Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 30, pp. 1–12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu S-ASYMPTOTICALLY ω-PERIODIC MILD SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS DARIN BRINDLE, GASTON M. N’GUÉRÉKATA Abstract. This article concerns the existence of mild solutions to the semi- linear fractional differential equation Dαt u(t) = Au(t) +Dα−1 t f(t, u(t)), t ≥ 0 with nonlocal conditions u(0) = u0 + g(u) where Dαt (·) (1 < α < 2) is the Riemann-Liouville derivative, A : D(A) ⊂ X → X is a linear densely defined operator of sectorial type on a complex Banach space X, f : R+ × X → X is S-asymptotically ω-periodic with respect to the first variable. We use the Krsnoselskii’s theorem to prove our main theorem. The results obtained are new even in the context of asymptotically ω-periodic functions. An application to fractional relaxation-oscillation equations is given. 1. Introduction Consider the semilinear fractional differential equation with non-local conditions, Dα t u(t) = Au(t) +Dα−1 t f(t, u(t)), 1 < α < 2, t ≥ 0, (1.1) u(0) = u0 + g(u) , (1.2) where A : D(A) ⊂ X → X is a linear densely defined operator of sectorial type on a complex Banach space X,u0 ∈ X and Dα t (·) is the Riemann-Liouville derivative, and g : C → C is a continuous mapping. In 2012, Zhao, Chang and N’Guérékata [33] showed that there exists a mild solu- tion u(t) that is asymptotically almost automorphic. We assume that the semilinear function f is asymptotically almost automorphic. We show here that there exists a mild solution u(t) that is S-asymptotically ω-periodic, if the semilinear function. We assume that f is S-asymptotically ω-periodic function, a concept introduced in 2008, by Henriquez, Pierri and Tabos [19]. Both sets containing each of these type functions also contains the set of asymptotically ω-periodic functions. Cuevas and de Souza [9] proved the existence and uniqueness of an S-asymptotically ω-periodic solution of an equivalent problem with local conditions assuming a Lipschitz condi- tion. Our results consider non-local conditions and provide assumptions where the Lipschitz condition is not necessary. 2010 Mathematics Subject Classification. 34G20, 34G10. Key words and phrases. S-asymptotically ω-periodic sequence; fractional semilinear differential equation. c©2020 Texas State University. Submitted August 11, 2019. Published April 7, 2020. 1 2 D. BRINDLE, G. M. N’GUÉRÉKATA EJDE-2020/30 Many real world phenomena can be described very successfully by models using mathematical tools of fractional calculus, such as dielectric polarization, electrode- electrolyte polarization, electromagnetic waves, modeling of earthquakes, fluid dy- namics, traffic models, measurements of viscoelastic material properties and vis- coplasticity; see [1, 9] and references therein. A fractional oscillator equation is a generalization of the classical harmonic os- cillator equation by replacing the second-order derivative by a fractional order de- rivative; that is Dα t u(t) + c2u(t) = f(t), 1 < α < 2, t ≥ 0, c ∈ R. Damping effects can be expanded to fractional relaxation-oscillation and diffusion- wave phenomena, which include generalized equations (1.1) and (1.2); see [3, 8, 22] The paper is organized as follows. In Section 2, we recall some properties of S-asymptotically ω-periodic functions and derive a variation of constants formula. In Section 3 we prove our main results and present an example in Section 4. 