Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 210, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
S-ASYMPTOTICALLY ω-PERIODIC SOLUTIONS FOR
ABSTRACT NEUTRAL DIFFERENTIAL EQUATIONS
MICHELLE PIERRI, DONAL O’REGAN
Abstract. In this article we study the existence of S-asymptotically ω-periodic solutions for abstract neutral functional differential equations recently,
introduced in the literature. An application involving a partial neutral differential equation is presented.
1. introduction
In this work we study the existence of S-asymptotically ω-periodic solutions for
a class of abstract neutral differential equations of the form
u0 (t) = Au(t) + f (t, ut , u0t ),
t ∈ [0, ∞),
u0 = ϕ ∈ B,
(1.1)
(1.2)
where A : D(A) ⊂ X → X is the infinitesimal generator of an analytic semigroup of
bounded linear operators (T (t))t≥0 defined on a Banach space (X, k · k), the history
ut belongs to an abstract Banach space (B, k · kB ) defined axiomatically, u0t denotes
the derivative at t of the function s → us and f (·) is a suitable function.
The neutral system (1.1)–(1.2) was introduced recently by Hernández and O’
Regan [21]. As pointed out in [21], the study of the existence of solutions for this
class of problems via semigroup methods and fixed point techniques is highly nontrivial since the temporal derivative of the solution appears in the integral equation
used to define the concept of mild solution of (1.1)–(1.2), see Definition 2.4. As a
consequence, it is necessary to work on spaces of differentiable functions which is a
complex problem under the semigroup framework.
To the best of our knowledge, the paper [21] is the first and only work treating
neutral problems described in the abstract form (1.1)–(1.2). On the current state
of the theory of abstract neutral differential equations we cite [1, 7, 10, 11, 15, 16,
17, 18, 19, 20, 22, 23, 33] and the references therein.
The concept of S-asymptotically ω-periodic was introduced recently in the literature, see [13, 14]. We note that a continuous function u(·) defined on [0, ∞) is
said to be S-asymptotically periodic if there exists ω ∈ R such that limt→∞ [f (t +
ω) − f (t)] = 0.
2010 Mathematics Subject Classification. 34K30, 34K40, 34K14, 35B15, 47D06.
Key words and phrases. Neutral equation; S-asymptotically ω-periodic function;
analytic semigroup; strict solutions; partial delayed differential equation.
c
2015
Texas State University - San Marcos.
Submitted January 30, 2014. Published August 10, 2015.
1
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M. PIERRI, D. O’REGAN
EJDE-2015/210
For qualitative properties of S-asymptotically ω-periodic functions, we cite [13,
28, 14]. Concerning the problem of the existence of S-asymptotically ω-periodic
solutions for differential equations, we cite [8, 12, 31, 32, 34] for ordinary differential
equations on finite dimensional spaces and [4, 5, 6, 13, 14] for ordinary differential
equations defined on abstract Banach spaces.
The abstract problem (1.1)–(1.2) arises, for example, in the theory of heat conduction in fading memory material. In the classical theory of heat conduction, it
is assumed that the internal energy and the heat flux depends linearly on the temperature u(·) and on its gradient ∇u(·). Under these conditions, the classical heat
equation describes sufficiently well the evolution of the temperature in different
types of materials. However, this description is not satisfactory in materials with
fading memory. In the theory developed in [9, 29], the internal energy and the heat
flux are described as functionals of u and ux . The next system, see [2, 3, 25], has
been frequently used to describe these phenomena,
d
(u(t, x) +
dt
Z
t
Z
t
k1 (t − s)u(s, x)ds) = c∆u(t, x) +
−∞
k2 (t − s)∆u(s, x)ds,
−∞
x ∈ ∂Ω.
u(t, x) = 0,
(1.3)
In this system, Ω ⊂ Rn is open, bounded and has smooth boundary, (t, x) ∈
[0, ∞) × Ω, u(t, x) denotes the temperature in x at time t, c is a physical constant
and ki : R → R, i = 1, 2, are the internal energy and the heat flux relaxation
respectively. If we assume that k1 = γ1 + γ2 and the solution u(·) is known on
(−∞, 0], we can study the above system via the initial-value problem
d
(u(t, x) +
dt
Z
t
γ1 (t − s)u(s, x)ds)
−∞
Z
t
Z
t
k2 (t − s)∆u(s, x)ds −
= c∆u(t, x) +
−∞
γ2 (t − s)u0 (s, x)ds,
(1.4)
−∞
u(s, x) = ϕ(s, x),
s ≥ 0, x ∈ Ω,
which can be represented in the abstract form (1.1)–(1.2). For additional applications and examples on neutral differential equations we cite our recent papers
[18, 21] and the references therein.
Next, we include some notations, definitions and technicalities. Let (Z, k · kZ )
and (W, k · kW ) be Banach spaces. In this paper, L(Z, W ) denotes the space of
bounded linear operators from Z into W endowed with the norm of operators
denoted k · kL(Z,W ) and we write L(Z) and k · kL(Z) when Z = W . We use the
notation Z ,→ W to indicate that Z is continuously included in W .
Let I ⊂ R. As usual, C(I; Z) is the space formed by all the bounded continuous functions from I into Z endowed with the sup-norm denoted by k · kC(I;Z)
and C 1 (I; Z) the space formed by all the functions u ∈ C(I; Z) such that u0 ∈
C(I; Z) endowed with the norm kukC 1 (I;Z) = kukC(I;Z) + ku0 kC(I;Z) . In addition, C γ ([0, ∞); Z) (with γ ∈ (0, 1)) is the space formed by all the functions
ξ ∈ C([0, ∞); Z) such that
[ξ]C γ ([0,∞);Z) =
kξ(s) − ξ(t)kZ
| t − s |γ
t,s∈[0,∞),t6=s
sup
EJDE-2015/210
S-ASYMPTOTICALLY ω-PERIODIC SOLUTIONS
3
is finite, provided with the norm kξkC γ ([0,∞);Z) = kξkC([0,∞);Z) + [ξ]C γ ([0,∞);Z) .
