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 β§β¨β©
π’β²(π‘) = π΄π’(π‘) + π(π‘, π’(π‘), π’(π(π‘))), π‘ β πΌ = [0, π] or [0,β),
π’(0) = π’0,
Ξπ’(π‘π) = π½π(π’(π‘π)), π β π½ β β,
by using Leray-Schauderβs alternative theorem and in [4], the existence of solutions for impulsive neutral functionaldifferential equations of the formβ§β¨β©
πππ‘ (π’(π‘) + πΉ (π‘, π’π‘)) = π΄(π‘)π’(π‘) +πΊ(π‘, π’π‘), π‘ β πΌ, π‘ β= π‘π,
Ξπ’(π‘π) = πΌπ(π’π‘π),
π’0 = π β β¬,by using Leray-Schauderβs alternative theorem.
In this paper, we establish the existence of mild solutions for a class of impulsive neutral functional differentialequations described by
π
ππ‘(π’(π‘) + π(π‘, π’π‘)) = π΄π’(π‘) + π(π‘, π’π‘), π‘ β πΌ, π‘ β= π‘π, (1)
π’0 = π β β¬, (2)
Ξπ’(π‘π) = πΌπ(π’π‘π), (3)βCorresponding author Email address: [email protected]
Copyright cβWorld 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 β β β 413
where π΄ is the infinitesimal generator of a πΆ0-semigroup of bounded linear operators ((π (π‘))π‘β₯0 defined on a Banachspace (π, β₯ β β₯); πΌ is an interval of the form [0, π ); 0 < π‘1 < π‘2 < β β β < π‘π < β β β < π are prefixed numbers; the historyπ’π‘ : (ββ, 0] β π, π’π‘(π) = π’(π‘+ π), belongs to some abstract phase space β¬ defined axiomatically; π, π : πΌ Γ β¬ β π,πΌπ : β¬ β π are appropriate functions and the symbol Ξπ(π‘) represents the jump of the function π at π‘, which is definedby Ξπ(π‘) = π(π‘+)β π(π‘β).
The existence of global solutions for impulsive ordinary and partial differential system has been considered recentlyin the literature, see among others, [4, 8, 10β15, 23β26, 32β34]. To the best of our knowledge, the study of the existenceof global solutions for the impulsive neutral functional differential equations described in the general βabstractβ 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 πΌ = [0, π]. 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 (π, β₯ β β₯π) and(π, β₯ β β₯π) be Banach spaces, and β(π,π) be the Banach space of bounded linear operators from π into π equipped withits natural topology; in particular, we use the notation β(π) when π = π.
Throughout this work, π΄ is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators(π (π‘))π‘β₯0 defined on a Banach space (π, β₯ β β₯) and οΏ½οΏ½ is a positive constant such that β₯π (π‘)β₯ β€ οΏ½οΏ½ for every π‘ β πΌ . Forthe theory of strongly continuous semigroup, refer the reader to Pazy [27].
In this paper, we employ an axiomatic definition of the phase space β¬, which is similar to the one introduced in Hinoet al [16] and suitably modified to treat retarded impulsive differential equations. More precisely, β¬ is a linear space offunctions mapping (ββ, 0] into π endowed with a seminorm β₯ β β₯β¬ and we assume that β¬ satisfies the following axioms:
(A) If π₯ : (ββ, π + π] β π, π > 0, π β β such that π₯π β β¬, and π₯β£[π, π + π] β π«π([π, π + π],π), then for everyπ‘ β [π, π + π) the following conditions hold:
(i) π₯π‘ is in β¬;
(ii) β₯π₯(π‘)β₯π β€ π»β₯π₯π‘β₯β¬;
(iii) β₯π₯π‘β₯β¬ β€ πΎ(π‘β π) sup{β₯π₯(π )β₯π : π β€ π β€ π‘}+π(π‘β π)β₯π₯πβ₯β¬,
where π» > 0 is a constant; πΎ,π : [0,β) β [1,β), πΎ is continuous, π is locally bounded and π»,πΎ,π areindependent of π₯(β ).
(B) The space β¬ is complete.
Remark 1 In impulsive functional differential systems, the map [π, π + π] β β¬, π‘β π₯π‘ is in general discontinuous. Forthis reason, this property has been omitted from the description of phase space β¬.
Example 1 The Phase Space π«ππ Γ πΏ2(π,π) .Let π > 0 and π : (ββ,βπ] β β be a non-negative, locally Lebesgue integrable function. Assume that there is a
non-negative measurable, locally bounded function π(β ) on (ββ, 0] such that π(π + π) β€ π(π)π(π) for all π β (ββ, 0]and π β (ββ,βπ] β ππ, where ππ β (ββ,βπ] is a set with Lebesgue measure zero. We denote by π«ππ Γ πΏ2(π,π)the set of all functions π : (ββ, 0] β π such that πβ£[βπ, 0] β π«π([βπ, 0],π) and
β« βπ
ββ π(π)β₯π(π)β₯2πππ < +β. Inπ«ππ Γ πΏ2(π,π), we consider the seminorm defined by
β₯πβ₯β¬ = supπβ[βπ,0]
β₯π(π)β₯π +
(β« βπ
ββπ(π)β₯π(π)β₯2πππ
)1/2
.
From the proceeding conditions, the space π«ππ Γ πΏ2(π,π) satisfies the axioms (A) and (B). Moreover, when π = 0, we
can take π» = 1, πΎ(π‘) =(1 +
β« 0
βπ‘π(π)
)1/2
and π(π‘) = π(βπ‘) for π‘ β₯ 0.
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414 International Journal of Nonlinear Science, Vol.11(2011), No.4, pp. 412-426
Let 0 < π‘1 < β β β < π‘π < π be pre-fixed numbers. We introduce the space π«π = π«π([0, π];π) formed by all functionsπ’ : [0, π] β π such that are continuous at π‘ β= π‘π, π’(π‘βπ ) = π’(π‘π) and π’(π‘+π ) exists, for all π = 1, ....π. In this paper, weassume that π«π is endowed with the norm β₯π’β₯π«π = supπ β[0,π] β₯π’(π )β₯π. It is clear that (π«π, β₯ β β₯π«π) is a Banach space;see [9] for details.
In what follows, we put π‘0 = 0, π‘π+1 = π, and for π’ β π«π, we denote by οΏ½οΏ½π β πΆ([π‘π, π‘π+1],π), π = 0, 1, 2, ..., π, thefunction given by
οΏ½οΏ½π(π‘) =
{π’(π‘), for π‘ β (π‘, π‘π+1],
π’(π‘+π ), for π‘ = π‘π.
Moreover, for β¬ β π«π, we employ the notation οΏ½οΏ½π, π = 0, 1, 2, ..., π, for the sets οΏ½οΏ½π = {οΏ½οΏ½π : π’ β π΅}.
Lemma 2 ([9]). A Set π΅ β π«π is relatively compact in π«π if and only if the set π΅π is relatively compact in the spaceπΆ([π‘π, π‘π+1];π), for every π = 0, 1, ..., π.