2. Preliminaries In what follows, (X, ‖ · ‖) will denote a complex Banach space, BC(R+, X) will be the space of all bounded and continuous functions f : R+ → X, C0(R+, X) the space of all continuous functions f : R+ → X such that limt→∞ ‖f(t)‖ = 0. Both spaces are Banach spaces equipped with the supremum norm. 2.1. S-asymptotically ω-periodic functions. Definition 2.1 (Fréchet). Let g ∈ BC(R+, X) and ω > 0. We say that a continu- ous and bounded function f : [0,∞)→ X is asymptotically ω-periodic if it admits the decomposition f = g + h, where g ∈ Pω(X) and h ∈ C0(R+, X). The set of all such functions is denoted: APω(X) := Pω(X)⊕ C0(R+, X). Definition 2.2 ([19]). A function f ∈ BC(R+, X) is said to be S-asymptotically ω-periodic if there exists ω > 0 such that lim t→∞ (f(t+ ω)− f(t)) = 0 In this case we say that ω is an asymptotic period of f . The set of all such functions is denoted by SAPω(X). Additionally, if we set the shift operator Πω : BC(R+, X) → BC(R+, X) with Πωf(t) = f(t+ ω), then SAPω(X) = (Πω − I)−1C0(R+, X). Remark 2.3 ([19]). It is easy to check that APω(X) ⊂ SAPω(X). The inclusion is strict. Indeed we have the following example. Example 2.4 ([19]). Let f : R+ → c0 where c0 = {x = (xn)n∈N : limn→∞ xn = 0} equipped with the norm ‖x‖ = supn∈N |x(n)|, and( f(t) = 2tn n2 + t2 ) n∈N . It is clear that f(t) is uniformly continuous and f ∈ SAPω(X). But f /∈ APω(X), because even though each coordinate fn ∈ APω(X), fn(n) = 1 ⇒ ‖f(n)‖ = 1 EJDE-2020/30 S-ASYMPTOTICALLY ω-PERIODIC SECTORIAL SOLUTIONS 3 for all n ∈ N on this infinite dimensional space. Therefore, there does not exist h(t) such that limt→∞ ‖h(t)‖ = 0. The function f above is a piecewise continuous function that is bounded and non-convergent. Other examples of S-asymptotically ω-periodic functions can be found in [5, 31]. It is proved in [19] that SAPω(X) the space of all S-asymptotically ω-periodic functions on X is a Banach space if equipped with the supremum norm. Definition 2.5 ([19, 31]). A continuous function f : [0,∞) × X → X is said to be uniformly S-asymptotically ω-periodic on bounded sets if for every bounded set K ⊂ X, the set {f(t, x) : t ≥ o, x ∈ K} is bounded and limt→∞ ‖f(t + ω, x) − f(t, x)‖ = 0 uniformly in x ∈ K. Definition 2.6 ([19, 31]). A continuous function f : [0,∞)×X → X is said to be asymptotically uniformly continuous on bounded sets if for every ε > 0 and every bounded set K ⊂ X, there exist Lε,K > 0, δε,K > 0 such that for every t > Lε,K ‖f(t, x)− f(t, y)‖ < ε and for every x, y ∈ K such that ‖x− y‖ < δε,K . Lemma 2.7 ([5, 19]). If f : [0,∞) × X → X is a function which is uniformly S-asymptotically ω-periodic and asymptotically uniformly continuous on bounded sets and u(t) ∈ SAPω(X), then the Nemytski operator N (·) := f(·, u(·)) is also in SAPω(X). Proof. Let K = R(u) be the closure of the range of the function u. Since R(u) is a bounded set, it follows that ∑ (·) is a bounded function. It is also obviously continuous. Let ε > 0. From 2.5, there exists T > 0 such that for all t > T , ‖f(t+ ω, u(t+ ω))− f(t, u(t+ ω))‖ < ε 2 . From