- Email: [email protected]

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces Khalil Ezzinbi a , Samir Fatajou a , Gaston Mandata N’Guérékata b,∗ a b

Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, B.P. 2390, Marrakech, Morocco Department of Mathematics, Morgan State University, 1700 East Cold Spring Lane, Baltimore, MD 21251, USA

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 10 March 2008 Available online 12 November 2008 Submitted by A. Daniilidis

This work aims to study the existence and uniqueness of pseudo compact almost automorphic solution for some dissipative ordinary and functional differential equations. We prove the existence and uniqueness of pseudo compact almost automorphic solution for dissipative differential equations in Banach spaces and then we apply this result to show the existence of pseudo compact almost automorphic solutions for some functional differential equations. © 2008 Elsevier Inc. All rights reserved.

Keywords: Dissipativeness Pseudo compact almost automorphy Differential equation Attractiveness

1. Introduction In this work, we investigate the existence and uniqueness of a pseudo compact almost automorphic solution for the following ordinary differential equation

x (t ) = f t , x(t ) ,

for t ∈ R,

(1.1)

where f : R × E → E is pseudo almost automorphic in t and dissipative with respect to x, here E is a Banach space. We study also the existence and uniqueness of pseudo compact almost automorphic solution for the following functional differential equation

x (t ) = F t , x(t ), x(t − r ) ,

for t ∈ R,

(1.2)

where F is a continuous function from R × E × E into E, pseudo compact almost automorphic in t, dissipative with respect to the second argument and Lipschitzian with respect to the third argument. The problem of the existence of almost periodic solutions has been extensively studied in the literature. In [1,7,8], the authors investigated the existence and uniqueness of an almost periodic solution for the following ordinary differential equation in a Banach space E

α

x (t ) = −x(t ) x(t ) + h(t ),

for t ∈ R,

(1.3)

where α 0 and h : R → E is a continuous function, they showed that if the input function h is almost periodic then Eq. (1.3) has a unique bounded solution on R which is also almost periodic. Recall that pseudo almost automorphic functions are more general than pseudo almost periodic functions. They were introduced recently in [5] and [6], where the authors studied some fundamental properties of pseudo almost automorphic functions. Almost automorphic functions were

*

Corresponding author. E-mail addresses: [email protected] (K. Ezzinbi), [email protected] (S. Fatajou), gaston.n’[email protected] (G.M. N’Guérékata).

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2008.11.017

© 2008 Elsevier Inc.

All rights reserved.

766

K. Ezzinbi et al. / J. Math. Anal. Appl. 351 (2009) 765–772

introduced at ﬁrst by S. Bochner [2], for more details about this topics we refer to the book [10] where an important overview is given on almost automorphic functions. In the literature, there are many works devoted to the existence of almost automorphic solutions in the context of differential equations. Recently, in [3], we extended the works [1,7,8] to almost automorphic case. In fact, we have proved that Eq. (1.1) has a unique bounded solution on R which is compact almost automorphic. Here we propose to extend this result to pseudo almost automorphic case. We will show that the unique bounded solution of Eq. (1.1) is pseudo compact almost automorphic if the input function f is pseudo compact almost automorphic in t. As application, we study the following differential equation

x (t ) = g x(t ) − γ x(t ) + θ(t ),

for t ∈ R,

(1.4)

where γ > 0 and g : H → H is a monotone function in a Hilbert space H . We will show if the input function θ is pseudo compact almost automorphic, then Eq. (1.4) has a unique bounded solution on R which is pseudo compact almost automorphic. This work is organized as follows. In Section 2, we recall some preliminary results on pseudo almost automorphic functions. In Section 3, we prove the existence and uniqueness of almost automorphic solutions for Eq. (1.1). In Section 4, we prove the existence and uniqueness pseudo almost automorphic solution for functional differential equation (1.2). The last Section is devoted to study Eq. (1.4). 2. Pseudo almost automorphic functions In this section, we give some properties about pseudo-almost automorphic functions. Let BC (R, X ) be the space of all bounded and continuous functions from R to a Banach space X , equipped with the uniform norm topology. Let x ∈ BC (R, X ) and τ ∈ R. We deﬁne the function xτ by xτ (s) = x(τ + s),

for s ∈ R.

