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J Appl Math Comput (2011) 37:625–634 DOI 10.1007/s12190-010-0455-y

JAMC

Pseudo-almost periodic solutions of a class of semilinear fractional differential equations Ravi P. Agarwal · Claudio Cuevas · Herme Soto

Received: 17 August 2010 / Published online: 27 October 2010 © Korean Society for Computational and Applied Mathematics 2010

Abstract We study existence and uniqueness of a pseudo-almost periodic (of class infinity) mild solution to the semilinear fractional equation ∂tα u = Au + ∂tα−1 f (·, u), 1 < α < 2, where A is a linear operator of sectorial negative type. Keywords Fractional order differential equations · Pseudo-almost periodic (of class infinity) function · Mild solution Mathematics Subject Classification (2000) 34A08 · 35R11 · 35B15

1 Introduction Pseudo-almost periodic functions have many applications in several problems for example in theory of functional differential equations, integral equations and partial differential equations. The concept of pseudo-almost periodicity, which is the central issue in this work, was introduced by Zhang [49–53] in the early nineties. Since then, such notion became of great interest to several mathematicians (see [1, 4–6,

The second author is partially supported by CNPQ/Brazil under Grant 300365/2008-0. The third author is partially supported by DIUFRO 120231. R.P. Agarwal () Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA e-mail: [email protected] C. Cuevas Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE 50540-740, Brazil e-mail: [email protected] H. Soto Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54-D, Temuco, Chile e-mail: [email protected]

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8, 11, 20, 22–27, 29–31] and references therein). The pseudo-almost periodicity is a natural generalization of the well-known (Bohr) almost periodicity. In Diagana and Hernández [28] a new pseudo-almost periodic space was introduced it is called the space of pseudo-almost periodic functions of class p (p.a.p.p. in short), which are heavily linked with the presence of (finite) delay. Recently, Cuevas and Hernández [19] have discussed the existence and uniqueness of p.a.p.p. solutions for a first-order abstract functional differential equation with linear part dominated by a Hille-Yosida type operator with non-dense domain. In Agarwal et al. [8] the authors have studied the existence of p.a.p.p. solutions for abstract partial neutral differential equations. To deal with unbounded delays Hernández and Henríquez [36] have introduced the notion of pseudo-almost periodicity (of class infinity). They have studied the existence of such type of mild solutions for a class of non-autonomous first-order abstract partial neutral functional differential equations with unbounded delay described in the form d D(t, ut ) = A(t)D(t, xt ) + g(t, ut ), dt where A(t) : D(A(t)) ⊆ X → X is a family of densely defined closed linear operators; the history xt : (−∞, 0] → X, xt (θ ) = x(t + θ ), belongs to some abstract phase space B (see for instance Hino’s et al. [37]); D(t, ψ) = ψ(0) + f (t, ψ) and f, g : R × B → X are suitable functions. Partial neutral integro-differential equations with unbounded delay arises in the theory development in Gurtin and Pipkin [35] and Nunziato [48] for the description of heat conduction in materials with fading memory. In Agarwal et al. [8], the authors have treated the existence and uniqueness of pseudo-almost periodic (of class infinity) solutions to abstract functional-differential equations in the framework of Hille-Yosida operators. We study in this work some sufficient conditions for the existence and uniqueness of a pseudo-almost periodic (of class infinity) mild solution to the following semilinear fractional differential equation Dtα u(t) = Au(t) + Dtα−1 f (t, u(t)),

t ∈ R,

(1.1)

where 1 < α < 2, A : D(A) ⊆ X → X is a linear densely defined operator of sectorial type on a complex Banach space X and f : R × X → X is a pseudo-almost periodic (of class infinity) function (see Definition 2.3) satisfying suitable conditions in x. We note that the definitions of the fractional derivative, which generalizes the ordinary integral and differential operators, are diverse. Here the fractional derivative is understood in the Riemann-Liouville sense. We remark that there is much interest in developing the theoretical analysis and numerical methods to fractional equations, because they have recently proved to be valuable in various fields of sciences and engineering. For details, including some applications and recent results, see the monographs of Ahn and MacVinish [10] and Gorenflo and Mainardi [34], and the papers of Agarwal et al. [2–7], Cuevas et al. [17, 18, 21, 32], Nieto et al. [9, 12], Lakshmikantham et al. [38–41] and N’Guérékata et al. [13, 42–47]. Type (1.1) equations are attracting increasing interest (cf. [10, 15, 16, 33]).

