Local existence for retarded Volterra integrodifferential equations with Hille–Yosida operators

Nonlinear Analysis 66 (2007) 2814–2832 www.elsevier.com/locate/na

Local existence for retarded Volterra integrodifferential equations with Hille–Yosida operators✩ Jung-Chan Chang Department of Applied Mathematics, I-Shou University, Ta-Hsu, Kaohsiung 84008, Taiwan Received 25 November 2005; accepted 10 April 2006

Abstract This paper is devoted to finding some existence and uniqueness theorems for classical solutions of the integrodifferential equations with infinite delay. We give some sufficient conditions ensuring the existence and uniqueness of solutions. We assume the linear part is not necessary to be densely defined and satisfies a Hille–Yosida condition so that it is the generator of a nondegenerated, locally Lipschitz continuous integrated semigroup. By using matrix operators and fixed point theorems, we obtain new results for the retarded integrodifferential equations. c 2006 Elsevier Ltd. All rights reserved.  Keywords: Hille–Yosida operator; Retarded Volterra integrodifferential equation; Integrated semigroup; C0 -semigroup; Extrapolated semigroup

1. Introduction Let (X, ·) be an infinite dimension Banach space. Suppose that A is a closed linear operator on X with domain D(A), the phase space P is a linear space of functions mapping (−∞, 0] into X satisfying some axioms which will be described later, F is an X-valued function, and for every t ≥ 0, the function u t (·) ∈ P is defined by u t (θ ) = u(t + θ )

for θ ∈ (−∞, 0].

✩ This research is supported in part by the National Science Council of Taiwan.

E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter  doi:10.1016/j.na.2006.04.009

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

The following partial functional differential equation with infinite delay  du(t) = Au(t) + F(t, u t ), t ≥ 0, dt u 0 = ϕ ∈ P,

2815

(RACP)

had been considered by many authors (cf. [1–3] and [18] etc.). In the literature devoted to Eq. (RACP) with finite delay, the state space is the set of all continuous functions on [−r, 0], r > 0, endowed with the uniform norm topology. The investigation of functional differentials with infinite delay in an abstract admissible phase space was initiated by Hale and Kato [11], Kapple and Schappacher [16], and Schumacher [21]. The method of using admissible phase spaces enables one to treat a large class of functional differential equations with infinite delay in the same time and obtain general results. For a detailed discussion on this topic, we refer the reader to the book by Hino et al. [15] and Wu [23]. There has been a great deal of work contributed to the study of partial differential equations with delay by using different methods under different conditions. The most classical work is due to Travis and Webb [22]. In the study of Eq. (RACP), one assumes that the operator A generates a C0 -semigroup on X (see [10]). In this case, A must be densely defined and satisfies the Hille–Yosida condition. More recently, in [1–3] and [4], the authors show that the density of domain for A is not necessary if A satisfies the Hille–Yosida condition. We shall focus on the case that A is not densely defined but satisfies the Hille–Yosida condition. In [1,2] and [3], the authors treated Eq. (RACP) by using variation-of-constant formula and integrated semigroups, and they extended the results of Henriquez [12,13] and [14]. In their cases, F satisfies the local Lipschitz condition with respect to the phase space. Based on their ideas and use of the extrapolation approach which is introduced by Nagel and Sinestrari [20], we consider the following equation ⎧  t ⎨ du(t) = Au(t) + B(t, θ, u(θ ))dθ + F(t, u t ), 0 ≤ t ≤ T, (VID1) 0 ⎩ dt u 0 = ϕ ∈ P, where B ∈ C({(x, y); 0 ≤ y ≤ x ≤ T } × X, X) and F ∈ C([0, T ] × P, X). We point out that Eq. (VID1) can be transformed into  du(t) = Au(t) + G(t, u t ), 0 ≤ t ≤ T, dt u 0 = ϕ ∈ P, 0 by setting G(t, u t ) := −t B(t, t + θ, u t (θ ))dθ + F(t, u t ). If we use this transformation and apply the method in [3] to this equation, the different assumptions of B will be given. For example, we must choose B such that G ∈ C([0, T ] × P, X). We do not treat Eq. (VID1) by this transformation. In general, the conclusions in [3] cannot be applied to our cases. The purpose of this paper is not only to consider Eq. (VID1) but also to solve the following equation ⎧   t ⎨ du(t) = A u(t) + B(t, θ, u(θ ))dθ + F(t, u t ), 0 ≤ t ≤ T, (VID2) dt 0 ⎩ u 0 = ϕ ∈ P. The form of Eq. (VID2) had been considered in [6,7] and [9]. In these papers, operator matrices are main ideas which were introduced first by Miller [19]. We will generalize the method in [9] to solve it. The obtained results would be an extension of [9]. Finally, the following

2816

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

similar equations ⎧  t ⎨ du(t) = Au(t) + B(t, θ, u(θ ))dθ + F(t, u t ), dt −∞ ⎩ u0 = ϕ ∈ P and

0 ≤ t ≤ T,

⎧   t ⎨ du(t) = A u(t) + B(t, θ, u(θ ))dθ + F(t, u t ), dt −∞ ⎩ u0 = ϕ ∈ P

0 ≤ t ≤ T,

(VID3)

(VID4)

are also studied. In Section 2, we recall some preliminary results about the extrapolation spaces and semigroups. In Section 3, we prove the existence of solutions to Eqs. (VID1), (VID2), (VID3) and (VID4), which are the main results of this paper. In Section 4, we give some examples to show our results are valuable. 2. Preliminary In this section, we give some definitions and fundamental notations of extrapolation spaces. For a more recent account we refer to [8,9,17] and [20], where also the missing proofs can be found. First, let X be a Banach space and let A be a linear operator with domain D(A). We assume that L(X) denotes the space of bounded linear operators on X. In the whole work, we suppose that A is a Hille–Yosida operator, T > 0 is an extended real number and ϕ always denotes the initial data in the following equations. Let T1 , T2 ∈ R with T1 < T2 . We use Δ(T1 , T2 ) to denote the set {(x, y); x, y ∈ R, T1 ≤ y ≤ x ≤ T2 }. Moreover, Δ(−∞, T2 ) and Δ(T1 , ∞) denote {(x, y); x, y ∈ R, y ≤ x ≤ T2 , x ≥ 0} and {(x, y); x, y ∈ R, T1 ≤ y ≤ x, x ≥ 0}, respectively. Definition 2.1. The linear operator A is a Hille–Yosida operator on X if there exists w ∈ R and M ≥ 1 such that (ω, +∞) ⊂ ρ(A) (ρ(A) is the usual resolvent set of A) and satisfies M for all λ > ω and n ∈ N. (HY) (λ − ω)n The domain D(A) of A is not necessarily dense and we denote its closure in X by X 0 . The part A0 of A in X 0 is the linear operator with domain D(A0 ) = {x ∈ D(A); Ax ∈ X 0 } and is defined by A0 x = Ax for all x ∈ D(A0 ). Here is an elementary property of the Hille–Yosida operator. The proof can be found in [8]. (λ − A)−n  ≤

