REGULARITY OF SOLUTIONS OF PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

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1 Proyecciones Vol. 21, N o 1, pp , May 22. Universidad Católica del Norte Antofagasta - Cile REGULARITY OF SOLUTIONS OF PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY EDUARDO HERNÁNDEZ Universidade de Sao Paulo - Brasil Abstract We prove te existence of regular solutions for a class of quasi-linear partial neutral functional differential equations wit unbounded delay tat can be described as te abstract retarded functional differential equation d dt (x(t) + F (t, x t)) = Ax(t) + G(t, x t ), were A is te infinitesimal generator of a strongly continuous semigroup of bounded linear operators on a Banac space X and F, G are appropriated functions. Keywords: Retarded functional differential equations, abstract Caucy problem, semigroup of bounded linear operators, regularity of solutions. Researc Partially Supported by FAPESP-SP-Brazil, grant #

2 66 Eduardo Hernández 1. Introduction Te purpose of tis paper is to establis some results of regularity, in a sense to be specified later, for solutions of a class of quasi-linear neutral functional differential equations wit unbounded delay tat can be described in te form (1.1) { d dt (x(t) + F (t, x t)) = Ax(t) + G(t, x t ), t > σ, x σ = ϕ B, were A is te infinitesimal generator of an uniformly bounded analytic semigroup of bounded linear operators, (T (t)) t, on a Banac space X, te istory x t : (, ] X, x t (θ) = x(t + θ), belongs to some abstract pase space B defined axiomatically, Ω B is open, σ < a and F, G : [σ, a] Ω X are appropriate continuous functions. Neutral differential equations arise in many areas of applied matematics and suc equations ave received muc attention in recent years. A good guide to te literature for neutral functional differential equations is te Hale & lunel book [3] and te references terein. Te work in partial neutral functional differential equations wit unbounded delay was initiated by Hernández & Henríquez in [4, 5]. In tese papers, Hernández & Henríquez proved te existence of mild, strong and periodic solutions for te neutral equation (1.1). In general, te results were obtained using te semigroup teory and te Sadovskii fixed point teorem ( see [11] ). Te results obtained in tis paper are te continuation of papers [4], [5] on te existence of mild, strong and periodical solutions for te neutral system (1.1) and generalization of te results reported by Henriquez in [6]. Trougout tis paper, X will be a Banac space provided wit norm and A : D(A) X will be te infinitesimal generator of an uniformly bounded analytic semigroup, T = (T (t)) t, of linear operators on X. For te teory of strongly continuous semigroup, we refer to Pazy [1] and Krein [9]. We will point out ere some notations and properties tat will be used in tis work. It is well know tat tere exist constants M and w IR suc tat T (t) Me wt, t.

3 Regularity of Solutions of Partial Neutral Functional 67 If T is a uniformly bounded and analytic semigroup suc tat ρ(a), ten it is possible to define te fractional power ( A) α, for < α 1, as a closed linear operator on its domain D( A) α. Furtermore, te subspace D( A) α is dense in X, and te expression x α = ( A) α x defines a norm in D( A) α. If X α represents te space D( A) α endowed wit te norm α, ten te following properties are well known ([1], pp. 74 ): Lemma 1. If te previous conditions old: 1. Let < α 1. Ten X α is a Banac space. 2. If < β α ten X α X β is continuous. 3. For every constant a >, tere exists C a > suc tat ( A) α T (t) C a t α, < t a. 4. For every a > tere exists a positive constant C a suc tat (T (t) I)( A) α C a t α, < t a. In tis work we will employ an axiomatic definition of te pase space B, introduced by Hale and Kato [2]. To establis te axioms of te space B we follows te terminology used in Hino-Murakami-Naito [8], and tus, B will be a linear space of functions mapping (, ] into X, endowed wit a seminorm B. We will assume tat B satisfies te following axioms: (A) If x : (, σ +ω) X, ω >, is continuous on [σ, σ +ω) and x σ B, ten for every t [σ, σ + ω) te following conditions old: i) x t is in B. ii) x(t) H x t B.

4 68 Eduardo Hernández iii) x t B K(t σ) sup{ x(s) : σ s t} + M(t σ) x σ B. Were H > is a constant; K, M : [, ) [, ), K is continuous, M is locally bounded and H, K, M are independent of x ( ). (A 1) For te function x ( ) in (A), x t is a B-.valued continuous function on [σ, σ + w). (B) Te space B is complete. For te literature on pase space, we refer te reader to [8]. Wile noting ere tat from te axiom (A 1), it follows, tat te operator function W ( ) dfined by (1.2)[W (t) ϕ] (θ) := { T (t + θ) ϕ () for t θ, ϕ (t + θ) for < θ < t, is an strongly continuous semigroup of bounded linear operators on B. In tis paper, A w wit domain D (A w ) will be te infinitesimal generator of W ( ). To obtain some of our results we will require additional properties for te pase space B, in particular we consider te following axiom (see [6], pp. 526 for details) ; (C 3 ) Let ρ >. Let x : (, σ + ρ] X be a continuous function suc tat x σ and te rigt derivative, denoted x (σ + ), exists. If te function ψ defined by ψ (θ) = for θ < and ψ () := x (σ + ) belongs to B ten ( 1 ) xσ+ ψ B as +. On te oter and, for a linear map P : D (P ) X X and ϕ B suc tat ϕ (θ) D (P ) for every θ, we denote by P ϕ : (, ] X, defined by P ϕ = P (ϕ(θ)). For any < α 1 we use te notation B α for te vector space B α = {( A) α ϕ : ϕ B }. It is easily to prove tat B α, endowed wit te seminorm defined by ψ Bα := ( A) α ψ B,

