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Pseudo almost periodic solutions to integro-differential equations of heat conduction in materials with memory
Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
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Pseudo almost periodic solutions to integro-differential equations of heat conduction in materials with memory Hui-Sheng Ding a , Jin Liang b , Ti-Jun Xiao c,∗ a
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, People’s Republic of China
b
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
c
Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
article
abstract
info
Article history: Received 26 October 2011 Accepted 18 March 2012
We investigate the existence of pseudo almost periodic solutions for a class of integrodifferential equation with nonlocal initial conditions arising in the study of heat conduction in materials with memory. We first establish some existence theorems for abstract semilinear integro-differential equations with nonlocal initial conditions, and then, we apply our abstract results to the addressed integro-differential equation arising in the study of heat conduction in materials with memory. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Pseudo almost periodic Nonlocal Cauchy problems heat conduction in materials with memory Integro-differential equations
1. Introduction This paper is mainly concerned with the existence of pseudo almost periodic solutions to the following integrodifferential equation arising from the study of heat conduction in materials with memory (see, e.g., [1])
t ∂ 2 θ (t , x) ∂θ (t , x) ∂θ (s, x) + β( 0 ) = α( 0 ) 1 θ ( t , x ) − β ′ (t − s) ds 2 ∂t ∂t ∂s 0 t + α ′ (t − s)1θ (s, x)ds + λ(t )µ[θ (t , x)],
(1.1)
0
where t ≥ 0, x ∈ Ω and Ω is a bounded open connected subset of R3 with C ∞ boundary. This equation and its variants have been investigated by many authors (see, e.g., [2,3]). In this paper, we will consider Eq. (1.1) with the following nonlocal initial conditions:
∂θ (t , x) = g2 (θ ). ∂ t t =0
θ (0, x) = g1 (θ ), As in [3,4], we assume that
(A1) α, β ∈ C 2 ([0, +∞), R) with α(0) and β(0) positive. Denote X = H01 (Ω ) × L2 (Ω ),
A=
∗
0 α(0)∆
I
−β(0)I
,
D (A) = (H 2 (Ω ) ∩ H01 (Ω )) × H01 (Ω );
Corresponding author. Tel.: +86 21 65642341; fax: +86 21 65646073. E-mail addresses: [email protected] (H.-S. Ding), [email protected] (J. Liang), [email protected] (T.-J. Xiao).
1468-1218/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2012.03.009
(1.2)
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θ (t ) [θ (t )](x) = θ (t , x), u( t ) = ; ′ θ (t ) 0 u1 ∈ X; [f (t , u)](x) = for u = u2 λ(t )µ[u1 (x)] g1 (u1 ) u1 ∈ X; u0 = 0, g (u) = for u = u2 g2 (u1 ) and B(t ) = F (t )A, where F (t ) = [Fij (t )] is defined by F11 (t ) = F12 (t ) = 0,
F21 (t ) = −β ′ (t )I + β(0)
α ′ (t ) I, α(0)
F22 (t ) =
α ′ (t ) I. α(0)
Then, Eqs. (1.1) and (1.2) is transformed into the following abstract integro-differential equation u′ (t ) = Au(t ) +
t
B(t − s)u(s)ds + f (t , u(t )),
t ≥ 0,
(1.3)
0
with a nonlocal initial condition u(0) = u0 + g (u),
(1.4)
in the Banach space X . Recall that the integro-differential equation (1.3) with a nonlocal initial condition (1.4) has been a topic of interest for many researchers. There is a lot of literature on the existence of solutions for various forms of Eqs. (1.3)–(1.4) on a finite interval (see, e.g, [5,6,4,7–12] and references therein). On the other hand, very recently, the existence of almost periodic type solutions and almost automorphic type solutions for Eqs. (1.3)–(1.4) and its variants has attracted more and more attention. We refer the reader to [13–17] for some recent development on this topic. In this paper, we will make further study on this topic, i.e., we will establish some existence theorems on pseudo almost periodic solutions for Eqs. (1.3)–(1.4), and then, apply our abstract results to Eqs. (1.1)–(1.2). 2. Pseudo almost periodic functions on R+ Throughout this paper, R denotes the set of real numbers; R+ denotes the set of nonnegative real numbers; X , Y denote two Banach spaces; Ω denotes a subset of X ; BC (R+ , X ) denotes the Banach space of bounded continuous functions from R+ to X with supremum norm; for p ≥ 1, C p (R+ × Ω , Y ) denotes the set of all BS p -uniformly continuous functions f : R+ × Ω → Y , i.e., there is a nonnegative function L ∈ BS p (R) satisfying that ∀ε > 0, there exists δ > 0 such that
∥f (t , u) − f (t , v)∥ ≤ L(t )ε for all t ∈ R+ and u, v ∈ Ω with ∥u − v∥ < δ . Next, let us recall and introduce some definitions and notations about almost periodicity and pseudo almost periodicity. For more details, we refer the reader to [18,19]. Definition 2.1. A continuous function f : R → X is called almost periodic if for each ε > 0 there exists l(ε) > 0 such that every interval I of length l(ε) contains a number τ with the property that sup ∥f (t + τ ) − f (t )∥ < ε. t ∈R
We denote by AP (X ) the set of all such functions. Definition 2.2. A continuous function f : R × Ω → Y is called almost periodic in t uniformly for x ∈ Ω if for each ε > 0 and for each compact subset K ⊂ Ω there exists l(ε) > 0 such that every interval I of length l(ε) contains a number τ with the property that sup sup ∥f (t + τ , x) − f (t , x)∥ < ε. t ∈R x∈K
We denote by AP (R × Ω , Y ) the set of all such functions. In this paper, we denote PAP0 (R , X ) = +
h ∈ BC (R , X ) : lim +
T →+∞
1 T
T
∥h(t )∥dt = 0 . 0
Moreover, we denote by PAP0 (R+ × Ω , Y ) the space of all continuous functions h : R+ × Ω → Y such that for each x ∈ Ω , h(·, x) ∈ PAP0 (R+ , Y ).
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Definition 2.3. A function f : R+ → X is called pseudo almost periodic if f (t ) = g (t ) + h(t ),
t ∈ R+ ,
where g ∈ AP (X ) and h ∈ PAP0 (R+ , X ). We denote by PAP (R+ , X ) the set of all such functions. Definition 2.4. A function f : R+ × Ω → Y is called pseudo almost periodic if f (t , x) = g (t , x) + h(t , x),
t ∈ R+ , x ∈ Ω ,
where g ∈ AP (R × Ω , Y ) and h ∈ PAP0 (R+ × Ω , Y ). We denote by PAP (R+ × Ω , Y ) the set of all such functions. Lemma 2.5. Let f = g + h ∈ PAP (R+ , X ) with g ∈ AP (X ) and h ∈ PAP0 (R+ , X ). Then
{g (t ) : t ∈ R} ⊂ {f (t ) : t ∈ R+ }. Proof. The proof is similar to that of pseudo almost periodic functions on R (cf. [19]). So we omit the details.
By Lemma 2.5, it is easy to see that the decomposition of every pseudo almost periodic function is unique; also, it is not difficult to prove the following result. Lemma 2.6. PAP (R+ , X ) is a Banach space under the supremum norm
∥f ∥ = sup ∥f (t )∥. t ∈R+
Next, we have the following lemma. Lemma 2.7. Let f ∈ BC (R+ , X ). Then f ∈ PAP0 (R+ , X ) if and only if ∀ε > 0, lim
mesMT ,ε (f ) T
T →+∞
= 0,
where MT ,ε (f ) = {t ∈ [0, T ] : ∥f (t )∥ ≥ ε}. Proof. The proof is also similar to that of pseudo almost periodic functions on R (cf. [19]). So we omit the details.
We also need to recall some definitions about Stepanov bounded functions (for more details, see [20]). Definition 2.8. The Bochner transform f b (t , s), t ∈ R, s ∈ [0, 1], of a function f (t ) on R, with values in X , is defined by f b (t , s) := f (t + s). Definition 2.9. Let p ∈ [1, ∞). The space BS p (X ) of all Stepanov bounded functions, with the exponent p, consists of all measurable functions f on R with values in X such that f b : R → Lp (0, 1; X ) is bounded. We denote
t +1
∥f ∥S p = sup t ∈R
∥f (τ )∥p dτ
1/p
.
t
Now, we are ready to establish the following composition theorem. Theorem 2.10. Let x ∈ PAP (R+ , X ) and Ω = {x(t ) : t ∈ R+ }. Suppose that f ∈ PAP (R+ × Ω , Y ) f (R+ × Ω ) is bounded. Then f (·, x(·)) ∈ PAP (R+ , Y ).
C 1 (R+ × Ω , Y ) and
Proof. Let x = α + β and f = g + h, where α ∈ AP (X ), β ∈ PAP0 (R+ , X ), g ∈ AP (R × Ω , Y ) and h ∈ PAP0 (R+ × Ω , Y ). Denote I (t ) = g (t , α(t )),
J (t ) = f (t , x(t )) − f (t , α(t )),
K (t ) = h(t , α(t )).
Then f (t , x(t )) = I (t ) + J (t ) + K (t ). Since f (R+ × Ω ) is bounded, f (·, x(·)) ∈ BC (R+ , Y ). In addition, we denote M =
sup t ∈R+ ,u∈Ω
∥f (t , u)∥.
By Lemma 2.5, {α(t ) : t ∈ R} ⊂ Ω . Then it is easy to see that I ∈ AP (Y ). Next, let us show that J ∈ PAP0 (R+ , Y ). Since f ∈ C 1 (R+ × Ω , Y ), there is a nonnegative function L ∈ BS 1 (R) satisfying that ∀ε > 0, there exists a constant δ > 0 such that ∥β(t )∥ < δ implies that
∥f (t , x(t )) − f (t , α(t ))∥ ≤ L(t )ε.
