Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces

Nonlinear Analysis 70 (2009) 2761–2771

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Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces Khalil Ezzinbi a,∗ , Hamidou Toure b , Issa Zabsonre b a Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, B.P. 2390 Marrakech, Morocco b Université de Ouagadougou, Département de Mathématiques, Unité de Recherche et de Formation en Sciences Exactes et Appliquées,

B.P.7021 Ouagadougou 03, Burkina Faso

article

info

Article history: Received 25 February 2008 Accepted 1 April 2008 Keywords: Resolvent operator Mild and strict solutions Partial functional integrodifferential equations c0 -semigroup

a b s t r a c t In this work, we study the existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces. We suppose that the undelayed part admits a resolvent operator in the sense given by Grimmer in [R. Grimmer, Resolvent operators for integral equations in a Banach space, Transaction of American Mathematical Society 273 (1982) 333–349]. The delayed part is assumed to be locally Lipschitz. Firstly, we show the existence of the mild solutions. Secondly, we give sufficient conditions ensuring the existence of the strict solutions. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction In this work, we study the existence and regularity of solutions for the following partial functional integrodifferential equation  Z t  0 u (t) = Au(t) + B(t − s)u(s)ds + f (t, ut ) for t ≥ 0 (1.1) 0  u0 = ϕ ∈ C = C ([−r, 0]; X ), where A : D(A) → X is the infinitesimal generator of a c0 -semigroup on a Banach space X , for t ≥ 0, B(t) is a closed linear operator with domain D(B) ⊃ D(A), C denotes the space of continuous functions from [−r, 0] to X endowed with the uniform norm topology, for every t ≥ 0, ut denotes the history function of C defined by ut (θ) = u(t + θ)

for − r ≤ θ ≤ 0,

+

f : R × C → X is a continuous function. In the case where B = 0, Eq. (1.1) has been studied by several authors. It is well known by the Hille–Yosida theorem that A is the infinitesimal generator of a c0 -semigroup of bounded linear operators in X

if and only if (i) D(A) = X , (ii) there exist β0 ≥ 1, ω0 ∈ R such that for λ > ω0 , (λI − A)−1 ∈ B (X ) and

|(λ − ω)−n (λI − A)−n | ≤ β0 for n ∈ N,

∗ Corresponding author. Tel.: +212 4 43 46 49; fax: +212 24 43 74 09. E-mail address: [email protected] (K. Ezzinbi). 0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.04.001

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where B (X ) is the space of bounded linear operators on X . In fact, Travis and Webb in [9] established the existence, regularity and stability of solutions of Eq. (1.1) when B = 0, the mild solutions of Eq. (1.1) are given by the following variation of constants formula Z t u(t) = T (t)ϕ(0) + T (t − s)f (s, us )ds for t ≥ 0, (1.2) 0

where (T (t))t≥0 is the semigroup generated by A. The authors proved the existence of solutions by using the strict contraction principle. In this case, the classical semigroup theory ensures the well posedness of Eq. (1.1) when B = 0, for more details we refer to [8] and [10]. Here we propose to extend the results of [9] when B 6= 0. In [2], the author investigated the existence and regularity of solutions of the following integrodifferential equation  Z t  0 u (t) = Au(t) + B(t − s)u(s)ds + g(t) for t ≥ 0 (1.3) 0  u(0) = u0 ∈ X , and the existence, uniqueness, representation of solutions via a variation of constants formula and other properties of the resolvent operator were studied. Recall that the resolvent operator plays an important role in solving Eq. (1.3) in the weak and strict sense, it replaces the role of the c0 -semigroup theory, for more details we refer to [1,3–7]. In this work, we will use the basic theory developed in [2] for Eq. (1.3) to build basic results on the existence of the so-called mild and strict solutions of Eq. (1.1). Firstly, we use the strict contraction principle to show the local existence of mild solutions which may blow up in finite time. Then, we reformulate some sufficient conditions to get the existence of strict solutions. The results obtained in this work generalize the well-known results developed in [9] when B = 0. The organization of this work is as follows, in Section 2, we collect some useful results on the resolvent operator. In Section 3, we study the existence of local mild solutions of Eq. (1.1) and we show the global continuation of solutions. We prove that in the case of local existence, the solutions blow up and we also show the dependence continuous with the initial data. In Section 4, we show the existence of strict solutions for Eq. (1.1) by using the regularity theorem due to Grimmer [2]. In Section 5, we prove the existence of mild solutions in the nonautonomous case. For illustration, we propose to study the existence of solutions for some partial functional integrodifferential equations with diffusion. 2. Preliminary results In this section, we collect some basic results about the resolvent operators for the following linear homogeneous equation  Z t  0 v (t) = Av(t) + B(t − s)v(s)ds for t ≥ 0 (2.1) 0  v(0) = v0 ∈ X , where A and B(t) are closed linear operators on a Banach space X . In the following Y denotes the Banach space D(A) equipped with the graph norm defined by,

|y|Y = |Ay| + |y| for y ∈ Y. C ([0, +∞[; Y ) is the space of continuous functions from [0, +∞[ to Y .