Proposition 2.2. Suppose that A is a Hille–Yosida operator and satisfies the condition (HY) on X. Then the part A0 of A in X 0 generates a C0 -semigroup T0 (·) on X 0 . Moreover, T0 (t) ≤ Meωt for t ≥ 0. Without loss of generalization, we suppose that ω < 0 in this paper. For a fixed λ0 ∈ ρ(A), we introduce on X 0 a new norm defined by x−1 = R(λ0 , A0 )x for x ∈ D(A0 ). The completion X −1 of (X 0 ,  · −1 ) is called the extrapolation space of X associated with A. Note that  · −1 and the norm on X 0 given by R(λ, A)x for different λ ∈ ρ(A) and x ∈ X 0

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

2817

are equivalent. The operator T0 (t) has a bounded linear extension T−1 (t) to the Banach space X −1 , T−1 (·) is a strongly continuous semigroup on X −1 . and T−1 (·) is called the extrapolated semigroup of T0 (·). We have some fundamental results. Proposition 2.3 ([20], Proposition 2.1). The following properties hold: (i) The space X 0 is dense in (X −1 ,  · −1 ). Hence the extrapolation space X −1 is also the completion of (X,  · −1 ). (ii) For f ∈ L 1 (R+ , X) and t > 0, let  t (T−1 ∗ f )(t) := T−1 (t − s) f (s)ds. 0

Then (T−1 ∗ f )(t) ∈ X 0 and (T−1 ∗ f )(t) ≤ M1  f  L 1 ((0,t ),X ) where M1 is a constant independent of f and t. In the following, M and M1 always denote the constants in Propositions 2.2 and 2.3, respectively. Finally, we give the basic definition and properties of integrated semigroups which can be found in [17] and its references. Definition 2.4. A family {S(t); t ∈ [0, ∞)} ⊂ L(X) is called an integrated semigroup if the following conditions are satisfied. (i) S(0) = 0. (ii) For every x ∈ X, t → S(t)x is a continuous function of t ≥ 0 with values in X. s (iii) S(t)S(s) = 0 (S(t + r ) − S(r ))dr for each t, s ≥ 0. S(·) is said to be nondegenerate if S(t)x = 0 for all t > 0 implies x = 0. The generator B of a nondegenerate integrated semigroup S(·) is defined as follows:  t S(u)ydu + t x for t ≥ 0. x ∈ D(B) and Bx = y if and only if S(t)x = 0

Proposition 2.5. Let B generate an integrated semigroup S(·). Then for all x ∈ X and t ≥ 0, we have  t  t S(s)xds ∈ D(B) and S(t)x = B S(s)xds + t x. 0

0

 t In [17], it is shown that A generates an integrated semigroup S(·) on X and S(t)x = 0 T0 (s)xds for x ∈ X 0 and t ≥ 0. Hence, by Proposition 2.3(i) we derive that  t S(t)x = T−1 (s)xds (2.1) 0

for all x ∈ X and t ≥ 0. (2.1) will be used later. 3. Local existence of solutions In this section, we discuss the local existence and the uniqueness of solutions. We prove them in an integrated form using a variation-of-constants formula in the sense of extrapolated semigroups.

2818

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

Definition 3.1. We say that a function u : (−∞, T ] → X is a classical solution of Eq. (VID1) on [0, T ] if u satisfies the following conditions (i) u ∈ C 1 ([0, T ], X) ∩ C([0, T ], D(A)). (ii) u satisfies (VID1) on [0, T ]. (iii) u(t) = ϕ(t) for −∞ < t ≤ 0. If x ∈ D(A), h ∈ W 1,1 (R+ , X) and Ax + h(0) ∈ X 0 , the following equation  du(t) = Au(t) + h(t), t ≥ 0, dt u(0) = x

(ACP)

has a unique solution that has been shown in [20]. Furthermore, the mild solution of Eq. (ACP) is given by  t u(t) = T0 (t)x + T−1 (t − s)h(s)ds 0

for t > 0. So, we give the following definition. Definition 3.2. We say that a continuous function u : (−∞, T ] → X (resp. u : (−∞, T ) → X) is a mild solution of Eq. (VID1) on [0, T ] (resp. [0, T )) if u satisfies the following condition  t  s ⎧ ⎪ ⎪ u(t) = T0 (t)ϕ(0) + T−1 (t − s) B(s, θ, u(θ ))dθ ds ⎪ ⎨ 0 0  t (IE) T−1 (t − s)F(s, u s )ds, + ⎪ ⎪ ⎪ 0 ⎩ u0 = ϕ on (−∞, T ] (resp. (−∞, T )). In all this paper, we assume that the phase space (P,  · P ) is a seminormed linear space consisting of functions from R− into X satisfying the following axioms introduced at first by Hale and Kato in [11]. (A1) There exist a positive constant H and functions K (·), M(·) : R+ → R+ , with K (·) continuous and M(·) locally bounded, such that for any σ ∈ R and a ≥ 0, if x : (−∞, σ + a) → X, x σ ∈ P and x(·) is continuous on [σ, σ + a], then for every t ∈ [σ, σ + a] the following conditions hold: (i) x t ∈ P, (ii) x(t) ≤ H x t P , (iii) x t P ≤ K (t − σ ) supσ ≤s≤t x(s) + M(t − σ )x σ P . (A2) For each function x(·) in (A1), t → x t is a P-value continuous function for t ∈ [σ, σ + a]. (B) The space P is complete. Note that the axiom (B) is equivalent to saying that the space of equivalence classes P/ · P is a Banach space. Let C00 be the set of continuous functions ψ : (−∞, 0] → X with compact support supp(ψ). Remark 3.3 ([15]). Any ψ ∈ C00 belongs to P.

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

2819

If u ∈ C([0, T ], X), we define u[0,a] := sup u(s) 0≤s≤a

for a ∈ [0, T ). (H1) B ∈ C(Δ(0, T ) × X, X). Let a ∈ [0, T ). There exists Ma1 dependent of a such that t 1 0 B(t, θ, u(θ )) − B(t, θ, v(θ ))dθ ≤ Ma u − v[0,t ] for u, v ∈ C([0, a], X) and t ∈ [0, a]. (H2) F ∈ C([0, T ] × P, X) and F satisfies the local Lipschitz condition, i.e. for each α > 0 and T > τ > 0 there is a constant L(τ, α) such that F(t, ψ1 ) − F(t, ψ2 ) ≤ L(τ, α)ψ1 − ψ2 P for ψ1 P , ψ2 P ≤ α and t ∈ [0, τ ]. Theorem 3.4. Suppose that (H1) and (H2) hold. If ϕ ∈ P and ϕ(0) ∈ X 0 , then there is a constant τ (ϕ) such that Eq. (VID1) has a unique mild solution u(·, ϕ) on [0, τ (ϕ)]. Proof. Set α := ϕP + 1. Define the function  T (t)ϕ(0), t ∈ [0, T ], y(t) = 0 ϕ(t), t < 0. By axioms (A1-i) and (A2), we derive that yt ∈ P and also the map t → yt is continuous. Then, for ε ∈ (0, 1), there is a δ ∈ (0, T ) such that yt − ϕP ≤ ε for t ∈ (0, δ). Let 0 < τ (ϕ) ≤ δ such that K τ (ϕ) (τ (ϕ)M1 Mτ1(ϕ) H α + τ (ϕ)M1 β + τ (ϕ)M1 L 1 ) < 1 − ε