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Thus, we have 1
T
T
∥J (t )∥dt = 0
1
T
MT ,δ (β)
≤ 2M · ≤ 2M · ≤ 2M ·
∥J (t )∥dt +
mesMT ,δ (β) T mesMT ,δ (β) T mesMT ,δ (β) T
+
1
T
1
[0,T ]\MT ,δ (β) T
T
∥J (t )∥dt
L(t )ε dt 0
+ε·
1
[T ]+1
T
L(t )dt 0
+ ε∥L∥S 1 ·
[T ] + 1 T
.
On the other hand, by Lemma 2.7, mesMT ,δ (β)
lim
T
T →+∞
= 0.
Thus, we get lim sup T →+∞
1
T
T
∥J (t )∥dt ≤ ε∥L∥S 1 , 0
which means that lim sup T →+∞
1
T
T
∥J (t )∥dt = 0, 0
i.e., 1
lim
T
T →+∞
T
∥J (t )∥dt = 0. 0
It remains to prove that K ∈ PAP0 (R+ , Y ). Let Σ = {α(t ) : t ∈ R}. It is well-known that Σ ⊂ Ω is compact. Since g is uniformly continuous on R × Σ , g ∈ C 1 (R+ × Σ , Y ). Thus, h ∈ C 1 (R+ × Σ , Y ), i.e., there is a function L ∈ BS 1 (R) satisfying that ∀ε > 0, there exists δ > 0 such that
∥h(t , u) − h(t , v)∥ ≤ L(t )ε
(2.1)
for all t ∈ R+ and u, v ∈ Σ with ∥u − v∥ < δ . For any fixed ε > 0, let δ be such that (2.1) holds. Then there exist x1 , . . . , xk ∈ Σ such that
Σ⊂
k
B(xi , δ).
i=1
For each t ∈ R+ , there is an xi such that ∥α(t ) − xi ∥ < δ . Then, we get
∥K (t )∥ = ∥h(t , α(t ))∥ = ∥h(t , α(t )) − h(t , xi )∥ + ∥h(t , xi )∥ k ≤ L(t )ε + ∥h(t , xi )∥, i =1
which gives that 1 T
T
∥K (t )∥dt ≤ 0
≤
1 T
T
1 L(t )ε dt +
0
T
T
0
k
k 1 [T ] + 1 ∥L∥S 1 · ε +
T
i =1
∥h(t , xi )∥dt
i =1
T
T
∥h(t , xi )∥dt . 0
Noting that h(·, xi ) ∈ PAP0 (R+ , X ), i = 1, . . . , k, we get lim sup T →+∞
1 T
T
∥K (t )∥dt ≤ ∥ L∥S 1 · ε. 0
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Thus, lim
1
T →+∞
T
T
∥K (t )∥dt = 0, 0
i.e., K ∈ PAP0 (R+ , Y ). This completes the proof.
3. Abstract results In this section, we will establish some abstract results for Eqs. (1.3)–(1.4). First, let us recall some notations about resolvent operator. Definition 3.1 ([21]). A family {R(t ) : t ≥ 0} of continuous linear operators on X is called a resolvent operator (to Eq. (1.3)), iff (R1) R(0) = I, the identity map on X , (R2) for all x ∈ X , the map t → R(t )x is a continuous X -valued function on [0, +∞), (R3) for all t ≥ 0, R(t ) is a continuous operator on Y , and for all y ∈ Y , the map t → R(t )y belongs to C ([0, +∞), Y ) ∩ C 1 ([0, +∞), X ) and satisfies d dt
R(t )y = AR(t )y +
t
B(t − s)R(s)yds 0
= R(t )Ay +
t
R(t − s)B(s)yds, 0
where Y = D(A) = D(B(t )) for all t ≥ 0 and it is equipped with the graph norm. For more details about resolvent operator, we refer the reader to [3,21]. If the resolvent operator R(·) of Eq. (1.3) exists, then we can define the mild solution of Eqs. (1.3)–(1.4) as the following. Definition 3.2. A function u ∈ C ([0, +∞), X ) is called a mild solution of Eqs. (1.3)–(1.4) if u(t ) = R(t )[u0 + g (u)] +
t
R(t − s)f (s, u(s)),
t ≥ 0.
0
Next, for convenience, we list some assumptions. (H1) A is a densely defined, closed linear operator in X and generates a compact C0 -semigroup T (t ). Hence D(A) endowed with the graph norm |u| = ∥u∥ + ∥Au∥ is a Banach space which will be denoted by (D(A), | · |). (H2) {B(t ) : t ≥ 0} is a family of continuous linear operators from (D(A), | · |) into (X , ∥ · ∥), and for each y ∈ D(A), the map 1,1 t → B(t )y belongs to Wloc (R+ , X ). Moreover, there exists a locally integrable function b : R+ → R+ such that
d B(t )y ≤ b(t )|y|, y ∈ D(A), t ∈ R+ . dt (H3) f ∈ PAP (R+ × X , X ) C 1 (R+ × X , X ) and f (R+ × B) is bounded for each bounded subset B ⊂ X . (H4) For each bounded sequence {un } ⊂ PAP (R+ , X ), if {un } is uniformly convergent in every compact subsets of R+ , then {f (·, un (·))} is precompact in BC (R+ , X ). (H5) g : PAP (R+ , X ) → X is a continuous and compact mapping. In the proof of our main result, we will need the following theorem. Theorem 3.3 ([21,8]). Assume that (H1) and (H2) hold. Then (a) there exists a resolvent operator R(·) of Eq. (1.3), and R(t ) is compact for each t > 0; (b) for any fixed T > 0, there exists a constant H = H (T ) such that
∥R(t + h) − R(h)R(t )∥ ≤ Hh for all h, t ∈ [0, T ] with h ≤ t; (c) R(·) is operator norm continuous on (0, +∞). Lemma 3.4. Let {R(t ) : t ≥ 0} be a strongly continuous family of bounded and linear operators on X satisfying ∥R(t )∥ ≤ Me−ωt for all t ≥ 0, where M , ω > 0 are fixed constants. Then, for each f ∈ PAP (R+ , X ), fR ∈ PAP (R+ , X ), fR (t ) =
t
R(t − s)f (s)ds, 0
t ≥ 0.
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Proof. Let f = g + h, where g ∈ AP (X ), h ∈ PAP0 (R+ , X ). We observe that t
fR (t ) =
R(t − s)g (s)ds +
R(t − s)h(s)ds 0
0 t
t
R(t − s)g (s)ds +
=
t
R(t − s)h(s)ds −
R(t − s)g (s)ds −
0
R(t − s)h(s)ds. −∞
−∞
−∞
−∞
0
Let t
G(t ) =
R(t − s)g (s)ds,
t ∈ R,
−∞
and H (t ) =
t
R(t − s)h(s)ds,
I (t ) = −
0
R(t − s)g (s)ds −
0
R(t − s)h(s)ds,
t ≥ 0.
−∞
−∞
−∞
By the strong continuity of {R(t )}, one can prove that G, H, and I are all continuous functions. It is easy to see that G ∈ AP (X ). By a direct calculation, one can get 1
T
T
1
∥H (t )∥dt ≤
T
T
0
+∞
0
+∞
Me−ωs ∥h(t − s)∥dsdt 0
Me−ωs H (s, T )ds,
= 0
where
H (s, T ) =
1 T
T
∥h(t − s)∥dt .
0
Noting that h ∈ PAP0 (R+ , X ), for each s ≥ 0, we have lim H (s, T ) = 0.
T →+∞
Then, by using Lebesgue’s dominated convergence theorem, we get lim
1
T →+∞
T
T
∥H (t )∥dt = 0, 0
i.e., H ∈ PAP0 (R+ , X ). In addition, we have
∥I (t )∥ ≤ [∥g ∥AP (X ) + ∥h∥BC (R+ ,X ) ]
0
Me−ω(t −s) ds
−∞
M = [∥g ∥AP (X ) + ∥h∥BC (R+ ,X ) ] e−ωt → 0,
ω
t → +∞,
which yields that I ∈ PAP0 (R+ , X ). This completes the proof.
Now, we are ready to establish one of our main results. Theorem 3.5. Assume that (H1)–(H5) hold. Moreover, the resolvent operator of Eq. (1.3) satisfies ∥R(t )∥ ≤ Me−ωt for all t ≥ 0, where M , ω > 0 are some fixed constants. Then there exists a pseudo almost periodic mild solution to Eqs. (1.3)–(1.4) provided that there exists a constant C > 0 such that
M·
∥u0 ∥ + sup ∥g (u)∥ + ∥u∥≤C
M
ω
·
sup t ∈R+ , ∥u∥≤C
∥f (t , u)∥ ≤ C .
(3.1)
Proof. Let E = {u ∈ PAP (R+ , X ) : ∥u∥ ≤ C }. Define an operator R on E by
(R u)(t ) = R(t )[u0 + g (u)] +
t
R(t − s)f (s, u(s))ds,
t ≥ 0.
0
It suffices to prove that R has a fixed point in E. First, by Theorem 3.3, we know that R(t ) is compact for each t > 0 and R(·) is operator norm continuous on (0, +∞). Next, we divide the proof into five steps.