The notation C 1+γ ([0, ∞); Z) is used for the space of all the differentiable functions ξ ∈ C γ ([0, ∞); Z) such that ξ 0 ∈ C γ ([0, ∞); Z) endowed with the norm
kξkC 1+γ ([0,∞);Z) = kξkC γ ([0,∞);Z) + kξ 0 kC γ ([0,∞);Z) .
From [13] we note the following concept.
Definition 1.1. A function f ∈ C([0, ∞), Z) is said to be S-asymptotically periodic
if there exists ω ∈ R such that limt→∞ [f (t + ω) − f (t)] = 0. In this case, we say
that ω is an asymptotic period of f (·) and that f (·) is S-asymptotically ω-periodic.
Here SAPω (Z) denotes the space formed by the Z-valued S-asymptotically ωperiodic functions provided with the norm k · kC([0,∞);Z) .
In this article, A : D(A) ⊂ X → X is the generator of an uniformly stable
analytic semigroup of bounded linear operators (T (t))t≥0 on X and γ, Ci , i ∈ N,
are positive constants such that
Ci e−γt
ti
for all t > 0 and each i ∈ N. For β > 0, we represent by Xβ the domain of
the fractional power (−A)β of −A endowed with the norm kxkβ = k(−A)β xk. In
addition, for β > 0 we assume that Cβ > 0 is such that
kAi T (t)kL(X) ≤
Cβ e−γt
, ∀t > 0.
tβ
The notation DA (η, ∞), η ∈ (0, 1), stands for the space
kAT (t)kL(Xβ ,X) ≤
DA (η, ∞) = {x ∈ X : [x]η,∞ = sup kt1−η AT (t)xk < ∞},
t∈(0,1)
with the norm kxkη,∞ = [x]η,∞ + kxk and we assume that for all k ∈ N ∪ {0} there
exits a constant Ck,η such that
Ck,η e−γt
t1−η
for all t > 0. For additional details on analytic semigroups and interpolation spaces
we cite [26].
In this article, (B, k · kB ) is a Banach space formed by functions defined from a
connected interval {0} ⊂ J ⊂ (−∞, 0] into X, satisfying the following conditions.
(A1) If x : (J + {σ}) ∪ [σ, σ + b) → X, b > 0, σ ∈ R, is continuous on [σ, σ + b)
and xσ ∈ B, then for every t ∈ [σ, σ + b) the following conditions hold:
(i) the function s → xs belongs to C([σ, σ + b), B),
(ii) kx(t)k ≤ Hkxt kB ,
(iii) kxt kB ≤ K(t − σ) sup{kx(s)k : σ ≤ s ≤ t} + M (t − σ)kxσ kB , where
H > 0 is a constant; K, M ∈ C([0, ∞); R+ ) and H, K(·), M (·) are
independent of x(·).
(iv) If (ψ n )n∈N is a sequence in C(J, X) ∪ B and ψ n → ψ uniformly on
compact subsets of J, then ψ ∈ B and kψ n − ψkB → 0 as n → ∞.
kAk T (t)kL(DA (η,∞),X) ≤
Remark 1.2. For β > 0, we represent by Bβ the space Bβ = {(−A)−β ψ : ψ ∈ B}
endowed with the norm kψkBβ = k(−A)β ψkBβ . We note that Bβ verifies the axiom
(A1) with Xβ in place X.
Remark 1.3. In the remainder of this paper, to simplify, we assume that K > 0
is a constant such that max{K(t), M (t)} ≤ K for all t ≥ 0.
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M. PIERRI, D. O’REGAN
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From from [21] we note the following result. In this result, Pu is the function
defined by Pu (t) = ut . We will use this notation for the remainder of this article.
Lemma 1.4. If u ∈ C 1 (J ∪ [0, b]; X), then Pu ∈ C 1 ([0, b]; B) and
for all t ∈ [0, b].
d
dt Pu (t)
= P du (t)
dt
This article has three sections. In the next section we study the existence of
S-asymptotically ω-periodic strict solutions for the problem (1.1)–(1.2). In the last
section, an application involving a partial neutral differential equation is presented.
2. Existence results
In this section we study the existence of S-asymptotically ω-periodic strict solution for the abstract neutral problem (1.1)–(1.2). To prove our results, we consider
the following conditions.
(H1) There are α ∈ (0, 1), β ∈ (0, 1] and functions f1 ∈ C α ([0, ∞); L(Bβ , X)),
f2 ∈ C α ([0, ∞); L(B, X)) such that f (t, ψ1 , ψ2 ) = f1 (t, ψ1 ) + f2 (t, ψ2 ) for
all t ≥ 0, ψ1 ∈ Bβ and ψ2 ∈ B.
(H2) There is a Banach space (Y, k · kY ) ,→ (X, k · k), a integrable function
H ∈ L1 ([0, ∞); R+ ), a continuous function Lf ∈ C([0, ∞); R+ ) and ω > 0
such that f ∈ C([0, ∞) × Bβ × B; Y ), kAT (s)kL(Y,X) ≤ H(s) for all s > 0,
and
kf (t, ψ1 , ζ1 ) − f (t, ψ2 , ζ2 )kY ≤ Lf (t)(kψ1 − ψ2 kBβ + kζ1 − ζ2 kB ),
(2.1)
for all ψi ∈ Bβ , ζi ∈ B, i = 1, 2, and every t ≥ 0.
(H3) There is a Banach space (Y, k · kY ) ,→ (X, k · k), such that the function f (·)
belongs to C([0, ∞) × Bβ × B; Y ) and f (·) is uniformly S-asymptotically
ω-periodic on bounded sets; that is,
lim
t→∞ kψk
kf (t + ω, ψ, ξ) − f (t, ψ, ξ)kY = 0.
sup
Bβ ≤r,kξkB ≤r
To prove our results, is convenient to include some comments on the problem
u0 (t) = Au(t) + ξ(t),
u(0) = x
t ∈ I,
∈ X,
(2.2)
(2.3)
with I = [0, a] or I = [0, ∞) and ξ ∈ L1 (I; X). We note that the function u : I → X
Rt
given by u(t) = T (t)x + 0 T (t − s)ξ(s)ds is called a mild solution of (2.2)-(2.3) on
I and that a function v ∈ C(I; X) is said to be a strict solution (2.2)-(2.3) on I if
v ∈ C 1 (I; X) ∩ C(I; X1 ) and v(·) satisfies (2.2)-(2.3).
