Key words: semilinear functional dierential equations

FOR NEUTRAL FUNCTIONAL EVOLUTION EQUATIONS

ABDESSALAM BALIKI and MOUFFAK BENCHOHRA Communicated by Gabriela Marinoschi

In this paper, we prove the global existence and attractivity of mild solutions for neutral semilinear evolution equations with delay in a Banach space.

AMS 2010 Subject Classication: 34G20, 34G25, 34K20, 34K30.

Key words: semilinear functional dierential equations. mild solution, attractiv- ity, evolution system, xed-point, innite delay, innite interval.

1. INTRODUCTION

Dierential equations with delay are used in many elds of science specif- ically equations with delay. The latter are called neutral dierential equations which the delayed argument occurs in the derivative of the state variable as well as in the independent variable. We refer the reader to the books Hale and Lunel [17], Lakshmikantham et al. [24], Kolmanovskii and Myshkis [23]. On the other hand, the neutral dierential equations have become more important in some mathematic models of real phenomena, especially in control, biological, and medical domains. For more details, we refer the reader to [9, 14, 16, 17].

In the literature there are many papers study the problems of neutral dierential equations using dierent methods. Among them, the xed point method combined by semigroup theory in Frechet space, see for exemple Baghli and Benchohra [4, 5, 6] and Hernandez et al. [18, 19, 20].

In [26, 27], Travis and Webb are the rst who considered the existence and stability of the dierential equations with delay, recently the study of equations of this type attracted the attention of many authors for their consideration in the study of dierent real phenomena see [2, 7, 8, 13, 21].

Inspired by the above-mentioned works, we consider in this paper some sucient conditions for the existence and attractivity of mild solutions of the following neutral evolution equation

(1)

_d

dt[y(t)−g(t, y_t)]−A(t)y(t) =f(t, y_t), t∈J := [0,∞)

y(t) =φ(t) t∈(−∞,0],

REV. ROUMAINE MATH. PURES APPL. 60 (2015), 1, 7182

where B is an abstract phase space to be specied later, f and g is a given function fromJ× B intoE, andφ∈ B is a given functions and {A(t)}_0≤t<+∞

is a family of linear closed (not necessarily bounded) operators fromE intoE that generate an evolution system of operators {U(t, s)}_(t,s)∈J×J for 0 ≤ s≤ t <+∞.

For any continuous functionyand anyt≥0, we denote byytthe element of B dened by y_t(θ) = y(t+θ) for θ ∈ (−∞,0] : Here y_t(·) represents the history of the state up to the present time t. We assume that the histories y_t belong to B.

As far as we know, there are few papers dealing with global existence results for the problem (1). Most of these results are stated in the Frechet space setting. The present paper provides sucient conditions for the existence and attractivity mild solutions to problem (1) in the Banach space setting.

2. PRELIMINARY

LetE a Banach space with the norm| · | andBC(J, E) the Banach space of all bounded and continuous functions y mapping J into E with the usual supremum norm

kyk= sup

t∈J

|y(t)|.

Let X be the space dened by

X ={y:R→E such thaty|_J ∈BC(J, E) and y0∈ B}, we denote byy|_J the restriction ofy to J.

In this paper, we will employ an axiomatic denition of the phase space Bintroduced by Hale and Kato in [15] and follow the terminology used in [22].

Thus,(B,k·k_B)will be a seminormed linear space of functions mapping(−∞,0]

intoE, and satisfying the following axioms:

(A₁) If y : (−∞, b) → E, b > 0, is continuous on [0, b] and y₀ ∈ B, then for everyt∈[0, b) the following conditions hold:

(i) yt∈ B ;

(ii) There exists a positive constant H such that|y(t)| ≤Hky_tk_B ; (iii) There exist two functions K(·), M(·) : R+ → R+ independent of y withK continuous andM locally bounded such that :

ky_tkB≤K(t) sup{ |y(s)|: 0≤s≤t}+M(t)ky₀kB.

(A2) For the functiony in(A1),ytis aB−valued continuous function on[0, b].

(A₃) The spaceB is complete.

