Ding and Ahmad Advances in Difference Equations (2016) 2016:203 DOI 10.1186/s13662-016-0927-y

RESEARCH

Open Access

Analytical solutions to fractional evolution equations with almost sectorial operators Xiao-Li Ding1 and Bashir Ahmad2* *

Correspondence: [email protected] Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article 2

Abstract In this paper, with the aid of functional analysis, for almost sectorial operators and some fixed point theorems, we study the existence and uniqueness of mild solutions to fractional neutral evolution equations with almost sectorial operators. We also show that mild solutions can become strong and classical solutions under appropriate assumptions. Finally, we present an example to illustrate the applicability of our results. Keywords: fractional neutral evolution equations; almost sectorial operators; fixed point theorems; existence and uniqueness; mild solutions; strong solutions and classical solutions

1 Introduction Throughout this paper, by (X,  · ) we denote a Banach space. As usual, for a linear operator A, D(A), R(A), and σ (A) stand for the domain, range, and spectrum of A, respectively. Moreover, L(X) denotes the space of all bounded linear operators on X. A sectorial operator is a linear operator A in a Banach space whose spectrum lies in a closed sector Sω = {z ∈ C\{} | | arg z| ≤ μ} ∪ {} for some  ≤ ω < π and whose resolvent (z – A)– satisfies the estimate   (z – A)–  ≤ M|z|–

for all z ∈/ Sω .

(.)

Several elliptic differential operators considered in the spaces of continuous functions or Lebesgue spaces belong to the class of sectorial operators. Therefore, many PDEs with elliptic operators can be transformed into evolution equations with sectorial operators in a Banach space; for example, see [–]. In , Wahl [] first pointed out that the resolvent estimates of elliptic differential operators considered in spaces of regular functions, such as the spaces of Hölder continuous functions, do not satisfy estimate (.). However, such operators satisfy the following estimate for some – < γ < :   (z – A)–  ≤ M|z|γ

for all z ∈/ Sω .

Now we recall the following definition [–]. © 2016 Ding and Ahmad. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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γ

Definition . Let – < γ <  and  ≤ ω < π . By ω (X) we denote the set of all closed linear operators A : D(A) ⊆ X → X such that (i) the spectral set σ (A) of A is in the sector Sμ = {z ∈ C\{} | | arg z| ≤ μ} ∪ {}, that is, σ (A) ⊆ Sμ ; (ii) for every ω < μ < π , there exists a constant Cμ >  such that   (z – A)–  ≤ Cμ |z|γ

for all z ∈/ Sμ . γ

A linear operator is called an almost sectorial operator on X if it belongs to ω (X). Concerning the relationship between sectorial and almost sectorial operators, it has been found that a sectorial operator is an almost sectorial operator, but the converse is not true [, , –]. Some recent results on almost sectorial operators can be found in [, –]. The use of fractional calculus in the mathematical modeling of engineering and physical problems has become increasingly popular in recent years. Examples include material sciences, mechanics, wave propagation, signal processing, system identification, and so on. In consequence, the topic of fractional (ordinary, partial, functional) differential equations has developed into a hot research area; for example, see [–]. Wang et al. [] studied a fractional-order Cauchy problem with almost sectorial operators. In [], the author discussed mild solutions for abstract fractional differential equations with almost sectorial operators and infinite delay. More recently, in [], the authors investigated fractional Cauchy problems with almost sectorial operators. Fractional functional differential equations are used to describe anomalous diffusion processes with memory or hereditary properties. For details and some recent results on functional fractional differential equations, we refer the reader to a series of papers [–, , , , , , , ]. To the best of our knowledge, the study of fractional neutral evolution equations (FNEEs) with almost sectorial operators is yet to be initiated. The aim of this paper is to investigate the existence and uniqueness of solutions of FNEEs. The rest of the paper is organized as follows. In Section , we introduce some notation, definitions, and basic properties about fractional derivatives and functional analysis associated with almost sectorial operators. In Section , we prove the existence and uniqueness of mild solutions to FNEEs with almost sectorial operators. Under some suitable assumptions, we also show that mild solutions correspond to strong and classical solutions. Section  contains an example for illustration of our results, and we conclude our work in Section .

2 Preliminaries Here, we recall some preliminary material related to our work [, –]. 2.1 Fractional integrals and derivatives In this subsection, we give some basic definitions and properties of the fractional integral and derivatives. Definition . The Riemann-Liouville integral a Itα x and the Riemann-Liouville fractional derivative a Dαt x are respectively defined as 

 α a It x (t) =

 (α)



t

(t – τ )α– x(τ ) dτ , a

t > a,

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and 

  m+ m+–α  α x (t) = a It a Dt x (t) = D

 dm+ (m +  – α) dt m+



t

(t – τ )m–α x(τ ) dτ ,

t > a,

a

provided that the right-hand side is pointwise defined on [a, ∞). Here,  denotes the gamma function, and m ∈ N. Definition . The Caputo fractional derivative Liouville derivative as C

α a Dt x

  m   x(k) (a) α k (t – a) , (t) = a Dt x(t) – k!

C α a Dt x

is defined via the Riemann-

t > a, m ≤ α < m + , m ∈ N.

k=

Note that if x(k) (a) = , k = , , . . . , m, then (Ca Dαt x)(t) coincides with (a Dαt x)(t). Now we enlist some properties of the Riemann-Liouville fractional integral and the Caputo derivative (see [, –]). Proposition . Let α, β > . Then the following properties hold: β α+β (i) (a Itα a It x)(t) = (a It x)(t) for all x ∈ L (a, b); (ii) (a Itα (x ∗ y))(t) = ((a Itα x) ∗ y)(t) for all x, y ∈ L (a, b), where ∗ denotes convolution; (iii) The Caputo derivative Ca Dαt is a left inverse of a Itα , that is, (Ca Dαt a Itα x)(t) = x(t) for all x ∈ L (a, b); in general, it is not a right inverse. In fact, for m ≤ α < m + , m ∈ N, and x(k) (a) k x ∈ C m+ ([a, b]), we have (a Itα Ca Dαt x)(t) = x(t) – m k= k! (t – a) .

2.2 Special functions Here we present some basic definitions and properties of two special functions that we need in the sequel (see [, , , ]). Definition . The two-parameter Mittag-Leffler function is defined by

Eα,β (z) =

∞  k=

zk , (kα + β)

α, β > , z ∈ C.

In particular, if β = , then Eα,β coincides with the one-parameter Mittag-Leffler function Eα (z), that is, Eα, (z) = Eα (z). If α = β = , then E, (z) = ez . Definition . The Wright-type function is defined by

α (z) =

∞   (–z)k (kα) sin(kπα), π (k – )!

 < α < , z ∈ C.

k=

The following properties of the Wright-type function (cf. []) are useful in establishing the definition of mild solutions to FNEEs with almost sectorial operators. Proposition . Let – < ν < ∞, λ > . Then the following properties hold: (i) α (t) ≥  for all t > ;

∞ α α (ii)  tα+ α ( tα )e–λt dt = e–λ ;

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∞

∞ (+ν) (iii)  α (t)t ν dt = (+αν) ; in particular,  α (t) dt = ;

∞ (iv)  α (t)e–zt dt = Eα (–z), z ∈ C;

∞ (v)  αtα (t)e–zt dt = Eα,α (–z), z ∈ C.