In the following definition, we introduce the concept of a mild solution for the problem (1.1)-(1.3).
Definition 1 ([28]). A function π’ : (ββ, 0]βͺπΌ β π is called a mild solution of the abstract Cauchy problem (1.1)-(1.3),if π’0 = π β β¬; π’β£πΌ β π«π(πΌ;π); the function π β π΄π (π‘β π )π(π , π’π ) is integrable in [0, π‘) for all π‘ β πΌ and
π’(π‘) = π (π‘)(π(0) + π(0, π)
)β π(π‘, π’π‘)ββ« π‘
0
π΄π (π‘β π )π(π , π’π )ππ +
β« π‘
0
π (π‘β π )π(π , π’π )ππ
+βπ‘π<π‘
π (π‘β π‘π)πΌπ(π’π‘π) +βπ‘π<π‘
[π(π‘, π’π‘)β£π‘+π
β π(π‘, π’π‘)β£π‘βπ
], π‘ β πΌ.(4)
Motivated by the previous definition, we introduce the following assumptions. There exists a Banach space (π, β₯ β β₯π)continuously included in π such that
(S1) The function π β π (π‘β π )π¦ β πΆ([π‘,+β);π), π‘ β₯ 0.
(S2) The function π β π΄π (π‘ β π ) defined from (π‘,+β), π‘ > 0 into πΏ(π,π) is continuous and there is a functionβ(β ) β πΏ1([0,β);β+) such that
β₯π΄π (π‘β π )β₯β(π,π) β€ β(π‘β π ), for all π‘ > π .
Remark 3 Assume that (S1) and (S2) hold and π’ β πΆ([0, π‘];π), then from the Bochnerβs criterion for integrable func-tions and the estimate
β₯π΄π (π‘β π )π’(π )β₯π β€ β₯π΄π (π‘β π )β₯β(π,π)β₯π’(π )β₯π β€ β(π‘β π )β₯π’(π )β₯π,
we have that the function π β π΄π (π‘ β π )π’(π ) is integrable over [0, π‘) for all π‘ > 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 π β π΄π (π‘β π ) 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-Schauderβs Alternative Theorem). Let π· be a closed convex subset of a Banachspace (β€, β₯ β β₯β€) and assume that 0 β π·. If πΉ : π· β π· is a completely continuous map, then the set {π₯ β π· : π₯ =ππΉ (π₯), 0 < π < 1} is unbounded or the map πΉ has a fixed point in π·.
Theorem 5 ([22, Corollary 4.3.2]). Let π· be a closed, convex and bounded subset of a Banach space (β€, β₯ β β₯β€). Ifπ΅,πΆ : π· β β€ are continuous functions such that
(a) π΅π§ + πΆπ§ β π· for all π§ β β€;
(b) πΆ(π·) is compact;
(c) there exists 0 β€ πΎ < 1 such that β₯π΅π§ βπ΅π€β₯ β€ πΎβ₯π§ β π€β₯ for all π§, π€ β π·;
then there exists π₯ β π· such that π΅π₯+ πΆπ₯ = π₯.
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S. Sivasankaran, V. Vijayakumar, M. M. Arjunan: Existence of Global Solutions for Impulsive Abstract Partial Neutral β β β 415
3 Local Existence
In this section, we study the existence of mild solutions for the impulsive systems (1.1)-(1.3), when πΌ = [0, π]. To obtainour results, we introduce the following conditions.
(H1) The function π : πΌ Γ β¬ β π is completely continuous, π(πΌ Γ β¬) β π , π β πΆ(πΌ Γ β¬,π) and there exist positiveconstants π1 and π2 such that
β₯π(π‘, π)β₯π β€ π1β₯πβ₯β¬ + π2, π‘ β πΌ, π β β¬.
(H2) The function π : πΌ Γ β¬ Γ π β π satisfies the following conditions.
(a) The function π(π‘, β ) : β¬ β π is continuous for all π‘ β πΌ .
(b) The function π(β , π) : πΌ β π is strongly measurable for all π β β¬.
(c) There exists a continuous function π : πΌ β [0,β) and a continuousnon-decreasing function π : [0,β) β (0,β) such that
β₯π(π‘, π)β₯ β€ π(π‘)π (β₯πβ₯β¬), for every (π‘, π) β πΌ Γ β¬.
(H3) The maps πΌπ : β¬ β π are completely continuous and uniformly bounded, π β π½ = {1, 2, β β β π}. In what follows,we use the notation: ππ = sup{β₯πΌπ(π)β₯ : π β β¬}.
(H4) There are positive constants πΏπ, such that
β₯πΌπ(π1)β πΌπ(π2)β₯ β€ πΏπβ₯π1 β π2β₯β¬, π1, π2 β β¬, π β π½.
(H5) The function π : πΌ Γ β¬ β π is Lipschitz constant; that is, there is a constant πΏπ > 0 such that
β₯π(π‘, π1)β π(π‘, π2)β₯π β€ πΏπβ₯π1 β π2β₯β¬, π‘ β πΌ, ππ β β¬, π = 1, 2.
(H6) Let π¦ : (ββ, 0] βͺ πΌ β π be the extension of π such that π¦(π‘) = π (π‘)π(0) for all π‘ β πΌ and π(πΌ) the spaceπ(πΌ) = {π₯ : (ββ, 0]βͺ πΌ β π : π₯0 = 0, π₯β£πΌ β π«π(πΌ;π)} endowed with uniform convergence topology in πΌ . Then
the set of functions{π‘β π(π‘, π₯π‘ + π¦π‘); π₯ β π
}is equicontinuous in πΌ for all bounded sets π β π(πΌ).
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 π‘, π β [0, π], π‘ > π and π > 0, the set {π (π‘β π )π(π , π) : π β [0, π‘], β₯πβ₯β¬ β€ π} is relatively compact in π.
(b) π = π1πΎπ
(β₯ππβ₯β(π,π) +
β« π
0β(π )ππ
)< 1, οΏ½οΏ½πΎπ
1βπ
β« π
0π(π )ππ <
β«βπ
ππ π (π ) , where ππ : π β π is the inclusion
operator and
π =πΎπ
1β π
[(οΏ½οΏ½π» + π1οΏ½οΏ½β₯ππβ₯β(π,π) +πΎβ1
π ππ
)β₯πβ₯β¬ + π2β₯ππβ₯β(π,π)(οΏ½οΏ½ + 1) + π2
β« π
0
β(π )ππ + οΏ½οΏ½πβπ=1
ππ
],
(5)
whereπ» is a constant given by Axiom (π΄),πΎπ andππ are given byπΎπ = sup0β€π‘β€ππΎ(π‘) andππ = sup0β€π‘β€ππ(π‘),respectively.
Then there exists a mild solution of the initial value problem (1.1)-(1.3).
