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Existence results for nondensely defined impulsive neutral functional differential equations with infinite delay
Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Existence results for nondensely defined impulsive neutral functional differential equations with infinite delay Meili Li Department of Applied Mathematics, Donghua University, Shanghai 201620, PR China
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Article history: Received 17 July 2010 Accepted 29 October 2010 Keywords: Integrated semigroup Hille–Yosida condition Integral solution Strict solution
abstract In this paper, we study a class of impulsive neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille–Yosida theorem. We give some sufficient conditions ensuring the existence of integral solutions and strict solutions. To illustrate our abstract results, we conclude this work with an example. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction In this work, we are concerned with the existence of integral solutions and strict solutions for the following impulsive neutral functional differential equation:
d [x(t ) − g (t , xt )] = A[x(t ) − g (t , xt )] + f (t , xt ), dt − 1x|t =tk = Ik (x(tk )), x0 = φ ∈ B h ,
k = 1 , . . . , m,
t ∈ J , t ̸= tk , (1.1)
where J = [0, b], x(·) takes values in Banach space X with the norm ‖ · ‖, and A : D(A) ⊆ X → X is a nondensely defined closed linear operator which generates an integrated semigroup {S (t )}t ≥0 . g , f : J × Bh → X are given functions, Ik ∈ C (X , X ) (k = 1, 2, . . . , m), 0 = t0 < t1 < t2 < · · · < tm < tm+1 = b. 1x|t =tk = x(tk+ ) − x(tk− ), where x(tk− ) and x(tk+ ) represent the left and right limits of x(t ) at t = tk , respectively. The histories xt : (−∞, 0] → X , xt (θ ) = x(t + θ ), θ ≤ 0, belong to an abstract space Bh . Neutral differential equations arise in many areas of applied mathematics, and for this reason these equations have received much attention in recent decades. The literature relative to ordinary neutral functional differential equations is very extensive, and we refer the reader to [1–3], and the Hale and Lunel book [4], and the references therein. Partial neutral differential equations with finite delay arise, for instance, from transmission line theory. Wu and Xia have shown in [5] that a ring array of identical resistively coupled lossless transmission lines leads to a system of neutral functional differential equations with discrete diffusive coupling which exhibits various types of discrete wave. By taking a natural limit, they obtain from this system of neutral equations a scalar partial neutral functional differential equation with finite delay defined on the unit circle. Such a partial neutral functional differential equation is also investigated by Hale in [6] under the form
∂2 D ut (x) + f (ut )(x), ∂ x2 udt = ϕ ∈ C ([− r , 0]; C (S 1 ; R)), 0 d
D ut (x) =
t ≥ 0,
E-mail address: [email protected]. 1751-570X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.10.012
M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
503
0
where k is a constant, D (ψ)(s) := ψ(0)(s) − −r [dη(θ )]ψ(θ )(s) for s ∈ S 1 , ψ ∈ C ([−r , 0]; C (S 1 ; R)), and η is a function of bounded variation which is nonatomic at zero. In [7], Desch et al. studied abstract functional differential equations of neutral type with infinite delay. They proved that the model proposed by Coleman and Gurtin [8], Gurtin and Pipkin [9], and Miller [10] can be regarded as the following abstract functional differential equation of neutral type with infinite delay: d dt
[
x( t ) +
∫
0
]
[
K (t − s)x(s)ds = A x(t ) +
0
∫
−∞
]
K (t − s)x(s)ds + F (t , xt ), −∞
where the operator A is a generator of a C0 -semigroup on a Banach space. Recently, several works reporting existence results of mild solutions for first-order abstract neutral functional differential have been published. See, for example, [11,12] for the case with finite delay and [13,14] for the case with infinite delay. In these works, the linear operator A is always defined densely in X and satisfies the Hille–Yosida condition so that it generates a C0 -semigroup or an analytic semigroup. However, as indicated in [15], we sometimes need to deal with nondensely defined operators. For example, when we look at a one-dimensional heat equation with the Dirichlet condition on [0, 1] and consider A = ∂∂2 x in C ([0, 1]; R), in order to measure the solution in the sup-norm we take the domain 2
D(A) = {x ∈ C 2 ([0, 1]; R)|x(0) = x(1) = 0}, and then it is not dense in C ([0, 1], R) with the sup-norm. The example presented in Section 4 also shows the advantages of nondensely defined operators in handling some practical problems. See [15–17] for more examples and remarks concerning nondensely defined operators. On the other hand, the theory of impulsive differential equations has become an important area of investigation in recent years, stimulated by their numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, etc. Relative to this matter, we refer the reader to Lakshmikantham et al. [18], Benchohra et al. [19], Abada et al. [20], and Li and Han [21]. In [20], Abada et al. have established sufficient conditions for the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in separable Banach spaces, and in [21] the existence of integral solutions and strict solutions for a class of neutral impulsive functional equations with nonlocal conditions has been considered. To the best of our knowledge, there are very few results for impulsive neutral functional differential equations with infinite delay. Our objective in this work is to establish the existence of integral solutions and strict solutions for nondensely defined impulsive neutral functional differential equations, modelled as the initial value problem (1.1), on a Banach space X of infinite dimension. The organization of this work is as follows. In Section 2, we recall some basic results on integrated semigroups and we collect some properties of phase space. Section 3 is devoted to the existence of integral solutions and strict solutions for (1.1). Finally, in Section 4, an example is presented to illustrate the applications of the results obtained. 2. Integrated semigroups and phase space In this section, we recall some results about integrated semigroups and we present the abstract phase space Bh . 2.1. Integrated semigroups Definition 2.1 ([22]). Let X be a Banach space. An integrated semigroup is a family {S (t )}t ≥0 of bounded linear operators S (t ) on X with the following properties: (i) S (0) = 0; (ii) t → S (t ) isstrongly continuous; t (iii) S (t )S (s) = 0 [S (s + τ ) − S (τ )]dτ , for all t , s ≥ 0. Definition 2.2 ([22]). An operator A is said to be the generator of an integrated semigroup if there exists ω ∈ R such that (ω, +∞) ⊂ ρ(A), and there exists a strongly continuous exponentially bounded family {S (t )}t ≥0 of bounded linear ∞ operators such that S (0) = 0 and (λI − A)−1 = λ 0 e−λt S (t )dt for all λ > ω. Definition 2.3 ([23]). (i) An integrated semigroup {S (t )}t ≥0 is said to be locally Lipschitz continuous if, for all b > 0, there exists a constant L > 0 such that ‖S (t ) − S (s)‖ ≤ L|t − s|, t , s ∈ [0, b]. (ii) An integrated semigroup {S (t )}t ≥0 is said to be nondegenerate if S (t )x = 0 for all t ≥ 0 implies that x = 0. Definition 2.4 ([23]). We say that a linear operator A satisfies the Hille–Yosida condition (HY) if there exists M ≥ 1 and ω ∈ R such that (ω, +∞) ⊂ ρ(A) and sup{(λ − ω) ¯ n ‖R(λ, A)n ‖, n ∈ N , λ > ω} ≤ M , where ρ(A) is the resolvent set of A and R(λ, A) = (λI − A)
(HY) −1
.
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M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
Theorem 2.5 ([23]). The following assertions are equivalent: (i) A is the generator of a non-degenerate, locally Lipschitz continuous integrated semigroup; (ii) A satisfies condition (HY). We know from [23] that, under condition (HY), A is the generator of a locally Lipschitz continuous integrated semigroup {S (t )}t ≥0 on X . In addition, the derivative {S ′ (t )}t ≥0 of {S (t )}t ≥0 generates a C0 -semigroup on D(A) such that ‖S ′ (t )x‖ ≤ Meωt ‖x‖, for all t ≥ 0 and x ∈ D(A). Furthermore, let A0 be the generator of the C0 -semigroup {S ′ (t )}t ≥0 ; then A0 is the part of A on D(A) defined by D(A0 ) = {x ∈ D(A) : Ax ∈ D(A)}, A0 x = Ax. Next, we give some general properties of the integrated semigroup {S (t )}t ≥0 . Proposition 2.6 ([22]). Let A be the generator of an integrated semigroup {S (t )}t ≥0 . Then, for all x ∈ X and t ≥ 0, D(A) and S (t )x = A
t 0
S (s)xds + tx. Moreover, for all x ∈ D(A) and t ≥ 0,
S (t )x ∈ D(A),
AS (t )x = S (t )Ax
S (t )x =
and
t 0
S (s)xds ∈
t
∫
S (s)Axds + tx. 0
Corollary 2.7 ([22]). Let A be the generator of an integrated semigroup {S (t )}t ≥0 ; then, for all x ∈ X and t ≥ 0, S (t )x ∈ D(A). Moreover, for any x ∈ X , S (·)x is right-hand-side differentiable in t ≥ 0 if and only if S (t )x ∈ D(A), and in that case we have S ′ (t )x = AS (t )x + x. In what follows, we give some results for the existence of solutions of the following Cauchy problem: d
x(t ) = Ax(t ) + f (t ), dt x(0) = x0 ∈ X ,
t ≥ 0,
(2.1)
where A satisfies the Hille–Yosida condition without being densely defined. Let {S (t )}t ≥0 be the integrated semigroup generated by A; then one has the following theorem. Theorem 2.8 ([24]). Let f : J → X be a continuous function. Then, for x0 ∈ D(A), there is a unique continuous function x : J → X such that (i)
t 0
x(s)ds ∈ D(A), t ∈ J;
t
t
(ii) x(t ) = x0 + A 0 x(s)ds + 0 f (s)ds, t ∈J; t (iii) ‖x(t )‖ ≤ Meωt ‖x0 ‖ + 0 e−ωs ‖f (s)‖ds , t ∈ J. Moreover, x satisfies the following variation of constant formula: d x(t ) = S ′ (t )x0 + dt
t
∫
S (t − s)f (s)ds,
t ≥ 0.
(2.2)
0
Proposition 2.9 ([25]). Let A : D(A) ⊆ X → X be a linear operator satisfying the Hille–Yosida condition, {S (t )}t ≥0 be the integrated semigroup generated by A, and f : [0, T ] → X , T > 0, be a Bochner-integrable function. Then the function K : [0, T ] → X t defined by K (t ) = 0 S (t − s)f (s)ds is continuously differentiable on [0, T ], and satisfies that, for λ > ω and t ∈ [0, T ], R(λ, A)K (t ) = ′
t
∫
S ′ (t − s)R(λ, A)f (s)ds. 0
Remark 1. Let Bλ = λR(λ, A); then, for all x ∈ D(A), Bλ x → x as λ → ∞. 2.2. Phase space In this work, we introduce an abstract phase space Bh which is similar to the one used in [26] and suitably modified to treat retarded impulsive differential equations [27,28]. Denote J0 = [0, t1 ], Jk = (tk , tk+1 ], k = 1, 2, . . . , m. Assume that h : (−∞, 0] → (0, +∞) is a continuous function
0
with l = −∞ h(s)ds < +∞. For any a > 0, we define
B = {φ : [−a, 0] → X : φ is continuous everywhere except a finite number of points t˜ at which φ(t˜+ ), φ(t˜− ) exist and φ(t˜) = φ(t˜− )},
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and equip the space B with the form
‖φ‖[−a,0] = sup ‖φ(θ )‖,
∀φ ∈ B .
