Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications

Nonlinear Analysis 70 (2009) 4079–4085

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Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applicationsI Ti-Jun Xiao a , Xing-Xing Zhu b , Jin Liang c,∗ a

School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China

b

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China

c

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, PR China

article

a b s t r a c t

info

Article history: Received 27 April 2008 Accepted 28 August 2008

In this paper, we introduce a new concept of bi-almost automorphic functions and obtain new existence and uniqueness theorems for pseudo-almost automorphic mild solutions to several nonautonomous differential equations. Moreover, two examples are given to illustrate the general theorems. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Pseudo-almost periodic Almost automorphic Pseudo-almost automorphic Bi-almost automorphic functions Nonautonomous evolution equations

1. Introduction Very recently, Xiao, Liang and Zhang [16] introduced a new concept of a function called a pseudo-almost automorphic function. They established a general existence and uniqueness theorem for pseudo-almost automorphic mild solutions to some semilinear abstract differential equations as well as solving a basic problem on the space (PAA(R, X), k · k∞ ). In this paper, we investigate the existence and uniqueness of pseudo-almost automorphic solutions to the following nonautonomous evolution equations: x0 (t ) = A(t )x(t ) + f (t , x(t )),

t ∈ R,

x (t ) = A(t )x(t ) + f (t , x(t − h)), 0

t ∈ R,

x (t ) = A(t )x(t ) + f (t , x(t ), x[α(t , x(t ))]), 0

(1.1) (1.2) t ∈ R,

(1.3)

in a Banach space X, where h ≥ 0 is a fixed constant. Almost periodic, pseudo-almost periodic, almost automorphic and asymptotically almost automorphic mild solutions to (1.1)–(1.3) have been studied by many researchers for the case where A(t ) ≡ A or A(t + T ) = A(t ) for some T ∈ R \ {0} in recent years (cf., e.g., [3,5–16] and references therein). However, to the best of our knowledge, results for pseudo-almost automorphic mild solutions to (1.1)–(1.3) in a general nonautonomous case are rare. In this paper, we introduce a new concept of bi-almost automorphic functions, in order to study the existence of pseudo-almost automorphic solutions to (1.1)–(1.3). Then, we give some new existence and uniqueness theorems for pseudo-almost automorphic mild solutions to (1.1)–(1.3). Finally, we investigate two heat equations with Dirichlet boundary conditions as applications of our main results. I The work was supported partly by the National Natural Science Foundation of China and the Specialized Research Fund for the Doctoral Program of Higher Education of China. ∗ Corresponding author. E-mail addresses: [email protected] (T.-J. Xiao), [email protected] (X.-X. Zhu), [email protected] (J. Liang).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.08.018

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2. Preliminaries Throughout this paper, we assume that (X, k · k), (Y, k · kY ) are two Banach spaces, BC (R, X) (resp. BC (R × Y, X)) is the space of bounded continuous function f : R 7→ X (resp. f : R × Y 7→ X) with supremum norm. Definition 2.1 (See (1) in [3], and See (2) in [16]). (1) A continuous function f : R 7→ X is called almost automorphic if for every sequence of real numbers {s0n }∞ n=1 , we can extract a subsequence {sn }∞ such that g ( t ) = lim f ( t + s ) is well defined in t ∈ R , and lim g ( t − sn ) = f ( t ) n →∞ n n →∞ n=1 for each t ∈ R. AA(R, X) stands for the set of all such functions. A continuous function f : R × Y 7→ X is called almost automorphic if f (t , x) is almost automorphic in t ∈ R uniformly for all x in any bounded subset of Y. AA(R × Y, X) is the set of all such functions. (2) A continuous function f : R 7→ X (resp. R × Y 7→ X) is called pseudo-almost automorphic if it can be decomposed as f =g +φ where g ∈ AA(R, X) (resp. AA(R × Y, X)) and φ is a bounded continuous function with vanishing mean value (resp. φ is a bounded continuous function with lim