Remark 2.1. In the sequel we assume that K and M are bounded on J and

γ:= max (

sup

t∈R+

{K(t)},sup

t∈R+

{M(t)}

) .

For other details we refer, for instance to the book by Hino et al [22].

In what follows, we assume that{A(t), t≥0}is a family of closed densely dened linear unbounded operators on the Banach space E and with domain D(A(t))independent of t.

Denition 2.1. A family of bounded linear operators

{U(t, s)}_(t,s)∈∆:U(t, s) :E→E(t, s)∈∆ :={(t, s)∈J×J : 0≤s≤t <+∞}

is called en evolution system if the following properties are satised:

1. U(t, t) =I whereI is the identity operator inE, 2. U(t, s) U(s, τ) =U(t, τ) for 0≤τ ≤s≤t <+∞,

3. U(t, s) ∈ B(E) the space of bounded linear operators on E, where for every (s, t) ∈ ∆ and for each y ∈ E, the mapping (t, s) → U(t, s) y is continuous.

More details on evolution systems and their properties could be found on the books of Ahmed [1], Engel and Nagel [12] and Pazy [25].

Lemma 2.1 ([10]). Let C ⊂ BC(J, E) be a set satisfying the following conditions:

(i) C is bounded in BC(J, E);

(ii) the functions belonging to C are equicontinuous on any compact interval of J;

(iii) the set C(t) := {y(t) : y ∈ C} is relatively compact on any compact interval of J;

(iv) the functions from C are equiconvergent, i.e., given ε > 0, there corre- sponds T(ε) > 0 such that |y(t)−y(+∞)| < ε for any t ≥ T(ε) and y∈C.

Then C is relatively compact in BC(J, E).

Theorem 2.1 ([3] Buton-Kirk's xed point theorem). Let X Banach space, and A, B : X → X two operators. Suppose that B is a contraction and A a compact operator. Then either

(i) x=λB ^x_λ

+λAx has a solution forλ= 1, or (ii) the set {x∈X:x=λB ^x_λ

+λAx, λ∈(0,1)} is unbounded.

3. MAIN RESULT

Denition 3.1. A function y ∈ X is said to be a mild solution of the problem (1), if

(2) y(t) =















φ(t), if t≤0

U(t,0)(φ(0)−g(0, φ)) +g(t, yt) +

Z t 0

U(t, s)A(s)g(s, yt)ds+ Z t

0

U(t, s)f(s, yt)ds, if t∈J.

To prove our results we introduce the following conditions:

(H₁) There exists a constantMc≥1 and ω >0 such that

kU(t, s)k_B(E)≤M ec ^−ω(t−s) for every (s, t)∈∆.

(H2) There exists a functionp∈L¹(J,R+) such that:

|f(t, u)| ≤p(t)(kuk_B+ 1)for a.e. t∈J and eachu∈ B.

(H₃) For each (t, s)∈∆we have: lim

t→+∞

Z t 0

e^−w(t−s)p(s)ds= 0. (H₄) There exists a constantM >f 0 such that:

kA⁻¹(t)k_B(E)≤Mf for all t∈J.

(H5) There exists a constant` >0such that

|A(t)g(t, φ)−A(s)g(s, ϕ)| ≤`(|t−s|+kφ−ϕkB) for all t, s∈J and φ, ϕ∈ B.

(H6) There exists a bounded continuous functionζ :J →R+ such that:

|A(t)g(t, φ)| ≤ζ(t)kφk_B for all t∈J, φ∈ B.

Theorem 3.1. Assume (H₁)−(H₆) are satised, and if γ `Mf+Mc

ω

!

<1, and

M ζf ^∗γ+ M ζc ^∗

ω +M γ <c 1.

Then the problem (1) admits at least one mild solution.

Proof. It is clear that we will obtain the results if we show that the operator T :X → X dened by:

(3) T y(t) =























φ(t), if t≤0

U(t,0)(φ(0)−g(0, φ)) +g(t, yt) +

Z t 0

U(t, s)A(s)g(s, yt)ds +

Z t 0

U(t, s)f(s, yt)ds, if t∈J.

has a xed point.

For φ ∈ B, we can introduce the following function x : (−∞,+∞) → E by

x(t) =







φ(t), if t∈(−∞,0]

U(t,0)φ(0) if t∈J.