−a≤θ≤0
Let us define
Bh = φ : (−∞, 0] → X : for any c > 0, φ|[−c ,0] ∈ B and
∫
0
h(s) sup ‖φ(θ )‖ds < +∞ . −∞
s≤θ≤0
If Bh is endowed with the norm
∫
0
h(s) sup ‖φ(θ )‖ds,
‖φ‖Bh = −∞
∀φ ∈ Bh ,
s≤θ≤0
then it is easy to prove that (Bh , ‖ · ‖Bh ) is a Banach space. Now we consider the space
Bb = {x : (−∞, b] → X : xk ∈ C (Jk , X ), k = 0, . . . , m and there exist x(tk+ ), x(tk− ), k = 1, . . . , m with x(tk ) = x(tk− ), x0 = φ ∈ Bh }. Set ‖ · ‖b be a semimorm in Bb defined by
‖x‖b = ‖φ‖Bh + sup ‖x(s)‖,
x ∈ Bb .
s∈J
Lemma 2.10 ([27]). Suppose that x ∈ Bb ; then, for t ∈ J, xt ∈ Bh . Moreover, l‖x(t )‖ ≤ ‖xt ‖Bh ≤ l sup ‖x(s)‖ + ‖x0 ‖Bh , s∈[0,t ]
0
where l = −∞ h(s)ds < +∞. In addition, we define the following classes of functions: PC (J , X ) = {x : J → X : xk ∈ C (Jk , X ), k = 0, 1, . . . , m and there exist x(tk+ ), x(tk− ), k = 1, . . . , m with x(tk ) = x(tk− )}, PC 1 (J , X ) = {x ∈ PC (J , X ) : x′k ∈ C (Jk , X ), k = 0, 1, . . . , m and there exist x′ (tk+ ), x′ (tk− ), k = 1, . . . , m with x′ (tk ) = ′ − x (tk )}, where xk and x′k represent the restriction of x and x′ to Jk , respectively (k = 0, . . . , m). Obviously, PC (J , X ) is a Banach space with the norm ‖x‖PC = supt ∈J ‖x(t )‖, and PC 1 (J , X ) is also a Banach space with the norm ‖x‖PC 1 = max{‖x‖PC , ‖x′ ‖PC }. 3. Main results Definition 3.1. We say that x : (−∞, b] → X is an integral solution of Eq. (1.1) if (i) x is continuous on Jk (k = 0, 1, . . . , m); (ii) x(t ) = φ(t ), t ∈ (−∞, 0]; t (iii) 0 (x(s) − g (s, xs ))ds ∈ D(A) for t ∈ J;
(iv) x(t ) = g (t , xt ) + S ′ (t )[φ(0) − g (0, φ)] + s)f (s, xs )ds, t ∈ J.
∑
0
S ′ (t − tk ){[x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )]} + k
d dt
t 0
S (t −
Definition 3.2. Let φ ∈ Bh . We say that a function x : (−∞, b] → X is a strict solution of Eq. (1.1) if the following conditions hold: (i) t → x(t ) − g (t , xt ) ∈ PC 1 (J , X ) ∩ PC (J , D(A)); (ii) x satisfies Eq. (1.1) on (−∞, b]. Remark 2. (A) It is not difficult to prove that, if x is an integral solution of Eq. (1.1) on (−∞, b], then, for almost all t ∈ J, x(t ) − g (t , xt ) ∈ D(A). In particular, φ(0) − g (0, φ) ∈ D(A). (B) If x is an integral solution of Eq. (1.1) on (−∞, b], such that t → x(t ) − g (t , xt ) belongs to PC 1 (J , X ) or PC (J , D(A)), then x is a strict solution. To obtain the existence and uniqueness of the integral solutions, we make the following assumptions.
(H0 ) The operator A satisfies condition (HY) and generates an integrated semigroup {S (t )}t ≥0 on X . (H1 ) g ∈ C (J × Bh , X ), and there exists constant α0 > 0 such that ‖g (t , φ1 ) − g (t , φ2 )‖ ≤ α0 ‖φ1 − φ2 ‖Bh for φ1 , φ2 ∈ Bh and t ∈ J . (H2 ) f ∈ C (J × Bh , X ), and there exists constant β0 > 0 such that ‖f (t , φ1 ) − f (t , φ2 )‖ ≤ β0 ‖φ1 − φ2 ‖Bh for φ1 , φ2 ∈ Bh and t ∈ J . (H3 ) Ik ∈ C (X , X ), and there exist constants αk > 0, k = 1, . . . , m such that ‖Ik (x) − Ik (y)‖ ≤ αk ‖x − y‖,
x, y ∈ X .
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M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
(H4 ) The semigroup {S ′ (t )}t ≥0 is compact on (D(A), ‖ · ‖), and there is a constant M ′ ≥ 1 such that ‖S ′ (t )‖ ≤ M ′ ,
for all t ∈ J .
Theorem 3.3. Assume that conditions (H0 )–(H4 ) are satisfied, and that
α0 l + M
m −
′
αk + 2α0 lm + bM β0 l ≡ γ0 < 1.
(3.1)
k=1
Let φ(0) − g (0, φ) ∈ D(A) and [x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )] ∈ D(A); then Eq. (1.1) has a unique integral solution k
x(·, φ) defined on (−∞, b].
Proof. Consider the nonempty closed subset of PC (J , X ) defined by Zb (φ) := {z ∈ PC (J , X ) : z (0) = φ(0)}. For z ∈ Zb (φ), we define z˜ : (−∞, b] → X by z˜ (t ) =
z (t ), t ∈ J , φ(t ), t ≤ 0.
Consider the operator J : Zb (φ) → Zb (φ) defined by
−
(Jz )(t ) = g (t , z˜t ) + S ′ (t )[φ(0) − g (0, φ)] +
S ′ (t − tk ){Ik (z (tk− )) − [g (tk , z˜t + ) − g (tk , z˜tk )]} k
0
+
d dt
t
∫
S (t − s)f (s, z˜s )ds. 0
From Proposition 2.9 and condition (HY), we get
∫ t ∫ d t ′ Bλ S ( t − s ) f ( s , x ) ds = S ( t − s )λ R (λ, A ) f ( s , x ) ds s s dt 0 0 ∫ b λ ≤ MM ′ ‖f (s, xs )‖ds. λ−ω 0 Letting λ → ∞, we obtain that ∫ t ∫ b d ≤ MM ′ S ( t − s ) f ( s , x ) ds ‖f (s, xs )‖ds. s dt
0
(3.2)
0
By the hypotheses, Lemma 2.10, and (3.2), we can see that, for every z 1 , z 2 ∈ Zb (φ) and t ∈ J,
‖(Jz 1 )(t ) − (Jz 2 )(t )‖ ≤ α0 l sup ‖z 1 (s) − z 2 (s)‖ + s∈[0,t ]
−
S ′ (t − tk )[αk ‖z 1 (tk− ) − z 2 (tk− )‖
0 < tk < t
+ 2α0 l sup ‖z 1 (s) − z 2 (s)‖] + MM ′ s∈[0,t ]
≤
α0 l + M ′
b
∫
β0 l sup ‖z 1 (s) − z 2 (s)‖ds 0
m −
s∈[0,t ]
αk + 2α0 lm + bM β0 l
‖z 1 − z 2 ‖PC
k=1
= γ0 ‖z 1 − z 2 ‖PC . Then, J is a strict contraction in Zb (φ), and the fixed point of J gives a unique integral solution x(·, φ) on (−∞, b]. This ends the proof. To prove the existence of strict solutions, we add the following assumptions.
(H5 ) g , f ∈ C 1 (J × Bh , X ) and their partial derivatives are locally Lipschitzian with respect to the second argument in the sense that, for any compact set Q ⊂ J × Bh , there exists a constant β1 > 0 such that ‖D f (t , φ) − Dφ f (t , ψ)‖ ≤ β1 ‖φ − ψ‖Bh φ ‖Dt f (t , φ) − Dt f (t , ψ)‖ ≤ β1 ‖φ − ψ‖Bh ‖Dφ g (t , φ) − Dφ g (t , ψ)‖ ≤ β1 ‖φ − ψ‖Bh ‖Dt g (t , φ) − Dt g (t , ψ)‖ ≤ β1 ‖φ − ψ‖Bh , for all (t , φ), (t , ψ) ∈ Q and t ∈ J, where Dt f , Dφ f , Dt g , Dφ g denote the derivatives with respect to t and φ . (H6 ) Dx Ik (x) ∈ C (X , X ), k = 1, 2, . . . , m, where Dx Ik denote the derivatives with respect to x.
M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
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Theorem 3.4. Let x be the unique integral solution of Eq. (1.1) obtained by Theorem 3.3. If the hypotheses (H0 )–(H6 ) are satisfied, then x is also a strict solution of the Cauchy problem (1.1), provided that
φ ′ ∈ Bh ,
φ(0) − g (0, φ) ∈ D(A0 ),
φ − Dt g (0, φ) − Dφ g (0, φ)φ ′ ∈ D(A), ′
[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] − [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ] ∈ D(A), k
k
k
(3.3)
φ ′ − Dt g (0, φ) − Dφ g (0, φ)φ ′ = A[φ(0) − g (0, φ)] + f (0, φ), A{[x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )]} = [y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] k
k
k
k
− [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]. Proof. By Theorem 3.3, we know that Eq. (1.1) has a unique integral solution x := x(·, φ), which is also the unique solution of
−
x(t ) = g (t , xt ) + S ′ (t )[φ(0) − g (0, φ)] +
S ′ (t − tk ){[x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )]} k
0
+
d
t
∫
dt
S (t − s)f (s, xs )ds,
t ∈ J.
(3.4)
0
By Corollary 2.7, the assumption that φ(0) − g (0, φ) ∈ D(A0 ) implies that S ′ (t )[φ(0) − g (0, φ)] = S (t )A[φ(0) − g (0, φ)] + [φ(0) − g (0, φ)]. Then Eq. (3.4) can be rewritten as x(t ) = g (t , xt ) + S (t )A[φ(0) − g (0, φ)] + [φ(0) − g (0, φ)] +
−
S ′ (t − tk ){[x(tk+ ) − g (tk , xt + )] − [x(tk ) k
0
− g (tk , xtk )]} +
d dt
t
∫
S (t − s)f (s, xs )ds,
t ∈ J.
(3.5)
0
Consider the following equation:
d dt [y(t ) − Dt g (t , xt ) − Dφ g (t , xt )yt ] = A[y(t ) − Dt g (t , xt ) − Dφ g (t , xt )yt ] + Dt f (t , xt ) + Dφ f (t , xt )yt , t ∈ J , t ̸= tk , k = 1, . . . , m. − − 1 y |t =tk′ = Dx Ik (x(tk ))y(tk ), k = 1, . . . , m. y0 = φ , t ∈ (−∞, 0].