The proof of Proposition 2.1 is similar to the proof of [26, Theorem 4.3.1]. We
include some details of the proof for completeness.
Proposition 2.1. Assume ξ ∈ C α ([0, ∞); X), x ∈ X1 and Ax + ξ(0) ∈ DA (α, ∞).
If u(·) is the mild solution of (2.2)-(2.3) on [0, ∞), then u(·) is a strict solution,
u ∈ C α ([0, ∞); X1 ) and
Z t
0
u (t) = AT (t)x +
AT (t − s)(ξ(s) − ξ(t))ds + T (t)ξ(t), ∀t ≥ 0.
(2.4)
0
Moreover,
[u]C α ([0,∞);X1 ) ≤
C1,α
kAx + ξ(0)kα,∞ + ΛkξkC α ([0,∞);X) ,
α
(2.5)
S-ASYMPTOTICALLY ω-PERIODIC SOLUTIONS
EJDE-2015/210
C1,α
kAx + ξ(0)kα,∞ + (Λ + 1)kξkC α ([0,∞);X) ,
α
≤ C0 kAxk + Λ1 [ξ]C α ([0,∞);X) + Λ2 kξkC([0,∞);X) ,
5
[u0 ]C α ([0,∞);X) ≤
(2.6)
kukC([0,∞);X1 )
(2.7)
0
ku kC([0,∞);X) ≤ C0 kAxk + Λ1 [ξ]C α ([0,∞);X) + (Λ2 + 1)kξkC([0,∞);X) ,
1
where Λ = ( 2C
α + 3C0 + 1 +
C2
α(1−α) ),
(2.8)
Λ1 = C1 ( γ1 + α1 ) and Λ2 = (C0 + 1).
Proof. Let T > 0. From [26, Theorem 4.3.1] we know that u|[0,T ] is a strict solution
of the problem (2.2)-(2.3) on [0, T ], u ∈ C α ([0, T ]); X1 ) and that the representation
(2.4) is valid on [0, T ]. Moreover, a review of the proof of [26, Theorem 4.3.1] permit
us to assert that
C1,α
2C1
C2
[u]C α ([0,T ];X1 ) ≤
kAx + ξ(0)kα,∞ + (
+ 3C0 + 1 +
)[ξ]C α ([0,T ];X) .
α
α
α(1 − α)
From the above, we infer that u(·) is a strict solution of the problem (2.2)-(2.3)
on [0, ∞), the representation (2.4) is satisfied on [0, ∞) and (2.5) is valid with
C2
1
Λ = ( 2C
α + 3C0 + 1 + α(1−α) ). In addition, from the representation
Z t
Au(t) = T (t)Ax +
AT (t − s)(ξ(s) − ξ(t))ds + (T (t) − I)ξ(t),
0
it is easy to see that
kukC([0,∞);X1 ) ≤ C0 kAxk + C1 (
1
1
+ )[ξ]C α ([0,∞);X) + (C0 + 1)kξkC([0,∞);X) ,
γ
α
which establish (2.7). Finally, from (2.5) and (2.7), and the fact that u(·) is a strict
solution we obtain (2.6) and (2.8). This completes the proof.
For completeness, from [21] we quote the followings result.
Lemma 2.2. Assume the condition (H2) is satisfied, x ∈ X1 and ξ ∈ C([0, b]; Y ).
Then, the mild solution w(·) of (2.2)-(2.3) is a strict solution and
Z t
w0 (t) = T (t)Ax +
AT (t − s)ξ(s)ds + ξ(t), ∀t ∈ I.
0
Remark 2.3. In the remainder of this paper, for u ∈ C(J ∪ [0, ∞); Xβ ), v ∈
C(J ∪ [0, ∞); X) with u0 ∈ Bβ and v0 ∈ B, we use the notations Pv , Pu and fu,v
for the functions Pv : [0, ∞) → B, Pu : [0, ∞) → Bβ and fu,v : [0, ∞) → X given
by Pv (t) = vt , Pu (t) = ut and fu,v (t) = f (t, ut , vt ).
By considering the above remarks, we adopt the following concepts of solution.
Definition 2.4. A function u : J ∪ [0, ∞) → X is called a mild solution of the
abstract problem (1.1)–(1.2) if u0 = ϕ, Pu ∈ C 1 ([0, ∞); B) ∩ C([0, ∞); Bβ ) and
Z t
u(t) = T (t)ϕ(0) +
T (t − s)fu,u0 (s)ds, ∀t ∈ [0, ∞).
0
Definition 2.5. A function u : J ∪ [0, ∞) → X is said to be a strict solution
of (1.1)–(1.2) if Pu ∈ C 1 ([0, ∞); B) ∩ C([0, ∞); Bβ ), u ∈ C([0, ∞); X1 ) and u(·)
satisfies (1.1)–(1.2).
Next, we include some results on S-asymptotically ω-periodic functions. In the
next results, (Z, k · kZ ) is a Banach space.
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M. PIERRI, D. O’REGAN
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Lemma
2.6. Assume (Z, k · kZ ) ,→ (X, k · k), u ∈ C(J ∪ [0, ∞); Z) and u0 ∈ B. If
u[0,∞) ∈ SAPω (Z) and M (t) → 0 as t → ∞, then Pu ∈ SAPω (B). Similarly, if
(Z, k · kZ ) ,→ (Xβ , k · k), w ∈ C(J ∪ [0, ∞); Z), w0 ∈ Bβ , w
∈ SAPω (Z) and
[0,∞)
M (t) → 0 as t → ∞, then Pw ∈ SAPω (Bβ ).