Pseudo-almost periodic solutions of a class of semilinear fractional

627

The study of pseudo-almost periodic mild solution of (1.1) was initiated recently by Agarwal et al. [6]. To the knowledge of the authors no results yet exist for pseudoalmost periodic (of class infinity) solutions of (1.1).

2 Preliminaries and basic results In this section, we introduce notations, definitions and preliminary facts which are used throughout this work. Let (X, · ) and (Y, · ) be two Banach spaces. The notation B(X, Y ) stands for the space of bounded linear operators from X into Y endowed with the uniform operator topology, and we abbreviate to B(X), whenever X = Y. 2.1 Sectorial linear operators and their associated solution operator A closed and linear operator A is said to be sectorial of type ω if there are 0 < θ < π2 , M > 0 and ω ∈ R such that its resolvent exists outside the sector ω + θ := M {ω + λ : λ ∈ C, | arg(−λ)| < θ } and (λ − A)−1 ≤ |λ−ω| ,λ ∈ / ω + θ . Sectorial operator are well studied in the literature. In order to give an operator theoretical approach we recall the following definition. Definition 2.1 [4] Let A be a closed and linear operator with domain D(A) defined on a Banach space X. We call A is the generator of a solution operator if there are ω ∈ R and a strongly continuous function Sα : R+ → B(X) such that {λα : ∞ −λt α−1 α −1 Re λ > ω} ⊆ ρ(A) and λ (λ − A) x = 0 e Sα (t)x dt, Re λ > ω, x ∈ X. In this case, Sα (t) is called the solution operator generated by A. We note that if A is sectorial of type ω with 0 ≤ θ ≤ π(1 − α2 ), then A is the gen λt α−1 α 1 erator of a solution operator given by Sα (t) := 2πi (λ − A)−1 dλ, where γ e λ γ is a suitable path lying outside the sector ω + θ (cf. [14]). Recently, Cuesta [14, Theorem 1] has proved that if A is a sectorial operator of type ω < 0 for some M > 0 and 0 ≤ θ ≤ π(1 − α2 ), then there is C > 0 such that Sα (t)

B(X)

≤

CM , 1 + |ω|t α

t ≥ 0.

(2.1)

2.2 Pseudo-almost periodic functions of class p We recall the following definition (see [8, 28, 36]). Definition 2.2 (1) A continuous function f : R → X is called (Bohr) almost periodic in t ∈ R if for every > 0 there is l( ) > 0 such that every interval of length l( ) contains a number τ with the property that f (t + τ ) − f (t) ≤ for each t ∈ R. The number τ is called an -translation number of f and the collection of all such functions will be denoted AP(X).

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(2) A continuous function f : R × Y → X is called (Bohr) almost periodic in t ∈ R uniformly in y ∈ Y if for every > 0 and any compact K ⊂ Y there is l( ) > 0 such that every interval of length l( ) contains a number τ with the property that f (t + τ, y) − f (t, y) ≤ for each t ∈ R and y ∈ K. The collection of those functions is denoted by AP(Y, X). (3) A bounded continuous function f : R → X (resp., f : R × Y → X) is called pseudo-almost periodic of class p if f = g + ϕ, where g ∈ AP(X) (resp., AP(Y, r X)) and ϕ is bounded continuous function such that limr→∞ 2r1 −r (supθ∈[t−p,t] ϕ(θ )X ) dt = 0 (resp., ϕ : R × Y → X is a bounded r continuous function such that limr→∞ 2r1 −r (supθ∈[t−p,t] ϕ(θ, z)X ) dt = 0, uniformly in compact subset of Y ). Denote by PAPp (X) (resp., PAPp (Y, X)) the set of all such functions. For p > 0. We define the spaces 1 r p PAP0 (X) := u ∈ BC(R, X) : lim sup u(θ )X dt = 0 . r→∞ 2r −r θ∈[t−p,t] p PAP0 (Y, X) := u ∈ BC(R × Y, X) : 1 r→∞ 2r lim

r −r

sup

u(θ, z)X dt = 0 ,

θ∈[t−p,t]

where the limit (as r → ∞) is uniform in compact subsets of Y . p

p

Remark 2.1 [36] The spaces PAP0 (X) and PAP0 (Y, X) endowed with the uniform convergence topology are Banach spaces. To deal with unbounded delay in [36] the authors have introduced the following new spaces of functions PAP∞ 0 (X) :=

p>0

PAP0 (X) and PAP∞ 0 (Y, X) := p

p

PAP0 (Y, X).