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416 International Journal of Nonlinear Science, Vol.11(2011), No.4, pp. 412-426
Proof.Let π(πΌ) and π¦ : (ββ, π] β π be introduced in (H6) and Ξ : π(πΌ) β π(πΌ) be the operator defined by (Ξπ’)0 = 0
and
Ξπ’(π‘) = π (π‘)π(0, π)βπ(π‘, π’π‘+π¦π‘)ββ« π‘
0
π΄π (π‘βπ )π(π , π’π +π¦π )ππ +β« π‘
0
π (π‘βπ )π(π , π’π +π¦π )ππ +β
0<π‘π<π‘
π (π‘βπ‘π)πΌπ(π’π‘π+π¦π‘π),(6)
To prove that the function Ξπ’ β π(πΌ), we need to show that the following properties are satisfied:(i) The function Ξπ’ is continuous in π‘ β= π‘π.(ii) The limit limπ‘βπ‘βπ
Ξπ’(π‘) = Ξπ’(π‘π), for all π = 1, ..., π .(iii) The limit limπ‘βπ‘+π
Ξπ’(π‘), exists, for all π = 1, ..., π .Taking into the account that π and π‘ β π (π‘)π(0, π) are continuous, it follows from (S1), (H2) and the Remark 2
that the function Ξπ’(π‘) is continuous in π‘ β= π‘π. On the other hand, from the definition of Ξ, the Dominated ConvergenceTheorem and the condition (S2), we have that limπ‘βπ‘βπ
Ξπ’(π‘) = Ξπ’(π‘π) and that limπ‘βπ‘+πΞπ’(π‘) = Ξπ’(π‘π) + πΌπ(π’π‘π).
Next, we show that Ξ is a continuous operator. Let (π’π)πββ be a sequence in π(πΌ) and π’ β π(πΌ) such that π’π β π’in π(πΌ). By using the continuity of π and conditions ((H2)β (a)), (H3) and (H6), we prove that π(π‘, π’ππ + π¦ππ ) βπ(π‘, π’π + π¦π ) uniformly in πΌ , π(π , π’ππ + π¦π ) β π(π , π’π + π¦π ) for all π β πΌ , and πΌπ(π’ππ‘ + π¦π‘) β πΌ(π’π‘ + π¦π‘) uniformly on πΌ .Using conditions (S2), ((H2)β (b, c)) and Lebesgueβs Dominated Convergence Theorem, we conclude thatβ« π‘
0
π΄π (π‘βπ )π(π , π’ππ +π¦ππ )ππ ββ« π‘
0
π΄π (π‘βπ )π(π , π’π +π¦π )ππ ,β« π‘
0
π (π‘βπ )π(π , π’ππ +π¦ππ )ππ ββ« π‘
0
π (π‘βπ )π(π , π’π +π¦π )ππ ,
as πβ β, which clearly implies that Ξ is a continuous operator.In order to use Theorem 2, we need to obtain a ππππππ bound for the solutions of the integral equations πΞπ’ = π’,
π β (0, 1). Let π β (0, 1) and let π₯π be a solution of πΞπ’ = π’, 0 < π < 1; taking into account that β₯π¦π‘β₯β¬ β€(πΎποΏ½οΏ½π» +ππ)β₯πβ₯β¬ and using Remark 2, (H2) and (H4), we find that
β₯π₯π(π‘)β₯ β€οΏ½οΏ½(β₯ππβ₯β(π,π)(π1β₯πβ₯β¬ + π2)) + β₯ππβ₯β(π,π)
(π1(πΎπβ₯π₯πβ₯π‘ + (πΎποΏ½οΏ½π» +ππ)β₯πβ₯β¬) + π2
)+
β« π‘
0
β(π‘β π )(π1(πΎπβ₯π₯πβ₯π + (πΎποΏ½οΏ½π» +ππ)β₯πβ₯β¬) + π2
)ππ + οΏ½οΏ½
β« π‘
0
π(π )π (πΎπβ₯π₯πβ₯π
+(πΎποΏ½οΏ½π» +ππ)β₯πβ₯β¬)ππ +πβπ=1
οΏ½οΏ½ππ,
where we are using the following notation: β₯πβ₯π‘ = sup0β€π β€π‘ β₯π(π )β₯π. Next, putting
πΏπ(π‘) = πΎπβ₯π₯πβ₯π‘ + (πΎποΏ½οΏ½π» +ππ)β₯πβ₯β¬,it follows that
β₯π₯π(π‘)β₯ β€ β₯ππβ₯β(π,π)(οΏ½οΏ½(π1β₯πβ₯β¬ + π2)) + β₯ππβ₯β(π,π)(π1πΏπ(π‘) + π2) +
β« π‘
0
β(π‘β π )(π1πΏπ(π ) + π2)ππ
+οΏ½οΏ½
β« π‘
0
π(π )π (πΏπ(π ))ππ +
πβπ=1
οΏ½οΏ½ππ.
We obtain after some simplification and a rearrangement of terms
πΏπ(π‘) β€[πΎποΏ½οΏ½π» + π1πΎποΏ½οΏ½β₯ππβ₯β(π,π) +ππ
]β₯πβ₯β¬ + π2πΎπβ₯ππβ₯β(π,π)(οΏ½οΏ½ + 1) +πΎππ2
β« π
0
β(π )ππ
+π1πΎπ
(β₯ππβ₯β(π,π) +
β« π
0
β(π )ππ )πΏπ(π‘) +πΎποΏ½οΏ½
β« π‘
0
π(π )π (πΏπ(π ))ππ +πΎποΏ½οΏ½
πβπ=1
ππ.
From the hypothesis (b), we have that
πΏπ(π‘) β€ π+οΏ½οΏ½πΎπ
1β π
β« π‘
0
π(π )π (πΏπ(π ))ππ .
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S. Sivasankaran, V. Vijayakumar, M. M. Arjunan: Existence of Global Solutions for Impulsive Abstract Partial Neutral β β β 417
Denote by π½π(π‘) the right-hand side of the previous inequality. Computing π½β²π(π‘) and observing that πΏπ(π‘) β€ π½π(π‘)
for π‘ β πΌ , we arrive at
π½β²π(π‘) β€
οΏ½οΏ½πΎπ
1β ππ(π‘)π (π½π(π‘)),
hence β« π½π(π‘)
π½π(0)
ππ
π (π )β€ οΏ½οΏ½πΎπ
1β π
β« π
0
π(π )ππ β€β« β
π
ππ
π (π ).
This allow us to conclude that the set of functions {π½π : π β (0, 1)} is bounded. This implies that {π’π : π’π = Ξπ’π, π β(0, 1)} is bounded in π(πΌ).
It remains to show that Ξ is completely continuous. To this end, we introduce the decomposition Ξ = Ξ1 +Ξ2, where(Ξππ’)0 = 0, π = 1, 2, and
Ξ1π’(π‘) = π (π‘)π(0, π)β π(π‘, π’π‘ + π¦π‘)ββ« π‘
0
π΄π (π‘β π )π(π , π’π + π¦π )ππ +
β« π‘
0
π (π‘β π )π(π , π’π + π¦π )ππ , π‘ β [0, π],
Ξ2π’(π‘) =β
0<π‘π<π‘
π (π‘β π‘π)πΌπ(π’π‘π + π¦π‘π), π‘ β [0, π].