Proof. We only prove the assertion involving the space X. Let iZ,X : Z → X be
the inclusion map from Z into X. For 1 > ε > 0 there exists Lε > 0 such that
ku(t + ω) − u(t)kZ ≤ ε and M (t) ≤ ε for all t ≥ Lε . Then, for t ≥ 2Lε we obtain
kPu (t + ω) − Pu (t)kB
= kut+ω − ut kB
≤ M (t − Lε )kuLε kB + KkiZ,X kL(Z,X) sup ku(s + ω) − u(s)kZ
s≥Lε
≤ M (t − Lε ) M (Lε )ku0 kB + KkiZ,X kL(Z,X) kukC([0,∞);Z) + KkiZ,X kL(Z,X) ε
≤ ε εku0 kB + KkiZ,X kL(Z,X) kukC([0,∞);Z) + KkiZ,X kL(Z,X) ,
which implies that limt→∞ kPu (s + ω) − Pu (s)kB = 0 and Pu ∈ SAPω (B).
Lemma 2.7. Assume Q ∈ C([0, ∞); L(Z, X)) ∩ L1 ([0, ∞); L(Z, X)), v ∈ SAPω (Z)
Rt
and let u : [0, ∞) → X be the function given by u(t) = 0 Q(t − s)v(s)ds. Then
u ∈ SAPω (X).
Proof. From Bochner’s criteria for integrable functions and the estimate
Z t
ku(t)k ≤
kQ(t − s)kL(Z,X) kv(s)kZ ds ≤ kQkL1 ([0,∞),L(Z,X)) kvkC([0,∞);Z) ,
0
it follows that v ∈ C([0, ∞); X). On the other hand, for ε > 0 given there is Lε > 0
such that kv(s + ω) − v(s)kZ ≤ ε and kQkL1 ([Lε ,∞),L(Z,X)) ≤ ε for all s ≥ Lε . Then,
for t ≥ 2Lε we obtain
ku(t + ω) − u(t)k
Z ω
≤
kQ(t + ω − s)kL(Z,X) kv(s)kZ ds
0
Z t
+
kQ(t − s)kL(Z,X) kv(s + ω) − v(s)kZ ds
0
Z t+ω
≤ kvkC([0,∞);Z)
kQ(s)kL(Z,X) ds
t
Z
Lε
Z
t
kQ(t − s)kL(Z,X) ds + ε
+ 2kvkC([0,∞);Z)
0
kQ(t − s)kL(Z,X) ds
Lε
≤ kvkC([0,∞);Z) kQkL1 ([Lε ,∞);L(Z,X)) + 2kvkC([0,∞);Z) kQkL1 ([Lε ,∞);L(Z,X))
+ εkQkL1 ([0,∞);L(Z,X))
≤ ε(3kvkC([0,∞);Z) + kQkL1 ([0,∞);L(Z,X)) ),
which implies that limt→∞ [u(s + ω) − u(s)] = 0 and u ∈ SAPω (X).
In addition to the above, from [13] we note (without proof) the following corollary.
S-ASYMPTOTICALLY ω-PERIODIC SOLUTIONS
EJDE-2015/210
7
Corollary 2.8 ([13, Corollary 3.2]). Let w : [0, ∞) → Z be a S-asymptotically ωperiodic function and assume that w0 is bounded and uniformly continuous. Then
w0 is S-asymptotically ω-periodic.
We include now some Lemmas on α-Hölder functions. We omit the proofs.
Lemma 2.9. Assume that u ∈ C α ([0, b]; X), ψ ∈ B ∩ C α (J; X) and u(0) = ψ(0).
Let v be the function v : J ∪ [0, b] → X given by v = ψ on J and v = u on
[0, b]. Then v ∈ C α (J ∪ [0, b]; X) and [v]C α (J∪[0,b];X) ≤ [ψ]C α (J;X) + [u]C α ([0,b];X) .
Moreover, the assertion is valid replacing X by Xβ .
Lemma 2.10. Assume that the condition (H1) is satisfied, z ∈ C α ([0, ∞); Bβ ) and
w ∈ C α ([0, ∞); B). Then f1 (·, w(·)) ∈ C α ([0, ∞); X), f2 (·, w(·)) ∈ C α ([0, ∞); X)
and
[f1 (·, z)]C α ([0,∞);X) ≤ [f1 ]C α ([0,∞);L(Bβ ,X)) kzkC([0,∞);Bβ )
+ kf1 kC([0,∞);L(Bβ ,X)) [z]C α ([0,∞);Bβ ) ,
[f2 (·, w)]C α ([0,∞);X) ≤ [f2 ]C α ([0,∞);L(B,X)) kwkC([0,∞);B)
+ kf2 kC([0,∞);L(B,X)) [w]C α ([0,∞);B) .
We can establish now our first result on the existence of an S-asymptotically
ω-periodic strict solution for (1.1)–(1.2). Next, Λ, Λ1 and Λ2 are the constant in
Proposition 2.1 and ϕ̃ : J ∪ [0, ∞) → X is the function given by ϕ̃(t) = ϕ(t) for
t ∈ J and ϕ̃(t) = T (t)ϕ(0) for t ≥ 0.
Theorem 2.11. Assume (H1) is satisfied, C(J; X) ,→ B, M (t) → 0 as t → ∞,
f1 ∈ SAPω (L(Bβ , X)) and f2 ∈ SAPω (L(B, X)). Suppose that the function ϕ(·)
−
belongs to C 1+α (J; X) ∩ C α (J; Xβ ), ϕ(0) ∈ X1 , ddtϕ (0) = Aϕ(0) ∈ DA (α, ∞),
f (0, ϕ, ϕ0 ) = 0, Pϕ̃ ∈ C 1+α ([0, ∞); B) ∩ C α ([0, ∞); Bβ ) and
Ξ = K (K + 1)(AΛ + 1) + A(Λ1 + Λ2 ) + 1 Θ(f1 , f2 ) < 1,
where
Θ(f1 , f2 ) = kf1 kC α ([0,∞);L(Bβ ,X)) + kf2 kC α ([0,∞);L(B,X))
−1
and A = kA k + k(−A)β−1 k + 1. Then there exists a unique S-asymptotically
ω-periodic strict solution u ∈ C 1+α (J ∪ R+ ; X) ∩ C(J ∪ R+ ; Xβ ) of (1.1)–(1.2).