Definition 2.6, there exists δε,K > 0, Lε,K > 0 such that for all t > Lε,K > 0, ‖f(t, u(t+ ω))− f(t, u(t))‖ < ε 2 , if ‖u(t+ω)−u(t)‖ < δε,K . Let t > max{T, Lε,K}. Then combining all of the above, gives ‖f(t+ ω, u(t+ ω))− f(t, u(t))‖ ≤ ‖f(t+ ω, u(t+ ω))− f(t, u(t+ ω))‖+ ‖f(t, u(t+ ω))− f(t, u(t))‖ < ε 2 + ε 2 . The proof is complete. � 2.2. A variation of constants formula. Let us recall sectorial operators: Definition 2.8 ([7, 32]). A closed and linear operator A is said to be sectorial if there exist 0 < θ < π 2 , M > 0 and τ ∈ R such that its resolvent exists outside the sector τ + Sθ := {τ + λ : λ ∈ C, | arg(−λ)| < τ} and ‖(λ−A)−1‖ ≤ M |λ− τ | , λ /∈ τ + Sθ, where A generates a family of strongly continuous operators Eα : R+ → B(X) defined as Eα(t) := 1 2πi ∫ φ etλ(λα −A)−1λα−1dλ are on a suitable path φ outside the sector τ + Sθ. 4 D. BRINDLE, G. M. N’GUÉRÉKATA EJDE-2020/30 Theorem 2.9 ([2, 7, 33]). The equation Dα t u(t) = Au(t) +Dα−1 t f(t, u(t)), 1 < α < 2, t ≥ 0, u(0) = u0 + g(u), where A is sectorial with 0 < θ < π(1− α 2 ) < π/2, has a mild solution generated by A: u(t) = Eα(t)[u0 + g(u)] + ∫ t 0 Eα(t− s)f(s, u(s)) ds, 0 ≤ t ≤ T, Proof. By applying the definition of the Riemann-Liouville derivative, Dα t (r(t)) = dm dtm ∫ t 0 (t− s)m−α−1 Γ(m− α) r(s) ds, m− 1 < α < m , to equation (1.1) after using the Riemann-Liouville derivative D1−α t (·) on both sides of equation (1.1) with β = 1− α and since m = 2 (m = 0 for β), Dα t u(t) = Au(t) +Dα−1 t f(t, u(t)), 1 < α < 2, t ≥ 0, u(0) = u0 + g(u) implies u′(t) = ∫ t 0 (t− s)−β−1 Γ(−β) Au(s) ds+ f(t, u(t)), −1 < β < 0, t ≥ 0, u(0) = u0 + g(u) , which implies u′(t) = ∫ t 0 (t− s)α−2 Γ(α− 1) Au(s)ds+ f(t, u(t)), 1 < α < 2, t ≥ 0 (2.1) u(0) = u0 + g(u) . (2.2) Then integrating by t, we have u(t) = u0 + g(u) + ∫ t 0 (t− s)α−1 Γ(α) Au(s)ds+ ∫ t 0 f(s, u(s))ds (2.3) for 1 < α < 2 and t ≥ 0. Now we use Laplace transforms r̂(λ) = ∫∞ 0 eiλtr(t)dt to find the sectorial resol- vent and its mild solution. The Laplace transform of equation 2.3 is û(λ) = u0 + g(u) λ + 1 λα Aû(λ) + 1 λ f̂(λ, û(λ)). Then û = [(λα −A)−1λα−1](u0 + g(u) + f̂). Let Êα(λ) = [(λα −A)−1λα−1]. Then there exists the mild solution u(t) = Eα(t)[u0 + g(u)] + ∫ t 0 Eα(t− s)f(s, u(s))ds, 0 ≤ t ≤ T, where the family of sectorial operators Eα(t) := 1 2πi ∫ φ etλ(λα −A)−1λα−1dλ are on a suitable path φ outside the sector τ + Sθ. � The previous proof connects theorems and lemmas from references [7, 33], and shows that (2.1), (2.2) is equivalent to (1.1), (1.2). EJDE-2020/30 S-ASYMPTOTICALLY ω-PERIODIC SECTORIAL SOLUTIONS 5 Lemma 2.10 ([2, 7, 33]). Let A : D(A) ⊂ X → X be a sectorial operator in a complex Banauch space x satisfying τ + Sθ := {τ + λ : λ ∈ C, | arg(−λ)| < τ} and ‖(λ−A)−1‖ ≤ M |λ− τ | , λ /∈ τ + Sθ for some M > 0, τ < 0 and 0 < θ < π(1− α 2 ) < π/2. Then there exists C > 0 such that ‖Eα(t)‖B(X) ≤ CM 1 + |τ |tα , t ≥ 0 . Theorem 2.11 (Krasnosel’skii fixed point theorem). Let M be a closed convex and non-empty subset of a Banach space X and A,B two operators such that (i) Ax+By ∈M whenever x, y ∈M ; (ii) A is compact and continuous (iii) B is a contraction mapping. Then there exists z ∈M such that z = Az +Bz. 3. Main results Lemma 3.1. Suppose h(t) ∈ SAPω(X). Then the function F : [0,∞)→ X defined by F (t) := ∫ t 0 Eα(t− ξ)h(ξ)dξ, is also in SAPω(X), where the family of operators generated by the sectorial oper- ator A, Eα(t) := 1 2πi ∫ φ etλ(λα −A)−1λα−1dλ, 1 < α < 2, are on a suitable path φ outside the sector τ + Sθ, (as in Definition 2.5). Proof. Let us write F (t+ ω)− F (t) = ∫ t+ω 0 Eα(t+ ω − ξ)h(ξ)dξ − ∫ t 0 Eα(t− ξ)h(ξ)dξ = ∫ t −ω Eα(t− ξ)h(ξ + ω)dξ − ∫ t 0 Eα(t− ξ)h(ξ)dξ = ∫ t −ω Eα(t− ξ)[h(ξ + ω)− h(ξ)]dξ + ∫ 0 −ω Eα(t− ξ)h(ξ)dξ. Let ε > 0 be given. Since h(t) ∈ SAPω(X), there exists T > 0 such that for every ξ > T , we have ‖h(ξ + ω)− h(ξ)‖ < ε. This implies ‖F (t+ ω)− F (t)‖ ≤ ∫ T −ω ‖Eα(t− ξ)[h(ξ + ω)− h(ξ)]‖dξ + ∫ t T ‖Eα(t− ξ)[h(ξ + ω)− h(ξ)]‖dξ + ∫ 0 −ω ‖Eα(t− ξ)h(ξ)‖dξ ≤ 2‖h‖∞ ∫ T −ω ‖Eα(t− ξ)‖dξ + ε ∫ t T ‖Eα(t− ξ)‖dξ + ‖h‖∞ ∫ 0 −ω ‖Eα(t− ξ)‖dξ 6 D. BRINDLE, G. M. N’GUÉRÉKATA EJDE-2020/30 ≤ 2‖h‖∞ ∫ t+ω t−T ‖Eα(ξ)‖dξ + ε ∫ t−T 0 ‖Eα(ξ)‖dξ + ‖h‖∞ ∫ t+ω t ‖Eα(ξ)‖dξ ≤ 3‖h‖∞ ∫ t+ω t−T ‖Eα(ξ)‖dξ + ε ∫ ∞ 0 ‖Eα(ξ)‖dξ ≤ 3‖h‖∞ ∫ t+ω t−T CM 1 + |τ |ξα dξ + ε ∫ ∞ 0 CM 1 + |τ |ξα dξ ≤ 3‖h‖∞(T + ω) CM 1 + |τ |(t− T )α + ε CM |τ | −1 α π/α sin(π/α) , where the constants C > 0, M > 0, and τ < 0 are given by Lemma 2.10. Thus ‖F (t+ ω)− F (t)‖ → 0 as t→∞. The proof is now complete. � We use the following assumptions: (A1) The operator A is of sectorial of type τ < 0, which generates a strongly continuous family of linear operators Eα(t)t≥0 ⊂ B(X). (A2) f : [0,∞) × X → X is a function which is uniformly S-asymptotically ω-periodic and asymptotically uniformly continuous on bounded sets. (A3) There exists Lf > 0 such that ‖f(t, x) − f(t, y)‖ < Lf‖x − y‖, for all t ≥ 0, x, y ∈ X. (A3’) There exists cf > 0 such that ‖f(t, x)‖ < cf (1 + ‖x‖) for all t ≥ 0, (A4) There exists Lg > 0 such that for all u, v ∈ C := BC([0,∞), X) → C, ‖g(u)− g(v)‖ < Lg‖u− v‖∞. We assume CMLg < 1. Remark 3.2. It is clear that (A3) implies (A3’). Indeed by (A3), we obtain ‖f(x)‖ ≤ ‖f(x)− f(0)‖+ ‖f(0)‖ ≤ Lf‖x‖+ ‖f(0)‖ ≤ cf (‖x‖+ 1) where cf = max{Lf , ‖f(0)‖}. Now we state and prove our first result. Theorem 3.3. Under assumptions (A1)–(A4), (1.1)-(1.2) possesses a unique so- lution in SAPω(X) provided CM ( Lg + Lf |τ | −1 α π/α sin(π/α) ) < 1. Proof. Consider the operator Ω : SAPω(X)→ SAPω(X) defined by Ωu(t) := Eα(t)[u0 + g(u)] + ∫ t 0 Eα(t− ξ)f(ξ, u(ξ))dξ. In view of Lemmas 2.7 and 3.1, Ω is well-defined. Now if u, v ∈ SAPω(X), we obtain ‖(Ωu)(t)− (Ωv)(t)‖ ≤ ‖Eα(t)‖‖g(u)− g(v)‖+ ∫ t 0 ‖Eα(t− ξ)‖‖f(ξ, u(ξ))− f(ξ, v(ξ))‖dξ ≤ ( CM 1 + |τ |tα Lg + Lf ∫ t 0 CM 1 + |τ |ξα dξ ) ‖u− v‖∞ ≤ CM ( Lg + Lf |τ | −1 α π/α sin(π/α) ) ‖u− v‖∞ . Therefore ‖Ωu− Ωv‖∞ ≤ γf,g,α‖u− v‖∞, where γf,g,α = CM ( Lg + Lf |τ | −1 α π/α sin(π/α) ) < 1 . EJDE-2020/30 S-ASYMPTOTICALLY ω-PERIODIC SECTORIAL SOLUTIONS 7 We conclude the existence of a unique solution using the Banach’s fixed point theorem. � Remark 3.4. When equation (1.2) is the local condition g(u) = 0, we recover the results by Cuevas and de Souza [9]. Theorem 3.5. Assume (A1), (A2), (A3’), (A4). Then problem (1.1)-(1.2) has at least one mild solution u(t) ∈ SAPω(X) if we assume that Eα(t) is compact for any t > 0. Proof. Note that (A4) implies the existence a constant cg > 0 such that ‖g(u)‖ ≤ cg(1 + ‖u‖) for any u ∈ BC([0,∞), X), as in Remark 3.2. We consider the same operator Ω as in the previous theorem and use several