5 Regularity of Solutions of Partial Neutral Functional 69 is a pase space of functions wit values in X α. Te paper is organized as follows. In section 2, we define te different concepts used in tis work and establis te existence of N-classical and classical solutions for te initial value problem (1.1). Our results are based on te properties of analytic semigroups and te ideas contained in Pazy [1] and Henríquez [6]. Trougout tis work we assume tat X is an abstract Banac space. Te terminology and notations are tose generally used in operator teory. In particular, if X and Y are Banac spaces, we indicate by L (X : Y ) te Banac space of te bounded linear operators of X in Y and we abbreviate tis notation to L (X) wen ever X = Y. In addition B r (x : X) will denote te closed ball in te space X wit center at x and radius r. For some bounded function ξ : [σ, a] X and σ s < t a we employ te notation (1.3) ξ( ) [s,t] = sup{ ξ(θ) : θ [s, t]} and we will write simply ξ t for ξ( ) [σ,t] wen no confusion arises. If x X, we will use te notation X x for te function X x : (, ] X were X x = for θ < and X x () = x. Finally, a function f : I IR X is α-hölder continuous, < α 1, if tere exists a constant L > suc tat f(s) f(t) L t s α, s, t I. We represent by C, α (I; X) te space of α-hölder continuous function from I into X. Similarly, C k, α (I; X) consist of tose functions from I into X, tat are k-times continuously differentiable and wose k t -derivative is α-hölder continuous.

6 7 Eduardo Hernández 2. Regularity of Mild Solutions In tis section we will study te regularity of mild solutions of te abstract Caucy problem (1.1). Hencefort we will assume tat A is te infinitesimal generator of a uniformly bounded analytic semigroup, T = (T (t)) t, on X, tat Ω B is open and tat F, G : [σ, a] Ω X are continuous functions. Furter, to avoid unnecessary notation, we suppose tat ρ(a) and tat T (t) M, for some constant M 1 and every t. Our regularity results are based on tose of regularity of mild solutions for te abstract Caucy problem (2.1) { x (t) = Ax(t) + f(t), x() = x. By analogy wit te abstract Caucy problem (2.1) we adopt te following definitions: Definition 1. We will say tat a function x : (, σ +b) X, σ + b a, is a mild solution of te abstract Caucy problem (1.1) if: x σ = ϕ; te restriction of x( ) to te interval [σ, σ + b) is continuous; for eac σ t < σ+b te function AT (t s)f (s, x s ), s [σ, t), is integrable and (2.2) x(t) = T (t σ)(ϕ() + F (σ, ϕ)) F (t, x t ) t σ AT (t s)f (s, x s)ds + t σ T (t s)g(s, x s)ds, for every t [σ, σ + b). Te existence and uniqueness of mild solution of system (1.1) was establised in [5] as consequence of te contraction principle. More precisely: Teorem 1. Let ϕ Ω and assume tat te following conditions old: a) Tere exist β (, 1) and L suc tat te function F is X β -valued and satisfies te Lipscitz condition ( A) β F (t, ψ 1 ) ( A) β F (s, ψ 2 ) (2.3) L ( t s + ψ 1 ψ 2 B )

7 Regularity of Solutions of Partial Neutral Functional 71 for every σ s, t a, ψ 1, ψ 2 Ω and (2.4) K()L ( A) β < 1. b) Te function G is continuous and tere exist N > suc tat G(t, ψ 1 ) G(s, ψ 2 ) (2.5) N( t s + ψ 1 ψ 2 B ) for every σ s, t a and ψ 1, ψ 2 Ω. Ten tere exists a unique mild solution x(, ϕ) of te abstract Caucy problem (1.1) defined on (, σ + r), for some < r < a σ. Furtermore, if Ω = B ten r can be cosen independent of ϕ. Considering te concepts of mild and classical solutions adopted by Henriquez in [6], we introduce te followings definitions. Definition 2. We will say tat a function x : (, σ +b) X, b >, is a classical solution of te abstract Caucy problem (1.1) if: x σ = ϕ; x( ) C([σ, σ + b); X) C 1 ((σ, σ + b); X); x(t) D(A) for every t (σ, σ + b); for eac σ t < σ + b te function AT (t s)f (s, x s ), s [σ, t), is integrable and x( ) satisfies equation (1.1) on [σ, σ + b). Definition 3. We will say tat a function x : (, σ +b) X, b >, is an N-classical solution of te abstract Caucy problem (1.1) if: x σ = ϕ, x( ) C([σ, σ + b); X); x(t) D(A) for every d t (σ, σ + b); dt (x(t) + F (t, x t)) is continuous on (σ, σ + b); for eac σ t < σ + b te function AT (t s)f (s, x s ), s [σ, t), is integrable and x( ) satisfies equation (1.1) on [σ, σ + b). In relation wit te previous definitions, we consider te following result. Proposition 1. Te following properties old.

8 72 Eduardo Hernández (a) If x( ) : (, σ+b) X is a classical or N-classical solution of (1.1) ten x( ) is a mild solution. (b) If x( ) : (, σ + b) X is a N-classical solution of (1.1) and d dt F (t, x t) is continuous on (σ, σ + b), ten x( ) is a classical solution. (c) Every classical solution is a N-classical solution. Proof: We only prove (a). Using tat (T (t)) t is analytic; for t, s [σ, σ + b) wit t > s, we find tat d [T (t s)(x(s) + F (s, x ds s))] = AT (t s)(x(s) + F (s, x s )) +T (t s) d (x(s) + F (s, x ds s)) = AT (t s)f (s, x s + T (t s)g(s, x s ), wic in turn implies tat x(t) + F (t, x t ) = T (t σ)(ϕ() + F (σ, ϕ)) t σ AT (t s)f (s, x s)ds + t σ T (t s)g(s, x s)ds, since s AT (t s)f (s, x s ) is integrable on [σ, t). Tus x( ) is a mild solution. Te proof is complete To prove our first regularity Teorem, we need previously some tecnical results. Next, we study te regularity of mild solutions of te abstract Caucy problem ẋ(t) = Ax(t) + ( A) 1 β g(t), (2.6) x() =, were g( ) C([, a]; X 1 β ) C, ϑ ([, a]; X); β, ϑ (, 1) and β + ϑ > 1. To tis end, for a mild solution, u( ), of (2.6) we introduce te decomposition u (t) = t ( A)1 β T (t s)(g(s) g(t))ds + t ( A)1 β T (t s)g(t)ds = u 1 (t) + u 2 (t).