Proof. Let Y = C 1+α (J ∪ R+ ; X) ∩ C α (J ∪ R+ ; Xβ ) endowed with the norm k · kY
given by k · kY = k · kC 1+α (J∪R+ ;X) + k · kC(J∪R+ ;Xβ ) and S be the space
S = {u ∈ Y : u[0,∞) ∈ SAPω (X), Pu ∈ C 1+α ([0, ∞); B) ∩ C α ([0, ∞); Bβ )}, (2.9)
endowed with the metric
Φ(u, v) = kPu − Pv kC 1+α ([0,∞);B) + kPu − Pv kC α ([0,∞);Bβ ) .
Let Γ be the map Γ : S → S given by (Γu)0 = ϕ and
Z t
Γu(t) = T (t)ϕ(0) +
T (t − s)fu,u0 (s)ds,
t ≥ 0,
(2.10)
0
where fu,u0 : [0, ∞) → X is the function defined by fu,u0 (t) = f (t, ut , u0t ).
In the remainder of this proof we show that Γ is a contraction on S. To this
end, next we assume that u, v ∈ S.
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M. PIERRI, D. O’REGAN
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Step 1. Γu|[0,∞) ∈ SAPω (X) ∩ C α (R+ ; X1 ) and Γu ∈ C 1+α (J ∪ R+ ; X) ∩ C α (J ∪
R+ ; Xβ ).
From the definition of Y we have that u0 (·) is uniformly continuous on [0, ∞),
which implies via Corollary 2.8 that u0 ∈ SAPω (X). Moreover, from Lemma 2.6
and Lemma 2.10 we infer that f1 ◦ Pu and f2 ◦ Pu0 belong to SAPω (X) and from
Lemma 2.7 (with Q(s) = T (s) and Z = X) we obtain that Γu|[0,∞) ∈ SAPω (X).
On the other hand, from Lemma 2.10 we have that fu,u0 ∈ C α ([0, ∞); X) which
implies via Proposition 2.1 that Γu|[0,∞) ∈ C α ([0, ∞); X1 ) ∩ C 1+α ([0, ∞); X) ∩
C α ([0, ∞); Xβ ) and
C1,α
kAϕ(0)kα,∞ + Λ[fu,u0 ]C α ([0,∞);X) ,
α
kΓukC([0,∞);X1 ) ≤ C0 kAϕ(0)k + (Λ1 + Λ2 )kfu,u0 kC α ([0,∞);X) ,
C1,α
kAxkα,∞ + (Λ + 1)[fu,u0 ]C α ([0,∞);X) ,
[(Γu)0 ]C α ([0,∞);X) ≤
α
0
k(Γu) kC([0,∞);X) ≤ C0 kAϕ(0)k + (Λ1 + Λ2 + 1)kfu,u0 kC α ([0,∞);X) ,
[Γu]C α ([0,∞);X1 ) ≤
kΓukC([0,∞);Xβ ) ≤ k(−A)β−1 kΓukC([0,∞);X1 ) ,
β−1
[Γu]C α ([0,∞);Xβ ) ≤ k(−A)
k[Γu]C α ([0,∞);X1 ) .
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
Moreover, by noting that (Γu)0 (0) = Aϕ(0)+fu,u0 (0) = Aϕ(0) = ϕ0 (0) and Γu(0) =
ϕ(0), from the properties of the function ϕ̃ and Lemma 2.9 we infer that Γu ∈
C 1+α (J ∪ [0, b]; X) ∩ C α (J ∪ [0, b]; Xβ ), which completes the proof of this step.
Step 2. PΓu ∈ C 1+α ([0, ∞); B) and
[PΓu − PΓv ]C α ([0,b];B) ≤ K(K + 1)kA−1 kΛ[fu,u0 − fv,v0 ]C α ([0,∞);X) ,
kPΓu − PΓv kC([0,b];B) ≤ KkA
−1
k(Λ1 + Λ2 )kfu,u0 − fv,v0 kC α ([0,∞);X) ,
(2.17)
(2.18)
0
0
(2.19)
0
0
(2.20)
[(PΓu ) − (PΓv ) ]C α ([0,b];B) ≤ K(K + 1)(Λ + 1)[fu,u0 − fv,v0 ]C α ([0,∞);X) ,
k(PΓu ) − (PΓv ) kC([0,b];B) ≤ K(Λ1 + Λ2 + 1)kfu,u0 − fv,v0 kC α ([0,∞);X) ).
From the properties of the space B it is easy to see that PΓu ∈ C([0, ∞); B). To
show that PΓu ∈ C α ([0, ∞); B) is convenient to estimate kPΓu (t) − ϕkB . By using
the properties of Pϕ̃ , for t ≥ 0 we obtain
kPΓu (t) − ϕkB
≤ kPΓu (t) − Pϕe(t)kB + kPϕe(t) − ϕkB
≤ K sup kΓu(s) − ϕ(0)k
e
+ K sup kϕ(0)
e − ϕ(s)k
e
+ tα [Pϕe]C α ([0,∞);B)
s∈[0,t]
s∈[0,t]
≤ tα K[Γu]C α ([0,∞);X) + tα (KH + 1)[Pϕe]C α ([0,∞);B) .