We now prove that Ξ1 is a completely continuous operator. We consider π’ in π΅π(0, π(πΌ)), an open ball of radius π.Applying now the Mean Value Theorem for the Bochner integral (see [22, Lemma 2.1.3]), we infer that
(Ξ1π’)(π‘) β π (π‘)π(0, π)β π(π‘, π’π‘ + π¦π‘)β π‘ππ{π΄π (π‘β π)π(π, π’π + π¦π), π β [0, π‘], π’ β π΅π(0, π(πΌ))}π
+π‘ππ{π (π‘β π)π(π, π’π + π¦π), π β [0, π‘], π’ β π΅π(0, π(πΌ))}.Taking the advantage of assumptions (a) and (H1), we conclude that
{(Ξ1π’)(π‘) : π’ β π΅π(0, π(πΌ))} is a compact subset of π.Now, we prove that the set of functions {(Ξ1(π’);π’ β π΅π(0, π(πΌ))} is equicontinuous in πΌ . To this end, consider
0 < π < π‘ < π‘β² for π‘, π‘β² β [0, π]; then there is 0 < πΏ < π such that
β₯π (π‘)π(0, π)β π (π‘β²)π(0, π)β₯ < π,
β₯π(π‘, π’π‘ + π¦π‘)β π(π‘β², π’π‘β² + π¦π‘β²)ππ )β₯ < π,
β₯π (π‘β π )π(π’π + π¦π )β π (π‘β² β π )πΊ(π’π + π¦π , )β₯ < π,
for all π β [0, π‘], β£π‘β π‘β²β£ < πΏ and π’ β π΅π(0, π(πΌ)). On the other-hand, using the continuity of the map (π‘, π ) β π΄π (π‘β π )for π‘ > π , we arrive at
β₯π΄π (π‘β π )π(π , π’π + π¦π )βπ΄π (π‘β² β π )π(π , π’π + π¦π )β₯ < π,
for all π β [0, π‘β π], β£π‘β π‘β²β£ < πΏ and π’ β π΅π(0, π(πΌ)). Under these conditions, for π’ β π΅π(0, π(πΌ)), β£π‘β π‘β²β£ < πΏ, we inferthat
β₯Ξ1π’(π‘)β Ξ1π’(π‘β²)β₯ β€ β₯(π (π‘)β π (π‘β²))π(0, π)β₯+ β₯π(π‘, π’π‘ + π¦π‘)β π(π‘β², π’π‘β² + π¦π‘β²)β₯+
β« π‘β²
π‘
β₯π΄π (π‘β² β π )π(π , π’π + π¦π )β₯ππ
+
β« π‘βπ
0
β₯(π΄π (π‘β π )βπ΄π (π‘β² β π ))π(π , π’π + π¦π )β₯ππ +β« π‘
π‘βπ
β₯(π΄π (π‘β π )βπ΄π (π‘β² β π ))π(π , π’π + π¦π )β₯ππ
+
β« π‘
0
β₯(π (π‘β π )β π (π‘β² β π ))π(π , π’π + π¦π )β₯ππ +β« π‘β²
π‘
β₯π (π‘β² β π )π(π , π’π + π¦π )β₯ππ .
From the above estimate, one can deduce the following inequality:
β₯Ξ1π’(π‘)β Ξ1π’(π‘β²)β₯ β€ 2π+ π(π‘β π) + ππ‘+ (π1(πΎππ +πΎποΏ½οΏ½π» +ππ))
β« π‘
π‘βπ
β(π‘β π )ππ +
β« π‘β²
π‘
β(π‘β π )ππ
+οΏ½οΏ½π (π1(πΎππ +πΎποΏ½οΏ½π» +ππ))
β« π‘β²
π‘
π(π )ππ ,
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which shows the equicontinuity at π‘ β πΌ . So, to conclude the proof, we show that Ξ2 is completely continuous. We observethat the continuity of Ξ2 is obvious. On the other hand, for π > 0, π‘ β [π‘π, π‘π+1], π β₯ 1, and π’ β π΅π = π΅π(0,π«π([0, π];π)),we have that there exists π > 0 such that
[ ΛΞ2π’]π(π‘) β
β§β¨β©βπ
π=1 π (π‘β π‘π)πΌπ(π΅π(0;β¬)), π‘ β (π‘π, π‘π+1),βππ=1 π (π‘π+1 β π‘π)πΌπ(π΅π(0;β¬)), π‘ = π‘π+1,βπβ1π=1 π (π‘π β π‘π)πΌπ(π΅π(0;β¬)) + πΌπ(π΅π(0;β¬)), π‘ = π‘π,
where π΅π(0;β¬) is an open ball of radius π. From condition (H4) it follows that [ ΛΞ2(π΅π)]π(π‘) is relatively compact in π,for all π‘ β [π‘π, π‘π+1], π β₯ 1. Moreover, using the fact that the operators {πΌπ}π are compact and the strong continuity of(π (π‘))π‘β₯0, we conclude that for π > 0, there exists πΏ > 0 such that
β₯π (π‘β π‘π)π§ β π (π β π‘π)π§β₯ β€ π, π§ βπβͺπ=1
πΌπ(π΅π(0,β¬)), (3.3)
for all π‘π, π = 1, ..., π , π‘, π β (π‘π, π‘π+1] with β£π‘ β π β£ < πΏ. Thus for π’ β π΅π, π‘ β [π‘π, π‘π+1], π β₯ 0, and 0 < β£ββ£ < πΏ withπ‘+ β β [π‘π, π‘π+1], we have that
β₯[ ΛΞ2π’]π(π‘+ β)β [ ΛΞ2π’]π(π‘)β₯ β€ ππ, π = 1, ..., π. (3.4)
We have shown that the set Λ[Ξ2(π΅π)]π is uniformly equicontinuous for π β₯ 0. Now, from Lemma 1, we conclude that Ξ2
is completely continuous.We have proven that Ξ 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 πΉ and πΌπ, π = 1, ..., π 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
[πΎππΏπ
(β₯ππβ₯β(π,π) +
β« π
0
β(π )ππ )+ οΏ½οΏ½πΎπ
πβπ=1
πΏπ + οΏ½οΏ½πΎπ limπββ inf
π (π)
π
β« π
0
π(π )ππ ]< 1,
where ππ denotes the inclusion operator from π into π, then there exists a mild solution of the impulsive problem (1.1)-(1.3).
Proof. Let Ξ be the function given in the proof of Theorem 4 and consider the decomposition Ξ = Ξ1 + Ξ2, where(Ξππ’) = 0 on (ββ, 0], π = 1, 2, and
Ξ1π’(π‘) = π (π‘)π(0, π)β π(π‘, π’π‘ + π¦π‘)ββ« π‘
0
π΄π (π‘β π )π(π , π’π + π¦π )ππ +β
0<π‘π<π‘
π (π‘β π‘π)πΌπ(π’π‘π + π¦π‘π), π‘ β [0, π],
Ξ2π’(π‘) =
β« π‘
0
π (π‘β π )π(π , π’π + π¦π )ππ , π‘ β [0, π].