(3.1)

where K τ (ϕ) := max0≤s≤τ  s(ϕ) K (s), L τ (ϕ) := max0≤s≤τ (ϕ) F(s, 0), L 1 := L(τ (ϕ), α)α + L τ (ϕ) , β := sups∈[0,τ (ϕ)] 0 B(s, θ, 0)dθ , Mτ1(ϕ) is the constant in (H1). By (H2), we know that F(t, ψ1 ) − F(t, ψ2 ) ≤ L(τ (ϕ), α)ψ1 − ψ2 P for t ∈ [0, τ (ϕ)] and ψ1 P , ψ2 P ≤ α. Furthermore, for t ∈ [0, τ (ϕ)], by (H1) and (H2) we can derive that  t  t  t B(t, θ, u(θ ))dθ ≤ B(t, θ, u(θ )) − B(t, θ, 0)dθ + B(t, θ, 0)dθ 0

0

0

≤ Mτ1(ϕ) u[0,t ] + β

(3.2)

for u ∈ C([0, τ (ϕ)], X) and F(t, ψ)P ≤ L(τ (ϕ), α)α + F(t, 0) ≤ L 1 for ψP ≤ α. Define the following set Z (τ (ϕ)) := {u : (−∞, τ (ϕ)] → X; u 0 ∈ P and u : [0, τ (ϕ)] → X is continuous} endowed with the seminorm  ·  Z defined by u Z = u 0 P +

sup

0≤s≤τ (ϕ)

u(s).

(3.3)

2820

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

From axiom (B), it follows that (Z (τ (ϕ)),  ·  Z ) is complete. Furthermore, we define Z ϕ (τ (ϕ)) := {u ∈ Z (τ (ϕ)); u 0 − ϕP = 0 and

sup

0≤s≤τ (ϕ)

u s − ϕP ≤ 1}.

In [3], the authors had shown that Z ϕ (τ (ϕ)) is a nonempty closed subset of Z (τ (ϕ)). Define the following mapping P : Z ϕ (τ (ϕ)) → Z ϕ (τ (ϕ)) by  t  s ⎧ ⎪ T (t)ϕ(0) + T (t − s) B(s, θ, u(θ ))dθ ds ⎪ 0 −1 ⎪ ⎨ 0 0  t (Pu)(t) = ⎪ + T−1 (t − s)F(s, u s )ds, t ∈ [0, τ (ϕ)], ⎪ ⎪ 0 ⎩ ϕ(t), t ∈ (−∞, 0]. We show P well-defined by showing Pu ∈ Z ϕ (τ (ϕ)) for every u ∈ Z ϕ (τ (ϕ)). Let u ∈ Z ϕ (τ (ϕ)). By (H2) and axiom (A2), we know that s → F(s, u s ) is a continuous function. It follows that v := Pu is continuous on [0, τ (ϕ)] and v belongs to Z (τ (ϕ)) from Proposition 2.3. Next, let w := v − y, we have vt − ϕP ≤ wt P + yt − ϕP ≤ wt P + ε for t ∈ [0, τ (ϕ)]. Since u ∈ Z ϕ (τ (ϕ)), we deduce that u s P ≤ α for s ∈ [0, τ (ϕ)]. Hence, F(s, u s ) ≤ L 1

(3.4)

for s ∈ [0, τ (ϕ)] from (3.3). We rewrite  t  s  t w(t) = T−1 (t − s) B(s, θ, u(θ ))dθ ds + T−1 (t − s)F(s, u s )ds 0

:= P1 (t) + P2 (t)

0

0

for t ∈ [0, τ (ϕ)]. For P1 , by Proposition 2.3, (H1), (A1-i) and (3.2), we have  t  s P1 (t) ≤ M1 B(s, θ, u(θ ))dθ ds 0

0

≤ τ (φ)(M1 Mτ1(ϕ) u[0,τ (ϕ)] + M1 β) ≤ τ (φ)(M1 Mτ1(ϕ) H α + M1 β) for t ∈ [0, τ (ϕ)]. For P2 , by Proposition 2.3 and (3.4), we have P2 (t) ≤ M1 L 1 τ (ϕ) for t ∈ [0, τ (ϕ)]. On the other hand, by wt P ≤ K τ (ϕ) w[0,τ (ϕ)] and (3.1), we have vt − ϕP ≤ wt P + ε ≤ K τ (ϕ) (P1 (t)[0,τ (ϕ)] + P2 (t)[0,τ (ϕ)] ) + ε < 1−ε+ε =1 for t ∈ [0, τ (ϕ)]. It implies v ∈ Z ϕ (τ (ϕ)).

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

2821

Next, we show that P is a strict contraction mapping. Suppose that u, v ∈ Z ϕ (τ (ϕ)) and t ∈ [0, τ (ϕ)], by (H1), (H2) and (3.1), we have  t  s T−1 (t − s) B(s, θ, u(θ )) − B(s, θ, v(θ ))dθ ds (Pu − Pv)(t) ≤ 0 0  t + T−1 (t − s)(F(s, u s ) − F(s, vs ))ds 0

≤ K τ (ϕ) (τ (φ)M1 Mτ1(ϕ) H + τ (φ)M1 L(τ (ϕ), α))u − v[0,τ (ϕ)] ≤ K τ (ϕ) (τ (φ)M1 Mτ1(ϕ) H α + τ (ϕ)M1 β + τ (φ)M1 L 1 ) × u − v Z < u − v Z .