T →∞

1

Z

2T

T

kφ(σ , x)k = 0 −T

uniformly for x in any bounded subset of X). Denote by PAA(R, X) (resp. PAA(R × Y, X)) the set of all such functions. We define PAAL (R, X) := {x ∈ PAA(R, X); kx(t ) − x(s)k ≤ L|t − s|(t , s ∈ R)}, AA0 (R, X) :=



x(t ) ∈ BC (R, X); lim

AA0 (R × Y, X) :=

T →∞



1 2T

Z

T

L > 0,



kx(t )kdt = 0 ,

−T

1

Z

T

kf (t , x)kdt = 0  uniformly for x in any bounded subset of Y . f (t , x) ∈ BC (R × Y, X); lim

T →∞

2T

−T

Lemma 2.2 ([16]). (PAA(R, X), k · k∞ ) is a Banach space, where k · k∞ is the supremum norm. Remark 2.3. It is clear that PAAL (R, X) is also a Banach space, as it is a closed subspace of PAA(R, X). Lemma 2.4 ([11]). Let f = g + φ ∈ PAA(R × X, X) with g (t , x) ∈ AA(R × X, X), φ(t , x) ∈ AA0 (R × X, X), and let f (t , x) be uniformly continuous in any bounded subset K ⊂ X uniformly for t ∈ R. If x(t ) ∈ PAA(R, X), then f (·, x(·)) ∈ PAA(R, X). Remark 2.5. From the proof of [16, Theorem 2.2], we can see that

kg (t , x) − g (t , y)k∞ ≤ kf (t , x) − f (t , y)k∞ for any x, y ∈ X. Therefore, the hypotheses of Lemma 2.4 imply that the conditions of [16, Theorem 2.2] are satisfied. So Lemma 2.4 is a consequence of [16, Theorem 2.2]. In particular, when f (t , x) is Lipschitz continuous in x uniformly for t ∈ R, the conclusion of Lemma 2.4 holds. Next, we introduce a new concept of a function which will be used to obtain our main result. Definition 2.6. A continuous function f (t , s) : R × R 7→ X is called bi-almost automorphic if for every sequence of real ∞ numbers {τn0 }∞ n=1 , we can extract a subsequence {τn }n=1 such that g (t , s) = limn→∞ f (t + τn , s + τn ) is well defined in t , s ∈ R, and limn→∞ g (t − τn , s − τn ) = f (t , s) for each t , s ∈ R. bAA(R × R, X) stands for the set of all such functions. Remark 2.7. If f ∈ C (R × R, X) and f (t , s) = g (t − s) for some g ∈ C (R, X), then f ∈ bAA(R × R, X). On the other hand, the concept of bi-almost automorphic function is a natural generalization of the function f (t , s) having the same period in the two arguments, that is f (t + T , s + T ) = f (s, t ) for all t , s ∈ R for some T ∈ R \ {0}. Example 2.8. f (t , s) = sin t cos s is a bi-almost automorphic function from R × R to R as f (t + 2π , s + 2π ) = f (t , s) for all t , s ∈ R. We also need to recall the following notation concerning exponential dichotomy. An evolution family U is called hyperbolic (or has exponential dichotomy) if there are projections P (t ), t ∈ R, uniformly bounded and strongly continuous in t, and constants M, ω > 0 such that

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(a) U (t , s)P (s) = P (t )U (t , s) for all t ≥ s, (b) the restriction UQ (t , s) : Q (s)X → Q (t )X is invertible for all t ≥ s (and we set UQ (s, t ) = UQ (t , s)−1 ), (c) kU (t , s)P (s)k ≤ Me−ω(t −s) and kUQ (s, t )Q (t )k ≤ Me−ω(t −s) for all t ≥ s. Here and below Q := I − P. If U is hyperbolic, then

Γ (t , s) :=



U (t , s)P (s), t ≥ s, t , s ∈ R, −UQ (t , s)Q (s), t < s, t , s ∈ R

is called Green’s function corresponding to U and P (·). 3. Main results In this paper, we assume that {A(t )}t ∈R satisfies the ‘Acquistapace–Terreni’ conditions introduced in [1,2], that is