We claim that there exists π > 0 such that Ξ(π΅π(0, π(πΌ))) β π΅π(0, π(πΌ)). In fact, if we assume that this assertion is false,then for all π > 0 we can choose π₯π β π΅π = π΅π(0, π(πΌ)) and π‘π β [0, π] such that β₯Ξπ₯π(π‘π)β₯ > π. Observe that standardcomputations involving the phase space axioms yield
π β€ β₯Ξπ₯π(π‘π)β₯ β€ οΏ½οΏ½β₯π(0, π)β₯π + β₯ππβ₯πΏ(π,π) sup0β€π‘β€π
β₯π(π‘, 0)β₯π + πΏπβ₯ππβ₯β(π,π)(πΎππ + (πΎποΏ½οΏ½π» +ππ)β₯πβ₯β¬)
+
β« π
0
β(π )ππ (πΏπ(πΎππ + (πΎποΏ½οΏ½π» +ππ)β₯πβ₯β¬) + sup
0β€πβ€πβ₯π(π, 0)β₯π
)+οΏ½οΏ½
β« π
0
π(π )π ((πΎππ +πΎποΏ½οΏ½π» +ππ)β₯πβ₯β¬)ππ + οΏ½οΏ½πβπ=1
(πΏπ(πΎππ + (πΎποΏ½οΏ½π» +ππ)β₯πβ₯β¬) + β₯πΌπ(0)β₯).
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Thus,
1 β€[πΎππΏπ
(β₯ππβ₯β(π,π) +
β« π
0
β(π )ππ )+ οΏ½οΏ½πΎπ
πβπ=1
πΏπ + οΏ½οΏ½πΎπ limπββ inf
π (π)
π
β« π
0
π(π )ππ ],
which is contrary to our assumptions.Let π > 0 such that Ξ(π΅π(0, π(πΌ))) β π΅π(0, π(πΌ)). It follows from the proof of Lemma 3.1 in [10] that Ξ2 is
completely continuous in π΅π(0, π(πΌ)). Moreover, the estimate
β₯Ξ1π’β Ξ1π£β₯β β€[πΎππΏπΉ
(β₯ππβ₯β(π,π) +
β« π
0
β(π )ππ )+ οΏ½οΏ½πΎπ
πβπ=1
πΏπ
]β₯π’β π£β₯β,
π’, π£ β π΅π, shows that Ξ1 is a contraction on π΅π. Consequently, Ξ is a condensing operator from π΅π into π΅π. 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
π
ππ‘(π’(π‘) + π(π‘, π’π‘)) = π΄π’(π‘) + π(π‘, π’π‘), π‘ β πΌ = [0,β), (7)
π’0 = π β β¬ (8)
Ξπ’(π‘π) = πΌπ(π’π‘π), 0 < π‘1 < π‘2 < β β β < π‘π < β β β (9)
where (π‘π)πββ is an increasing sequence of positive numbers.Let β : [0,β) β [0,β) be a positive, non-decreasing, continuous function such that β(0) = 1 and limπ‘ββ β(π‘) =
+β. In addition, suppose that the map (π‘, π ) β π (π‘βπ ) is uniformly bounded. Moreover, π«π([0,β);π) and (π«πβ)0(π)
denote the spaces
π«π([0,β);π) =
β§β¨β©π₯ : [0,β) β π : π₯β£[0,π]
β π«π, βπ β (0,β) β {π‘π : π β β},π₯0 = 0,
β₯π₯β₯β = supπ‘β₯0 β₯π₯(π‘)β₯ <β
β«β¬β(π«π)0β(π) =
{π₯ β π«π([0,β);π) : lim
π‘βββ₯π₯(π‘)β₯β(π‘)
= 0
},
endowed with the norms β₯π₯β₯β = supπ‘β₯0 β₯π₯(π‘)β₯ and β₯π₯β₯π«πβ= supπ‘β₯0
β₯π₯(π‘)β₯β(π‘) , respectively.
To get the next results, we need a very detailed knowledge of the relativity compact sets of the space (π«π)0β(π). Wewill use the following result.
Lemma 8 A bounded set β¬ β (π«π)0β(π) is relatively compact in (π«π)0β(π) if, and only if:
(a) The set π΅π = {π’β£[0,π]: π’ β π΅} is relatively compact in π«π([0, π];π), for all π β (0,β) β {π‘π; π β β}.
(b) limπ‘βββ₯π₯(π‘)β₯β(π‘) = 0, uniformly for π₯ β π΅.
We introduce the following concept of global solution of system (4.1)-(4.3).
Definition 2 A function π’ : β β π is called a global solution of the problem (4.1)-(4.3), if the conditions (4.2) and (4.3)are verified; π’β£[0,π]
β π«π([0, π];π) for all π β (0,+β) β {π‘π : π β β} and
π’(π‘) = π (π‘)(π(0)+π(0, π)
)βπ(π‘, π’π‘)ββ« π‘
0
π΄π (π‘βπ )π(π , π’π )ππ +β« π‘
0
π (π‘βπ )π(π , π’π )ππ +βπ‘π<π‘
π (π‘βπ‘π)πΌπ(π’π‘π), π‘ β πΌ = [0,β).
(10)
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420 International Journal of Nonlinear Science, Vol.11(2011), No.4, pp. 412-426
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 inπΌ = [0, π] for all π > 0. Suppose, in addition, that functions π(β ) and πΎ(β ) given by Axiom (A) are bounded and thatthe following conditions hold.
(a) For all π½ > 0, limπ‘ββ 1β(π‘)
β« π‘
0π(π )π (π½β(π ))ππ = 0.
(b) The function π : πΌ Γβ¬ β π is completely continuous and there exists a positive continuous function π1 : [0,β) β[0,β) with π1(π‘) β 0 when π‘ β β and a positive constant π2 such that β₯π(π‘, π)β₯π β€ π1(π‘)β₯πβ₯β¬ + π2 for allπ‘ > 0, π β β¬ and
limπ‘ββ
1
β(π‘)
β« π‘
0
β(π‘β π )β(π )ππ = 0.
(c) π =(β₯ππβ₯β(π,π)πΎβπ1,β + supπ‘β₯0πΎβ
β« π‘
0β(π‘β π )π1(π )ππ
)< 1.
(d) οΏ½οΏ½πΎβ1βπ
β«β0π(π )ππ <
β«βπ
1π (π )ππ , with
π = (1β π)β1((οΏ½οΏ½πΎβπ» +πβ + οΏ½οΏ½πΎββ₯ππβ₯β(π,π)π1,β)β₯πβ₯β¬ + β₯ππβ₯β(π,π)πΎβ(οΏ½οΏ½ + 1)π2 + οΏ½οΏ½πΎβ
ββπ=1
ππ
+π2πΎββ« β
0
β(π )ππ )<β,
where π1,β = supπ‘β₯0 π1(π‘), πΎβ = supπ‘β₯0πΎ(π‘), πβ = supπ‘β₯0π(π‘).