Deﬁnition 2.1. (See [10].) A bounded continuous function x : R → X is said to be almost periodic if

{xτ : τ ∈ R} is relatively compact in BC (R, X ). Deﬁnition 2.2. (See [10].) A continuous function x : R → X is said to be almost automorphic if for every sequence of real numbers (sn )n there exists a subsequence (sn )n such that y (t ) = lim x(t + sn ) exists for all t in R n→∞

and lim y (t − sn ) = x(t )

n→∞

for all t in R.

Remark. By the pointwise convergence, the function y is just measurable and not necessarily continuous. If the convergence in both limits is uniform, then x is almost periodic. The concept of almost automorphy is then larger than almost periodicity. √ If x is almost automorphic, then its range is relatively compact, thus bounded in norm. Let p (t ) = 2 + cos t + cos 2t and 1 x : R → R such that x(t ) = sin p (t ) . Then x is almost automorphic, but x is not uniformly continuous on R. It follows that x is not almost periodic. Deﬁnition 2.3. (See [10].) A continuous function h : R → X is said to be compact almost automorphic if for every sequence of real numbers (sn )n , there exists a subsequence (sn )n such that lim

lim h(t + sn − sm ) = h(t ) uniformly on any compact set in R.

m→∞ n→∞

Theorem 2.4. (See [10].) If we equip A A c ( X ), the space of all compact almost automorphic X -valued functions, with the sup norm, then A A c ( X ) turns out to be a Banach space. Deﬁnition 2.5. A continuous function ρ : R × X → X is said to be almost automorphic in t with respect to u if for every sequence of real numbers (sn )n , there exists a subsequence (sn )n such that lim

lim

m→∞ n→∞

ρ (t + sn − sm , u ) = ρ (t , u )

for all t and uniformly for u in any bounded set of X .

K. Ezzinbi et al. / J. Math. Anal. Appl. 351 (2009) 765–772

767

Deﬁnition 2.6. A continuous function ρ : R × X → X is said to be compact almost automorphic in t with respect to u if for every sequence of real numbers (sn )n , there exists a subsequence (sn )n such that lim

lim

m→∞ n→∞

ρ (t + sn − sm , u ) = ρ (t , u )

uniformly on each bounded set of R × X . Deﬁnition 2.7. (See [5].) A bounded continuous function x : R → X is said to be pseudo almost automorphic if x is decomposed as follows: x = x1 + x2 , where x1 is almost automorphic and x2 is ergodic: lim

r →+∞

1 2r

+r x2 (t ) dt = 0. −r

For the sequel, PAA(R, X ) will denote the set of all pseudo almost automorphic functions. Theorem 2.8. (See [6].) If we equip PAA(R, X ) with the sup norm, then PAA(R, X ) turns out to be a Banach space. Lemma 2.9. (See [6].) Let x be a pseudo almost automorphic function such that x = x1 + x2 , where x1 ∈ A A (R, X ) and x2 is ergodic. Then

x1 (t ): t ∈ R ⊂ x(t ): t ∈ R .

We introduce the following new concept of pseudo compact almost automorphic functions. Deﬁnition 2.10. A bounded continuous function x : R → X is said to be pseudo compact almost automorphic if x is decomposed as follows: x = x1 + x2 , where x1 is compact almost automorphic and x2 is ergodic. PAAc (R, X ) will denote the set of all pseudo compact almost automorphic functions. Since A A c ( X ) is a Banach space. Then by Lemma 2.9, we have the following result. Proposition 2.11. PAAc (R, X ) provided with the supremum norm is a Banach space. Deﬁnition 2.12. (See [5].) A continuous function f : R × X → X is said to be pseudo almost automorphic in t with respect to the second argument x if and only if f (t , x) = f 1 (t , x) + f 2 (t , x)

for t ∈ R and x ∈ X ,

where f 1 : R × X → X is almost automorphic in t uniformly with respect to the second argument x and lim