(H1 ) there exist constants λ0 ≥ 0, θ ∈ ( π2 , π ), K1 , K2 ≥ 0, and β1 , β2 ∈ (0, 1] with β1 + β2 > 1 such that X K1 ∪{0} ⊂ ρ(A(t ) − λ0 ), kR(λ, A(t ) − λ0 )k ≤ 1 + |λ| θ and

k(A(t ) − λ0 )R(λ, A(t ) − λ0 )[R(λ0 , A(t )) − R(λ0 , A(s))]k ≤ K2 |t − s|β1 |λ|−β2 P for t , s ∈ R, λ ∈ θ := {λ ∈ C \ {0} : | arg λ| ≤ θ }. We further suppose that:

(H2 ) the evolution family U (t , s) generated by A(t ) has an exponential dichotomy with constants M , ω > 0, dichotomy projections P (t ), t ∈ R, and Green’s function Γ . (H3 ) Γ (t , s)x ∈ bAA(R × R, X) uniformly for all x in any bounded subset of X. (H4 ) f (t , x) ∈ PAA(R × X, X) such that kf (t , x) − f (t , y)k ≤ L0 kx − yk ω for all t ∈ R, x, y ∈ X and some 0 < L0 < 2M .

Now, we are ready to state our first main result. Theorem 3.1. Assume that (H1 )–(H4 ) hold. Then (1.1) has a unique pseudo-almost automorphic mild solution given by x(t ) =

Z

Γ (t , s)f (s, x(s))ds,

t ∈ R.

(3.1)

R

Proof. From the proof of [4, Theorem 4.28], we know that (1.1) has a unique bounded mild solution x given by (3.1). If x(t ) ∈ PAA(R × X, X), then Lemma 2.4 ensures the existence of two functions g ∈ AA(R × X, X), φ ∈ AA0 (R × X, X) such that f (t , x(t )) = g (t ) + φ(t )

for t ∈ R.

Choose a bounded subset K of X such that g (t ), φ(t ) ∈ K for all t ∈ R, and define a nonlinear operator Υ by

(Υ x)(t ) :=

Z

Γ (t , s)f (s, x(s))ds = Υ1 (t ) + Υ2 (t ), R

where

Υ1 (t ) =

Z R

Γ (t , s)g (s)ds,

Υ2 (t ) =

Z

Γ (t , s)φ(s)ds. R

First we prove that Υ1 (t ) ∈ AA(R, X). Let {τn0 } be a sequence of real numbers. By (H3 ), we can extract a subsequence {τn } of {τn0 } such that

(P1 ) (P2 ) (P3 ) (P4 )

limn→∞ Γ (t + τn , s + τn )x = Γ1 (t , s)x for each t , s ∈ R, x ∈ K , limn→∞ Γ1 (t − τn , s − τn )x = Γ (t , s)x for each t, s ∈ R, x ∈ K , limn→∞ g (t + τn ) = g1 (t ) for each t ∈ R, limn→∞ g1 (t − τn ) = g (t ) for each t ∈ R.

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Write

Φ1 (t ) =

Z

Γ1 (t , s)g1 (s)ds.

R

Then

Z

Υ1 (t + τn ) − Φ1 (t ) =

[Γ (t + τn , s)g (s) − Γ1 (t , s)g1 (s)]ds

ZR = ZR =

[Γ (t + τn , s + τn )g (s + τn ) − Γ1 (t , s)g1 (s)]ds Z Γ (t + τn , s + τn )[g (s + τn ) − g1 (s)]ds + [Γ (t + τn , s + τn ) − Γ1 (t , s)]g1 (s)ds. R

R

From (H2 ) and (P1 ), it follows that

kΓ1 (t , s)xk ≤ Me−ω|t −s| kxk for x ∈ K . By the Lebesgue Dominated Convergence Theorem and (P1 ), (P3 ), we have lim Υ1 (t + τn ) = Φ1 (t ),

n→∞

t ∈ R.