From this estimate, for t > 0 and h > 0, we see that
kPΓu (t + h) − PΓu (t)kB
≤ KkPΓu (h) − ϕkB + K sup kΓu(s + h) − Γu(s)k
s∈[0,t]
2 α
≤ K h [Γu]C α ([0,∞);X) + hα K(KH + 1)[Pϕe]C α ([0,∞);B) + Khα [Γu]C α ([0,∞);X) ,
which implies that
[PΓu ]C α ([0,∞);B) ≤ K(K + 1)[Γu]C α ([0,∞);X) + K(KH + 1)[Pϕe]C α ([0,∞);B) ,
(2.21)
S-ASYMPTOTICALLY ω-PERIODIC SOLUTIONS
EJDE-2015/210
9
and PΓu ∈ C α ([0, ∞); B) since [Γu]C α ([0,∞);X) ≤ kA−1 k[Γu]C α ([0,∞);X1 ) < ∞.
Moreover, by noting that Γu − Γv is the mild solution of (2.2)-(2.3) with x = 0 and
ξ = fu,u0 − fv,v0 , from the above remarks, the estimative (2.21) and Proposition 2.1
we infer
[PΓu − PΓv ]C α ([0,∞);B) ≤ K(K + 1)[Γu − Γv]C α ([0,∞);X)
≤ K(K + 1)kA−1 k[Γu − Γv]C α ([0,∞);X1 )
≤ K(K + 1)kA−1 kΛ[fu,u0 − fv,v0 ]C α ([0,∞);X) ,
which establish the estimate (2.17).
Proceeding as above, we also can prove that (PΓu )0 ∈ C α ([0, ∞); B). Since ϕ0 ∈ B
and (Γu)00 = ϕ0 (note that (Γu)0 (0) = Aϕ(0) + fu,u0 (0) = Aϕ(0) = ϕ0 (0)), from the
properties of B and the estimate (2.14) we obtain that (PΓu )0 ∈ C([0, ∞); B). In
addition, for t ≥ 0 it is easy to see that
kP(Γu)0 (t) − ϕ0 kB ≤ tα K[(Γu)0 ]C α ([0,∞);X) + tα (KH + 1)[Pϕe0 ]C α ([0,∞);B) .
(2.22)
Using this estimate, we have that
kP(Γu)0 (t + h) − P(Γu)0 (t)kB
≤ KkP(Γu)0 (h) − ϕ0 kB + K sup k(Γu)0 (s + h) − (Γu)0 (s)k
s∈[0,t]
0
α
≤ K(h K[(Γu) ]C α ([0,∞);X) + hα (KH + 1)[Pϕe0 ]C α ([0,∞);B) )
+ Khα [(Γu)0 ]C α ([0,∞);X)
≤ hα K(K + 1)[(Γu)0 ]C α ([0,∞);X) + hα K(KH + 1)[Pϕe0 ]C α ([0,∞);B) ),
which implies
[P(Γu)0 ]C α ([0,∞);B) ≤ K(K+1)[(Γu)0 ]C α ([0,∞);X) +K(KH +1)[Pϕe0 ]C α ([0,∞);B) , (2.23)
and P(Γu)0 ∈ C α ([0, ∞); B) since [(Γu)0 ]C α ([0,∞);X) is finite. This completes the
proof that PΓu ∈ C 1+α ([0, ∞); B). Proceeding as above, we also note that
[P(Γu)0 − P(Γv)0 ]C α ([0,∞);B) ≤ K(K + 1)[(Γu)0 − (Γv)0 ]C α ([0,∞);X)
≤ K(K + 1)(Λ + 1)[fu,u0 − fv,v0 ]C α ([0,∞);X) ,
which establish (2.19).
Concerning the estimate (2.18), from Proposition 2.1 we have
kPΓu (t) − PΓv (t)kB
≤ K sup kΓu(s) − Γv(s)k
s∈[0,t]
≤ KkA−1 k sup kΓu(s) − Γv(s)kX1
s∈[0,t]
≤ KkA
−1
k(Λ1 [fu,u0 − fv,v0 ]C α ([0,∞);X) + Λ2 kfu,u0 − fv,v0 kC([0,∞);X) ),
which proves (2.18). Finally, from the estimate
k(PΓu )0 − (PΓv )0 kC([0,b];B)
≤ Kk(Γu)0 − (Γv)0 kC([0,b];X)
≤ K(kΓu − ΓvkC([0,b];X1 ) + kfu,u0 − fv,v0 kC([0,∞);X) )
≤ K(Λ1 [fu,u0 − fv,v0 ]C α ([0,∞);X) + Λ2 kfu,u0 − fv,v0 kC([0,∞);X) )
+ Kkfu,u0 − fv,v0 kC([0,∞);X) ,
10
M. PIERRI, D. O’REGAN
EJDE-2015/210
we obtain (2.20).
Step 3. The function PΓu belongs to C α ([0, ∞); Bβ ) and
kPΓu − PΓv kC([0,b];Bβ ) ≤ Kk(−A)β−1 k(Λ1 + Λ2 )kfu,u0 − fv,v0 kC α ([0,∞);X) ), (2.24)
[PΓu − PΓv ]C α ([0,b];Bβ ) ≤ K(K + 1)k(−A)β−1 kΛ[fu,u0 − fv,v0 ]C α ([0,∞);X) .
(2.25)
From Proposition 2.1 it follows that
kPΓu − PΓv kC([0,b];Bβ ) ≤ KkΓu − ΓvkC([0,b];Xβ )
≤ Kk(−A)β−1 kkΓu − ΓvkC([0,b];X1 )
≤ Kk(−A)β−1 k(Λ1 + Λ2 )kfu,u0 − fv,v0 kC α ([0,∞);X) ,
which establish (2.24). Concerning (2.25) we note that
kPΓu (t + h) − PΓv (t + h) − (PΓu (t) − PΓv (t))kBβ
≤ KkPΓu (h) − PΓv (h)kBβ + K sup kΓu(s + h) − Γu(s + h) − (Γu(s) − Γu(s))kβ
s∈[0,h]
≤K
2
sup kΓu(s) − Γu(h) − (Γu(0) − Γu(0))kβ
s∈[0,h]
+ K sup kΓu(s + h) − Γu(s + h) − (Γu(s) − Γu(s))kβ
s∈[0,t]
2
≤ K k(−A)β−1 k sup kΓu(s + h) − Γu(s + h) − (Γu(s) − Γu(s))kX1
s∈[0,t]
+ Kk(−A)
β−1
k sup kΓu(s + h) − Γu(s + h) − (Γu(s) − Γu(s))kX1
s∈[0,t]
≤ K(K + 1)k(−A)β−1 k[Γu − Γu]C α ([0,∞);X1 ) ,
which, via Proposition 2.1, implies
[PΓu − PΓv ]C α ([0,b];Bβ ) ≤ K(K + 1)k(−A)β−1 kΛ[fu,u0 − fv,v0 ]C α ([0,∞);X) .