9 Regularity of Solutions of Partial Neutral Functional 73 Te proofs of te following tree results follow from te proofs of Teorems 4.3.2, and Lemma in Pazy [1]. However tere are some differences tat require special attention and we include te principal ideas of te proofs for completeness. Lemma 2. Let ϑ, β (, 1) wit β + ϑ > 1 and g( ) C([, a]; X 1 β ) C, ϑ ([, a]; X). If u( ) is a mild solution of (2.6) ten u( ) C([, a]; X 1 ) C 1 ([, a]; X). Proof. Clearly Au 2 (t) C ([, a] ; X), since Au 2 (t) = T ( A) 1 β g (t) ( A) 1 β g (t). In order to study te function u 1, for ɛ > we define te function u 1,ɛ C ([, a] ; X) by u 1,ɛ (t) := (2.7) t ɛ ( A) 1 β T (t s) (g (s) g (t)) ds, for t [ɛ, a], for t [, ɛ), From Lemma in [1], u 1,ɛ (t) D(A) for every t [, a] and Au 1,ɛ (t) := (2.8) { t ɛ ( A) 2 β T (t s)(g(s) g(t))ds, t [ɛ, a],, t [, ɛ). Moreover, from te Lebesgue dominated convergence teorem, te estimate ( A) 2 β T (t s)(g(s) g(t)) ds (2.9) C 2 β, t > s, (t s) 2 β β and te assumption β + ϑ > 1, follow tat u 1 (t) D (A) for every t [, a] and tat (2.1) t ( A)2 β T (t s)(g(s) g(t))ds, for t >,, for t =,

10 74 Eduardo Hernández since A is a closed operator. Te continuity of Au 1 on [, a] follows from te Lebesgue dominated convergence teorem and inequality (2.9). Te property for u ( ) is proved in usual form. complete. Te proof is Lemma 3. Under te assumptions of Lemma 2, Au 1 C, ν ([, a]; X) for β + ϑ 1 + ν. Proof. At first we observe tat for t > s and ξ (, 1), ( A) 2 ξ T (t) ( A) 2 ξ T (s) t s ( A) 3 ξ T (τ) dτ and ence C 3 ξ t s dτ τ 3 ξ C 3 ξt 1 ξ (t s) t 2 ξ s 2 ξ, (2.11) ( A) 2 ξ T (t) ( A) 2 ξ T (s) C 3 ξ(t s). t 1 ξ s 2 ξ Let δ > and t (δ, a]. Using (2.1), for > we get Au 1 (t + ) Au 1 (t) t ( A)2 β (T (t + s) T (t s))(g(s) g(t)) ds + t ( A) 2 β (2.12) T (t + s)(g(t) g(t + )) ds + t+ t ( A) 2 β T (t + s)(g(s) g(t + )) ds = I 1 (t, ) + I 2 (t, ) + I 3 (t, ). Next we estimate eac I i (t, ) separately. From inequality (2.11)

11 Regularity of Solutions of Partial Neutral Functional 75 we find tat I 1 (t, ) t ( A) 2 β (T (t + s) T (t s))(g(s) g(t)) ds C 3 β ds (t + s) 1 β (t s) 2 β ϑ C 3 β ds (t + s) 1 β (t s) 2 β ϑ t + t t C 3 β β+ϑ 1 t +C 3 β β t t ds (t + s) 1 β ds (t s) 2 β ϑ (2.13) consequently (2.14) c 1 β+ϑ 1 + c 2 2β+ϑ 1, I 1 (t, ) c 3 β+ϑ 1. For te second term we see tat I 1 (t, ) t ( A) 2 β T (t + s)(g(t + ) g(t)) ds c 4 ϑ t ds (t + s) 2 β ϑ c 5, 1 β and ence (2.15) I 2 (t, ) c 5 β+ϑ 1. Similarly, for te tird term we find tat (2.16) I 3 (t, ) c 6 β+ϑ 1. Te assertion is now consequence of (2.12), (2.14), (2.15) and (2.16). Te proof is complete. Proposition 2. Assume tat te assumptions of Lemma 2 old. If u( ) is te mild solution of (2.6), ten te following properties are verified.

12 76 Eduardo Hernández (a) If g C, µ ([, a]; X 1 β ) ten u C, ϱ ([δ, a]; X 1 ) C 1, ϱ ([δ, a]; X) for every δ > and every < ϱ min{µ, β + ϑ 1}. (b) If g C, µ ([, a]; X 1 β ) and g() =, ten u C, ϱ ([, a]; X 1 ) C 1, ϱ ([, a]; X) for every ϱ min{µ, β+ ϑ 1}. (c) If g C([, a]; X 1 β+ν ) ten u C, ϱ ([, a]; X 1 ) C 1 ([, a]; X) for every ϱ < ν. Proof. From Lemma 2, Au C([, a]; X) and t Au(t) = ( A) 2 β T (t s)(g(s) g(t))ds = t t ( A) 2 β T (t s)g(t)ds ( A) 2 β T (t s)(g(s) g(t))ds ( A) 1 β T (t)g(t) + A 1 β g(t) := Au 1 (t) + v(t) + w(t). Next we discuss (a), (b), (c) separately. (a) Let δ >. From te assumptions and Lemma 3, we know tat w C, µ ([, a]; X) and tat Au 1 C, β+ϑ 1 ([, a]; X). On te oter and, since v(t + ) v(t) (T (t + ) T (t))( A) 1 β g(t + ) + ( A) 1 β T (t)(g(t + ) g(t)) te followings inequalities old v (t + ) v (t) c 1 δ c 3 + c 2 ϑ δ 1 β + c δ 4 µ, c 5 v + c 6 ϑ, ν < β δ 1 β+ϑ δ 1 β c 7 ν + c δ 1 β+ϑ 8 µ, ν < β

