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WEIGHTED PSEUDO ALMOST AUTOMORPHIC MILD SOLUTIONS TO A NEW CLASS OF
FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS
José Vanterler da Costa Sousa
To cite this version:
José Vanterler da Costa Sousa. WEIGHTED PSEUDO ALMOST AUTOMORPHIC MILD SOLU- TIONS TO A NEW CLASS OF FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS.
2021. �hal-03189228�
FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS
J. VANTERLER DA C. SOUSA
Abstract. In the paper, we investigate new results on the existence and uniqueness for pseudo almost automorphic and weighted pseudo almost automorphic of mild solutions of a fractional partial neutral functional differential equation in the sense of the Weyl fractional derivative in Banach space. To attack these results, we use the Banach fixed point theorem and fractional powers of operators.
1. Introduction
From the first ideas about the theory of almost automorphic, pseudo almost automorphic to Stepanov-type𝜇-pseudo almost automorphic, what has been noticed, is the growing and prominence of the theory in the field of differential and integro-differential equations [22,31,21,23,36] and references therein. Renowned researchers such as Diagana, N’Guereketa, Blot, Lizama, Cuevas, Liang, Morphou, Chang, among others, have been shown to be effective and have provided important results for the growth of the area [25,26,28,31,32,33,34,46].
Diagana and N’Guerekata [27], discussed Stepanov-like almost automorphic concept was extensively utilized to obtain the existence and uniqueness of almost automorphic solutions to the semilinear differential equations
𝑢0(𝑡)=𝐴𝑢(𝑡) +𝐹(𝑡 , 𝑢(𝑡)), 𝑡∈R
where 𝐴 : 𝐷(𝐴) ⊂ 𝑋 → 𝑋 is a densely defined closed linear operator on a Banach space X, which also is the infinitesimal generator of an exponentially stable𝐶0-semigroup(𝑇(𝑡)), 𝑡 ≥0 on𝑋and𝐹 :R×𝑋 → 𝑋is𝑆𝑝-almost automorphic for 𝑝 >1.
Blot et al. [24] have introduced the concept of𝜇-pseudo almost automorphic functions by defining the ergodicity of a function over a positive measure onR, which satisfies some hypothesis concerning translated sets inR. During these decades, numerous researchers have dedicated themselves to discussing results of existence for almost automorphic, pseudo-almost automorphic and weighted pseudo almost automorphic mild solutions of differential equations with infinite delay [31,25,29,30] and references therein.
In 2010, Zhao et al. [32] investigated the existence of almost automorphic and pseudo-almost automorphic mild solutions to the following equation
𝑑 𝑢(𝑡) 𝑑 𝑡
=𝐴𝑢(𝑡) + 𝑑 𝑑 𝑡
𝐹1(𝑡 , 𝑢(ℎ1(𝑡))) +𝐹2(𝑡 , 𝑢(ℎ2(𝑡)))
where𝐴:𝐷(𝐴) ⊂ 𝑋→𝑋is the infinitesimal generator of a𝐶0-semigroup{𝑇(𝑡), 𝑡 ≥0}on a Banach space𝑋. On the other hand, the theory of fractional calculus, once consolidated, has shown to be extremely important and of great interest in several areas, in particular, involving applications [6,10,11,12,13,14,15,16]. Furthermore, when it comes to addressing problems of existence and uniqueness of almost automorphic solutions of fractional differential and integro-differential differential equations, what has been noted is an exponential growth in the investigation of these results [4,5,7,8,9,17,18,19].
Alvarez [41], investigate the existence of Stepanov-like weighted pseudo almost automorphic mild solutions to the following fractional order integro-differential equation given by
𝐷𝛼
𝑡𝑥(𝑡)=𝐴𝑥(𝑡) +𝐷𝛼−1
𝑡 𝑓(𝑡 , 𝑥(𝑡), 𝐾 𝑥(𝑡)) 𝐾 𝑥(𝑡)=
∫ 𝑡
−∞
𝑘(𝑡−𝑠)ℎ(𝑠, 𝑥(𝑠))𝑑𝑠, 𝑡 ∈R, where𝐷𝛼
𝑡 is the Caputo fractional derivative of order 1< 𝛼 < 2 and𝐴:𝐷(𝐴) ⊂ 𝑋 → 𝑋is a linear densely defined operator of sectorial type on a complex Banach space(𝑋 ,k · k). Moreover, the function𝑘satisfies|𝑘(𝑡) | ≤𝑐𝑘𝑒−𝑏𝑡 for
2020Mathematics Subject Classification. 35R11, 44A35, 42A85, 42A75.
∗Correspondent author. J. Vanterler da C. Sousa.
Key words and phrases. Existence and uniqueness, Pseudo almost automorphic, Weighted pseudo almost automorphic, Fractional partial functional differential equations, Fractional powers of operators.
1
𝑡 ≥0 and𝑐𝑘,𝑏are positive constants, the function 𝑓 :𝑅×𝑋×𝑋 →𝑋andℎ:𝑅×𝑋 →𝑋are Stepanov-like weighted pseudo almost automorphic in𝑡for each𝑥 , 𝑦∈𝑋, satisfying suitable conditions.
Em 2017 Hattab [42], investigate the existence and uniqueness of𝜇-pseudo almost automorphic solutions of abstract fractional integro-differential neutral equations with an infinite delay given by
𝑑 𝑑 𝑡
𝐷(𝑡 , 𝑢𝑡)=
∫ 𝑡 0
(𝑡−𝑠)𝛼−2
Γ(𝛼−1) 𝐴 𝐷(𝑠, 𝑢𝑠)𝑑𝑠+𝑔(𝑡 , 𝑢𝑡), 𝑡 ≥0
with initial condition𝑢0 = 𝜑, where 1 < 𝛼 <2, 𝐷(𝑡 , 𝜑) = 𝜑(0) + 𝑓(𝑡 , 𝜑), 𝐴 : 𝐷(𝐴) ⊂ 𝑋 → 𝑋 is a linear densely defined operator of sectorial type on a Banach space𝑋, the history𝑢𝑡 : (−∞,0] → 𝑋, defined by𝑢𝑡(𝜃) =𝑢(𝑡+𝜃), belongs to an abstract phase space B defined axiomatically, and 𝑓 , 𝑔are functions subject to some additional conditions.
In 2018, Alvarez [51], established a new convolution and composition theorems for 𝜇-Stepanov pseudo almost periodic functions and investigated the existence and uniqueness of𝜇-pseudo almost periodic mild solutions for the fractional integro-differential equation given by
𝐷𝛼𝑢(𝑡)= 𝐴𝑢(𝑡) +
∫ 𝑡
−∞
𝑎(𝑡−𝑠)𝐴𝑢(𝑠)𝑑𝑠+ 𝑓(𝑡 , 𝑢(𝑡)) where𝐴generates an𝛼-resolvent family{𝑆𝛼(𝑡)}𝑡≥0on Banach space𝑋 , 𝛼∈𝐿1
𝑙 𝑜𝑐(R+ +),𝐷𝛼(·)is the Weyl fractional derivative of order𝛼 >0 and 𝑓 is𝜇-Stepanov pseudo almost periodic.