From Steps 2 and 3 we obtain that Φ(Γu, Γv) ≤ ΞΦ(u, v) which proves that Γ is a
contraction on S. Finally, from the contraction mapping principle and Proposition
2.1 we infer that there exists a unique S-asymptotically ω-periodic strict solution
u ∈ C 1+α (J ∪ R+ ; X) ∩ C(J ∪ R+ ; Xβ ) of the problem (1.1)–(1.2).
In the next theorem we prove the existence of a strict solution via the conditions
(H2) and (H3). In this result, Y is the space in condition (H2) and iY,X denotes
the inclusion map from Y into X.
Theorem 2.12. Assume the conditions (H2) and (H3) are satisfied, C(J; X) ,→ B,
−
M (t) → 0 as t → ∞, ϕ ∈ C 1 (J; X) ∩ Bβ , ϕ(0) ∈ X1 , ddtϕ (0) = Aϕ(0) and
f (0, ϕ, ϕ0 ) = 0. Suppose, in addition, there are ψ1 ∈ BB and ψ2 ∈ B such that
f (·, ψ1 , ψ2 ) ∈ C([0, ∞); Y ) and
Θ = K(k(−A)β−1 k + 1)kH ∗ Lf kC([0,∞);R+ )
(2.26)
C0
+ 1) < 1,
+ KkLf kC([0,∞);R+ ) kiY,X kL(Y,X) (
γ
Rt
where H ∗ Lf (t) = 0 H(t − s)Lf (s)ds. Then there exists a unique S-asymptotically
ω-periodic strict solution u ∈ C 1 (J ∪ R+ ; X) ∩ C(J ∪ R+ ; Xβ ) of (1.1)–(1.2).
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S-ASYMPTOTICALLY ω-PERIODIC SOLUTIONS
11
Proof. Let S = C 1 (J ∪ [0, ∞); X) ∩ C(J ∪ [0, ∞); Xβ ) and
F = {u ∈ S : u[0,∞) , u0 [0,∞) ∈ SAPω (X), Pu ∈ C([0, ∞); Bβ ) ∩ C 1 ([0, ∞); B)},
endowed with the metric Φ(u, v) = kPu − Pv kC 1 ([0,∞);B) + kPu − Pv kC([0,∞);Bβ ) .
Let Γ : F → F the map defined in the proof of Theorem 2.11. Next, we show that
Γ is a contraction on F. To begin, we show that Γ is a well defined function from
F into F. In the remainder of this proof we assume that u, v ∈ F.
By noting that fu,u0 ∈ C([0, ∞); Y ), from condition (H2) it is easy to see that
k(−A)β Γu(t)k
≤ C0 e−γt k(−A)β ϕ(0)k + k(−A)β−1 kL(X)
Z
t
kAT (t − s)kL(Y,X) kfu,u0 (s)kY ds
0
≤ C0 e−γt k(−A)β ϕ(0)k + k(−A)β−1 kL(X) kfu,u0 kC([0,∞);Y ) kHkL1 ([0,∞),R+ ) ,
which implies that Γu ∈ C(J ∪ [0, ∞); Xβ ) and PΓu ∈ C([0, ∞); Bβ ) since ϕ ∈ Bβ .
Moreover, from Lemma 2.2 we have that Γu is continuously differentiable and
k(Γu)0 (t)k ≤ e−γt kAϕ(0)k + kfu,u0 kC([0,∞);Y ) (kHkL1 ([0,∞),R+ ) + 1),
which shows that (Γu)0 ∈ C([0, ∞); X). In addition to the above, from Lemma 1.4,
−
the compatibility condition ddtϕ (0) = Aϕ(0)+f (0, ϕ, ϕ0 ) = Aϕ(0) and the fact that
Z t
(Γu)0 (t) = AT (t)ϕ(0) +
AT (t − s)fu,u0 (s)ds + f (t, ut , u0t ), ∀t ≥ 0,
0
we infer that PΓu ∈ C 1 ([0, ∞); B) and (PΓu )0 = P(Γu)0 .
To complete theproof that Γ has values in F, it remain to prove that the functions
Γu[0,∞) and (Γu)0 [0,∞) belong to SAPω (X). From Lemma 2.6 we have that Pu ∈
SAPω (Bβ ), Pu0 ∈ SAPω (B) and from the conditions (H2) and (H3) is easy to
show that f (·, Pu (·), Pu0 (·)) ∈ SAPω (Y ). In addition, by using Lemma 2.7 with
Z = X and Q(t) = T (t) we obtain that Γu ∈ SAPω (X). Moreover, arguing
as above,
but using Lemma 2.7 with Z = Y and Q(t) = AT (t) it follows that
(Γu)0 [0,∞) ∈ SAPω (X). This completes the proof that Γ is a F-valued function.
On the other hand, for u, v ∈ F and t ≥ 0 we obtain
Z t
β−1
kΓu(t) − Γv(t)kβ ≤ k(−A)
k
kAT (t − s)kL(Y,X) Lf (s)Φ(u, v)ds,
0
and hence,
kΓu − ΓvkC([0,∞);Xβ ) ≤ k(−A)β−1 kkH ∗ Lf kC([0,∞);R+ ) Φ(u, v).