Then there exists a global solution of (4.1)-(4.3).
Proof. Let π¦ : β β π be the extension of π such that π¦(π‘) = π (π‘)π(0), π‘ β₯ 0 and π(β) be the set defined by
π(β) ={π₯ : β β π : π₯0 = 0 πππ π₯β£[0,β)
β π«π0β(π)
},
endowed with the norm β₯π₯β₯π«πβ= supπ‘β₯0
β₯π₯(π‘)β₯β(π‘) and Ξ : π(β) β π(β) be the operator defined by (Ξπ’)0 = 0 and
Ξπ’(π‘) = π (π‘)π(0, π)β π(π‘, π¦π‘ + π’π‘)ββ« π‘
0
π΄π (π‘β π )π(π , π¦π + π’π )ππ +
β« π‘
0
π (π‘β π )π(π , π¦π + π’π )ππ
+βπ‘π<π‘
π (π‘β π‘π)πΌπ(π¦π‘π + π’π‘π), π‘ β₯ 0.
Since β₯π’(π‘)β₯ β€ β₯π’β₯π«πββ(π‘) for all π‘ β₯ 0, we have that
β₯Ξπ’(π‘)β₯β(π‘)
β€ οΏ½οΏ½β₯π(0, π)β₯β(π‘)
+β₯ππβ₯β(π,π)
[π1(π‘)πΎββ₯π’β₯π«πβ
β(π‘)]
β(π‘)+
β₯ππβ₯β(π,π)[π1(π‘)(οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯β¬ + π2]
β(π‘)+
οΏ½οΏ½
β(π‘)
ββπ=1
ππ
+1
β(π‘)
β« π‘
0
β(π‘β π )(π1(π )(πΎββ₯π’β₯π«πβ
+ (οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯β¬) + π2
)β(π )ππ
+οΏ½οΏ½
β(π‘)
β« π‘
0
π(π )π ((πΎββ₯π’β₯π«πβ+ (οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯β¬)β(π ))ππ ,
so by (a) and (b), we have that Ξπ’ β π(β).Next, we show that Ξ(π΅π), where π΅π = π΅π(0, (π«π)0β(π)), satisfies the conditions of Lemma 2. To do this, we show
the continuity of the operator Ξ. Let (π’π)πββ be a sequence in π(β) and π’ β π(β) such that π’π β π’ in π(β). TakeπΆ = sup{β₯π’πβ₯π«πβ
, β₯π’β₯π«πβ; π β β}, π½ = (οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯β¬ + πΆπΎβ and the function π : [0,β) β β defined
by π(π‘) = π1(π‘)π½ + π2. From the conditions (a), (b) and (c), there exists πΏ1 > 0 such that
2
β(π‘)
β« π‘
0
β(π‘β π )π(π )β(π )ππ +2οΏ½οΏ½
β(π‘)
β« π‘
0
π(π )π (π½β(π ))ππ +2οΏ½οΏ½
β(π‘)
ββπ=1
ππ <π
2,
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for π‘ β₯ πΏ1. From the proof of Theorem 4, we have that π(π‘, π’ππ‘ + π¦π‘) β π(π‘, π’π‘ + π¦π‘) uniformly for π‘ β [0, πΏ1], whenπβ β. From Lebesgueβs Dominated Convergence Theorem, we can fix a positive number ππ > 0 such that
1
β(π‘)β₯π(π‘, π’ππ‘ + π¦π‘)β π(π‘, π’π‘ + π¦π‘)β₯+ οΏ½οΏ½
β(π‘)
β« πΏ1
0
β₯π(π , π’ππ + π¦π )β π(π , π’π + π¦π )β₯ππ
+1
β(π‘)
β« π‘
0
β(π‘β π )β₯π(π , π’ππ + π¦π )β π(π , π’π + π¦π )β₯ππ
+οΏ½οΏ½
β(π‘)
βπ‘πβ€πΏ1
β₯πΌπ(π’ππ‘π + π¦π‘π)β πΌπ(π’π‘π + π¦π‘π)β₯ <π
2,
for all π‘ β [0, πΏ1] and π β₯ ππ. Using the above inequality, for π‘ β [0, πΏ1] and π β₯ ππ, we have that
sup
{β₯Ξπ’π(π‘)β Ξπ’(π‘)β₯β(π‘)
: π‘ β [0, πΏ1], π β₯ ππ
}β€ π (11)
On the other hand, for π‘ β₯ πΏ1 and π β₯ ππ, we get
β₯Ξπ’π(π‘)β Ξπ’(π‘)β₯β(π‘)
β€ β₯π(π‘, π’ππ‘ + π¦π‘)β π(π‘, π’π‘ + π¦π‘)β₯β(π‘)
+1
β(π‘)
β« πΏ1
0
β₯π΄π (π‘β π )(π(π , π’ππ + π¦π )β π(π , π’π + π¦π ))β₯ππ
+1
β(π‘)
β« π‘
πΏ1
β₯π΄π (π‘β π )(π(π , π’ππ + π¦π )β π(π , π’π + π¦π ))β₯ππ
+1
β(π‘)
β« πΏ1
0
β₯π (π‘β π )(π(π , π’ππ + π¦π )β π(π , π’π + π¦π ))β₯ππ + 1
β(π‘)
β« π‘
πΏ1
β₯π (π‘β π )(π(π , π’ππ + π¦π )β π(π , π’π + π¦π ))β₯ππ
+οΏ½οΏ½
β(π‘)
βπ‘πβ€πΏ1
β₯πΌπ(π’ππ‘π + π¦π‘π)β πΌπ(π’π‘π + π¦π‘π)β₯+2οΏ½οΏ½
β(π‘)
βπ‘πβ₯πΏ1
ππ
β€ 2
β(π‘)
β« π‘
0
β(π‘β π )π(π )β(π )ππ +2οΏ½οΏ½
β(π‘)
β« π‘
0
π(π )π (π½β(π ))ππ +2οΏ½οΏ½
β(π‘)
ββπ=1
ππ +π
2,
and thus
sup{β₯Ξπ’π(π‘)β Ξπ’(π‘)β₯
β(π‘): π‘ β₯ πΏ1, π β₯ ππ
}β€ π. (12)
Using inequalities (4.5) and (4.6), we conclude that Ξ is continuous.Next, we prove that the set Ξ(π΅π) is relatively compact. Let π > 0 be a positive real number, π½ = (οΏ½οΏ½πΎβπ» +
πβ)β₯πβ₯β¬ + ππΎβ. From the proof of Theorem 4, it follows that the set Ξ(π΅π)β£[0,π] = {Ξπ’β£[0,π] : π’ β π΅π} is relativelycompact in π«π([0, π];π) for all π β (0,β) β {π‘π, π β β}. Moreover, for π₯ β π΅π, we have that
β₯Ξπ’(π‘)β₯β(π‘)
β€ 1
β(π‘)[οΏ½οΏ½β₯ππβ₯β(π,π)(π1(0)β₯πβ₯β¬ + π2)] +
1
β(π‘)
(β₯ππβ₯β(π,π)π1(π‘)π½β(π‘) + π2)
+οΏ½οΏ½
β(π‘)
β« π‘
0
π(π )π (π½β(π ))π+οΏ½οΏ½
β(π‘)
ββπ=1
ππ +1
β(π‘)
β« π‘
0
β(π‘β π )(π1(π )π½β(π ) + π2
)ππ
β€ 1
β(π‘)[οΏ½οΏ½β₯ππβ₯β(π,π)(π1(0)β₯πβ₯β¬ + π2)] +
π½π1,ββ(π‘)
β« π‘
0
β(π‘β π )β(π )ππ +π2β(π‘)
β« π‘
0
β(π‘β π )ππ + β₯ππβ₯β(π,π)(π1(π‘)π½
+π2β(π‘)
)+
οΏ½οΏ½
β(π‘)
β« π‘
0
π(π )π (π½β(π ))ππ +οΏ½οΏ½
β(π‘)
ββπ=1
ππ,
(13)
which enables us to conclude that β₯Ξπ’(π‘)β₯β(π‘) β 0, when π‘β β, uniformly for π₯ β π΅π.