Definition 2.1 ([2]). A resolvent operator for Eq. (2.1) is a bounded operator valued function R(t) ∈ B (X ) for t ≥ 0 such that (i) R(0) = I and |R(t)| ≤ Neβt for some constants N and β. (ii) For all x ∈ X , R(t)x is strongly continuous for t ≥ 0. (iii) R(t) ∈ B (Y ) for t ≥ 0. For x ∈ Y , R(.)x ∈ C 1 ([0, +∞[; X ) ∩ C ([0, +∞[; Y ) and Z t R0 (t)x = AR(t)x + B(t − s)R(s)xds 0 Z t = R(t)Ax + R(t − s)B(s)xds for t ≥ 0. 0

The resolvent operator plays an interesting role in the investigation of the existence of solution for Eq. (1.3). Definition 2.2 ([2]). We say that a continuous function u : [0, +∞[→ X is a strict solution of Eq. (1.3) if the following conditions hold. (i) u ∈ C 1 ([0, +∞[; X ) ∩ C ([0, +∞[; Y ), (ii) u satisfies Eq. (1.3) for t ≥ 0.

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Remark. Condition (ii) means that u(t) ∈ D(A) for all t ≥ 0 and s → B(t − s)u(s) is integrable on [0, t], moreover the derivative u0 satisfies Eq. (1.3) for all t ≥ 0. The following theorem gives a relationship between the resolvent operator and the expression of strict solution. Theorem 2.3 ([2]). Assume that Eq. (2.1) has a resolvent operator. If u is a strict solution of Eq. (1.3), then Z t u(t) = R(t)u0 + R(t − s)g(s)ds for t ≥ 0.

(2.2)

0

The basic theory for the existence of a resolvent operator is given in [2]. In the following, we suppose that (H0 ) Eq. (2.1) has a resolvent operator. Here for simplicity, we suppose that the following assumptions ensure the existence of the resolvent operator. (H1 ) A generates a strongly continuous semigroup in a Banach space X . (H2 ) For all t ≥ 0, B(t) is a closed linear operator from D(A) to X and B(t) ∈ B (Y, X ), where B (Y, X ) is the space of all bounded linear operators from Y to X . For any y ∈ Y , the map t → B(t)y is bounded uniformly continuous, differentiable and the derivative t → B0 (t)y is bounded uniformly continuous on R+ . By Theorem 2.3 in [2], we can see that (H1 ) and (H2 ) imply (H0 ), in fact, we have the following result. Theorem 2.4 ([2]). Assume that (H1 ) and (H2 ) hold. Then there exists a unique resolvent operator for Eq. (2.1). The following theorem gives sufficient conditions ensuring the existence of strict solutions of Eq. (1.3) which generalizes the well-known results on the c0 -semigroup theory. Theorem 2.5 ([2]). Let g ∈ C 1 ([0, +∞[; X ) and v be defined by Z t v(t) = R(t)v0 + R(t − s)g(s)ds for t ≥ 0. 0

If v0 ∈ D(A), then v is a strict solution of Eq. (1.3). 3. Global existence and blowing up of the mild solutions Definition 3.1. We say that a continuous function u : [−r, +∞[→ X is a strict solution of Eq. (1.1) if the following conditions hold. (i) u ∈ C 1 ([0, +∞[; X ) ∩ C ([0, +∞[; Y ), (ii) u satisfies Eq. (1.1) on [0, +∞[, (iii) u(θ) = ϕ(θ) for −r ≤ θ ≤ 0. Proposition 3.2. Assume that (H0 ) holds. If u is a strict solution of Eq. (1.1) then Z t u(t) = R(t)ϕ(0) + R(t − s)f (s, us )ds for t ≥ 0.

(3.1)

0

Proof. It is just a consequence of Theorem 2.3 In fact, let us suppose g(t) = f (t, ut ) for t ≥ 0. Then we get the desired result.  Remark. The converse is not true. In fact if u satisfies Eq. (3.1), u may be not differentiable, that is why we distinguish between mild and strict solutions. Definition 3.3. We say that a continuous function u : [−r, +∞[→ X is a mild solution of Eq. (1.1) if u satisfies the following equation  Z t  u(t) = R(t)ϕ(0) + R(t − s)f (s, us )ds for t ≥ 0 0  u0 =

ϕ.

In the following, we give a local existence of mild solutions of Eq. (1.1). For this purpose, we make the following assumption. (H3 ) f is locally Lipschitz, that is, for each α > 0 there is a constant c0 (α) > 0 such that if ϕ1 , ϕ2 ∈ C with |ϕ1 |, |ϕ2 | ≤ α then

|f (t, ϕ1 ) − f (t, ϕ2 )| ≤ c0 (α)|ϕ1 − ϕ2 | for t ∈ [0, α].