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Step 1. R (E ) ⊂ E. Fix u ∈ E. Noting that ∥R(t )∥ ≤ Me−ωt for all t ≥ 0, by (3.1), we have ∥R u∥ ≤ C . On the other hand, by (H3) and Theorem 2.10, we get f (·, u(·)) ∈ PAP (R+ , X ), and then Lemma 3.4 yields that t
R(t − s)f (s, u(s))ds
t → 0
belongs to PAP (R+ , X ). In addition, it is easy to see that R(·)[u0 + g (u)] ∈ PAP0 (R+ , X ). So R u ∈ PAP (R+ , X ), and thus R u ∈ E . Step 2. R : E → E is continuous. Let un → u in PAP (R+ , X ). Fix ε > 0. Since f ∈ C 1 (R+ × X , X ), there exists a nonnegative function L ∈ BS 1 (R) and a constant δ > 0 such that
∥f (t , u) − f (t , v)∥ ≤ L(t )ε for all t ∈ R+ and u, v ∈ X with ∥u − v∥ < δ . Then, when n is sufficiently large, ∥un − u∥ < δ , which gives that
∥f (t , un (t )) − f (t , u(t ))∥ ≤ L(t )ε,
∀t ∈ R+ .
On the other hand, by (H4), g : PAP (R+ , X ) → X is continuous, and thus
∥g (un ) − g (u)∥ < ε when n is sufficiently large. Now, for sufficiently large n, we have
∥R un − R u∥ ≤ M ∥g (un ) − g (u)∥ + sup
t ∈R+
≤ M ε 1 + sup
e
t ∈R+
= M ε 1 + sup
≤ M ε 1 + sup
−ω(t −s)
≤ M ε 1 + sup
e
−ωs
L(t − s)ds
≤ Mε 1 +
+∞
0
[t ]
[t ]
k +1
e
−ωs
L(t − s)ds
k
e
−ωk
k+1
t ∈R+ k=0
L(s)ds
t
t ∈R+ k=0
Me−ω(t −s) L(s)ε ds
0
0
t ∈R+
t
t
L(t − s)ds k
e
−ωk
· ∥L∥S 1
k =0
∥L∥S 1 ≤ Mε 1 + . 1 − e−ω Thus, we conclude that R un → R u in PAP (R+ , X ), which means that R is continuous. Step 3. For each t ∈ R+ , {(R u)(t ) : u ∈ E } is precompact in X . For t = 0,
{R(t )[u0 + g (u)] : u ∈ E } = {u0 } + g (E ) is precompact in X since g is compact. For each t > 0, since R(t ) is compact and {u0 + g (u) : u ∈ E } is bounded, we know that {R(t )[u0 + g (u)] : u ∈ E } is precompact. It remains to show that t
R(t − s)f (s, u(s))ds : u ∈ E
0
is precompact for each fixed t > 0. It follows from (3.1) that sup t ∈R+ ,u∈E
∥f (t , u(t ))∥ ≤
ωC M
.
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Then, for each ε ∈ (0, t ) and u ∈ E, by (b) of Theorem 3.3, we have
t t −ε R(t − s)f (s, u(s))ds − R(ε) R(t − s − ε)f (s, u(s))ds 0 0 t t −ε ωC ≤ R(t − s)f (s, u(s))ds ∥R(t − s) − R(ε)R(t − s − ε)∥ds + M
t −ε
≤ ωC ε +
ωC
t −2 ε
t −ε
+
M
t −2ε
0
≤ (M + 2)ωC ε + ≤ (M + 2)ωC ε +
ωC
t −2ε
M
ωCt M
0
∥R(t − s) − R(ε)R(t − s − ε)∥ds
∥R(t − s) − R(ε)R(t − s − ε)∥ds
0
H (t )ε,
(3.2)
where H (t ) is a constant, which is only dependent on t. On the other hand, for each ε ∈ (0, t ),
R(ε)
t −ε
R(t − s − ε)f (s, u(s))ds : u ∈ E
0
is precompact since R(ε) is compact and
t −ε 0
R(t − s − ε)f (s, u(s))ds ≤ ωCt .
Then, by using (3.2), we conclude that t
R(t − s)f (s, u(s))ds : u ∈ E
0
is totally bounded, and thus precompact. Step 4. For each t ∈ R+ , {R u : u ∈ E } is equicontinuous at t. Since {u0 + g (u) : u ∈ E } is precompact, by the strong continuity of R(·),
{R(·)[u0 + g (u)] : u ∈ E } is equicontinuous at t. In addition, for each u ∈ E and 1t ≥ 0, we have
t +1 t t R ( t + ∆ t − s ) f ( s , u ( s )) ds − R ( t − s ) f ( s , u ( s )) ds 0 0 t +1 t t ≤ R(t + ∆t − s)f (s, u(s))ds ∥[R(t + ∆t − s) − R(t − s)]f (s, u(s))∥ds + t 0 t ≤ ωC |1t | + ∥[R(t + ∆t − s) − R(t − s)]f (s, u(s))∥ds 0 ωC t ≤ ωC |1t | + ∥R(t + ∆t − s) − R(t − s)∥ds, M
0
and for each u ∈ E and 1t ≤ 0, we have
t +1 t
R(t + ∆t − s)f (s, u(s))ds −
t
0
0
R(t − s)f (s, u(s))ds
t +∆t R(t − s)f (s, u(s))ds ∥[R(t + ∆t − s) − R(t − s)]f (s, u(s))∥ds + t +1 t 0 ω C t +∆ t ∥R(t + 1t − s) − R(t − s)∥ds. ≤ ωC |1t | + ≤
t
M
0
Then, by using Lebesgue’s dominated convergence theorem and the operator norm continuity of R(·) on (0, +∞), we conclude that
t
R(t − s)f (s, u(s))ds : u ∈ E
t → 0
is equicontinuous at each t ∈ R+ .
H.-S. Ding et al. / Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
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Step 5. R has a fixed point in E. Denote the closed convex hull of R (E ) by F . By Step 1, R (E ) ⊂ E. Since E is closed and convex, we conclude that F ⊂ E. Then, we have R (F ) ⊂ R (E ) ⊂ F .
Moreover, by Step 2, R : F → F is continuous. In addition, this follows from Steps 3 and 4; we know that for each t ∈ R+ , {u(t ) : u ∈ F } is precompact in X , and {u : u ∈ F } is equicontinuous at t. Next, let us show that R (F ) is precompact. Let {uk }∞ k=1 ⊂ F be an arbitrary sequence. Since g is compact, one can assume that g (uk ) → g (u) in X (by taking a subsequence, if necessary). For any fixed n ∈ N, noting that {uk (t ) : k ∈ N} is precompact in X for each t ∈ R+ and {uk }∞ k=1 is uniformly equicontinuous on [0, n], we conclude that {uk : k ∈ N} is precompact in C ([0, n], X ) by the abstract version of Arzela–Ascoli’s theorem. Then, by the diagonal method, one can choose ∞ ∞ a subsequence of {uk }∞ k=1 , which we also denote by {uk }k=1 for convenience, such that {uk }k=1 is uniformly convergent on + + every compact subset of R . Now, (H4) yields that f (·, uk (·)) is precompact in BC (R , X ). Thus, there is a subsequence um such that f (·, um (·)) is uniformly convergent on R+ , which gives that
t
R(t − s)f (s, um (s))ds : m ∈ N
t →
0
is uniformly convergent on R+ . Recalling that g (uk ) → g (u) in X , we conclude that R (um ) is convergent in PAP (R+ , X ), which means that R (F ) is precompact. Now, by using Schauder’s fixed point theorem, R has a fixed point in F . Corollary 3.6. Assume that all the assumptions of Theorem 3.5 hold except for (H4) and the compactness of T (·). Moreover, there is a nonnegative function L ∈ BS p (R) with p > 1 such that
∥f (t , u) − f (t , v)∥ ≤ L(t )∥u − v∥,
∀t ∈ R+ , ∀u, v ∈ X .
Then there exists a pseudo almost periodic mild solution to Eqs. (1.3)–(1.4) provided that
∥L∥S p <
1 − e−ω
M
ωq 1 − e−ωq
1/q
1
,
p
1
+
q
= 1.
Proof. It follows from [21] that there also exists a resolvent operator R(·) of Eq. (1.3), which is now not necessarily compact. Let E be as in the proof of Theorem 3.5. Define two operators on E by
(A u)(t ) = R(t )[u0 + g (u)],
t ≥0
and
(B u)(t ) =
t
R(t − s)f (s, u(s))ds,
t ≥ 0.
0
Then, similarly as in the proof of Step 1 of Theorem 3.5, we can get A (E ) + B (E ) ⊂ E. By (H5), it is easy to see that A is continuous and A (E ) is precompact in PAP (R+ , X ). In addition, for u, v ∈ E, we have
∥B u − B v∥ ≤ sup
t ∈R+
t
Me−ω(t −s) L(s)ds · ∥u − v∥
0
≤ M ∥u − v∥ · sup
t ∈R+
≤ M ∥u − v∥ · sup
t
e−ωs L(t − s)ds 0
[t ]
t ∈R+ k=0
≤ M ∥u − v∥ · sup
k+1
e−ωs L(t − s)ds k
[t ]
t ∈R+ k=0
≤ M ∥L∥S p ∥u − v∥ ·
k+1
e
≤
−ωq 1/q
1−e
ωq
1/q
1 − e−ω
k+1
1/q
e−ωqs ds k
∥L∥S p · ∥u − v∥,
1/p
L (t − s)ds p
k
+∞
M
k+1
ds
k
k =0
−ωqs
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H.-S. Ding et al. / Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
where
1 − e−ωq
1/q
ωq
M 1 − e−ω
∥L∥S p < 1.
Then, by using Krasnoselskii’s fixed point theorem, we conclude that A + B has a fixed point in E, which is just a pseudo almost periodic mild solution to Eqs. (1.3)–(1.4). Remark 3.7. In Corollary 3.6, if L ∈ BS 1 (R), then we can obtain the same conclusion provided that
∥L∥S 1 <
1 − e−ω M
.