2.3 Complex powers In this subsection, we give the definition and properties of complex powers of almost sectorial operators, which will be used in the next section. For more details, see [, ]. We define the path ϒθ := {R+ eiθ } ∪ {R+ e–iθ } ( < θ < π ) oriented so that Sθ lies to the left γ γ γ of ϒθ . In the forthcoming analysis, we write ω instead of ω (X). Let β ∈ C and A ∈ ω β with – < γ <  and  < ω < π/. Then the complex power A of A is defined by  A := z (A) = πi β

 zβ (z – A)– dz,

β

z ∈ C\(–∞, ].

ϒθ

Now we list some properties of Aβ (see []). γ

Proposition . Let A ∈ ω with – < γ <  and  < ω < π/. Then, for all α, β ∈ C, the following statements hold. (i) The operator Aβ is closed. (ii) Aα Aβ ⊆ Aα+β . Moreover, if Aβ is bounded, then Aα Aβ = Aα+β . (iii) Aβ is injective, and (Aβ )– = A–β . (iv) An = A · · A for all n ∈ N and A = I.  · n times

(v) If Re(β) >  + γ , then A–β is bounded. Based on Proposition ., we now prove a lemma, which will be used in the next section. γ

Lemma . Let A ∈ ω with – < γ <  and  < ω < π/, and let Re(β) >  + γ . Then A–β Aβ = Aβ A–β = A. Proof Since Re(β) >  + γ , by Proposition .(ii), (v) we get Aβ– A–β = A– . On the other hand, by Proposition .(iii) we can verify that (Aβ– A–β )– = Aβ A–β . Therefore, we have Aβ A–β = A, and the proof is completed.  It is worth noting that Proposition .(v) implies that the operator A–β belongs to L(X) whenever Re(β) >  + γ . So, in this situation, the linear space X β := D(Aβ ) (Re(β) >  + γ ) is a Banach space with the graph norm xβ = Aβ x, x ∈ X β . Of particular interest is that these spaces X β will provide the basic topology for analyzing the solutions of FNEEs with almost sectorial operators.

2.4 Properties of the operators Sα (t) and Pα (t) We introduce families of operators {T(t)}t∈S , {Sα (t)}t∈S , and {Pα (t)}t∈S assoπ /–ω π /–ω π /–ω ciated with the operator A as follows: T(t) := e–tz (A) = 

 πi

 α

 e–tz (z – A)– dz, ϒθ

Sα (t) := Eα –zt (A) =

 πi



t ∈ Sπ /–ω , z ∈ C\(–∞, ],

t ∈ Sπ /–ω , z ∈ C\(–∞, ],

  Eα –zt α (z – A)– dz, ϒθ

(.)

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Pα (t) := Eα,α t



 –zt (A) = πi α

∈ Sπ /–ω , z



Page 5 of 25

  Eα,α –zt α (z – A)– dz, ϒθ

∈ C\(–∞, ].

These operators appear in representations of solutions for FNEEs with almost sectorial operators. Further details about these operators can be found in [, ]. is a semigroup in view of the semigroup Notice that the operator family {T(t)}t∈S π /–ω  property: T(s + t) = T(s)T(t) for all s, t ∈ Sπ /–ω . Also, the operator T(t) can characterize the resolvent (z + A)– of –A as  (z + A)– =

∞

e–zt T(t) dt,

z ∈ C, Re(z) > .

(.)



From (.) and (.) it follows that there is a one-to-one correspondence between A and the semigroup T(t). By Proposition .(iv), (v) and definition (.) we can obtain the third property, that is, the operators Sα (t) and Pα (t) can be represented by T(t) as 

∞

Sα (t)x =

  α (s)T st α x ds,





Pα (t)x =

∞

  αsα (s)T st α x ds,



  t ∈ Sπ /–ω , x ∈ D Sα (t) ,   t ∈ Sπ /–ω , x ∈ D Pα (t) .

(.) (.)

For the reader’s convenience, we recall some more properties of the operator T in the following proposition. γ

Proposition . Let A ∈ ω with – < γ <  and  < ω < π/. Then the following properties hold. n  n (i) T(t) is analytic in Sπ /–ω , and d dtT(t) n = (–A) T(t), t ∈ Sπ /–ω , n ∈ N. (ii) There is a constant C = C (γ ) such that T(t) ≤ C t –γ – , t > . (iii) The range R(T(t)) of T(t), t ∈ Sπ /–ω , is contained in D(A∞ ). In particular,

R(T(t)) ⊆ D(Aβ ) for all β ∈ C with Re(β) > , Aβ T(t)x = π i ϒθ zβ e–tz (z – A)– dz for all x ∈ X, and there exists a constant C ∗ = C ∗ (β, γ ) >  such that Aβ T(t) ≤ C ∗ t –γ –Re(β)– for all t > . (iv) If β >  + γ , then D(Aβ ) ⊆ T = {x ∈ X| limt→+ T(t)x = x}. In the following, we describe the properties of the operators Sα (t) and Pα (t) [, ]. γ

Proposition . Let A ∈ ω with – < γ <  and  < ω < π/. The following statements hold. (i) For each fixed t ∈ Sπ /–ω , Sα (t) and Pα (t) are linear and bounded operators on X. Moreover, there exist constants Cs = C(α, γ ) >  and Cp = C(α, γ ) >  such that for all t > , Sα (t) ≤ Cs t –α(+γ ) and Pα (t) ≤ Cp t –α(+γ ) . (ii) For t > , Sα (t) and Pα (t) are continuous in the uniform operator topology. Moreover, for every r > , the continuity is uniform on [r, ∞).

t (iii) For each fixed t ∈ Sπ /–ω and all x ∈ D(A), (Sα (t) – I)x =  –sα– APα (s)x ds. (iv) For all x ∈ D(A) and t > , C Dαt Sα (t)x = –ASα (t)x. (v) For all t > , Sα (t) =  Itα (t α– Pα (t)). (vi) Let β >  + γ . For all x ∈ D(Aβ ), limt→+ Sα (t)x = x.

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γ

Proposition . Let A ∈ ω with – < γ <  and  < ω < π/, and let  < β <  – γ . Then (i) the range R(Pα (t)) of Pα (t) for t >  is contained in D(Aβ ); (ii) Sα (t)x = –t α– APα (t)x, and Sα (t)x for x ∈ D(A) is locally integrable on (, ∞); (iii) for all x ∈ D(A) and t > , ASα (t)x ≤ Ct –α(+γ ) Ax, where C is a constant depending on γ , α. γ

Lemma . Let A ∈ ω with – < γ <  and  < ω < π/, and let  < β <  – γ . Then, for each fixed t ∈ Sπ /–ω , Pα (t) is a bounded linear operator on X β . Moreover, there exists a positive constant C such that for all t > ,   β A Pα (t)x ≤ αC ( – γ – β) t –α(γ +β+) x. ( – α(γ + β)) Proof By relation (.), Proposition .(iii), and Proposition .(iii) we get   β A Sα (t)x ≤



∞

   αsα (s)Aβ T st α  dsx





≤ αC t

–α(γ +β+)

∞

s–γ –β α (s) dsx



= αC

( – γ – β) –α(γ +β+) x. t ( – α(γ + β)) 

The proof is completed.

3 Main results Consider a problem of fractional neutral evolution equations (FNEEs) with almost sectorial operator given by 

C α  Dt (x(t) – g(t, xt )) + Ax(t) = f (t, xt ),

x(t) = ϕ(t),

 < α < , t ∈ [, T],

t ∈ [–h, ],

(.)