(3.6)
Then, using the same reasoning as in the proof of Theorem 3.3, one can show that Eq. (3.6) has a unique integral solution y : (−∞, b] → X given by y(t ) = φ ′ (t ), t ∈ (−∞, 0] and y(t ) = S ′ (t )[φ ′ − Dt g (0, φ) − Dφ g (0, φ)φ ′ ] + Dt g (t , xt ) + Dφ g (t , xt )yt +
−
S ′ (t − tk ){[y(tk+ ) − Dt g (tk , xt + )
0
− Dφ g (tk , xt + )yt + ] − [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]} k k ∫ t d + S (t − s)[Dt f (s, xs ) + Dφ f (s, xs )ys ]ds, t ∈ J . dt
k
(3.7)
0
Let w : (−∞, b] → X be the function defined by
w(t ) =
φ(t )
φ(0) +
for t ∈ (−∞, 0],
t
∫
y(s)ds + 0
−
Ik (x(tk )) −
for t ∈ J .
0
Next, we will show that x = w on (−∞, b]. Integrating both sides of Eq. (3.7) from 0 to t, we have t
∫
y(s)ds = S (t ){A[φ(0) − g (0, φ)] + f (0, φ)} +
t
∫
0
Dt g (s, xs )ds + 0
+
− 0
∫
t
Dφ g (s, xs )ys ds
0
S (t − tk ){[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] k
− [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]} +
k
k
t
∫
S (t − s)(Dt f (s, xs ) + Dφ f (s, xs )ys )ds. 0
(3.8)
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M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
On the other hand, for t ∈ J, t
∫
d dt
S (t − s)f (s, ws )ds = S (t )f (0, φ) +
t
∫
S (t − s)(Dt f (s, ws ) + Dφ f (s, ws )ys )ds. 0
0
Consequently, t
∫
d
S (t )f (0, φ) =
dt
S (t − s)f (s, ws )ds −
t
∫
S (t − s)(Dt f (s, ws ) + Dφ f (s, ws )ys )ds.
(3.9)
0
0
Since
−
w(t ) − g (t , wt ) − [φ(0) − g (0, φ)] −
{[w(tk+ ) − g (tk , wt + )] − [w(tk ) − g (tk , wtk )]} k
0
∫ = 0
d dt
[w(s) − g (s, ws )]ds =
t
∫
[y(s) − Dt g (s, ws ) − Dφ g (s, ws )ys )ds 0
and (3.8) holds, we get that (noting that x0 = w0 = φ ) t
∫
w(t ) = g (t , wt ) +
[y(s) − Dt g (s, ws ) − Dφ g (s, ws )ys ]ds + [φ(0) − g (0, φ)] ∫ t − + {[w(tk+ ) − g (tk , wt + )] − [w(tk ) − g (tk , wtk )]} − y(s)ds + S (t ){A[φ(0) − g (0, φ)] + f (0, φ)} 0
k
0
∫
Dt g (s, xs )ds +
+ 0
0
t
∫
−
Dφ g (s, xs )ys ds +
S (t − tk ){[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] k
0
0
k
k
∫ t − [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]} + S (t − s)[Dt f (s, xs ) + Dφ f (s, xs )ys ]ds 0 ∫ t ∫ t = g (t , wt ) + [Dt g (s, xs ) − Dt g (s, ws )]ds + [Dφ g (s, xs ) − Dφ g (s, ws )]ys ds + S (t ){A[φ(0) − g (0, φ)] 0 0 − + f (0, φ)} + [φ(0) − g (0, φ)] + {[w(tk+ ) − g (tk , wt + )] − [w(tk ) − g (tk , wtk )]} k
0 < tk < t
−
+
S (t − tk ){[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] − [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]} k
0
k
k
t
∫
S (t − s)[Dt f (s, xs ) + Dφ f (s, xs )ys ]ds.
+ 0
Therefore, x(t ) − w(t ) = g (t , xt ) − g (t , wt ) +
−
S ′ (t − tk ){[x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )]} k
0
+
d
t
∫
dt
S (t − s)f (s, xs )ds − S (t )f (0, φ) − 0
[Dφ g (s, xs ) − Dφ g (s, ws )]ys ds −
−
{[w(tk+ ) − g (tk , wt + )] − [w(tk ) − g (tk , wtk )]} k
0
0
−
[Dt g (s, xs ) − Dt g (s, ws )]ds 0
t
∫ −
t
∫
−
S (t − tk ){[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] − [y(tk ) − Dt g (tk , xtk ) k
0 < tk < t
− Dφ g (tk , xtk )ytk ]} −
k
k
t
∫
S (t − s)[Dt f (s, xs ) + Dφ f (s, xs )ys ]ds. 0
Eq. (3.9) yields x(t ) − w(t ) = g (t , xt ) − g (t , wt ) +
−
S ′ (t − tk ){[x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )]} k
0
+
d
t
∫
dt
S (t − s)[f (s, xs ) − f (s, ws )]ds − 0
t
∫
[Dφ g (s, xs ) − Dφ g (s, ws )]ys ds −
− 0
t
∫
[Dt g (s, xs ) − Dt g (s, ws )]ds
0
− 0
{[w(tk+ ) − g (tk , wt + )] − [w(tk ) − g (tk , wtk )]} k
M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
−
−
509
S (t − tk ){[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] k
k
k
0
∫ t S (t − s)[Dt f (s, ws ) − Dt f (s, xs )]ds − [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]} + 0 ∫ t S (t − s)[Dφ f (s, ws ) − Dφ f (s, xs )]ys ds. + 0
Consequently, by Proposition 2.6, Corollary 2.7, and Lemma 2.10, we deduce that
‖x(t ) − w(t )‖ ≤ α0 ‖xt − wt ‖Bh + σ (b)
t
∫
−
‖xs − ws ‖Bh ds +
0
0
{[(x(tk+ ) − g (tk , x+ tk )) − (x(tk ) − g (tk , xtk ))]
− [(w(tk+ ) − g (tk , wt+k )) − (w(tk ) − g (tk , wtk ))]} ∫ t ‖xs − ws ‖Bh ds + 2mα0 ‖xt − wt ‖Bh + 2m sup ‖x(t ) − w(t )‖ ≤ α0 ‖xt − wt ‖Bh + σ (b) 0≤s≤t
0
≤ [(2m + 1)α0 l + 2m] sup ‖x(t ) − w(t )‖ + σ (b) s∈[0,t ]
t
∫
‖xs − ws ‖Bh ds, 0
where
σ (b) = MM ′ β0 + β1 + β1 b0 + β1 b0 + β1 b20 ,
b0 = max{ sup ‖S (s)‖, sup ‖ys ‖Bh }. 0≤s≤b
0≤s≤b
Hence,
‖xt − wt ‖Bh ≤ l sup ‖x(s) − w(s)‖ ≤ [1 − 2m − (2m + 1)α0 l]−1 lσ (b) s∈[0,t ]
t
∫
‖xs − ws ‖Bh ds. 0
Using Gronwall’s lemma, we conclude that
‖xt − wt ‖Bh = 0,
t ∈ J.
Hence, x(t ) = w(t ) for all t ∈ (−∞, b]. Therefore, we conclude that x ∈ PC 1 (J , X ), since w has obviously this property. Thus, by Remark 2, x is a strict solution of Eq. (1.1). The proof of Theorem 3.4 is complete. 4. Application As applications of the obtained results of this paper, we study the following impulsive partial functional differential system with infinite delay:
] [ ] ∫ 0 ∫ 0 [ ∂2 ∂ u(t , ξ ) − K1 (θ , u(t + θ , ξ ))dθ = u ( t , ξ ) − K (θ , u ( t + θ , ξ )) d θ 1 ∂ x2 −∞ −∞ ∂t ∫ 0 + K2 (θ , u(t + θ , ξ ))dθ , t ∈ J = [0, 1], ξ ∈ [0, π], −∞ u(t , 0) = u(t , π ) = 0, u(tk+ , ξ ) − u(tk− , ξ ) = Ik (u(tk− , ξ )), k = 1, . . . , m, u(θ , ξ ) = u0 (θ , ξ ), θ ∈ (−∞, 0], ξ ∈ [0, π],
(4.1)
where K1 , K2 : (−∞, 0] × R → R and u0 : (−∞, 0] × [0, π] → R are continuous functions. Let X = C ([0, π]; R) be the space of continuous functions from [0, π] into R endowed with the uniform norm topology. Define A : D(A) ⊂ X → X by Aw = w ′′ with domain D(A) = {w ∈ C 2 ([0, π]) : w(0) = w(π ) = 0}. We have D(A) = {w ∈ X : w(0) = w(π ) = 0} ̸= X and
ρ(A) ⊇ (0, +∞),
‖(λI − A)−1 ‖ ≤
1
λ
,
for λ > 0.
This implies that A satisfies condition (HY) on X . It is well known that A generates an integrated semigroup {S (t )}t ≥0 and that ‖S ′ (t )‖ ≤ e−t for t ≥ 0.
510
M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
Let h(s) = eγ s , γ > 0, s < 0; then 0
∫
1
h(s)ds =
l=
γ
−∞
,
and let
∫
0
h(s) sup ‖φ(θ )‖ds.
‖φ‖Bh =
s≤θ≤0
−∞
Thus, for (t , φ) ∈ J × Bh , where φ(θ )(ξ ) = u0 (θ , ξ ), (θ , ξ ) ∈ (−∞, 0] × [0, π], let x(t )(ξ ) = u(t , ξ ),
g (φ)(ξ ) =
∫
0
K1 (θ , φ(θ )(ξ ))dθ , −∞
f (φ)(ξ ) =
0
∫
K2 (θ , φ(θ )(ξ ))dθ . −∞
Then Eq. (4.1) takes the following abstract form:
d [x(t ) − g (xt )] = A[x(t ) − g (xt )] + f (xt ), dt − 1x|t =tk = Ik (x(tk )), x0 = φ ∈ Bh .
t ∈ J , t ̸= tk , (4.2)
k = 1 , . . . , m,
In order to study the existence of solutions of Eq. (4.2), we make the following assumptions. (i) For each θ ≤ 0, −∞ eγ θ supξ ∈[0,π ] ‖u0 (θ , ξ )‖dθ < ∞, u0 (θ , 0) = u0 (θ , π ) = 0 and K1 (·, 0) = 0. (ii) For each i = 1, 2, θ ≤ 0 and ζ1 , ζ2 ∈ R, ‖Ki (θ , ζ1 ) − Ki (θ , ζ2 )‖ ≤ ki (θ )‖ζ1 − ζ2 ‖, where k1 , k2 are measurable nonnegative functions on (−∞, 0] such that
0
∫
0
(1 + γ )ki (θ )dθ = ρi < ∞,
i = 1, 2.
−∞
(iii) Ik ∈ C (X , X ), and there exist constants αk > 0, k = 1, . . . , m such that
‖Ik (x1 ) − Ik (x2 )‖ ≤ αk ‖u1 − u2 ‖,
x1 , x2 ∈ X .