r →+∞

1 2r

+r f 2 (s, x) ds = 0, −r

uniformly for x in any bounded subset of X . PAA(R × X , X ) denotes the space of all pseudo almost automorphic functions in t with respect to the second argument. Deﬁnition 2.13. A continuous function f : R × X → X is said to be pseudo compact almost automorphic in t with respect to the second argument x if and only if f (t , x) = f 1 (t , x) + f 2 (t , x)

for t ∈ R and x ∈ X ,

768

K. Ezzinbi et al. / J. Math. Anal. Appl. 351 (2009) 765–772

where f 1 : R × X → X is compact almost automorphic in t uniformly with respect to the second argument x and 1

lim

r →+∞

2r

+r f 2 (s, x) ds = 0, −r

uniformly for x in any bounded subset of X . PAAc (R × X , X ) denotes the space of all pseudo compact almost automorphic functions in t with respect to the second argument. Theorem 2.14. (See [10].) Let f ∈ A A (R × X , X ) and x ∈ A A (R, X ). We assume also that f is lipschitzian with respect to the second variable uniformly in the ﬁrst variable. Then f (., x(.)) ∈ A A (R, X ). The pseudo almost automorphic version of the above theorem has been recently obtained, in fact, we have Theorem 2.15. (See [5].) Let f ∈ PAA(R × X , X ) be such that f = f 1 + f 2 where f 1 and f 2 are respectively the almost automorphic part and the ergodic part of f . Assume that (i) f 1 is uniformly continuous on any bounded set K ⊂ X uniformly in t ∈ R, (ii) f 2 is uniformly continuous on any bounded set K ⊂ X uniformly in t ∈ R. If x ∈ PAA(R, X ), then f (., x(.)) ∈ PAA(R, X ). Moreover the almost automorphic part of the function f (., x(.)) is given by f 1 (., x1 (.)) where x1 is the almost automorphic part of x. Using Lemma 2.9, we deduce from the above theorem the following result. Corollary 2.16. Let f ∈ PAA(R × X , X ) be a Lipschitzian function with respect to the second argument. If x ∈ PAA(R, X ), then f (., x(.)) ∈ PAA(R, X ). Moreover the almost automorphic part of the function f (., x(.)) is given by f 1 (., x1 (.)) where x1 and f 1 are respectively the almost automorphic parts of x and f . 3. Pseudo compact almost automorphic solutions for dissipative differential equations Let E be a Banach space endowed with the norm |.|, we deﬁne the following functional

[x, y ] = lim

|x + hy | − |x|

h→0+

for x, y ∈ E .

h

The following lemma on the functional [,] is well known. Lemma 3.1. (See [4].) Let x, y and z be in E. Then the functional [,] has the following properties: (i) (ii) (iii) (iv)

[x, y ] = infh>0 |x+hyh|−|x| , |[x, y ]| | y |, [x, y + z] [x, y ] + | z|,

Let u be a function from a real interval J into E such that u (t 0 ) exists for an interior point t 0 of J . Then D + |u (t 0 )| exists and

D + u (t 0 ) = u (t 0 ), u (t 0 ) , where D + |u (t 0 )| denotes the right derivative of |u (t )| at t 0 . In this section we suppose that (H1 ) f : R × E → E is continuous and lipschitzian with respect to the second argument. (H2 ) f is dissipative with respect to the second argument: there exists a positive constant p such that

x − y , f (t , x) − f (t , y ) − p |x − y | for t ∈ R and x, y ∈ E .

Theorem 3.2. (See [1,8].) Assume that (H1 ) and (H2 ) hold. If M = supt ∈R | f (t , 0)| < ∞, then Eq. (1.1) has a unique bounded solution x f on R such that

supx f (t ) t ∈R

M p

.