13 Regularity of Solutions of Partial Neutral Functional 77 and ence v(t + ) v(t) c 3 max{ϑ,µ} Consequently, u C, ϱ ([δ, a]; X) C 1, ϱ ([δ, a]; X) for ϱ min{µ, β + ϑ 1}. (b) As in te previous case, w C, µ ([, a]; X) and Au 1 C β+ϑ 1 ([, a]; X). Moreover, for > we find tat v(t + ) v(t) T (t + )(( A) 1 β g(t + ) ( A) 1 β g(t)) + t+ t AT (s)[( A) 1 β g(t) ( A) 1 β g()] ds T (t + )(( A) 1 β g(t + ) ( A) 1 β g(t)) t+ t c 1 s µ 1 ds c 1 ϑ + c 2 µ, or 1 β c 3 µ, c 4 µ, tus u C, ϱ ([, a]; X 1 ) C 1, ϱ ([, a]; X) for ϱ min{β + ϑ 1, µ}. (c) From Lemma 2, we know tat (2.17) Au(t) = t ( A) 2 β T (t s)g(s)ds, t. Under tis remark, for t [, a], > and ϱ < ν we get Au(t + ) Au(t) t (T () I)( A) 1 ν T (t s)( A) 1 β+ν g(s) ds + t+ t ( A) 1 ν T (t + s)( A) 1 β+ν g(s) ds C ξ ς ( A) 1 β+ν g t ds a (t s) ϱ+1 ν +C ( A) 1 β+ν 1 ν g t+ ds a t ds (t+ s) 1 ν c 1 ϱ + c 2 ν. Tus, u C, ϱ ([, a]; X 1 ) C 1 ([, a]; X) for ϱ < ν. Tis complete te proof. Te following result can be proved using te steps in te proof of Proposition 2. We will omit te proof. δ

14 78 Eduardo Hernández Corollary 1. Let ϑ, β, η (, 1) wit β + ϑ > 1 ; f( ) C, η ([, a]; X) and g( ) C, ϑ ([, a]; X) C([, a]; X 1 β ). If u( ) is te mild solution of (2.18) { x (t) = Ax(t) + A 1 β g(t) + f(t), t [, a], x() =, ten te following properties are verified: (a) u C([, a]; X 1 ) C 1 ([, a], X), (b) If g C, µ ([, a]; X 1 β ) ten u C, ϱ ([δ, a]; X 1 ) C 1, ϱ ([δ, a]; X) for every < ϱ min{µ, β + ϑ 1, η}, (c) If g C, µ ([, a]; X 1 β ) and g() = f() = ten u C,ϱ ([, a]; X 1 ) C 1,ϱ ([, a]; X) for eac < ϱ min{µ, β + ϑ 1, η}, (d) If g C([, a]; X 1 β+ν ) and f C([, a]; X µ ) ten u C, ϱ ([, a]; X 1 ) C 1 ([, a]; X) for every < ϱ < min{µ, ν}. In te rest of tis paper, we always assume tat te functions F, G verifies te ypotesis in Teorem 1.1. Moreover, to simplify our notations, we only consider te case σ =. Now we establis a first result about te existence of regular solutions; specifically we prove existence of N-classical solutions. Teorem 2. Assume tat tere exist constants < α < β < 1; < γ 1, γ 2 1 and an open subset Ω α B α suc tat F : [, a] Ω α X 1 and G : [, a] Ω α X are continuous functions and tat te following conditions old:

15 Regularity of Solutions of Partial Neutral Functional 79 (a) Te function F is X β -valued and tere exist positive constants L i, L i, i = 1, 2, suc tat ( A) β F (t, ψ 1 ) ( A) β F (s, ψ 2 ) L 1 t s γ 1 +L 1 ψ 1 ψ 2 Bα, G(t, ψ 1 ) G(s, ψ 2 ) L 2 t s γ 2 +L 2 ψ 1 ψ 2 Bα, for every s, t a, ψ 1, ψ 2 Ω α. (b) K()L 1 ( A) α β < 1. (c) ϕ Ω α and tere exists < ξ 1 suc tat te function W ( )ϕ is ξ-hölder on [, a]. If β + Min{β α, ξ, γ 1, γ 2 } > 1, ten tere exists a unique N- classical solution x(, ϕ) of te abstract Caucy problem (1.1) defined on (, b), for some < b < a. Proof: From our assumptions on te operator A, see lemma (1), we fix positive constants C α and C α+1 β suc tat for all t (, T ] ( A) α T (t) C α t α and ( A) 1 β+α T (t) C α+1 β t (1 β+α). Let b 1 > and δ > suc tat V = {(s, ψ) : s b 1, ψ ϕ Bα < δ} [, a] Ω α and µ = K b1 ( A) α β L 1 < 1. Now we coose < b < b 1 suc tat (2.19) sup W (θ) ( A) α (ϕ) ( A) α (ϕ) β < [,b] (1 µ) δ, 4 (2.2) K b sup T (θ) ( A) α ϕ () ( A) α ϕ () [,b] < (1 µ) δ, 8