Motivated by the papers above and by a limited number of works in the area involving fractional operators, in this present we consider the fractional partial neutral functional differential equations abstract
(1.1) 𝑊D
𝛽
−∞(𝑢(𝑡) + 𝑓 (𝑡 , 𝑢(ℎ1(𝑡))))=𝒜𝑢(𝑡) +𝑔(𝑡 , 𝑢(ℎ2(𝑡))), 𝑡∈R where𝑊D
𝛽
−∞(·)is the Weyl fractional derivative of order 0 < 𝛽 <1,𝒜:𝐷(𝒜) ⊂X→Xis called the generator of an𝛽-resolvent familyS𝛽(𝑡), 𝑡≥0 on Banach space.
Now we list the following basic assumptions of this paper:
(H1) 1. There exists a positive number𝛼 ∈ (0,1) such that 𝑓 : R×X → X𝛼 is continuous and (−𝒜)𝛼 𝑓 ∈ PAA (R×X,X). LetL(1)
𝑓
>0 be such that for each(𝑡 , 𝑥),(𝑡 , 𝑦) ∈R×X
| (−𝒜)𝛼𝑓 (𝑡 , 𝑥) − (−𝒜)𝛼𝑓 (𝑡 , 𝑦) | ≤ L(1)𝑓 |𝑥−𝑦|.
2. There exists a positive number 𝛼 ∈ (0,1) such that R×X → Xis continuous and (−𝒜)𝛼𝑓 = 𝜑1 +𝜑1 ∈ WPAA (R×X, 𝜌)and there exists positive numbersL(2)𝑓 ,L𝜑such that for each(𝑡 , 𝑥),(𝑡 , 𝑦) ∈R×X
| (−𝒜)𝛼𝑓 (𝑡 , 𝑥) − (−𝒜)𝛼𝑓 (𝑡 , 𝑦) | ≤ L(2)
𝑓 |𝑥−𝑦| and
|𝜑1(𝑡 , 𝑥) −𝜑2(𝑡 , 𝑦) | ≤ L𝜑1|𝑥−𝑦|.
(H2)1.𝑔 ∈ PAA (R×X,X)and these exists a positive numberL𝑔(1)such that for each(𝑡 , 𝑥),(𝑡 , 𝑦) ∈R×X
|𝑔(𝑡 , 𝑥) −𝑔(𝑡 , 𝑦) | ≤ L(1)
𝑓 |𝑥−𝑦|.
2. 𝑔=𝜑2+𝜓2∈ WPAA (R×X, 𝜌)and there exist positive numbersL𝑔(2),L𝜑2such that for each(𝑡 , 𝑥),(𝑡 , 𝑦) ∈ R×X
|𝑔(𝑡 , 𝑥) −𝑔(𝑡 , 𝑦) | ≤ L𝑔(2)|𝑥−𝑦|. and
|𝜑2(𝑡 , 𝑥) −𝜑2(𝑡 , 𝑦) | ≤ L𝜑2|𝑥−𝑦|.
(H3) The functions ℎ𝑖 : R → R, ℎ𝑖(R) = R are continuously differentiable on R, and for 𝑢(·) ∈ AA (X), 𝑢(ℎ𝑖(·)) ∈ AA (X), ℎ0
𝑖(𝑡)>0,𝑖=1,2,are nondecreasing with lim sup
𝑟→∞
|ℎ𝑖(−𝑟) + |ℎ𝑖(𝑟) ||
𝑟 ℎ𝑖(−𝑟)
<∞.
(H4) The function ℎ𝑖 : R → R, ℎ𝑖(R) = R are continuously differentiable on R, and for 𝑢(·) ∈ AA (X), 𝑢(ℎ𝑖(·)) ∈ AA (X),ℎ0
𝑖(𝑡)>0,𝑖=1,2,are nondecreasing with lim sup
𝑟→∞
𝑚 𝑟∗
𝑖, 𝜌 𝑚(𝑟 , 𝜌)ℎ0
𝑖(−𝑟)
<∞ and
0<sup
𝑡∈R
𝜌(𝑡) 𝜌(ℎ𝑖(𝑡)) <∞
where𝑟0
𝑖 =|ℎ𝑖(−𝑟) | + |ℎ𝑖(𝑟) |for𝑖=1,2.
(H5)Assume thatAgenerates an𝛽-resolvent family S𝛽(𝑡)
𝑡≥0such that S𝛽(𝑡)
≤𝚯𝑒−𝛿 𝑡(𝚯>0, 𝛿 >0) for all 𝑡 ≥0.
The main contributions of this paper are to investigate the existence and uniqueness of pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to Eq.(1.1) solutions. In other words, by means of the basic assumptions(H1) − (H5), we will attack the following results:
Theorem 1.1. Assume the conditions (H1).2, (H2).2 and (H4) are satisfied, then the problem (1.1) has a unique weighted pseudo almost automorphic mild solution onRprovide that
(1.2) 𝐿:=k (−𝒜)𝛼k L(2)𝑓 +𝚯1−𝛼𝛿−𝛼Γ(𝛼) L(2)𝑓 +𝚯 𝛿
L𝑔(2)<1 whereΓ(·)is the gamma functions.
Theorem 1.2. Assume the conditions(H1) (1),(H2) (2)and(H3)hold, then the problem(1.1)has a unique pseudo almost automorphic mild solution onRprovide that
(1.3) k (−𝒜)−𝛼k L(1)
𝑓 +𝚯1−𝛼𝛿−𝛼Γ(𝛼) L(1)
𝑓 +𝚯L𝑔(1)𝛿−1 <1.
In the rest, the paper is organized as follows: Section 2, we present some fundamental concepts about: almost automorphic functions, pseudo almost automorphic functions, definition of Weyl fractional derivative and fractional integral, among other essential definitions. In this sense, some important results are presented that we used throughout the paper. In section 3, we will attack the main results of this paper, i.e., about some conditions (see (H1) −H5) and through the results presented in the preliminary section, in particular, via Lebesgue contraction and dominated convergence theorem, we prove new results on existence and uniqueness for pseudo almost automorphic and weighted almost automorphic of mild solutions of a fractional partial neutral functional differential equations of the form Eq.(1.1).
2. Preliminaries and basic results
In this section, we introduce some basic definitions, notations, lemmas, which will be used in what follows.