K. Ezzinbi et al. / J. Math. Anal. Appl. 351 (2009) 765–772

769

The existence of an almost periodic solution of Eq. (1.1) has been studied in [1,8] when (H1 ) and (H2 ) are satisﬁed. More precisely, the authors proved the following result. Theorem 3.3. (See [1,8].) Assume that (H1 ), (H2 ) hold and f is almost periodic in t uniformly in x, then x f is almost periodic. This theorem has been recently extended in [3] to the case where f is almost automorphic in t with respect to x. In fact, we have the following result. Theorem 3.4. (See [3].) Assume that (H1 ), (H2 ) hold and f is compact almost periodic in t with respect to x. Then x f is compact almost automorphic. We propose to prove Theorem 3.4 in pseudo almost automorphic case. For this goal, we assume furthermore the following assumption: (H3 ) f is pseudo compact almost automorphic in t with respect to x. Theorem 3.5. Assume that (H1 )–(H3 ) hold. Then x f is pseudo compact almost automorphic. Moreover x f is globally attractive. The proof will be done through several lemmas. Let f be a pseudo compact almost automorphic function. Let f 1 and f 2 denote respectively the almost automorphic part and the ergodic part of f . Lemma 3.6. Assume that (H1 )–(H3 ). Then f 1 is lipschitzian and dissipative with respect to the second argument

x − y , f 1 (t , x) − f 1 (t , y ) − p |x − y | for t ∈ R and x, y ∈ E .

Proof. Lemma 2.9 implies that for any x and y in E, we have

f 1 (t , x) − f 1 (t , y ): t ∈ R ⊂ f (t , x) − f (t , y ): t ∈ R .

Let (sn )n be a real sequence such that f 1 (t , x) − f 1 (t , y ) = lim f (sn , x) − f (sn , y ).

(3.1)

n→∞

Then it follows that f 1 is lipschitzian. Moreover,

which implies by Lemma 3.1 that

x − y , f 1 (t , x) − f 1 (t , y ) = x − y , f 1 (t , x) − f (sn , x) + f (sn , x) − f (sn , y ) + f (sn , y ) − f 1 (t , y )

x − y , f 1 (t , x) − f 1 (t , y ) − p |x − y | + f (sn , x) − f (sn , y ) − f 1 (t , x) + f 1 (t , y ).

Letting n go to ∞, by (3.1), we deduce that

x − y , f 1 (t , x) − f 1 (t , y ) − p |x − y | for t ∈ R and x, y ∈ E ,

which implies that f 1 is dissipative. Consider the following differential equation

x (t ) = f 1 t , x(t ) ,

for t ∈ R.

(3.2)

Since f 1 is dissipative, we deduce by Theorem 3.4 that Eq. (3.2) has a unique compact almost automorphic solution x f 1 such that

supx f 1 (t ) t ∈R

M p

2

.

Lemma 3.7. Assume that (H1 )–(H3 ) hold. Then x f − x f 1 is ergodic. Proof.

D + x f (t ) − x f 1 (t ) = x f (t ) − x f 1 (t ), f t , x f (t ) − f 1 t , x f 1 (t )

= x f (t ) − x f 1 (t ), f t , x f (t ) − f t , x f 1 (t ) + f t , x f 1 (t ) − f 1 t , x f 1 (t ) .

This implies that

D + x f (t ) − x f 1 (t ) − p x f (t ) − x f 1 (t ) + f 2 t , x f 1 (t ) ,

(3.3)

770

K. Ezzinbi et al. / J. Math. Anal. Appl. 351 (2009) 765–772

where f 2 is the ergodic part of f . Let V (.) denote the scalar function deﬁned on R by

V (t ) = x f (t ) − x f 1 (t ). By integrating the differential inequality (3.3), we deduce that V (r ) − V (−r ) 2r

−p

+r +r V (s) ds + 1 f 2 s, x f (s) ds. 1

1 2r

2r

−r

−r

Since V (.) is bounded, it follows that V (r ) − V (−r )

lim

r →+∞

2r

= 0.

Moreover by Corollary 2.16, we have 1

lim

r →+∞

2r

+r f 2 s, x f (s) ds = 0. 1 −r

Consequently, lim

1

r →+∞

r

+r V (s) ds = 0. −r

We deduce that for all t ∈ R,

x f (t ) = x f 1 (t ) + x f (t ) − x f 1 (t ) . Since x f − x f 1 is ergodic, we deduce that x f ∈ PAA(R, X ). To show the attractiveness of x f , let y be another solution of Eq. (1.1) on R+ . Then for t 0, we have

D + x f (t ) − y (t ) = x f (t ) − y (t ), f t , x f (t ) − f t , y (t ) and

D + x f (t ) − y (t ) − p x f (t ) − y (t ). We need to use the following lemma in order to solve the differential inequality (3.4).