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Theorem 3.4. Assume that (H0 ) and (H3 ) hold. Let ϕ ∈ C . Then, there exist a maximal interval of existence [−r, bϕ [ and a unique mild solution u(., ϕ) of Eq. (1.1) defined on [−r, bϕ [ and either bϕ = +∞

or

limt→b−ϕ |u(t, ϕ)| = +∞.

Moreover, u(t, ϕ) is a continuous function of ϕ in the sense that if ϕ ∈ C and t ∈ [0, bϕ [, then there exist positive constants k and ε such that, for ψ ∈ C and |ϕ − ψ| < ε, we have t ∈ [0, bψ [

and |u(s, ϕ) − u(s, ψ)| ≤ k|ϕ − ψ|

for all s ∈ [−r, t].

Proof. Let b1 > 0 and α > 0. Then from the local Lipschitz condition on f there exists c0 (α) such that for ϕ ∈ C with |ϕ| < α, and t ∈ [0, b1 ] we have

|f (t, ϕ)| ≤ c0 (α)|ϕ| + |f (t, 0)| ≤ c0 (α)α + sup |f (s, 0)|. s∈[0,b1 ]

b1 will be chosen sufficiently small enough to get the local existence of the mild solutions. Let ϕ ∈ C , α = |ϕ| + 1 and c1 (α) = c0 (α)α + sup |f (s, 0)|. s∈[0,b1 ]

Consider the following set ( Zϕ =

u ∈ C ([−r, b1 ]; X ) : u(s) =

)

ϕ(s) if s ∈ [−r, 0] and sup |u(s) − ϕ(0)| ≤ 1 , s∈[0,b1 ]

where C ([−r, b1 ]; X ) is endowed with the uniform convergence topology, then Zϕ is a closed set of C ([−r, b1 ]; X ). Consider the mapping

K : Zϕ → C ([−r, b1 ]; X ) defined by  Z t  R(t − s)f (s, us )ds R(t)ϕ(0) + K (u)(t) = 0  ϕ(t) for t ∈ [−r, 0].

for t ∈ [0, b1 ]

We will show that K (Zϕ ) ⊂ Zϕ . Let u ∈ Zϕ and t ∈ [0, b1 ], then Z t |K (u)(t) − ϕ(0)| ≤ |R(t)ϕ(0) − ϕ(0)| + R(t − s)f (s, us )ds 0 Z t ≤ |R(t)ϕ(0) − ϕ(0)| + Neβt e−βs |f (s, us )|ds. 0

Without loss of generality, we assume that β > 0. Then, Z t |K (u)(t) − ϕ(0)| ≤ |R(t)ϕ(0) − ϕ(0)| + Neβt |f (s, us )|ds. 0

Since |u(s) − ϕ(0)| ≤ 1 for s ∈ [0, b1 ] and α = |ϕ| + 1, we deduce that |us | ≤ α for s ∈ [0, b1 ]. Then

|f (s, us )| ≤ c0 (α)|us | + |f (s, 0)| ≤ c1 (α). If we choose b1 to be sufficiently small such that sup {|R(s)ϕ(0) − ϕ(0)| + Neβs c1 (α)s} < 1,

s∈[0,b1 ]

consequently

|K (u)(t) − ϕ(0)| ≤ |R(t)ϕ(0) − ϕ(0)| + Neβt c1 (α)t < 1 for t ∈ [a, b1 ]. Hence K (Zϕ ) ⊂ Zϕ . On the one hand, let u, v ∈ Zϕ and t ∈ [0, b1 ]. Then Z t |K (u)(t) − K (v)(t)| = R(t − s)(f (s, us ) − f (s, vs ))ds 0 Z t ≤ N eβ t |f (s, us ) − f (s, vs )|ds 0 Z t ≤ Neβb1 c0 (α) |us − vs |ds 0

≤ Neβb1 c0 (α)b1 |u − v|.

(3.2)

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Since Neβb1 c0 (α)b1 ≤ Neβb c1 (α)b1

< sup {|R(s)ϕ(0) − ϕ(0)| + Neβs c1 (α)s}. s∈[0,b1 ]

Condition (3.2) implies that Neβb1 c0 (α)b1

< 1.