(2.27)
Similarly, from the inequality
Z t
kΓu(t) − Γv(t)k ≤ C0
e−γ(t−s) kiY,X kL(Y,X) kfu,u0 (s) − fv,v0 (s)kY ds
0
we obtain
C0
kiY,X kL(Y,X) kLf kC([0,∞);R+ ) Φ(u, v).
(2.28)
γ
In addition, from Lemma 2.2, it follows that
Z t
0
0
k(Γu) (t) − (Γv) (t)k ≤
H(t − s)Lf (s)(kus − vs kBβ + ku0s − vs0 kB )ds
kΓu − ΓvkC([0,∞);X) ≤
0
+ Lf (t)kiY,X kL(Y,X) (kut − vt kBβ + ku0t − vt0 kB ),
12
M. PIERRI, D. O’REGAN
EJDE-2015/210
from which we infer that
k(Γu)0 − (Γv)0 kC([0,∞);X)
≤ kH ∗ Lf kC([0,∞);R+ ) + kLf kC([0,∞);R+ ) kiY,X kL(Y,X) Φ(u, v).
(2.29)
From inequalities (2.27)-(2.29) we obtain that Φ(Γu, Γv) ≤ ΘΦ(u, v) which proves
that Γ is a contraction on F and there exits a unique mild solution u ∈ F of (1.1)–
(1.2). Finally, from Lemma 2.2 it follows that u(·) is a strict solution of the problem
(1.1)–(1.2). This completes the proof.
3. Applications
Motivated by the examples presented in the introduction, in this section we
discuss the existence of a S-asymptotically ω-periodic strict solution for the partial
neutral differential problem
Z t
u0 (t, ξ) = ∆u(t, ξ) −
a(t)k(t − s)u0 (s, ξ)ds + g(t),
(3.1)
−∞
u(t, 0) = u(t, π) = 0,
u(s, ξ) = ϕ(s, ξ),
s ≤ 0,
(3.2)
(3.3)
for (t, ξ) ∈ [0, ∞) × [0, π] where g ∈ SAPω (R) and ϕ(·) is a function defined from
(−∞, 0] × [0, π] into R.
To treat the problem (3.1)-(3.3) under the abstract framework in Section 1, we
take X = L2 ([0, π]) and A : D(A) ⊂ X → X be the operator Ax = x00 on D(A) =
{x ∈ X : x00 ∈ X, x(0) = x(π) = 0}. It is well known that A is the generator of
an analytic semigroup (T (t))t≥0 on X, A has discrete spectrum with eigenvalues
−n2 , n ∈ N, and associated normalized eigenvectors zn (ξ) = ( π2 )1/2 sin(nξ). We
t
t
t
note that kT (t)k ≤ e− 2 , kAT (t)k ≤ e− 2 t−1 and kA2 T (t)k ≤ 4e− 2 t−2 for t > 0.
As a phase space we consider the space B = Cr ×Lp (ρ, X). Let r ≥ 0, 1 ≤ p < ∞
and ρ : (−∞, −r] → R be a nonnegative measurable function which satisfies the
conditions (g-5)-(g-7) in the terminology of [24]. The space Cr × Lp (ρ, X) is formed
by all classes of functions ψ : (−∞, 0] → X such that ψ|[−r,0] ∈ C([−r, 0], X), ψ(·)
1
is Lebesgue-measurable and ρ p ψ ∈ Lp ((−∞, −r], X). The norm in Cr × Lp (ρ, D) is
1
given by kψkB = kψkC([−r,0];X) + kρ p ψkLp ((−∞,−r],X) . From [24], we know that B
satisfy the conditions in Section 2 and B is a uniform fading memory space, which
implies that M (t) → 0 as t → ∞ and K(·) is bounded, see [24, pp.190] for details.
Next we assume k ∈ C([0, ∞); R), a ∈ C α ([0, ∞), R) ∩ SAPω (R) for some α ∈
(0, 1) and ω > 0, and
Z 0 k 2 (−τ ) 1/2
Θ = 2kakC([0,∞);R)
dτ
< ∞.
−∞ ρ(τ )
Under these conditions, the function f : [0, ∞) × B → X given by f (t, ψ)(ξ) =
R0
a(t)k(−τ )ψ(τ, ξ)dτ is well defined, f ∈ C α ([0, ∞); L(B, X)) ∩ SAPω (L(B))
−∞
and kf kC α ([0,∞);L(B,X)) ≤ Θ.
Next, we say that a function u ∈ C 1 (J ∪ R+ ; X) ∩ C(J ∪ R+ ; X1 ) is a strict
solution of (3.1)-(3.3) if u(·) is a strict solution of the associated problem (1.1)–
(1.2). In the next result, which follows directly from Theorem 2.11, Λ is the number
in Proposition 2.1 and DA (α, ∞), K are as in Section 1. We also note that Λ appears
EJDE-2015/210
S-ASYMPTOTICALLY ω-PERIODIC SOLUTIONS
13
explicitly in the end of the proof of Proposition 2.1 and that in the current case
4
Λ = ( α2 + 4 + α(1−α)
), Λ1 = (2 + α1 ) and Λ2 = 2.
−
Proposition 3.1. Assume that ϕ(0, ·) ∈ X1 , ddtϕ (0, ·) = Aϕ(0, ·) ∈ DA (α, ∞),
f (0, ϕ) + f2 (0, ϕ0 ) = 0, ϕ ∈ C 1+α (J; X) ∩ C α (J; Xβ ) and Pϕ̃ ∈ C 1+α ([0, ∞); B). If
1
K (K + 1)(AΛ + 1) + A(4 + ) + 1 Θ < 1,
(3.4)
α
where A = k(−A)β−1 k + kA−1 k + 1, then there exists a unique S-asymptotically
ω-periodic strict solution of the problem (3.1)–(3.3).
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Michelle Pierri
Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, CEP 14040-901 Ribeirão Preto,
SP, Brazil
E-mail address: michellepierri@ffclrp.usp.br
Donal O’Regan
School of Mathematics, Statistics and Applied Mathematics, National University of
Ireland, Galway, Ireland
E-mail address: donal.oregan@nuigalway.ie