steps to achieve our conclusion. Step 1. Let Bρ := {u ∈ SAPτ (X) : ‖u‖∞ ≤ ρ}, where ρ > max { CM ( α sin(π/α)cg + cf |τ |−1/απ ) α sin(π/α)− CM ( α sin(π/α) + α sin(π/α)cg + cf |τ |−1/απ ) , 0 } Define the operators P,Q : SAPτ (X)→ SAPτ (X) by (Pv)(t) : Eα(t)[v0 + g(v)], (Qu)(t) := ∫ t 0 Eα(t− ξ)f(ξ, u(ξ))dξ . Using (A3’) we obtain ‖(Pv)(t) + (Qu)(t)‖ ≤ ‖Eα(t)‖‖u0 + g(v)]‖+ ∫ t 0 ‖Eα(t− ξ)f(ξ, u(ξ))‖dξ ≤ CM ( 1 1 + |τ |tα (‖v0‖+ ‖g(v)‖) + ∫ t 0 1 1 + |τ |(t− ξ)α ‖f(ξ, u(ξ))‖dξ ) ≤ CM ( ‖v0‖+ ‖g(v)‖+ cf (1 + ‖u‖) ∫ t 0 1 1 + |τ |ξα dξ ) ≤ CM ( ‖v0‖+ cg(1 + ‖v‖) + cf (1 + ‖u‖)|τ | −1 α π/α sin(π/α) ) ≤ CM [ ρ+ ( cg + cf |τ |−1/απ α sin(π/α) ) (1 + ρ) ] ≤ ρ. We conclude that For all u, v ∈ Bρ, Pv +Qv ∈ Bρ. Step 2. The operator P is contractive. Indeed, for u, v ∈ SAPτ (X) we have ‖(Pu)(t) + (Pv)(t)‖ ≤ ‖Eα(t)‖‖g(u)− g(v)‖ ≤ CM 1 1 + |τ |tα Lg‖u− v‖∞ . Therefore ‖Pu− Pv‖∞ ≤ CMLg‖u− v‖∞ . We conclude by using the assumption CMLg < 1. Step 3. The operator Q is continuous on Bρ. Let (un) ⊂ Bρ such that un → u in Bρ. Then in view of Definition 2.6, f(ξ, un(ξ)) → f(ξ, u(ξ)) as n → ∞ for all 8 D. BRINDLE, G. M. N’GUÉRÉKATA EJDE-2020/30 ξ ∈ [0,∞). Now we have ‖(Qun)(t)− (Qu)(t)‖ = ‖ ∫ t 0 Eα(t− ξ)[f(ξ, un(ξ))− f(ξ, u(ξ))]dξ‖ ≤ CM ∫ t 0 1 1 + |τ |(t− ξ)α [‖f(ξ, un(ξ))‖+ ‖f(ξ, u(ξ))‖]dξ ≤ CMcf ∫ t 0 1 1 + |τ |ξα [2 + ‖un(ξ)‖+ ‖u(ξ)‖]dξ ≤ 2CMcf (1 + ρ)|τ |−1/α π α sin(π/α) ≤ 2CMcf (1 + ρ)|τ |−1/απ α sin(π/α) <∞ . Therefore, Qun → Qu as n → ∞ by the Lebesgues’s Dominated Convergence Theorem. Step 4. The set (Qun) where (un) ⊂ Bρ is uniformly bounded. Indeed for all n, we have ‖(Qun)(t)‖ = ‖ ∫ t 0 Eα(t− ξ)f(ξ, un(ξ)dξ‖ ≤ CM ∫ t 0 1 1 + |τ |(t− ξ)α ‖f(ξ, un(ξ))‖dξ ≤ CMcf ∫ t 0 1 1 + |τ |ξα [1 + ‖un(ξ)‖]dξ ≤ CMcf (1 + ρ)|τ |−1/α π α sin(π/α) ≤ CMcf (1 + ρ)|τ |−1/απ α sin(π/α) . This shows that (Qun) is uniformly bounded. Step 5. (Qun) with (un) ⊂ Bρ is equicontinuous. Indeed taking t1, t2 such that 0 ≤ t1 < t2, we have ‖(Qun)(t1)− (Qun)(t2)‖ = ‖ ∫ t1 0 Eα(t1 − ξ)f(ξ, un(ξ))dξ − ∫ t2 0 Eα(t2 − ξ)f(ξ, un(ξ))dξ‖ = ‖ ∫ t1 0 [Eα(t2 − ξ)− Eα(t1 − ξ)]f(ξ, un(ξ))dξ − ∫ t2 t1 Eα(t2 − ξ)f(ξ, un(ξ))dξ‖ ≤ ‖ ∫ t1 0 [Eα(t2 − ξ)− Eα(t1 − ξ)]f(ξ, un(ξ))dξ‖+ ‖ ∫ t2 t1 Eα(t2 − ξ)f(ξ, un(ξ))dξ‖ ≤ CMcf (∫ t1 0 ( 1 1 + |τ |(t2 − ξ)α − 1 1 + |τ |(t1 − ξ)α )[1 + ‖un(ξ)‖]dξ + ∫ t2 t1 1 1 + |τ |(t2 − ξ)α [1 + ‖un(ξ)‖]dξ ) ≤ CMcf (1 + ρ) (∫ t1 0 1 1 + |τ |(t2 − ξ)α dξ − ∫ t1 0 1 1 + |τ |(t1 − ξ)α dξ EJDE-2020/30 S-ASYMPTOTICALLY ω-PERIODIC SECTORIAL SOLUTIONS 9 + ∫ t2 t1 1 1 + |τ |(t2 − ξ)α dξ ) ≤ CMcf (1 + ρ) (∫ t2 0 1 1 + |τ |(ξ)α dξ − ∫ t1 0 1 1 + |τ |(ξ)α dξ ) ≤ CMcf (1 + ρ) ∫ t1 t2 1 1 + |τ |(ξ)α dξ < 2CMcf (1 + ρ) α |τ |−1/α π α sin(π/α) <∞ . Since lim t1→t2 [ CMcf (1 + ρ) α ∫ t1 t2 1 1 + |τ |(ξ)α dξ] = 0, we conclude the equicontinuity of (Qun). Step 6. Q is compact. First, we show that the set {(Qu)(t) : u(t) ∈ Bρ} is relatively compact in X for each t > 0. To this end, fix t > 0 and ε0 such that 0 < ε0 < t. We have{ (Qε0u)(t) := ∫ t−ε0 0 Eα(t− ε0 − ξ)f(ξ, u(ξ))dξ } is uniformly bounded for u ∈ Bρ. This with the assumption that Eα(ε0) is compact yield the set {Eα(ε0)(Qε0u)(t) : u ∈ Bρ} is relatively compact. Since from Definition 2.5, Eα(0) = I and Eα(t)x is continuous for every x ∈ X, we obtain R(ε0)(Qε0u)(t) = Eα(ε0) ∫ t−ε0 0 Eα(t− ε0 − ξ)f(ξ, u(ξ))dξ}, which shows that lim ε0→0 Eα(ε0)(Qε0u)(t) = (Qu)(t) . We conclude that {(Qu)(t) : u(t) ∈ Bρ} is relatively compact in X. Finally, Q is compact as claimed. From all of the above, we conclude that problem (1.1)-(1.2) has at least one mild solution u(t) ∈ SAPω(X), using the Krasnosel’ski’s fixed point theorem. � These results are new even in the context of asymptotically ω-periodic functions. 