This means that K is a strict contraction in Zϕ . Thus by a fixed point theorem, K has a unique fixed point u in Zϕ . We conclude that Eq. (1.1) has one and only one mild solution which is defined on [−r, b1 ] and denoted by u(., ϕ). Using the same arguments, we can show that u(., ϕ) can be extended to a maximal interval of existence [0, bϕ [. If we assume that bϕ < +∞ and limt→b− |u(t, ϕ)| < +∞, then there exists a constant α > 0 such that |u(t, ϕ)| ≤ α for all t ∈ [0, bϕ [. We claim ϕ that u(., ϕ) is uniformly continuous. Consequently lim u(t, ϕ) exists,

− t→bϕ

which contradicts the maximality of [0, bϕ [. Let us show the uniform continuity of u(., ϕ). Let t, t + h ∈ [0, bϕ [, h > 0 and θ ∈ [−r, 0]. If t + θ ≥ 0, then the map t 7→ R(t + θ)ϕ(0) is uniformly continuous. On the other hand, we have u(t + h + θ, ϕ) − u(t + θ, ϕ)

= R(t + h + θ)ϕ(0) − R(t + θ)ϕ(0) +

Z

= R(t + h + θ)ϕ(0) − R(t + θ)ϕ(0) +

Z

Z

+

t +θ +h

t +θ

+

t +θ +h

t +θ

0 t +θ

0

R(t + θ + h − s)f (s, us (., ϕ))ds −

t +θ

Z 0

Z

t +θ

0

R(t + θ − s)f (s, us (., ϕ))ds

R(s)f (t + θ + h − s, ut+θ+h−s (., ϕ))ds

R(s)f (t + θ + h − s, ut+θ+h−s (., ϕ))ds −

= R(t + h + θ)ϕ(0) − R(t + θ)ϕ(0) + Z

t+θ+h

Z 0

t+θ

R(s)f (t + θ − s, us (., ϕ))ds

R(s) [f (t + θ + h − s, ut+θ+h−s (., ϕ)) − f (t + θ − s, ut+θ−s (., ϕ))] ds

R(s)f (t + θ + h − s, ut+θ+h−s (., ϕ))ds.

Thus

|u(t + h + θ, ϕ) − u(t + θ, ϕ)| ≤ |R(t + h + θ)ϕ(0) − R(t + θ)ϕ(0)| + Neβbϕ c1 (α)h + Neβbϕ c0 (α)

Z 0

t

|us+h (., ϕ) − us (., ϕ)|ds.

If t + θ < 0. Let h0 > 0 be sufficiently small such that for h ∈]0, h0 [

|ut+h (θ, ϕ) − ut (θ, ϕ)| ≤ sup |u(σ + h, ϕ) − u(σ, ϕ)| = |uh − ϕ|. −r≤σ ≤0

Since the map t 7→ R(t)ϕ(0) is uniformly continuous, consequently, for t, t + h ∈ [0, bϕ [ and h ∈]0, h0 [, we have Z t |ut+h (., ϕ) − ut (., ϕ)| ≤ δ1 (h) + δ2 (h) + c1 (α)Neβbϕ h + Neβbϕ c0 (α) |us+h (., ϕ) − us (., ϕ)|ds, 0

where

δ1 (h) = |uh − ϕ| and δ2 (h) =

sup t

|R(t + h)ϕ(0) − R(t)ϕ(0)|.

such that t+h∈[0,bϕ [

By Gronwall’s lemma, it follows that

|ut+h (., ϕ) − ut (., ϕ)| ≤ γ(h) exp[c0 (α)Neβbϕ bϕ ], with

γ(h) = δ1 (h) + δ2 (h) + c1 (α)Neβbϕ h. This completes the proof that u(., ϕ) is uniformly continuous and u(., ϕ) can be extended over [0, bϕ + η], which contradicts the maximality of [0, bϕ [. Using the same reasoning, one can show a similar result for h < 0. Now, we want to prove that the solution depends continuously on the initial data. Let ϕ ∈ C and t ∈ [0, bϕ [ be fixed. Set

α = 1 + sup |us (., ϕ)| −r≤s≤t

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and c(t) = Neβt exp(Neβt c0 (α)t).

Let ε ∈]0, 1[ be such that εc(t) < 1. Take ψ ∈ C such that |ϕ − ψ| < ε. Then

|ψ| ≤ |ϕ| + ε < α. We define

> 0 : |uσ (., ψ)| ≤ α for σ ∈ [0, s] .

 b0 := sup s



If we suppose that b0 < t, we obtain for s ∈ [0, b0 ]

|us (., ϕ) − us (., ψ)| ≤ Neβt |ϕ − ψ| + Neβt c0 (α)

Z

s

0

|uσ (., ϕ) − uσ (., ψ)|ds.

By Gronwall’s lemma, we deduce that

|us (., ϕ) − us (., ψ)| ≤ c(t)|ϕ − ψ|.

(3.3)

This implies that

|us (., ψ)| ≤ c(t)ε + α − 1 < α for all s ∈ [0, b0 ]. It follows that b0 cannot be the largest number s > 0 such that |us (., ψ)| < α, for σ ∈ [0, s]. Thus b0 ≥ t and t < bψ . Furthermore, |us (., ϕ)| < α for s ∈ [0, t], then using the inequality (3.3), we deduce the continuous dependence on the initial data.  Corollary 3.5. Assume that (H0 ) and (H3 ) hold. Let k1 be a continuous function on R+ and k2 ∈ L1loc (R+ ; R+ ) be such that

|f (t, ϕ)| ≤ k1 (t)|ϕ| + k2 (t)

for t ≥ 0 and ϕ ∈ C .