(3.4)

2

Lemma 3.8. (See [9].) Let D be an open set of R2 and ϑ is a continuous function from D into R. Consider the following scalar differential equation

w (t ) = ϑ t , w (t ) ,

(3.5)

w (t 0 ) = w 0

and is a solution of Eq. (3.5) which is deﬁned on [t 0 , t 1 ]. Let z be a continuous function from [t 0 , t 1 ] into R such that (t , z(t )) ∈ D , for t ∈ [t 0 , t 1 ], z(t 0 ) w 0 and

D + z(t ) ϑ t , z(t )

for t ∈ [t 0 , t 1 ].

Then z(t ) (t ) for t ∈ [t 0 , t 1 ]. Then, we deduce for t 0 that

x f (t ) − y (t ) e − pt x f (0) − y (0).

This implies that x f is globally attractive. This completes the proof of the theorem. 4. Functional differential equations In this section, we consider the existence of almost automorphic solution for Eq. (1.2). In the sequel, we assume that (H4 ) F : R × E × E → E is a function such that if y : R → E is pseudo compact almost automorphic then the function

h y (t , x) = F t , x, y (t − r )

satisﬁes assumptions (H1 )–(H3 ).

for t ∈ R and x ∈ E

K. Ezzinbi et al. / J. Math. Anal. Appl. 351 (2009) 765–772

771

Remark. If for example F (t , x, y ) = F 1 (t , x) + F 2 ( y ), with F 1 satisﬁes (H1 )–(H3 ) and F 2 is Lipschitzian continuous, then (H4 ) is true. In fact if y is pseudo compact almost automorphic, then the history function t → y (t − r ) is also pseudo compact almost automorphic form R to E. In that case, we have

h y (t , x) = F 1 (t , x) + F 2 y (t − r )

for t ∈ R and x ∈ E .

(H5 ) supt ∈R, y ∈ E | F (t , 0, y )| < ∞. (H6 ) There exist positive constants q, L such that

x − y , F (t , x, z1 ) − F (t , y , z2 ) −q|x − y | + L | z1 − z2 |,

for all t ∈ R, x, y,z1 , z2 ∈ E . Proposition 4.1. Assuming that (H4 )–(H6 ) hold with

L q

< 1. Then Eq. (1.2) has a unique pseudo compact almost automorphic solution.

= supt ∈R, y∈ E | F (t , 0, y )|. Consider the following set Proof. Let M

M . Γ = x ∈ PAAc (R, E ): supx(t ) q

t ∈R

For h ∈ Γ, we consider the following equation

x (t ) = F t , x(t ), h(t − r )

for t ∈ R.

(4.1)

Since the function

(t , x) → F t , x(t ), h(t − r ) for t ∈ R and x ∈ E

satisﬁes all assumptions (H1 )–(H3 ), it follows by Theorem 3.4, that Eq. (4.1) has a unique pseudo compact almost automorphic solution which is denoted by xh and

supxh (t ) t ∈R

M q

.

Consequently, we obtain that xh ∈ Γ. Deﬁne the following operator G on Γ by Gh = xh . We will use the strict contraction principle to show that G has a unique ﬁxed point in Γ. In fact, let h, k ∈ Γ and xh = Gh and xk = Gk. Then

D + xh (t ) − xk (t ) = xh (t ) − xk (t ), xh (t ) − xk (t )

= xh (t ) − xk (t ), F t , xh (t ), h(t − r ) − F t , xk (t ), k(t − r ) .

By hypothesis (H6 ) we obtain that

D + xh (t ) − xk (t ) −qxh (t ) − xk (t ) + L |h − k|, which implies that

D + xh (t ) − xk (t ) −qxh (t ) − xk (t ) + L |h − k|.