Throughout the paper , let(X,|·|)be a Banach space andC (R,X)stand for the collection of continuous functions from Rinto toX. We denote by BC (R,X) the Banach space of all bounded continuous functions fromRinto X endowed with the supremum norm defined byk𝑥kB C (R,X) :=sup
𝑡∈R
|𝑥(𝑡) |. Furthermore,BC (R×X,X)is the space of all bounded continuous functionsℱ:R×X→X[36].
Let𝑈denote the set of all functions (weights)𝜌:R→ (0,∞),which are locally integrable overRsuch that𝜌 >0 almost everywhere. For a given𝑟 >0 and for each𝜌∈𝑈, we set [37]
𝑚(𝑟 , 𝜌)=
∫ 𝑟
−𝑟
𝜌(𝑥)𝑑𝑥 . We denote by𝑈∞the set of all𝜌∈𝑈with lim
𝑟→∞
𝑚(𝑟 , 𝜌)=∞and𝑈𝑏 the set of all𝜌 ∈𝑈∞such that𝜌is bounded and inf
𝑥∈R
𝜌(𝑥)>0.
Is is clear that𝑈𝑏 ⊂𝑈∞ ⊂𝑈, with strict inclusions.
Let 0∈ 𝜌(𝒜), then it is possible to define the fractional power(−𝒜)𝛼, for 0< 𝛼 ≤1, as a closed linear operator on its domain𝐷( (−𝒜)𝛼). Furthermore, the subspace𝐷( (−𝒜)𝛼)is dense inXand the expression
|X|𝛼 =| (−𝒜)𝛼𝑥|, 𝑥∈𝐷( (−𝒜)𝛼)
defines a norm on𝐷( (−𝒜)𝛼). Hereafter we denote byX𝛼the Banach space𝐷( (−𝒜)𝛼)with norm|𝑥|𝛼.
Definition 2.1. [25,36]A continuous functionℱ:R→ Xis said to be almost automorphic if for every sequence of real numbers 𝑠𝑛0
𝑛∈Nthere exists a subsequence(𝑠𝑛)𝑛∈Nsuch that 𝐺(𝑡):= lim
𝑛→∞ℱ(𝑡+𝑠𝑛) is well defined for each𝑡 ∈Rand
𝑛→∞lim
𝐺(𝑡−𝑠𝑛)=ℱ(𝑡) for each𝑡 ∈R. The collection of such functions will be denote byAA (X).
Definition 2.2. [36]A continuous functionℱ : R×𝑥 → Xis said to be almost automorphic ifℱ(𝑡 ,X) is almost automorphic for each 𝑡 ∈ Runiformly for all𝑥 ∈ B, whereB is any bounded subset ofX. The collection of such functions will be denote byAA (R×X,X).
Lemma 2.3. [21]
AA (X),k·kA A (
X)
is Banach endowed with the supremum norm given by kℱkA A (X)=sup
𝑡∈R
|ℱ(𝑡) |. The notationPAA0(X)stands for the spaces of functions
PAA0(X)=
ℱ∈ BC (R,X): lim
𝑟→∞
1 2𝑟
∫ 𝑟
−𝑟
|ℱ(𝑡) |𝑑 𝑡 =0
. Similarly, the notationPAA0(R×X,X)stand for the spaces of functions
PAA0(R×X,X)=
ℱ∈𝐵𝐶(R×X,X): lim
𝑟→∞
1 2𝑟
∫ 𝑟
−𝑟
|ℱ(𝑡) |𝑑 𝑡=0, uniformly in𝑥in any bounded subset ofX
.
Definition 2.4. [36] A continuous function ℱ : R → X (resp. R×X → 𝑋) is called pseudo almost auto- morphic if it can be decomposed as ℱ = k+ 𝜙, where k ∈ AA (X) (resp. AA (R×X,X)) and 𝜙 ∈ PAA0 (X) (resp. PAA0(R×X,X)). The class of all such functions will be denote byPAA (X) (resp. PAA (R×X,X)).
Lemma 2.5. [34]
PAA (X),k·kP A A (X)
is a Banach endowed with the supremum norm given by kℱkP A A (X) =sup
𝑡∈R
|ℱ(𝑡) |.
Lemma 2.6. [34] Assume ℱ = k + 𝜙 ∈ PAA (R×X,X), where k(𝑡 , 𝑥) ∈ AA (R×X,X) and 𝜙(𝑡 , 𝑥) ∈ PAA0(R×X,X), and suppose that ℱ(𝑡 , 𝑥) is uniformly continuous in any bounded subset 𝐾 ⊂ X uniformly for𝑡∈R. If𝑥(𝑡) ∈ PAA (R,X), thenℱ(·, 𝑥(·)) ∈ PAA (R, 𝜌).
Now for𝜌∈𝑈∞we define PAA0(R, 𝜌)=
ℱ∈ BC (R,X): lim
𝑟→∞
1 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
|ℱ(𝑡) |𝜌(𝑡)𝑑 𝑡=0
;
PAA0(R×X, 𝜌)=
ℱ∈ BC (R×X,X): lim
𝑟→∞
1 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
|ℱ(𝑡 , 𝑥) |𝜌(𝑡)𝑑 𝑡=0 uniformly in𝑥∈X
.
Definition 2.7. [24]A bounded continuous functionℱ : R → X (resp. R×X → X) is called weighted pseudo almost automorphic if it can be decomposed as ℱ = k +𝜙, where k ∈ AA (X) (resp. AA (R×X,X)) and 𝜙∈ PAA0(R, 𝜌)(resp.PAA0(R×X, 𝜌)). The class of all such functions will be denote byWPAA (R, 𝜌)(resp.
WPAA (R×X, 𝜌)).
Lemma 2.8. [35]Let𝜌∈𝑈∞.Suppose thatPAA0(R, 𝜌)is translation invariant. Then the decomposition of weighted pseudo almost automorphic functions is unique.
Lemma 2.9. [46]Let𝜌∈𝑈∞. IfPAA0(R, 𝜌)is translation invariant, then (WPAA (R, 𝜌),k·kW P A A (R, 𝜌)) is a Banach space endowed with the supremum norm given by
kℱkW P P A (R, 𝜌)=sup
𝑡∈R
|ℱ(𝑡) |.
Lemma 2.10. [24]Letℱ=k+𝜙∈ WPAA (R×X, 𝜌)where𝜌∈𝑈∞,k∈ AA (R×X,X)and𝜙∈ PAA0(R×X, 𝜌).
Assume both ℱand k are Lipschitzian in 𝑥 ∈ X uniformly in𝑡 ∈ R. If𝑥(𝑡) ∈ WP PA (R, 𝜌) then the function ℱ(·, 𝑥(·)) ∈ WPAA (R, 𝜌).
Given a function𝑔:R→ 𝑋, the Weyl fractional integral of order𝛼 >0 is defined by [1,3]
𝐼−∞𝛼 𝑢(𝑡):= 1 Γ(𝛼)
∫ 𝑡
−∞
(𝑡−𝑠)𝛼−1𝑢(𝑠)𝑑𝑠, 𝑡 ∈R when this integral is convergent.