Then Eq. (1.1) has a unique mild solution which is defined for all t ≥ 0. Proof. Let [−r, bϕ [ denote the maximal interval of existence of the mild solution u(t, ϕ) of Eq. (1.1). Then bϕ = +∞

or limt→b−ϕ |u(t, ϕ)| = +∞.

If bϕ < +∞, then limt→b−ϕ |u(t, ϕ)| = +∞. Since R(t) ∈ B (X ) for t ≥ 0, then there exists M > 0 such that |R(t)| ≤ M for all t ∈ [0, bϕ ]. Thus, we have Z t |u(t, ϕ)| ≤ |R(t)ϕ(0)| + |R(t − s)| |f (s, us )|ds 0 Z t Mk1 (s)|us |ds for t ∈ [0, bϕ [, ≤ k0 + 0

where k0 = M|ϕ| +

Z 0

bϕ

Mk2 (s)ds.

By Gronwall’s lemma, we deduce that   Z t k1 (s) ds < +∞ for t ∈ [0, bϕ [, |ut (ϕ)| ≤ k0 exp M 0

and limt→b−ϕ |u(t, ϕ)| < +∞, which gives a contradiction.



As an immediate consequence, we get the following result. Corollary 3.6. Assume that (H0 ) holds and f is Lipschitzian with respect to the second argument, namely,

|f (t, ϕ1 ) − f (t, ϕ2 )| ≤ L |ϕ1 − ϕ2 |

for t ≥ 0 and ϕ1 , ϕ2 ∈ C .

Then Eq. (1.1) has a unique mild solution which is defined for all t ≥ 0.

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In the following, we give some estimations of solutions. Proposition 3.7. Assume that (H0 ) holds and f is Lipschitzian with respect to the second argument. Let u and b u be the mild solutions of Eq. (1.1) corresponding respectively to ϕ and b ϕ ∈ C . Then ( |ut − b ut | ≤ N|ϕ − b ϕ|e(β+NL)t if β ≥ 0 −βr − β r |ut − b ut | ≤ Ne |ϕ − b ϕ|e(β+NLe )t if β < 0, where L is the Lipschitz constant of f . Proof. Let u and b u be the mild solutions of Eq. (1.1) corresponding respectively to ϕ and b ϕ ∈ C , we have Z t |u(t) − b u(t)| ≤ Neβt |ϕ(0) − b ϕ(0)| + NL eβ(t−s) |us − b us |ds. 0

If β ≥ 0, then

|ut − b ut | ≤ Neβt |ϕ − b ϕ| + NLeβt

Z 0

t

e−βs |us − b us |ds.

If β < 0, then

|ut − b ut | ≤ Ne−βr eβt |ϕ − b ϕ| + NLe−βr eβt By Gronwall’s lemma, the result follows.

Z

t

e−βs |us − b us |ds.

0



4. Existence of strict solutions Theorem 4.1. Assume that (H0 ) and (H3 ) hold and f is continuously differentiable. Moreover assume that the partial derivatives Dt f and Dϕ f are locally Lipschitz in the classical sense. Let ϕ be in C 1 ([−r, 0], X ) such that

˙ 0) = Aϕ(0) + f (0, ϕ). ϕ(0) ∈ D(A) and ϕ( Then the corresponding mild solution u becomes a strict solution of Eq. (1.1). Proof. Let ϕ be in C 1 ([−r, 0]; X ) such that

˙ 0) = Aϕ(0) + f (0, ϕ). ϕ(0) ∈ D(A) and ϕ( Let u be the corresponding mild solution of Eq. (1.1) which is defined on some maximal interval [0, bϕ [. Let a < bϕ . Then by using the strict contraction principle, we can show that there exists a unique continuous function v solution of the following equation on [0, a]  Z t Z t    v(t) = R(t)(Aϕ(0) + f (0, ϕ)) + R(t − s) Dt f (s, us ) + Dϕ f (s, us )vs ds + R(t − s)B(s)ϕ(0)ds. 0 0  0 v0 =

ϕ.

We introduce the function w defined by  Z t  w(t) = ϕ(0) + v(s)ds if t ≥ 0 0  w(t) = ϕ(t) if − r ≤ t ≤ 0.

(4.1)

Then, we can see that Z t vs ds for t ∈ [0, a]. wt = ϕ + 0

Rt Consequently, the maps t → wt and t → 0 R(t − s)f (s, ws )ds are continuously differentiable and the following formula holds Z Z d t d t R(t − s)f (s, ws )ds = R(s)f (t − s, wt−s )ds dt 0 dt 0 Z t = R(t)f (0, w0 ) + R(t − s)(Dt f (s, ws ) + Dϕ f (s, ws )vs )ds 0 Z t = R(t)f (0, ϕ) + R(t − s)(Dt f (s, ws ) + Dϕ f (s, ws )vs )ds, 0

which implies Z t 0

R(s)f (0, ϕ)ds =

Z 0

t

R(t − s)f (s, ws )ds −

Z tZ 0

0

s

R(s − τ)(Dt f (τ, wτ ) + Dϕ f (τ, wτ )vτ )dτ ds.