By Lemma 4.1, we infer that Ξ(π΅π) is relatively compact in (π«π)0β(π). Thus, Ξ is completely continuous. To finishthe proof, we need to obtain an π ππππππ estimate for the solutions of the integral equations πΞπ’ = π’, π β (0, 1). To this
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422 International Journal of Nonlinear Science, Vol.11(2011), No.4, pp. 412-426
end, let π’π be the solution of the equation πΞπ’π = π’π. For π‘ β₯ 0, we have that
β₯π’π(π‘)β₯ β€ β₯ππβ₯β(π,π)[π1,β
((οΏ½οΏ½(1 +πΎβπ») +πβ)β₯πβ₯β¬ +πΎββ₯π’πβ₯π‘
)]+ β₯ππβ₯β(π,π)(οΏ½οΏ½ + 1)π2
+
β« π‘
0
β(π‘β π )(π1(π )((οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯β¬ +πΎββ₯π’πβ₯π ) + π2
)ππ
+οΏ½οΏ½
β« π‘
0
π(π )π ((οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯β¬ +πΎββ₯π’πβ₯π )ππ + οΏ½οΏ½ββπ=1
ππ.
PuttingπΏπ(π‘) = (οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯β¬ +πΎββ₯π’πβ₯π‘,
we have that
πΏπ(π‘) β€ (1β π)β1((οΏ½οΏ½πΎβπ» +πβ + οΏ½οΏ½πΎββ₯ππβ₯β(π,π)π1,β)β₯πβ₯β¬β₯ππβ₯β(π,π)πΎβ(οΏ½οΏ½ + 1)π2 + π2πΎβ
β« β
0
β(π )ππ
+οΏ½οΏ½πΎββ« π‘
0
π(π )π (πΏπ(π ))ππ + οΏ½οΏ½πΎβββπ=1
ππ
),
where π =(β₯ππβ₯β(π,π)πΎβπ1,β + supπ‘β₯0πΎβ
β« π‘
0β(π‘ β π )π1(π )ππ
). Denoting the right-hand side of the previous
expression as π½π(π‘), we see that
π½β²π(π‘) β€
οΏ½οΏ½πΎβ1β π
π(π‘)π (π½π(π‘)),
and subsequently, upon integrating over [0, π‘], we obtainβ« π½π(π‘)
π½π(0)
ππ
π (π )β€ οΏ½οΏ½πΎβ
1β π
β« π
0
π(π )ππ β€β« β
π
ππ
π (π )
The above inequality together with condition (d) enables us to conclude that the set of functions {π’π : π’π = πΞπ’π} isbounded in π«π([0,β);π). Note that, if π₯ β π«π([0,β);π), then β₯π₯β₯π«πβ
β€ β₯π₯β₯β. This allows us to conclude that theset {π’π : π’π = πΞπ’π, π β (0, 1)} is bounded in π(β). Finally, from Theorem 2 we infer the existence of a fixed pointof Ξ, and consequently, the existence of a global solution of (4.1)-(4.3). This completes the proof.
If we suppose that the functions πΌπ, π β β 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 π > 0. Assume also thatcondition (a) of Theorem 4 and condition (b) of Theorem 6 are satisfied and that
ββπ=1 β₯πΌπ(0)β₯ < +β. If
π1,βπΎβ(β₯ππβ₯β(π,π) +
β« β
0
β(π )ππ )+πΎβοΏ½οΏ½
ββπ=1
πΏπ + lim infπββ οΏ½οΏ½
β« β
0
π(π )π (π (π, π ))
πππ < 1 (14)
where π (π, π ) = (οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯π΅ +πΎβπβ(π ), then there exists a global solution of the problem (4.1)-(4.3).
Proof. Let Ξ be the operator introduced in the Theorem 6 and consider the decomposition Ξ = Ξ1+Ξ2, where (Ξππ’)0 = 0,π = 1, 2, and
Ξ1π’(π‘) = π (π‘)π(0, π)β π(π‘, π¦π‘ + π’π‘)ββ« π‘
0
π΄π (π‘β π )π(π , π¦π + π’π )ππ +
β« π‘
0
π (π‘β π )π(π , π¦π + π’π )ππ , π‘ β₯ 0,
Ξ2π’(π‘) =βπ‘π<π‘
π (π‘β π‘π)πΌπ(π¦π‘π + π’π‘π), π‘ β₯ 0.