On the other hand, the Weyl fractional derivative𝑊−∞D𝑡𝛼𝑢of order𝛼 >0 is defined by [1,3]
𝑊
−∞D𝑡𝛼𝑢(𝑡):= 𝑑𝑛 𝑑 𝑡𝑛
𝐷−(𝑛−𝛼)𝑢(𝑡), 𝑡∈R where𝑛=[𝛼] +1.
Definition 2.11. [47]Let𝒜be a closed and linear operator with domain𝐷(𝒜)defined on a Banach space Ωand 𝛼 >0. Given𝑎∈ 𝐿1
𝑙 𝑜𝑐(R+), we say that𝒜is the generator of an𝛼-resolvent family, if there exists𝜔≥0and a strongly continuous functionS𝛼:[0,∞) → B (Ω)such that
𝜆𝛼
1+b𝑎(𝜆) :𝑅 𝑒(𝜆)>
⊂𝜌(𝒜)and for all𝑥∈Ω
(𝜆𝛼− (1+b𝑎(𝜆))𝒜)−1𝑥 = 1 1+b𝑎(𝜆)
𝜆𝛼 1+b𝑎(𝜆) −𝒜
−1
𝑥
=
∫ ∞ 0
𝑒−𝜆𝑡S𝛼(𝑡)𝑥 𝑑 𝑡 , 𝑅 𝑒(𝜆) > 𝜔
whereb𝑎denotes the Laplace transform of𝑎. In this case,{S𝛼(𝑡)}𝑡≥0is called the𝛼−resolvent family generated by𝒜.
Remark 2.12. [47]Observe that if 𝑏(𝑡) = 𝑔𝛼(𝑡) + (𝑔𝛼∗𝑎) (𝑡) 𝑡 ≥ 0, where 𝑔𝛼(𝑡) = 𝑡𝛼−1
Γ(𝛼) and (𝑔𝛼∗𝑎) (𝑡) =
∫ 𝑡
0
𝑔𝛼(𝑡−𝑠)𝑎(𝑠)𝑑𝑠, then we have that the family𝛼-resolvent{S𝛼(𝑡)}𝑡≥0is a(𝑏, 𝑔𝛼)-regularized family. In partic- ular, if𝑎=0, an 1-resolvent family is the same as a𝑐0-semigroup, whereas that a 2-resolvent family corresponds to the concepts of sine family. Therefore, if𝒜is the generator of an𝛼-resolvent family{S𝛼(𝑡)}𝑡≥0then have that the family {S𝛼(𝑡)}𝑡≥0verifies the following properties:
1. S𝛼(0)=𝑔𝛼(0);
2. S𝛼(𝑡)𝜁 ∈𝐷(𝒜)andS𝛼(𝑡)𝒜𝜁 =𝒜S𝛼(𝑡)𝜁for all𝜁 ∈𝐷(𝒜)and𝑡≥0;
3. S𝛼(𝑡)𝜁 =𝑔𝛼(𝑡)𝜁+
∫ 𝑡 0
𝑏(𝑡−𝑠)𝐴S𝛼(𝑡)𝜁 𝑑𝑠, for all𝑥∈𝐷(𝐴)and𝑡 ≥0;
4.
∫ 𝑡 0
𝑏(𝑡−𝑠)S𝛼(𝑡)𝜁 𝑑𝑠∈𝐷(𝒜)andS𝛼(𝑡)𝜁 =𝑔𝛼(𝑡)𝜁+𝒜
∫ 𝑡 0
𝑏(𝑡−𝑠)S𝛼(𝑠)𝜁 𝑑𝑠,for all𝜁 ∈Ωand𝑡≥0.
Sufficient conditions implying that{S𝛼(𝑡)}𝑡≥0 ⊂𝐵(Ω)is an𝛼-resolvent family.
Lemma 2.13. Letℱ=k+𝜙∈ WPAA (R, 𝜌)where𝜌∈𝑈∞and
S𝛽(𝑡) 𝑡≥0is an𝛽-resolvent family stable. Then F(𝑡):=
∫ 𝑡
−∞S𝛽(𝑡−𝑠)ℱ(𝑠)𝑑𝑠∈ WPAA (R, 𝜌). Proof. Let𝐹(𝑡)=𝐺(𝑡) +Φ(𝑡)where
(2.1) 𝐺(𝑡):=
∫ 𝑡
−∞S𝛽(𝑡−𝑠)𝑔(𝑠)𝑑𝑠
(2.2) Φ(𝑡):=
∫ 𝑡
−∞S𝛽(𝑡−𝑠)𝜙(𝑠)𝑑𝑠.
Then by [4]𝐺(𝑡) ∈ AA (R, 𝑋). Now let us show thatΦ(𝑡) ∈ PAA (R, 𝜌). We have 1
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
kΦ(𝑠) k𝜌(𝑠)𝑑𝑠 = 1 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
∫ 𝑠
−∞S𝛽(𝑠−𝜎)𝜙(𝜎)𝑑𝜎
𝜌(𝑠)𝑑𝑠
= 1
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
∫ −𝑟
−∞ S𝛽(𝑠−𝜎)𝜙(𝜎)𝑑𝜎+
∫ 𝑠
−𝑟 S𝛽(𝑠−𝜎)𝜙(𝜎)𝑑𝜎
𝜌(𝑠)𝑑𝑠
≤ 1
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
∫ −𝑟
−∞ S𝛽(𝑠−𝜎)𝜙(𝜎)𝑑𝜎
𝜌(𝑠)𝑑𝑠
+ 1
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
∫ 𝑠
−𝑟 S𝛽(𝑠−𝜎)𝜙(𝜎)𝑑𝜎
𝜌(𝑠)𝑑𝑠.
Then, we have 𝐼1:= 1
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
∫ −𝑟
−∞ S𝛽(𝑠−𝜎)𝜙(𝜎)𝑑𝜎
𝜌(𝑠)𝑑𝑠 ≤ 1 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
∫ −𝑟
−∞
S𝛽(𝑠−𝜎)
k𝜙(𝜎) k𝑑𝜎
𝜌(𝑠)𝑑𝑠
≤ 𝚯
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
𝑒−𝛿 𝑠𝜌(𝑠)𝑑𝑠 ∫ −𝑟
−∞
𝑒𝛿 𝜎k𝜙(𝜎) k𝑑𝜎
≤ 𝚯
𝛿𝑚(𝑟 , 𝜌) k𝜌k𝐿1
𝑙 𝑜𝑐(R)𝑒−2𝛿𝑟sup
𝑡∈R
k𝜙(𝑡) k
∫ −𝑟
−∞
𝑒𝛿 𝜎𝑑𝜎
≤ 𝚯
𝛿2𝑚(𝑟 , 𝜌) k𝜌k𝐿1
𝑙 𝑜𝑐(R)sup
𝑡∈R
k𝜙(𝑡) k.