Then x0 =φ. For each functionz∈ X, set y(t) =x(t) +z(t).

It is obvious thaty satises (2) if and only ifzsatisesz₀= 0 and for all t∈J

z(t) = U(t,0)g(0, φ) +g(t, xt+zt) + Z t

0

U(t, s)A(s)g(s, xs+zs)ds +

Z t 0

U(t, s)f(s, x_s+z_s)ds.

Let

X₀ ={z∈ X : z0= 0}.

The X₀ is a Banach space with norm kzk_X0 = sup

t∈J

|z(t)|+kz₀k_B = sup

t∈J

|z(t)|

Now, dene the operators F, L:X₀→ X₀ by F z(t) =

Z t 0

U(t, s)f(s, zs+xs)ds, for t∈J, and

Lz(t) =U(t,0)g(0, φ) +g(t, xt+zt) + Z t

0

U(t, s)A(s)g(s, xs+zs)ds.

Obviously, the problem (1) has a solution is equivalent to F +L has a xed point. To prove this end, we start with the following estimation.

For each z∈ X₀ andt∈J, we have kz_t+xtk_B ≤ kz_tk_B+kx_tk_B

≤ K(t)|z(t)|+K(t)kU(t,0)k_B(E)kφk_B+M(t)kφk_B

≤ γkzkX₀ +γM ec ^−ωtkφkB+γkφkB

≤ γkzk_X0 +γ(Mc+ 1)kφk_B. (4)

Now, we prove that the operators F, L satised conditions of Theorem 2.1.

Step 1. F is continuous and compact.

• F is continuous. Let (z^k)k∈N be a sequence in X₀ such that z^k → z inX₀. We get for every t∈J

|F(z^k)(t)−F(z)(t)| ≤ Z t

0

kU(t, s)k_B(E)|f(s, z_s^k+x_s)−f(s, z_s+x_s)|ds

≤ Mc Z t

0

e^−ω(t−s)|f(s, z_s^k+xs)−f(s, zs+xs)|ds.

Since f is continuous, we obtain by the Lebesgue dominated convergence theorem that

kF z_k−F zk_X0 →0 as k→+∞.

Thus, F is continuous.

• F(D) relatively compact. LetD is a bounded sub set of X₀. Then, we will Lemma 2.1.

Let η≥0 such thatD={z∈ X₀:kxk_X0 ≤η}, then for z∈Dwe have

|F(z)(t)| ≤ Z _t

0

kU(t, s)k_B(E)|f(s, z_s+x_s)|ds

≤ Mc Z t

0

e^−ω(t−s)p(s)(kz_s+xsk_B+ 1)ds

≤ Mc(γkzkX₀ +γ(Mc+ 1)kφkB+ 1) Z t

0

e^−ω(t−s)p(s)ds

≤ M ξkpkc _L1, withξ :=γη+γ(Mc+ 1)kφk_B+ 1.

Thus, F(D) is bounded.

• F(D) is equicontinuous. Let s, t∈[0, b]witht > s and z∈D. Then

|(F z)(t)−(F z)(s)| =

Z s 0

(U(t, τ)−U(s, τ))f(τ, zτ +xτ)dτ

+ Z t

s

U(t, τ)f(τ, z_τ+x_τ)dτ

≤ Z s

0

kU(t, τ)−U(s, τ)k_B(E) p(τ)(kz_τ +x_τk_B+ 1)dτ + Mc

Z _t

s

e^{−ω(t−τ)} p(τ)(kz_τ +x_τk_B+ 1)dτ.

Using (4) we get

|(F z)(t)−(F z)(s)| ≤ ξ Z s

0

kU(t, τ)−U(s, τ)k_B(E) p(τ)dτ +M ξc Z t

s

p(τ)dτ.

The right-hand side of the above inequality tends to zero as t−s → 0, thenF(D) is equicontinuous.

Now, we will prove that Λ :={(F z)(t) : z∈D} is relatively compact in E. Let t∈J be a xed and let0< ε < t≤b. For z∈Dwe dene

F_ε(z)(t) =U(t, t−ε) Z t−ε

0

U(t−ε, s)f(s, z_s+x_s)ds.