4. Integro-differential equations of heat conduction in materials with memory In this section, by virtue of our abstract results in Section 3, we will study the nonlocal problem of heat conduction in materials with memory, i.e., Eqs. (1.1)–(1.2). Let X , A, B, f , u0 , g be as in Section 1. First, it follows from [22,4] that A generates a C0 -semigroup {T (t )}t ≥0 with
∥T (t )∥ ≤ Me−γ t ,
∀t ≥ 0
for some constants M , γ > 0. Next, let us present an existence theorem about pseudo almost periodic solution for Eqs. (1.1)–(1.2). Theorem 4.1. Assume that (A1) and the following assumptions hold. (A2) α ′ (t )eγ t , α ′′ (t )eγ t , β ′ (t )eγ t , β ′′ (t )eγ t are bounded and uniformly continuous, and for all t ≥ 0, max{∥F21 (t )∥, ∥F22 (t )∥} ≤
γ e−γ t 2M
,
′ ′ max{∥F21 (t )∥, ∥F22 (t )∥} ≤
γ 2 e−γ t 4M 2
.
(A3) λ ∈ PAP (R+ , R); µ : R → R is bounded and there exists a constant L > 0 such that
|µ(x) − µ(y)| ≤ L|x − y|,
x, y ∈ R .
(A4) g1 : PAP (R+ , H01 (Ω )) → H01 (Ω ) and g2 : PAP (R+ , L2 (Ω )) → L2 (Ω ) are both continuous and compact. (A5) sup ∥g1 (u)∥ + sup ∥g2 (u)∥ lim sup
∥u∥1 ≤C
∥u∥2 ≤C
<
C
C →+∞
1 M
,
where ∥ · ∥1 denotes the norm of PAP (R+ , H01 (Ω )) and ∥ · ∥2 the norm of PAP (R+ , L2 (Ω )). e−ω Then Eqs. (1.1)–(1.2) has a pseudo almost periodic mild solution if ∥λ∥S 1 < 1−ML .
Proof. As noted in Section 1, Eqs. (1.1)–(1.2) can be transformed into the abstract nonlocal problem Eqs. (1.3)–(1.4) in the Banach space H01 (Ω ) × L2 (Ω ). We will show the conclusion by Corollary 3.6. So it remains to prove that all the assumptions of Corollary 3.6 hold. Since A generates a C0 -semigroup {T (t )}t ≥0 , (H1) holds except for the compactness of T (t ). Moreover, combining (A2) and the fact that
∥T (t )∥ ≤ Me−γ t ,
∀t ≥ 0
gives, by Theorem 4.1 in [3], that there exists a resolvent operator R(t ) satisfying γt
∥R(t )∥ ≤ Me− 2 ,
t ≥ 0.
Note that B(t ) = F (t )A. Then, (H2) holds by (A1). Since λ ∈ PAP (R+ , R), f ∈ PAP (R+ × X , X ). Moreover, we can get
∥f (t , u) − f (t , v)∥ ≤ L(t )∥u − v∥, where L(t ) = |λ(t )|L. As λ is bounded, it is easy to see that L ∈ BS p (R) for each p ≥ 1. In addition, since µ is also bounded,
∥f (t , u)∥ ≤ sup L(t ) · sup |µ(x)| · t ∈R+
x∈R
√
mesΩ < +∞,
for all t ∈ R+ and u ∈ X , which means that f (R+ × X ) is bounded. Thus, (H3) holds. By (A4), one can show that g : PAP (R+ , X ) → X is a continuous and compact mapping, i.e., (H5) holds.
H.-S. Ding et al. / Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
By (A5), there exists C ′ > 0 and k ∈ (0,
1 M
) such that
sup ∥g1 (u)∥ + sup ∥g2 (u)∥ < kC ,
∀C ≥ C ′ .
∥u∥1 ≤C
2669
∥u∥2 ≤C
Combining this with the fact that u0 = 0, f (R+ × X ) is bounded, and
∥g (u)∥ ≤ sup ∥g1 (u)∥ + sup ∥g2 (u)∥,
sup ∥u∥PAP (R+ ,X ) ≤C
∥u∥1 ≤C
∥u∥2 ≤C
we conclude that
M·
M 2M ∥u0 ∥ + sup ∥g (u)∥ + · sup ∥f (t , u)∥ ≤ kMC + · sup ∥f (t , u)∥ ≤ C , ω t ∈R+ , ∥u∥≤C γ t ∈R+ , u∈X ∥u∥≤C
for sufficiently large C , where ω = Finally, we note that ∥λ∥S 1 < 1 − e−ω
∥L(·)∥S 1 <
M
γ
. So (3.1) holds.
2 1−e−ω ML
, which yields that
.
Then, by Remark 3.7, we know that Eqs. (1.1)–(1.2) has a pseudo almost periodic mild solution.
Moreover, we give one more example to illustrate our abstract results. Example 4.2. Consider the following integro-differential equation with nonlocal initial condition:
t sin[u(t , x)] 1 −(t −s) e uxx (s, x)ds + + a(t ), ut (t , x) = uxx (t , x) + 1 + t2 1 +∞ 0 4 2 u(0, x) = xe−t u(t , η)dtdη, x ∈ (0, 1), 0
t ≥ 0, x ∈ (0, 1), (4.1)
0
where a(t ) = cos t + cos π t + max{exp[−(t ± k2 )2 ]}. k∈Z
Let X = L2 (0, 1);
(Au)(x) = u′′ (x) for x ∈ (0, 1) and u ∈ D(A) with D(A) = {u ∈ C 1 [0, 1] : u′ is absolutely continuous on [0, 1], u′′ ∈ X , u(0) = u(1) = 0}; 1 −t ′′ e u (x), 4 sin[u(x)]
[B(t )u](x) = f (t , u)(x) = g (u)(x) =
1 + t2
0
+ a(t ) t ≥ 0, u ∈ X , x ∈ (0, 1);
+∞
1
t ≥ 0, u ∈ D(A), x ∈ (0, 1);
2
xe−t u(t , η)dtdη,
u ∈ PAP (R+ , X ), x ∈ (0, 1).
0
It is well-known that A generates a compact C0 semigroup T (t ) satisfying 2
∥T (t )∥ ≤ e−π t ,
t ≥ 0,
which means that (H1) holds. In addition, it is easy to verify that (H2) and (H3) hold. Moreover, by Grimmer [3, Theorem 4.1], the resolvent operator of Eq. (4.1) satisfies 1
∥R(t )∥ ≤ e− 2 t ,
t ≥ 0.
Now, let us verify that (H4) holds. Take a bounded sequence
{un } ⊂ PAP (R+ , X ) such that {un } is uniformly convergent in every compact subsets of R+ . ∀ε > 0, there exists a constant M > 0 such that
2 1 + t2
2
< ε,
t > M.
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H.-S. Ding et al. / Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
Combining this with the fact that {un } is uniformly convergent in [0, M ], we can show that for ε > 0, there exists N > 0 such that 1
|f (t , un (t ))(x) − f (t , um (t ))(x)|2 dx ≤
1
0
0
1
|un (t )(x) − um (t )(x)|2 dx
≤ ≤ε
sin[un (t )(x)] − sin[um (t )(x)] 2 dx 1 + t2
0
for all t ∈ R and n, m > N. This means that {f (·, un (·))} is convergent in BC (R+ , X ). Next, let us show that (H5) holds. Let E ⊂ PAP (R+ , X ) be a bounded subset. By some direct calculations, we can get +
∥g (u)∥C [0,1] ≤
√ π 2
sup ∥u∥,
u ∈ E,
u∈E
and
{g (u) : u ∈ E } are equi-continuous on [0, 1]. Then, by using Arzela–Ascoli’s theorem, we know that g (E ) is precompact in C [0, 1], and thus precompact in X . Also, the continuity of g can be verified. Noting that M = 1, ω = 21 and u0 = 0, by a direct calculation, one can get
M·
∥u0 ∥ + sup ∥g (u)∥ + ∥u∥≤C
M
ω
·
sup t ∈R+ , ∥u∥≤C
∥f (t , u)∥ ≤
√ π 2
C,
∀C > 0,
which yields that (3.1) holds. So all the assumptions of Theorem 3.5 hold. Thus, the nonlocal problem (4.1) has a pseudo almost periodic mild solution. Acknowledgments H.S. Ding acknowledges support from the NSF of China (11101192), Chinese Ministry of Education (211090), and the NSF of Jiangxi Province. J. Liang and T.J. Xiao acknowledge support from the NSF of China (11071042), (11171210), the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900) and the Laboratory of Mathematics for Nonlinear Science at Fudan University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal. 31 (1968) 113–126. R.K. Miller, An integrodifferential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978) 313–332. R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc. 273 (1982) 333–349. J. Liang, T.J. Xiao, Semilinear integrodifferential equations with nonlocal initial conditions, Comput. Math. Appl. 47 (2004) 863–875. J. Liang, J.H. Liu, T.J. Xiao, Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal. 5 (2006) 529–535. J. Liang, J. van Casteren, T.J. Xiao, Nonlocal Cauchy problems for semilinear evolution equations, Nonlinear Anal. 50 (2002) 173–189. T.J. Xiao, J. Liang, Existence of classical solutions to nonautonomous nonlocal parabolic problems, Nonlinear Anal. 63 (2005) 225–232. J. Liang, J.H. Liu, T.J. Xiao, Nonlocal problems for integrodifferential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 15 (2008) 815–824. Y. Lin, J.H. Liu, Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear Anal. 26 (1996) 1023–1033. J.H. Liu, K. Ezzinbi, Non-autonomous integrodifferential equations with nonlocal conditions, J. Integral Equations Appl. 15 (2003) 79–93. S. Aizicovici, M. McKibben, Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear Anal. 39 (2000) 649–668. S. Aizicovici, H. Lee, Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. Math. Lett. 18 (2005) 401–407. A. Caicedo, C. Cuevas, H.R. Henríquez, Asymptotic periodicity for a class of partial integrodifferential equations, ISRN Math. Anal. Volume (2011) 18. Article ID 537890. J.P.C. dos Santos, C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. Math. Lett. 23 (2010) 960–965. T. Diagana, E. Hernández, J.P.C. dos Santos, Existence of asymptotically almost automorphic solutions to some abstract partial neutral integrodifferential equations, Nonlinear Anal. 71 (2009) 248–257. H.S. Ding, J. Liang, T.J. Xiao, Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, J. Math. Anal. Appl. 338 (2008) 141–151. E. Hernández, J.P.C. dos Santos, Asymptotically almost periodic and almost periodic solutions for a class of partial integrodifferential equations, Electron. J. Differ. Eq. 38 (2006) 1–8. G.M. N’Guérékata, Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plnum Publishers, New York, Berlin, Moscow, 2001. C. Zhang, Almost Periodic Type Functions and Ergodicity, Kluwer Academic Publishers, Dordrecht, Boston, London, 2003. G.M. N’Guérékata, A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal. 68 (2008) 2658–2667. W. Desch, R. Grimmer, W. Schappacher, Some considerations for linear integro-differential equations, J. Math. Anal. Appl. 104 (1984) 219–234. G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control 17 (1979) 66–81.