Similarly we can prove that lim Φ1 (t − τn ) = Υ1 (t ),

n→∞

t ∈ R.

Hence Υ1 (t ) ∈ AA(R, X). Furthermore, from the proof of [8, Theorem 2.4], we see that Υ2 (t ) ∈ AA0 (R, X). Therefore (Υ x)(t ) ∈ PAA(R, X). Next, we prove that Υ is a contraction mapping from PAA(R, X) into itself. Note that we have already proved Υ : PAA(R, X) 7→ PAA(R, X). Moreover, we have

Z

kΓ (t , s)kkf (s, x(s)) − f (s, y(s))kds Z t Z ∞ ≤ L0 kx − yk∞ sup + kΓ (t , s)kds t ∈R −∞ t Z t  Z ∞ ≤ L0 kx − yk∞ sup Me−ω(t −s) ds + Me−ω(s−t ) ds

kΥ x − Υ yk∞ ≤ sup t ∈R

R

t ∈R

≤

2L0 M

ω

−∞

t

kx − yk∞ .

Thus Υ is a contraction on PAA(R, X). Since PAA(R, X) is a Banach space (Lemma 2.2), by the contraction mapping principle, Υ has a unique fixed point x(t ) ∈ PAA(R, X), which ends the proof.  Lemma 3.2. If x(·) ∈ PAA(R, X), then x(· − h) ∈ PAA(R, X), where h ≥ 0 is a fixed constant. The proof is similar to the proof of [8, Theorem 2.6], and we omit the details here. Theorem 3.3. Assume that (H1 )–(H4 ) hold. Then (1.2) has a unique pseudo-almost automorphic mild solution. Proof. Consider the nonlinear operator Υ given by

(Υ x)(t ) :=

Z

Γ (t , s)f (s, x(s − h))ds. R

Then, from the proof of Theorem 3.1, we can see that Υ maps PAA(R, X) into itself. Moreover,

kΥ x − Υ yk∞ ≤

2L0 M

ω

kx − yk∞ .

Therefore, Υ has a unique fixed point in PAA(R, X) which is the pseudo-almost automorphic mild solution of (1.2).



In order to investigate the solution to (1.3), we need to prove a composition theorem. Theorem 3.4. Assume that:

(H04 ) f (t , x, y) ∈ PAA(R × X × X, X), and for each u1 , u2 , v1 , v2 ∈ X, t ∈ R, kf (t , u1 , v1 ) − f (t , u2 , v2 )k ≤ C1 (ku1 − u2 k + kv1 − v2 k).

(3.2)

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(H5 ) α(t , x) ∈ PAA(R × X, R), and for each u, v ∈ X, t ∈ R, |α(t , u) − α(t , v)| ≤ C2 ku − vk. If x(t ) ∈ PAAL (R, X), then f (·, x(·), x[α(·, x(·))]) ∈ PAA(R, X). Proof. Since α ∈ PAA(R × X, R) and satisfies (H5 ), by Lemma 2.4, the function β(·) := α(·, x(·)) ∈ PAA(R, R). Define g (s, t ) = x(t ),

s, t ∈ R .

Then g (s, t ) ∈ PAA(R × R, X) as it is independent of s. Moreover, for each s, t1 , t2 ∈ R, we have

kg (s, t1 ) − g (s, t2 )k ≤ L|t1 − t2 |. Using Lemma 2.4 again, we see that y(·) := x[α(·, x(·))] = g (·, β(·)) ∈ PAA(R, X). Note that f ∈ PAA(R × X × X, X) and satisfies (3.2). Reasoning exactly as in the proof of Lemma 2.4, we get f (·, x(·), x[α(·, x(·))]) ∈ PAA(R, X).  If P (t ) = I in (H2 ), then U is exponentially stable, that is

kU (t , s)k ≤ Me−ω(t −s) ,

t ≥ s.