SinceU(t, s)is a compact operator, and the setΛε:={(F_εz)(t) : z∈D}

is the image of bounded set of E by U(t, s) then Λ_ε is precompact in E for every0< ε < t. Furthermore, for z∈D, we have

|F(z)(t)−Fε(z)(t)| ≤ Z t

t−ε

kU(t, s)k_B(E)|f(s, zs+xs)|ds

≤ Z t

t−ε

kU(t, s)k_B(E)p(s)(kz_s+xsk_B+ 1)ds

≤ ξMc Z t

t−ε

e^−ω(t−s)p(s)ds.

The right-hand side tends to zero asε→0, thenFε(z)converge uniformly to F(z)which implies thatD(t)is precompact inE.

Finally, F is equiconvergent.

Let z∈D, then from(H₁),(H₂) and (4) we have

|(F z)(t)| ≤M ξc Z t

0

e^−ω(t−s)p(s)ds, it follows immediately by (4) that |(F z)(t)| −→0ast→+∞.

Then

t→+∞lim |(F z)(t)−(F z)(+∞)|= 0 which implies thatF is equiconvergent.

Step 2. L is a contraction. Takez,z¯∈ X₀, then for each t ∈J and by (H₁),(H₄),(H₅) and (4)

|(Lz)(t)−(L¯z)(t)| ≤ |g(t, x_t+z_t)−g(t, x_t+ ¯z_t)|

+ Z _t

0

kU(t, s)k_B(E)|A(s)g(s, x_s+z_s)−A(s)g(s, x_s+ ¯z_s)|ds

≤ kA⁻¹(t)k_B(E)|A(t)g(t, x_t+z_t)−A(t)g(t, x_t+ ¯z_t)|

+ Mc Z t

0

e^−ω(t−s)|A(s)g(s, x_s+zs)−A(s)g(s, xs+ ¯zs)|ds

≤ `fMkz_t−z¯_tk_B+Mc Z t

0

e^−ω(t−s)kz_s−z¯_sk_Bds

≤ γ

`Mf+Mc Z t

0

e^−ω(t−s)ds

kz−zk¯ _X0

≤ γ `fM +Mc ω

!

kz−zk¯ _X0. Therefore,

kLz−L¯zk_X0 ≤γ `Mf+ Mc ω

!

kz−zk¯ _X0. Thus, the operator Lis a contraction.

For applying the Theorem 2.1, we must check hypothesis (ii) is not hold, i.e. prove that the set

D_λ= n

z∈ X₀ :z=λL z

λ

+λF(z) for λ∈(0,1) o

, is bounded. Letz∈D_λ then for eacht∈J,z(t) =λL _λ^z

(t) +λF(z)(t) then we obtain

|z(t)| ≤ λkU(t,0)k_B(E)kA⁻¹(t)k_B(E)|A(t)g(0, φ)|

+ kA⁻¹(t)k_B(E)|A(t)g(t, x_t+z_t λ)|

+ λ

Z t 0

kU(t, s)k_B(E)|A(s)g(s, x_s+z_s λ)|ds

+ λ

Z _t

0

kU(t, s)k_B(E)|f(s, x_s+z_s)|ds

≤ λMcM ζ(t)kφkf _B+λfM ζ(t)kx_t+ zt

λk_B + λMc

Z t 0

e^−ω(t−s)ζ(s)kx_s+z_s λk_Bds + λMc

Z t 0

e^−ω(t−s)p(s)(kx_s+zsk_B+ 1)ds

≤ McM ζf ^∗kφk_B+M ζf ^∗γ(kzk_X0 +µ) + M ζc ^∗

ω (kzk_X0 +µ) +M γ(kzkc _X0 +µ+ 1)kpk_L1, withµ:= (Mc+ 1)kφk_B and ζ^∗ := sup_t∈J|ζ(t)|.

Therefore,

kzk_X0 ≤ McM ζf ^∗kφk_B+M ζf ^∗γµ+^{M ζc}_ω^∗µ+M γ(µc + 1)kpk_L1

(1−M ζf ^∗γ− ^{M ζc}_ω^∗ −M γ)c

:=c, which implies thatD_λ is bounded.