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Pseudo almost periodic solutions to integro-differential equations of heat conduction in materials with memory Hui-Sheng Ding a , Jin Liang b , Ti-Jun Xiao c,∗ a
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, People’s Republic of China
b
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
c
Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
article
abstract
info
Article history: Received 26 October 2011 Accepted 18 March 2012
We investigate the existence of pseudo almost periodic solutions for a class of integrodifferential equation with nonlocal initial conditions arising in the study of heat conduction in materials with memory. We first establish some existence theorems for abstract semilinear integro-differential equations with nonlocal initial conditions, and then, we apply our abstract results to the addressed integro-differential equation arising in the study of heat conduction in materials with memory. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Pseudo almost periodic Nonlocal Cauchy problems heat conduction in materials with memory Integro-differential equations
1. Introduction This paper is mainly concerned with the existence of pseudo almost periodic solutions to the following integrodifferential equation arising from the study of heat conduction in materials with memory (see, e.g., [1])
t ∂ 2 θ (t , x) ∂θ (t , x) ∂θ (s, x) + β( 0 ) = α( 0 ) 1 θ ( t , x ) − β ′ (t − s) ds 2 ∂t ∂t ∂s 0 t + α ′ (t − s)1θ (s, x)ds + λ(t )µ[θ (t , x)],
(1.1)
0
where t ≥ 0, x ∈ Ω and Ω is a bounded open connected subset of R3 with C ∞ boundary. This equation and its variants have been investigated by many authors (see, e.g., [2,3]). In this paper, we will consider Eq. (1.1) with the following nonlocal initial conditions:
∂θ (t , x) = g2 (θ ). ∂ t t =0
θ (0, x) = g1 (θ ), As in [3,4], we assume that
(A1) α, β ∈ C 2 ([0, +∞), R) with α(0) and β(0) positive. Denote X = H01 (Ω ) × L2 (Ω ),
A=
∗
0 α(0)∆
I
−β(0)I
,
D (A) = (H 2 (Ω ) ∩ H01 (Ω )) × H01 (Ω );
Corresponding author. Tel.: +86 21 65642341; fax: +86 21 65646073. E-mail addresses: [email protected] (H.-S. Ding), [email protected] (J. Liang), [email protected] (T.-J. Xiao).
1468-1218/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2012.03.009
(1.2)
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θ (t ) [θ (t )](x) = θ (t , x), u( t ) = ; ′ θ (t ) 0 u1 ∈ X; [f (t , u)](x) = for u = u2 λ(t )µ[u1 (x)] g1 (u1 ) u1 ∈ X; u0 = 0, g (u) = for u = u2 g2 (u1 ) and B(t ) = F (t )A, where F (t ) = [Fij (t )] is defined by F11 (t ) = F12 (t ) = 0,
F21 (t ) = −β ′ (t )I + β(0)
α ′ (t ) I, α(0)
F22 (t ) =
α ′ (t ) I. α(0)
Then, Eqs. (1.1) and (1.2) is transformed into the following abstract integro-differential equation u′ (t ) = Au(t ) +
t
B(t − s)u(s)ds + f (t , u(t )),
t ≥ 0,
(1.3)
0
with a nonlocal initial condition u(0) = u0 + g (u),
(1.4)
in the Banach space X . Recall that the integro-differential equation (1.3) with a nonlocal initial condition (1.4) has been a topic of interest for many researchers. There is a lot of literature on the existence of solutions for various forms of Eqs. (1.3)–(1.4) on a finite interval (see, e.g, [5,6,4,7–12] and references therein). On the other hand, very recently, the existence of almost periodic type solutions and almost automorphic type solutions for Eqs. (1.3)–(1.4) and its variants has attracted more and more attention. We refer the reader to [13–17] for some recent development on this topic. In this paper, we will make further study on this topic, i.e., we will establish some existence theorems on pseudo almost periodic solutions for Eqs. (1.3)–(1.4), and then, apply our abstract results to Eqs. (1.1)–(1.2). 2. Pseudo almost periodic functions on R+ Throughout this paper, R denotes the set of real numbers; R+ denotes the set of nonnegative real numbers; X , Y denote two Banach spaces; Ω denotes a subset of X ; BC (R+ , X ) denotes the Banach space of bounded continuous functions from R+ to X with supremum norm; for p ≥ 1, C p (R+ × Ω , Y ) denotes the set of all BS p -uniformly continuous functions f : R+ × Ω → Y , i.e., there is a nonnegative function L ∈ BS p (R) satisfying that ∀ε > 0, there exists δ > 0 such that
∥f (t , u) − f (t , v)∥ ≤ L(t )ε for all t ∈ R+ and u, v ∈ Ω with ∥u − v∥ < δ . Next, let us recall and introduce some definitions and notations about almost periodicity and pseudo almost periodicity. For more details, we refer the reader to [18,19]. Definition 2.1. A continuous function f : R → X is called almost periodic if for each ε > 0 there exists l(ε) > 0 such that every interval I of length l(ε) contains a number τ with the property that sup ∥f (t + τ ) − f (t )∥ < ε. t ∈R
We denote by AP (X ) the set of all such functions. Definition 2.2. A continuous function f : R × Ω → Y is called almost periodic in t uniformly for x ∈ Ω if for each ε > 0 and for each compact subset K ⊂ Ω there exists l(ε) > 0 such that every interval I of length l(ε) contains a number τ with the property that sup sup ∥f (t + τ , x) − f (t , x)∥ < ε. t ∈R x∈K
We denote by AP (R × Ω , Y ) the set of all such functions. In this paper, we denote PAP0 (R , X ) = +
h ∈ BC (R , X ) : lim +
T →+∞
1 T
T
∥h(t )∥dt = 0 . 0
Moreover, we denote by PAP0 (R+ × Ω , Y ) the space of all continuous functions h : R+ × Ω → Y such that for each x ∈ Ω , h(·, x) ∈ PAP0 (R+ , Y ).
H.-S. Ding et al. / Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
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Definition 2.3. A function f : R+ → X is called pseudo almost periodic if f (t ) = g (t ) + h(t ),
t ∈ R+ ,
where g ∈ AP (X ) and h ∈ PAP0 (R+ , X ). We denote by PAP (R+ , X ) the set of all such functions. Definition 2.4. A function f : R+ × Ω → Y is called pseudo almost periodic if f (t , x) = g (t , x) + h(t , x),
t ∈ R+ , x ∈ Ω ,
where g ∈ AP (R × Ω , Y ) and h ∈ PAP0 (R+ × Ω , Y ). We denote by PAP (R+ × Ω , Y ) the set of all such functions. Lemma 2.5. Let f = g + h ∈ PAP (R+ , X ) with g ∈ AP (X ) and h ∈ PAP0 (R+ , X ). Then
{g (t ) : t ∈ R} ⊂ {f (t ) : t ∈ R+ }. Proof. The proof is similar to that of pseudo almost periodic functions on R (cf. [19]). So we omit the details.
By Lemma 2.5, it is easy to see that the decomposition of every pseudo almost periodic function is unique; also, it is not difficult to prove the following result. Lemma 2.6. PAP (R+ , X ) is a Banach space under the supremum norm
∥f ∥ = sup ∥f (t )∥. t ∈R+
Next, we have the following lemma. Lemma 2.7. Let f ∈ BC (R+ , X ). Then f ∈ PAP0 (R+ , X ) if and only if ∀ε > 0, lim
mesMT ,ε (f ) T
T →+∞
= 0,
where MT ,ε (f ) = {t ∈ [0, T ] : ∥f (t )∥ ≥ ε}. Proof. The proof is also similar to that of pseudo almost periodic functions on R (cf. [19]). So we omit the details.
We also need to recall some definitions about Stepanov bounded functions (for more details, see [20]). Definition 2.8. The Bochner transform f b (t , s), t ∈ R, s ∈ [0, 1], of a function f (t ) on R, with values in X , is defined by f b (t , s) := f (t + s). Definition 2.9. Let p ∈ [1, ∞). The space BS p (X ) of all Stepanov bounded functions, with the exponent p, consists of all measurable functions f on R with values in X such that f b : R → Lp (0, 1; X ) is bounded. We denote
t +1
∥f ∥S p = sup t ∈R
∥f (τ )∥p dτ
1/p
.
t
Now, we are ready to establish the following composition theorem. Theorem 2.10. Let x ∈ PAP (R+ , X ) and Ω = {x(t ) : t ∈ R+ }. Suppose that f ∈ PAP (R+ × Ω , Y ) f (R+ × Ω ) is bounded. Then f (·, x(·)) ∈ PAP (R+ , Y ).