γ

where h, T > , A is an almost sectorial operator, that is, A ∈ ω (– < γ < ,  < ω < π/), f (t, xt ), g(t, xt ) : [, T] × C([–h, ], X) → X are given functions, ϕ(t) : [–h, ] → X is an initial function, and xt is defined by xt (s) = x(t + s) for s ∈ [–h, ]. To study problem (.), we need the following assumption. (H) x(t) ∈ C([–h, T], X), x(t) ∈ D(A) for all t ∈ [, T], Ax ∈ L ((, T), X), f (t, xt ) ∈ L ((, T), X), and there exists a constant β such that β >  + γ and Aβ g(t, xt ) ∈ L ((, T), X). To define a mild solution of (.), we prove the following lemma. Lemma . Assume that condition (H) holds and x(t) satisfies problem (.). Then, for every ϕ(t) ∈ C([–h, ], X β ), x(t) satisfies the integral equation ⎧

t ⎪ – g(, x )) + g(t, xt ) –  (t – s)α– APα (t – s)g(s, xs ) ds ⎨Sα (t)(ϕ()

t x(t) = +  (t – s)α– Pα (t – s)f (s, xs ) ds, t ∈ [, T], ⎪ ⎩ ϕ(t), t ∈ [–h, ].

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Proof In view of condition (H), we know that (Theorem . in []) problem (.) is equivalent to the fractional integral equation ⎧

t  α– ⎪ ⎨x(t) = ϕ() – g(, x ) + g(t, xt ) – (α)  (t – s) Ax(s) ds t  α– + (α) f (s, xs ) ds, t ∈ [, T],  (t – s) ⎪ ⎩ x(t) = ϕ(t), t ∈ [–h, ]. Applying the Laplace transform to this integral equation, we get   –   –  – ϕ() – g(, x ) + sα sα + A G(s) + sα + A F(s), X(s) = sα– sα + A

(.)

where 

∞

X(s) =

 e–st x(t) dt,

∞

G(s) =



 e–st g(t, xt ) dt,

F(s) =



∞

e–st f (t, xt ) dt.



Using (.), integration by parts, and Proposition .(i), we have  – sα sα + A G(s)  ∞ ∞ α α =s e–s t T(t)e–sτ g(τ , xτ ) dt dτ 





∞ ∞

= sα 

e–s

  T λα e–sτ g(τ , xτ )αλα– dλ dτ



∞  ∞



α λα

=



α



 –sτ

–T λ e 





= e–(sλ)

α

 



∞ ∞

– 



∞

=

∞

g(τ , xτ ) dτ de–(sλ)

α

∞    –T λα e–sτ g(τ , xτ ) dτ 

λ=

 α



–(sλ)α –sτ

αλα– AT λ e



∞ ∞



e–sτ g(τ , xτ ) dτ –



e



g(τ , xτ ) dτ dλ

  α αλα– AT λα e–(sλ) e–sτ g(τ , xτ ) dτ dλ.

(.)



Furthermore, by using Proposition .(ii) and relation (.) the right-hand side of (.) can be written as  ∞ ∞  ∞   α e–sτ g(τ , xτ ) dτ – αλα– AT λα e–(sλ) e–sτ g(τ , xτ ) dτ dλ 





∞

=

–sτ

e 



× α 

∞

= 





∞ ∞ ∞

g(τ , xτ ) dτ – 





  α αλα– AT λα α+ θ

  e–sλθ e–sτ g(τ , xτ ) dθ dτ dλ θα

e–sτ g(τ , xτ ) dτ



∞ ∞ ∞

– 

× α



τ





ω–τ α θ

α–

 AT

  e–sω g(τ , xτ ) dθ dω dτ θα

ω–τ θ

α 

α θ α+

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∞

=

–sτ

e

Page 8 of 25

∞ ∞ ∞

g(τ , xτ ) dτ –





τ

  α(ω – τ )α– uAT u(ω – τ )α



× α (u)e g(τ , xτ ) du dω dτ  ∞ ∞  ∞ e–sτ g(τ , xτ ) dτ – (ω – τ )α– APα (ω – τ )e–sω g(τ , xτ ) dω dτ = –sω









∞

=

–sτ

e

τ ∞

g(τ , xτ ) dτ –





(ω – τ )

e 



ω

–sω

α–

APα (ω – τ )g(τ , xτ ) dτ dω.

(.)



By a similar argument we get  –    – ϕ() – g(, x ) + sα + A F(s) sα– sα + A  ∞   = e–st Sα (t) ϕ() – g(, x ) dt 



∞

+



(t – τ )

e 



t

–st

α–

Pα (t – τ )f (τ , xτ ) dτ dt.

(.)



Finally, combining (.)-(.), we conclude   x(t) = Sα (t) ϕ() – g(, x ) + g(t, xt ) –  +



t

(t – s)α– APα (t – s)g(s, xs ) ds

 t

(t – s)α– Pα (t – s)f (s, xs ) ds.



The proof is completed.



By Lemma . we define a mild solution to problem (.) as follows. Definition . By a mild solution to problem (.) on the interval [–h, T] we mean a function x(t) ∈ C([–h, T], X) satisfying ⎧

t ⎪ – g(, x )) + g(t, xt ) –  (t – s)α– APα (t – s)g(s, xs ) ds ⎨Sα (t)(ϕ()

t x(t) = +  (t – s)α– Pα (t – s)f (s, xs ) ds, t ∈ [, T], ⎪ ⎩ ϕ(t), t ∈ [–h, ]. In the sequel, we use |w| = maxs∈[–h,] w(s), where  ·  is an arbitrary norm in X. To study the existence and uniqueness of a mild solution to problem (.), we require the following assumptions. (H ) The resolvent (λI + A)– of –A is compact for every λ > . (H ) The function g(t, xt ) : [, T] × C([–h, ], X) → D(Aβ ) is a continuous function with respect to t ∈ [, T], and there exists a positive constant Mg such that for any xt ∈ C([–h, ], X), Aβ g(t, xt ) is strongly measurable and satisfies the inequality   β   A g(t, xt ) ≤ Mg  + |xt | , and there exist positive constants Lg and θ with θ > α( + γ ) such that for any t, s ∈ [, T] and xt , ys ∈ C([–h, ], X), Aβ g(t, xt ) satisfies the Lipschitz condition   β   A g(t, xt ) – Aβ g(s, ys ) ≤ Lg |t – s|θ + |xt – ys | .

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(H ) For almost all t ∈ [, T], the function f (t, xt ) : [, T]×C([–h, ], X) → X is continuous; for each xt ∈ C([–h, ], X), f (t, xt ) is strongly measurable; and there exists a function m(t) ∈ Lp ((, T), R+ ) with p > – αγ such that f (t, xt ) ≤ m(t) for all t ∈ [, T] and xt ∈ C([–h, ], X). We now show the existence of a mild solution to problem (.) via Krasnoselskii’s fixed point theorem. γ

Theorem . Let A ∈ ω with – < γ <  and  < ω < π/, and ϕ(t) ∈ C([–h, ], X β ) for β >  + γ . Assume that conditions (H )-(H ) hold. Then there exists T such that problem (.) has a mild solution on the interval [–h, T ]. Proof For any fixed r > , we set       B = x(t) ∈ C [–h, T], X : x(t) = ϕ(t), t ∈ [–h, ]; max x(t) – ϕ() ≤ r . t∈[,T]

Obviously, B is a closed convex subset of C([–h, T], X). Choose T ∈ (, T] such that –α(γ +–β)   αr∗ C Mg (β – γ ) T K + r∗ Mg A–β  + · ( – α(γ +  – β)) –α(γ +  – β)  –q(+αγ )  q T mLp ((,T),R+ ) ≤ r + Cp  – q( + αγ )

and   Lg A–β  +

–α(γ +–β) αC Lg (β – γ ) T · < ( – α(γ +  – β)) –α(γ +  – β)

(.)

with q = p/(p – ) and      r∗ =  + max r + ϕ(), max ϕ(t) , t∈[–h,]

(.)