16 8 Eduardo Hernández (2.21) K b sup T (θ) ( A) α f (, ϕ) ( A) α f (, ϕ) [,b] < (1 µ) δ, 4 ( A) α β K b L 1 b γ 1 (1 µ) (2.22) < δ, 8 b β α β α K b C α+1 β {L 1 b γ 1 + L 1 δ+ ( A) β F (, ϕ) < (2.23) b 1 α (2.24) K b 1 α C α{l 2 b γ 2 + L 2 δ+ G(, ϕ) } < (2.25) K b {C α+1 β L 1 (1 µ) δ, 4 b β α β α + C α L b 1 α (1 µ) 2 } < 1 α 2 (1 µ) δ, 4 In te space Y = C([, b] : X) provided wit te topology of uniform convergence, we define; A(ϕ, α, b) = {u Y : u() = ( A) α ϕ(), ( A) α ϕ ũ t B δ, t [, b] }, were ũ is te extension of u to (, b] wit ũ = ( A) α ϕ. From (2.19), it follows tat A(ϕ, α, b) is a nonempty, convex and closed subset of Y. On A(ϕ, α, b) we define te operator Φ by te expression Φ(u)(t) = T (t)(( A) α (ϕ() + F (, ϕ)) ( A) α F (t, ( A) α ũ t ) + t ( A)α+1 β T (t s)( A) β F (s, ( A) α ũ s )ds + t ( A)α T (t s)g(s, ( A) α ũ s )ds In order to use te contraction mapping principle, we now sow tat te range of Φ is included in A(ϕ, α, b). To tis end we introduce te functions y α, z i : (, b] X, i = 1, 2, 3, were y α (t) = T (t)( A) α ϕ for t, (y α ) = ( A) α ϕ and z 1 (t) = T (t)( A) α F (, ϕ) ( A) α F (t, ( A) α ũ t ), z 2 (t) = t ( A)α+1 β T (t s)( A) β F (s, ( A) α ũ s )ds, z 3 (t) = t ( A)α T (t s)g(s, ( A) α ũ s )ds, for t >, z i = for i {1, 2, 3}. Clearly (2.26) Φ(u)(t) = y α (t) + z 1 (t) + z 2 (t) + z 3 (t)

17 Regularity of Solutions of Partial Neutral Functional 81 on [, b] and Φ(u) t = (y α ) t + zt 1 + zt 2 + zt 3. Wit te previous notations, for t [, b] we ave, (2.27) Φ(u) t ( A) α ϕ B (y α ) t ( A) α ϕ B + zt 1 B + zt 2 B + zt 3 B. Using axiom (A) concerning te pase space, we estimate eac term on te rigt and side of (2.27) separately. Directly from te coice of b we get te estimate (2.28) (y α ) t ( A) α ϕ B W (t)( A) α ϕ ( A) α ϕ B On te oter and, for t [, b] (2.29) and for zt 1 (1 µ) δ. 4 K b sup z 1 (s) B s [,t] z 1 (s) T (s)( A) α F (, ϕ) ( A) α F (, ϕ) + ( A) α F (, ϕ) ( A) α F (s, ( A) α ũ s ) (1 µ) δ + L 1 ( A) α β b γ 1 8K b +L 1 ( A) α β ( A) α ϕ ũ s B ence (2.3) z 1 (s) and substituting (2.3) into (2.29) (2.31) 2(1 µ) δ + L 1 ( A) α β δ, 8K b z 1 t B (1 µ) δ + µδ. 4 Now, for te function z 2 ( ) we ave tat (2.32) zt 2 B K b sup z 2 (s) s [,t]

18 82 Eduardo Hernández and for s [, t] z 2 (s) s ( A) α+1 β T (s θ)(( A) β F (θ, ( A) α ũ θ ) ( A) β F (, ϕ)) dθ + s s ( A) α+1 β T (s θ)( A) β F (, ϕ)) dθ C α+1 β (s θ) α+1 β {L 1θ γ1 ũ θ ( A) α ϕ B }}dθ + bβ α β α C α+1 β ( A) β F (, ϕ) (L 1 b γ 1 + L 1 δ) bγ α β α C α+1 β + bβ α β α C α+1 β ( A) β F (, ϕ). Employing te last inequality in (2.32) we obtain tat; (2.33) z 2 t bβ α K { } β α bc α+1 β L1 b γ 1 + L 1δ + ( A) β F (, ϕ) (1 µ) 4 δ. Similarly for z 3, zt 3 b 1 α B K b 1 α C α {L 2 b γ 2 + L (1 µ) 2δ+ G(, ϕ) } δ. 4 (2.34) Combining (2.27), (2.28), (2.31), (2.33) and (2.34), we conclude tat Φ(u) A(ϕ, α, b). Now we prove tat Φ is a contraction. Let u, v A(ϕ, α, b), so tat Φ(u)(t) Φ(v)(t) ( A) α F (t, ( A) α ũ t ) ( A) α F (t, ( A) α ṽ t ) + t ( A) α+1 β T (t s) ( ( A) β F ( s, ( A) α ) ũ s

19 Regularity of Solutions of Partial Neutral Functional 83 ( A) β F ( s, ( A) α ṽ s )) ds + t ( A)α T (t s)(g(s, ( A) α ũ s ) G(s, ( A) α ṽ s )) ds ( A) α β L 1 ũ t ṽ t B + ( t Cα+1 B L 1 (t s) α+1 B + C αl ) 2 (t s) α ũ s ṽ s B ds, tus (2.35) Φ(u) Φ(v) b K b { ( A) α β L 1 +C α+1 β L 1 bβ α + C β α αl 2 b1 α } u v 1 α B. From (2.25), (2.35) and te contraction mapping principle, we can conclude tat Φ as a unique fixed point x( ) in A(ϕ, α, b). From te assumptions on F and G, it follows tat te functions t G(t, ( A) α x t ) and t ( A) β F (t, ( A) α x t ) are continuous and bounded on [, b]. In te following N will be a positive constant suc tat ( A) β F (t, ( A) α x t ) N, G(t, ( A) α x t ) N, for all t [, b]. Next we will prove tat te functions t ( A) β F (t, ( A) α x t ) and t G(t, ( A) α x t ) are Hölder continuous on [, b]. From Lemma 1 and te condition β +Min{β α, ξ, γ 1, γ 2 } > 1, we fix < ϑ < min{β α, ξ, γ 1, γ 2 } wit ϑ + β > 1 and C > suc tat for all < s < t < b 1 and < < 1 (2.36) (2.37) (T () I)( A) α T (t s) C ϑ (t s) (ϑ+α), (T () I)( A) α+1 β T (t s) C ϑ (t s) (α+1 β+ϑ). For t [, b) and < < 1 wit t + < b we get