4. An Example As an application, we investigate the following fractional relaxation-oscillation equations, that are similar to those introduced in [2, 9, 33]. Example 4.1. Dα t u(t, x) = ∂2 ∂x2 u(t, x)− µv(t, x) +Dα−1 t ( βu(t, x)(cos t+ cos(3t)) + β(−1)n[ln(1 + t)− (2n+ 1)] sin(u(t, x)) ) , for e2n − 1 ≤ t ≤ e2n+2 − 1, n ∈ N , u(t, 0) = u(t, π) = 0, 1 < α < 2, t ≥ 0, x ∈ [0, π], u(0, η) = u0(η) + g(u), η ∈ [0, π], 10 D. BRINDLE, G. M. N’GUÉRÉKATA EJDE-2020/30 where u0 ∈ L2[0, π]. Let X = (L2[0, π]; ‖ · ‖2), define the linear operator A defined on X by Au = u′′ − µu, (µ > 0) with domain D(A) := {u ∈ X : u′′ ∈ X,u(0) = u(π) = 0} . Also, let g(u) be a function that satisfies (A4). It is well-known that ∆u = u′′ is an infinitesimal generator of a analytic semigroup on L2[0, π]; then A is a sectorial of type τ = −µ. The equations above can be formulated into (1.1)-(1.2) where u(t) = u(t, ·). Let us consider the nonlinearity, for all u ∈ X, t ≥ 0, s ∈ [0, π] and β ∈ R with u ∈ SAP2π. Therefore two cases follow. Case 1. ‖f(t, u(s))− f(t, v(s))‖ = ‖β(u(s)− v(s))(cos t+ cos(3t)) + β(−1)n[ln(1 + t)− (2n+ 1)](sin(u(s))− sin(v(s)))‖ ≤ |β| (2‖u(s)− v(s)‖∞ + ‖ sin(u(s))− sin(v(s))‖∞). Therefore, ‖f(t, u(s))− f(t, v(s))‖ ≤ 3 |β| ‖u(s)− v(s)‖∞, or ‖f(t, u(s))− f(t, v(s))‖ ≤ 3 |β| ‖ sin(u(s))− sin(v(s))‖∞ . In either inequality, we assume |β| < |µ| 1α sin(π/α) π/α 1− CMLg 3CM ; when by Theorem 3.3, problem (1.1)-(1.2) has a unique S-asymptotically 2π-periodic solution. Case 2. Since ‖f(t, u(s))‖ = ‖β(u(s))(cos t+ cos(3t)) + β(−1)n[ln(1 + t)− (2n+ 1)](sin(u(s))‖ ≤ |β| (2‖u(s)‖∞ + ‖ sin(u(s))‖∞) ≤ 3|β|(1 + ‖u(s)‖∞) ⇒ ∃cf = 3|β|, by Theorem 3.5, problem (1.1)-(1.2) has at least one S-asymptotically 2π-periodic solution. References [1] R. P. Agarwal, B. Andrade, C. Cuevas; On Type of Periodicity and Ergodicity to a Class of Fractional Order Differential Equations, Advances in Difference Equations, Vol. 2010, Article number: 179750 (2010). [2] R. P. Agarwal, B. Andrade, C. Cuevas; Weighted psuedo-almost periodic solutions of semilin- ear fractional differential equations, Nonlinear Anal. Real World Appl., 11 (2010), 3532–3554. [3] K. Balachandran, V. Govindaraj, M. Rivero, J. J. Trujillo; Controllability of fractional damped dynamical systems, Math. of Comp., 257 (2015), 66–73. [4] J. Blot, P. Cieutat, G. M. N’Guérékata; S-asymptotically ω-periodic functions and appli- cations to evolution equations, African Diaspora Journal of Mathematics, New Series, 12 (2009), 113–121. [5] D. Brindle, G. M. N’Guérékata; Existence results of S-asymptotically τ -periodic mild solu- tions to some integrodifferential equations, PanAmerican Mathematics Journal, 29 (2019) No. 2, 63–74. [6] J. Cao, Z. Huang, G. M. N’Guérékata; Existence of asymptotically almost automorphic mild solutions for nonautonomous semilinear evolution equations, Elect. J. Diff. Equ., Vol. 2018 (2018), No. 