(iv) u0 (0, ·) −
∫
0
K1 (θ , u0 (θ , ·))dθ ∈ D(A), −∞
and
[u(tk+ , ξ ) −
∫
0
K1 (θ , u((tk + θ )+ , ξ ))dθ ] − [u(tk , ξ ) − −∞
∫
0
K1 (θ , u(tk + θ , ξ ))dθ ] ∈ D(A). −∞
Assumption (ii) implies that, for φ1 , φ2 ∈ Bh ,
‖g (φ1 ) − g (φ2 )‖ = sup ‖g (φ1 )(ξ ) − g (φ2 )(ξ )‖ ≤ ρ1 ‖φ1 − φ2 ‖Bh ξ ∈[0,π ]
and
‖f (φ1 ) − f (φ2 )‖ = sup ‖f (φ1 )(ξ ) − f (φ2 )(ξ )‖ ≤ ρ2 ‖φ1 − φ2 ‖Bh . ξ ∈[0,π]
Consequently, (H1 ) and (H2 ) are true. Then, from Theorem 3.3, we know that, if
ρ1 γ −1 +
m −
αk + 2ρ1 γ −1 m + ρ2 γ −1 < 1,
k=1
then system (4.2) admits a unique integral solution x. To assert that x is a strict solution, we have to make more assumptions.
M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
511
(v) For each i = 1, 2, Ki is C 2 -smooth, and the second derivative of Ki with respect to the second variable satisfies the following estimate:
2 ∂ ≤ β˜ i (θ )‖ζ ‖ for θ ≤ 0 and ζ ∈ R, K (θ , ζ ) ∂ζ 2 i where β˜ i is a measurable nonnegative function on (−∞, 0] such that 0
∫
(1 + γ )2 β˜ i (θ )dθ < ∞. −∞
(vi) Dx Ik (x) ∈ C (X , X ), k = 1, 2, . . . , m, where Dx Ik denote the derivatives with respect to x. By assumption (v), g and f are continuously differentiable and satisfy, for φ, ψ ∈ Bh and ξ ∈ [0, π],
∫ 0 ∂ ′ g (φ)(ψ)(ξ ) = K1 (θ, φ(θ )(ξ ))(ψ)(θ)(ξ )dθ , ∂ζ −∞ ∫ 0 ∂ f ′ (φ)(ψ)(ξ ) = K2 (θ, φ(θ )(ξ ))(ψ)(θ)(ξ )dθ . −∞ ∂ζ Moreover, as a consequence of assumption (v), g ′ and f ′ are Lipschitz continuous in Bh . In fact, this is a consequence of the following:
∫
∂ Ki (θ , φ1 (θ )(ξ ))(ψ)(θ)(ξ ) − ∂ Ki (θ , φ2 (θ)(ξ ))(ψ)(θ )(ξ ) dθ ∂ζ ∂ζ −∞ ∫ 0 ≤ β˜ i (θ )‖φ1 (θ)(ξ ) − φ2 (θ )(ξ )‖ ‖ψ(θ )(ξ )‖dθ, 0
−∞
which implies that
∫
∂ Ki (θ , φ1 (θ )(ξ ))(ψ)(θ)(ξ ) − ∂ Ki (θ , φ2 (θ)(ξ ))(ψ)(θ )(ξ ) dθ ∂ζ ∂ζ −∞ ∫ 0 ≤ (1 + γ )2 β˜ i (θ)dθ ‖φ1 − φ2 ‖Bh ‖ψ‖Bh . 0
−∞
Then, (H5 ) is fulfilled. Consider the following equation:
] ∫ 0 [ ∂ ∂ v( t, ξ ) − K1 (θ, u(t + θ , ξ ))v(t + θ, ξ )dθ ∂t −∞ ∂ζ [ ] ∫ 0 ∂2 ∂ = v( t , ξ ) − K (θ, u ( t + θ , ξ ))v( t + θ , ξ ) d θ 1 ∂ x∫2 −∞ ∂ζ 0 ∂ K2 (θ, u(t + θ, ξ ))v(t + θ, ξ )dθ , t ∈ J , ξ ∈ [0, π], + −∞ ∂ζ v(t , 0) = v(t , π) = 0, + v(t , ξ ) − v(tk− , ξ ) = Du Ik (u(tk− , ξ ))v(tk− , ξ ), k = 1, . . . , m, k v(θ , ξ ) = ∂ u (θ, ξ ), θ ∈ (−∞, 0], ξ ∈ [0, π]. 0 ∂θ We assume that
∂ ∂θ
u0 ∈ Bh ,
u0 (0, ·) −
∫
0
K1 (θ, u0 (θ, ·))dθ ∈ D(A0 ), −∞
∫ 0 ∂ ∂ ∂ u0 (0, ·) − K1 (θ, u0 (θ , ·)) u0 (θ, ·)dθ ∈ D(A), ∂θ ∂θ −∞ ∂ζ [ ] ∫ 0 ∂ v(tk+ , ξ ) − K1 (θ , u((tk + θ)+ , ξ ))v((tk + θ)+ , ξ )dθ −∞ ∂ζ [ ] ∫ 0 ∂ − v(tk , ξ ) − K1 (θ, u(tk + θ, ξ ))v(tk + θ, ξ )dθ ∈ D(A). −∞ ∂ζ
(4.3)
512
M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
By Theorem 3.3, it is enough to show that Eq. (4.3) has a unique integral solution v on (−∞, 1]. In addition, we assume that
∫ 0 ∂ ∂ ∂ u0 (0, ξ ) − K1 (θ , u0 (θ , ξ )) u0 (θ , ξ )dθ ∂θ ∂θ −∞ ∂ζ ] ∫ 0 [ ∫ 0 ∂2 K2 (θ , u0 (θ , ξ ))dθ , K1 (θ , u0 (θ , ξ ))dθ + = 2 u0 (0, ξ ) − ∂ξ −∞ −∞ and
∂2 ∂ξ 2
[
u(tk+ , ξ ) −
0
∫
K1 (θ , u((tk + θ )+ , ξ ))dθ
]
−∞
∂ K1 (θ , u((tk + θ )+ , ξ ))v((tk + θ )+ , ξ )dθ = v(tk+ , ξ ) − −∞ ∂ζ [ ] ∫ 0 ∂ − v(tk , ξ ) − K1 (θ , u(tk + θ , ξ ))v(tk + θ , ξ )dθ . −∞ ∂ζ [
∫
0
[ ∫ − u(tk , ξ ) −
0
K1 (θ , u(tk + θ , ξ ))dθ
]
−∞
]
Hence, all the assumptions of Theorem 3.4 are satisfied. Then we obtain that the integral x of Eq. (4.2) is strict. Consequently, the function u, defined by u(t , ξ ) = x(t )(ξ ) for t ∈ J and ξ ∈ [0, π], is a solution of Eq. (4.1). Acknowledgement This work is supported by NNSF of China (No. 10971139). References [1] S.K. Ntouyas, Y.G. Sficas, C.P. Tsamatos, Existence results for initial value problems for neutral functional differential equations, J. Differential Equations 114 (2) (1994) 527–537. [2] M. Benchohra, A. Ouanab, Impulsive neutral functional differential equations with variable times, Nonlinear Anal. 55 (6) (2003) 679–693. [3] M. Benchohra, E. Gatsori, S.K. Ntouyas, Existence results for functional and neutral functional integrodifferential inclusions with lower semicontinuous right-hand side, J. Math. Anal. Appl. 281 (2) (2003) 525–538. [4] J.K. Hale, Sjoerd M. Verduyn Lunel, Introduction to Functional Differential Equations, in: Appl. Math. Sci., vol. 99, Springer-Verlag, New York, 1993. [5] J. Wu, H. Xia, Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Differential Equations 124 (1) (1996) 247–278. [6] J.K. Hale, Partial neutral functional differential equations, Rev. Roumaine Math. Pures Appl. 39 (4) (1994) 339–344. [7] W. Desch, R. Grimmer, W. Schapacher, Wellposedness and wave propagation for a class of integrodifferential equations in Banach space, J. Differential Equations 74 (1988) 391–411. [8] B.D. Coleman, M.E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18 (1967) 199–208. [9] M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with infinite wave speeds, Arch. Ration. Mech. Anal. 31 (1968) 113–126. [10] R.K. Miller, An integrodifferential equation for rigid heat conductions with memory, J. Math. Anal. Appl. 66 (1978) 313–332. [11] X. Fu, K. Ezzinbi, Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Anal. 54 (2) (2003) 215–227. [12] K. Ezzinbi, X. Fu, K. Hilal, Existence and regularity in the α -norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal. 67 (5) (2007) 1613–1622. [13] E. Hernríquez, Hernán R. Henríquez, Existence results for partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl. 221 (2) (1998) 499–522. [14] K. Ezzinbi, S. Ghnimib, M.A. Taoudi, Existence and regularity of solutions for neutral partial functional integrodifferential equations with infinite delay, Nonlinear Anal. Hybrid Syst. 4 (1) (2010) 54–64. [15] G. Da Prato, E. Sinestrari, Differential operators with nondense domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. 14 (1987) 285–344. [16] M. Adimy, H. Bouzahir, K. Ezzinbi, Existence and stability for some partial neutral functional differential equations with infinite delay, J. Math. Anal. Appl. 294 (2004) 438–461. [17] M. Adimy, K. Ezzinbi, A. Ouhinou, Behavior near hyperbolic stationary solutions for partial functional differential equations with infinite delay, Nonlinear Anal. 68 (2008) 2280–2302. [18] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [19] M. Benchohra, J. Henderson, S.K. Ntouyas, An existence result for first-order impulsive functional differential equations in Banach spaces, Comput. Math. Appl. 42 (2001) 1303–1310. [20] N. Abada, M. Benchohra, H. Hammouche, Existence and controllability for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations 246 (2009) 3834–3863. [21] M. Li, M. Han, Existence for neutral impulsive functional differential equations with nonlocal conditions, Indag. Math. (NS) 20 (3) (2009) 435–451. [22] W. Arendt, Resolvent positive operators and integrated semigroup, Proc. Lond. Math. Soc. 54 (1987) 321–349. [23] H. Kellermann, M. Hieber, Integrated semigroup, J. Funct. Anal. 15 (1989) 160–180. [24] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987) 327–352. [25] H. Thiems, Integrated semigroup and integral solutions to abstract Cauchy problems, J. Math. Anal. Appl. 152 (1990) 416–447. [26] Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, Springer, New York, 1991, p. 1473. [27] B. Liu, Controllability of impulsive neutral functional differential inclusion with infinite delay, Nonlinear Anal. 60 (2005) 1533–1552. [28] Y.K. Chang, A. Anguraj, M. Mallika Arjunan, Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear Anal. Hybrid Syst. 2 (2008) 209–218.