20 84 Eduardo Hernández x(t + ) x(t) C 1 ξ + C 2 1 α + ( A) α F (t +, ( A) α x t+ ) ( A) α F (t, ( A) α x t ) + t ( A) α+1 β (T () I)T (t s)( A) β F (s, ( A) α x s ) ds + t+ t ( A) α+1 β T (t + s)( A) β F (s, ( A) α x s ) ds + t+ t ( A) α (T () I) T (t s) G (s, ( A) α x s ) ds + t+ t ( A) α T (t + s)g(s, ( A) α x s ) ds. (2.38) We estimate eac term on te rigt and side of te last inequality separately. For te tird term we ave I 3 ( A) α β ( A) β F (t +, ( A) α x t+ ) ( A) β F (t, ( A) α x t ) ( A) α β {L1 γ 1 + L 1 x t+ x t B } ( A) α β L1 γ 1 + ( A) α β L 1M b x ( A) α ϕ B +( A) α β L 1 K b equivalently, sup x(θ + ) x(θ) θ [,t] I 3 C 3 γ 1 + ( A) α β L 1 M b x ( A) α ϕ B (2.39) + ( A) α β L 1 K b sup θ [,t] x(θ + ) x(θ), for some constant C 3, independent of t and. Wit respect to te fourt term, we get I 4 = t (T () I)( A) α+1 β T (t s)( A) β F (s, ( A) α x s ) ds t ϑ N C ds (t s) α+1 β+ϑ tβ α ϑ C N ϑ, β α ϑ

21 Regularity of Solutions of Partial Neutral Functional 85 wic can be abbreviated as (2.4) I 4 C 4 ϑ, were C 4 is a constant independent of t and. For I 5 we find tat I 5 t+ t ten ( A) α+1 β T (t + s)( A) β F (s, ( A) α x s ) ds C α+1 β N β α β α C 5 ϑ (2.41) I 5 C 5 ϑ were C 5 is a constant independent of t [, b) and < < 1. In a similar manner we can prove tat (2.42) I 6 C 6 ϑ and I 7 C 7 ϑ, were C 6 and C 7 are positive constants independents of t [, b) and < < 1. Using te estimates (2.39)-(2.42), tere exists a constant C 1 >, independent of t [, b) suc tat for < < 1 wit < t+ < b, x(t + ) x(t) C 1 ϑ + ( A) α β L 1 M b x ( A) α ϕ B + ( A) α β L 1 K b sup x(θ + ) x (θ) θ [,t] consequently (2.43) x(θ + ) x(θ) [,t] C 1 ϑ 1 µ + M bl 1 1 µ ( A)α β x ( A) α ϕ B since < µ = K b1 ( A) α β L 1 < 1. Moreover, using te definition of y α and te decomposition of x(t) = Φ(x)(t) as

22 86 Eduardo Hernández indicated in (2.26), it is possible to prove tat x ( A) α ϕ B W ()( A) α ϕ ( A) α ϕ B + 3 i=1 z i B C 8 ξ + z 1 B +C 9 ϑ. (2.44) Now from axiom (A) K sup T (s)( A) α F (, ϕ) ( A) α F (s, ( A) α x s ) B z 1 s [,] (2.45) and for s T (s)( A) α F (, ϕ) ( A) α F (s, ( A) α x s ) (2.46) T (s)( A) α F (, ϕ) ( A) α F (, ϕ) +L 1 s γ 1 ( A) α β +L 1 ( A) α β ( A) α ϕ x s B C 1 ( 1 α + γ 1 ) + L 1 ( A) α β ( A) α ϕ x s B. Employing tis last inequality in (2.45), it follows tat: (2.47) z 1 B C 1 K b ( 1 α + γ 1 ) +K b L 1 A α β A α ϕ x τ τ [,]. From (2.47), (2.44), te coice of b and te fact tat ϑ < Min{β α, ξ, γ 1, γ 2 }; for < < 1 we find tat tus (2.48) x τ ( A) α ϕ τ [,] C 11 ϑ +K b L 1 ( A) α β ( A) α ϕ x τ τ [,] x ( A) α ϕ B C 11 1 µ ϑ. Te inequalities (2.48) and (2.43) sow tat tere exists C 12 >, independent of θ [, b) and >, suc tat (2.49) x(θ + ) x(θ) C 12 ϑ