p>0

∞ Remark 2.2 [36] The spaces PAP∞ 0 (X) and PAP0 (Y, X) are, respectively, closed p p subspaces of PAP

0 (X) and PAP0 (Y, X) and hence

both∞are Banach spaces. Furthermore, AP(X) PAP∞ (X) = {0} and AP(Y, X) PAP0 (Y, X) = {0}. 0

Definition 2.3 A bounded continuous function f : R → X (resp., f : R × Y → X) is called pseudo-almost periodic (of class infinity) function if f = g + ϕ, where ∞ g ∈ AP(X) (resp., AP(Y, X)) and ϕ ∈ PAP∞ 0 (X) (resp., PAP0 (Y, X)). Denote by ∞ ∞ PAP (X) (resp., PAP (Y, X)) the set of all such functions. It is well known that the study of composition of two functions with special properties is so important. Results in such direction provide us with a sensible tool to make deep investigations.


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Lemma 2.1 [8] Let F ∈ PAP∞ (Y, X) so that F (t, y) is uniformly continuous on any bounded subset K ⊆ Y uniformly in t ∈ R and h ∈ PAP∞ (Y ). Then the function t → F (t, h(t)) belongs to PAP∞ (X). Lemma 2.2 Assume that A is sectorial of type ω < 0. If u : R → X is a pseudoalmost periodic function of class infinity and vαu (·) is given by t u Sα (t − s)u(s) ds, t ∈ R, vα (t) = −∞

then

vαu

∞

∈ PAP (X).

Proof If u = u1 + u2 , where u1 ∈ AP(X) and u2 ∈ PAP∞ 0 (X). By [4, Lemma 2.2], 1 2 u u vα ∈ AP(X). To complete the proof, we show that vα ∈ PAP∞ 0 (X). For p > 0, ∞ 1 > 0; p # ≥ p large enough so that p# 1+|ω|t α dt < and θ ∈ [t − p, t], we have the following estimates t−p 1 2 vαu (θ ) ≤ CM u2 (s) ds α 1 + |ω|(θ − s) −∞ θ 1 u2 (s) ds + CM 1 + |ω|(θ − s)α t−p t−p 1 ≤ CM u2 (s) ds 1 + |ω|(t − p − s)α −∞ θ 1 ds sup u2 (τ ) + CM α t−p≤τ ≤t t−p 1 + |ω|(θ − s) t−p 1 u2 (s) ds ≤ CM α −∞ 1 + |ω|(t − p − s) ∞ 1 + CM ds sup u2 (τ ). 1 + |ω|s α t−p≤τ ≤t 0 Therefore sup θ∈[t−p,t]

2 vαu (θ )

≤ CM

t−p

−∞

1 u2 (s) ds 1 + |ω|(t − p − s)α −1

CM|ω| α π + α sin( πα )

sup

t−p≤τ ≤t

u2 (τ ).

Using (2.2) we can infer that 1 r 2 sup vαu (θ ) dt 2r −r θ∈[t−p,t] CM r t−p 1 ≤ u2 (s) ds dt 2r −r −∞ 1 + |ω|(t − p − s)α

(2.2)

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−1 CM|ω| α π 1 r 2 sup u (τ ) dt α sin( πα ) 2r −r t−p≤τ ≤t 1 CM r−p ∞ u2 (t − s) ds dt = 2r −r−p 0 1 + |ω|s α −1 CM|ω| α π 1 r 2 sup u (τ ) dt + α sin( πα ) 2r −r t−p≤τ ≤t # CM r−p p 1 ≤ u2 (t − s) ds dt 2r −r−p 0 1 + |ω|s α 1 CM r−p ∞ u2 (t − s) ds dt + 2r −r−p p# 1 + |ω|s α −1 CM|ω| α π 1 r 2 sup + u (τ ) dt α sin( πα ) 2r −r t−p≤τ ≤t −1 CM|ω| α π 1 r 2 u (τ ) dt ≤ sup α sin( πα ) 2r −r t−p≤τ ≤t −1 p 1 CM|ω| α π 1 + + π α sin( α ) r 2(r + p) r+p sup u2 (τ ) dt + CMu2 ∞ . × +

−r−p

t−p # ≤τ ≤t

Since u2 ∈ PAP∞ 0 (X), we obtain that 1 r u2 sup v (θ ) dt = 0. lim r→∞ 2r −r θ∈[t−p,t] α The proof is now completed.