    K = max Sα (t) ϕ() – g(, x ) – ϕ(). t∈[,T ]

Now we consider two operators F and F on C([–h, T], X): ⎧ ⎪ – g(, x )) + g(t, xt ) ⎨Sα (t)(ϕ()

t (F x)(t) = +  (t – s)α– APα (t – s)g(s, xs ) ds, ⎪ ⎩ ϕ(t), t ∈ [–h, ],

t ∈ [, T],

and  t (F x)(t) =

 (t

,

– s)α– Pα (t – s)f (s, xs ) ds, t ∈ [–h, ].

t ∈ [, T],

Obviously, x(t) is a mild solution to equation (.) if and only if the operator equation F x + F x = x has a solution x ∈ B. Therefore, the existence of a mild solution is equivalent

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to the existence of a function x ∈ B such that F x + F x = x. To prove the latter, we divide the proof into four steps. Step . The operators F and F map the set B into C([–h, T], X), respectively. First, we show that for any fixed x(t) ∈ B, (F x)(t) is continuous for all t ∈ [–h, T]. It is clear that (F x)(t) is continuous for t ∈ [–h, ). For the case t = , we have        (F x)(t) – (F x)() ≤  Sα (t) – I ϕ() + g(t, xt ) – Sα (t)g(, x )   t   α– . (t – s) A P (t – s)g(s, x ) ds + α s   

Noting that ϕ(), g(t, xt ) ∈ X β , by Proposition .(vi), we get (Sα (t) – I)ϕ() →  and g(t, xt ) – Sα (t)g(, x ) →  as t → + , respectively. On the other hand, by Lemmas . and . we obtain  t     (t – s)α– APα (t – s)g(s, xs ) ds     t   ≤ r∗ Mg (t – s)α– A–β Pα (t – s) ds 

≤

∗

αr C Mg (β – γ ) ( – α(γ +  – β))



t

(t – s)α– (t – s)–α(γ –β+) ds 

αr∗ C Mg (β – γ ) t –α(γ –β+) = · . ( – α(γ +  – β)) –α(γ – β + )

t This shows that  (t – s)α– APα (t – s)g(s, xs ) ds →  as t → + . So, (F x)(t) is continuous at t = . For the case  < t < t ≤ T, we have        (F x)(t ) – (F x)(t ) ≤  Sα (t ) – Sα (t ) x() – g(, x )  + g(t , xt ) – g(t , xt )     t  (t – s)α– APα (t – s)g(s, xs ) ds +   – 



t

  (t – s)α– APα (t – s)g(s, xs ) ds  = I + I + I ,

where      I = g(t , xt ) – g(t , xt ), I =  Sα (t ) – Sα (t ) x() – g(, x ) ,  t   t   α– α–  (t – s) APα (t – s)g(s, xs ) ds – (t – s) APα (t – s)g(s, xs ) ds I =  . 



By condition (H ) and Proposition .(vi) we have I →  and I →  as t → t . Now it remains to show that, in this case, I →  as t → t . Using condition (H ) and Lemmas . and ., we have  t   t   α– α–  I =  s APα (s)g(t – s, xt –s ) ds – s APα (s)g(t – s, xt –s ) ds      t     α–  s A P (s) g(t – s, x ) – g(t – s, x ) ds ≤ α  t –s  t –s   

Ding and Ahmad Advances in Difference Equations (2016) 2016:203

  +   

s t

α–

  APα (s)g(t – s, xt –s ) ds 

   sα– A–β Pα (s)Aβ g(t – s, xt –s ) – Aβ g(t – s, xt –s ) ds

t

≤

t



t

+ t

   sα– A–β Pα (s)Aβ g(t – s, xt –s ) ds

αC Lg (β – γ ) ≤ ( – α(γ +  – β))



t 

∗

+ ≤

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αr C Mg (β – γ ) ( – α(γ +  – β))



  s–αγ +αβ–α– |t – t |θ + |xt –s – xt –s | ds t

s–αγ +αβ–α– ds

t

–α(γ –β+)   αC Lg (β – γ ) t · |t – t |θ + max |xt –s – xt –s | s∈[,t ] ( – α(γ +  – β)) –α(γ – β + )

+

αr∗ C Mg (β – γ ) t–α(γ –β+) – t–α(γ –β+) · . ( – α(γ +  – β)) –α(γ – β + )

Moreover, since x(t) is continuous and β >  + γ , we have I →  as t → t . It follows that (F x)(t) is continuous for all t ∈ (, T]. Hence, the operator F maps the set B into C([–h, T], X). Next, we show that for any fixed x(t) ∈ B, (F x)(t) is continuous for all t ∈ [–h, T]. Obviously, (F x)(t) is continuous for any t ∈ [–h, ). For the case t = , by the Hölder inequality we get   (F x)(t) – (F x)() ≤



t

(t – s)

 q Pα (t – s) ds

α– 

 q

mLp ((,T),R+ )



≤

t –q(+αγ ) mLp ((,T),R+ )  – q( + αγ )

p  with  < q = p– < +αγ . This shows that (F x)(t) – (F x)() →  as t → + . Thus, it follows that the function (F x)(t) is continuous at t = . For  < t < t ≤ T, we have

    t (F x)(t ) – (F x)(t ) =  (t – s)α– Pα (t – s)f (s, xs ) ds     t  α– – (t – s) Pα (t – s)f (s, xs ) ds  

 t      α–  ≤ (t – s) Pα (t – s) – Pα (t – s) f (s, xs ) ds     t     α– α–  (t – s) – (t – s) P (t – s)f (s, x ) ds +   α  s     + 



t

(t – s) t

α–

  Pα (t – s)f (s, xs ) ds  = I + I + I ,

where   I =  



t

    (t – s)α– Pα (t – s) – Pα (t – s) f (s, xs ) ds ,

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 t      α– α–  (t – s) – (t – s) I =  Pα (t – s)f (s, xs ) ds ,   t    α–  (t – s) Pα (t – s)f (s, xs ) ds I =  . t

For I , we choose a small δ >  and get 

t –δ

I ≤ 



   (t – s)α– Pα (t – s) – Pα (t – s)f (s, xs ) ds    (t – s)α– Pα (t – s) – Pα (t – s)f (s, xs ) ds

t

+ t –δ

     ≤ max Pα (t – s) – Pα (t – s) s∈[,t –δ]



t –δ

(t – s)

q(α–)

 q ds

mLp ((,T),R+ )



   (t – s)α– Cp (t – s)–α(+γ ) + Cp (t – s)–α(+γ ) f (s, xs ) ds

t

+ t –δ

   ≤ max Pα (t – s) – Pα (t – s) s∈[,t –δ]

t –δ

(t – s)q(α–) ds

 q

mLp ((,T),R+ )





t

+ Cp

(t – s)