23 Regularity of Solutions of Partial Neutral Functional 87 were ϑ < Min{β α, ξ, γ 1, γ 2 } and ϑ + β > 1. From (2.48), (2.49) and axiom (A) follows tat te functions ( A) β F (t, ( A) α x t ) and G(t, ( A) α x t ) are ϑ-hölder wit ϑ+ β > 1. We know from Corollary (1), tat te abstract Caucy problem (2.5) G(t, ( A) α x t ), w (t) = Aw(t) + ( A) 1 β (( A) β F (t, ( A) α x t )) + x() = ϕ() + F (, ϕ), as a unique classical solution y C((, b]; X 1 ) wic is given by y(t) = T (t)(ϕ() + F (, ϕ)) + t ( A)1 β T (t s)( A) β (2.51) F (s, ( A) α x s )ds + t σ T (t s)g(s, ( A) α x s )ds. Operating on (2.51) wit ( A) α and using te ideas in te proof of Lemma 2, follows tat ( A) α y(t) = x(t)+( A) α F (t, ( A) α x t ), wic in turn implies tat z = ( A) α x C((, b]; X 1 ) since t F (t, x t ) is continuous wit values in X 1. Let z : (, b] X a extension of z suc tat z = ϕ. Clearly, z is a N-classical solution of te neutral problem (1.1). Te proof is complete. Now we turn our attention to te problem of existence of classical solutions. In te rest of tis paper, for a function j : [, a] B X and IR we use te notation j for te function j(t) := j(t +, ψ) j(t, ψ). Moreover, if j is differentiable we will employ te following decomposition (2.52) j(t + s, ψ + ψ 1 ) j(t, ψ) = (D 1 j(t, ψ), D 2 j(t, ψ))(s, ψ 1 ) + (s, ψ 1 ) R(j, t, ψ, s, ψ 1 )

24 88 Eduardo Hernández were (2.53) R(j, t, ψ, s, ψ 1 ) as (s, ψ 1 ). To prove Teorem 3 below, we will employ te following property. Lemma 4. Let X, Y be Banac spaces, Ω X open, K Ω compact and f : Ω X Y be a continuously differentiable function. Ten, for every ɛ > tere exists δ > suc tat f(x) f(y) (D x f)(x y) ɛ x y, for every x, y K suc tat x y < δ. Te next result establises te existence of classical solutions for te neutral system (1.1), making use of usual regularity assumptions for te functions ( A) β F and G. Teorem 3. Let assumptions in Teorem 1 be satisfied. Assume tat ϕ D(A W ), tat ( A) β F and G are continuously differentiable on [, a] Ω, tat F is continuous wit values in X 1 and tat D(( A) β F )(, ϕ). If X G(,ϕ) or X G(,ϕ) B and B satisfies axiom C 3, ten tere exists a unique classical solution of te system (1.1) defined on [, b] for some < b < a. Proof. Let u := u(, ϕ) te mild solution of (1.1). In te following we assume tat u( ) is defined on (, 2b] were < 2b < a and µ = K 2b [ D 2 F (s, u s ) + bβ C 1 β β (2.54) +b M ] D 2 G(s, u s ) 2b < 1. D 2 ( A) β F (s, u s ) Let z( ) be te solution of te integral equation

25 Regularity of Solutions of Partial Neutral Functional 89 (2.55) z(t) = T (t)aϕ() + p(t) D 2 F (t, u t )(z t ) + + t t wit initial condition ( A) 1 β T (t s)[d 2 ( A) β F (s, u s )](z s )ds T (t s)d 2 G(s, u s )(z s )ds, t, (2.56) z = A W (ϕ) + X G(,ϕ), and were p(t) = D 1 F (t, u t ) + + t t ( A) 1 β T (t s)d 1 ( A) β F (s, u s )ds T (t s)d 1 G(s, u s )ds + T (t)g(, ϕ). Te existence and uniqueness of local solution to te integral equation (2.55)-(2.56) is clear and we omit te proof. In wat follows we assume tat z( ) C([, b] : X). We affirm tat u ( ) = z( ) on [, b]. In order to prove te assertion, for t [, b] and < < 1 sufficiently small we ave u (t) z (t) [ T () I T (t) ϕ() Aϕ()] ( ) T () I + T (t) F (, ϕ) + 1 ( A)1 β T (t + s)( A) β F (s, u s ) ds F (t +, u t+ ) + F (t, u t ) + + D 1 F (t, u t ) + D 2 F (t, u t )(z t ) + t C 1 β (t s) 1 β ( A) β F (s, u s ) D 1 ( A) β F (s, u s ) D 2 ( A) β F (s, u s )(z s ds

26 9 Eduardo Hernández 1 + T (t + s)g(s, u s)ds T (t)g(, ϕ) + t M G(s, u s ) D 1 G(s, u s ) D 2 G(s, u s )(z s ) ds Next we use te notations I i (t, ), i = 1,...6, for te terms of te rigt and side of te last inequality. It is clear tat (2.57) I i (t, ), as, i {1, 2, 5} uniformly for t [, b]. On te oter and for te tird term I 3 (t, ) = F (t, u t ) + D 1 F (t, u t ) + D 2 F (t, u t )(z t ) D 2 F (t, u t )( u t+ u t z t ) + (, u t+ u t ) R(F, t, u t,, u t+ u t ). Since te function t x t is Lipscitz continuous on [, b], see Proposition (3.1) in [5], follows from Lemma 4 we get (, u t+ u t )) (2.58) (2.59) R(F, t, u t,, u t+ u t ) as, uniformly for t [, b]. Consequently, we can rewrite te last inequality in te form (2.6) I 3 (t, ) ξ 3 (t, )+ D 2 F (t, u t ) u t+ u t B z t were ξ 3 (t, ) if uniformly for t [, b]. Using similar arguments, we ave for I 4 tat I 4 (t, ) t C 1 β D2 ( A) β F (s, u (t s) 1 β s ) u s+ u s z B s ds + t R(( A) β F, s, u s,, u s+ u s ) ds (,u s+ u s)