So, it follows that P is a strict contraction mapping on Z ϕ (τ (ϕ)). We conclude that Eq. (VID1) has a mild solution u in Z ϕ (τ (ϕ)) by contraction mapping principle. Let x be another fixed point of P. Then u − x Z = 0 and u 0 = x 0 = ϕ. By the definition of seminorm of  ·  Z , we know that u(t) = x(t) for t ∈ [0, τ (ϕ)]. We derive that Eq. (VID1) has only one mild solution on [0, τ (ϕ)].  Theorem 3.5. Suppose that (H1) and (H2) hold for the case T = ∞. Assume that ϕ ∈ P, ϕ(0) ∈ X 0 and define T (ϕ) := sup{t > 0; Eq. (VID1) has a unique mild solution u(·) on [0, t)}. If T (ϕ) < ∞, then lim supt ↑T (ϕ) u(t) = ∞. Proof. Theorem 3.4 indicates T (ϕ) is well-defined. Let u(·) be the corresponding solution to Eq. (IE) on [0, T (ϕ)), and suppose that T (ϕ) < ∞ and lim supt ↑T (ϕ) u(t) < ∞. According to T (ϕ) < ∞, lim supt ↑T (ϕ) u(t) < ∞ and axiom (A1-iii), we know that there is a γ > 0 such that u s P ≤ γ for all s ∈ [0, T (ϕ)). For each 0 < t < T (ϕ) and 0 < η < 1, we set Z u t (t + η) = {x ∈ C((−∞, t + η], X); x|(−∞,t ] = u|(−∞,t ] }. Define Pt +η : Z u t (t + η) → C((−∞, t + η], X) by  s  r ⎧ ⎪ T−1 (s − r ) B(r, θ, x(θ ))dθ dr ⎪T0 (s − t)x(t) + ⎪ ⎨ t 0  s (Pt +η x)(s) = T−1 (s − r )F(r, xr )dr, s ∈ [t, t + η], + ⎪ ⎪ ⎪ t ⎩ xt = u t . We put α := γ + 1, L T (ϕ) := max0≤s≤T (ϕ) F(s, 0), L 1 = L(T (ϕ), α)α + L T (ϕ) and β = s sup0≤s≤T (ϕ) 0 B(s, θ, 0)dθ . Using a similar argument as the one in the proof of Theorem 3.4, we can deduce that there is a 0 < τ < 1, which is independent of t ∈ [0, T (ϕ)) such that Pt +τ has a unique fixed point x(·) in Z u t (t + τ ) and x(·) is the unique mild solution of Eq. (VID1) on [0, t + τ ]. Taking t such that 0 < T (ϕ) − t < τ , then t + τ > T (ϕ). This is in contradiction with the definition of T (ϕ). So, it follows that T (ϕ) < ∞, then lim supt ↑T (ϕ) u(t) = ∞.  Corollary 3.6. Suppose the assumptions in Theorem 3.5 hold. In addition, assume that t (1) B ∈ C(Δ(0, ∞) × X, X) satisfies (H1) and  0 B(t, θ, x(θ ))dθ  ≤ h 1 (t)x[0,t ] + h 2 (t), where h 1 and h 2 are locally integrable functions from R+ into R+ for each x ∈ C([0, t], X) and t > 0.

2822

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

(2) F ∈ C([0, ∞) × P, X) satisfies (H2) and t ∈ R+ , ψ ∈ P,

F(t, ψ) ≤ h 3 (t)ψP + h 4 (t),

where h i (i = 3, 4) are locally integrable functions from R+ into R+ . Then T (ϕ) = ∞. Proof. Let u be the unique solution of Eq. (IE) on [0, T (ϕ)). By Assumptions (1) and (2), we obtain that

  t t u(t) ≤ Mϕ(0) + M1 (h 1 (s)u[0,s] + h 2 (s))ds + M1 h 3 (s)K (s)u[0,s] ds 0



t

+ 0

 h 3 (s)( sup M(τ ))ϕP ds + 0≤τ ≤t

t

0

 h 4 (s)ds

0

for any t ∈ [0, T (ϕ)). Therefore, for each 0 < T0 < T (ϕ) and t ∈ [0, T0 ),  t  t h 1 (s)u[0,s] ds + M1 h 2 (s)ds u[0,t ] ≤ Mϕ(0) + M1 0

+ M1 sup M(τ )ϕP 0≤τ ≤T0



t

+ M1 0

0



T0

h 3 (s)ds

0



h 3 (s)K (s)u[0,s] ds + M1

T0

h 4 (s)ds.

0

Hence, by Gronwall’s inequality, for any 0 < T0 < T (ϕ) and t ∈ [0, T0 ),

 T0  h 2 (s)ds + M1 sup M(τ )ϕP u[0,t ] ≤ Mϕ(0) + M1 0



T0

+ M1

 h 4 (s)ds × e

T0

0 0≤τ ≤T0 t t M1 ( sup K (s)( 0 h 3 (r)dr+ 0 h 1 (r)dr))

h 3 (s)ds

.

0≤s≤T0

0

It implies that T (ϕ) = ∞ by arbitrariness of T0 and Theorem 3.5.



Theorem 3.7. Suppose the assumptions in Theorem 3.5 hold. If u(·, ϕ) is the corresponding mild solution to Eq. (VID1) with maximal interval of existence [0, T (ϕ)) and t ∈ [0, T (ϕ)), then there exist positive constants C and δ such that for ψ ∈ P, ψ(0) ∈ X 0 and ϕ − ψP ≤ δ, we have u(s, ϕ) − u(s, ψ) ≤ Cϕ − ψP ,

for s ∈ [0, t].

Proof. Let t ∈ [0, T (ϕ)). We put α := 1 + sup0≤s≤t u s (·, ϕ)P and c(t) := (Mt + K t M H )e K t (M1 Mt

1 H +M

1 L(t,α))t

where K t = sup0≤λ≤t K (λ) and Mt = sup0≤λ≤t M(λ). Let δ ∈ (0, 1) be such that c(t)δ < 1 and let Bδ := {ψ ∈ P; ψ(0) ∈ X 0 and ϕ − ψP ≤ δ}. It will be shown that u(s, ϕ) − u(s, ψ) ≤ Cϕ − ψP for s ∈ [0, t] and ψ ∈ Bδ but first we prove the following.

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

2823

Claim. Let ψ ∈ Bδ and let T0 = sup{s > 0; u τ (·, ψ)P ≤ α for τ ∈ [0, s]}. Then t < T0 . Suppose the claim is false, i.e. T0 ≤ t. Let τ ≤ T0 and let c(T0 ) := (MT0 + K T0 M H )e

K T0 (M1 MT1 H +M1 L(t,α))t 0

where K T0 = sup0≤λ≤T0 K (λ) and MT0 = sup0≤λ≤T0 M(λ). By axioms (A1), (A2) and hypothesis (H1), we have u τ (·, ϕ) − u τ (·, ψ)P ≤ MT0 ϕ − ψP + K T0 sup u(ξ, ϕ) − u(ξ, ψ) 0≤ξ ≤τ

≤ MT0 ϕ − ψP + K T0 M H ϕ − ψP  τ 1 + K T0 M1 MT0 H sup u ξ (·, ϕ) − u ξ (·, ψ)P dr 0 0≤ξ ≤r τ

 + K T0 M1 L(t, α)

sup u ξ (·, ϕ) − u ξ (·, ψ)P dr.

0 0≤ξ ≤r

By Gronwall’s inequality, we deduce that u τ (·, ϕ) − u τ (·, ψ)P ≤ c(T0 )ϕ − ψP .

(3.5)

It is easy to see that c(t) ≥ c(T0 ). So, we obtain that u s (·, ψ)P ≤ c(t)δ + α − 1 < α for s ∈ [0, T0 ]. By the continuity of u · (·, ψ), it implies that T0 cannot be the largest number s > 0 such that u τ (·, ψ) ≤ α for each τ ∈ [0, s]. So, it follows that T0 > t. We complete the proof of the claim. Finally, we know that u s (·, ψ)P ≤ α for s ∈ [0, t] and ψ ∈ Bδ from the claim. By a similar argument to inequality (3.5) and axiom (A1-i), we deduce the continuous dependence with the initial data and u(s, ϕ) − u(s, ψ) ≤ H u s (·, ϕ) − u s (·, ψ)P ≤ Cϕ − ψP for s ∈ [0, t], C := H c(t) and ψ ∈ Bδ .