–q(αγ +)

 q ds

mLp ((,T),R+ )

t –δ

    t q(α–)+ – δ q(α–)+ q = max Pα (t – s) – Pα (t – s)  mLp ((,T),R+ ) s∈[,t –δ] q(α – ) +   –q(αγ +)  q δ + Cp mLp ((,T),R+ ) .  – q(αγ +  This, together with Proposition .(ii), leads to I →  as t → t and δ → . For I , by the Hölder inequality we have  I ≤ Cp

t 

 (t – s)α– – (t – s)α– q (t – s)–αq(+γ ) ds

 q

mLp ((,T),R+ )



 ≤ Cp

t 

 (t – s)–q(+αγ ) – (t – s)–q(+αγ ) ds

 q

mLp ((,T),R+ )



 = Cp

–q(+αγ )

t

–q(+αγ )

– t  – q( + αγ )

(t – t )–q(+αγ ) +  – q( + αγ )

 q

mLp ((,T),R+ ) .

p  Moreover, since  < q = p– < +αγ , we get I →  as t → t . For I , by the same reasoning we have

 I ≤

t 

(t – s)

 q Pα (t – s) ds

α– 

 q

mLp ((,T),R+ )

t

 ≤ Cp

(t – t )–q(+αγ )  – q( + αγ )

 q

mLp ((,T),R+ ) .

This gives I →  as t → t . It follows that (F x)(t) is continuous for any t ∈ (, T]. Hence, F maps the set B into C([–h, T], X).

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Step . We show that F x + F y ∈ B for every pair x, y ∈ B. By the definitions of the operators F and F we have ⎧

t ⎪ – g(, x )) + g(t, xt ) –  (t – s)α– APα (t – s)g(s, xs ) ds ⎨Sα (t)(ϕ()

t (F x)(t) + (F y)(t) = +  (t – s)α– Pα (t – s)f (s, ys ) ds, t ∈ [, T], ⎪ ⎩ ϕ(t), t ∈ [–h, ], so we only need to verify that maxt∈[,T] F x + F y – ϕ() ≤ r. According to assumptions (H )-(H ) and Lemmas . and ., we get         F x + F y – ϕ() ≤ Sα (t) ϕ() – g(, x ) – ϕ() + g(t, xt )   t   α–  (t – s) A P (t – s)g(s, x ) ds + α s   

 t    α–  +  (t – s) Pα (t – s)f (s, ys ) ds  

–α(γ +–β)   αr∗ C Mg (β – γ ) T ≤ K + r∗ Mg A–β  + · ( – α(γ +  – β)) –α(γ +  – β)  –q(+αγ )  q T mLp ((,T),R+ ) + Cp  – q( + αγ )

≤ r. Therefore, F x + F y ∈ B for every pair x, y ∈ B. Step . The mapping F is contractive. For any x, y ∈ B, by Lemmas . and . we have   (F x)(t) – (F y)(t)

     t    α–    ≤ g(t, xt ) – g(t, yt ) +  (t – s) APα (t – s) g(s, xs ) – g(s, ys ) ds       αC Lg (β – γ ) t –α(γ +–β) · x – y. ≤ Lg A–β  + ( – α(γ +  – β)) –α(γ +  – β)

which, together with (.), shows that the mapping F is contractive on B. Step . The operator F is compact. First, we show that F is continuous. Let {xn } ⊆ B with xn → x on B. Then, by assumption (H ) and the fact that xnt → xt for t ∈ [, T] we get   f t, xnt → f (t, xt ),

a.e., t ∈ [, T], as n → ∞.

On the other hand, by assumption (H ) and Proposition .(i) we have   t –αγ      (t – s)α– Pα (t – s)f s, xn ds ≤ Cp t mLp ((,T),R+ ) . s   –αγ 

This implies that (t – s)α– Pα (t – s)f (s, xns ) ∈ L ((, t), X). Thus, by the Lebesgue dominated convergence theorem we get  t         (t – s)α– Pα (t – s) f s, xn – f (s, xs ) ds →  as n → ∞. s   

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Hence, F is continuous. It remains to show that F (B) is relatively compact. According to assumption (H ) and Theorem . in [], the family of functions {F x : x ∈ B} is uniformly bounded. From Step  we observe that {F x : x ∈ B} is a family of equicontinuous functions. So, by the known Ascoli-Arzelà theorem, the family {F x : x ∈ B} is relatively compact. Hence, F is compact. Therefore, by Krasnoselskii’s fixed point theorem, we deduce that problem (.) has a  mild solution on the interval [–h, T ]. The proof is completed. Next, we discuss the uniqueness of mild solutions to problem (.). For that, we need an additional condition. (H ) There exists a constant Lf >  such that for all t ∈ [, T] and xt , yt ∈ C([–h, ], X), the function f satisfies the Lipschitz condition   f (t, xt ) – f (t, yt ) ≤ Lf |xt – yt | . The uniqueness result is based on the Banach contraction principle. γ

Theorem . Let A ∈ ω with – < γ <  and  < ω < π/, and ϕ(t) ∈ C([–h, ], X β ) for β >  + γ . Assume that conditions (H )-(H ) hold. Then there exists T ∈ (, T] such that problem (.) has a unique mild solution on the interval [–h, T ]. Proof For any fixed r > , we set       B = x ∈ C [–h, T], X : x(t) = ϕ(t), t ∈ [–h, ]; max x(t) – ϕ() ≤ r . t∈[,T]

Obviously, B is a closed convex subset of C([–h, T], X). Choose T ∈ (, T] such that –α(γ +–β)   αr∗ C Mg (β – γ ) T · K + r∗ Mg A–β  + ( – α(γ +  – β)) –α(γ +  – β)  –q(+αγ )  q T mLp ((,T),R+ ) ≤ r + Cp  – q( + αγ )

and   Lg A–β  +

–αγ

–α(γ +–β) αC Lg (β – γ ) Cp Lf T T · + ( – α(γ +  – β)) –α(γ +  – β) –αγ

< ,

where r∗ and K are the constants defined by (.). Now we consider the operator F defined by ⎧

t ⎪ – g(, x )) + g(t, xt ) –  (t – s)α– APα (t – s)g(s, xs ) ds ⎨Sα (t)(ϕ()

t (Fx)(t) = +  (t – s)α– Pα (t – s)f (s, xs ) ds, t ∈ [, T], ⎪ ⎩ ϕ(t), t ∈ [–h, ]. Similarly to the proof of Theorem ., we can show that F maps the subset B into itself. Moreover, for any x(t), y(t) ∈ B, we have

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  (Fx)(t) – (Fy)(t)      t    α–  ≤ g(t, xt ) – g(t, yt ) +  (t – s) A P (t – s) g(s, x ) – g(s, y ) ds α s s   

  t     α–  +  (t – s) Pα (t – s) f (s, xs ) – f (s, ys ) ds      –β  αC Lg (β – γ ) Cp Lf t –αγ t –α(γ +–β)   ≤ Lg A + · + x – y. ( – α(γ +  – β)) –α(γ +  – β) –αγ So the mapping F is contractive. It follows from the Banach contractive principle that  problem (.) has a unique mild solution on the interval [–h, T ]. In particular, if g(t, xt ) ≡ c (c is a constant), then we have the following corollary. γ

Corollary . Let A ∈ ω with – < γ <  and  < ω < π/, and ϕ(t) ∈ C([–h, ], X β ) for β >  + γ . Assume that conditions (H )-(H ) hold. Then there exists T ∈ (, T] such that the abstract fractional functional equation  (C Dαt x)(t) + Ax(t) = f (t, xt ), x(t) = ϕ(t), t ∈ [–h, ]