Since sup
𝑡∈R
k𝜙(𝑡) k<∞and lim
𝑟→∞
𝑚(𝑟 , 𝜌)=∞, then lim
𝑟→∞
𝐼1 =0. On the other hand, we have
𝐼2 : = 1
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
∫ 𝑠
−𝑟 S𝛽(𝑠−𝜎)𝜙(𝜎)𝑑𝜎
𝜌(𝑠)𝑑𝑠
≤ 1
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
∫ 𝑠
−𝑟
S𝛽(𝑠−𝜎)
k𝜙(𝜎) k𝑑𝜎 𝜌(𝑠)𝑑𝑠
≤ 𝚯
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
∫ 𝑠
−𝑟
𝑒−𝛿(𝑠−𝜎)k𝜙(𝜎) k𝑑𝜎
𝜌(𝑠)𝑑𝑠
≤ 𝚯
𝛿𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
k𝜙(𝑠) k𝜌(𝑠)𝑑𝑠
.
Since𝜙∈ PAA0(R, 𝜌)then lim
𝑟→∞
∫ 𝑟
−𝑟
k𝜙(𝑠) k𝜌(𝑠)𝑑𝑠=0.Thus lim
𝑟→∞
𝐼2 =0.
The next result is a straightforward consequence of Lemma2.13when𝜌=1.
Lemma 2.14. Letℱ =k+𝜙 ∈ PAA (X)and
S𝛽(𝑡) 𝑡≥
0 is an 𝛽-resolvent family stable. IfF : R → Xbe the function defined by
(2.3) F(𝑡)=
∫ 𝑡
−∞S𝛽(𝑡−𝑠)ℱ(𝑠)𝑑𝑠, 𝑡 ≥𝑠∈R thenF(·) ∈ PAA (X).
Definition 2.15. A function𝑢 ∈ BC (R,X)is called a pseudo almost automorphic mild solution ofEq.(1.1)onRif 𝑢 ∈ PAA (X) and the function𝑠→ 𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠))) is integrable on(−∞, 𝑡)for each𝑡 ∈ Rand𝑢(𝑡) satisfies
𝑢(𝑡) = S𝛽(𝑡−𝑎) [𝑢(𝑎) + 𝑓 (𝑎, 𝑢(ℎ1(𝑎)))] − 𝑓 (𝑡 , 𝑢(ℎ1(𝑡)))
−
∫ 𝑡 𝑎
AS𝛽(𝑡−𝑠) 𝑓 (𝑡 , 𝑢(ℎ1(𝑠)))𝑑𝑠+
∫ 𝑡 𝑎
S𝛽(𝑡−𝑠)𝑔(𝑡 , 𝑢(ℎ2(𝑠)))𝑑𝑠 (2.4)
for all𝑡 ≥𝑎and all𝑎 ∈R.
Definition 2.16. A function𝑢 ∈ BC (R,X)is called a weighted pseudo almost automorphic mild solution ofEq.(1.1) onRif𝑢∈ WPAA (R, 𝜌)and the function𝑠→𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))is integrable on(−∞, 𝑡)for each𝑡∈R and𝑢(𝑡)satisfies
𝑢(𝑡) = S𝛽(𝑡−𝑎) [𝑢(𝑎) + 𝑓 (𝑎, 𝑢(ℎ1(𝑎)))] − 𝑓 (𝑡 , 𝑢(ℎ1(𝑡)))
−
∫ 𝑡
𝑎
𝒜𝑆𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠+
∫
S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠 (2.5)
for all𝑡 ≥𝑎and all𝑎 ∈R.
Lemma 2.17. [37]Let0< 𝛾≤𝜇≤1and0< 𝛽≤1. Then the following properties hold:
1)X𝜇is a Banach space andX𝜇 ↩→X𝛾is continuous.
2) The functions 𝑠 → (−𝒜)𝜇S𝛽(𝑡) is continuous in the uniform operator topology on (0,∞) and there exists 𝑀𝜇 >0such that
(−𝒜)𝜇S𝛽(𝑡)
≤𝑀𝜇𝑒−𝛿 𝑡𝑡−𝜇 for each𝑡 >0.
3) For each𝑥∈𝐷( (−𝒜)𝜇)and𝑡 ≥0,(−𝒜)𝜇S𝛽𝑥=S𝛽(𝑡) (−𝒜)𝜇𝑥 .
4)(−𝒜)−𝜇is a bounded linear operator inXwith𝐷( (−𝒜)𝜇)=𝐼 𝑚( (−𝒜)−𝜇).
3. Existence and uniqueness of mil solution
In this section, we investigate some results of the existence and uniqueness of mild solutions (pseudo almost automorphic and weighted pseudo almost automorphic) to a new class of fractional partial functional differential equation in Banach space.
Lemma 3.1. Let0< 𝛼≤1,0< 𝛽 ≤1and(−𝒜)𝛼𝑣∈ WPAA (R, 𝜌). If𝑢(·):R→𝑋be the function defined by
(3.1) 𝑢(𝑡)=
∫ 𝑡
−∞
𝒜S𝛽(𝑡−𝑠)𝑣(𝑠)𝑑𝑠, 𝑡 ≥𝑠 then𝑢(·) ∈ WPAA (R, 𝜌).
Proof. The proof of this result will be investigated in two steps.
Step 1. 𝑢(·)is well defined.
Since (−𝒜)𝛼𝑣∈ WPAA (R, 𝜌)then(−𝒜)𝛼𝑣is bounded. Then we can assume that there exists𝚯1 > 0 such thatk (−𝒜)𝛼𝑣kW P A A (R, 𝜌) ≤𝚯1. In this sense, we have
|𝑢(𝑡) | ≤
∫ 𝑡
−∞
(−𝒜)1−𝛼S𝛽(𝑡−𝑠) 𝚯1𝑑𝑠
≤ 𝚯1𝚯1−𝛼
∫ 𝑡
−∞
𝑒−𝛿(𝑡−𝑠)(𝑡−𝑠)𝛼−1𝑑𝑠.
(3.2)
By making the following variable change,𝜇=𝑡−𝑠in integral of Eq.(3.2), yields
|𝑢(𝑡) | ≤ 𝚯1𝚯1−𝛼
∫ ∞ 0
𝑒−𝛿 𝜇𝜇𝛼−1𝑑 𝜇
≤ 𝚯1𝚯1−𝛼𝛿−𝛼Γ(𝛼) (3.3)
whereΓ(𝛼)=
∫ ∞ 0
𝑥𝛼−1𝑒−𝑥𝑑𝑥is the gamma function.
Then𝒜S𝛽(𝑡−𝑠)𝑣(𝑠) is integrable on(−∞, 𝑡) for each𝑡 ∈ Rand therefore we conclude that𝑢(𝑡) is bounded continuous functions.