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On the one hand, we have Z t Z t Z tZ s w(t) = ϕ(0) + R(s)Aϕ(0) ds + R(s)f (0, ϕ) ds + R(s − τ)(Dt f (τ, uτ ) 0 0 0 0 Z tZ s + Dϕ f (τ, uτ )vτ )dτ ds + R(s − τ)B(τ)ϕ(0)dτ ds, 0

0

and from Definition 2.1, we have Z t Z tZ s R(s)Aϕ(0)ds = R(t)ϕ(0) − ϕ(0) − R(s − τ)B(τ)ϕ(0)dτ ds, 0

0

0

then it follows that t

Z tZ s R(t − s)f (s, ws )ds + R(s − τ) [(Dτ f (τ, uτ ) − Dτ f (τ, wτ )) 0 0 0
 + Dϕ f (τ, uτ )vτ − Dϕ f (τ, wτ )vτ dτ ds.

w(t) = R(t)ϕ(0) +

Z

On the other hand u(t) = R(t)ϕ(0) +

t

Z 0

R(t − s)f (s, us )ds

Then for t ∈ [0, a], we have Z t

|u(t) − w(t)| ≤

0

for t ≥ 0.

|R(t − s)||f (s, us ) − f (s, ws )|ds +

Z tZ

s

+ 0

0

Z tZ 0

0

s

|R(s − τ)||Dτ f (τ, uτ ) − Dτ f (τ, wτ )|dτ ds

|R(s − τ)||Dϕ f (τ, uτ )vτ − Dϕ f (τ, wτ )vτ |dτ ds.

(4.2)

Let H = {us , ws : s ∈ [0, a]}. Then H is a compact set in C , it follows that f , Dt f and Dϕ f are globally Lipschitz on H. Let d1 be such that for t ∈ [0, a] and x, y ∈ H, we have

|f (t, x) − f (t, y)| ≤ d1 |x − y| |Dt f (t, x) − Dt f (t, y)| ≤ d1 |x − y| |Dϕ f (t, x) − Dϕ f (t, y)| ≤ d1 |x − y|. Consequently, using Eq. (4.2) we can find a positive constant k(a) such that Z t |uτ − wτ | ≤ k(a) |us − ws |ds for s ∈ [0, a], 0

which implies that u = w. Consequently, we deduce that the mild solution is continuously differentiable from [−r, a] to X and the function t → f (t, ut ) is continuously differentiable on [0, a]. According to Theorem 2.5, we deduce that u is a strict solution of Eq. (1.1) on [0, a]. This holds for any a < bϕ .  5. General framework In this section, we consider the nonautonomous case  Z t  0 u (t) = A(t)u(t) + B(t, s)u(s)ds + f (t, ut ) for t ≥ 0 0  u0 =

ϕ ∈ C,

(5.1)

for each t ≥ 0, A(t) is a closed linear operator with dense domain D(A) which is independent of t, and for 0 ≤ s ≤ t, B(t, s) is closed linear operator with domain D(B) ⊃ D(A). We use the approach developed in [2] to show the existence at least of a mild solution. Firstly, we give the sense of the resolvent operator given in [2]. Consider the following linear homogeneous equation  Z t  0 v (t) = A(t)v(t) + B(t, s)v(s)ds for t ≥ 0 (5.2) 0  v(0) = v0 ∈ X . Suppose that Z is the Banach space D(A) provided with the following norm

|y|Z = |A(0)y| + |y| for y ∈ Z . Definition 5.1 ([2]). A resolvent operator for Eq. (5.2) is a bounded operator valued function R(t, s) ∈ B (X ), for 0 ≤ s ≤ t, having the following properties. (i) R(t, s) is strongly continuous in s and t, R(s, s) = I for 0 ≤ s ≤ t and |R(t, s)| ≤ N1 eβ1 (t−s) for some constants N 1 and β1 .

K. Ezzinbi et al. / Nonlinear Analysis 70 (2009) 2761–2771

(ii) R(t, s)Z ⊂ Z , R(t, s) is strongly continuous in s and t on Z . (iii) For every x ∈ Z , R(t, s)x is strongly continuously differentiable in s and t and  Z t ∂R    (t, s)x = A(t)R(t, s)x + B(t, r)R(r, s)xdr, ∂t sZ t ∂R    (t, s)x = −R(t, s)A(s)x − R(t, r)B(r, s)xdr.