 < α < , t ∈ [, T],

(.)

has a unique mild solution on the interval [–h, T ]. In particular, when β = , then we have  >  + γ (– < γ < ). So X  = D(A) is a Banach space with the graph norm x = Ax, where x ∈ D(A). In this case, condition (H ) can be written as follows: (H∗ ) The function g(t, xt ) : [, T] × C([–h, ], X) → D(A) is a continuous function with respect to t ∈ [, T]; there exists a positive constant Mg such that for any xt ∈ C([–h, ], X), Ag(t, xt ) is strongly measurable and satisfies the inequality     Ag(t, xt ) ≤ Mg  + |xt | ; and there exist positive constants Lg and θ with θ > α( + γ ) such that for any t, s ∈ [, T] and xt , ys ∈ C([–h, ], X), Ag(t, xt ) satisfies the Lipschitz condition     Ag(t, xt ) – Ag(s, ys ) ≤ Lg |t – s|θ + |xt – ys | . Consequently, we arrive at the following corollary, which is a particular case of Theorem .. γ

Corollary . Let A ∈ ω with – < γ <  and  < ω < π/, and ϕ(t) ∈ C([–h, ], X  ). Assume that conditions (H∗ ), (H ), and (H ) hold. Then there exists T ∈ (, T] such that problem (.) has a unique mild solution on the interval [–h, T ]. Now we turn our attention to further conditions on f and g so that the mild solution becomes a strong solution and a classical solution. We first give the definitions of the strong solution and the classical solution to problem (.).

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Page 16 of 25

Definition . A function x(t) : [–h, T] → X is said to be a strong solution to problem (.) if (i) x(t) is continuous on [–h, T], and (C Dαt x)(t) ∈ L ((, T), X); and (ii) x(t) takes values in D(A) and satisfies problem (.). Definition . A function x(t) : [–h, T] → X is said to be a classical solution to problem (.) if (i) x(t) is continuous on [–h, T], and (C Dαt x)(t) ∈ C([, T], X); and (ii) x(t) takes values in D(A) and satisfies problem (.). In the following, we prove that a mild solution can become a strong solution to problem (.) under some assumptions. To do this, we require stronger conditions than conditions (H ) and (H ): (H∗ ) For almost all t ∈ [, T], the function f (t, xt ) : [, T] × C([–h, ], X) → X is continuous; for each xt ∈ C([–h, ], X), f (t, xt ) is strongly measurable; and there exists a func+α– such that f (t, xt ) ≤ m(t) for all t ∈ [, T] tion m(t) ∈ Lp ((, T), R+ ) with p > αγ α(γ +) and xt ∈ C([–h, ], X). (H∗ ) There exist constants Lf >  and θ with θ > α( + γ ) such that for all t, s ∈ [, T] and xt , ys ∈ C([–h, ], X), the function f satisfies the Lipschitz condition     f (t, xt ) – f (s, ys ) ≤ Lf |t – s|θ + |xt – yt | . γ

Theorem . Let A ∈ ω with – < γ < –  and  < ω < π/, and ϕ(t) ∈ C([–h, ], X  ). Suppose that conditions (H∗ )-(H∗ ) hold. In addition, suppose that the following conditions are satisfied: (Ha ) For almost all t ∈ [, T] and xt ∈ C([–h, ], X), C Dαt g(t, xt ) ∈ L ([, T], X). (Hb ) In condition (H∗ ),  < Lg < . (Hc ) For almost all t ∈ [, T] and xt ∈ C([–h, ], X), Ag(t, xt ) ∈ L ((, T), X  ) and A g(t, xt ) ∈ L∞ ((, T), X). (Hd ) For almost all t ∈ [, T] and xt ∈ C([–h, ], X), f (t, xt ) ∈ L ((, T), X  ) and Af (t, xt ) ∈ L∞ ((, T), X). Then the mild solution x is the unique strong solution to problem (.), provided that A(ϕ() – g(, x )) ∈ L∞ ((, T), X). +α– > – αγ . Hence, the conclusion of Proof Using the assumption – < γ < –  , we have αγ α(γ +) Theorem . is also true if we replace conditions (H )-(H ) by (H∗ )-(H∗ ). Now, we will follow the argument of Wang et al. in (see [], Theorem .) to prove that the mild solution x is a strong solution to problem (.). First, we show that x(t) is Hölder continuous with an exponent ϑ with ϑ > α( + γ ) on the interval [–h, T ]. For any t ∈ [, T ], taking t >  such that t + t ≤ T , we have

  x(t + t) – x(t)      ≤  Sα (t + t) – Sα (t) ϕ() – g(, x )  + g(t + t, xt+t ) – g(t, xt )  t+t  + (t + t – s)α– APα (t + t – s)g(s, xs ) ds  

Ding and Ahmad Advances in Difference Equations (2016) 2016:203



t

–

(t – s)

α–



  + 

Page 17 of 25

  APα (t – s)g(s, xs ) ds  

t+t

(t + t – s)

α–

t

Pα (t + t – s)f (s, xs ) ds –

(t – s)



α–



  Pα (t – s)f (s, xs ) ds 

= I + I + I + I . For I , by Proposition .(i), (iii) we get    I =  Sα (t + t) – Sα (t) ϕ() – g(, x )    t+t     α–  –s A P (s) ϕ() – g(, x ) ds = α    t

  (t + t)–αγ – t –αγ  A ϕ() – g(, x ) . –αγ

≤ Cp

For I , using Lemma ., condition (H∗ ), and Proposition .(i), we get  t+t  (t + t – s)α– APα (t + t – s)g(s, xs ) ds I =      t  α– – (t – s) APα (t – s)g(s, xs ) ds  

  t     α–  (t – s) A P (t – s) g(s + t, x ) – g(s, x ) ds ≤ α s+t s     t    α–  + (t + t – s) A P (t + t – s)g(s, x ) ds α s    ≤ Lg

 t

   (t – s)α– Pα (t – s) (t)θ + |xs+t – xs | ds





∗

+ r Mg ≤ C p Lg

 –αγ T

–αγ

t

  (t + t – s)α– Pα (t + t – s) ds 

t

(t)θ + Cp Lg

(t – s)–αγ – |xs+t – xs | ds + r∗ Mg



(t + t)–αγ – t –αγ . –αγ

Similarly to I , we have   I =  



t+t

(t + t – s)



α–

Pα (t + t – s)f (s, xs ) ds –

t

(t – s)

α–



  Pα (t – s)f (s, xs ) ds 

–αγ

Lf Cp T (t + t)–q(+αγ ) – t –q(+αγ ) mLp ((,T),R+ ) +  – q( + αγ ) –αγ  t + Lf Cp (t – s)–αγ – |xs+t – xs | ds.