27 Regularity of Solutions of Partial Neutral Functional 91 and so, from (2.58) we get I 4 (t, ) ξ 4 (t, ) + t C 1 β D2 ( A) β F (s, u (t s) 1 β s ) (2.61) ( u s+ u s z s ) ds B were ξ 4 (t, ) as uniformly for t [, b]. Analogously, for I 6 we see tat t I 6 (t, ) ξ 6 (t, )+ M D 2 G(s, u s ) (u s+ u s z s ) ds B (2.62) were ξ 6 (t, ) as uniformly for t [, b]. Combining (2.57), (2.6), (2.61), (2.62) and te first inequality we get u(t) z(t) ( ) ut+ u ξ 7 (t, ) + D 2 F (t, u t ) t B z t + t C ( ) 1 β D (t s) 1 β 2 ( A) β us+ u F (s, u s ) s B z s ds + t M ( ) us+ u D 2 G(s, u s ) s B z s ds, were ξ 7 (t, ) as uniformly for t [, b]. Using axiom (A) and (2.54) we infer tat, u( ) z( ) [,t] 1 max 1 µ s [,t] ξ 7 (s, ) + M 2bµ + u ϕ B z. (1 µ)k 2b Clearly u ( ) = z( ) if 1 (u ϕ) z as. Next we will prove tis convergence. For > we consider te decomposition (2.63) ( u ϕ B A W (ϕ) X G(,ϕ) W ()ϕ ϕ z 1 + z 2 A W (ϕ) + B B z 3 + X G(,ϕ) B

28 92 Eduardo Hernández were z i = on (, ] and z 1 (θ) = T (θ)f (, ϕ) F (θ, u θ ) z 2 (θ) = z 3 (θ) = θ θ ( A) 1 β T (θ s)( A) β F (s, u s )ds T (θ s)g(s, u s )ds, for θ [, b]. Let I i (), i = 1, 2, 3, be te terms of te rigt and side of (2.63). Clearly (2.64) I 1 () as, since ϕ D(A W ). On te oter and, in bot X G(,ϕ) = or X G(,ϕ), te ypotesis imply tat (2.65) I 3 () as. For te second term we get (2.66) I 2 () 1 K b (T (θ) I)F (, ϕ) + θ ( A)1 β T (θ s)( A) β F (s, u s )ds 1 +K b F (, ϕ) F (θ, u θ) θ [,]. θ [,] Moreover for θ [, ], 1 (T (θ) I) F (, ϕ) + θ ( A)1 β T (θ s)( A) β F (s, u s )ds 1 θ ( A) 1 β T (θ s){( A) β F (s, u s ) ( A) β F (, ϕ)} ds 1 θ C 1 β L (θ s) (s+ ϕ u 1 β s B )ds 1 C 1 βl( + C) β β

29 Regularity of Solutions of Partial Neutral Functional 93 were we use te Lipscitz continuity of s u s, see Proposition (3.1) in [5]. Consequently 1 (T (θ) I)F (, ϕ) + θ (2.67) Similarly, we can to prove tat (2.68) ( A)1 β T (θ s)( A) β F (s, u s )ds, as. F (, ϕ) F (θ, u θ ) [,] as, since DF (, ϕ). Using (2.67) and (2.68) in (2.66), we conclude tat (2.69) I 2 (). Now, te convergence of 1 (u ϕ) to z follows from (2.64), (2.65), (2.69) and (2.63). We know from Corollary 1, tat te unique mild solution, y( ), of te Caucy problem w (t) = Aw(t) + ( A) 1 β (( A) β F (t, u t )) + G(t, u t ), w () = ϕ() + F (, ϕ), t (, b), is a classical solution. Consequently, y = d dt (u(t) + F (t, u t)) is continuous on [, b] and u(t) D(A) for every t [, b], since ϕ() D(A) and F ([, a] Ω) D(A). Tus u( ) is a classical solution of (1.1). Te proof is complete. Acknowledgement: Te autor wises to tank to te referees for teir comments and suggestions.

30 94 Eduardo Hernández References [1] Goldstein, Jerome A. Semigroups of linear operators and applications. Oxford Matematical Monograps. Te Clarendon Press, Oxford University Press, New York, (1985). [2] Hale, Jack K.; Kato, Junji.(1978) Pase space for retarded equations wit infinite delay. Funkcial. Ekvac. 21, No. 1, pp [3] Hale, Jack K.; Verduyn Lunel, Sjoerd M. Introduction to functional-differential equations. Applied Matematical Sciences, 99. Springer-Verlag, New York, (1993). [4] Hernández, Eduardo; Henríquez, Hernán R. Existence of periodic solutions of partial neutral functional-differential equations wit unbounded delay. J. Mat. Anal. Appl. 221, No. 2, PP , (1998). [5] Hernández, Eduardo; Henríquez, Hernán R. Existence results for partial neutral functional-differential equations wit unbounded delay. J. Mat. Anal. Appl. 221 (1998), no. 2, [6] Henríquez, Hernán R. Regularity of solutions of abstract retarded functional-differential equations wit unbounded delay. Nonlinear Anal. 28, No. 3, pp , (1997). [7] Henríquez, Hernán R. Periodic solutions of quasi-linear partial functional-differential equations wit unbounded delay. Funkcial. Ekvac. 37, No. 2, pp , (1994). [8] Hino, Yosiyuki; Murakami, Satoru; Yosizawa, Taro. Existence of almost periodic solutions of some functionaldifferential equations wit infinite delay in a Banac space. Tôoku Mat. J. (2) 49, No. 1, pp , (1997). [9] Krein, S. G. Linear differential equations in Banac space. Translated from te Russian by J. M. Danskin. Translations of Matematical Monograps, Vol. 29. American Matematical Society, Providence, R. I., (1971).

31 Regularity of Solutions of Partial Neutral Functional 95 [1] Pazy, A. Semigroups of linear operators and applications to partial differential equations. Applied Matematical Sciences, 44. Springer-Verlag, New York-Berlin, (1983). [11] B. N. Sadovskii. On a fixed point principle. Funct. Anal. Appl. 1, pp , (1967). [12] Travis, C. C.; Webb, G. F. Existence and stability for partial functional differential equations. Trans. Amer. Mat. Soc. 2, pp , (1974). Received : December 21.. Eduardo Hernández Instituto de Ciências Matemáticas e de computação Universidade de São Paulo Campus de São Carlos Caixa Postal 668, , São Carlos, SP Brazil lalom@icmc.sc.usp.br.