37, pp. 1–16. EJDE-2020/30 S-ASYMPTOTICALLY ω-PERIODIC SECTORIAL SOLUTIONS 11 [7] E. Cuesta; Asymptotic behavior of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst. (Suppl.) (2007), 277–285. [8] E. Cuesta, C. Lubich, C. Palencia; Convolution Quadrature Time Discretion of Fractional Diffusion-wave Equations, Math. of Comp., 254 (2006) ,673–696. [9] C. Cuevas, J. C. de Souza; Existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Analysis: Theory, Methods and Applications, Vol. 72, 3–4, Feb (2010), 1683–1689. [10] K. Deng; Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993) 630–637. [11] W. Desch, R. Grimmer, W.Schappacher; Some Considerations for linear integro-differential equations, J. Math. Anal. Appl., 104 (1984) 219–234. [12] W. Dimbour, G. M. N’Guérékata; On S-asymptotically ω-periodic solutions to some classes of partial evolution equations, J. Math. Anal. Appl., 343 (2008), 1119–1130. [13] W. Dimbour, S.M. Manou-Abi; S-asymptotically ω-periodic solution for a nonlinear differ- ential equation with piecewise constant argument via S-asymptotically ω-periodic functions in the Stepanov sense, J. Nonlinear Syst. Appl., 7 (2018), no. 1, 14–20. [14] W. Dimbour, S. M. Manou-Abi; Asymptotically ω-periodic functions in the Stepanov sense and its application for an advanced differential equation with piecewise constant argument in a Banach space, Mediterr. J. Math., 15 (2018), no. 1, Art. 25, 18. [15] W. Dimbour, J.-C. Mado; S-asymptotically ω-periodic solution for a nonlinear differential equation with piecewise constant argument in a Banach space, Cubo 16 (2014), no. 3, 55–65. [16] W. Dimbour, G. Mophou, G. M. N’Guérékata; S-asymptotically ω-periodic solutions for partial differential equations with finite delay, Electronic J. Diff. Equ., (2011), no. 117, 1–12. [17] H.-S. Ding, T.-J. Xiao, J. Liang; Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, J. Math. Anal. Appl., 338 (2008), 141–151. [18] R. Grimmer; Resolvent operators for integral equations in a Banach Space, Trans. Amer. Math. Soc., 273 (1982), 333–349. [19] H. R. Henriquez, M. Pierri, P. Tabos; On S-asymptotically ω-periodic functions on Banach spaces and applications, J. Math. Anal. Appl., 343 (2008), 1119–1130. [20] F. Li, J. Liang, H. Wang; S-asymptotically ω-periodic solutions for fractional differential equations of order q ∈ (0, 1) with finite delay, Adv. Difference Equ. 217, Paper No.83, 14 pp. [21] C. Lizama, G. N’Guérékata; Bounded Mild Solutions for Semilinear Integro Differential Equations in Banach Spaces, Integr. Equ. Oper. Theory, 68 (2010), 207–227. [22] F. Mainari; Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena, Chaos, Solitons and Fractals, Vol.7 No.9, (1996), 1461–1477. [23] R. K. Miller; Nonlinear Volterra Equations in Banach Spaces, W. A. Benjamin Inc. Philip- pines (1971). [24] V. N. Minh, G. M. N’Guérékata, R. Yuan; Lectures on the asymptotic behavior of solutions of differential equations, Nova Science Publishers Inc. New York (2008). [25] G. M. N’Guérékata; Quelques remarques sur les fonctions asymptotiquement presqu’automorphes, Ann. Math. Sci. Québec, VII (1983), 185–191. [26] G. M. N’Guérékata; A Cauchy problem for some fractional abstract differential equation with non local conditions, Nonlinear Analysis, 