C 1 (R+ × Ω , Y ) and
Proof. Let x = α + β and f = g + h, where α ∈ AP (X ), β ∈ PAP0 (R+ , X ), g ∈ AP (R × Ω , Y ) and h ∈ PAP0 (R+ × Ω , Y ). Denote I (t ) = g (t , α(t )),
J (t ) = f (t , x(t )) − f (t , α(t )),
K (t ) = h(t , α(t )).
Then f (t , x(t )) = I (t ) + J (t ) + K (t ). Since f (R+ × Ω ) is bounded, f (·, x(·)) ∈ BC (R+ , Y ). In addition, we denote M =
sup t ∈R+ ,u∈Ω
∥f (t , u)∥.
By Lemma 2.5, {α(t ) : t ∈ R} ⊂ Ω . Then it is easy to see that I ∈ AP (Y ). Next, let us show that J ∈ PAP0 (R+ , Y ). Since f ∈ C 1 (R+ × Ω , Y ), there is a nonnegative function L ∈ BS 1 (R) satisfying that ∀ε > 0, there exists a constant δ > 0 such that ∥β(t )∥ < δ implies that
∥f (t , x(t )) − f (t , α(t ))∥ ≤ L(t )ε.
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Thus, we have 1
T
T
∥J (t )∥dt = 0
1
T
MT ,δ (β)
≤ 2M · ≤ 2M · ≤ 2M ·
∥J (t )∥dt +
mesMT ,δ (β) T mesMT ,δ (β) T mesMT ,δ (β) T
+
1
T
1
[0,T ]\MT ,δ (β) T
T
∥J (t )∥dt
L(t )ε dt 0
+ε·
1
[T ]+1
T
L(t )dt 0
+ ε∥L∥S 1 ·
[T ] + 1 T
.
On the other hand, by Lemma 2.7, mesMT ,δ (β)
lim
T
T →+∞
= 0.
Thus, we get lim sup T →+∞
1
T
T
∥J (t )∥dt ≤ ε∥L∥S 1 , 0
which means that lim sup T →+∞
1
T
T
∥J (t )∥dt = 0, 0
i.e., 1
lim
T
T →+∞
T
∥J (t )∥dt = 0. 0
It remains to prove that K ∈ PAP0 (R+ , Y ). Let Σ = {α(t ) : t ∈ R}. It is well-known that Σ ⊂ Ω is compact. Since g is uniformly continuous on R × Σ , g ∈ C 1 (R+ × Σ , Y ). Thus, h ∈ C 1 (R+ × Σ , Y ), i.e., there is a function L ∈ BS 1 (R) satisfying that ∀ε > 0, there exists δ > 0 such that
∥h(t , u) − h(t , v)∥ ≤ L(t )ε
(2.1)
for all t ∈ R+ and u, v ∈ Σ with ∥u − v∥ < δ . For any fixed ε > 0, let δ be such that (2.1) holds. Then there exist x1 , . . . , xk ∈ Σ such that
Σ⊂
k
B(xi , δ).
i=1
For each t ∈ R+ , there is an xi such that ∥α(t ) − xi ∥ < δ . Then, we get
∥K (t )∥ = ∥h(t , α(t ))∥ = ∥h(t , α(t )) − h(t , xi )∥ + ∥h(t , xi )∥ k ≤ L(t )ε + ∥h(t , xi )∥, i =1
which gives that 1 T
T
∥K (t )∥dt ≤ 0
≤
1 T
T
1 L(t )ε dt +
0
T
T
0
k
k 1 [T ] + 1 ∥L∥S 1 · ε +
T
i =1
∥h(t , xi )∥dt
i =1
T
T
∥h(t , xi )∥dt . 0
Noting that h(·, xi ) ∈ PAP0 (R+ , X ), i = 1, . . . , k, we get lim sup T →+∞
1 T
T
∥K (t )∥dt ≤ ∥ L∥S 1 · ε. 0
H.-S. Ding et al. / Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
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Thus, lim
1
T →+∞
T
T
∥K (t )∥dt = 0, 0
i.e., K ∈ PAP0 (R+ , Y ). This completes the proof.
3. Abstract results In this section, we will establish some abstract results for Eqs. (1.3)–(1.4). First, let us recall some notations about resolvent operator. Definition 3.1 ([21]). A family {R(t ) : t ≥ 0} of continuous linear operators on X is called a resolvent operator (to Eq. (1.3)), iff (R1) R(0) = I, the identity map on X , (R2) for all x ∈ X , the map t → R(t )x is a continuous X -valued function on [0, +∞), (R3) for all t ≥ 0, R(t ) is a continuous operator on Y , and for all y ∈ Y , the map t → R(t )y belongs to C ([0, +∞), Y ) ∩ C 1 ([0, +∞), X ) and satisfies d dt
R(t )y = AR(t )y +
t
B(t − s)R(s)yds 0
= R(t )Ay +
t
R(t − s)B(s)yds, 0
where Y = D(A) = D(B(t )) for all t ≥ 0 and it is equipped with the graph norm. For more details about resolvent operator, we refer the reader to [3,21]. If the resolvent operator R(·) of Eq. (1.3) exists, then we can define the mild solution of Eqs. (1.3)–(1.4) as the following. Definition 3.2. A function u ∈ C ([0, +∞), X ) is called a mild solution of Eqs. (1.3)–(1.4) if u(t ) = R(t )[u0 + g (u)] +
t
R(t − s)f (s, u(s)),
t ≥ 0.
0
Next, for convenience, we list some assumptions. (H1) A is a densely defined, closed linear operator in X and generates a compact C0 -semigroup T (t ). Hence D(A) endowed with the graph norm |u| = ∥u∥ + ∥Au∥ is a Banach space which will be denoted by (D(A), | · |). (H2) {B(t ) : t ≥ 0} is a family of continuous linear operators from (D(A), | · |) into (X , ∥ · ∥), and for each y ∈ D(A), the map 1,1 t → B(t )y belongs to Wloc (R+ , X ). Moreover, there exists a locally integrable function b : R+ → R+ such that
d B(t )y ≤ b(t )|y|, y ∈ D(A), t ∈ R+ . dt (H3) f ∈ PAP (R+ × X , X ) C 1 (R+ × X , X ) and f (R+ × B) is bounded for each bounded subset B ⊂ X . (H4) For each bounded sequence {un } ⊂ PAP (R+ , X ), if {un } is uniformly convergent in every compact subsets of R+ , then {f (·, un (·))} is precompact in BC (R+ , X ). (H5) g : PAP (R+ , X ) → X is a continuous and compact mapping. In the proof of our main result, we will need the following theorem. Theorem 3.3 ([21,8]). Assume that (H1) and (H2) hold. Then (a) there exists a resolvent operator R(·) of Eq. (1.3), and R(t ) is compact for each t > 0; (b) for any fixed T > 0, there exists a constant H = H (T ) such that
∥R(t + h) − R(h)R(t )∥ ≤ Hh for all h, t ∈ [0, T ] with h ≤ t; (c) R(·) is operator norm continuous on (0, +∞). Lemma 3.4. Let {R(t ) : t ≥ 0} be a strongly continuous family of bounded and linear operators on X satisfying ∥R(t )∥ ≤ Me−ωt for all t ≥ 0, where M , ω > 0 are fixed constants. Then, for each f ∈ PAP (R+ , X ), fR ∈ PAP (R+ , X ), fR (t ) =
t
R(t − s)f (s)ds, 0
t ≥ 0.
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Proof. Let f = g + h, where g ∈ AP (X ), h ∈ PAP0 (R+ , X ). We observe that t
fR (t ) =
R(t − s)g (s)ds +
R(t − s)h(s)ds 0
0 t
t
R(t − s)g (s)ds +
=
t
R(t − s)h(s)ds −
R(t − s)g (s)ds −
0
R(t − s)h(s)ds. −∞
−∞
−∞
−∞
0
Let t
G(t ) =
R(t − s)g (s)ds,
t ∈ R,
−∞
and H (t ) =
t
R(t − s)h(s)ds,
I (t ) = −
0
R(t − s)g (s)ds −
0
R(t − s)h(s)ds,
t ≥ 0.
−∞
−∞
−∞
By the strong continuity of {R(t )}, one can prove that G, H, and I are all continuous functions. It is easy to see that G ∈ AP (X ). By a direct calculation, one can get 1
T
T
1
∥H (t )∥dt ≤
T
T
0
+∞
0
+∞
Me−ωs ∥h(t − s)∥dsdt 0
Me−ωs H (s, T )ds,
= 0
where
H (s, T ) =
1 T
T
∥h(t − s)∥dt .
0
Noting that h ∈ PAP0 (R+ , X ), for each s ≥ 0, we have lim H (s, T ) = 0.
T →+∞
Then, by using Lebesgue’s dominated convergence theorem, we get lim
1
T →+∞
T
T
∥H (t )∥dt = 0, 0
i.e., H ∈ PAP0 (R+ , X ). In addition, we have
∥I (t )∥ ≤ [∥g ∥AP (X ) + ∥h∥BC (R+ ,X ) ]
0
Me−ω(t −s) ds
−∞
M = [∥g ∥AP (X ) + ∥h∥BC (R+ ,X ) ] e−ωt → 0,
ω
t → +∞,
which yields that I ∈ PAP0 (R+ , X ). This completes the proof.
Now, we are ready to establish one of our main results. Theorem 3.5. Assume that (H1)–(H5) hold. Moreover, the resolvent operator of Eq. (1.3) satisfies ∥R(t )∥ ≤ Me−ωt for all t ≥ 0, where M , ω > 0 are some fixed constants. Then there exists a pseudo almost periodic mild solution to Eqs. (1.3)–(1.4) provided that there exists a constant C > 0 such that
M·
∥u0 ∥ + sup ∥g (u)∥ + ∥u∥≤C
M
ω
·
sup t ∈R+ , ∥u∥≤C
∥f (t , u)∥ ≤ C .