Step 2. 𝑢(·) ∈ WPAA (R, 𝜌).
Indeedd, let(−𝒜)𝛼𝑣(𝑡)=𝑚(𝑡) +𝑝(𝑡)where𝑚(·) ∈ AA (X)and𝑝(·) ∈ PAA0(R, 𝜌). Then, we have 𝑢(𝑡) = −
∫ 𝑡
−∞
(−𝒜)1−𝛼S𝛽(𝑡−𝑠) (−𝒜)𝛼𝑣(𝑠)𝑑𝑠
= −
∫ 𝑡
−∞
(−𝒜)1−𝛼S𝛽(𝑡−𝑠)𝑚(𝑠)𝑑𝑠−
∫ 𝑡
−∞
(−𝒜)1−𝛼S𝛽(𝑡−𝑠)𝑝(𝑠)𝑑𝑠.
(3.4)
Now, we consider B𝛽(𝑡) = −
∫ 𝑡
−∞
(−𝒜)1−𝛼S𝛽(𝑡−𝑠)𝑚(𝑠)𝑑𝑠 andC𝛽(𝑡) = −
∫ 𝑡
−∞
(−𝒜)1−𝛼S𝛽(𝑡−𝑠)𝑝(𝑠)𝑑𝑠. Note that,𝑢(𝑡)=B𝛽(𝑡) +C𝛽(𝑡).
(A).B𝛽(𝑡) ∈ AA (X).
Considering the following arbitrary sequence of real numbers given by 𝑠0𝑛
𝑛∈N. Since𝑚(𝑡) ∈ AA (X), there exists a subsequence(𝑠𝑛)𝑛∈Nof 𝑠0𝑛
𝑛∈Nsuch that𝜑(𝑡):= lim
𝑛→∞
𝑚(𝑡+𝑠𝑛)and𝑚(𝑡)= lim
𝑛→∞
𝜑(𝑡−𝑠𝑛)for each𝑡 ∈R, are well defined. Using the same idea of Eq.(3.3), yields
B𝛽(𝑡+𝑠𝑛) =
−
∫ 𝑡
−∞
(−𝒜)1−𝛼S𝛽(𝑡−𝑠)𝑚(𝑠+𝑠𝑛)𝑑𝑠
≤ 𝚯1−𝛼
∫ 𝑡
−∞
𝑒−𝛼(𝑡−𝑠)(𝑡−𝑠)𝛼−1|𝑚(𝑠+𝑠𝑛) |𝑑𝑠
≤ k𝑚kA A (X)𝚯1−𝛼
∫ 𝑡
−∞
𝑒−𝛼(𝑡−𝑠)(𝑡−𝑠)𝛼−1𝑑𝑠
≤ 𝚯1−𝛼𝛿−𝛼Γ(𝛼) k𝑚kA A (X)
for𝑛 = 1,2, .... Using Lemma??(ii), it follows that (−𝒜)1−𝛼S𝛽(𝑡−𝑠)𝑚(𝑠+𝑠𝑛) → (−𝒜)1−𝛼S𝛽(𝑡−𝑠)𝜑(𝑠) as 𝑛→ ∞, for each𝑠∈Rfixed and any𝑡≥𝑠. In this sense, by the Lebesgue dominated convergence theorem, we have
(3.5) lim
𝑛→∞B𝛽(𝑡+𝑠𝑛)=−
∫ 𝑡
−∞
(−𝒜)1−𝛼S𝛽(𝑡−𝑠)𝜑(𝑠)𝑑𝑠.
Following the steps of the test above, it follows that
𝑛→∞lim
−
∫ 𝑡−𝑠𝑛
−∞
(−𝒜)1−𝛼S𝛽(𝑡−𝑠𝑛−𝑠)𝜑(𝑠)𝑑𝑠
=B𝛽(𝑡). Therefore,B𝛽(𝑡) ∈ AA (X).
(B). C𝛽(𝑡) ∈ PAA0(R, 𝜌).
Indeed, we have
𝑟→∞lim 1 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
C𝛽(𝑡)
𝜌(𝑡)𝑑 𝑡 ≤ lim
𝑟→∞
1 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
∫ −𝑟
−∞
+
∫ 𝑡
−𝑟
(−𝒜)1−𝛼S𝛽(𝑡−𝑠)𝑝(𝑠)
𝜌(𝑠)𝑑𝑠 𝑑 𝑡
≤ lim
𝑟→∞
1 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
𝑑 𝑡
∫ −𝑟
−∞
𝚯1−𝛼𝑒−𝛿(𝑡−𝑠)(𝑡−𝑠)𝛼−1|𝑝(𝑠) |𝜌(𝑠)𝑑𝑠 +lim
𝑟→∞
1 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
𝑑 𝑡
∫ 𝑡
−𝑟
𝚯1−𝛼𝑒−𝛿(𝑡−𝑠)(𝑡−𝑠)𝛼−1|𝑝(𝑠) |𝜌(𝑠)𝑑𝑠
≤ lim
𝑟→∞
𝚯1−𝛼k𝑝k𝐵𝐶(
R,X)
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
𝜌(𝑡)𝑑 𝑡
∫ ∞ 𝑡+𝑟
𝜎𝛼−1𝑒−𝛿 𝜎𝑑𝜎 +lim
𝑟→∞
𝚯1−𝛼 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
|𝑝(𝑡) |𝜌(𝑡)𝑑 𝑡
∫ ∞ 0
𝜎𝛼−1𝑒−𝛿 𝜎𝑑𝜎
≤ lim
𝑟→∞
𝚯1−𝛼k𝑝k𝐵𝐶(
R,X)
𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
𝜌(𝑡)𝑑 𝑡
∫ ∞ 2𝑟
𝜎𝛼−1𝑒−𝛿 𝜎𝑑𝜎 +lim
𝑟→∞
𝚯1−𝛼𝛿−𝛼Γ(𝛼) 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
|𝑝(𝑡) |𝜌(𝑡)𝑑 𝑡
≤ 𝚯1−𝛼k𝑝k𝐵𝐶(R,X)
(2𝑟)1−𝛼𝑒2𝛿𝑟𝛿
| {z }
(𝐼)
+ lim
𝑟→∞
𝚯1−𝛼𝛿−𝛼Γ(𝛼) 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
|𝑝(𝑡) |𝜌(𝑡)𝑑 𝑡
| {z }
(𝐼 𝐼)
.
Note that𝐼 →0 when𝑟 → ∞and since𝑝(·) ∈ PAA0(R,X), then lim
𝑟→∞
1 𝑚(𝑟 , 𝜌)
∫ 𝑟
−𝑟
|𝑝(𝑡) |𝜌(𝑡)𝑑 𝑡 =0.In this sense, we have𝐼 𝐼→0.In view of the above it is clear that𝑢∈ WPAA (R, 𝜌). We concluded the proof.