∂s

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(5.3)

s

Definition 5.2. Let (A(t))t≥0 be a family of generators of c0 -semigroups. (A(t))t≥0 is said to be stable if there are real constants N0 ≥ 1 and α0 so that k Y (A(tj ) − λI)−1 ≤ N0 (λ − α0 )−k j=1 for all λ > α0 , 0 ≤ t1 ≤ t2 ≤ ... ≤ tk < +∞ for k = 1, 2, . . .. To obtain the existence of the resolvent operator of Eq. (5.2), we assume the following hypotheses due to Grimmer [2]. (H4 ) (A(t))t≥0 is a stable family of generators such that A(t)x is strongly continuously differentiable on [0, +∞[ for x ∈ Z . In addition, B(t)x is strongly continuously differentiable on [0, +∞[ for x ∈ Z . (H5 ) B(t) is continuous on [0, +∞[ into B (Z , F ), where F is a subspace of the set of bounded uniformly continuous functions from R+ into X denoted by BUC (R+ ; X ), F is a Banach space with a norm stronger than the sup norm on BUC (R+ ; X ), where B(.) is defined by

(B(t)x)(s) = B(t + s, t)x for x ∈ Z and t, s ≥ 0. (H6 ) B(t) : Z → D(Ds ) for all t ≥ 0, where Ds is the generator of c0 -semigroup (S(t))t≥0 on F defined by S(t)f (s) = f (t + s) for t, s ≥ 0. (H7 ) Ds B(t) is continuous on [0, +∞[ into B (Z , F ). Theorem 5.3 ([2]). Assume (H4 ), (H5 ), (H6 ) and (H7 ), then Eq. (5.2) has a unique resolvent operator. Definition 5.4. We say that a continuous function u : [−r, +∞[→ X is a strict solution of Eq. (5.1) if the following conditions hold. (i) u ∈ C 1 ([0, +∞[; X ) ∩ C ([0, +∞[; Z ). (ii) u satisfies Eq. (1.1) on [0, +∞[. (iii) u(θ) = ϕ(θ) for −r ≤ θ ≤ 0. Theorem 5.5. Assume that (H4 ), (H5 ), (H6 ) and (H7 ) hold. If u is strict solution of Eq. (5.1), then Z t R(t, s)f (s, us )ds for t ≥ 0. u(t) = R(t, 0)ϕ(0) +

(5.4)

0

Proof. If u is a strict solution of Eq. (5.1), then u is a strict solution of the following problem  Z t  0 B(t, s)v(s)ds for + h(t) for t ≥ 0 v (t) = A(t)v(t) + 0  v(0) = ϕ(0) ∈ X . By [2], we know also that we have the following variation of constants formula Z t v(t) = R(t, 0)ϕ(0) + R(t, s)h(s)ds for t ≥ 0. 0

This completes the proof.



Remark. The converse is not true. In fact if u satisfies formula (5.4), u is not differentiable in general, that is why we have to distinguish between mild and strict solutions. Definition 5.6. We say that a continuous function u : [−r, +∞[→ X is a mild solution of Eq. (5.1) if u satisfies the following equation  Z t  u(t) = R(t, 0)ϕ(0) + R(t, s)f (s, us )ds for t ≥ 0 0  u0 =

ϕ.

K. Ezzinbi et al. / Nonlinear Analysis 70 (2009) 2761–2771

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Theorem 5.7. Assume that (H4 ), (H5 ), (H6 ) and (H7 ) hold and f is Lipschitzian with respect to the second argument. Then for any ϕ ∈ C , Eq. (5.1) has a unique mild solution which is defined for t ≥ 0. Proof. Let ϕ ∈ C and fix b > 0. Consider the set Λ defined by Λ = {u ∈ C ([0, b]; X ) : u(0) = ϕ(0)}

and let e u be a function defined by  u(t) if t ∈ [0, b] e u(t) = ϕ(t) if t ∈ [−r, 0]. We introduce the operator K defined by Z t Ku(t) = R(t, 0)ϕ(0) + R(t, s)f (s, e us )ds for t ∈ [0, b]. 0

Since for every x ∈ X , (t, s) → R(t, s)x is continuous for t ≥ s, it follows that by the Banach–Steinhauss theorem that Ma =

sup |R(t, s)| < ∞.

0≤s≤t≤a

Let L be the Lipschitz constant of f , then we have for t ∈ [0, b] Z t |Ku(t) − Kv(t)| ≤ |R(t, s)||f (s, e us ) − f (s, e vs )|ds 0

≤ Ma Lt|u − v|. We have also

|K 2 u(t) − K 2 v(t)| = |K (Ku(t)) − K (Kv(t)) | Z

≤ 0

t

|R(t, s)||f (s, Ke us ) − f (s, Ke v s ) |d s Z

≤ Ma L ≤

0

t

|Ke us − Ke vs |ds

(Ma Lt)2 2!

|u − v |.

Consequently, we deduce for all n ≥ 2 that

|K n u − K n v | ≤

(Ma Lb)n n!

|u − v |.