Next, we want to give a sufficient condition for the existence of a classical solution to Eq. (VID1). To do this, we need the differentiability of mild solutions. We give the following more restrictive conditions. (C) If (φn ) is a Cauchy sequence in P and if (φn ) converges compactly to φ on (−∞, 0], then φ ∈ P and φn − φP → 0, as n → ∞. (D) For a sequence (φn ) in P, if φn P → 0 as n → ∞, then φn (θ ) → 0, as n → ∞, for each θ ∈ (−∞, 0]. (H3) F is continuously differentiable and the derivatives D1 F, D2 F satisfy the following Lipschitz conditions: there is a constant L  (τ, α) > 0 such that D1 F(t, ψ1 ) − D1 F(t, ψ2 ) ≤ L  (τ, α)ψ1 − ψ2 P , D2 F(t, ψ1 ) − D2 F(t, ψ2 ) ≤ L  (τ, α)ψ1 − ψ2 P ,

for τ ∈ [0, T ), t ∈ [0, τ ), α > 0 and ψ1 P , ψ2 P ≤ α.

2824

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

Lemma 3.8 ([15]). Let P satisfy axiom (C) and let f : [0, a] → P, a > 0, be a continuous function such that f (t)(θ ) is continuous for (t, θ ) ∈ [0, a] × (−∞, 0]. Then  a  a f (t)dt (θ ) = f (t)(θ )dt 0

0

for θ ∈ (−∞, 0]. Lemma 3.9 ([5]). Let P satisfy axiom (D) and let f : [0, a] → P, a > 0, be a continuous function. Then for all θ ∈ (−∞, 0], the function f (·)(θ ) is continuous and  a  a f (t)dt (θ ) = f (t)(θ )dt 0

0

for θ ∈ (−∞, 0]. Theorem 3.10. Let P satisfy axiom (C) or (D). Assume that (H1), (H2) and (H3) hold. In addition, assume that D1 B(·, ·, ·) ∈ C(Δ(0, T ) × X, X), ϕ ∈ P is continuously differentiable with ϕ  ∈ P, ϕ(0) ∈ D(A) and ϕ  (0) = Aϕ(0) + F(0, ϕ) ∈ X 0 . If u(·, ϕ) is a mild solution of Eq. (VID1) on [0, T (ϕ)), then u is continuously differentiable on [0, T (ϕ)). Furthermore, u(·, ϕ) is a classical solution of Eq. (VID1) on [0, T (ϕ)). Proof. Let a ∈ [0, T (ϕ)). Consider the equation  t ⎧ ⎪ ⎪ y(t) = T (t)(Aϕ(0) + F(0, ϕ)) + T−1 (t − s)B(t, s, u(s))ds 0 ⎪ ⎪ ⎪ 0   ⎪ t s ⎪ ⎪ ⎨ T−1 (t − s) D1 B(s, θ, u(θ ))dθ ds + 0 0  t ⎪ ⎪ ⎪ ⎪ + T−1 (t − s)(D1 F(s, u s ) + D2 F(s, u s )ys )ds, t ∈ [0, a], ⎪ ⎪ ⎪ 0 ⎪ ⎩ y(t) = ϕ  (t), t ∈ (−∞, 0],

(3.6)

where u(s) = u(s, ϕ). A similar argument as Theorem 19 in [2], it can be shown that Eq. (3.6) has a unique solution y(·, ϕ  ). Define the function z by ⎧  t ⎨ y(s)ds, t ∈ [0, a]. ϕ(0) + z(t) = 0 ⎩ ϕ(t), t ∈ (−∞, 0]. We shall show u = z. Recall the integrated semigroup S(·) generated by A. By (2.1) and Fubini’s Theorem, we obtain  t s T−1 (s − r )(D1 F(r, u r ) + D2 F(r, u r )yr )dr ds 0 0  t t T−1 (s − r )(D1 F(r, u r ) + D2 F(r, u r )yr )dsdr = 0 r  t = S(t − s)(D1 F(s, u s ) + D2 F(s, u s )ys )ds, t ∈ [0, a]. (3.7) 0

Moreover, by the elementary properties of S(·), ϕ  ∈ P, ϕ(0) ∈ D(A) and ϕ  (0) = Aϕ(0) + F(0, ϕ) ∈ X 0 , we know that S(t)ϕ  (0) = T0 (t)ϕ(0) − ϕ(0) + S(t)F(0, ϕ)

for t ∈ [0, a].

(3.8)

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

Therefore, z becomes



t



2825

s

z(t) = T0 (t)ϕ(0) + S(t)F(0, ϕ) + T−1 (t − s) B(s, θ, u(θ ))dθ ds 0 0  t S(t − s)(D1 F(s, u s ) + D2 F(s, u s )ys )ds, t ∈ [0, a] +

(3.9)

0

by (2.1) and (3.6)–(3.8). By Lemma 3.8 or Lemma 3.9, we obtain  t ys ds for t ∈ [0, a]. zt = ϕ +

(3.10)

0

Using integral by parts, we have  t T−1 (t − s)F(s, z s )ds 0  t S(t − s)(D1 F(s, z s ) + D2 F(s, z s )ys )ds = S(t)F(0, ϕ) +

for t ∈ [0, a].

0

So, we deduce that



S(t)F(0, ϕ) = − 0 t

 + 0

t

 S(t − s)(D1 F(s, z s ) + D2 F(s, z s )ys )ds

T−1 (t − s)F(s, z s )ds

for t ∈ [0, a].

Consequently, by (3.9) and (3.11), z satisfies  t  s T−1 (t − s) B(s, θ, u(θ ))dθ ds z(t) = T0 (t)ϕ(0) + 0 0  t S(t − s)(D1 F(s, u s ) + D2 F(s, u s )ys )ds + 0  t  − S(t − s)(D1 F(s, z s ) + D2 F(s, z s )ys )ds 0  t + T−1 (t − s)F(s, z s )ds for t ∈ [0, a]. 0

Therefore, for α := max(sup0≤t ≤a u s P , sup0≤t ≤a z s P ), we have  t u(t) − z(t) ≤ T−1 (t − s)(F(s.u s ) − F(s, z s ))ds 0  t S(t − s)(D1 F(s, u s ) − D1 F(s, z s ))ds + 0 t S(t − s)(D2 F(s, u s )ys − D2 F(s, z s )ys )ds + 0  t u s − z s P ds ≤ M1 L(a, α) 0  t   t u s − z s P ds + u s − z s P ys P ds + M1 a L  (a, α) 0

0

(3.11)

2826

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

≤ K a (M1 L(a, α) + M1 a L  (a, α) + M1 a L  (a, α) × max ys P ) 0≤s≤a  t sup u(ζ ) − z(ζ )ds. × 0 0≤ζ ≤s