A natural consequence of Lemma3.1, is when we choose𝜌=1, given by the following result:
Lemma 3.2. Let0< 𝛼≤1,0< 𝛽 ≤1and(−𝒜)𝛼𝑣∈ PAA (X). If𝑢(·):R→Xbe the function defined by 𝑢(𝑡)=
∫ 𝑡
−∞
𝒜S𝛽(𝑡−𝑠)𝑣(𝑠)𝑑𝑠, 𝑡 ≥𝑠 then𝑢(·) ∈ PAA (X).
Proof. The proof follows the same steps as the Lemma3.1test with𝜌=1.
Lemma 3.3. [32]Assume that bothℎ1(·)andℎ2(·)satisfy(H3). If𝑢 ∈ PAA (X), then𝑢(ℎ𝑖(·)) ∈ PAA (X)for 𝑖=1,2.
Lemma 3.4. [36] Assume that both ℎ1(·) and ℎ2(·) satisfy (H4). If 𝑢 ∈ WPAA (R, 𝜌), then 𝑢(ℎ𝑖(·)) ∈ WPAA (R, 𝜌)for𝑖=1,2.
Theorem 3.5. Assume the conditions(H1) (1),(H2) (2)and(H3)hold, then the problem(1.1)has a unique pseudo almost automorphic mild solution onRprovide that
(3.6) k (−𝒜)−𝛼k L(𝑓1)+𝚯1−𝛼𝛿−𝛼Γ(𝛼) L(𝑓1)+𝑀 𝐿𝑔(1)𝛿−1 <1.
The proof of this result will be discussed in two stages. First, consider𝚵 : PAA (X) →𝐶(X)be the operator defined by
𝚵𝑢(𝑡)=−𝑓 (𝑡 , 𝑢(ℎ1(𝑡))) −
∫ 𝑡
−∞
𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠+
∫ 𝑡
−∞S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠 𝑡 ∈R.
Step 1. 𝚵𝑢(𝑡)is well defined.
From the continuity of𝑠→𝒜S𝛽(𝑡−𝑠)and𝑠→S𝛽(𝑡−𝑠)in the uniform operator topology on(−∞, 𝑡)for each 𝑡 ∈Rand the estimate
(3.7)
𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))
≤𝚯1−𝛼𝑒−𝛿(𝑡−𝑠)(𝑡−𝑠)𝛼−1k (−𝒜)𝛼 𝑓 (𝑠, 𝑢(ℎ1(𝑠))) kB C (R×X,X)
it follows that𝑠→𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))and𝑠→S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))are integrable on(−∞, 𝑡)for every 𝑡 ∈ R. Therefore, we have thatΛ𝑢is well defined and continuous. Moreover, from Lemma3.2, Lemma3.3, Lemma 2.6and Lemma2.14we infer that𝚵𝑢(𝑡) ∈ PAA (X).
Step 2.𝚵is a continuous onPAA (X).
Indeed, for each𝑡∈R,𝑢, 𝑣∈ PAA (X), yields
|𝚵𝑢(𝑡) −𝚵𝑣(𝑡) | ≤ k (−𝒜)−𝛼k k (−𝒜)𝛼 𝑓 (𝑡 , 𝑢(ℎ1(𝑡))) − (−𝒜)𝛼 𝑓 (𝑡 , 𝑣(ℎ1(𝑡))) k +
∫ 𝑡
−∞
(−𝒜)1−𝛼𝑆𝛽(𝑡−𝑠)
| (−𝒜)𝛼 𝑓 (𝑠, 𝑢(ℎ1(𝑠))) − (−𝒜)𝛼 𝑓 (𝑠, 𝑣(ℎ1(𝑠))) |𝑑𝑠 +
∫ 𝑡
−∞
𝚯𝑒−𝛿(𝑡−𝑠)|𝑔(𝑠, 𝑢(ℎ2(𝑠))) −𝑔(𝑠, 𝑣(ℎ2(𝑠))) |𝑑𝑠
≤ k (−𝒜)−𝛼k L(𝑓1)k𝑢−𝑣kP A A (X)+ L(𝑓1)k𝑢−𝑣kP A A (X)
∫ 𝑡
−∞
𝚯1−𝛼𝑒−𝛿(𝑡−𝑠)(𝑡−𝑠)𝛼−1𝑑𝑠 +L𝑔(1)k𝑢−𝑣kP A A (X)
∫ 𝑡
−∞
𝚯𝑒−𝛿(𝑡−𝑠)𝑑𝑠
≤ k (−𝒜)−𝛼k L(1)
𝑓 k𝑢−𝑣kP A A (
X)+ L(1)
𝑓 k𝑢−𝑣kP A A (
X)𝚯1−𝛼𝛿−1Γ(𝛼) +L𝑔(1)k𝑢−𝑣kP A A (X)𝚯
𝛿
=
k (−𝒜)−𝛼k L(𝑓1)+ L(𝑓1)𝚯1−𝛼𝛿−1Γ(𝛼) + L𝑔(1)
𝚯 𝛿
k𝑢−𝑣kP A A (X). In this sense, we have
k𝚵𝑢(𝑡) −𝚵𝑣(𝑡) k ≤
k (−𝒜)−𝛼k L(1)𝑓 + L(1)𝑓 𝚯1−𝛼𝛿−1Γ(𝛼) + L𝑔(1)
𝚯 𝛿
k𝑢−𝑣kP A A (X)
which implies that𝚵is a contraction by inequality (3.6). Therefore, we conclude by the Banach fixed point theorem, that there is a single fixed point𝑢(·) for𝚵inPAA (X)such that𝚵𝑢=𝑢, i.e.,
𝑢(𝑡) = −𝑓 (𝑡 , 𝑢(ℎ1(𝑡))) −
∫ 𝑡
−∞
𝒜S𝛽(𝑡−𝑠)𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠+
∫ 𝑡
−∞
S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠 for all𝑡 ∈R. If we let
𝑢(𝑎) = −𝑓 (𝑎, 𝑢(ℎ1(𝑎))) −
∫ 𝑎
−∞
𝒜S𝛽(𝑎−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠+
∫ 𝑎
−∞S𝛽(𝑎−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠 then
S𝛽(𝑡−𝑎)𝑢(𝑎) = −S𝛽(𝑡−𝑎)𝑓 (𝑎, 𝑢(ℎ1(𝑎))) −
∫ 𝑎
−∞
𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠 +
∫ 𝑎
−∞S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠.