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Existence results for nondensely defined impulsive neutral functional differential equations with infinite delay Meili Li Department of Applied Mathematics, Donghua University, Shanghai 201620, PR China
article
info
Article history: Received 17 July 2010 Accepted 29 October 2010 Keywords: Integrated semigroup Hille–Yosida condition Integral solution Strict solution
abstract In this paper, we study a class of impulsive neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille–Yosida theorem. We give some sufficient conditions ensuring the existence of integral solutions and strict solutions. To illustrate our abstract results, we conclude this work with an example. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction In this work, we are concerned with the existence of integral solutions and strict solutions for the following impulsive neutral functional differential equation:
d [x(t ) − g (t , xt )] = A[x(t ) − g (t , xt )] + f (t , xt ), dt − 1x|t =tk = Ik (x(tk )), x0 = φ ∈ B h ,
k = 1 , . . . , m,
t ∈ J , t ̸= tk , (1.1)
where J = [0, b], x(·) takes values in Banach space X with the norm ‖ · ‖, and A : D(A) ⊆ X → X is a nondensely defined closed linear operator which generates an integrated semigroup {S (t )}t ≥0 . g , f : J × Bh → X are given functions, Ik ∈ C (X , X ) (k = 1, 2, . . . , m), 0 = t0 < t1 < t2 < · · · < tm < tm+1 = b. 1x|t =tk = x(tk+ ) − x(tk− ), where x(tk− ) and x(tk+ ) represent the left and right limits of x(t ) at t = tk , respectively. The histories xt : (−∞, 0] → X , xt (θ ) = x(t + θ ), θ ≤ 0, belong to an abstract space Bh . Neutral differential equations arise in many areas of applied mathematics, and for this reason these equations have received much attention in recent decades. The literature relative to ordinary neutral functional differential equations is very extensive, and we refer the reader to [1–3], and the Hale and Lunel book [4], and the references therein. Partial neutral differential equations with finite delay arise, for instance, from transmission line theory. Wu and Xia have shown in [5] that a ring array of identical resistively coupled lossless transmission lines leads to a system of neutral functional differential equations with discrete diffusive coupling which exhibits various types of discrete wave. By taking a natural limit, they obtain from this system of neutral equations a scalar partial neutral functional differential equation with finite delay defined on the unit circle. Such a partial neutral functional differential equation is also investigated by Hale in [6] under the form
∂2 D ut (x) + f (ut )(x), ∂ x2 udt = ϕ ∈ C ([− r , 0]; C (S 1 ; R)), 0 d
D ut (x) =
t ≥ 0,
E-mail address: [email protected]. 1751-570X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.10.012
M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
503
0
where k is a constant, D (ψ)(s) := ψ(0)(s) − −r [dη(θ )]ψ(θ )(s) for s ∈ S 1 , ψ ∈ C ([−r , 0]; C (S 1 ; R)), and η is a function of bounded variation which is nonatomic at zero. In [7], Desch et al. studied abstract functional differential equations of neutral type with infinite delay. They proved that the model proposed by Coleman and Gurtin [8], Gurtin and Pipkin [9], and Miller [10] can be regarded as the following abstract functional differential equation of neutral type with infinite delay: d dt
[
x( t ) +
∫
0
]
[
K (t − s)x(s)ds = A x(t ) +
0
∫
−∞
]
K (t − s)x(s)ds + F (t , xt ), −∞
where the operator A is a generator of a C0 -semigroup on a Banach space. Recently, several works reporting existence results of mild solutions for first-order abstract neutral functional differential have been published. See, for example, [11,12] for the case with finite delay and [13,14] for the case with infinite delay. In these works, the linear operator A is always defined densely in X and satisfies the Hille–Yosida condition so that it generates a C0 -semigroup or an analytic semigroup. However, as indicated in [15], we sometimes need to deal with nondensely defined operators. For example, when we look at a one-dimensional heat equation with the Dirichlet condition on [0, 1] and consider A = ∂∂2 x in C ([0, 1]; R), in order to measure the solution in the sup-norm we take the domain 2
D(A) = {x ∈ C 2 ([0, 1]; R)|x(0) = x(1) = 0}, and then it is not dense in C ([0, 1], R) with the sup-norm. The example presented in Section 4 also shows the advantages of nondensely defined operators in handling some practical problems. See [15–17] for more examples and remarks concerning nondensely defined operators. On the other hand, the theory of impulsive differential equations has become an important area of investigation in recent years, stimulated by their numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, etc. Relative to this matter, we refer the reader to Lakshmikantham et al. [18], Benchohra et al. [19], Abada et al. [20], and Li and Han [21]. In [20], Abada et al. have established sufficient conditions for the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in separable Banach spaces, and in [21] the existence of integral solutions and strict solutions for a class of neutral impulsive functional equations with nonlocal conditions has been considered. To the best of our knowledge, there are very few results for impulsive neutral functional differential equations with infinite delay. Our objective in this work is to establish the existence of integral solutions and strict solutions for nondensely defined impulsive neutral functional differential equations, modelled as the initial value problem (1.1), on a Banach space X of infinite dimension. The organization of this work is as follows. In Section 2, we recall some basic results on integrated semigroups and we collect some properties of phase space. Section 3 is devoted to the existence of integral solutions and strict solutions for (1.1). Finally, in Section 4, an example is presented to illustrate the applications of the results obtained. 2. Integrated semigroups and phase space In this section, we recall some results about integrated semigroups and we present the abstract phase space Bh . 2.1. Integrated semigroups Definition 2.1 ([22]). Let X be a Banach space. An integrated semigroup is a family {S (t )}t ≥0 of bounded linear operators S (t ) on X with the following properties: (i) S (0) = 0; (ii) t → S (t ) isstrongly continuous; t (iii) S (t )S (s) = 0 [S (s + τ ) − S (τ )]dτ , for all t , s ≥ 0. Definition 2.2 ([22]). An operator A is said to be the generator of an integrated semigroup if there exists ω ∈ R such that (ω, +∞) ⊂ ρ(A), and there exists a strongly continuous exponentially bounded family {S (t )}t ≥0 of bounded linear ∞ operators such that S (0) = 0 and (λI − A)−1 = λ 0 e−λt S (t )dt for all λ > ω. Definition 2.3 ([23]). (i) An integrated semigroup {S (t )}t ≥0 is said to be locally Lipschitz continuous if, for all b > 0, there exists a constant L > 0 such that ‖S (t ) − S (s)‖ ≤ L|t − s|, t , s ∈ [0, b]. (ii) An integrated semigroup {S (t )}t ≥0 is said to be nondegenerate if S (t )x = 0 for all t ≥ 0 implies that x = 0. Definition 2.4 ([23]). We say that a linear operator A satisfies the Hille–Yosida condition (HY) if there exists M ≥ 1 and ω ∈ R such that (ω, +∞) ⊂ ρ(A) and sup{(λ − ω) ¯ n ‖R(λ, A)n ‖, n ∈ N , λ > ω} ≤ M , where ρ(A) is the resolvent set of A and R(λ, A) = (λI − A)
(HY) −1
.
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Theorem 2.5 ([23]). The following assertions are equivalent: (i) A is the generator of a non-degenerate, locally Lipschitz continuous integrated semigroup; (ii) A satisfies condition (HY). We know from [23] that, under condition (HY), A is the generator of a locally Lipschitz continuous integrated semigroup {S (t )}t ≥0 on X . In addition, the derivative {S ′ (t )}t ≥0 of {S (t )}t ≥0 generates a C0 -semigroup on D(A) such that ‖S ′ (t )x‖ ≤ Meωt ‖x‖, for all t ≥ 0 and x ∈ D(A). Furthermore, let A0 be the generator of the C0 -semigroup {S ′ (t )}t ≥0 ; then A0 is the part of A on D(A) defined by D(A0 ) = {x ∈ D(A) : Ax ∈ D(A)}, A0 x = Ax. Next, we give some general properties of the integrated semigroup {S (t )}t ≥0 . Proposition 2.6 ([22]). Let A be the generator of an integrated semigroup {S (t )}t ≥0 . Then, for all x ∈ X and t ≥ 0, D(A) and S (t )x = A
t 0
S (s)xds + tx. Moreover, for all x ∈ D(A) and t ≥ 0,
S (t )x ∈ D(A),
AS (t )x = S (t )Ax
S (t )x =
and
t 0
S (s)xds ∈
t
∫
S (s)Axds + tx. 0
Corollary 2.7 ([22]). Let A be the generator of an integrated semigroup {S (t )}t ≥0 ; then, for all x ∈ X and t ≥ 0, S (t )x ∈ D(A). Moreover, for any x ∈ X , S (·)x is right-hand-side differentiable in t ≥ 0 if and only if S (t )x ∈ D(A), and in that case we have S ′ (t )x = AS (t )x + x. In what follows, we give some results for the existence of solutions of the following Cauchy problem: d
x(t ) = Ax(t ) + f (t ), dt x(0) = x0 ∈ X ,
t ≥ 0,
(2.1)
where A satisfies the Hille–Yosida condition without being densely defined. Let {S (t )}t ≥0 be the integrated semigroup generated by A; then one has the following theorem. Theorem 2.8 ([24]). Let f : J → X be a continuous function. Then, for x0 ∈ D(A), there is a unique continuous function x : J → X such that (i)
t 0
x(s)ds ∈ D(A), t ∈ J;
t
t
(ii) x(t ) = x0 + A 0 x(s)ds + 0 f (s)ds, t ∈J; t (iii) ‖x(t )‖ ≤ Meωt ‖x0 ‖ + 0 e−ωs ‖f (s)‖ds , t ∈ J. Moreover, x satisfies the following variation of constant formula: d x(t ) = S ′ (t )x0 + dt
t
∫
S (t − s)f (s)ds,
t ≥ 0.
(2.2)
0
Proposition 2.9 ([25]). Let A : D(A) ⊆ X → X be a linear operator satisfying the Hille–Yosida condition, {S (t )}t ≥0 be the integrated semigroup generated by A, and f : [0, T ] → X , T > 0, be a Bochner-integrable function. Then the function K : [0, T ] → X t defined by K (t ) = 0 S (t − s)f (s)ds is continuously differentiable on [0, T ], and satisfies that, for λ > ω and t ∈ [0, T ], R(λ, A)K (t ) = ′
t
∫
S ′ (t − s)R(λ, A)f (s)ds. 0
Remark 1. Let Bλ = λR(λ, A); then, for all x ∈ D(A), Bλ x → x as λ → ∞. 2.2. Phase space In this work, we introduce an abstract phase space Bh which is similar to the one used in [26] and suitably modified to treat retarded impulsive differential equations [27,28]. Denote J0 = [0, t1 ], Jk = (tk , tk+1 ], k = 1, 2, . . . , m. Assume that h : (−∞, 0] → (0, +∞) is a continuous function
0
with l = −∞ h(s)ds < +∞. For any a > 0, we define
B = {φ : [−a, 0] → X : φ is continuous everywhere except a finite number of points t˜ at which φ(t˜+ ), φ(t˜− ) exist and φ(t˜) = φ(t˜− )},
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and equip the space B with the form
‖φ‖[−a,0] = sup ‖φ(θ )‖,
∀φ ∈ B .