By Gronwall’s Lemma, we get u = z. So, we derive that u is continuously differentiable on [0, a]. By the arbitrary nature of a, we complete the proof.  Next, we consider the solutions of Eqs. (VID2) and (VID3). Definition 3.11. We say that a function u : (−∞, T ] → X is a classical solution of Eq. (VID2) on [0, T ] if u satisfies the following conditions t (i) u(t) + 0 B(t, θ, u(θ ))dθ ∈ D(A) for t ∈ [0, T ]. (ii) u ∈ C 1 ([0, T ], X). (iii) u satisfies Eq. (VID2) on [0, T ] and u(t) = ϕ(t) for −∞ < t ≤ 0. Lemma 3.12. Suppose that x 1 and x 2 are two vectors of X, then there is a function ψ ∈ P such that ψ is continuously differentiable with ψ(0) = x 1 and ψ  (0) = x 2 . Proof. Let Cc∞ (R) := { f : R → R; f is infintely differentiable with compact support}. By Urysohn’s Lemma, there is a function h ∈ Cc∞ such that h(t) = 1 for t ∈ [−1, 0]. It follows that the function ψ defined by ψ(t) = h(t −0.5)x 1 +th(t −0.5)x 2 for t ∈ (−∞, 0] is the desired function from Remark 3.3.  Theorem 3.13. Let P satisfy axiom (C) or (D). Assume that (1) B(·, ·, ·), D1 B(·, ·, ·), D12 B(·, ·, ·) ∈ C(Δ(0, T )× X, X). D1 B(·, ·, ·) satisfies the hypothesis (H1) and F satisfies the hypothesis (H2) and (H3). Moreover, H : [0, T ] × P → X defined by H (t, ψ) = B(t, t, ψ(0)) also satisfies the hypotheses (H2) and (H3). (2) ϕ ∈ P is continuously differentiable with ϕ  ∈ P, ϕ(0) ∈ D(A), ϕ  (0) = Aϕ(0) + F(0, ϕ) ∈ X 0 and Aϕ(0) + B(0, 0, ϕ(0)) + F(0, ϕ) ∈ X 0 . Then there exists a T (ϕ) ∈ R+ such that Eq. (VID2) has a unique classical solution on [0, T (ϕ)).   Proof. Let A = 00 AA . Grimmer and Liu in [9] show that A is a Hille–Yosida operator on

X × X. From Lemma 3.12, it follows that there is a ϕ2 ∈ P such that ϕ2 ∈ P, ϕ2 (0) = ϕ(0) and ϕ2 (0) = Aϕ(0) + B(0, 0, ϕ(0)) + F(0, ϕ). Let ϕ = ϕϕ12 where ϕ1 = ϕ. Therefore, ϕ ∈ P × P. We consider the following equation in the Banach space X × X and phase space P × P ⎧  t d ⎪ ⎪  θ, w(θ  w  = Aw(t)  + B(t,  ))dθ + F(t,  t ), 0 ≤ t ≤ T, ⎨ w(t) dt 0  (3.12) ϕ ⎪ ⎪ 0 = 1 , ⎩w ϕ2 where



u(·) , w(·)   ·, w(·)) B(·,  :=

w(·)  :=

0 D1 B(·, ·, u(·))



J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

and  w F(·,  · ) :=



2827

F(·, u · ) . H (·, u ·) + F(·, u · )

We show that Eq. (3.12) satisfies the hypothesis of Theorem 3.10. By assumption (1), we know  ·, ·) satisfies hypothesis (H1). Moreover, it is easy to see that F satisfies the hypotheses that B(·, (H2) and (H3) from assumption (1). Finally,      0 A ϕ1 (0) F(0, ϕ) Aϕ(0) + F(0, ϕ) + = 0 A ϕ2 (0) H (0, ϕ) + F(0, ϕ) Aϕ(0) + B(0, 0, ϕ(0)) + F(0, ϕ)   ϕ (0) ∈ X 0 × X 0. = 1 ϕ2 (0)  and w  ϕ) ∈ X 0 × X 0 . Hence, there exists So, w  0 ∈ P × P, w  0 (0) ∈ D( A)  0 (0) = Aϕ(0)  + F(0, + a T (ϕ) ∈ R such that Eq. (3.12) has a unique classical solution on [0, T (ϕ)) by Theorem 3.10. On the other hand, we rewrite Eq. (3.12) in the following component form u  (t) = Aw(t) + F(t, u t ),  t  w (t) = Aw(t) + D1 B(t, θ, u(θ ))dθ + B(t, t, u(t)) + F(t, u t ) 0 d t = u  (t) + B(t, θ, u(θ ))dθ. dt 0 Since w(0) = ϕ1 (0) = ϕ(0) = ϕ2 (0) = u(0), it follows that w(t) = u(t) + So, u is the unique classical solution Eq. (VID2) on [0, T (ϕ)). 

t 0

B(t, θ, u(θ ))dθ .

Definition 3.14. We say that a function u : (−∞, T ] → X is a classical solution of Eq. (VID3) on [0, T ] if u satisfies the following conditions (i) u ∈ C 1 ([0, T ], X) ∩ C([0, T ], D(A)). (ii) u satisfies Eq. (VID3) on [0, T ]. (iii) u(t) = ϕ(t) for −∞ < t ≤ 0. Definition 3.15. We say that a function u : (−∞, T ] → X is a classical solution of Eq. (VID4) on [0, T ] if u satisfies the following conditions t (i) u(t) + −∞ B(t, θ, u(θ ))dθ ∈ D(A) for t ∈ [0, T ]. (ii) u ∈ C 1 ([0, T ], X). (iii) u satisfies Eq. (VID4) on [0, T ] and u(t) = ϕ(t) for −∞ < t ≤ 0. t 0 t Since −∞ B(t, θ, u(θ ))dθ = −∞ B(t, θ, u(θ ))dθ + 0 B(t, θ, u(θ ))dθ , we can rewrite Eq. (VID3) as ⎧  t  0 ⎨ du(t) = Au(t) + B(t, θ, u(θ ))dθ + F(t, u t ) + B(t, θ, ϕ(θ ))dθ, t ∈ [0, T ], 0 −∞ ⎩ dt u 0 = ϕ ∈ P. Theorem 3.16. Let P satisfy axiom (C) or (D). Assume that 1. B(·, ·, ·) ∈ C(Δ(−∞, T ) × X, X) satisfies the hypothesis (H1) and D1 B(·, ·, ·) ∈ C(Δ(0, T ) × X, X). F satisfies the condition (H2) and (H3).

2828

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

2. ϕ ∈ P is continuously differentiable with ϕ  ∈ P, D(A), and ϕ  (0)

0

B(·, θ, ϕ(θ ))dθ ∈ X is continuously 0 = Aϕ(0)+F(0, ϕ)+ −∞ B(0, θ, ϕ(θ ))dθ ∈ −∞

differentiable on [0, T ], ϕ(0) ∈ X 0. Then there exists a T (ϕ) ∈ R+ such that Eq. (VID3) has a unique classical solution on [0, T (ϕ)).