70 (2009), 1873–1876. [27] G. M. N’Guérékata; Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer, Amsterdam, 2001. [28] G. M. N’Guérékata; Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, Semigroup Forum, Vol. 69 (2004), No. 1, 80–89. [29] G. M. N’Guérékata; Topics in Almost Automorphy, Springer, New York, 2005. [30] G. M. N’Guérékata; Spectral Theory for Bounded Functions and Applications to Evolution Equations, Nova Science Publishers, Inc., New York, 2017. [31] E. R. Oueama-Guengai, G. M. N’Guérékata; S-asymptotically ω-periodic mild solutions to some fractional diferential equations in abstract spaces, Math. Meth. Appl. Sci. (2018), 1–7, https://doi.org/10.1002/mma.5062. [32] A. Pazy; Semigroups of Linear Operators and Applications to Differential Equations, (Ap- plied Mathematical Sciences; vol. 44), Springer-Verlag, New York, 1983. [33] J. Q. Zhao, Y. K. Chang, G. M. N’Guérékata; Asymptotically Behavior of Mild Solutions to Semilinear Fractional Differential Equations., J. Optim. Theory Appl., 156 (2013), 106–114 12 D. BRINDLE, G. M. N’GUÉRÉKATA EJDE-2020/30 5. Addendum posted on April 18, 2020 In response to a reader’s comments, we want to make the following corrections: (1) Change the title of subsection 2.2 to “Application of the Laplace transform and subsequent sectorial solutions” (2) Page 3 line -3: change “defined as” to “defined for {λα : Reλ > µ} ⊂ ρ(A) as” (3) Page 3 line -1: change “are on a suitable” to “which are on a suitable” (4) Page 4: line -2: delete “The previous proof . . . equivalent to (1.1), (1.2)” (5) Add the condition g(u) = −u0 to the assumptions of Theorem 2.9, and replace its proof by the following. Proof of Theorem 2.9. By applying the Riemann-Liouville derivative, D1−α t (·), to both sides of (1.1) with β = 1− α, and since m = 2 (m = 0 for β), from Dα t u(t) = Au(t) +Dα−1 t f(t, u(t)), 1 < α < 2, t ≥ 0, u(0) = u0 + g(u) = 0 we obtain u′(t) = ∫ t 0 (t− s)−β−1 Γ(−β) Au(s) ds+ f(t, u(t)), −1 < β < 0, t ≥ 0, u(0) = u0 + g(u) = 0 . Recall that the Riemann-Loiuville derivative is Dβ t (r(t)) = dm dtm ∫ t 0 (t− s)m−β−1 Γ(m− β) r(s) ds, m− 1 < β < m . Therefore, u′(t) = ∫ t 0 (t− s)α−2 Γ(α− 1) Au(s)ds+ f(t, u(t)), 1 < α < 2, t ≥ 0, (5.1) u(0) = u0 + g(u) = 0 . (5.2) Now we use Laplace transforms to find the sectorial resolvent and its mild solution. Since ∫ t 0 (t−s)α−2 Γ(α−1) Au(s)ds is a convolution, the Laplace transform of (5.1)–(5.2) is λû(λ)− ( u0 + g(u) ) = Aû(λ) λα−1 + f̂(λ, û(λ)) which implies û = [(λα −A)−1λα−1](u0 + g(u) + f̂). Let Êα(λ) = [(λα −A)−1λα−1]. Then we obtain the mild solution u(t) = Eα(t)[u0 + g(u)] + ∫ t 0 Eα(t− s)f(s, u(s))ds, t ≥ 0, where the family of sectorial operators Eα(t) := 1 2πi ∫ φ etλ(λα −A)−1λα−1dλ are defined on a suitable path φ outside the sector τ + Sθ. � End of addendum Darin Brindle Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA Email address: Darin.Brindle@morgan.edu EJDE-2020/30 S-ASYMPTOTICALLY ω-PERIODIC SECTORIAL SOLUTIONS 13 Gaston M. N’Guérékata Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA Email address: Gaston.N’Guerekata@morgan.edu 1. Introduction 2. Preliminaries 2.1. S-asymptotically -periodic functions 2.2. A variation of constants formula 3. Main results 4. An Example References 5. Addendum posted on April 18, 2020