≤ Cp

(t)θ



As a consequence, we get   x(t + t) – x(t) ≤ Cp

  (t + t)–αγ – t –αγ  A ϕ() – g(, x )  + Lg (t)θ + Lg |xt+t – xt | –αγ

Ding and Ahmad Advances in Difference Equations (2016) 2016:203



–αγ

+

Cp Lg T –αγ

(t)θ + Cp Lg

t

Page 18 of 25

(t – s)–αγ – |xs+t – xs | ds



(t + t)–αγ – t –αγ (t + t)–q(+αγ ) – t –q(+αγ ) + Cp mLp ((,T),R+ ) –αγ  – q( + αγ )  t –αγ Lf Cp T θ (t) + Lf Cp (t – s)–αγ – |xs+t – xs | ds. + –αγ  + r∗ Mg

Using the inequality bc – ac ≤ (b – a)c ( < a < b,  < c < ), we have   x(t + t) – x(t) ≤

 –αγ  Cp A(ϕ() – g(, x )) + r∗ Mg Cp Lg T (t)–αγ + Lg + (t)θ –αγ –αγ –αγ

+ Lg |xt+t – xt | + 

t

+ Cp (Lg + Lf )

Lf Cp T –αγ

Cp mLp ((,T),R+ ) (t)–q(+αγ )  – q( + αγ )

(t)θ +

(t – s)–αγ – |xs+t – xs | ds.

(.)



Putting   ϑ = min –αγ , θ , θ ,  – q( + αγ ) > α(γ + ), Cp Lg T Cp A(ϕ() – g(, x )) + r∗ Mg + Lg + –αγ –αγ

–αγ

M=

+

–αγ

+

Lf Cp T –αγ

Cp mLp ((,T),R+ ) ,  – q( + αγ )

we can rewrite (.) in the form   x(t + t) – x(t) ≤ M(t)ϑ + Lg |xt+t – xt |  t + Cp (Lg + Lf ) (t – s)–αγ – |xs+t – xs | ds. 

Then it follows from the definition of | · | that the inequality  |xt+t – xt | ≤ M(t)ϑ + Lg |xt+t – xt | + Cp (Lg + Lf )

t

(t – s)–αγ – |xs+t – xs | ds



holds. Then, in view of condition (Hb ), we have |xt+t – xt | ≤

Cp (Lg + Lf ) M (t)ϑ +  – Lg  – Lg



t

(t – s)–αγ – |xs+t – xs | ds.

(.)



Applying the generalized Gronwall inequality [] to (.), we get the estimate |xt+t – xt | ≤ Q(t)ϑ

(.)

with Q=

  Cp (Lg + Lf ) M –αγ . E–αγ (–αγ )T  – Lg  – Lg

(.)

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It follows from (.) that x(t) is Hölder continuous with an exponent ϑ > α( + γ ) on [–h, T ]. Next, we show that x(t) satisfies problem (.). To do this, let 

t

u(t) =

(t – s)α– APα (t – s)g(s, xs ) ds.

(.)



By the assumption A g(t, xt ) ∈ L∞ ((, T), X) and Proposition .(i), we have 

t

Au ≤ 

     t –αγ  A g  ∞ (t – s)α– Pα (t – s) dsA g L∞ ((,T),X) ≤ Cp . L ((,T),X) –αγ

This implies that u(t) ∈ D(A) for all t ∈ [, T]. Note that u() = . So by Proposition .(ii) and Proposition .(v) we get C



α  Dt u

(t) =

 d  –α  (t) = u (t) I dt t



α  Dt u

 d  –α α– It t APα (t)(t) ∗ g(t, xt )  dt  d ASα (t) ∗ g(t, xt ) . = dt =

Now we need to calculate the first derivative of v(t) := ASα (t) ∗ g(t, xt ). Let t >  and t + t ≤ T . Then, by Proposition .(ii) we obtain 

ASα (t + t – s) – ASα (t – s) g(s, xs ) ds t   t+t  ASα (t + t – s)g(s, xs ) ds + t t  t = –A (t – s)α– APα (t – s)g(s, xs ) ds

v(t + t) – v(t) = t

t



 + t



t+t

ASα (t + t – s)g(s, xs ) ds

t

= –Au(t) + I, where I=

 t

 = t  = t +



t+t

ASα (t + t – s)g(s, xs ) ds

t



t

ASα (τ )g(t + t – τ , xt+t–τ ) dτ







 t

t



  ASα (τ ) g(t + t – τ , xt+t–τ ) – g(t – τ , xt–τ ) dτ t



= I + I + I .

   ASα (τ ) g(t – τ , xt–τ ) – g(t, xt ) dτ + t

 

t

ASα (τ )g(t, xt ) dτ

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Page 20 of 25

For I , combining Proposition .(i) and relation (.), we have  t         A S (τ ) g(t + t – τ , x ) – g(t – τ , x ) dτ α t+t–τ t–τ  t    t     Sα (τ )Ag(t + t – τ , xt+t–τ ) – Ag(t – τ , xt–τ ) dτ ≤ t    Cs Lf t –α(+γ )  ≤ (t)θ + |xt+t–τ – xt–τ | dτ τ t  ≤ C s Lf

(t)ϑ–α(γ +) (t)θ –α(γ +) + C s Lf Q .  – α(γ + )  – α(γ + )

Moreover, since ϑ > α(γ + ) and  – α(γ + ) > , we get I →  as t → . Similarly, we can show that I →  as t → . For I , by Proposition .(vi) we have limt→ I = Ag(t, xt ). Hence, we get v (t) = –Au(t) + Ag(t, xt ). It follows that 

C

α  Dt u

(t) = –Au(t) + Ag(t, xt ).

(.)

Analogously, taking 

t

w(t) =

(t – s)α– Pα (t – s)f (s, xs ) dt,

(.)



we can show that C



α  Dt w

(t) = –Aw(t) + f (t, xt ).

(.)

Combining Proposition .(iv), (.), and (.), we obtain that the mild solution x(t) satisfies problem (.). To complete the proof, it remains to show that (C Dαt x)(t) ∈ L ((, T ), X). In other words,

T we need to prove that   |(C Dαt x)(t)| dt < ∞. As before, we have C α   D x 

L ((,T ),X)

t

  ≤ C Dαt g L ((,T





T



t

+ 

T



t

+ 





T 

    Sα (t) dt A ϕ() – g(, x )  ∞ L ((,T

 ),X)

    (t – s)α– Pα (t – s) ds dt A g L∞ ((,T

 ),X)





 + ),X)

+ AgL ((,T ),X  )

  (t – s)α– Pα (t – s) ds dtAf L∞ ((,T ),X) + f L ((,T ),X  )

  ≤ C Dαt g L ((,T



+ Cs ),X)

–α(+γ )    T A ϕ() – g(, x )  ∞ L ((,T ),X)  – α( + γ )

    Cp T A g  ∞ + Af L∞ ((,T ),X) + AgL ((,T ),X  ) L ((,T ),X)  –αγ ( – αγ ) –αγ

+

+ f L ((,T ),X  ) , which shows that (C Dαt x)(t) ∈ L ((, T ), X). Hence, the mild solution x(t) is a strong solution to problem (.). The proof is completed. 