On the other hand, for𝑡≥𝑎we obtain
∫ 𝑡 𝑎
S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠 = −
∫ 𝑡
−∞S𝛽(𝑎−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠−
∫ 𝑎
−∞S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠
= 𝑢(𝑠) + 𝑓 (𝑡 , 𝑢(ℎ1(𝑡))) +
∫ 𝑡
−∞
𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠
−S𝛽(𝑡−𝑎) [𝑢(𝑎) + 𝑓 (𝑎, 𝑢(ℎ1(𝑎)))] −
∫ 𝑎
−∞
𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠
= 𝑢(𝑠) + 𝑓 (𝑡 , 𝑢(ℎ1(𝑡))) +
∫ 𝑡
𝑎
𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠
−S𝛽(𝑡−𝑎) [𝑢(𝑎) + 𝑓 (𝑎, 𝑢(ℎ1(𝑎)))]. Therefore, we concluded that
𝑢(𝑡) = S𝛽(𝑡−𝑎) [𝑢(𝑎) + 𝑓 (𝑎, 𝑢(ℎ1(𝑎)))] −𝑓 (𝑡 , 𝑢(ℎ1(𝑡))) −
∫ 𝑡
𝑎
𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠 +
∫ 𝑡 𝑎
S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠 is a mild solution of Eq.(1.1) and𝑢∈ PAA (X).
Theorem 3.6. Assume the conditions (H1).2, (H2).2 and (H4) are satisfied, then the problem (1.1) has a unique weighted pseudo almost automorphic mild solution onRprovide that
(3.8) 𝐿:=k (−𝒜)𝛼k L(2)𝑓 +𝚯1−𝛼𝛿−𝛼Γ(𝛼) L(2)𝑓 +𝚯 𝛿
L𝑔(2)<1 whereΓ(·)is the gamma functions.
Proof. For proof of this result, we consider the following𝚵:WPAA (R, 𝜌) →𝐶(R,X)as 𝚵𝑢(𝑡) = − (−𝒜)−𝛼(−𝒜)𝛼 𝑓 (𝑡 , 𝑢(ℎ1(𝑡))) +
∫ 𝑡
−∞
(−𝒜)1−𝛼S𝛽(𝑡−𝑠) (−𝒜)𝛼 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠 +
∫ 𝑡
−∞S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠
= −𝑓 (𝑡 , 𝑢(ℎ1(𝑡))) −
∫ 𝑡
−∞
𝒜S𝛽(𝑡−𝑠)𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠+
∫ 𝑡
−∞S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠 for𝑡∈R.
Step 1. 𝚵𝑢(𝑡)is well defined and continuous.
The same argument used in the proof Theorem3.6we can prove that𝚵𝑢(𝑡)is well defined and continuous. Moreover, from Lemma3.4, Lemma2.13, Lemma3.1and Lemma2.14, we infer that𝚵𝑢(𝑡) ∈ WPAA (R, 𝜌), that is,𝚵maps WPAA (R, 𝜌)into itself.
Step 2. 𝚵is a contraction onWPAA (R, 𝜌).
Indeed, for each𝑡∈R,𝑢, 𝑣∈ WPAA (R, 𝜌), yields
|𝚵𝑢(𝑡) −𝚵𝑣(𝑡) | ≤ |𝑓 (𝑡 , 𝑢(ℎ1(𝑡))) − 𝑓 (𝑡 , 𝑣(ℎ1(𝑡))) | +
∫ 𝑡
−∞
𝒜S𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠))) −𝒜S𝛽(𝑡−𝑠)𝑓 (𝑠, 𝑣(ℎ1(𝑠))) 𝑑𝑠 +
∫ 𝑡
−∞
S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠))) −S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑣(ℎ2(𝑠))) 𝑑𝑠
≤ k (−𝒜)−𝛼k k (−𝒜)−𝛼 𝑓 (𝑡 , 𝑢(ℎ1(𝑡))) − (−𝒜)−𝛼 𝑓 (𝑡 , 𝑢(ℎ2(𝑡))) k +
∫ 𝑡
−∞
(−𝒜)1−𝛼𝑆𝛽(𝑡−𝑠)
k (−𝒜)𝛼 𝑓 (𝑠, 𝑢(ℎ1(𝑠))) − (−𝒜)𝛼 𝑓 (𝑠, 𝑣(ℎ1(𝑠))) k𝑑𝑠 +
∫ 𝑡
−∞
𝚯𝑒−𝛿(𝑡−𝑠)|𝑔(𝑠, 𝑢(ℎ2(𝑠))) −𝑔(𝑠, 𝑣(ℎ2(𝑠))) |𝑑𝑠
≤ k (−𝒜)−𝛼k𝐿(2)
𝑓 |𝑢(ℎ1(𝑡)) −𝑣(ℎ1(𝑡)) | +L(2)𝑓
∫ 𝑡
−∞
𝚯1−𝛼𝑒−𝛿(𝑡−𝑠)|𝑢(ℎ1(𝑠)) −𝑣(ℎ1(𝑠)) |𝑑𝑠 +L𝑔(2)
∫ 𝑡
−∞
𝑀 𝑒−𝛿(𝑡−𝑠)|𝑢(ℎ2(𝑠)) −𝑣(ℎ2(𝑠)) |𝑑𝑠
≤ k (−A)−𝛼k L(𝑓2)k𝑢−𝑣kW P A A (R, 𝜌)+ L(𝑓2)𝚯1−𝛼𝛿−𝛼Γ(𝛼) k𝑢−𝑣kW P A A (R, 𝜌) +L𝑔(2)𝑀 𝛿−1k𝑢−𝑣kW P A A (R, 𝜌)
= n
k (−𝒜)−𝛼k L(𝑓2)+ L(𝑓2)𝚯1−𝛼𝛿−𝛼Γ(𝛼) + L𝑔(2)𝚯𝛿−1 o
k𝑢−𝑣kW P A A (R, 𝜌)
= 𝐿k𝑢−𝑣kW P A A (R, 𝜌). In this sense, we obtain
k𝚵𝑢(𝑡) −𝚵𝑣(𝑡) kW P A A (
R, 𝜌)≤𝐿k𝑢−𝑣kW P A A (
R, 𝜌).
It follows that𝚵is a contraction from (3.8). By the contraction principle, there exists a unique fixed point𝑢(·)for 𝚵inWPAA (R, 𝜌)such that𝚵𝑢=𝑣. Moreover, using the same proof as in Theorem3.6, we can see that
𝑢(𝑡) = S𝛽(𝑡−𝑎) (𝑢(𝑎) + 𝑓 (𝑎, 𝑢(ℎ1(𝑎)))) −𝑓 (𝑎, 𝑢(ℎ1(𝑡)))
−
∫ 𝑡
𝑎
AS𝛽(𝑡−𝑠) 𝑓 (𝑠, 𝑢(ℎ1(𝑠)))𝑑𝑠+
∫ 𝑡
𝑎
S𝛽(𝑡−𝑠)𝑔(𝑠, 𝑢(ℎ2(𝑠)))𝑑𝑠 is a mild solution of Eq.(1.1) and𝑢∈ WPAA (R, 𝜌).
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