−a≤θ≤0
Let us define
Bh = φ : (−∞, 0] → X : for any c > 0, φ|[−c ,0] ∈ B and
∫
0
h(s) sup ‖φ(θ )‖ds < +∞ . −∞
s≤θ≤0
If Bh is endowed with the norm
∫
0
h(s) sup ‖φ(θ )‖ds,
‖φ‖Bh = −∞
∀φ ∈ Bh ,
s≤θ≤0
then it is easy to prove that (Bh , ‖ · ‖Bh ) is a Banach space. Now we consider the space
Bb = {x : (−∞, b] → X : xk ∈ C (Jk , X ), k = 0, . . . , m and there exist x(tk+ ), x(tk− ), k = 1, . . . , m with x(tk ) = x(tk− ), x0 = φ ∈ Bh }. Set ‖ · ‖b be a semimorm in Bb defined by
‖x‖b = ‖φ‖Bh + sup ‖x(s)‖,
x ∈ Bb .
s∈J
Lemma 2.10 ([27]). Suppose that x ∈ Bb ; then, for t ∈ J, xt ∈ Bh . Moreover, l‖x(t )‖ ≤ ‖xt ‖Bh ≤ l sup ‖x(s)‖ + ‖x0 ‖Bh , s∈[0,t ]
0
where l = −∞ h(s)ds < +∞. In addition, we define the following classes of functions: PC (J , X ) = {x : J → X : xk ∈ C (Jk , X ), k = 0, 1, . . . , m and there exist x(tk+ ), x(tk− ), k = 1, . . . , m with x(tk ) = x(tk− )}, PC 1 (J , X ) = {x ∈ PC (J , X ) : x′k ∈ C (Jk , X ), k = 0, 1, . . . , m and there exist x′ (tk+ ), x′ (tk− ), k = 1, . . . , m with x′ (tk ) = ′ − x (tk )}, where xk and x′k represent the restriction of x and x′ to Jk , respectively (k = 0, . . . , m). Obviously, PC (J , X ) is a Banach space with the norm ‖x‖PC = supt ∈J ‖x(t )‖, and PC 1 (J , X ) is also a Banach space with the norm ‖x‖PC 1 = max{‖x‖PC , ‖x′ ‖PC }. 3. Main results Definition 3.1. We say that x : (−∞, b] → X is an integral solution of Eq. (1.1) if (i) x is continuous on Jk (k = 0, 1, . . . , m); (ii) x(t ) = φ(t ), t ∈ (−∞, 0]; t (iii) 0 (x(s) − g (s, xs ))ds ∈ D(A) for t ∈ J;
(iv) x(t ) = g (t , xt ) + S ′ (t )[φ(0) − g (0, φ)] + s)f (s, xs )ds, t ∈ J.
∑
0
S ′ (t − tk ){[x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )]} + k
d dt
t 0
S (t −
Definition 3.2. Let φ ∈ Bh . We say that a function x : (−∞, b] → X is a strict solution of Eq. (1.1) if the following conditions hold: (i) t → x(t ) − g (t , xt ) ∈ PC 1 (J , X ) ∩ PC (J , D(A)); (ii) x satisfies Eq. (1.1) on (−∞, b]. Remark 2. (A) It is not difficult to prove that, if x is an integral solution of Eq. (1.1) on (−∞, b], then, for almost all t ∈ J, x(t ) − g (t , xt ) ∈ D(A). In particular, φ(0) − g (0, φ) ∈ D(A). (B) If x is an integral solution of Eq. (1.1) on (−∞, b], such that t → x(t ) − g (t , xt ) belongs to PC 1 (J , X ) or PC (J , D(A)), then x is a strict solution. To obtain the existence and uniqueness of the integral solutions, we make the following assumptions.
(H0 ) The operator A satisfies condition (HY) and generates an integrated semigroup {S (t )}t ≥0 on X . (H1 ) g ∈ C (J × Bh , X ), and there exists constant α0 > 0 such that ‖g (t , φ1 ) − g (t , φ2 )‖ ≤ α0 ‖φ1 − φ2 ‖Bh for φ1 , φ2 ∈ Bh and t ∈ J . (H2 ) f ∈ C (J × Bh , X ), and there exists constant β0 > 0 such that ‖f (t , φ1 ) − f (t , φ2 )‖ ≤ β0 ‖φ1 − φ2 ‖Bh for φ1 , φ2 ∈ Bh and t ∈ J . (H3 ) Ik ∈ C (X , X ), and there exist constants αk > 0, k = 1, . . . , m such that ‖Ik (x) − Ik (y)‖ ≤ αk ‖x − y‖,
x, y ∈ X .
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M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
(H4 ) The semigroup {S ′ (t )}t ≥0 is compact on (D(A), ‖ · ‖), and there is a constant M ′ ≥ 1 such that ‖S ′ (t )‖ ≤ M ′ ,
for all t ∈ J .
Theorem 3.3. Assume that conditions (H0 )–(H4 ) are satisfied, and that
α0 l + M
m −
′
αk + 2α0 lm + bM β0 l ≡ γ0 < 1.
(3.1)
k=1
Let φ(0) − g (0, φ) ∈ D(A) and [x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )] ∈ D(A); then Eq. (1.1) has a unique integral solution k
x(·, φ) defined on (−∞, b].
Proof. Consider the nonempty closed subset of PC (J , X ) defined by Zb (φ) := {z ∈ PC (J , X ) : z (0) = φ(0)}. For z ∈ Zb (φ), we define z˜ : (−∞, b] → X by z˜ (t ) =
z (t ), t ∈ J , φ(t ), t ≤ 0.
Consider the operator J : Zb (φ) → Zb (φ) defined by
−
(Jz )(t ) = g (t , z˜t ) + S ′ (t )[φ(0) − g (0, φ)] +
S ′ (t − tk ){Ik (z (tk− )) − [g (tk , z˜t + ) − g (tk , z˜tk )]} k
0
+
d dt
t
∫
S (t − s)f (s, z˜s )ds. 0
From Proposition 2.9 and condition (HY), we get
∫ t ∫ d t ′ Bλ S ( t − s ) f ( s , x ) ds = S ( t − s )λ R (λ, A ) f ( s , x ) ds s s dt 0 0 ∫ b λ ≤ MM ′ ‖f (s, xs )‖ds. λ−ω 0 Letting λ → ∞, we obtain that ∫ t ∫ b d ≤ MM ′ S ( t − s ) f ( s , x ) ds ‖f (s, xs )‖ds. s dt
0
(3.2)
0
By the hypotheses, Lemma 2.10, and (3.2), we can see that, for every z 1 , z 2 ∈ Zb (φ) and t ∈ J,
‖(Jz 1 )(t ) − (Jz 2 )(t )‖ ≤ α0 l sup ‖z 1 (s) − z 2 (s)‖ + s∈[0,t ]
−
S ′ (t − tk )[αk ‖z 1 (tk− ) − z 2 (tk− )‖
0 < tk < t
+ 2α0 l sup ‖z 1 (s) − z 2 (s)‖] + MM ′ s∈[0,t ]
≤
α0 l + M ′
b
∫
β0 l sup ‖z 1 (s) − z 2 (s)‖ds 0
m −
s∈[0,t ]
αk + 2α0 lm + bM β0 l
‖z 1 − z 2 ‖PC
k=1
= γ0 ‖z 1 − z 2 ‖PC . Then, J is a strict contraction in Zb (φ), and the fixed point of J gives a unique integral solution x(·, φ) on (−∞, b]. This ends the proof. To prove the existence of strict solutions, we add the following assumptions.
(H5 ) g , f ∈ C 1 (J × Bh , X ) and their partial derivatives are locally Lipschitzian with respect to the second argument in the sense that, for any compact set Q ⊂ J × Bh , there exists a constant β1 > 0 such that ‖D f (t , φ) − Dφ f (t , ψ)‖ ≤ β1 ‖φ − ψ‖Bh φ ‖Dt f (t , φ) − Dt f (t , ψ)‖ ≤ β1 ‖φ − ψ‖Bh ‖Dφ g (t , φ) − Dφ g (t , ψ)‖ ≤ β1 ‖φ − ψ‖Bh ‖Dt g (t , φ) − Dt g (t , ψ)‖ ≤ β1 ‖φ − ψ‖Bh , for all (t , φ), (t , ψ) ∈ Q and t ∈ J, where Dt f , Dφ f , Dt g , Dφ g denote the derivatives with respect to t and φ . (H6 ) Dx Ik (x) ∈ C (X , X ), k = 1, 2, . . . , m, where Dx Ik denote the derivatives with respect to x.
M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
507
Theorem 3.4. Let x be the unique integral solution of Eq. (1.1) obtained by Theorem 3.3. If the hypotheses (H0 )–(H6 ) are satisfied, then x is also a strict solution of the Cauchy problem (1.1), provided that
φ ′ ∈ Bh ,
φ(0) − g (0, φ) ∈ D(A0 ),
φ − Dt g (0, φ) − Dφ g (0, φ)φ ′ ∈ D(A), ′
[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] − [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ] ∈ D(A), k
k
k
(3.3)
φ ′ − Dt g (0, φ) − Dφ g (0, φ)φ ′ = A[φ(0) − g (0, φ)] + f (0, φ), A{[x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )]} = [y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] k
k
k
k
− [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]. Proof. By Theorem 3.3, we know that Eq. (1.1) has a unique integral solution x := x(·, φ), which is also the unique solution of
−
x(t ) = g (t , xt ) + S ′ (t )[φ(0) − g (0, φ)] +
S ′ (t − tk ){[x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )]} k
0
+
d
t
∫
dt
S (t − s)f (s, xs )ds,
t ∈ J.
(3.4)
0
By Corollary 2.7, the assumption that φ(0) − g (0, φ) ∈ D(A0 ) implies that S ′ (t )[φ(0) − g (0, φ)] = S (t )A[φ(0) − g (0, φ)] + [φ(0) − g (0, φ)]. Then Eq. (3.4) can be rewritten as x(t ) = g (t , xt ) + S (t )A[φ(0) − g (0, φ)] + [φ(0) − g (0, φ)] +
−
S ′ (t − tk ){[x(tk+ ) − g (tk , xt + )] − [x(tk ) k
0
− g (tk , xtk )]} +
d dt
t
∫
S (t − s)f (s, xs )ds,
t ∈ J.
(3.5)
0
Consider the following equation:
d dt [y(t ) − Dt g (t , xt ) − Dφ g (t , xt )yt ] = A[y(t ) − Dt g (t , xt ) − Dφ g (t , xt )yt ] + Dt f (t , xt ) + Dφ f (t , xt )yt , t ∈ J , t ̸= tk , k = 1, . . . , m. − − 1 y |t =tk′ = Dx Ik (x(tk ))y(tk ), k = 1, . . . , m. y0 = φ , t ∈ (−∞, 0].