Proof. It is obvious from Theorem 3.10 if we replace F(t, u t ) in Theorem 3.10 with F(t, u t ) + 0  −∞ B(t, θ, ϕ(θ ))dθ . The following is the existence and uniqueness theorem for Eq. (VID4). Theorem 3.17. Let P satisfy axiom (C) or (D). Assume that (1) B(·, ·, ·), D1 B(·, ·, ·) ∈ C(Δ(−∞, T ) × X, X). D12 B(·, ·, ·) ∈ C(Δ(0, T ) × X, X) and D1 B(·, ·, ·) satisfies the hypothesis (H1). Moreover, F and H : [0, T ] × P → X defined by H (t, ψ) = B(t, t, ψ(0)) satisfy the hypotheses (H2) and(H3). 0 (2) ϕ ∈ P is continuously differentiable with ϕ  ∈ P and −∞ B(t, θ, ϕ(θ ))dθ ∈ X for each t ∈ [0, T ]. 0 (3) G ∈ C 2 ([0, T ], X) where G is defined by G(t) := −∞ B(t, θ, ϕ(θ ))dθ . 0 (4) ϕ(0) + −∞ B(0, θ, ϕ(θ ))dθ ∈ D(A),    0  ϕ (0) = A ϕ(0) + B(0, θ, ϕ(θ ))dθ + F(0, ϕ) ∈ X 0 −∞

and





A ϕ(0) +

0 −∞

 B(0, θ, ϕ(θ ))dθ + B(0, 0, ϕ(0)) + F(0, ϕ) + G  (0) ∈ X 0 .

Then there exists a T (ϕ) ∈ R+ such that Eq. (VID4) has a unique classical solution on [0, T (ϕ)).   Proof. Let A = 00 AA . According to Lemma 3.12, we choose ϕ2 ∈ P such that ϕ2 ∈ P, 0 0 ϕ2 (0) = ϕ(0) + −∞ B(0, θ, ϕ(θ ))dθ and ϕ2 (0) = A [ ϕ(0) + −∞ B(0, θ, ϕ(θ ))dθ ] +   B(0, 0, ϕ(0)) + F(0, ϕ) + G  (0). Let ϕ = ϕϕ12 where ϕ1 = ϕ. Therefore, ϕ ∈ P × P. We consider the following equation in the Banach space X × X and phase space P × P ⎧  t d ⎪ ⎪  θ, w(θ  w  = Aw(t)  + B(t,  ))dθ + F(t,  t ), 0 ≤ t ≤ T, ⎨ w(t) dt 0  ϕ ⎪ ⎪ 0 = 1 , ⎩w ϕ2 where



u(·) w(·)  := , w(·)   B(·, ·, w(·))  :=

and  w F(·,  · ) :=



0 D1 B(·, ·, u(·))



F(·, u · ) . H (·, u · ) + F(·, u · ) + G  (·)

The rest of the proof is similar to Theorem 3.13.



J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

2829

Remark 3.18. Eqs. (VID2), (VID3) and (VID4) also have similar results as Theorem 3.5, 3.6, 3.7. The details are omitted. 4. Examples In this section, we apply our abstract results to treat the following partial differential equation with infinite delay  0 ∂2 ∂ ω(t, ξ ) = a 2 ω(t, ξ ) + E(t, θ, ω(t + θ, ξ ))dθ ∂t ∂ξ −∞ + h(t, ω(t, ξ ), ω(t − τ, ξ )), ω(t, 0) = w(t, π) = 0, ω(θ, ξ ) = ω0 (θ, ξ ),

0 ≤ t ≤ T, 0 ≤ ξ ≤ π,

(PDED)

t ∈ [0, T ],

−∞ < θ ≤ 0, 0 ≤ ξ ≤ π,

where a, τ are positive constants, E is a function from [0, T ] × R− × R into R and h is a function from [0, T ] × R × R into R. The special cases, E(t, θ, ω(t +θ, ξ )) = cG(θ )ω(t +θ, ξ ), G is a positive integrable function on (−∞, 0], c is a positive constant, h(t, ω(t, ξ ), ω(t − τ, ξ )) = bω(t, ξ ) + f (ω(t − τ, ξ )), b is a positive constant and f : R → R is a local Lipschitz continuous function, had been considered in [3]. In this paper, the authors choose X = C([0, π], R) and the Hille–Yosida operator  D(A) = {y ∈ C 2 ([0, π], R); y(0) = y(π) = 0}, Ay = ay . Noting that A is not densely defined and X 0 = {y ∈ C([0, π], R); y(0) = y(π) = 0}. Let γ > 0. The phase space in [3] is P := {φ ∈ C((−∞, 0], X); lim eγ θ φ(θ ) exists in X} θ→−∞

with ψP := supθ≤0 eγ θ ψ(θ ) for ψ ∈ P. Furthermore, P also satisfies the axiom (D). The partial differential equation is transformed into  du(t) = Au(t) + F(u t ), 0 ≤ t ≤ T, (4.1) dt u0 = ϕ ∈ P by setting u(t)(ξ ) = ω(t, ξ ),

0 ≤ t ≤ T, 0 ≤ ξ ≤ π,  0 F(φ)(ξ ) = bφ(0)(ξ ) + f (φ(−τ )(ξ )) + c G(θ )φ(θ )(ξ )dθ,

ϕ(θ )(ξ ) = ω0 (θ, ξ ),

−∞

ξ ∈ [0, π], φ ∈ P,

−∞ < θ ≤ 0, 0 ≤ ξ ≤ π.

Under some certain conditions, they obtain the mild solution and classical solution to this delay equation. In our situations, we also choose the same Banach space, Hille–Yosida operator and phase space as [3]. We suppose equation (PDED) satisfies the following conditions

2830

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

(i) B ∈ C(Δ(−∞, T ) × X, X), where B(t, θ, u)(ξ ) = E(t, θ − t, u(θ )(ξ )),

(t, θ ) ∈ Δ(−∞, T ), u ∈ X, ξ ∈ [0, π]

and B satisfies (H1). (ii) ω0 ∈ C((−∞, 0] × [0, π], R), with limθ→−∞ (eγ θ sup0≤ξ ≤π |w0 (θ, ξ )|) exists, ω0 (0, 0) = 0 ω0 (0, π) = 0 and −∞ B(t, θ, ϕ)dθ exists for t ∈ [0, T ]. (iii) Let β > 0 and let α > 0. h ∈ C([0, T ] × R × R, R) and there is a positive constant L(β, α) > 0 such that |h(t, x 1 , y1 ) − h(t, x 2 , y2 )| ≤ L(β, α)(|x 1 − x 2 | + |y1 − y2 |) for t ∈ [0, β] and x 1 , x 2 , y1 , y2 ∈ R with |x 1 |, |x 2 |, |y1 |, |y2 | ≤ α. We transform equation (PDED) into ⎧  t ⎨ du(t) = Au(t) + B(t, θ, u(θ ))dθ + F(t, u t ), dt 0 ⎩ u0 = ϕ ∈ P

0 ≤ t ≤ T,

(4.2)

by setting u(t)(ξ ) = ω(t, ξ ),

t ∈ [0, T ], 0 ≤ ξ ≤ π,

B(t, θ, u(θ ))(ξ ) = E(t, θ − t, u(θ )(ξ )),

(t, θ ) ∈ Δ(−∞, T ), ξ ∈ [0, π],

ϕ(θ )(ξ ) = ω0 (θ, ξ ), −∞ < θ ≤ 0, 0 ≤ ξ ≤ π, F(t, φ)(ξ ) = h(t, φ(0)(ξ ), φ(−τ )(ξ ))  0 + B(t, θ, ϕ(θ ))(ξ )dθ, ξ ∈ [0, π], t ∈ [0, T ], φ ∈ P. −∞