(4.2)

Using Lemma 3.8, we get for all t a

xh (t ) − xk (t ) xh (a) − xk (a)e −q(t −a) + L |h − k|. q

Letting a go to −∞, then we deduce that L

|Gh − Gk| |h − k|. q

Consequently, G is a strict contraction and it has a unique ﬁxed point in Γ which gives that Eq. (1.2) has a unique pseudo compact almost automorphic solution. 2 5. Application 5.1. Example 1 For illustration, let H be a real Hilbert space. Let e 0 be a nontrivial vector of H , we propose to study the following differential equation in H

x (t ) = g x(t ) − γ x(t ) + sin

1 2 + cos t + cos

√

2t

+

1 1 + t2

e0

for t ∈ R,

(5.1)

772

K. Ezzinbi et al. / J. Math. Anal. Appl. 351 (2009) 765–772

where

γ > 0 and the function g : H → H is monotone in the sense that

g (x) − g ( y ), x − y 0

for x, y ∈ H ,

where .,. denotes the scalar product in H . Let

ξ(t , x) = g (x) − γ x + sin

1 2 + cos t + cos

√

2t

+

1 1 + t2

for t ∈ R and x ∈ H .

e0

Then ξ is pseudo compact almost automorphic in t with respect to x. Moreover, we have Lemma 5.1. (See [3].)

x − y , ξ(t , x) − ξ(t , y ) −γ |x − y | for t ∈ R and x, y ∈ H .

By Theorem 3.4, we get the following result. Proposition 5.2. Eq. (5.1) has a unique pseudo compact almost automorphic solution. 5.2. Example 2 Here we consider the following functional differential equation

x (t ) = g x(t ) − γ x(t ) + υ x(t − r ) + sin

1 2 + cos t + cos

√

2t

+

1 1 + t2

e0

for t ∈ R,

(5.2)

υ : H → H is bounded and lipschitzian continuous. Let δ be such that υ (u ) − υ ( v ) δ|u − v | for u , v ∈ H .

where r > 0 and

Then we get the following result. Proposition 5.3. Let δ < γ . Then Eq. (5.2) has a unique pseudo compact almost automorphic solution. Proof. Let F : R × H × H → H be deﬁned by

F (t , x, y ) = g (x) − γ x + υ ( y ) + sin Then, using Lemma 3.1, we obtain

1 2 + cos t + cos

√

2t

+

1 1 + t2

e0

for t ∈ R and x, y ∈ H .

x − y , F (t , x, y 1 ) − F (t , z, y 2 ) −γ |x − z| + δ| y 1 − y 2 |,

for all t ∈ R, x, y , y 1 , y 2 ∈ H . Consequently assumptions (H4 )–(H6 ) are true and by applying Theorem 4.1, we get the desired result. 2 References d [1] O. Arino, E. Hanebaly, Solutions presque-périodiques de ( dt x(t ) + |x(t )|α x(t ) = h(t ), α 0), sur les espaces de Banach, C. R. Math. Acad. Sci. Paris 306 (1988) 707–710. [2] S. Bochner, Continuous mappings of almost automorphic and almost automorphic functions, Proc. Natl. Acad. Sci. USA 52 (1964) 907–910. [3] K. Ezzinbi, S. Fatajou, G.M. N’Guérékata, Almost automorphic solutions for dissipative ordinary and functional differential equations in Banach spaces, Commun. Math. Anal. 4 (2) (2008) 8–18. [4] S. Kato, Almost periodic solutions of functional differential equations with inﬁnite delay, Funkcial. Ekvac. 38 (3) (1995) 505–517. [5] J. Liang, J. Zhang, T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2008) 1493–1499. [6] T.J. Xiao, J. Liang, J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Nonlinear Anal. (2007), in press. [7] E. Hanebaly, Solutions presque périodiques d’équations différentielles monotones, C. R. Acad. Sci. Paris 296 (1983) 263–265. [8] E. Hanebaly, Contribution à l’etude des solutions périodiques et presque périodiques d’equations différentielles non linéaires sur les espaces de Banach, Thèse de Doctorat d’Etat, Université de Pau et des Pays de l’Adour, 1988. [9] V. Lakshmikantham, V. Leela, Differential and Integral Inequalities, Theory and Applications, vol. 1, Academic Press, 1969. [10] G.M. N’Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer, 2001.