Proceeding as in the proof of the Theorem 6, we infer that the map Ξ1 is completely continuous. Moreover, it is easy tosee that
β₯Ξ2π’β Ξ2π£β₯π«πββ€ οΏ½οΏ½πΎβ
ββπ=1
πΏπβ₯π’β π£β₯π«πβ, π’, π£ β π(β),
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S. Sivasankaran, V. Vijayakumar, M. M. Arjunan: Existence of Global Solutions for Impulsive Abstract Partial Neutral β β β 423
which implies that Ξ2 is a contraction in π(β). We prove that there is π > 0 such that Ξπ΅π β π΅π, where π΅π =π΅π(0, π(β)). In fact, if we assume that the assertion is false, then, for every π > 0, there exists π’π β π΅π and π‘π β₯ 0 suchthat β₯Ξπ’π(π‘
π)β(π‘π) β₯ > π. This yields that
π β€ 1
β(π‘π)οΏ½οΏ½β₯π(0, π)β₯+ οΏ½οΏ½
β(π‘π)
βπ‘π<π‘π
β₯πΌπ(π’ππ‘π + π¦π‘π)β₯+β₯π(π‘π, π’ππ‘π + π¦π‘π )β₯
β(π‘π)
+1
β(π‘π)
β« π‘π
0
β₯π΄π (π‘π β π )β₯β(π,π)β₯π(π , π’ππ + π¦π )β₯πππ + 1
β(π‘π)
β« π‘π
0
οΏ½οΏ½β₯π(π , π’ππ + π¦π )β₯ππ
β€ 1
β(π‘π)οΏ½οΏ½β₯π(0, π)β₯+ β₯ππβ₯β(π,π)π1,β((οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯β¬ +πΎβπβ(π‘π))
β(π‘π)
+1
β(π‘π)
β« π‘π
0
β(π‘π β π )[π1,β((οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯β¬ +πΎβπβ(π )) + π2
]ππ
+1
β(π‘π)
β« π‘π
0
οΏ½οΏ½π(π )π ((οΏ½οΏ½πΎβπ» +πβ)β₯πβ₯β¬ +πΎβπβ(π ))ππ
+οΏ½οΏ½
β(π‘π)
ββπ=1
πΏπ((οΏ½οΏ½πΎβπ»)β₯πβ₯+πΎβπβ(π‘π)) +β₯ππβ₯β(π,π)π2
β(π‘π)+
οΏ½οΏ½
β(π‘π)
ββπ=1
β₯πΌπ(0)β₯,
which implies that
1 β€ π1,βπΎβ(β₯ππβ₯β(π,π) +
β« β
0
β(π )ππ )+πΎβοΏ½οΏ½
ββπ=1
πΏπ + lim infπββ οΏ½οΏ½
β« β
0
π(π )π (π (π, π ))
πππ ,
which is contrary to our assumptions. This prove that there exists π > 0 such that Ξ is a condensing operator from π΅π intoπ΅π. This completes the proof.
5 An applicationLet π = πΏ2([0, π]) be the space of functions which are square integrable and β¬ = π«ππ Γ πΏ2(π,π). Now, we considerthe operator π΄ : π·(π΄) β π β π defined by π΄π₯ = π₯β²β², where
π·(π΄) = {π₯ β π : π₯β²β² β π, π₯(0) = π₯(π) = 0}.Is well known that π΄ is the infinitesimal generator of an analytic and compact semigroup (π (π‘))π‘β₯0 on π. Furthermore,π΄ has a discrete spectrum, the eigenvalues are βπ2, π β β, with corresponding normalized eigenfunctions π§π(π) =( 2π )
1/2π ππ(ππ), and the following properties hold.
(a) {π§π : π β β} is an orthonormal basis for π.
(b) If π₯ β π·(π΄) then π΄π₯ = βββπ=1 π
2β¨π₯, π§πβ©π§π.
(c) For π₯ β π, π (π‘)π₯ =ββ
π=1 πβπ2π‘β¨π₯, π§πβ©π§π. In particular, we have that (π (π‘))π‘β₯0 is a uniformly stable semigroup
with β₯π (π‘)β₯ β€ πβπ‘ for all π‘ β₯ 0. Moreover, it is possible to define the fractional power of π΄; see e.g. [5, 15, 20].
(d) For each π₯ β π and each π½ β (0, 1), (βπ΄)βπ½π₯ =βπ
π=11
π2π½ β¨π₯, π§πβ©π§π. In particular, β₯(βπ΄)β1/2β₯ = 1.
(e) For each π₯ β π and π½ β (0, 1), (βπ΄)π½π₯ =βπ
π=1 π2π½β¨π₯, π§πβ©π§π. Moreover,
π·((βπ΄)π½) ={π₯ : π₯ β π,
πβπ=1
π2π½β¨π₯, π§ππ§π β π
}.
Consider the impulsive partial neutral differential system,
β
βπ‘
[π€(π‘, π) +
β« π‘
ββ
β« π
0
π1(π‘)π(π β π‘, π, π)π€(π, π )ππππ ]=
β2
βπ2π€(π‘, π) + π£(π‘, π€(π‘, π))π‘ β₯ 0, 0 β€ π β€ π, (15)
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424 International Journal of Nonlinear Science, Vol.11(2011), No.4, pp. 412-426
π€(π‘, 0) = π€(π‘, π) = 0, π‘ β₯ 0, (16)
π€(π, π) = π(π, π), π β€ 0, 0 β€ π β€ π, (17)
Ξπ€(π‘π, β ) = π€(π‘+π , β )β π€(β , π‘βπ ) =β« π
0
ππ(π, π€(π‘π, π ))ππ , (18)
where (π‘π)πββ is a strictly increasing sequence of positive real numbers. To treat this system, we assume the followingconditions.
(a)β The functions π£ : β Γ β β β, is continuous and there is a continuous and integrable function π : β β [0,β)such that
β£π£(π‘, π₯)β£ β€ π(π‘)(β£π₯β£), π‘ β₯ 0, π₯ β β.
(b)β The function π1 : β β [0,β) is continuous and limπ‘β+β π1(π‘) = 0.
(c)β The functions π(π , π, π), ββπ π(π , π, π) are integrable, π(π , π, π) = π(π , π, 0) = 0 and
πΏπΉ = sup
{β« π
0
β« 0
ββ
β« π
0
(1
π(π )
βππ(π , π, π)
βππππππ ππ
)2
: π = 0, 1
}< +β.
(d)β The functions ππ : [0, π]Γ β β β, π β β, are continuous and there are positive constants πΏπ such that
β£ππ(π, π )β ππ(π, π )β£ β€ πΏπβ£π β π β£, π β [0, π], π , π β β.
We now define the functions π, π : [0,β)Γ β¬ β π, πΌπ : π β π by
π(π‘, π)(π) =
β« 0
ββ
β« π
0
π1(π‘)π(π , π, π)π(π , π)ππππ , π‘ β₯ 0, π β [0, π],
π(π‘, π)(π) = π£(π‘, π(0, π)) π‘ β₯ 0, π β [0, π],
πΌπ(π)(π) =
β« π
0
ππ(π, π(0, π ))ππ , π β β, π β [0, π],
and
π = π·(π΄1/2) =
{π₯(β ) β π :
ββπ=1
π < π₯, π§π > π§π β π
}.
Using the properties of semigroup (π (π‘))π‘β₯0, we have that π verifies the conditions (H1) (H2) and (H3). Moreover, inthis case
β₯(βπ΄) 12π (π‘)β₯π β€ π
βπ‘2 π‘
β12β
2, πππ π‘ > 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 thatββ
π=1[β« π
0β£ππ(π, 0)β£2ππ]1/2 <β. If
πΎβ[πΏπ
(1 + Ξ(1/2)
)+ οΏ½οΏ½
β« β
0
π(π )ππ +
ββπ=1
πΏπ
]< 1,
where Ξ(β ) : (0,β) β β denotes the Gamma function and π1,β = πΏ1/2 supπ‘β₯0 π1(π‘), then there is a global solution ofthe impulsive system (5.1)-(5.4).
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S. Sivasankaran, V. Vijayakumar, M. M. Arjunan: Existence of Global Solutions for Impulsive Abstract Partial Neutral β β β 425
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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