Now we present another main result. Theorem 3.5. Assume that (H1 ) –(H3 ), (H04 ) and (H5 ) hold, U is exponentially stable, M0 =

sup

t ∈R,x,y∈X

kf (t , x, y)k < ∞,

and define L = MM0 . If C1 M (C2 L + 2)/ω < 1, then (1.3) has a unique pseudo-almost automorphic mild solution in the Banach space PAAL (R, X) := {x ∈ PAA(R, X) : kx(t ) − x(s)k ≤ L|t − s|}. Proof. Given x(t ) ∈ PAAL (R, X), Theorem 3.4 implies that f (·, x(·), x[α(·, x(·))]) ∈ PAA(R, X). Consider the following nonlinear operator:

(Υ x)(t ) :=

t

Z

U (t , s)f (s, x(s), x[α(s, x(s))])ds. −∞

We know from the proof of Theorem 3.1 that Υ maps PAAL (R, X) into PAA(R, X). Moreover,

Z t



≤ MM0 |t − s|. k(Υ x)(t ) − (Υ x)(s)k = U ( t , r ) f ( r , x ( r ), x [α( r , x ( r ))]) dr

s

Hence Υ maps PAAL (R, X) into itself. Let x1 (t ), x2 (t ) ∈ PAAL (R, X); we have

k(Υ x1 )(t ) − (Υ x2 )(t )k Z t ≤ kU (t , s)kkf (r , x1 (r ), x1 [α(r , x1 (r ))]) − f (r , x2 (r ), x2 [α(r , x2 (r ))])kdr −∞

Z

t

≤ C1 M

e−ω(t −r ) (kx1 (r ) − x2 (r )k + kx1 [α(r , x1 (r ))] − x2 [α(r , x2 (r ))]k)dr

−∞

Z

t

≤ C1 M

e−ω(t −r ) (kx1 (r ) − x2 (r )k + kx1 [α(r , x1 (r ))] − x1 [α(r , x2 (r ))]k

−∞

+ kx1 [α(r , x2 (r ))] − x2 [α(r , x2 (r ))]k)dr Z t ≤ C1 M e−ω(t −r ) ((C2 L + 1)kx1 (r ) − x2 (r )k + kx1 [α(r , x2 (r ))] − x2 [α(r , x2 (r ))]k)dr −∞

Z

t

≤ C1 M −∞

e−ω(t −r ) (C2 L + 2)kx1 − x2 k∞ dr .

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T.-J. Xiao et al. / Nonlinear Analysis 70 (2009) 4079–4085

Therefore

k(Υ x1 ) − (Υ x2 )k∞ ≤ C1 M (C2 L + 2)kx1 − x2 k∞ sup t ∈R

Z

t

e−ω(t −r ) dr

−∞

C1 M (C2 L + 2)

kx 1 − x 2 k∞ , ω i.e., Υ is a contraction on PAAL (R, X). Note that PAAL (R, X) is a Banach space (Remark 2.3). By the contraction mapping principle, Υ has a unique fixed point x(t ) ∈ PAAL (R, X). It is clear that the fixed point is the mild solution to (1.3). On the other hand, let y(t ) be a pseudo-almost automorphic mild solution to (1.3). Then Z t U (t , s)f (s, y(s), y[α(s, y(s))])ds. y(t ) = U (t , a)y(a) + ≤

a

Letting a → ∞ yields y(t ) =

t

Z

U (t , s)f (s, y(s), y[α(s, y(s))])ds, −∞

since y(t ) is bounded and (U (t , s)t ≥s ) is exponentially stable. Thus the uniqueness of mild solution is proved, which ends the proof.  Remark 3.6. Theorem 3.5 generalizes the corresponding result of Gal [10]. As one can see, we do not assume the boundedness of A even in the case of A(t ) ≡ A, which is needed in [10, Theorem 2.1]. On the other hand, M = sup kf (s, φ(s), φ[α(s, φ(s))])k, s∈R

is dependent on the solution φ , and it is not a constant generally. Therefore, in order to make Gφ(t ) ∈ AAL (R, X) as well as G be a contraction on AAL (R, X), we assume M = supt ∈R,x,y∈X kf (t , x, y)k < ∞. Such an assumption seems to be also needed in the proof of [10, Theorem 2.1]. 4. Applications Example 4.1. Consider the following heat equation with Dirichlet boundary conditions:

 ∂ 2u 1  ∂u u(t , x) = 2 u(t , x) + u(t , x) sin √ + f1 (t , u(t , x)), ∂ t ∂ x 2 + cos t + cos 2t  u(t , 0) = u(t , 1) = 0, t ∈ R. Let X = L2 (0, 1), and D(B) := {x ∈ C 1 [0, 1]; x0 is absolutely continuous on [0, 1], x00 ∈ X, x(0) = x(1) = 0}, Bx(r ) = x00 (r ),

r ∈ (0, 1), x ∈ D(B).

Then B generates a C0 -semigroup T (t ) on X given by

(T (t )x)(r ) =

∞ X

2 2 e−n π t hx, en iL2 en (r )

n =1

√

2 where en (r ) = 2 sin nπ r, n = 1, 2, . . .. Moreover, kT (t )k ≤ e−π t , t ≥ 0. Define a family of linear operators A1 (t ) by

 D(A1 (t )) =  D(B), A1 (t )x =

B + sin

t ∈R 1 2 + cos t + cos

 √ 2t

x,

x ∈ D(A1 (t )).

Then, {A1 (t ), t ∈ R} generates an evolution family {U1 (t , s)}t ≥s such that U1 (t , s)x = T (t − s)e

Rt

1 √ s sin 2+cos τ +cos 2τ dτ

Hence

kU1 (t , s)k ≤ e−(π

2 −1)(t −s)

,

t ≥ s.

x.

(4.1)

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It is easy to see that U1 (t , s) satisfies (H1 )–(H3 ) with M = 1, ω = π 2 − 1. Write f1 (t , u) = u sin

1

2 )2

+ max{e−(t ±k

√

2 + cos t + cos

3t

k∈Z

} cos u,

t ∈ R.

Then, f1 satisfies (H4 ) clearly. By Theorem 3.1, (4.1) has a unique pseudo-almost automorphic mild solution. Example 4.2. Consider the following Dirichlet boundary problem:

  ∂u ∂ 2u 1 u(t , x) = 2 u(t , x) + u(t , x) cos + f2 (t , u(t , x), u((α(t , u(t , x))), x)), ∂ t ∂ x 2 + sin t + sin π t u(t , 0) = u(t , 1) = 0, t ∈ R.

(4.2)

Define a family of linear operators A2 (t ) by

 D(A2 (t )) =  D(B),

A2 (t )x = B + cos

t ∈R



1 2 + sin t + sin π t

x,

x ∈ D(A2 (t )).

Then, {A2 (t ), t ∈ R} generates an evolution family {U2 (t , s)}t ≥s such that U2 (t , s)x = T (t − s)e

Rt

1 s cos 2+sin τ +sin πτ dτ

x.

Hence

kU2 (t , s)k ≤ e−(π

2 −1)(t −s)

,

t ≥ s.

It is easy to see that U2 (t , s) satisfies (H1 )–(H3 ) with M = 1, ω = π 2 − 1. Write



f2 (t , x, y) = [cos x + cos y] cos

1 2 + sin t + sin

√ 5t

2 )2

+ max{e−(t ±k

 }

k∈Z

and

α(t , x) =

kx k 4

sin

1

√

2 + cos t + cos

7t

.

Then, f2 , α satisfy (H04 ) and (H5 ) respectively with C1 = 2, C2 = C1 M (C2 L + 2)

ω

=

2 × ( 14 × 4 + 2)

π2 − 1

1 . 4

Notice that ω = π 2 − 1, M = 1, M0 = 4, L = 4. Therefore,

< 1.

By Theorem 3.5, (4.2) has a pseudo-almost automorphic mild solution. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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