Thus, by Theorem 3.1 the operator T has at least one xed point which is a mild solution of problem (1).

4. ATTRACTIVITY OF SOLUTIONS

In this section, we study the attractivity of solutions the problem (1) Denition 4.1 ([11]). We say that solutions of (1) are locally attractive if there exists a closed ball B(z^∗, ρ) in the space X₀ for some z^∗ ∈ X such that for arbitrary solutionsz and ezof (1) belonging to B(z^∗, ρ) we have that

t→+∞lim (z(t)−z(t)) = 0.e

Under the assumption of Section 3, let z^∗ a solution of (1) and B(z^∗, ρ) the closed ball in X₀ witch ρ satises the following inequality

ρ≥ 2Mkpkc _L1

1−M `γf −<sup>M `γc</sup>_ω −2M γkpkc _L1

. Moreover, we assume that

(5) lim

t→∞ζ(t) = 0 and lim

t→∞

Z t 0

e^−ω(t−s)ζ(s)ds= 0,

which ζ is the function in (H6). Then, forz ∈B(z^∗, ρ) by (H1)(H2) and (4) we have

|(T z)(t)−z^∗(t)| = |(T z)(t)−(T z^∗)(t)|

≤ kA⁻¹(t)k_B(E)|A(t)g(t, z_t+xt)−A(t)g(t, z_t^∗+xt)|

+ Z t

0

kU(t, s)k_B(E)|f(s, z_s+x_s)−f(s, z_s^∗+x_s)|ds +

Z t 0

kU(t, s)k_B(E)|A(s)g(s, z_s+xs)−A(s)g(s, z_s^∗+xs)|ds

≤ M `kzf <sub>t</sub>−z<sup>∗</sup><sub>t</sub>kB+M `c Z t

0

e^−ω(t−s)kz_s−z_s^∗kBds + Mc

Z t 0

e^−ω(t−s)p(t)(kz_s+x_sk_B+kz_s^∗+x_sk_B+ 2)ds

≤ M `γρf + M `γρc

ω + 2Mc(γρ+ 1)kpk_L1

≤ M `γρf + M `γρc

ω + 2Mc(γρ+ 1)kpk_L1 ≤ρ.

Therefore, we get T(B(z^∗, ρ)) ⊂ B(z^∗, ρ). So, for each z ∈ B(z^∗, ρ) solution of problem (1) andt∈J, we have

|z(t)−z^∗(t)| = |(T z)(t)−(T z^∗)(t)|

≤ kA⁻¹(t)k_B(E)|A(t)g(t, z_t+xt)−A(t)g(t, z_t^∗+xt)|

+ Z _t

0

kU(t, s)k_B(E)|f(s, zs+xs)−f(s, z^∗_s+xs)|ds +

Z t 0

kU(t, s)k_B(E)|A(s)g(s, z_s+xs)−A(s)g(s, z_s^∗+xs)|ds

≤ M ζ(t)(kzf _t+xtk_B+kz^∗_t +xtk_B) + Mc

Z t 0

e^−ω(t−s)p(s)(ψ(kz_s+xsk_B) +ψ(kz_s^∗+xsk_B))ds + Mc

Z t 0

e^−ω(t−s)ζ(s)(kz_s+xsk_B+kz_s^∗+xsk_B)ds

≤ 2γMf(ρ+µ)ζ(t) + 2M ψ(γρc +γµ) Z t

0

e^−ω(t−s)p(s)ds + 2γMc(ρ+µ)

Z _t

0

e^−ω(t−s)ζ(s)ds.

Hence, from (H₃) and (5), we conclude that

t→∞lim |z(t)−ez(t)|= 0.

Consequently, the solutions of equation (1) are locally attractive.

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Received 14 November 2013 University of Sidi Bel-Abbes Laboratory of Mathematics, PO Box 89, 22000 Sidi Bel-Abbes, Algeria

[email protected] [email protected] King Abdulaziz University,

Faculty of Science, Department of Mathematics, P.O. Box 80203, Jeddah 21589

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