(3.1)
Proof. Let E = {u ∈ PAP (R+ , X ) : ∥u∥ ≤ C }. Define an operator R on E by
(R u)(t ) = R(t )[u0 + g (u)] +
t
R(t − s)f (s, u(s))ds,
t ≥ 0.
0
It suffices to prove that R has a fixed point in E. First, by Theorem 3.3, we know that R(t ) is compact for each t > 0 and R(·) is operator norm continuous on (0, +∞). Next, we divide the proof into five steps.
H.-S. Ding et al. / Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
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Step 1. R (E ) ⊂ E. Fix u ∈ E. Noting that ∥R(t )∥ ≤ Me−ωt for all t ≥ 0, by (3.1), we have ∥R u∥ ≤ C . On the other hand, by (H3) and Theorem 2.10, we get f (·, u(·)) ∈ PAP (R+ , X ), and then Lemma 3.4 yields that t
R(t − s)f (s, u(s))ds
t → 0
belongs to PAP (R+ , X ). In addition, it is easy to see that R(·)[u0 + g (u)] ∈ PAP0 (R+ , X ). So R u ∈ PAP (R+ , X ), and thus R u ∈ E . Step 2. R : E → E is continuous. Let un → u in PAP (R+ , X ). Fix ε > 0. Since f ∈ C 1 (R+ × X , X ), there exists a nonnegative function L ∈ BS 1 (R) and a constant δ > 0 such that
∥f (t , u) − f (t , v)∥ ≤ L(t )ε for all t ∈ R+ and u, v ∈ X with ∥u − v∥ < δ . Then, when n is sufficiently large, ∥un − u∥ < δ , which gives that
∥f (t , un (t )) − f (t , u(t ))∥ ≤ L(t )ε,
∀t ∈ R+ .
On the other hand, by (H4), g : PAP (R+ , X ) → X is continuous, and thus
∥g (un ) − g (u)∥ < ε when n is sufficiently large. Now, for sufficiently large n, we have
∥R un − R u∥ ≤ M ∥g (un ) − g (u)∥ + sup
t ∈R+
≤ M ε 1 + sup
e
t ∈R+
= M ε 1 + sup
≤ M ε 1 + sup
−ω(t −s)
≤ M ε 1 + sup
e
−ωs
L(t − s)ds
≤ Mε 1 +
+∞
0
[t ]
[t ]
k +1
e
−ωs
L(t − s)ds
k
e
−ωk
k+1
t ∈R+ k=0
L(s)ds
t
t ∈R+ k=0
Me−ω(t −s) L(s)ε ds
0
0
t ∈R+
t
t
L(t − s)ds k
e
−ωk
· ∥L∥S 1
k =0
∥L∥S 1 ≤ Mε 1 + . 1 − e−ω Thus, we conclude that R un → R u in PAP (R+ , X ), which means that R is continuous. Step 3. For each t ∈ R+ , {(R u)(t ) : u ∈ E } is precompact in X . For t = 0,
{R(t )[u0 + g (u)] : u ∈ E } = {u0 } + g (E ) is precompact in X since g is compact. For each t > 0, since R(t ) is compact and {u0 + g (u) : u ∈ E } is bounded, we know that {R(t )[u0 + g (u)] : u ∈ E } is precompact. It remains to show that t
R(t − s)f (s, u(s))ds : u ∈ E
0
is precompact for each fixed t > 0. It follows from (3.1) that sup t ∈R+ ,u∈E
∥f (t , u(t ))∥ ≤
ωC M
.
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Then, for each ε ∈ (0, t ) and u ∈ E, by (b) of Theorem 3.3, we have
t t −ε R(t − s)f (s, u(s))ds − R(ε) R(t − s − ε)f (s, u(s))ds 0 0 t t −ε ωC ≤ R(t − s)f (s, u(s))ds ∥R(t − s) − R(ε)R(t − s − ε)∥ds + M
t −ε
≤ ωC ε +
ωC
t −2 ε
t −ε
+
M
t −2ε
0
≤ (M + 2)ωC ε + ≤ (M + 2)ωC ε +
ωC
t −2ε
M
ωCt M
0
∥R(t − s) − R(ε)R(t − s − ε)∥ds
∥R(t − s) − R(ε)R(t − s − ε)∥ds
0
H (t )ε,
(3.2)
where H (t ) is a constant, which is only dependent on t. On the other hand, for each ε ∈ (0, t ),
R(ε)
t −ε
R(t − s − ε)f (s, u(s))ds : u ∈ E
0
is precompact since R(ε) is compact and
t −ε 0
R(t − s − ε)f (s, u(s))ds ≤ ωCt .
Then, by using (3.2), we conclude that t
R(t − s)f (s, u(s))ds : u ∈ E
0
is totally bounded, and thus precompact. Step 4. For each t ∈ R+ , {R u : u ∈ E } is equicontinuous at t. Since {u0 + g (u) : u ∈ E } is precompact, by the strong continuity of R(·),
{R(·)[u0 + g (u)] : u ∈ E } is equicontinuous at t. In addition, for each u ∈ E and 1t ≥ 0, we have
t +1 t t R ( t + ∆ t − s ) f ( s , u ( s )) ds − R ( t − s ) f ( s , u ( s )) ds 0 0 t +1 t t ≤ R(t + ∆t − s)f (s, u(s))ds ∥[R(t + ∆t − s) − R(t − s)]f (s, u(s))∥ds + t 0 t ≤ ωC |1t | + ∥[R(t + ∆t − s) − R(t − s)]f (s, u(s))∥ds 0 ωC t ≤ ωC |1t | + ∥R(t + ∆t − s) − R(t − s)∥ds, M
0
and for each u ∈ E and 1t ≤ 0, we have
t +1 t
R(t + ∆t − s)f (s, u(s))ds −
t
0
0
R(t − s)f (s, u(s))ds
t +∆t R(t − s)f (s, u(s))ds ∥[R(t + ∆t − s) − R(t − s)]f (s, u(s))∥ds + t +1 t 0 ω C t +∆ t ∥R(t + 1t − s) − R(t − s)∥ds. ≤ ωC |1t | + ≤
t
M
0
Then, by using Lebesgue’s dominated convergence theorem and the operator norm continuity of R(·) on (0, +∞), we conclude that
t
R(t − s)f (s, u(s))ds : u ∈ E
t → 0
is equicontinuous at each t ∈ R+ .
H.-S. Ding et al. / Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
2667
Step 5. R has a fixed point in E. Denote the closed convex hull of R (E ) by F . By Step 1, R (E ) ⊂ E. Since E is closed and convex, we conclude that F ⊂ E. Then, we have R (F ) ⊂ R (E ) ⊂ F .
Moreover, by Step 2, R : F → F is continuous. In addition, this follows from Steps 3 and 4; we know that for each t ∈ R+ , {u(t ) : u ∈ F } is precompact in X , and {u : u ∈ F } is equicontinuous at t. Next, let us show that R (F ) is precompact. Let {uk }∞ k=1 ⊂ F be an arbitrary sequence. Since g is compact, one can assume that g (uk ) → g (u) in X (by taking a subsequence, if necessary). For any fixed n ∈ N, noting that {uk (t ) : k ∈ N} is precompact in X for each t ∈ R+ and {uk }∞ k=1 is uniformly equicontinuous on [0, n], we conclude that {uk : k ∈ N} is precompact in C ([0, n], X ) by the abstract version of Arzela–Ascoli’s theorem. Then, by the diagonal method, one can choose ∞ ∞ a subsequence of {uk }∞ k=1 , which we also denote by {uk }k=1 for convenience, such that {uk }k=1 is uniformly convergent on + + every compact subset of R . Now, (H4) yields that f (·, uk (·)) is precompact in BC (R , X ). Thus, there is a subsequence um such that f (·, um (·)) is uniformly convergent on R+ , which gives that
t
R(t − s)f (s, um (s))ds : m ∈ N
t →
0
is uniformly convergent on R+ . Recalling that g (uk ) → g (u) in X , we conclude that R (um ) is convergent in PAP (R+ , X ), which means that R (F ) is precompact. Now, by using Schauder’s fixed point theorem, R has a fixed point in F . Corollary 3.6. Assume that all the assumptions of Theorem 3.5 hold except for (H4) and the compactness of T (·). Moreover, there is a nonnegative function L ∈ BS p (R) with p > 1 such that
∥f (t , u) − f (t , v)∥ ≤ L(t )∥u − v∥,
∀t ∈ R+ , ∀u, v ∈ X .
Then there exists a pseudo almost periodic mild solution to Eqs. (1.3)–(1.4) provided that
∥L∥S p <
1 − e−ω
M
ωq 1 − e−ωq
1/q
1
,
p
1
+
q
= 1.
Proof. It follows from [21] that there also exists a resolvent operator R(·) of Eq. (1.3), which is now not necessarily compact. Let E be as in the proof of Theorem 3.5. Define two operators on E by
(A u)(t ) = R(t )[u0 + g (u)],
t ≥0
and
(B u)(t ) =
t
R(t − s)f (s, u(s))ds,
t ≥ 0.