3 Pseudo-almost periodic mild solutions Before starting our main results in this section, we recall the definition of the mild solution to (1.1). Definition 3.1 Suppose A generates an integrable solution operator Sα (t). A continuous function u : R → X satisfying the integral equation t Sα (t − s)f (s, u(s)) ds, ∀t ∈ R, (3.1) u(t) = −∞

is called a mild solution on R to (1.1).


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Theorem 3.1 Assume that A is sectorial of type ω < 0. Let f ∈ PAP∞ (X, X) and that there is a bounded continuous function Lf : R → [0, ∞) such that f (t, x) − f (t, y) ≤ Lf (t)x − y,

∀t ∈ R, ∀x, y ∈ X.

(3.2)

t Lf (s) If CM(supt∈R −∞ 1+|ω|(t−s) α ds) < 1, where C and M are the constants in (2.1). Then (1.1) has a unique pseudo-almost periodic (of class infinity) mild solution. Proof We define the operator F : PAP∞ (X) → PAP∞ (X) by t (F u)(t) := Sα (t − s)f (s, u(s)) ds, t ∈ R. −∞

(3.3)

Given u ∈ PAP∞ (X), in view of composition result, Lemma 2.1, we have that s → f (s, u(s)) is a pseudo-almost periodic (of class infinity) function. Now by Lemma 2.2, we obtain F u ∈ PAP∞ (X) and hence F is well defined. It suffices now to show that the operator F has a unique fixed point in PAP∞ (X). For this, let u1 , u2 be in PAP∞ (X), we have t Sα (t − s)Lf (s)u1 (s) − u2 (s) ds F u1 (t) − F u2 (t) ≤ −∞

≤ CM

t −∞

Lf (s) ds u1 − u2 ∞ . 1 + |ω|(t − s)α

Hence

F u1 − F u2 ∞ ≤ CM sup

t

t∈R −∞

Lf (s) ds u1 − u2 ∞ . 1 + |ω|(t − s)α

This proves that F is a contraction, so by the Banach fixed point theorem, F has a unique fixed point, which gives rise to a unique u ∈ PAP∞ (X). This completes the proof. The following consequence is now immediate. Corollary 3.1 Assume that A is sectorial of type ω < 0. Let f ∈ PAP∞ (X, X) satisfying the Lipschitz condition f (t, x) − f (t, y) ≤ Lf x − y,

∀t ∈ R, ∀x, y ∈ X.

(3.4)

−1

If CM|ω| α πLf < α sin( πα ), then (1.1) has a unique pseudo-almost periodic (of class infinity) mild solution. Example 1 We take X = L2 [0, π] and let A be the operator given by Au = u − μu, (μ > 0) with domain D(A) = {u ∈ X : u ∈ X, u(0) = u(π) = 0}. It is well known that u = u is the infinitesimal generator of an analytic semigroup on L2 [0, π]. Hence, A is sectorial of type ω = −μ < 0. We examine the existence and

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uniqueness of a pseudo-almost periodic (of class infinity) mild solution to the fractional relaxation-oscillation equation given by √ ∂tα u(t, x) = ∂x2 u(t, x) − μu(t, x) + ∂tα−1 (β sin(u(t, x))(cos(t) + cos( 2t)) + βa(t) sin(u(t, x))),

t ∈ R, x ∈ [0, 1]

(3.5)

with boundary conditions u(t, 0) = u(t, π) = 0,

t ∈ R.

(3.6)

We have the following result. Proposition 3.1 Suppose that a ∈ PAP∞ (R) and that |β| is small enough, then the problem (3.5)–(3.6) has a unique pseudo-almost periodic (of class infinity) mild solution. Proof Problem (3.5)–(3.6) can be expressed as an abstract fractional differential equation of the form (1.1) in X = L2 [0, π] where u(t)(x) = u(t, x), for√ t ∈ R, x ∈ [0, π]. Let us consider the nonlinearity f (t, φ)(s) = β(cos(t) + cos( 2t)) × sin(φ(s)) + βa(t) sin(φ(s)), for all φ ∈ X, t ∈ R, s ∈ [0, π]. We note that f ∈ PAP∞ (X, X). On the other hand, we have the following estimate f (t, φ) − α sin( πα ) f (t, ψ)L2 ≤ |β|(2 + a∞ )φ − ψL2 , φ, ψ ∈ X. If |β| < , by −1 CM|ω|

α

π(2+a∞ )

Corollary 3.1, problem (3.5)–(3.6) has a unique pseudo-almost periodic (of class infinity) mild solution. Acknowledgements This work was completed when the second author was visiting the Universidad de La Frontera (March–July 2010). It is a pleasure for him to express gratitude for the warm hospitality.

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