(3.6)
Then, using the same reasoning as in the proof of Theorem 3.3, one can show that Eq. (3.6) has a unique integral solution y : (−∞, b] → X given by y(t ) = φ ′ (t ), t ∈ (−∞, 0] and y(t ) = S ′ (t )[φ ′ − Dt g (0, φ) − Dφ g (0, φ)φ ′ ] + Dt g (t , xt ) + Dφ g (t , xt )yt +
−
S ′ (t − tk ){[y(tk+ ) − Dt g (tk , xt + )
0
− Dφ g (tk , xt + )yt + ] − [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]} k k ∫ t d + S (t − s)[Dt f (s, xs ) + Dφ f (s, xs )ys ]ds, t ∈ J . dt
k
(3.7)
0
Let w : (−∞, b] → X be the function defined by
w(t ) =
φ(t )
φ(0) +
for t ∈ (−∞, 0],
t
∫
y(s)ds + 0
−
Ik (x(tk )) −
for t ∈ J .
0
Next, we will show that x = w on (−∞, b]. Integrating both sides of Eq. (3.7) from 0 to t, we have t
∫
y(s)ds = S (t ){A[φ(0) − g (0, φ)] + f (0, φ)} +
t
∫
0
Dt g (s, xs )ds + 0
+
− 0
∫
t
Dφ g (s, xs )ys ds
0
S (t − tk ){[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] k
− [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]} +
k
k
t
∫
S (t − s)(Dt f (s, xs ) + Dφ f (s, xs )ys )ds. 0
(3.8)
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M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
On the other hand, for t ∈ J, t
∫
d dt
S (t − s)f (s, ws )ds = S (t )f (0, φ) +
t
∫
S (t − s)(Dt f (s, ws ) + Dφ f (s, ws )ys )ds. 0
0
Consequently, t
∫
d
S (t )f (0, φ) =
dt
S (t − s)f (s, ws )ds −
t
∫
S (t − s)(Dt f (s, ws ) + Dφ f (s, ws )ys )ds.
(3.9)
0
0
Since
−
w(t ) − g (t , wt ) − [φ(0) − g (0, φ)] −
{[w(tk+ ) − g (tk , wt + )] − [w(tk ) − g (tk , wtk )]} k
0
∫ = 0
d dt
[w(s) − g (s, ws )]ds =
t
∫
[y(s) − Dt g (s, ws ) − Dφ g (s, ws )ys )ds 0
and (3.8) holds, we get that (noting that x0 = w0 = φ ) t
∫
w(t ) = g (t , wt ) +
[y(s) − Dt g (s, ws ) − Dφ g (s, ws )ys ]ds + [φ(0) − g (0, φ)] ∫ t − + {[w(tk+ ) − g (tk , wt + )] − [w(tk ) − g (tk , wtk )]} − y(s)ds + S (t ){A[φ(0) − g (0, φ)] + f (0, φ)} 0
k
0
∫
Dt g (s, xs )ds +
+ 0
0
t
∫
−
Dφ g (s, xs )ys ds +
S (t − tk ){[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] k
0
0
k
k
∫ t − [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]} + S (t − s)[Dt f (s, xs ) + Dφ f (s, xs )ys ]ds 0 ∫ t ∫ t = g (t , wt ) + [Dt g (s, xs ) − Dt g (s, ws )]ds + [Dφ g (s, xs ) − Dφ g (s, ws )]ys ds + S (t ){A[φ(0) − g (0, φ)] 0 0 − + f (0, φ)} + [φ(0) − g (0, φ)] + {[w(tk+ ) − g (tk , wt + )] − [w(tk ) − g (tk , wtk )]} k
0 < tk < t
−
+
S (t − tk ){[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] − [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]} k
0
k
k
t
∫
S (t − s)[Dt f (s, xs ) + Dφ f (s, xs )ys ]ds.
+ 0
Therefore, x(t ) − w(t ) = g (t , xt ) − g (t , wt ) +
−
S ′ (t − tk ){[x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )]} k
0
+
d
t
∫
dt
S (t − s)f (s, xs )ds − S (t )f (0, φ) − 0
[Dφ g (s, xs ) − Dφ g (s, ws )]ys ds −
−
{[w(tk+ ) − g (tk , wt + )] − [w(tk ) − g (tk , wtk )]} k
0
0
−
[Dt g (s, xs ) − Dt g (s, ws )]ds 0
t
∫ −
t
∫
−
S (t − tk ){[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] − [y(tk ) − Dt g (tk , xtk ) k
0 < tk < t
− Dφ g (tk , xtk )ytk ]} −
k
k
t
∫
S (t − s)[Dt f (s, xs ) + Dφ f (s, xs )ys ]ds. 0
Eq. (3.9) yields x(t ) − w(t ) = g (t , xt ) − g (t , wt ) +
−
S ′ (t − tk ){[x(tk+ ) − g (tk , xt + )] − [x(tk ) − g (tk , xtk )]} k
0
+
d
t
∫
dt
S (t − s)[f (s, xs ) − f (s, ws )]ds − 0
t
∫
[Dφ g (s, xs ) − Dφ g (s, ws )]ys ds −
− 0
t
∫
[Dt g (s, xs ) − Dt g (s, ws )]ds
0
− 0
{[w(tk+ ) − g (tk , wt + )] − [w(tk ) − g (tk , wtk )]} k
M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
−
−
509
S (t − tk ){[y(tk+ ) − Dt g (tk , xt + ) − Dφ g (tk , xt + )yt + ] k
k
k
0
∫ t S (t − s)[Dt f (s, ws ) − Dt f (s, xs )]ds − [y(tk ) − Dt g (tk , xtk ) − Dφ g (tk , xtk )ytk ]} + 0 ∫ t S (t − s)[Dφ f (s, ws ) − Dφ f (s, xs )]ys ds. + 0
Consequently, by Proposition 2.6, Corollary 2.7, and Lemma 2.10, we deduce that
‖x(t ) − w(t )‖ ≤ α0 ‖xt − wt ‖Bh + σ (b)
t
∫
−
‖xs − ws ‖Bh ds +
0
0
{[(x(tk+ ) − g (tk , x+ tk )) − (x(tk ) − g (tk , xtk ))]
− [(w(tk+ ) − g (tk , wt+k )) − (w(tk ) − g (tk , wtk ))]} ∫ t ‖xs − ws ‖Bh ds + 2mα0 ‖xt − wt ‖Bh + 2m sup ‖x(t ) − w(t )‖ ≤ α0 ‖xt − wt ‖Bh + σ (b) 0≤s≤t
0
≤ [(2m + 1)α0 l + 2m] sup ‖x(t ) − w(t )‖ + σ (b) s∈[0,t ]
t
∫
‖xs − ws ‖Bh ds, 0
where
σ (b) = MM ′ β0 + β1 + β1 b0 + β1 b0 + β1 b20 ,
b0 = max{ sup ‖S (s)‖, sup ‖ys ‖Bh }. 0≤s≤b
0≤s≤b
Hence,
‖xt − wt ‖Bh ≤ l sup ‖x(s) − w(s)‖ ≤ [1 − 2m − (2m + 1)α0 l]−1 lσ (b) s∈[0,t ]
t
∫
‖xs − ws ‖Bh ds. 0
Using Gronwall’s lemma, we conclude that
‖xt − wt ‖Bh = 0,
t ∈ J.
Hence, x(t ) = w(t ) for all t ∈ (−∞, b]. Therefore, we conclude that x ∈ PC 1 (J , X ), since w has obviously this property. Thus, by Remark 2, x is a strict solution of Eq. (1.1). The proof of Theorem 3.4 is complete. 4. Application As applications of the obtained results of this paper, we study the following impulsive partial functional differential system with infinite delay:
] [ ] ∫ 0 ∫ 0 [ ∂2 ∂ u(t , ξ ) − K1 (θ , u(t + θ , ξ ))dθ = u ( t , ξ ) − K (θ , u ( t + θ , ξ )) d θ 1 ∂ x2 −∞ −∞ ∂t ∫ 0 + K2 (θ , u(t + θ , ξ ))dθ , t ∈ J = [0, 1], ξ ∈ [0, π], −∞ u(t , 0) = u(t , π ) = 0, u(tk+ , ξ ) − u(tk− , ξ ) = Ik (u(tk− , ξ )), k = 1, . . . , m, u(θ , ξ ) = u0 (θ , ξ ), θ ∈ (−∞, 0], ξ ∈ [0, π],
(4.1)
where K1 , K2 : (−∞, 0] × R → R and u0 : (−∞, 0] × [0, π] → R are continuous functions. Let X = C ([0, π]; R) be the space of continuous functions from [0, π] into R endowed with the uniform norm topology. Define A : D(A) ⊂ X → X by Aw = w ′′ with domain D(A) = {w ∈ C 2 ([0, π]) : w(0) = w(π ) = 0}. We have D(A) = {w ∈ X : w(0) = w(π ) = 0} ̸= X and
ρ(A) ⊇ (0, +∞),
‖(λI − A)−1 ‖ ≤
1
λ
,
for λ > 0.
This implies that A satisfies condition (HY) on X . It is well known that A generates an integrated semigroup {S (t )}t ≥0 and that ‖S ′ (t )‖ ≤ e−t for t ≥ 0.
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M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
Let h(s) = eγ s , γ > 0, s < 0; then 0
∫
1
h(s)ds =
l=
γ
−∞
,
and let
∫
0
h(s) sup ‖φ(θ )‖ds.
‖φ‖Bh =
s≤θ≤0
−∞
Thus, for (t , φ) ∈ J × Bh , where φ(θ )(ξ ) = u0 (θ , ξ ), (θ , ξ ) ∈ (−∞, 0] × [0, π], let x(t )(ξ ) = u(t , ξ ),
g (φ)(ξ ) =
∫
0
K1 (θ , φ(θ )(ξ ))dθ , −∞
f (φ)(ξ ) =
0
∫
K2 (θ , φ(θ )(ξ ))dθ . −∞
Then Eq. (4.1) takes the following abstract form:
d [x(t ) − g (xt )] = A[x(t ) − g (xt )] + f (xt ), dt − 1x|t =tk = Ik (x(tk )), x0 = φ ∈ Bh .
t ∈ J , t ̸= tk , (4.2)
k = 1 , . . . , m,
In order to study the existence of solutions of Eq. (4.2), we make the following assumptions. (i) For each θ ≤ 0, −∞ eγ θ supξ ∈[0,π ] ‖u0 (θ , ξ )‖dθ < ∞, u0 (θ , 0) = u0 (θ , π ) = 0 and K1 (·, 0) = 0. (ii) For each i = 1, 2, θ ≤ 0 and ζ1 , ζ2 ∈ R, ‖Ki (θ , ζ1 ) − Ki (θ , ζ2 )‖ ≤ ki (θ )‖ζ1 − ζ2 ‖, where k1 , k2 are measurable nonnegative functions on (−∞, 0] such that
0
∫
0
(1 + γ )ki (θ )dθ = ρi < ∞,
i = 1, 2.
−∞
(iii) Ik ∈ C (X , X ), and there exist constants αk > 0, k = 1, . . . , m such that
‖Ik (x1 ) − Ik (x2 )‖ ≤ αk ‖u1 − u2 ‖,
x1 , x2 ∈ X .