Ding and Ahmad Advances in Difference Equations (2016) 2016:203

Page 21 of 25

γ

Corollary . Let A ∈ ω with – < γ < –  and  < ω < π/, and ϕ(t) ∈ C([–h, ], X  ). Suppose that conditions (H∗ ), (H∗ ), and (Hd ) hold. Then the mild solution x to equation (.) is its unique strong solution, provided that Aϕ() ∈ L∞ ((, T), X). Finally, under suitable conditions, we show that the mild solution becomes a classical solution to problem (.). γ

Theorem . Let A ∈ ω with – < γ < –  and  < ω < π/, and ϕ(t) ∈ C([–h, ], X  ). Suppose that conditions (H∗ )-(H∗ ) hold. In addition, suppose that the following conditions hold: (Ha ) For all t ∈ [, T] and xt ∈ C([–h, ], X), C Dαt g(t, xt ) ∈ C([, T], X). (Hb ) In condition (H∗ ),  < Lg < . (Hc ) For almost all t ∈ [, T] and xt ∈ C([–h, ], X), Ag(t, xt ) ∈ L ((, T), X  ) and A g(t, xt ) ∈ L∞ ((, T), X). (Hd ) For almost all t ∈ [, T] and xt ∈ C([–h, ], X), f (t, xt ) ∈ L ((, T), X  ) and Af (t, xt ) ∈ L∞ ((, T), X). Then the mild solution is a classical solution to problem (.), provided that A(ϕ() – g(, x )) ∈ D(Aβ ) with β >  + γ . Proof To establish the conclusion, we observe from the proof of Theorem . that it is sufficient to establish that (C Dαt x)(t) ∈ C([, T ], X). We now define u(t) and w(t) as in (.) and (.). We first prove that (C Dαt u)(t) ∈ C([, T ], X). By (.) we only need to prove that –Au(t) + Ag(t, xt ) ∈ C([, T ], X). According to the assumption, Ag(t, xt ) is continuous for all t ∈ [, T ] and xt ∈ C([–h, ], X). So, it remains to prove that I(t) = –Au(t) is continuous for all t ∈ [, T ]. For that, we express I(t) as I(t) = I (t) + I (t), where 

t

I (t) = –A

  (t – s)α– APα (t – s) g(s, xs ) – g(t, xt ) ds,





t

I (t) = A

(t – s)α– APα (t – s)g(t, xt ) ds.



Using Proposition .(iii), we get I (t) = –(Sα (t) – I)Ag(t, xt ). So, by Proposition .(v) and condition (Hc ) we have that I (t) is continuous for t ∈ [, T ]. Next, we prove that I (t) is continuous for t ∈ [, T ]. Let t >  be such that t + t ≤ T . Then   I (t + t) – I (t)   t   (t + t – s)α– APα (t + t – s) – (t – s)α– ≤ A 

  × APα (t – s) g(s, xs ) – g(t, xt ) ds     t     α–  (t + t – s) A P (t + t – s) g(t, x ) – g(t + t, x ) ds + A α t t+t   

   + A



t+t t



    (t + t – s)α– APα (t + t – s) g(s, xs ) – g(t + t, xt+t ) ds 

= h (t) + h (t) + h (t).

Ding and Ahmad Advances in Difference Equations (2016) 2016:203

Page 22 of 25

For h (t), on the one hand, we have    t     A (t + t – s)α– APα (t + t – s) g(s, xs ) – g(t, xt ) ds   

≤ Lg

t ϑ–α(+γ ) t θ –α(+γ ) + Lg Q . θ – α( + γ ) ϑ – α( + γ )

Moreover, since θ > α( + γ ) and ϑ > α( + γ ), we get A(t + t – s)α– APα (t + t – s)(g(s, xs ) – g(t, xt )) ∈ L ((, t), X). On the other hand, by Proposition .(v) we have   lim A(t + t – s)α– APα (t + t – s) g(s, xs ) – g(t, xt )

t→

  = A(t – s)α– APα (t – s) g(s, xs ) – g(t, xt ) ,

so that by the Lebesgue dominated convergence theorem we get h (t) →  as t → . For h (t), we have the estimate   t      α–  h (t) ≤ A (t + t – s) APα (t + t – s) g(t, xt ) – g(t + t, xt+t ) ds  



≤ Lg C p

t

  (t + t – s)–α–αγ – (t)θ + Q(t)ϑ ds



= Lg C p

 (t)–α(+γ ) – (t + t)–α(+γ )  (t)θ + Q(t)ϑ . –α( + γ )

Moreover, since θ > α( + γ ) and ϑ > α( + γ ), we get h (t) →  as t → . For h (t), by Proposition .(i) and condition (H∗ ) we have the estimate h (t) ≤

 (t)–αγ  A g  ∞ , L ((,T ),X) –αγ

which implies that h (t) →  as t → . Hence, –Au(t) + Ag(t, xt ) is continuous for all t ∈ [, T ]. By the argument used earlier we have that (C Dαt w)(t) is continuous for all t ∈ [, T ]. Combining Proposition .(v) and the assumption A(ϕ() – g(, x )) ∈ D(Aβ ) with β >  + γ , it follows that (C Dαt x)(t) is continuous for all t ∈ [, T ], and the proof is completed.  γ

Corollary . Let A ∈ ω with – < γ < –  and  < ω < π/, and ϕ(t) ∈ C([–h, ], X  ). Suppose that conditions (H∗ ), (H∗ ), and (Hd ) hold. Then the mild solution x to equation (.) is its unique classical solution, provided that Aϕ() ∈ D(Aβ ) with β >  + γ .

4 Application In this section, we demonstrate the applicability of the obtained results to the following problem: ⎧C α

t  –t  ⎪  ∂t (u(t, y) –  e sin(y(t – h))) = ∂y u(t, y) + t–h χ(s – t)u(s, y) ds, ⎪ ⎪ ⎨ t ∈ [, T], y ∈ [, π], ⎪u(t, ) = u(t, π) = , t ∈ [, T], ⎪ ⎪ ⎩ u(t, y) = (ϕ(t))(y), t ∈ [–h, ], y ∈ [, π],

(.)

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in the space of Hölder continuous functions X := C l ([, π], R) ( < l < ), where C ∂tα is the Caputo fractional partial derivative of order  < α <  with respect to t, that is, C



α  ∂t u

  t  ∂  –α –α (t, y) = (t – s) u(s, y) ds – t u(, y) . ( – α) ∂t 

We now introduce the operator     D( A) = u ∈ C +l [, π] : u(t, ) = u(t, π) =  ,

 A := –∂y ,

in the space C l ([, π], R) ( < l < ) of Hölder continuous functions. It follows from [] that l – there exist ν,  >  such that  A + ν ∈  π (X).  –

To represent this system in the abstract form (.), we introduce the function f : [, T] × C([–h, ], X) → X given by 



f (t, )(y) =

χ(s)(s, y) ds. –h

If χ ∈ L ([, T], R), then f ∈ C([, T], X), and it follows that there exists a function m(t) ∈ L ([, T], R) such that   f (t, ·) ≤ m(t). It follows from Theorem . that there exists T such that equation (.) has a unique mild solution on [–h, T ].

5 Conclusions The research on almost sectorial operators has been of significant interest during the past years. However, it has been found that there is no published material addressing the existence and uniqueness of solutions for fractional neutral evolution equations with almost sectorial operators. To enrich the literature on the topic, we have investigated the existence and uniqueness of mild solutions to fractional neutral evolution equations with almost sectorial operators in this article. Our study relies on some fixed point theorems. Under some suitable assumptions, we have shown that a mild solution can become a strong solution and a classical solution. As an illustration of our work, we have discussed the existence and uniqueness of a mild solution for a fractional partial differential equation. Competing interests The authors declare that they have no competing interests. Authors’ contributions Both authors XLD and BA contributed to each part of this work equally and read and approved the final version of the manuscript. Author details 1 Department of Mathematics, Xi’an Polytechnic University, Xi’an, Shaanxi 710048, China. 2 Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. Acknowledgements This work was supported by the Natural Science Foundation of China (NSFC) under grant 11501436 and the Science and Technology Planning Project (2014JQ1041) of Shaanxi Province. Received: 10 June 2016 Accepted: 27 July 2016

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