Let n0 be such that (ManLb!) < 1. Then K n0 is a strict contraction and consequently K n0 has a unique fixed point u. This implies 0 that Eq. (5.1) has a unique mild solution on [0, b], this holds for any b > 0.  n0

6. Application For illustration, we propose to study the existence of solutions for the following model  Z t Z 0 ∂ ∂2 ∂2   z ( t , x ) = z ( t , x ) + α( t − s ) z ( s , x ) d s + g(t, z(t + θ, x))dθ    ∂t Z ∂x2 ∂x2 0 −r   0 g(t, z(t + θ, x))dθ for t ≥ 0 and x ∈ [0, π], +   −r   z(t, 0) = z(t, π) = 0 for t ≥ 0,   z(θ, x) = ϕ0 (θ, x) for θ ∈ [−r, 0] and x ∈ [0, π],

(6.1)

where g : R+ × R → R is continuous and Lipschitzian with respect to the second argument, α : R+ → R is bounded uniformly continuous, continuously differentiable and α0 is bounded uniformly continuous, the function ϕ0 : [−r, 0] × [0, π] → R will be specified later. To rewrite Eq. (6.1) in the abstract form, we introduce the space X = C0 ([0, π]; R), the space of continuous functions from [0, π] to R+ vanishing at 0 and π, equipped with the uniform norm topology. Let A : D(A) → X be defined by ( D(A) = {y ∈ X ∩ C 2 ([0, π], R) : y0 , y00 ∈ X } Ay = y00 .

Let B : D(A) → X be defined by B(t)(y) =

α(t)Ay for t ≥ 0.

K. Ezzinbi et al. / Nonlinear Analysis 70 (2009) 2761–2771

Let f : C → X be defined by Z 0 f (t, ψ)(x) =

−r

g(t, ψ(θ)(x))dθ

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for x ∈ [0, π] and t ≥ 0.

The initial data ϕ ∈ C is defined by

ϕ(θ)(x) = ϕ0 (θ, x) for θ ∈ [−r, 0] and x ∈ [0, π]. Let us suppose v(t) = z(t, x). Then Eq. (6.1) takes the following abstract form  Z t d v(t) = Av(t) + B(t − s)v(s)ds + f (t, vt ) for t ≥ 0 0  dt v0 =

ϕ.

(6.2)

It is well known that A is the generator of c0 -semigroup, which implies that (H1 ) is satisfied. Moreover (H2 ) is true, it follows that the linear Eq. (2.1) has a resolvent operator. Since f is continuous and Lipschitzian with the second argument, then by Theorem 3.4, we deduce that Eq. (6.2) has a unique mild solution which is defined for t ≥ 0. For the regularity, we impose the following conditions which imply the hypotheses of Theorem 4.1. (H8 ) g ∈ C 1 (R+ × R; R), such that ∂∂gt and ∂∂gx are locally Lipschitz continuous. (H9 )  1  ϕ0 ∈ C ([−r, 0] × [0, π]) such that ϕ0 (0, .) ∈ D(A) Z 0 ∂2 ∂  g(0, ϕ0 (θ, x))dθ for x ∈ [0, π].  ϕ0 (0, x) = 2 ϕ0 (0, x) +

∂θ

∂x

−r

Consequently, by Theorem 4.1, we obtain the following existence result. Proposition 6.1. Under the above assumptions, Eq. (6.1) has a unique strict solution v and the solution u defined by u(t, x) = v(t)(x) for t ≥ 0 and x ∈ [0, π] is a solution Eq. (6.1). Acknowledgment The authors would like to thank the referee for his careful reading of the paper. His valuable suggestions and critical remarks led to numerous improvements throughout the paper. References [1] G. Chen, R. Grimmer, Semigroup and integral equations, Journal of Integral Equations 2 (1980) 133–154. [2] R. Grimmer, Resolvent operators for integral equations in a Banach space, Transaction of American Mathematical Society 273 (1982) 333–349. [3] R. Grimmer, J. Lui, in: A. Fink, R. Miller, W. Kliemann (Eds.), Liapunov–Razumikhin Methods for Integrodifferential Equations in Hilbert Space, in Delay and Differential Equations, World scientific, London, 1992, pp. 9–24. [4] G. Gripenberg, S.-O. Londen, O. Staffans, Volterra Integral and Functional Equation, Cambridge University Press, Cambridge, 1990, pp. 12–13. [5] S.I. Grossman, R.K. Miller, Perturbation theory for Volterra system, Journal of Differential Equations 8 (1970) 457–474. [6] D. Jackson, Existence and uniqueness of a solution to semilinear nonlocal parabolic equations, Journal of Mathematical Analysis and Applications 172 (1993) 256–265. [7] R.K. Miller, Volterra integral equations in a Banach, Funkcial Ekvack 18 (1975) 163–193. [8] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. [9] C.C. Travis, G.F. Webb, Existence and stability for partial functional differential equations, Transaction of American Mathematical Society 200 (1974) 395–418. [10] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer Verlag, 1996.