From assumption (iii) and definition of P, we know that F(t, ψ1 ) − F(t, ψ2 ) = sup |h(t, ψ1 (0)(ξ ), ψ1 (−τ )(ξ )) − h(t, ψ2 (0)(ξ ), ψ2 (−τ )(ξ ))| ξ ∈[0,π] γτ

≤ 2e

(4.3)

L(β, eγ τ α)ψ1 − ψ2 P

for t ∈ [0, β] and ψ1 P , ψ2 P ≤ α. We conclude that the assumptions (i), (ii) and (iii) imply that the assumptions in Theorem 3.4 hold. So, it follows that there is a T (ϕ) ∈ R+ such that Eq. (4.2) has an unique mild solution on [0, T (ϕ)). Remark 4.1. In [3], the authors assume that E(t, θ, u(θ, ξ )) = G(θ )u(θ, ξ ) and G(·)eγ · is integrable on (−∞, 0]. It is easy to see that E satisfies assumption (i) if G is continuous. Moreover, we can remove the integrality of G(·)eγ · on (−∞, 0] when applying our method 0 to this situation. We only need to assume the continuity of G and the existence of −∞ G(θ − t)ϕ(θ )dθ for each t ∈ [0, T ]. Moreover, if Eq. (4.2) also satisfies (iv) ω0

∂ C 2 ((−∞, 0] × [0, π], R), with limθ→−∞ (eγ θ sup0≤ξ ≤π | ∂θ w0 (θ, ξ )|) exists, ∂ = ∂θ ω0 (0, π) = 0 and  0 ∂ ∂2 ω0 (0, ξ ) = a 2 ω0 (0, ξ ) + h(0, ω0 (0, ξ ), ω0 (−τ, ξ )) + E(0, θ, ω0 (θ, ξ ))dθ ∂θ ∂ξ −∞

∈

∂ ∂θ ω0 (0, 0)

for ξ ∈ [0, π].

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

(v)

2831

0

−∞ B(·, θ, ϕ(θ ))dθ is continuously differentiable on [0, T ] and D1 B ∈ C(Δ(0, T ) × X, X), D1 h, D2 h, D3 h exist and are continuous on [0, T ] × R × R. Moreover, D1 h, D2 h and D3 h satisfy the following local Lipschitz condition: there is a constant L  (β, α) > 0 such that

|D1 h(t, x 1 , y1 ) − D1 h(t, x 2 , y2 )| + |D2 h(t, x 1 , y1 ) − D2 h(t, x 2 , y2 )| + |D3 h(t, x 1 , y1 ) − D3 h(t, x 2 , y2 )| ≤ L  (β, α)(|x 1 − x 2 | + |y1 − y2 |) for t ∈ [0, β] and x 1 , x 2 , y1 , y2 ∈ R with |x 1 |, |x 2 |, |y1 |, |y2 | ≤ α. From assumption (v), we know that both D1 F and D2 F exist on [0, T ] × P. Moreover, it is also easy to see that  d 0 B(t, θ, ϕ(θ ))dθ (ξ ) + D1 h(t, φ(0)(ξ ), φ(−τ )(ξ )) D1 F(t, φ)(ξ ) = dt −∞ and D2 (F(t, φ))(ψ)(ξ ) = D2 h(t, φ(0)(ξ ), φ(−τ )(ξ ))ψ(0)(ξ ) + D3 h(t, φ(0)(ξ ), φ(−τ )(ξ ))ψ(−τ )(ξ ) for φ, ψ ∈ P. By assumption (v) and a similar computation for (4.3), we know that F satisfies hypothesis (H3). It follows that Eq. (4.2) has a unique classical solution on [0, T (ϕ)) by Theorem 3.10. References [1] M. Adimy, K. Ezzinbi, Local existence and linearized stability for partial function differential equations, Dyn. Systems Appl. 7 (1998) 389–404. [2] M. Adimy, H. Bouzahir, K. Ezzinbi, Existence for a class of partial functional differential equations with infinite delay, Nonlinear Anal. 46 (2001) 91–112. [3] M. Adimy, H. Bouzahir, K. Ezzinbi, Local existence and stability for some partial functional differential equations with infinite delay, Nonlinear Anal. 48 (2002) 323–348. [4] M. Adimy, K. Ezzinbi, A class of linear partial neutral functional differential equations with non-dense domain, J. Differential Equations 147 (1998) 285–332. [5] K. Ezzinbi, Existence and stability for some partial functional differential equations, Electron. J. Differential Equations 116 (2003) 1–13. [6] J.-C. Chang, On the Volterra integrodifferential equations and applications, Semigroup Forum 66 (2003) 68–80. [7] W. Desch, R. Grimmer, W. Schappacher, Wellposedness and wave propagation for a class of integrodifferential equations in Banach space, J. Differential Equations 74 (1988) 391–441. [8] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, 2000. [9] R. Grimmer, J.H. Liu, Integrated semigroups and integrodifferential equations, Semigroup Forum 48 (1994) 79–95. [10] J.A. Goldstein, Semigroups of Linear Operators and Applications, in: Oxford Mathematical Monographs, Oxford University Press, 1985. [11] J.K. Hale, J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978) 11–41. [12] H.R. Henriquez, Regularity of solutions of abstract retarded functional differential equations with unbounded delay, Nonlinear Anal. 28 (3) (1997) 513–531. [13] H.R. Henriquez, Periodic solutions of quasi-linear partial functional differential equations with unbounded delay, Funkcial. Ekvac. 37 (2) (1994) 329–343. [14] H.R. Henriquez, Approximation of abstract functional differential equations with unbounded delay Indian, J. Pure Appl. Math. 27 (4) (1996) 357–368. [15] Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, vol. 1473, Springer, Berlin, 1991. [16] F. Kappel, W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations 37 (1980) 141–183.

2832

J.-C. Chang / Nonlinear Analysis 66 (2007) 2814–2832

[17] H. Kellermann, M. Hieber, Integrated semigroups, J. Funct. Anal. 15 (1989) 160–180. [18] J. Liang, T.J. Xiao, A note on semilinear abstract functional differential and integrodifferential equations with infinite delay, Appl. Math. Lett. 17 (2004) 473–477. [19] R.K. Miller, Volterra integral equations in Banach space, Funkcial. Ekvac. 18 (1975) 163–194. [20] R. Nagel, E. Sinestrari, Inhomogenous Volterra integrodifferential equations for Hille–Yosida operators, Lecture Notes in Pure and Appl. Math., Dekker 150 (1994) 51–70. [21] K. Schumacher, Existence and continuous dependence for differential equations with unbounded delay, Arch. Ratio. Mech. Anal. 64 (1978) 313–335. [22] C.C. Travis, G.F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974) 395–418. [23] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, 1996.