0
Then, similarly as in the proof of Step 1 of Theorem 3.5, we can get A (E ) + B (E ) ⊂ E. By (H5), it is easy to see that A is continuous and A (E ) is precompact in PAP (R+ , X ). In addition, for u, v ∈ E, we have
∥B u − B v∥ ≤ sup
t ∈R+
t
Me−ω(t −s) L(s)ds · ∥u − v∥
0
≤ M ∥u − v∥ · sup
t ∈R+
≤ M ∥u − v∥ · sup
t
e−ωs L(t − s)ds 0
[t ]
t ∈R+ k=0
≤ M ∥u − v∥ · sup
k+1
e−ωs L(t − s)ds k
[t ]
t ∈R+ k=0
≤ M ∥L∥S p ∥u − v∥ ·
k+1
e
≤
−ωq 1/q
1−e
ωq
1/q
1 − e−ω
k+1
1/q
e−ωqs ds k
∥L∥S p · ∥u − v∥,
1/p
L (t − s)ds p
k
+∞
M
k+1
ds
k
k =0
−ωqs
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where
1 − e−ωq
1/q
ωq
M 1 − e−ω
∥L∥S p < 1.
Then, by using Krasnoselskii’s fixed point theorem, we conclude that A + B has a fixed point in E, which is just a pseudo almost periodic mild solution to Eqs. (1.3)–(1.4). Remark 3.7. In Corollary 3.6, if L ∈ BS 1 (R), then we can obtain the same conclusion provided that
∥L∥S 1 <
1 − e−ω M
.
4. Integro-differential equations of heat conduction in materials with memory In this section, by virtue of our abstract results in Section 3, we will study the nonlocal problem of heat conduction in materials with memory, i.e., Eqs. (1.1)–(1.2). Let X , A, B, f , u0 , g be as in Section 1. First, it follows from [22,4] that A generates a C0 -semigroup {T (t )}t ≥0 with
∥T (t )∥ ≤ Me−γ t ,
∀t ≥ 0
for some constants M , γ > 0. Next, let us present an existence theorem about pseudo almost periodic solution for Eqs. (1.1)–(1.2). Theorem 4.1. Assume that (A1) and the following assumptions hold. (A2) α ′ (t )eγ t , α ′′ (t )eγ t , β ′ (t )eγ t , β ′′ (t )eγ t are bounded and uniformly continuous, and for all t ≥ 0, max{∥F21 (t )∥, ∥F22 (t )∥} ≤
γ e−γ t 2M
,
′ ′ max{∥F21 (t )∥, ∥F22 (t )∥} ≤
γ 2 e−γ t 4M 2
.
(A3) λ ∈ PAP (R+ , R); µ : R → R is bounded and there exists a constant L > 0 such that
|µ(x) − µ(y)| ≤ L|x − y|,
x, y ∈ R .
(A4) g1 : PAP (R+ , H01 (Ω )) → H01 (Ω ) and g2 : PAP (R+ , L2 (Ω )) → L2 (Ω ) are both continuous and compact. (A5) sup ∥g1 (u)∥ + sup ∥g2 (u)∥ lim sup
∥u∥1 ≤C
∥u∥2 ≤C
<
C
C →+∞
1 M
,
where ∥ · ∥1 denotes the norm of PAP (R+ , H01 (Ω )) and ∥ · ∥2 the norm of PAP (R+ , L2 (Ω )). e−ω Then Eqs. (1.1)–(1.2) has a pseudo almost periodic mild solution if ∥λ∥S 1 < 1−ML .
Proof. As noted in Section 1, Eqs. (1.1)–(1.2) can be transformed into the abstract nonlocal problem Eqs. (1.3)–(1.4) in the Banach space H01 (Ω ) × L2 (Ω ). We will show the conclusion by Corollary 3.6. So it remains to prove that all the assumptions of Corollary 3.6 hold. Since A generates a C0 -semigroup {T (t )}t ≥0 , (H1) holds except for the compactness of T (t ). Moreover, combining (A2) and the fact that
∥T (t )∥ ≤ Me−γ t ,
∀t ≥ 0
gives, by Theorem 4.1 in [3], that there exists a resolvent operator R(t ) satisfying γt
∥R(t )∥ ≤ Me− 2 ,
t ≥ 0.
Note that B(t ) = F (t )A. Then, (H2) holds by (A1). Since λ ∈ PAP (R+ , R), f ∈ PAP (R+ × X , X ). Moreover, we can get
∥f (t , u) − f (t , v)∥ ≤ L(t )∥u − v∥, where L(t ) = |λ(t )|L. As λ is bounded, it is easy to see that L ∈ BS p (R) for each p ≥ 1. In addition, since µ is also bounded,
∥f (t , u)∥ ≤ sup L(t ) · sup |µ(x)| · t ∈R+
x∈R
√
mesΩ < +∞,
for all t ∈ R+ and u ∈ X , which means that f (R+ × X ) is bounded. Thus, (H3) holds. By (A4), one can show that g : PAP (R+ , X ) → X is a continuous and compact mapping, i.e., (H5) holds.
H.-S. Ding et al. / Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
By (A5), there exists C ′ > 0 and k ∈ (0,
1 M
) such that
sup ∥g1 (u)∥ + sup ∥g2 (u)∥ < kC ,
∀C ≥ C ′ .
∥u∥1 ≤C
2669
∥u∥2 ≤C
Combining this with the fact that u0 = 0, f (R+ × X ) is bounded, and
∥g (u)∥ ≤ sup ∥g1 (u)∥ + sup ∥g2 (u)∥,
sup ∥u∥PAP (R+ ,X ) ≤C
∥u∥1 ≤C
∥u∥2 ≤C
we conclude that
M·
M 2M ∥u0 ∥ + sup ∥g (u)∥ + · sup ∥f (t , u)∥ ≤ kMC + · sup ∥f (t , u)∥ ≤ C , ω t ∈R+ , ∥u∥≤C γ t ∈R+ , u∈X ∥u∥≤C
for sufficiently large C , where ω = Finally, we note that ∥λ∥S 1 < 1 − e−ω
∥L(·)∥S 1 <
M
γ
. So (3.1) holds.
2 1−e−ω ML
, which yields that
.
Then, by Remark 3.7, we know that Eqs. (1.1)–(1.2) has a pseudo almost periodic mild solution.
Moreover, we give one more example to illustrate our abstract results. Example 4.2. Consider the following integro-differential equation with nonlocal initial condition:
t sin[u(t , x)] 1 −(t −s) e uxx (s, x)ds + + a(t ), ut (t , x) = uxx (t , x) + 1 + t2 1 +∞ 0 4 2 u(0, x) = xe−t u(t , η)dtdη, x ∈ (0, 1), 0
t ≥ 0, x ∈ (0, 1), (4.1)
0
where a(t ) = cos t + cos π t + max{exp[−(t ± k2 )2 ]}. k∈Z
Let X = L2 (0, 1);
(Au)(x) = u′′ (x) for x ∈ (0, 1) and u ∈ D(A) with D(A) = {u ∈ C 1 [0, 1] : u′ is absolutely continuous on [0, 1], u′′ ∈ X , u(0) = u(1) = 0}; 1 −t ′′ e u (x), 4 sin[u(x)]
[B(t )u](x) = f (t , u)(x) = g (u)(x) =
1 + t2
0
+ a(t ) t ≥ 0, u ∈ X , x ∈ (0, 1);
+∞
1
t ≥ 0, u ∈ D(A), x ∈ (0, 1);
2
xe−t u(t , η)dtdη,
u ∈ PAP (R+ , X ), x ∈ (0, 1).
0
It is well-known that A generates a compact C0 semigroup T (t ) satisfying 2
∥T (t )∥ ≤ e−π t ,
t ≥ 0,
which means that (H1) holds. In addition, it is easy to verify that (H2) and (H3) hold. Moreover, by Grimmer [3, Theorem 4.1], the resolvent operator of Eq. (4.1) satisfies 1
∥R(t )∥ ≤ e− 2 t ,
t ≥ 0.
Now, let us verify that (H4) holds. Take a bounded sequence
{un } ⊂ PAP (R+ , X ) such that {un } is uniformly convergent in every compact subsets of R+ . ∀ε > 0, there exists a constant M > 0 such that
2 1 + t2
2
< ε,
t > M.
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H.-S. Ding et al. / Nonlinear Analysis: Real World Applications 13 (2012) 2659–2670
Combining this with the fact that {un } is uniformly convergent in [0, M ], we can show that for ε > 0, there exists N > 0 such that 1
|f (t , un (t ))(x) − f (t , um (t ))(x)|2 dx ≤
1
0
0
1
|un (t )(x) − um (t )(x)|2 dx
≤ ≤ε
sin[un (t )(x)] − sin[um (t )(x)] 2 dx 1 + t2
0
for all t ∈ R and n, m > N. This means that {f (·, un (·))} is convergent in BC (R+ , X ). Next, let us show that (H5) holds. Let E ⊂ PAP (R+ , X ) be a bounded subset. By some direct calculations, we can get +
∥g (u)∥C [0,1] ≤
√ π 2
sup ∥u∥,
u ∈ E,
u∈E
and
{g (u) : u ∈ E } are equi-continuous on [0, 1]. Then, by using Arzela–Ascoli’s theorem, we know that g (E ) is precompact in C [0, 1], and thus precompact in X . Also, the continuity of g can be verified. Noting that M = 1, ω = 21 and u0 = 0, by a direct calculation, one can get
M·
∥u0 ∥ + sup ∥g (u)∥ + ∥u∥≤C
M
ω
·
sup t ∈R+ , ∥u∥≤C
∥f (t , u)∥ ≤
√ π 2
C,
∀C > 0,
which yields that (3.1) holds. So all the assumptions of Theorem 3.5 hold. Thus, the nonlocal problem (4.1) has a pseudo almost periodic mild solution. Acknowledgments H.S. Ding acknowledges support from the NSF of China (11101192), Chinese Ministry of Education (211090), and the NSF of Jiangxi Province. J. Liang and T.J. Xiao acknowledge support from the NSF of China (11071042), (11171210), the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900) and the Laboratory of Mathematics for Nonlinear Science at Fudan University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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