(iv) u0 (0, ·) −
∫
0
K1 (θ , u0 (θ , ·))dθ ∈ D(A), −∞
and
[u(tk+ , ξ ) −
∫
0
K1 (θ , u((tk + θ )+ , ξ ))dθ ] − [u(tk , ξ ) − −∞
∫
0
K1 (θ , u(tk + θ , ξ ))dθ ] ∈ D(A). −∞
Assumption (ii) implies that, for φ1 , φ2 ∈ Bh ,
‖g (φ1 ) − g (φ2 )‖ = sup ‖g (φ1 )(ξ ) − g (φ2 )(ξ )‖ ≤ ρ1 ‖φ1 − φ2 ‖Bh ξ ∈[0,π ]
and
‖f (φ1 ) − f (φ2 )‖ = sup ‖f (φ1 )(ξ ) − f (φ2 )(ξ )‖ ≤ ρ2 ‖φ1 − φ2 ‖Bh . ξ ∈[0,π]
Consequently, (H1 ) and (H2 ) are true. Then, from Theorem 3.3, we know that, if
ρ1 γ −1 +
m −
αk + 2ρ1 γ −1 m + ρ2 γ −1 < 1,
k=1
then system (4.2) admits a unique integral solution x. To assert that x is a strict solution, we have to make more assumptions.
M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
511
(v) For each i = 1, 2, Ki is C 2 -smooth, and the second derivative of Ki with respect to the second variable satisfies the following estimate:
2 ∂ ≤ β˜ i (θ )‖ζ ‖ for θ ≤ 0 and ζ ∈ R, K (θ , ζ ) ∂ζ 2 i where β˜ i is a measurable nonnegative function on (−∞, 0] such that 0
∫
(1 + γ )2 β˜ i (θ )dθ < ∞. −∞
(vi) Dx Ik (x) ∈ C (X , X ), k = 1, 2, . . . , m, where Dx Ik denote the derivatives with respect to x. By assumption (v), g and f are continuously differentiable and satisfy, for φ, ψ ∈ Bh and ξ ∈ [0, π],
∫ 0 ∂ ′ g (φ)(ψ)(ξ ) = K1 (θ, φ(θ )(ξ ))(ψ)(θ)(ξ )dθ , ∂ζ −∞ ∫ 0 ∂ f ′ (φ)(ψ)(ξ ) = K2 (θ, φ(θ )(ξ ))(ψ)(θ)(ξ )dθ . −∞ ∂ζ Moreover, as a consequence of assumption (v), g ′ and f ′ are Lipschitz continuous in Bh . In fact, this is a consequence of the following:
∫
∂ Ki (θ , φ1 (θ )(ξ ))(ψ)(θ)(ξ ) − ∂ Ki (θ , φ2 (θ)(ξ ))(ψ)(θ )(ξ ) dθ ∂ζ ∂ζ −∞ ∫ 0 ≤ β˜ i (θ )‖φ1 (θ)(ξ ) − φ2 (θ )(ξ )‖ ‖ψ(θ )(ξ )‖dθ, 0
−∞
which implies that
∫
∂ Ki (θ , φ1 (θ )(ξ ))(ψ)(θ)(ξ ) − ∂ Ki (θ , φ2 (θ)(ξ ))(ψ)(θ )(ξ ) dθ ∂ζ ∂ζ −∞ ∫ 0 ≤ (1 + γ )2 β˜ i (θ)dθ ‖φ1 − φ2 ‖Bh ‖ψ‖Bh . 0
−∞
Then, (H5 ) is fulfilled. Consider the following equation:
] ∫ 0 [ ∂ ∂ v( t, ξ ) − K1 (θ, u(t + θ , ξ ))v(t + θ, ξ )dθ ∂t −∞ ∂ζ [ ] ∫ 0 ∂2 ∂ = v( t , ξ ) − K (θ, u ( t + θ , ξ ))v( t + θ , ξ ) d θ 1 ∂ x∫2 −∞ ∂ζ 0 ∂ K2 (θ, u(t + θ, ξ ))v(t + θ, ξ )dθ , t ∈ J , ξ ∈ [0, π], + −∞ ∂ζ v(t , 0) = v(t , π) = 0, + v(t , ξ ) − v(tk− , ξ ) = Du Ik (u(tk− , ξ ))v(tk− , ξ ), k = 1, . . . , m, k v(θ , ξ ) = ∂ u (θ, ξ ), θ ∈ (−∞, 0], ξ ∈ [0, π]. 0 ∂θ We assume that
∂ ∂θ
u0 ∈ Bh ,
u0 (0, ·) −
∫
0
K1 (θ, u0 (θ, ·))dθ ∈ D(A0 ), −∞
∫ 0 ∂ ∂ ∂ u0 (0, ·) − K1 (θ, u0 (θ , ·)) u0 (θ, ·)dθ ∈ D(A), ∂θ ∂θ −∞ ∂ζ [ ] ∫ 0 ∂ v(tk+ , ξ ) − K1 (θ , u((tk + θ)+ , ξ ))v((tk + θ)+ , ξ )dθ −∞ ∂ζ [ ] ∫ 0 ∂ − v(tk , ξ ) − K1 (θ, u(tk + θ, ξ ))v(tk + θ, ξ )dθ ∈ D(A). −∞ ∂ζ
(4.3)
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M. Li / Nonlinear Analysis: Hybrid Systems 5 (2011) 502–512
By Theorem 3.3, it is enough to show that Eq. (4.3) has a unique integral solution v on (−∞, 1]. In addition, we assume that
∫ 0 ∂ ∂ ∂ u0 (0, ξ ) − K1 (θ , u0 (θ , ξ )) u0 (θ , ξ )dθ ∂θ ∂θ −∞ ∂ζ ] ∫ 0 [ ∫ 0 ∂2 K2 (θ , u0 (θ , ξ ))dθ , K1 (θ , u0 (θ , ξ ))dθ + = 2 u0 (0, ξ ) − ∂ξ −∞ −∞ and
∂2 ∂ξ 2
[
u(tk+ , ξ ) −
0
∫
K1 (θ , u((tk + θ )+ , ξ ))dθ
]
−∞
∂ K1 (θ , u((tk + θ )+ , ξ ))v((tk + θ )+ , ξ )dθ = v(tk+ , ξ ) − −∞ ∂ζ [ ] ∫ 0 ∂ − v(tk , ξ ) − K1 (θ , u(tk + θ , ξ ))v(tk + θ , ξ )dθ . −∞ ∂ζ [
∫
0
[ ∫ − u(tk , ξ ) −
0
K1 (θ , u(tk + θ , ξ ))dθ
]
−∞
]
Hence, all the assumptions of Theorem 3.4 are satisfied. Then we obtain that the integral x of Eq. (4.2) is strict. Consequently, the function u, defined by u(t , ξ ) = x(t )(ξ ) for t ∈ J and ξ ∈ [0, π], is a solution of Eq. (4.1). Acknowledgement This work is supported by NNSF of China (No. 10971139). References [1] S.K. Ntouyas, Y.G. Sficas, C.P. Tsamatos, Existence results for initial value problems for neutral functional differential equations, J. Differential Equations 114 (2) (1994) 527–537. [2] M. Benchohra, A. Ouanab, Impulsive neutral functional differential equations with variable times, Nonlinear Anal. 55 (6) (2003) 679–693. [3] M. Benchohra, E. Gatsori, S.K. Ntouyas, Existence results for functional and neutral functional integrodifferential inclusions with lower semicontinuous right-hand side, J. Math. Anal. Appl. 281 (2) (2003) 525–538. [4] J.K. Hale, Sjoerd M. Verduyn Lunel, Introduction to Functional Differential Equations, in: Appl. Math. Sci., vol. 99, Springer-Verlag, New York, 1993. [5] J. Wu, H. Xia, Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Differential Equations 124 (1) (1996) 247–278. [6] J.K. Hale, Partial neutral functional differential equations, Rev. Roumaine Math. Pures Appl. 39 (4) (1994) 339–344. [7] W. Desch, R. Grimmer, W. Schapacher, Wellposedness and wave propagation for a class of integrodifferential equations in Banach space, J. Differential Equations 74 (1988) 391–411. [8] B.D. Coleman, M.E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18 (1967) 199–208. [9] M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with infinite wave speeds, Arch. Ration. Mech. Anal. 31 (1968) 113–126. [10] R.K. Miller, An integrodifferential equation for rigid heat conductions with memory, J. Math. Anal. Appl. 66 (1978) 313–332. [11] X. Fu, K. Ezzinbi, Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Anal. 54 (2) (2003) 215–227. [12] K. Ezzinbi, X. Fu, K. Hilal, Existence and regularity in the α -norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal. 67 (5) (2007) 1613–1622. [13] E. Hernríquez, Hernán R. Henríquez, Existence results for partial neutral functional differential equations with unbounded delay, J. Math. Anal. Appl. 221 (2) (1998) 499–522. [14] K. Ezzinbi, S. Ghnimib, M.A. Taoudi, Existence and regularity of solutions for neutral partial functional integrodifferential equations with infinite delay, Nonlinear Anal. Hybrid Syst. 4 (1) (2010) 54–64. [15] G. Da Prato, E. Sinestrari, Differential operators with nondense domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. 14 (1987) 285–344. [16] M. Adimy, H. Bouzahir, K. Ezzinbi, Existence and stability for some partial neutral functional differential equations with infinite delay, J. Math. Anal. Appl. 294 (2004) 438–461. [17] M. Adimy, K. Ezzinbi, A. Ouhinou, Behavior near hyperbolic stationary solutions for partial functional differential equations with infinite delay, Nonlinear Anal. 68 (2008) 2280–2302. [18] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [19] M. Benchohra, J. Henderson, S.K. Ntouyas, An existence result for first-order impulsive functional differential equations in Banach spaces, Comput. Math. Appl. 42 (2001) 1303–1310. [20] N. Abada, M. Benchohra, H. Hammouche, Existence and controllability for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations 246 (2009) 3834–3863. [21] M. Li, M. Han, Existence for neutral impulsive functional differential equations with nonlocal conditions, Indag. Math. (NS) 20 (3) (2009) 435–451. [22] W. Arendt, Resolvent positive operators and integrated semigroup, Proc. Lond. Math. Soc. 54 (1987) 321–349. [23] H. Kellermann, M. Hieber, Integrated semigroup, J. Funct. Anal. 15 (1989) 160–180. [24] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987) 327–352. [25] H. Thiems, Integrated semigroup and integral solutions to abstract Cauchy problems, J. Math. Anal. Appl. 152 (1990) 416–447. [26] Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, Springer, New York, 1991, p. 1473. [27] B. Liu, Controllability of impulsive neutral functional differential inclusion with infinite delay, Nonlinear Anal. 60 (2005) 1533–1552. [28] Y.K. Chang, A. Anguraj, M. Mallika Arjunan, Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear Anal. Hybrid Syst. 2 (2008) 209–218.